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The standard reference of formulas edited by Alan Jeffrey and Daniel Zwillinger, translated from Russian by Scripta Technica and published by Academic Press/Elsevier. The front matter and contents show chapters on finite sums, series and infinite products, elementary functions, indefinite and definite integrals, and later special-function material. It is a published book by others, kept in Phil's folder of integrals and series references.
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TABLE OF
INTEGRALS, SERIES,
AND PRODUCTS
EditedbyAlanJeffreyandDanielZwillinger
Table of Integrals, Series, and Products
Seventh Edition
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Table of Integrals, Series, and Products
Seventh Edition
I.S. Gradshteyn and I.M. Ryzhik
Alan Jeffrey, Editor
University of Newcastle upon Tyne, England
Daniel Zwillinger, Editor
Rensselaer Polytechnic Institute, USA
Translated from Russian by Scripta Technica, Inc.
AMSTERDAM •BOSTON •HEIDELBERG •LONDON
NEW YORK •OXFORD •PARIS •SAN DIEGO
SAN FRANCISCO •SINGAPORE •SYDNEY •TOKYO
Academic Press is an imprint of Elsevier
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ISBN-13: 978-0-12-373637-6
ISBN-10: 0-12-373637-4
PRINTED IN THE UNITED STATES OF AMERICA
0 70 80 91 01 1987654321
Contents
Preface to the Seventh Edition xxi
Acknowledgments xxiii
The Order of Presentation of the Formulas xxvii
Use of the Tables xxxi
Index of Special Functions xxxix
Notation xliii
Note on the Bibliographic References xlvii
0Introduction 1
0.1 Finite Sums ..................................... 1
0.11 Progressions .................................... 1
0.12 Sums of powers of natural numbers ........................ 1
0.13 Sums of reciprocals of natural numbers ...................... 3
0.14 Sums of products of reciprocals of natural numbers ............... 3
0.15 Sums of the binomial coefficients ......................... 3
0.2 Numerical Series and Infinite Products ...................... 6
0.21 The convergence of numerical series ....................... 6
0.22 Convergence tests ................................. 6
0.23–0.24 Examples of numerical series ........................... 8
0.25 Infinite products .................................. 1 4
0.26 Examples of infinite products ........................... 1 4
0.3 Functional Series .................................. 1 5
0.30 Definitions and theorems .............................. 1 5
0.31 Power series .................................... 1 6
0.32 Fourier series .................................... 1 9
0.33 Asymptotic series .................................. 2 1
0.4 Certain Formulas from Differential Calculus .................... 2 1
0.41 Differentiation of a definite integral with respect to a parameter ......... 2 1
0.42 Thenthderivative of a product (Leibniz’s rule) .................. 2 2
0.43 Thenthderivative of a composite function .................... 2 2
0.44 Integration by substitution ............................. 2 3
1Elementary Functions 25
1.1 Power of Binomials ................................. 2 5
1.11 Power series .................................... 2 5
1.12 Series of rational fractions ............................. 2 6
1.2 The Exponential Function ............................. 2 6
v
vi CONTENTS
1.21 Series representation ................................ 2 6
1.22 Functional relations ................................ 2 7
1.23 Series of exponentials ............................... 2 7
1.3–1.4 Trigonometric and Hyperbolic Functions ..................... 2 8
1.30 Introduction .................................... 2 8
1.31 The basic functional relations ........................... 2 8
1.32 The representation of powers of trigonometric and hyperbolic functions in terms
of functions of multiples of the argument (angle) ................. 3 1
1.33 The representation of trigonometric and hyperbolic functions of multiples of
the argument (angle) in terms of powers of these functions ........... 3 3
1.34 Certain sums of trigonometric and hyperbolic functions ............. 3 6
1.35 Sums of powers of trigonometric functions of multiple angles .......... 3 7
1.36 Sums of products of trigonometric functions of multiple angles ......... 3 8
1.37 Sums of tangents of multiple angles ........................ 3 9
1.38 Sums leading to hyperbolic tangents and cotangents ............... 3 9
1.39 The representation of cosines and sines of multiples of the angle as finite products 41
1.41 The expansion of trigonometric and hyperbolic functions in power series .... 4 2
1.42 Expansion in series of simple fractions ...................... 4 4
1.43 Representation in the form of an infinite product ................. 4 5
1.44–1.45 Trigonometric (Fourier) series ........................... 4 6
1.46 Series of products of exponential and trigonometric functions .......... 5 1
1.47 Series of hyperbolic functions ........................... 5 1
1.48 Lobachevskiy’s “Angle of Parallelism” Π(x) ................... 5 1
1.49 The hyperbolic amplitude (the Gudermannian) gdx............... 5 2
1.5 The Logarithm ................................... 5 3
1.51 Series representation ................................ 5 3
1.52 Series of logarithms (cf. 1.431) .......................... 5 5
1.6 The Inverse Trigonometric and Hyperbolic Functions ............... 5 6
1.61 The domain of definition .............................. 5 6
1.62–1.63 Functional relations ................................ 5 6
1.64 Series representations ............................... 6 0
2Indefinite Integrals of Elementary Functions 63
2.0 Introduction .................................... 6 3
2.00 General remarks .................................. 6 3
2.01 The basic integrals ................................. 6 4
2.02 General formulas .................................. 6 5
2.1 Rational Functions ................................. 6 6
2.10 General integration rules .............................. 6 6
2.11–2.13 Forms containing the binomial a+bxk...................... 6 8
2.14 Forms containing the binomial 1±xn...................... 7 4
2.15 Forms containing pairs of binomials: a+bxandα+βx............. 7 8
2.16 Forms containing the trinomial a+bxk+cx2k.................. 7 8
2.17 Forms containing the quadratic trinomial a+bx+cx2and powers of x.... 7 9
2.18 Forms containing the quadratic trinomial a+bx+cx2and the binomial α+βx 81
2.2 Algebraic Functions ................................ 8 2
2.20 Introduction .................................... 8 2
2.21 Forms containing the binomial a+bxkand√x................. 8 3
CONTENTS vii
2.22–2.23 Forms containingn/radicalbig
(a+bx)k........................... 8 4
2.24 Forms containing√
a+bxand the binomial α+βx............... 8 8
2.25 Forms containing√
a+bx+cx2......................... 9 2
2.26 Forms containing√
a+bx+cx2and integral powers of x............ 9 4
2.27 Forms containing√
a+cx2and integral powers of x............... 9 9
2.28 Forms containing√
a+bx+cx2and first- and second-degree polynomials . . . 103
2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals ....... 1 0 4
2.3 The Exponential Function ............................. 1 0 6
2.31 Forms containing eax............................... 1 0 6
2.32 The exponential combined with rational functions of x.............. 1 0 6
2.4 Hyperbolic Functions ................................ 1 1 0
2.41–2.43 Powers of sinhx,coshx,tanhx,a n dcothx................... 1 1 0
2.44–2.45 Rational functions of hyperbolic functions .................... 1 2 5
2.46 Algebraic functions of hyperbolic functions .................... 1 3 2
2.47 Combinations of hyperbolic functions and powers ................ 1 3 9
2.48 Combinations of hyperbolic functions, exponentials, and powers ......... 1 4 8
2.5–2.6 Trigonometric Functions .............................. 1 5 1
2.50 Introduction .................................... 1 5 1
2.51–2.52 Powers of trigonometric functions ......................... 1 5 1
2.53–2.54 Sines and cosines of multiple angles and of linear and more complicated func-
tions of the argument ............................... 1 6 1
2.55–2.56 Rational functions of the sine and cosine ..................... 1 7 1
2.57 Integrals containing√
a±bsinxor√
a±bcosx................. 1 7 9
2.58–2.62 Integrals reducible to elliptic and pseudo-elliptic integrals ............ 1 8 4
2.63–2.65 Products of trigonometric functions and powers ................. 2 1 4
2.66 Combinations of trigonometric functions and exponentials ............ 2 2 7
2.67 Combinations of trigonometric and hyperbolic functions ............. 2 3 1
2.7 Logarithms and Inverse-Hyperbolic Functions ................... 2 3 7
2.71 The logarithm ................................... 2 3 7
2.72–2.73 Combinations of logarithms and algebraic functions ............... 2 3 8
2.74 Inverse hyperbolic functions ............................ 2 4 0
2.8 Inverse Trigonometric Functions .......................... 2 4 1
2.81 Arcsines and arccosines .............................. 2 4 1
2.82 The arcsecant, the arccosecant, the arctangent, and the arccotangent ..... 2 4 2
2.83 Combinations of arcsine or arccosine and algebraic functions ........... 2 4 2
2.84 Combinations of the arcsecant and arccosecant with powers of x........ 2 4 4
2.85 Combinations of the arctangent and arccotangent with algebraic functions . . . 244
3–4Definite Integrals of Elementary Functions 247
3.0 Introduction .................................... 2 4 7
3.01 Theorems of a general nature ........................... 2 4 7
3.02 Change of variable in a definite integral ...................... 2 4 8
3.03 General formulas .................................. 2 4 9
3.04 Improper integrals ................................. 2 5 1
3.05 The principal values of improper integrals ..................... 2 5 2
3.1–3.2 Power and Algebraic Functions .......................... 2 5 3
3.11 Rational functions ................................. 2 5 3
viii CONTENTS
3.12 Products of rational functions and expressions that can be reduced to square
roots of first- and second-degree polynomials ................... 2 5 4
3.13–3.17 Expressions that can be reduced to square roots of third- and fourth-degree
polynomials and their products with rational functions .............. 2 5 4
3.18 Expressions that can be reduced to fourth roots of second-degree polynomials
and their products with rational functions ..................... 3 1 3
3.19–3.23 Combinations of powers of xand powers of binomials of the form (α+βx) . . 315
3.24–3.27 Powers of x,o fb i n o m i a l so ft h ef o r m α+βxpand of polynomials in x..... 3 2 2
3.3–3.4 Exponential Functions ............................... 3 3 4
3.31 Exponential functions ............................... 3 3 4
3.32–3.34 Exponentials of more complicated arguments ................... 3 3 6
3.35 Combinations of exponentials and rational functions ............... 3 4 0
3.36–3.37 Combinations of exponentials and algebraic functions .............. 3 4 4
3.38–3.39 Combinations of exponentials and arbitrary powers ................ 3 4 6
3.41–3.44 Combinations of rational functions of powers and exponentials ......... 3 5 3
3.45 Combinations of powers and algebraic functions of exponentials ......... 3 6 3
3.46–3.48 Combinations of exponentials of more complicated arguments and powers . . . 364
3.5 Hyperbolic Functions ................................ 3 7 1
3.51 Hyperbolic functions ................................ 3 7 1
3.52–3.53 Combinations of hyperbolic functions and algebraic functions .......... 3 7 5
3.54 Combinations of hyperbolic functions and exponentials ............. 3 8 2
3.55–3.56 Combinations of hyperbolic functions, exponentials, and powers ......... 3 8 6
3.6–4.1 Trigonometric Functions .............................. 3 9 0
3.61 Rational functions of sines and cosines and trigonometric functions of multiple
angles ........................................ 3 9 0
3.62 Powers of trigonometric functions ......................... 3 9 5
3.63 Powers of trigonometric functions and trigonometric functions of linear functions 397
3.64–3.65 Powers and rational functions of trigonometric functions ............. 4 0 1
3.66 Forms containing powers of linear functions of trigonometric functions ..... 4 0 5
3.67 Square roots of expressions containing trigonometric functions ......... 4 0 8
3.68 Various forms of powers of trigonometric functions ................ 4 1 1
3.69–3.71 Trigonometric functions of more complicated arguments ............. 4 1 5
3.72–3.74 Combinations of trigonometric and rational functions .............. 4 2 3
3.75 Combinations of trigonometric and algebraic functions .............. 4 3 4
3.76–3.77 Combinations of trigonometric functions and powers ............... 4 3 6
3.78–3.81 Rational functions of xand of trigonometric functions .............. 4 4 7
3.82–3.83 Powers of trigonometric functions combined with other powers ......... 4 5 9
3.84 Integrals containing/radicalbig
1−k2sin2x,√
1−k2cos2x, and similar expressions . . 472
3.85–3.88 Trigonometric functions of more complicated arguments combined with powers 475
3.89–3.91 Trigonometric functions and exponentials ..................... 4 8 5
3.92 Trigonometric functions of more complicated arguments combined with expo-
nentials ....................................... 4 9 3
3.93 Trigonometric and exponential functions of trigonometric functions ....... 4 9 5
3.94–3.97 Combinations involving trigonometric functions, exponentials, and powers . . . 497
3.98–3.99 Combinations of trigonometric and hyperbolic functions ............. 5 0 9
4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers . . . 516
4.13 Combinations of trigonometric and hyperbolic functions and exponentials .... 5 2 2
CONTENTS ix
4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and powers 525
4.2–4.4 Logarithmic Functions ............................... 5 2 7
4.21 Logarithmic functions ............................... 5 2 7
4.22 Logarithms of more complicated arguments .................... 5 2 9
4.23 Combinations of logarithms and rational functions ................ 5 3 5
4.24 Combinations of logarithms and algebraic functions ............... 5 3 8
4.25 Combinations of logarithms and powers ...................... 5 4 0
4.26–4.27 Combinations involving powers of the logarithm and other powers ........ 5 4 2
4.28 Combinations of rational functions of lnxand powers .............. 5 5 3
4.29–4.32 Combinations of logarithmic functions of more complicated arguments and powers 555
4.33–4.34 Combinations of logarithms and exponentials ................... 5 7 1
4.35–4.36 Combinations of logarithms, exponentials, and powers .............. 5 7 3
4.37 Combinations of logarithms and hyperbolic functions ............... 5 7 8
4.38–4.41 Logarithms and trigonometric functions ...................... 5 8 1
4.42–4.43 Combinations of logarithms, trigonometric functions, and powers ........ 5 9 4
4.44 Combinations of logarithms, trigonometric functions, and exponentials ..... 5 9 9
4.5 Inverse Trigonometric Functions .......................... 5 9 9
4.51 Inverse trigonometric functions .......................... 5 9 9
4.52 Combinations of arcsines, arccosines, and powers ................. 6 0 0
4.53–4.54 Combinations of arctangents, arccotangents, and powers ............. 6 0 1
4.55 Combinations of inverse trigonometric functions and exponentials ........ 6 0 5
4.56 A combination of the arctangent and a hyperbolic function ........... 6 0 5
4.57 Combinations of inverse and direct trigonometric functions ........... 6 0 5
4.58 A combination involving an inverse and a direct trigonometric function and a
power ........................................ 6 0 7
4.59 Combinations of inverse trigonometric functions and logarithms ......... 6 0 7
4.6 Multiple Integrals ................................. 6 0 7
4.60 Change of variables in multiple integrals ..................... 6 0 7
4.61 Change of the order of integration and change of variables ........... 6 0 8
4.62 Double and triple integrals with constant limits .................. 6 1 0
4.63–4.64 Multiple integrals .................................. 6 1 2
5Indefinite Integrals of Special Functions 619
5.1 Elliptic Integrals and Functions .......................... 6 1 9
5.11 Complete elliptic integrals ............................. 6 1 9
5.12 Elliptic integrals .................................. 6 2 1
5.13 Jacobian elliptic functions ............................. 6 2 3
5.14 Weierstrass elliptic functions ............................ 6 2 6
5.2 The Exponential Integral Function ........................ 6 2 7
5.21 The exponential integral function ......................... 6 2 7
5.22 Combinations of the exponential integral function and powers .......... 6 2 7
5.23 Combinations of the exponential integral and the exponential .......... 6 2 8
5.3 T h eS i n eI n t e g r a la n dt h eC o s i n eI n t e g r a l ..................... 6 2 8
5.4 The Probability Integral and Fresnel Integrals ................... 6 2 9
5.5 Bessel Functions .................................. 6 2 9
x CONTENTS
6–7Definite Integrals of Special Functions 631
6.1 Elliptic Integrals and Functions .......................... 6 3 1
6.11 Forms containing F(x, k) ............................. 6 3 1
6.12 Forms containing E(x, k) ............................. 6 3 2
6.13 Integration of elliptic integrals with respect to the modulus ........... 6 3 2
6.14–6.15 Complete elliptic integrals ............................. 6 3 2
6.16 The theta function ................................. 6 3 3
6.17 Generalized elliptic integrals ............................ 6 3 5
6.2–6.3 The Exponential Integral Function and Functions Generated by It ........ 6 3 6
6.21 The logarithm integral ............................... 6 3 6
6.22–6.23 The exponential integral function ......................... 6 3 8
6.24–6.26 The sine integral and cosine integral functions .................. 6 3 9
6.27 The hyperbolic sine integral and hyperbolic cosine integral functions ...... 6 4 4
6.28–6.31 The probability integral .............................. 6 4 5
6.32 Fresnel integrals .................................. 6 4 9
6.4 The Gamma Function and Functions Generated by It .............. 6 5 0
6.41 The gamma function ................................ 6 5 0
6.42 Combinations of the gamma function, the exponential, and powers ....... 6 5 2
6.43 Combinations of the gamma function and trigonometric functions ........ 6 5 5
6.44 The logarithm of the gamma function∗...................... 6 5 6
6.45 The incomplete gamma function ......................... 6 5 7
6.46–6.47 The function ψ(x)................................. 6 5 8
6.5–6.7 Bessel Functions .................................. 6 5 9
6.51 Bessel functions .................................. 6 5 9
6.52 Bessel functions combined with xandx2..................... 6 6 4
6.53–6.54 Combinations of Bessel functions and rational functions ............. 6 7 0
6.55 Combinations of Bessel functions and algebraic functions ............ 6 7 4
6.56–6.58 Combinations of Bessel functions and powers ................... 6 7 5
6.59 Combinations of powers and Bessel functions of more complicated arguments . 689
6.61 Combinations of Bessel functions and exponentials ................ 6 9 4
6.62–6.63 Combinations of Bessel functions, exponentials, and powers ........... 6 9 9
6.64 Combinations of Bessel functions of more complicated arguments, exponentials,
and powers ..................................... 7 0 8
6.65 Combinations of Bessel and exponential functions of more complicated argu-
ments and powers ................................. 7 1 1
6.66 Combinations of Bessel, hyperbolic, and exponential functions .......... 7 1 3
6.67–6.68 Combinations of Bessel and trigonometric functions ............... 7 1 7
6.69–6.74 Combinations of Bessel and trigonometric functions and powers ......... 7 2 7
6.75 Combinations of Bessel, trigonometric, and exponential functions and powers . 742
6.76 Combinations of Bessel, trigonometric, and hyperbolic functions ........ 7 4 7
6.77 Combinations of Bessel functions and the logarithm, or arctangent ....... 7 4 7
6.78 Combinations of Bessel and other special functions ................ 7 4 8
6.79 Integration of Bessel functions with respect to the order ............. 7 4 9
6.8 Functions Generated by Bessel Functions ..................... 7 5 3
6.81 Struve functions .................................. 7 5 3
6.82 Combinations of Struve functions, exponentials, and powers ........... 7 5 4
6.83 Combinations of Struve and trigonometric functions ............... 7 5 5
CONTENTS xi
6.84–6.85 Combinations of Struve and Bessel functions ................... 7 5 6
6.86 Lommel functions ................................. 7 6 0
6.87 Thomson functions ................................. 7 6 1
6.9 Mathieu Functions ................................. 7 6 3
6.91 Mathieu functions ................................. 7 6 3
6.92 Combinations of Mathieu, hyperbolic, and trigonometric functions ....... 7 6 3
6.93 Combinations of Mathieu and Bessel functions .................. 7 6 7
6.94 Relationships between eigenfunctions of the Helmholtz equation in different
coordinate systems ................................. 7 6 7
7.1–7.2 Associated Legendre Functions .......................... 7 6 9
7.11 Associated Legendre functions ........................... 7 6 9
7.12–7.13 Combinations of associated Legendre functions and powers ........... 7 7 0
7.14 Combinations of associated Legendre functions, exponentials, and powers . . . 776
7.15 Combinations of associated Legendre and hyperbolic functions ......... 7 7 8
7.16 Combinations of associated Legendre functions, powers, and trigonometric
functions ...................................... 7 7 9
7.17 A combination of an associated Legendre function and the probability integral . 781
7.18 Combinations of associated Legendre and Bessel functions ............ 7 8 2
7.19 Combinations of associated Legendre functions and functions generated by
Bessel functions .................................. 7 8 7
7.21 Integration of associated Legendre functions with respect to the order ..... 7 8 8
7.22 Combinations of Legendre polynomials, rational functions, and algebraic functions 789
7.23 Combinations of Legendre polynomials and powers ................ 7 9 1
7.24 Combinations of Legendre polynomials and other elementary functions ..... 7 9 2
7.25 Combinations of Legendre polynomials and Bessel functions ........... 7 9 4
7.3–7.4 Orthogonal Polynomials .............................. 7 9 5
7.31 Combinations of Gegenbauer polynomials Cν
n(x)and powers .......... 7 9 5
7.32 Combinations of Gegenbauer polynomials Cν
n(x)and elementary functions . . . 797
7.325∗Complete System of Orthogonal Step Functions ................. 7 9 8
7.33 Combinations of the polynomials Cν
n(x)and Bessel functions; Integration of
Gegenbauer functions with respect to the index ................. 7 9 8
7.34 Combinations of Chebyshev polynomials and powers ............... 8 0 0
7.35 Combinations of Chebyshev polynomials and elementary functions ....... 8 0 2
7.36 Combinations of Chebyshev polynomials and Bessel functions .......... 8 0 3
7.37–7.38 Hermite polynomials ................................ 8 0 3
7.39 Jacobi polynomials ................................. 8 0 6
7.41–7.42 Laguerre polynomials ................................ 8 0 8
7.5 Hypergeometric Functions ............................. 8 1 2
7.51 Combinations of hypergeometric functions and powers .............. 8 1 2
7.52 Combinations of hypergeometric functions and exponentials ........... 8 1 4
7.53 Hypergeometric and trigonometric functions ................... 8 1 7
7.54 Combinations of hypergeometric and Bessel functions .............. 8 1 7
7.6 Confluent Hypergeometric Functions ....................... 8 2 0
7.61 Combinations of confluent hypergeometric functions and powers ........ 8 2 0
7.62–7.63 Combinations of confluent hypergeometric functions and exponentials ..... 8 2 2
7.64 Combinations of confluent hypergeometric and trigonometric functions ..... 8 2 9
7.65 Combinations of confluent hypergeometric functions and Bessel functions . . . 830
xii CONTENTS
7.66 Combinations of confluent hypergeometric functions, Bessel functions, and powers 831
7.67 Combinations of confluent hypergeometric functions, Bessel functions, expo-
nentials, and powers ................................ 8 3 4
7.68 Combinations of confluent hypergeometric functions and other special functions 839
7.69 Integration of confluent hypergeometric functions with respect to the index . . 841
7.7 Parabolic Cylinder Functions ............................ 8 4 1
7.71 Parabolic cylinder functions ............................ 8 4 1
7.72 Combinations of parabolic cylinder functions, powers, and exponentials ..... 8 4 2
7.73 Combinations of parabolic cylinder and hyperbolic functions ........... 8 4 3
7.74 Combinations of parabolic cylinder and trigonometric functions ......... 8 4 4
7.75 Combinations of parabolic cylinder and Bessel functions ............. 8 4 5
7.76 Combinations of parabolic cylinder functions and confluent hypergeometric
functions ...................................... 8 4 9
7.77 Integration of a parabolic cylinder function with respect to the index ...... 8 4 9
7.8 Meijer’s and MacRobert’s Functions ( GandE)................. 8 5 0
7.81 Combinations of the functions GandEand the elementary functions ..... 8 5 0
7.82 Combinations of the functions GandEand Bessel functions .......... 8 5 4
7.83 Combinations of the functions GandEand other special functions ....... 8 5 6
8–9Special Functions 859
8.1 Elliptic Integrals and Functions .......................... 8 5 9
8.11 Elliptic integrals .................................. 8 5 9
8.12 Functional relations between elliptic integrals ................... 8 6 3
8.13 Elliptic functions .................................. 8 6 5
8.14 Jacobian elliptic functions ............................. 8 6 6
8.15 Properties of Jacobian elliptic functions and functional relationships between them 870
8.16 The Weierstrass function ℘(u) .......................... 8 7 3
8.17 The functions ζ(u) and σ(u)........................... 8 7 6
8.18–8.19 Theta functions .................................. 8 7 7
8.2 The Exponential Integral Function and Functions Generated by It ........ 8 8 3
8.21 The exponential integral function Ei (x)...................... 8 8 3
8.22 The hyperbolic sine integral shixand the hyperbolic cosine integral chix. . . 886
8.23 The sine integral and the cosine integral: sixandcix.............. 8 8 6
8.24 The logarithm integral li(x)............................ 8 8 7
8.25 The probability integral Φ(x), the Fresnel integrals S(x)andC(x), the error
function erf(x), and the complementary error function erfc(x) ......... 8 8 7
8.26 Lobachevskiy’s function L(x) ........................... 8 9 1
8.3 Euler’s Integrals of the First and Second Kinds .................. 8 9 2
8.31 The gamma function (Euler’s integral of the second kind): Γ(z) ........ 8 9 2
8.32 Representation of the gamma function as series and products .......... 8 9 4
8.33 Functional relations involving the gamma function ................ 8 9 5
8.34 The logarithm of the gamma function ....................... 8 9 8
8.35 The incomplete gamma function ......................... 8 9 9
8.36 The psi function ψ(x)............................... 9 0 2
8.37 The function β(x)................................. 9 0 6
8.38 The beta function (Euler’s integral of the first kind): B(x, y) .......... 9 0 8
8.39 The incomplete beta function Bx(p, q) ...................... 9 1 0
8.4–8.5 Bessel Functions and Functions Associated with Them .............. 9 1 0
CONTENTS xiii
8.40 Definitions ..................................... 9 1 0
8.41 Integral representations of the functions Jν(z)andNν(z)............ 9 1 2
8.42 Integral representations of the functions H(1)
ν(z)andH(2)
ν(z).......... 9 1 4
8.43 Integral representations of the functions Iν(z)andKν(z)............ 9 1 6
8.44 Series representation ................................ 9 1 8
8.45 Asymptotic expansions of Bessel functions .................... 9 2 0
8.46 Bessel functions of order equal to an integer plus one-half ............ 9 2 4
8.47–8.48 Functional relations ................................ 9 2 6
8.49 Differential equations leading to Bessel functions ................. 9 3 1
8.51–8.52 Series of Bessel functions ............................. 9 3 3
8.53 Expansion in products of Bessel functions ..................... 9 4 0
8.54 The zeros of Bessel functions ........................... 9 4 1
8.55 Struve functions .................................. 9 4 2
8.56 Thomson functions and their generalizations ................... 9 4 4
8.57 Lommel functions ................................. 9 4 5
8.58 Anger and Weber functions Jν(z)andEν(z)................... 9 4 8
8.59 Neumann’s and Schl¨ afli’s polynomials: On(z)andSn(z) ............ 9 4 9
8.6 Mathieu Functions ................................. 9 5 0
8.60 Mathieu’s equation ................................. 9 5 0
8.61 Periodic Mathieu functions ............................ 9 5 1
8.62 Recursion relations for the coefficients A(2n)
2r,A(2n+1)
2r+1,B(2n+1)
2r+1,B(2n+2)
2r+2.... 9 5 1
8.63 Mathieu functions with a purely imaginary argument ............... 9 5 2
8.64 Non-periodic solutions of Mathieu’s equation ................... 9 5 3
8.65 Mathieu functions for negative q......................... 9 5 3
8.66 Representation of Mathieu functions as series of Bessel functions ........ 9 5 4
8.67 The general theory ................................. 9 5 7
8.7–8.8 Associated Legendre Functions .......................... 9 5 8
8.70 Introduction .................................... 9 5 8
8.71 Integral representations .............................. 9 6 0
8.72 Asymptotic series for large values of |ν|...................... 9 6 2
8.73–8.74 Functional relations ................................ 9 6 4
8.75 Special cases and particular values ........................ 9 6 8
8.76 Derivatives with respect to the order ....................... 9 6 9
8.77 Series representation ................................ 9 7 0
8.78 The zeros of associated Legendre functions .................... 9 7 2
8.79 Series of associated Legendre functions ...................... 9 7 2
8.81 Associated Legendre functions with integer indices ................ 9 7 4
8.82–8.83 Legendre functions ................................. 9 7 5
8.84 Conical functions .................................. 9 8 0
8.85 Toroidal functions ................................. 9 8 1
8.9 Orthogonal Polynomials .............................. 9 8 2
8.90 Introduction .................................... 9 8 2
8.91 Legendre polynomials ............................... 9 8 3
8.919 Series of products of Legendre and Chebyshev polynomials ........... 9 8 8
8.92 Series of Legendre polynomials .......................... 9 8 8
8.93 Gegenbauer polynomials Cλ
n(t) .......................... 9 9 0
8.94 The Chebyshev polynomials Tn(x)andUn(x) .................. 9 9 3
xiv CONTENTS
8.95 The Hermite polynomials Hn(x) ......................... 9 9 6
8.96 Jacobi’s polynomials ................................ 9 9 8
8.97 The Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000
9.1 Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005
9.10 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005
9.11 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005
9.12 Representation of elementary functions in terms of a hypergeometric functions . 1006
9.13 Transformation formulas and the analytic continuation of functions defined by
hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008
9.14 A generalized hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . 1010
9.15 The hypergeometric differential equation . . . . . . . . . . . . . . . . . . . . . 1010
9.16 Riemann’s differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . 1014
9.17 Representing the solutions to certain second-order differential equations using
a Riemann scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017
9.18 Hypergeometric functions of two variables . . . . . . . . . . . . . . . . . . . . 1018
9.19 A hypergeometric function of several variables . . . . . . . . . . . . . . . . . . 1022
9.2 Confluent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . 1022
9.20 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022
9.21 The functions Φ(α,γ;z)andΨ(α,γ;z) . . . . . . . . . . . . . . . . . . . . . . 1023
9.22–9.23 The Whittaker functions Mλ,μ(z)andWλ,μ(z) . . . . . . . . . . . . . . . . . . 1024
9.24–9.25 Parabolic cylinder functions Dp(z) . . . . . . . . . . . . . . . . . . . . . . . . 1028
9.26 Confluent hypergeometric series of two variables . . . . . . . . . . . . . . . . . 1031
9.3 Meijer’s G-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032
9.30 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032
9.31 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033
9.32 A differential equation for the G-function . . . . . . . . . . . . . . . . . . . . . 1034
9.33 Series of G-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034
9.34 Connections with other special functions . . . . . . . . . . . . . . . . . . . . . 1034
9.4 MacRobert’s E-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035
9.41 Representation by means of multiple integrals . . . . . . . . . . . . . . . . . . 1035
9.42 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035
9.5 Riemann’s Zeta Functions ζ(z,q)andζ(z), and the Functions Φ(z,s,v)andξ(s) 1036
9.51 Definition and integral representations . . . . . . . . . . . . . . . . . . . . . . 1036
9.52 Representation as a series or as an infinite product . . . . . . . . . . . . . . . . 1037
9.53 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037
9.54 Singular points and zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038
9.55 The Lerch function Φ(z,s,v) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039
9.56 The function ξ(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040
9.6 Bernoulli Numbers and Polynomials, Euler Numbers . . . . . . . . . . . . . . . 1040
9.61 Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040
9.62 Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1041
9.63 Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043
9.64 The functions ν(x),ν(x, α),μ(x, β),μ(x, β, α ),a n dλ(x, y) . . . . . . . . . . 1043
9.65 Euler polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044
9.7 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045
9.71 Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045
9.72 Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045
CONTENTS xv
9.73 Euler’s and Catalan’s constants . . . . . . . . . . . . . . . . . . . . . . . . . . 1046
9.74 Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046
10Vector Field Theory 1049
10.1–10.8 Vectors, Vector Operators, and Integral Theorems . . . . . . . . . . . . . . . . 1049
10.11 Products of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049
10.12 Properties of scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049
10.13 Properties of vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049
10.14 Differentiation of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050
10.21 Operators grad, div, and curl . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050
10.31 Properties of the operator ∇. . . . . . . . . . . . . . . . . . . . . . . . . . . 1051
10.41 Solenoidal fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052
10.51–10.61 Orthogonal curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . 1052
10.71–10.72 Vector integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055
10.81 Integral rate of change theorems . . . . . . . . . . . . . . . . . . . . . . . . . 1057
11Algebraic Inequalities 1059
11.1–11.3 General Algebraic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059
11.11 Algebraic inequalities involving real numbers . . . . . . . . . . . . . . . . . . . 1059
11.21 Algebraic inequalities involving complex numbers . . . . . . . . . . . . . . . . . 1060
11.31 Inequalities for sets of complex numbers . . . . . . . . . . . . . . . . . . . . . 1061
12Integral Inequalities 1063
12.11 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
12.111 First mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
12.112 Second mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
12.113 First mean value theorem for infinite integrals . . . . . . . . . . . . . . . . . . 1063
12.114 Second mean value theorem for infinite integrals . . . . . . . . . . . . . . . . . 1064
12.21 Differentiation of Definite Integral Containing a Parameter . . . . . . . . . . . 1064
12.211 Differentiation when limits are finite . . . . . . . . . . . . . . . . . . . . . . . . 1064
12.212 Differentiation when a limit is infinite . . . . . . . . . . . . . . . . . . . . . . . 1064
12.31 Integral Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064
12.311 Cauchy-Schwarz-Buniakowsky inequality for integrals . . . . . . . . . . . . . . 1064
12.312 H¨older’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 1064
12.313 Minkowski’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.314 Chebyshev’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.315 Young’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.316 Steffensen’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.317 Gram’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.318 Ostrowski’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1066
12.41 Convexity and Jensen’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 1066
12.411 Jensen’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066
12.412 Carleman’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . 1066
12.51 Fourier Series and Related Inequalities . . . . . . . . . . . . . . . . . . . . . . 1066
12.511 Riemann-Lebesgue lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067
12.512 Dirichlet lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067
12.513 Parseval’s theorem for trigonometric Fourier series . . . . . . . . . . . . . . . . 1067
12.514 Integral representation of the nthpartial sum . . . . . . . . . . . . . . . . . . . 1067
xvi CONTENTS
12.515 Generalized Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067
12.516 Bessel’s inequality for generalized Fourier series . . . . . . . . . . . . . . . . . 1068
12.517 Parseval’s theorem for generalized Fourier series . . . . . . . . . . . . . . . . . 1068
13Matrices and Related Results 1069
13.11–13.12 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.111 Diagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.112 Identity matrix and null matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.113 Reducible and irreducible matrices . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.114 Equivalent matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.115 Transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.116 Adjoint matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.117 Inverse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.118 Trace of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.119 Symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.120 Skew-symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.121 Triangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.122 Orthogonal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.123 Hermitian transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.124 Hermitian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.125 Unitary matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.126 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.127 Nilpotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.128 Idempotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.129 Positive definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.130 Non-negative definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.131 Diagonally dominant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.21 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.211 Sylvester’s law of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072
13.212 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072
13.213 Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072
13.214 Positive definite and semidefinite quadratic form . . . . . . . . . . . . . . . . . 1072
13.215 Basic theorems on quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . 1072
13.31 Differentiation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073
13.41 The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074
3.411 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074
14Determinants 1075
14.11 Expansion of Second- and Third-Order Determinants . . . . . . . . . . . . . . 1075
14.12 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075
14.13 Minors and Cofactors of a Determinant . . . . . . . . . . . . . . . . . . . . . . 1075
14.14 Principal Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076
14.15*Laplace Expansion of a Determinant . . . . . . . . . . . . . . . . . . . . . . . 1076
14.16 Jacobi’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076
14.17 Hadamard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077
14.18 Hadamard’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077
14.21 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077
14.31 Some Special Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078
CONTENTS xvii
14.311 Vandermonde’s determinant (alternant) . . . . . . . . . . . . . . . . . . . . . . 1078
14.312 Circulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078
14.313 Jacobian determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078
14.314 Hessian determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079
14.315 Wronskian determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079
14.316 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079
14.317 Gram-Kowalewski theorem on linear dependence . . . . . . . . . . . . . . . . . 1080
15Norms 1081
15.1–15.9 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.11 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.21 Principal Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.211 The norm ||x||1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.212 The norm ||x||2(Euclidean or L2norm) . . . . . . . . . . . . . . . . . . . . . 1081
15.213 The norm ||x||∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.31 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.311 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.312 Induced norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.313 Natural norm of unit matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.41 Principal Natural Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.411 Maximum absolute column sum norm . . . . . . . . . . . . . . . . . . . . . . . 1082
15.412 Spectral norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.413 Maximum absolute row sum norm . . . . . . . . . . . . . . . . . . . . . . . . . 1083
15.51 Spectral Radius of a Square Matrix . . . . . . . . . . . . . . . . . . . . . . . . 1083
15.511 Inequalities concerning matrix norms and the spectral radius . . . . . . . . . . 1083
15.512 Deductions from Gerschgorin’s theorem (see 15.814 ). . . . . . . . . . . . . . 1083
15.61 Inequalities Involving Eigenvalues of Matrices . . . . . . . . . . . . . . . . . . . 1084
15.611 Cayley-Hamilton theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084
15.612 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084
15.71 Inequalities for the Characteristic Polynomial . . . . . . . . . . . . . . . . . . . 1084
15.711 Named and unnamed inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 1085
15.712 Parodi’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
15.713 Corollary of Brauer’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
15.714 Ballieu’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
15.715 Routh-Hurwitz theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
15.81–15.82 Named Theorems on Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 1087
15.811 Schur’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087
15.812 Sturmian separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087
15.813 Poincare’s separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087
15.814 Gerschgorin’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.815 Brauer’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.816 Perron’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.817 Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.818 Perron–Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.819 Wielandt’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.820 Ostrowski’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089
15.821 First theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . 1089
15.822 Second theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . 1089
xviii CONTENTS
15.823 Hermitian matrices and diophantine relations involving circular functions of
rational angles due to Calogero and Perelomov . . . . . . . . . . . . . . . . . . 1089
15.91 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
15.911 Rayleigh quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
15.912 Basic theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
16Ordinary Differential Equations 1093
16.1–16.9 Results Relating to the Solution of Ordinary Differential Equations . . . . . . . 1093
16.11 First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.111 Solution of a first-order equation . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.112 Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.113 Approximate solution to an equation . . . . . . . . . . . . . . . . . . . . . . . 1093
16.114 Lipschitz continuity of a function . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.21 Fundamental Inequalities and Related Results . . . . . . . . . . . . . . . . . . 1094
16.211 Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.212 Comparison of approximate solutions of a differential equation . . . . . . . . . 1094
16.31 First-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.311 Solution of a system of equations . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.312 Cauchy problem for a system . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.313 Approximate solution to a system . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.314 Lipschitz continuity of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.315 Comparison of approximate solutions of a system . . . . . . . . . . . . . . . . 1096
16.316 First-order linear differential equation . . . . . . . . . . . . . . . . . . . . . . . 1096
16.317 Linear systems of differential equations . . . . . . . . . . . . . . . . . . . . . . 1096
16.41 Some Special Types of Elementary Differential Equations . . . . . . . . . . . . 1097
16.411 Variables separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.412 Exact differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.413 Conditions for an exact equation . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.414 Homogeneous differential equations . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.51 Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.511 Adjoint and self-adjoint equations . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.512 Abel’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.513 Lagrange identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.514 The Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.515 Solutions of the Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.516 Solution of a second-order linear differential equation . . . . . . . . . . . . . . 1100
16.61–16.62 Oscillation and Non-Oscillation Theorems for Second-Order Equations . . . . . 1100
16.611 First basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1100
16.622 Second basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.623 Interlacing of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.624 Sturm separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.625 Sturm comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.626 Szeg¨o’s comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.627 Picone’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.628 Sturm-Picone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.629 Oscillation on the half line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.71 Two Related Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.711 Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
CONTENTS xix
16.712 Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.81–16.82 Non-Oscillatory Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.811 Kneser’s non-oscillation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.822 Comparison theorem for non-oscillation . . . . . . . . . . . . . . . . . . . . . . 1104
16.823 Necessary and sufficient conditions for non-oscillation . . . . . . . . . . . . . . 1104
16.91 Some Growth Estimates for Solutions of Second-Order Equations . . . . . . . . 1104
16.911 Strictly increasing and decreasing solutions . . . . . . . . . . . . . . . . . . . . 1104
16.912 General result on dominant and subdominant solutions . . . . . . . . . . . . . 1104
16.913 Estimate of dominant solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105
16.914 A theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105
16.92 Boundedness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
16.921 All solutions of the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
16.922 If all solutions of the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
16.923 Ifa(x)→∞ monotonically as x→∞, then all solutions of . . . . . . . . . . . 1106
16.924 Consider the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
16.93 Growth of maxima of |y|. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
17Fourier, Laplace, and Mellin Transforms 1107
17.1–17.4 Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107
17.11 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107
17.12 Basic properties of the Laplace transform . . . . . . . . . . . . . . . . . . . . . 1107
17.13 Table of Laplace transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . 1108
17.21 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117
17.22 Basic properties of the Fourier transform . . . . . . . . . . . . . . . . . . . . . 1118
17.23 Table of Fourier transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 1118
17.24 Table of Fourier transform pairs for spherically symmetric functions . . . . . . . 1120
17.31 Fourier sine and cosine transforms . . . . . . . . . . . . . . . . . . . . . . . . . 1121
17.32 Basic properties of the Fourier sine and cosine transforms . . . . . . . . . . . . 1121
17.33 Table of Fourier sine transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 1122
17.34 Table of Fourier cosine transforms . . . . . . . . . . . . . . . . . . . . . . . . . 1126
17.35 Relationships between transforms . . . . . . . . . . . . . . . . . . . . . . . . . 1129
17.41 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129
17.42 Basic properties of the Mellin transform . . . . . . . . . . . . . . . . . . . . . 1130
17.43 Table of Mellin transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131
18The z-Transform 1135
18.1–18.3 Definition, Bilateral, and Unilateral z-Transforms . . . . . . . . . . . . . . . . . 1135
18.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135
18.2 Bilateral z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136
18.3 Unilateral z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138
References 1141
Supplemental references 1145
Index of Functions and Constants 1151
General Index of Concepts 1161
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Preface to the Seventh Edition
Since the publication in 2000 of the completely reset sixth edition of Gradshteyn and Ryzhik, users of the
reference work have continued to submit corrections, new results that extend the work, and suggestions
for changes that improve the presentation of existing entries. It is a matter of regret to us that thestructure of the book makes it impossible to acknowledge these individual contributions, so, as usual,the names of the many new contributors have been added to the acknowledgment list at the front of thebook.
This seventh edition contains the corrections received since the publication of the sixth edition in
2000, together with a considerable amount of new material acquired from isolated sources. Following
our previous conventions, an amended entry has a superscript “11” added to its entry reference number,where the equivalent superscript number for the sixth edition was “10.” Similarly, an asterisk on anentry’s reference number indicates a new result. When, for technical reasons, an entry in a previousedition has been removed, to preserve the continuity of numbering between the new and older editions
the subsequent entries have not been renumbered, so the numbering will jump.
We wish to express our gratitude to all who have been in contact with us with the object of improving
and extending the book, and we want to give special thanks to Dr. Victor H. Moll for his interest inthe book and for the many contributions he has made over an extended period of time. We also wish toacknowledge the contributions made by Dr. Francis J. O’Brien Jr. of the Naval Station in Newport, inparticular for results involving integrands where exponentials are combined with algebraic functions.
Experience over many years has shown that each new edition of Gradshteyn and Ryzhik generates
a fresh supply of suggestions for new entries, and for the improvement of the presentation of existingentries and errata. In view of this, we do not expect this new edition to be free from errors, so all
users of this reference work who identify errors, or who wish to propose new entries, are invited tocontact the authors, whose email addresses are listed below. Corrections will be posted on the web sitewww.az-tec.com/gr/errata .
Alan Jeffrey
[email protected]
Daniel Zwillinger
[email protected]
xxi
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Acknowledgments
The publisher and editors would like to take this opportunity to express their gratitude to the following
users of the Table of Integrals, Series, and Products who, either directly or through errata published
inMathematics of Computation , have generously contributed corrections and addenda to the original
printing.
Dr. A. Abbas
Dr. P. B. Abraham
Dr. Ari Abramson
Dr. Jose AdachiDr. R. J. AdlerDr. N. AgmonDr. M. AhmadDr. S. A. Ahmad
Dr. Luis Alvarez-Ruso
Dr. Maarten H P AmbaumDr. R. K. AmietDr. L. U. AncaraniDr. M. Antoine
Dr. C. R. Appledorn
Dr. D. R. AppletonDr. Mitsuhiro Arikawa
Dr. P. AshoshauvatiDr. C. L. AxnessDr. E. Badralexe
Dr. S. B. Bagchi
Dr. L. J. BakerDr. R. BallDr. M. P. BarnettDr. Florian BaumannDr. Norman C. Beaulieu
Dr. Jerome Benoit
Mr. V. BentleyDr. Laurent BergerDr. M. van den BergDr. N. F. Berk
Dr. C. A. BertulaniDr. J. Betancort-Rijo
Dr. P. Bickerstaff
Dr. Iwo Bialynicki-Birula
Dr. Chris BidinostiDr. G. R. BiggDr. Ian BindlossDr. L. BlanchetDr. Mike Blaskiewicz
Dr. R. D. Blevins
Dr. Anders BlomDr. L. M. BlumbergDr. R. BlumelDr. S. E. Bodner
Dr. M. Bonsager
Dr. George BorosDr. S. Bosanac
Dr. B. Van den BosscheDr. A. Bostr¨ om
Dr. J. E. Bowcock
Dr. T. H. Boyer
Dr. K. M. BriggsDr. D. J. BroadhurstDr. Chris Van Den BroeckDr. W. B. BrowerDr. H. N. Browne
Dr. Christoph Bruegger
Dr. William J. BrunoDr. Vladimir BubanjaDr. D. J. BuchDr. D. J. Bukman
Dr. F. M. BurrowsDr. R. Caboz
Dr. T. Call oway
Dr. F. Calogero
Dr. D. Dal CappelloDr. David CardonDr. J. A. Carlson GallosDr. B. CarrascalDr. A. R. Carr
Dr. S. Carter
Dr. G. CavalleriM r .W .H .L .C a w t h o r n eDr. A. CecchiniDr. B. Chan
Dr. M. A. Chaudhry
Dr. Sabino Chavez-CerdaDr. Julian Cheng
Dr. H. W. ChewDr. D. ChinDr. Young-seek Chung
Dr. S. Ciccariello
Dr. N. S. ClarkeDr. R. W. ClearyDr. A. ClementDr. P. CochraneDr. D. K. Cohoon
Dr. L. Cole
Dr. Filippo ColomoDr. J. R. D. CopleyDr. D. CoxDr. J. Cox
Dr. J. W. Criss
xxiii
xxiv Acknowledgments
Dr. A. E. Curzon
Dr. D. Dadyburjor
Dr. D. DajaputraDr. C. Dal CappelloDr. P. DalyDr. S. Dasgupta
Dr. John Davies
Dr. C. L. DavisDr. A. DegasperisDr. B. C. DenardoDr. R. W. Dent
Dr. E. Deutsch
Dr. D. deVries
Dr. P. DitaDr. P. J. de DoelderDr. Mischa DohlerDr. G. Dˆ ome
Dr. Shi-Hai Dong
Dr. Balazs Dora
Dr. M. R. D’OrsognaDr. Adrian A. DragulescuDr. Eduardo DuenezMr. Tommi J. Dufva
Dr. E. B. Dussan, V
Dr. C. A. EbnerDr. M. van der EndeDr. Jonathan EngleDr. G. EngDr. E. S. Erck
Dr. Jan Erkelens
Dr. Olivier EspinosaDr. G. A. Est´ evez
Dr. K. EvansDr. G. EvendonDr. V. I. Fabrikant
Dr. L. A. Falkovsky
Dr. K. FarahmandDr. Richard J. FatemanDr. G. FedeleDr. A. R. Ferchmin
Dr. P. Ferrant
Dr. H. E. FettisDr. W. B. FichterDr. George FikiorisM r .J .C .S .S .F i l h oDr. L. Ford
Dr. Nicolao FornengoDr. J. France
Dr. B. Frank
Dr. S. FrasierDr. Stefan FredenhagenDr. A. J. FreemanDr. A. Frink
Dr. Jason M. Gallaspy
Dr. J. A. C. GallasDr. J. A. Carlson GallasDr. G. R. GamertsfelderDr. T. Garavaglia
Dr. Jaime Zaratiegui Garcia
Dr. C. G. Gardner
Dr. D. GarfinkleDr. P. N. GarnerDr. F. GasserDr. E. GathDr. P. Gatt
Dr. D. Gay
Dr. M. P. GelfandDr. M. R. GellerD r .A l iI .G e n cDr. Vincent Genot
Dr. M. F. George
Dr. P. GermainDr. Ing. Christoph GierullDr. S. P. GillDr. Federico GirosiDr. E. A. Gislason
Dr. M. I. Glasser
Dr. P. A. GlendinningDr. L. I. Goldfischer
Dr. Denis GolosovDr. I. J. GoodDr. J. Good
Mr. L. Gorin
Dr. Martin G¨ otz
Dr. R. GovindarajDr. M. De GraufDr. L. Green
Mr. Leslie O. Green
Dr. R. GreenwellDr. K. D. GrimsleyDr. Albert GroenenboomDr. V. GudmundssonDr. J. Guillera
Dr. K. GunnDr. D. L. Gunter
Dr. Julio C. Guti´ errez-Vega
Dr. Roger HaagmansDr. H. van HaeringenDr. B. HafiziDr. Bahman Hafizi
Dr. T. Hagfors
Dr. M. J. HaggertyDr. Timo HakulinenDr. Einar HalvorsenDr. S. E. Hammel
Dr. E. Hansen
Dr. Wes Harker
Dr. T. HarrettDr. D. O. HarrisDr. Frank HarrisMr. Mazen D. HasnaDr. Joel G. Heinrich
Dr. Sten Herlitz
Dr. Chris HerzogDr. A. HiguchiDr. R. E. HiseDr. Henrik Holm
Dr. Helmut H¨ olzler
Dr. N. HolteDr. R. W. HopperDr. P. N. HouleDr. C. J. HowardDr. J. H. Hubbell
Dr. J. R. Hull
Dr. W. HumphriesDr. Jean-Marc Hur´ e
Dr. Ben Yu-Kuang HuDr. Y. IksbeDr. Philip Ingenhoven
Mr. L. Iossif
Dr. Sean A. IrvineDr.´Ottar ´Isberg
Dr. Cyril-Daniel IskanderDr. S. A. Jackson
Dr. John David Jackson
Dr. Francois JaclotDr. B. JacobsDr. E. C. JamesDr. B. JancoviciDr. D. J. Jeffrey
Dr. H. J. Jensen
Acknowledgments xxv
Dr. Edwin F. Johnson
Dr. I. R. Johnson
Dr. Steven JohnsonDr. I. JohnstoneDr. Y. P. JoshiDr. Jae-Hun Jung
Dr. Damir Juric
Dr. Florian KaempferDr. S. KanmaniDr. Z. KapalDr. Dave Kasper
Dr. M. Kaufman
Dr. B. Kay
Dr. Avinash KhareDr. Ilki KimDr. Youngsun KimDr. S. KlamaDr. L. Klingen
Dr. C. Knessl
Dr. M. J. KnightDr. Mel KnightDr. Yannis KohninosDr. D. Koks
Dr. L. P. Kok
Dr. K. S. K¨ olbig
Dr. Y. KomninosDr. D. D. KonowalowDr. Z. KopalDr. I. Kostyukov
Dr. R. A. Krajcik
Dr. Vincent KrakoviackDr. Stefan Kramer
Dr. Tobias KramerDr. Hermann KrebsDr. J. W. Krozel
Dr. E. D. Krupnikov
Dr. Kun-Lin KuoDr. E. A. KuraevDr. Konstantinos KyritsisDr. Velimir Labinac
Dr. A. D. J. Lambert
Dr. A. LambertDr. A. LarrazaDr. K. D. LeeDr. M. Howard LeeDr. M. K. Lee
Dr. P. A. LeeDr. Todd Lee
Dr. J. Legg
Dr. Armando LemusDr. S. L. LevieDr. D. LeviDr. Michael Lexa
Dr. Kuo Kan Liang
Dr. B. LinetDr. M. A. LisaDr. Donald LivesayDr. H. Li
Dr. Georg Lohoefer
Dr. I. M. Longman
Dr. D. LongDr. Sylvie LorthoisDr. Y. L. LukeDr. W. LukoszDr. T. Lundgren
Dr. E. A. Luraev
Dr. R. LynchDr. R. MahurinDr. R. MallierDr. G. A. Mamon
Dr. A. Mangiarotti
Dr. I. ManningDr. J. MarmurDr. A. MartinSr. Yuzo MaruyamaDr. David J. Masiello
Dr. Richard Marthar
Dr. H. A. MavromatisDr. M. Mazzoni
Dr. K. B. MaDr. P. McCullaghDr. J. H. McDonnell
Dr. J. R. McGregor
Dr. Kim McInturffDr. N. McKinneyDr. David McA McKirdyDr. Rami Mehrem
Dr. W. N. Mei
Dr. Angelo MelinoMr. Jos´ e Ricardo Mendes
Dr. Andy MennimDr. J. P. MeunierDr. Gerard P. Michon
Dr. D. F. R. MildnerDr. D. L. Miller
Dr. Steve Miller
Dr. P. C. D. MillyDr. S. P. MitraDr. K. MiuraDr. N. Mohankumar
Dr. M. Moll
Dr. Victor H. MollDr. D. MonowalowMr. Tony MontagneseDr. Jim Morehead
Dr. J. Morice
Dr. W. Mueck
Dr. C. MuhlhausenDr. S. MukherjeeDr. R. R. M¨ uller
Dr. Pablo Parmezani MunhozDr. Paul Nanninga
Dr. A. Natarajan
Dr. Stefan NeumeierDr. C. T. NguyenDr. A. C. NicolDr. M. M. Nieto
Dr. P. Noerdlinger
Dr. A. N. NorrisDr. K. H. NorwichDr. A. H. NuttallDr. Frank O’BrienDr. R. P. O’Keeffe
Dr. A. Ojo
Dr. P. OlssonDr. M. Ortner
Dr. S. OstlundDr. J. OverduinDr. J. Pachner
Dr. John D. Paden
Mr. Robert A. PadgugDr. D. PapadopoulosDr. F. J. PappMr. Man Sik Park
Dr. Jong-Do Park
Dr. B. PattersonDr. R. F. PawulaDr. D. W. PeacemanDr. D. PelatDr. L. Peliti
Dr. Y. P. Pellegrini
xxvi Acknowledgments
Dr. G. J. Pert
Dr. Nicola Pessina
Dr. J. B. PetersonDr. Rickard PeterssonDr. Andrew PlumbDr. Dror Porat
Dr. E. A. Power
Dr. E. PredazziDr. William S. PriceDr. Paul RadmoreDr. F. Raynal
Dr. X. R. Resende
Dr. J. M. Riedler
Dr. Thomas RichardDr. E. RingelDr. T. M. RobertsDr. N. I. RobinsonDr. P. A. Robinson
Dr. D. M. Rosenblum
Dr. R. A. RosthalDr. J. R. RothDr. Klaus RottbrandDr. D. Roy
Dr. E. Royer
Dr. D. RudermannDr. Sanjib SabhapanditDr. C. T. SachradjaDr. J. SadikuDr. A. Sadiq
Dr. Motohiko Saitoh
Dr. Naoki SaitoDr. A. Salim
Dr. J. H. SamsonDr. Miguel A. Sanchis-LozanoDr. J. A. Sanders
Dr. M. A. F. Sanjun
Dr. P. SarquizDr. Avadh SaxenaDr. Vito ScarolaDr. O. Sch¨ arpf
Dr. A. Scherzinger
Dr. B. SchizerDr. Martin SchmidDr. J. ScholesDr. Mel SchopperDr. H. J. Schulz
Dr. G. J. SearsDr. Kazuhiko Seki
Dr. B. Seshadri
Dr. A. ShapiroDr. Masaki ShigemoriDr. J. S. ShengDr. Kenneth Ing Shing
Dr. Tomohiro Shirai
Dr. S. ShlomoDr. D. SiegelDr. Matthew StapletonDr. Steven H. Simon
Dr. Ashok Kumar Singal
Dr. C. Smith
Dr. G. C. C. SmithDr. Stefan Llewellyn SmithDr. S. SmithDr. G. SoltDr. J. Sondow
Dr. A. Sørenssen
Dr. Marcus SpradlinDr. Andrzej StaruszkiewiczDr. Philip C. L. StephensonDr. Edgardo Stockmeyer
Dr. J. C. Straton
Mr. H. SuraweeraDr. N. F. SvaiterDr. V. SvaiterDr. R. SzmytkowskiDr. S. Tabachnik
Dr. Erik Talvila
Dr. G. TanakaDr. C. Tanguy
Dr. G. K. TannahillDr. B. T. TanDr. C. Tavard
Dr. Gon¸ calo Tavares
Dr. Aba TelekiDr. Arash Dahi TaleghaniDr. D. TemperleyDr. A. J. Tervoort
Dr. Theodoros Theodoulidis
Dr. D. J. ThomasDr. Michael ThorwartDr. S. T. ThynellDr. D. C. TorneyDr. R. Tough
Dr. B. F. TreadwayDr. Ming Tsai
Dr. N. Turkkan
Dr. Sandeep TyagiDr. J. J. TysonDr. S. UeharaDr. M. Vadacchino
Dr. O. T. Valls
Dr. D. VandethMr. Andras VanyolosDr. D. VeitchMr. Jose Lopez Vicario
Dr. K. Vogel
Dr. J. M. M. J. Vogels
Dr. Alexis De VosDr. Stuart WalshDr. Reinhold WannemacherDr. S. WanzuraDr. J. Ward
Dr. S. I. Warshaw
Dr. R. WeberDr. Wei QianDr. D. H. WernerDr. E. Wetzel
Dr. Robert Whittaker
Dr. D. T. WiltonDr. C. WiufDr. K. T. WongM r .J .N .W r i g h tDr. J. D. Wright
Dr. D. Wright
Dr. D. WuDr. Michel Daoud Yacoub
Dr. Yu S. YakovlevDr. H.-C. YangDr. J. J. Yang
Dr. Z. J. Yang
Dr. J. J. WangDr. Peter WiderinMr. Chun Kin Au YeungDr. Kazuya Yuasa
Dr. S. P. Yukon
Dr. B. ZhangDr. Y. C. ZhangDr. Y. ZhaoDr. Ralf Zimmer
The Order of Presentation of the
Formulas
The question of the most expedient order in which to give the formulas, in particular, in what division
to include particular formulas such as the definite integrals, turned out to be quite complicated. Thethought naturally occurs to set up an order analogous to that of a dictionary. However, it is almostimpossible to create such a system for the formulas of integral calculus. Indeed, in an arbitrary formula
of the form
/integraldisplay
b
af(x)dx=A
one may make a large number of substitutions of the form x=ϕ(t) and thus obtain a number of
“synonyms” of the given formula. We must point out that the table of definite integrals by Bierens
de Haan and the earlier editions of the present reference both sin in the plethora of such “synonyms”
and formulas of complicated form. In the present edition, we have tried to keep only the simplest of the“synonym” formulas. Basically, we judged the simplicity of a formula from the standpoint of the simplicityof the arguments of the “outer” functions that appear in the integrand. Where possible, we have replaceda complicated formula with a simpler one. Sometimes, several complicated formulas were thereby reducedto a single, simpler one. We then kept only the simplest formula. As a result of such substitutions, we
sometimes obtained an integral that could be evaluated by use of the formulas of Chapter Two and the
Newton–Leibniz formula, or to an integral of the form
/integraldisplay
a
−af(x)dx,
where f(x) is an odd function. In such cases, the complicated integrals have been omitted.
Let us give an example using the expression
/integraldisplayπ/4
0(cotx−1)p−1
sin2xlntan xdx=−π
pcosecpπ. (0.1)
By making the natural substitution u=c o t x−1, we obtain
/integraldisplay∞
0up−1ln(1 + u)du=π
pcosecpπ. (0.2)
Integrals similar to formula (0.1) are omitted in this new edition. Instead, we have formula (0.2).
xxvii
xxviii The Order of Presentation of the Formulas
As a second example, let us take
I=/integraldisplayπ/2
0ln (tanpx+c o tpx)l nt a n xdx=0.
The substitution u=t a n xyields
I=/integraldisplay∞
0ln(up+u−p)l nu
1+u2du.
If we now set υ=l nu, we obtain
I=/integraldisplay∞
−∞υeυ
1+e2υln/parenleftbig
epυ+e−pυ/parenrightbig
dυ=/integraldisplay∞
−∞υln(2cosh pυ)
2c os h υdυ.
The integrand is odd, and, consequently, the integral is equal to 0.
Thus, before looking for an integral in the tables, the user should simplify as much as possible the
arguments (the “inner” functions) of the functions in the integrand.
The functions are ordered as follows: First we have the elementary functions:
1. The function f(x)=x.
2. The exponential function.
3. The hyperbolic functions.
4. The trigonometric functions.
5. The logarithmic function.
6. The inverse hyperbolic functions. (These are replaced with the corresponding logarithms in the
formulas containing definite integrals.)
7. The inverse trigonometric functions.
Then follow the special functions:
8. Elliptic integrals.
9. Elliptic functions.
10. The logarithm integral, the exponential integral, the sine integral, and the cosine integral functions.
11. Probability integrals and Fresnel’s integrals.
12. The gamma function and related functions.
13. Bessel functions.
14. Mathieu functions.
15. Legendre functions.
16. Orthogonal polynomials.
17. Hypergeometric functions.
18. Degenerate hypergeometric functions.
19. Parabolic cylinder functions.
20. Meijer’s and MacRobert’s functions.
21. Riemann’s zeta function.
The integrals are arranged in order of outer function according to the above scheme: the farther down
in the list a function occurs, (i.e., the more complex it is) the later will the corresponding formula appear
The Order of Presentation of the Formulas xxix
in the tables. Suppose that several expressions have the same outer function. For example, consider
sinex,s i nx, sin ln x. Here, the outer function is the sine function in all three cases. Such expressions are
then arranged in order of the inner function. In the present work, these functions are therefore arrangedin the following order: sin x,s i ne
x, sin ln x.
Our list does not include polynomials, rational functions, powers, or other algebraic functions. An
algebraic function that is included in tables of definite integrals can usually be reduced to a finite com-
bination of roots of rational power. Therefore, for classifying our formulas, we can conditionally treat a
power function as a generalization of an algebraic and, consequently, of a rational function.∗We shall
distinguish between all these functions and those listed above, and we shall treat them as operators.Thus, in the expression sin
2ex, we shall think of the squaring operator as applied to the outer function,
namely, the sine. In the expressionsinx+cos x
sinx−cosx, we shall think of the rational operator as applied to the
trigonometric functions sine and cosine. We shall arrange the operators according to the following order:
1. Polynomials (listed in order of their degree).
2. Rational operators.
3. Algebraic operators (expressions of the form Ap/q,w h e r e qandpare rational, and q>0; these are
listed according to the size of q).
4. Power operators.
Expressions with the same outer and inner functions are arranged in the order of complexity of the
operators. For example, the following functions [whose outer functions are all trigonometric, and whose
inner functions are all f(x)=x] are arranged in the order shown:
sinx,sinxcosx,1
sinx=c o s e c x,sinx
cosx=t a n x,sinx+c o s x
sinx−cosx,sinmx,sinmxcosx.
Furthermore, if two outer functions ϕ1(x)a n d ϕ2(x), where ϕ1(x) is more complex than ϕ2(x), appear
in an integrand and if any of the operations mentioned are performed on them, the corresponding integralwill appear [in the order determined by the position of ϕ
2(x) in the list] after all integrals containing
only the function ϕ1(x). Thus, following the trigonometric functions are the trigonometric and power
functions [that is, ϕ2(x)=x]. Then come
•combinations of trigonometric and exponential functions,
•combinations of trigonometric functions, exponential functions, and powers, etc.,
•combinations of trigonometric and hyperbolic functions, etc.
Integrals containing two functions ϕ1(x)a n d ϕ2(x) are located in the division and order corresponding
to the more complicated function of the two. However, if the positions of several integrals coincidebecause they contain the same complicated function, these integrals are put in the position defined bythe complexity of the second function.
To these rules of a general nature, we need to add certain particular considerations that will be easily
understood from the tables. For example, according to the above remarks, the function e
1
xcomes after
exas regards complexity, but ln xand ln1
xare equally complex since ln1
x=−lnx. In the section on
“powers and algebraic functions,” polynomials, rational functions, and powers of powers are formed from
power functions of the form ( a+bx)nand (α+βx)ν.
∗For any natural number n, the involution ( a+bx)nof the binomial a+bxis a polynomial. If nis a negative integer,
(a+bx)nis a rational function. If nis irrational, the function ( a+bx)nis not even an algebraic function.
This page intentionally left blank
Use of the Tables∗
For the effective use of the tables contained in this book, it is necessary that the user should first become
familiar with the classification system for integrals devised by the authors Ryzhik and Gradshteyn. This
classification is described in detail in the section entitled The Order of Presentation of the Formulas (see
page xxvii) and essentially involves the separation of the integrand into inner andouter functions. The
principal function involved in the integrand is called the outer function, and its argument, which is itself
usually another function, is called the inner function. Thus, if the integrand comprised the expression
lnsinx,t h eouter function would be the logarithmic function while its argument, the inner function,
would be the trigonometric function sin x. The desired integral would then be found in the section
dealing with logarithmic functions, its position within that section being determined by the position oftheinner function (here a trigonometric function) in Gradshteyn and Ryzhik’s list of functional forms.
It is inevitable that some duplication of symbols will occur within such a large collection of integrals,
and this happens most frequently in the first part of the book dealing with algebraic and trigonometric
integrands. The symbols most frequently involved are α,β,γ,δ,t,u,z,z
k, and Δ. The expressions
associated with these symbols are used consistently within each section and are defined at the start ofeach new section in which they occur. Consequently, reference should be made to the beginning of thesection being used in order to verify the meaning of the substitutions involved.
Integrals of algebraic functions are expressed as combinations of roots with rational power indices,
and definite integrals of such functions are frequently expressed in terms of the Legendre elliptic integrals
F(φ,k),E(φ,k)a n dΠ ( φ, n, k ), respectively, of the first, second, and third kinds.
The four inverse hyperbolic functions arcsinh z, arccosh z,a r c t a n h z, and arccoth zare introduced
through the definitions
arcsin z=1
iarcsinh( iz)
arccos z=1
iarccosh( z)
arctan z=1
iarctanh( iz)
arccot z=iarccoth( iz)
∗Prepared by Alan Jeffrey for the English language edition.
xxxi
xxxii Use of the Tables
or
arcsinh z=1
iarcsin( iz)
arccosh z=iarccos z
arctanh z=1
iarctan( iz)
arccoth z=1
iarccot( −iz)
The numerical constants CandGwhich often appear in the definite integrals denote Euler’s constant
and Catalan’s constant, respectively. Euler’s constant Cis defined by the limit
C= lim
s→∞/parenleftBiggs/summationdisplay
m=11
m−lns/parenrightBigg
=0.577215 ....
On occasion, other writers denote Euler’s constant by the symbol γ, but this is also often used instead
to denote the constant
γ=eC=1.781072 ....
Catalan’s constant Gis related to the complete elliptic integral
K≡K(k)≡/integraldisplayπ/2
0dx/radicalbig
1−k2sin2x
by the expression
G=1
2/integraldisplay1
0Kdk=∞/summationdisplay
m=0(−1)m
(2m+1 )2=0.915965 ....
Since the notations and definitions for higher transcendental functions that are used by different
authors are by no means uniform, it is advisable to check the definitions of the functions that occur inthese tables. This can be done by identifying the required function by symbol and name in the Index of
Special Functions and Notation on page xxxix, and by then referring to the defining formula or section
number listed there. We now present a brief discussion of some of the most commonly used alternative
notations and definitions for higher transcendental functions.
Bernoulli and Euler Polynomials and Numbers
Extensive use is made throughout the book of the Bernoulli and Euler numbers B
nandEnthat are
defined in terms of the Bernoulli and Euler polynomials of order n,Bn(x)a n dEn(x), respectively. These
polynomials are defined by the generating functions
text
et−1=∞/summationdisplay
n=0Bn(x)tn
n!for|t|<2π
and
2ext
et+1=∞/summationdisplay
n=0En(x)tn
n!for|t|<π .
The Bernoulli numbers are always denoted by Bnand are defined by the relation
Bn=Bn(0) for n=0,1,...,
when
B0=1,B 1=−1
2,B 2=1
6,B 4=−1
30,....
Use of the Tables xxxiii
The Euler numbers Enare defined by setting
En=2nEn/parenleftbigg1
2/parenrightbigg
forn=0,1,...
TheEnare all integral, and E0=1 ,E2=−1,E4=5 ,E6=−61,....
An alternative definition of Bernoulli numbers, which we shall denote by the symbol B∗
n,u s e st h e
same generating function but identifies the B∗
ndifferently in the following manner:
t
et−1=1−1
2t+B∗
1t2
2!−B∗
2t4
4!+....
This definition then gives rise to the alternative set of Bernoulli numbers
B∗
1=1/6,B∗
2=1/30,B∗
3=1/42,B∗
4=1/30,B∗
5=5/66,
B∗
6= 691 /2730,B∗
7=7/6,B∗
8= 3617 /510, ....
These differences in notation must also be taken into account when using the following relationships
that exist between the Bernoulli and Euler polynomials:
Bn(x)=1
2nn/summationdisplay
k=0/parenleftBign
k/parenrightBig
Bn−kEk(2x)n=0,1,...
En−1(x)=2n
n/braceleftbigg
Bn/parenleftbiggx+1
2/parenrightbigg
−Bn/parenleftBigx
2/parenrightBig/bracerightbigg
or
En−1(x)=2
n/braceleftBig
Bn(x)−2nBn/parenleftBigx
2/parenrightBig/bracerightBig
n=1,2,...
and
En−2(x)=2/parenleftBign
2/parenrightBig
−1n−2/summationdisplay
k=0/parenleftBign
k/parenrightBig/parenleftbig
2n−k−1/parenrightbig
Bn−kBn(x)n=2,3,...
There are also alternative definitions of the Euler polynomial of order n, and it should be noted that
some authors, using a modification of the third expression above, call/parenleftbigg2
n+1/parenrightbigg/braceleftBig
Bn(x)−2nBn/parenleftBigx
2/parenrightBig/bracerightBig
the Euler polynomial of order n.
Elliptic Functions and Elliptic Integrals
The following notations are often used in connection with the inverse elliptic functions sn u,c nu,a n d
dnu:
nsu=1
snuncu=1
cnundu=1
dnu
scu=snu
cnucsu=cnu
snudsu=dnu
snu
sdu=snu
dnucdu=cnu
dnudcu=dnu
cnu
xxxiv Use of the Tables
The elliptic integral of the third kind is defined by Gradshteyn and Ryzhik to be
Π/parenleftbig
ϕ, n2,k/parenrightbig
=/integraldisplayϕ
0da
/parenleftbig
1−n2sin2a/parenrightbig/radicalbig
1−k2sin2a
=/integraldisplaysinϕ
0dx
(1−n2x2)/radicalbig
(1−x2)(1−k2x2)/parenleftbig
−∞<n2<∞/parenrightbig
The Jacobi Zeta Function and Theta Functions
The Jacobi zeta function zn( u,k), frequently written Z(u), is defined by the relation
zn(u,k)=Z(u)=/integraldisplayu
0/braceleftbigg
dn2υ−E
K/bracerightbigg
dυ=E(u)−E
Ku.
This is related to the theta functions by the relationship
zn(u,k)=∂
∂ulnΘ(u)
giving
(i). zn( u,k)=π
2Kϑ/prime
1/parenleftBigπu
2K/parenrightBig
ϑ1/parenleftBigπu
2K/parenrightBig−cnudnu
snu
(ii). zn( u,k)=π
2Kϑ/prime
2/parenleftBigπu
2K/parenrightBig
ϑ2/parenleftBigπu
2K/parenrightBig−dnusnu
cnu
(iii). zn( u,k)=π
2Kϑ/prime
3/parenleftBigπu
2K/parenrightBig
ϑ3/parenleftBigπu
2K/parenrightBig−k2snucnu
dnu
(iv). zn( u,k)=π
2Kϑ/prime
4/parenleftBigπu
2K/parenrightBig
ϑ4/parenleftBigπu
2K/parenrightBig
Many different notations for the theta function are in current use. The most common variants are the
replacement of the argument uby the argument u/πand, occasionally, a permutation of the identification
of the functions ϑ1toϑ4with the function ϑ4replaced by ϑ.
The Factorial (Gamma) Function
In older reference texts, the gamma function Γ( z), defined by the Euler integral
Γ(z)=/integraldisplay∞
0tz−1e−tdt,
is sometimes expressed in the alternative notation
Γ(1 + z)=z!=Π ( z).
On occasions, the related derivative of the logarithmic factorial function Ψ( z)i su s e dw h e r e
d(lnz!)
dz=(z!)/prime
z!=Ψ (z).
Use of the Tables xxxv
This function satisfies the recurrence relation
Ψ(z)=Ψ ( z−1) +1
z−1
and is defined by the series
Ψ(z)=−C+∞/summationdisplay
n=0/parenleftbigg1
n+1−1
z+n/parenrightbigg
.
The derivative Ψ/prime(z) satisfies the recurrence relation
Ψ/prime(z+1 )=Ψ/prime(z)−1
z2
and is defined by the series
Ψ/prime(z)=∞/summationdisplay
n=01
(z+n)2.
Exponential and Related Integrals
The exponential integrals En(z) have been defined by Schloemilch using the integral
En(z)=/integraldisplay∞
1e−ztt−ndt (n=0,1,..., Rez>0).
They should not be confused with the Euler polynomials already mentioned. The function E1(z)i s
related to the exponential integral Ei( z) through the expressions
E1(z)=−Ei(−z)=/integraldisplay∞
ze−tt−1dt
and
li(z)=/integraldisplayz
0dt
lnt=E i( l n z)[ z>1].
The functions En(z) satisfy the recurrence relations
En(z)=1
n−1/braceleftbig
e−z−zEn−1(z)/bracerightbig
[n>1]
and
E/prime
n(z)=−En−1(z)
with
E0(z)=e−z/z.
The function En(z) has the asymptotic expansion
En(z)∼e−z
z/braceleftbigg
1−n
z+n(n+1 )
z2−n(n+1 ) (n+2 )
z3+···/bracerightbigg/bracketleftbigg
|argz|<3π
2/bracketrightbigg
while for large n,
En(x)=e−x
x+n/braceleftBigg
1+n
(x+n)2+n(n−2x)
(x+n)4+n/parenleftbig
6x2−8nx+n2/parenrightbig
(x+n)6+R(n, x)/bracerightBigg
,
where
−0.36n−4≤R(n, x)≤/parenleftbigg
1+1
x+n−1/parenrightbigg
n−4[x>0].
The sine and cosine integrals si( x) and ci( x) are related to the functions Si( x)a n dC i ( x)b yt h e
integrals
Si(x)=/integraldisplayx
0sint
tdt=s i (x)+π
2
and
xxxvi Use of the Tables
Ci(x)=C+l nx+/integraldisplayx
0(cost−1)
tdt.
The hyperbolic sine and cosine integrals shi( x)a n dc h i ( x) are defined by the relations
shi(x)=/integraldisplayx
0sinht
tdt
and
chi(x)=C+l nx+/integraldisplayx
0(cosht−1)
tdt.
Some authors write
Cin(x)=/integraldisplayx
0(1−cost)
tdt
so that
Cin(x)=−Ci(x)+l n x+C.
The error function erf( x) is defined by the relation
erf(x)=Φ ( x)=2√π/integraldisplayx
0e−t2dt,
and the complementary error function erfc( x) is related to the error function erfc( x)a n dt oΦ ( x)b yt h e
expression
erfc(x)=1−erf(x).
The Fresnel integrals S(x)a n dC(x) are defined by Gradshteyn and Ryzhik as
S(x)=2√
2π/integraldisplayx
0sint2dt
and
C(x)=2√
2π/integraldisplayx
0cost2dt.
Other definitions that are in use are
S1(x)=/integraldisplayx
0sinπt2
2dt, C1(x)=/integraldisplayx
0cosπt2
2dt,
and
S2(x)=1√
2π/integraldisplayx
0sint√
tdt, C2(x)=1√
2π/integraldisplayx
0cost√
tdt.
These are related by the expressions
S(x)=S1/parenleftBigg
x/radicalbigg
2
π/parenrightBigg
=S2/parenleftbig
x2/parenrightbig
and
C(x)=C1/parenleftBigg
x/radicalbigg
2
π/parenrightBigg
=C2/parenleftbig
x2/parenrightbig
Hermite and Chebyshev Orthogonal Polynomials
The Hermite polynomials Hn(x) are related to the Hermite polynomials Hen(x) by the relations
Hen(x)=2−n/2Hn/parenleftbiggx√
2/parenrightbigg
and
Hn(x)=2n/2Hen/parenleftBig
x√
2/parenrightBig
.
Use of the Tables xxxvii
These functions satisfy the differential equations
d2Hn
dx2−2xdHn
dx+2nHn=0
and
d2Hen
dx2−xdHen
dx+nHen=0.
They obey the recurrence relations
Hn+1=2xHn−2nHn−1
and
Hen+1=xHen−nHen−1.
The first six orthogonal polynomials Henare
He0=1,He1=x,He2=x2−1,He3=x3−3x,He4=x4−6x2+3,He5=x5−10x3+1 5x.
Sometimes the Chebyshev polynomial Un(x) of the second kind is defined as a solution of the equation
/parenleftbig
1−x2/parenrightbigd2y
dx2−3xdy
dx+n(n+2 )y=0.
Bessel Functions
A variety of different notations for Bessel functions are in use. Some common ones involve the replacement
ofYn(z)b yNn(z) and the introduction of the symbol
Λn(z)=/parenleftbigg1
2z/parenrightbigg−n
Γ(n+1 )Jn(z).
In the book by Gray, Mathews, and MacRobert, the symbol Yn(z) is used to denote1
2πYn(z)+
(ln 2−C)Jn(z) while Neumann uses the symbol Y(n)(z) for the identical quantity.
The Hankel functions H(1)
ν(z)a n d H(2)
ν(z) are sometimes denoted by Hsν(z)a n d Hiν(z), and some
authors write Gν(z)=/parenleftbigg1
2/parenrightbigg
πiH(1)
ν(z).
The Neumann polynomial On(t) is a polynomial of degree n+1i n1 /t, with O0(t)=1 /t.T h e
polynomials On(t) are defined by the generating function
1
t−z=J0(z)O0(t)+2∞/summationdisplay
k=1Jk(z)Ok(t),
giving
On(t)=1
4[n/2]/summationdisplay
k=0n(n−k−1)!
k!/parenleftbigg2
t/parenrightbiggn−2k+1
forn=1,2,...,
where/bracketleftbig1
2n/bracketrightbig
signifies the integral part of1
2n. The following relationship holds between three successive
polynomials:
(n−1)On+1(t)+(n+1 )On−1(t)−2/parenleftbig
n2−1/parenrightbig
tOn(t)=2n
tsin2nπ
2.
xxxviii Use of the Tables
The Airy functions Ai( z)a n dB i ( z) are independent solutions of the equation
d2u
dz2−zu=0.
The solutions can be represented in terms of Bessel functions by the expressions
Ai(z)=1
3√z/braceleftbigg
I−1/3/parenleftbigg2
3z3/2/parenrightbigg
−I1/3/parenleftbigg2
3z3/2/parenrightbigg/bracerightbigg
=1
π/radicalbiggz
3K1/3/parenleftbigg2
3z3/2/parenrightbigg
Ai(−z)=1
3√z/braceleftbigg
J1/3/parenleftbigg2
3z3/2/parenrightbigg
+J−1/3/parenleftbigg2
3z3/2/parenrightbigg/bracerightbigg
and by
Bi(z)=/radicalbiggz
3/braceleftbigg
I−1/3/parenleftbigg2
3z3/2/parenrightbigg
+I1/3/parenleftbigg2
3z3/2/parenrightbigg/bracerightbigg
,
Bi(−z)=/radicalbiggz
3/braceleftbigg
J−1/3/parenleftbigg2
3z3/2/parenrightbigg
−J1/3/parenleftbigg2
3z3/2/parenrightbigg/bracerightbigg
.
Parabolic Cylinder Functions and Whittaker Functions
The differential equation
d2y
dz2+/parenleftbig
az2+bz+c/parenrightbig
y=0
has associated with it the two equations
d2y
dz2+/parenleftbigg1
4z2+a/parenrightbigg
y=0a n dd2y
dz2−/parenleftbigg1
4z2+a/parenrightbigg
y=0,
the solutions of which are parabolic cylinder functions. The first equation can be derived from the second
by replacing zbyzeiπ/4andaby−ia.
The solutions of the equation
d2y
dz2−/parenleftbigg1
4z2+a/parenrightbigg
y=0
are sometimes written U(a,z)a n d V(a,z). These solutions are related to Whittaker’s function Dp(z)b y
the expressions
U(a,z)=D−a−1
2(z)
and
V(a,z)=1
πΓ/parenleftbigg1
2+a/parenrightbigg/braceleftBig
D−a−1
2(−z)+( s i n πa)D−a−1
2(z)/bracerightBig
.
Mathieu Functions
There are several accepted notations for Mathieu functions and for their associated parameters. The
defining equation used by Gradshteyn and Ryzhik is
d2y
dz2+/parenleftbig
a−2k2cos 2z/parenrightbig
y= 0 with k2=q.
Different notations involve the replacement of aandqin this equation by handθ,λandh2,a n d
bandc=2√q, respectively. The periodic solutions se n(z,q)a n dc e n(z,q) and the modified periodic
solutions Se n(z,q)a n dC e n(z,q) are suitably altered and, sometimes, re-normalized. A description of
these relationships together with the normalizing factors is contained in: Tables Relating to Mathieu
Functions . National Bureau of Standards, Columbia University Press, New York, 1951.
Index of Special Functions
NotationName of the function and the number of
the formula containing its definition
β(x) 8.37
Γ(z) Gamma function 8.31–8.33
γ(a,x),Γ(a,x) Incomplete gamma functions 8.35
Δ(n−k) Unit integer pulse function 18.1
ξ(s) 9.56
λ(x, y) 9.640
μ(x, β),μ(x, β, α ) 9.640
ν(x),ν(x, α) 9.640
Π(x) Lobachevskiy’s angle of parallelism 1.48
Π(ϕ, n, k ) Elliptic integral of the third kind 8.11
ζ(u) Weierstrass zeta function 8.17
ζ(z,q),ζ(z) Riemann’s zeta functions 9.51–9.54
Θ(u)=ϑ4/parenleftbigπu
2K/parenrightbig
,Θ1(u)=ϑ3/parenleftbigπu
2K/parenrightbig
Jacobian theta function 8.191–8.196⎧
⎪⎨
⎪⎩ϑ0(υ|τ)=ϑ4(υ|τ),
ϑ1(υ|τ),ϑ2(υ|τ),
ϑ3(υ|τ)⎫
⎪⎬
⎪⎭Elliptic theta functions 8.18, 8.19
σ(u) Weierstrass sigma function 8.17
Φ(x) See probability integral 8.25
Φ(z,s,υ) Lerch function 9.55
Φ(a,c;x)= 1F1(α;γ;x) Confluent hypergeometric function 9.21⎧
⎪⎨
⎪⎩Φ1(α,β,γ,x,y )
Φ2(β,β/prime,γ,x ,y )
Φ3(β,γ,x,y )⎫
⎪⎬
⎪⎭Degenerate hypergeometric series in two
variables9.26
ψ(x) Euler psi function 8.36
℘(u) Weierstrass elliptic function 8.16
am(u,k) Amplitude (of an elliptic function) 8.141
Bn Bernoulli numbers 9.61, 9.71
Bn(x) Bernoulli polynomials 9.620
B(x, y) Beta functions 8.38
Bx(p, q) Incomplete beta functions 8.39
bei(z),ber(z) Thomson functions 8.56
c o n t i n u e do nn e x tp a g e
xxxix
xl Index of Special Functions
continued from previous page
NotationName of the function and the number of
the formula containing its definition
C Euler constant 9.73, 8.367
C(x) Fresnel cosine integral 8.25
Cν(a) Young functions 3.76
Cλ
n(t) Gegenbauer polynomials 8.93
Cλ
n(x) Gegenbauer functions 8.932 1
ce2n(z,q),ce2n+1(z,q)Periodic Mathieu functions (Mathieu
functions of the first kind)8.61
Ce2n(z,q),Ce2n+1(z,q)Associated (modified) Mathieu functions of
the first kind8.63
chi(x) Hyperbolic cosine integral function 8.22
ci(x) Cosine integral 8.23
cn(u) Cosine amplitude 8.14
D(k)≡D Elliptic integral 8.112
D(ϕ, k) Elliptic integral 8.111
Dn(z),Dp(z) Parabolic cylinder functions 9.24–9.25
dnu Delta amplitude 8.14
e1,e2,e3 (used with the Weierstrass function) 8.162
En Euler numbers 9.63, 9.72
E(ϕ, k) Elliptic integral of the second kind 8.11–8.12/braceleftBigg
E(k)=E
E(k/prime)=E/prime/bracerightBigg
Complete elliptic integral of the second
kind8.11-8.12
E(p;ar:q;/rho1s:x) MacRobert’s function 9.4
Eν(z) Weber function 8.58
Ei(z) Exponential integral function 8.21
erf(x) Error function 8.25
erfc(x)=1−erf(x) Complementary error function 8.25
F(ϕ, k) Elliptic integral of the first kind 8.11–8.12
pFq(α1,...,α p;β1,...,β q;z) Generalized hypergeometric series 9.14
2F1(α,β;γ;z)=F(α,β;γ;z) Gauss hypergeometric function 9.10–9.13
1F1(α;γ;z)=Φ ( α,γ;z) Degenerate hypergeometric function 9.21
FΛ(α:β1,...,β n;
γ1,...... ,γ n:z1,...,z n)Hypergeometric function of several
variables9.19
F1,F2,F3,F4 Hypergeometric functions of two variables 9.18/braceleftBigg
fen(z,q),Fen(z,q)...
Feyn(z,q),Fekn(z,q).../bracerightBigg
Other nonperiodic solutions of Mathieu’s
equation8.64, 8.663
G Catalan constant 9.73
g2,g3 Invariants of the ℘(u)-function 8.161
gdx Gudermannian 1.49/braceleftBigg
gen(z,q),Gen(z,q)
Geyn(z,q),Gek n(z,q)/bracerightBigg
Other nonperiodic solutions of Mathieu’s
equation8.64, 8.663
Gm,n
p,q/parenleftBig
x/vextendsingle/vextendsingle/vextendsinglea1,... ,a p
b1,... ,b q/parenrightBig
Meijer’s functions 9.3
c o n t i n u e do nn e x tp a g e
Index of Special Functions xli
continued from previous page
NotationName of the function and the number of
the formula containing its definition
h(n) Unit integer function 18.1
heiν(z),herν(z) Thomson functions 8.56
H(1)
ν(z),H(2)
ν(z)Hankel functions of the first and second
kinds8.405, 8.42
H(u)=ϑ1/parenleftbigπu
2K/parenrightbigTheta function 8.192
H1(u)=ϑ2/parenleftbigπu
2K/parenrightbig
Theta function 8.192
Hn(z) Hermite polynomials 8.95
Hν(z) Struve functions 8.55
Iν(z) Bessel functions of an imaginary argument 8.406, 8.43
Ix(p, q) Normalized incomplete beta function 8.39
Jν(z) Bessel function 8.402, 8.41
Jν(z) Anger function 8.58
kν(x) Bateman’s function 9.210 3
K(k)=K,K(k/prime)=K/primeComplete elliptic integral of the first kind 8.11–8.12
Kν(z) Bessel functions of imaginary argument 8.407, 8.43
kei(z),ker(z) Thomson functions 8.56
L(x) Lobachevskiy’s function 8.26
Lν(z) Modified Struve function 8.55
Lα
n(z) Laguerre polynomials 8.97
li(x) Logarithm integral 8.24
Mλ,μ(z) Whittaker functions 9.22, 9.23
On(x) Neumann’s polynomials 8.59
Pμ
ν(z),Pμ
ν(x)Associated Legendre functions of the first
kind8.7, 8.8
Pν(z),Pν(x) Legendre functions and polynomials 8.82, 8.83, 8.91
P⎧
⎨
⎩abc
αβγδ
α/primeβ/primeγ/prime⎫
⎬
⎭Riemann’s differential equation 9.160
P(α,β)
n(x) Jacobi’s polynomials 8.96
Qμ
ν(z),Qμ
ν(x)Associated Legendre functions of the
second kind8.7, 8.8
Qν(z),Qν(x) Legendre functions of the second kind 8.82, 8.83
S(x) Fresnel sine integral 8.25
Sn(x) Schl¨afli’s polynomials 8.59
sμ,ν(z),Sμ,ν(z) Lommel functions 8.57
se2n+1(z,q),se2n+2(z,q) Periodic Mathieu functions 8.61
Se2n+1(z,q),Se2n+2(z,q)Mathieu functions of an imaginary
argument8.63
shi(x) Hyperbolic sine integral 8.22
si(x) Sine integral 8.23
snu Sine amplitude 8.14
Tn(x) Chebyshev polynomial of the 1stkind 8.94
Un(x) Chebyshev polynomials of the 2ndkind 8.94
c o n t i n u e do nn e x tp a g e
xlii Index of Special Functions
continued from previous page
NotationName of the function and the number of
the formula containing its definition
Uν(w,z),Vν(w,z) Lommel functions of two variables 8.578
Wλ,μ(z) Whittaker functions 9.22, 9.23
Yν(z) Neumann functions 8.403, 8.41
Zν(z) Bessel functions 8.401
Zν(z) Bessel functions
Notation
Symbol Meaning
⌊x⌋ The integral part of the real number x(also denoted by [ x])
/integraldisplay(b+)
a/integraldisplay(b−)
aContour integrals; the path of integration starting at the point aextends
to the point b(along a straight line unless there is an indication to the
contrary), encircles the point balong a small circle in the positive
(negative) direction, and returns to the point a, proceeding along the
original path in the opposite direction.
/integraltext
CLine integral along the curve C
PV/integraltextPrincipal value integral
z=x−iy The complex conjugate of z=x+iy
n! =1·2·3...n,0 ! = 1
(2n+1 ) ! ! =1·3...(2n+ 1). (double factorial notation)
(2n)!! =2·4...(2n). (double factorial notation)
0!! = 1 and ( −1)!! = 1 (cf. 3.372 for n=0 )
00=1 (cf. 0.112 and 0.113 for q=0 )
/parenleftBigp
n/parenrightBig=p(p−1)...(p−n+1 )
1·2...n=p!
n!(p−n)!,/parenleftbigp
0/parenrightbig
=1 ,/parenleftbigp
n/parenrightbig
=p!
n!(p−n)!
[n=1,2,...,p ≥n]
/parenleftBigx
n/parenrightBig
=x(x−1)...(x−n+1 )/n![n=0,1,...]
(a)n =a(a+1 )...(a+n−1) =Γ(a+n)
Γ(a)(Pochhammer symbol)
n/summationdisplay
k=muk =um+um+1+...+un.I fn<m , we definen/summationdisplay
k=muk=0
/summationdisplay/prime
n,/summationdisplay/prime
m,nSummation over all integral values of nexcluding n= 0, and summation
over all integral values of nandmexcluding m=n= 0, respectively.
/summationtext,/producttextAn empty/summationtexthas value 0, and an empty/producttexthas value 1
continued on next page
xliii
xliv Notation
continued from previous page
Symbol Meaning
δij=/braceleftBigg
1i=j
0i/negationslash=jKronecker delta
τ Theta function parameter (cf. 8.18)
×and∧ Vector product (cf. 10.11)
· Scalar product (cf. 10.11)
∇or “del” Vector operator (cf. 10.21)
∇2Laplacian (cf. 10.31)
∼ Asymptotically equal to
argz The argument of the complex number z=x+iy
curl or rot Vector operator (cf. 10.21)
div Vector operator (divergence) (cf. 10.21)
F Fourier transform (cf. 17.21)
Fc Fourier cosine transform (cf. 17.31)
Fs Fourier sine transform (cf. 17.31)
grad Vector operator (gradient) (cf. 10.21)
hiandgij Metric coefficients (cf. 10.51)
HHermitian transpose of a vector or matrix (cf. 13.123)
H(x)=/braceleftBigg
0x<0
1x≥0Heaviside step function
Imz≡y The imaginary part of the complex number z=x+iy
kThe letter k(when not used as an index of summation) denotes a number
in the interval [0, 1]. This notation is used in integrals that lead to ellipticintegrals. In such a connection, the number√
1−k2is denoted by k/prime.
L Laplace transform (cf. 17.11)
M Mellin transform (cf. 17.41)
N The natural numbers (0 ,1,2,...)
O(f(z))The order of the function f(z). Suppose that the point zapproaches z0.
If there exists an M>0 such that |g(z)|≤M|f(z)|in some sufficiently
small neighborhood of the point z0,w ew r i t e g(z)=O(f(z)).
continued on next page
Notation xlv
continued from previous page
Symbol Meaning
q The nome, a theta function parameter (cf. 8.18)
R The real numbers
R(x) A rational function
Rez≡x The real part of the complex number z=x+iy
Sm
n Stirling number of the first kind (cd. 9.74)
Sm
n Stirling number of the second kind (cd. 9.74)
signx=⎧
⎪⎨
⎪⎩+1x>0
0x=0
−1x<0The sign (signum) of the real number x
TTranspose of a vector or matrix (cf. 13.115)
Z The integers (0 ,±1,±2,...)
Zb Bilateral ztransform (cf. 18.1)
Zu Unilateral ztransform (cf. 18.1)
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Note on the Bibliographic References
The letters and numbers following equations refer to the sources used by Russian editors. The key to the
letters will be found preceding each entry in the Bibliography beginning on page 1141. Roman numerals
indicate the volume number of a multivolume work. Numbers without parentheses indicate page numbers,numbers in single parentheses refer to equation numbers in the original sources.
Some formulas were changed from their form in the source material. In such cases, the letter aappears
at the end of the bibliographic references.
As an example, we may use the reference to equation 3.354–5:
ET I 118 (1) a
The key on page 1141 indicates that the book referred to is:
Erd´elyi, A. et al., Tables of Integral Transforms .
The Roman numeral denotes volume one of the work; 118 is the page on which the formula will be
found; (1) refers to the number of the formula in this source; and the aindicates that the expression
appearing in the source differs in some respect from the formula in this book.
In several cases, the editors have used Russian editions of works published in other languages. Under
such circumstances, because the pagination and numbering of equations may be altered, we have referredthe reader only to the original sources and dispensed with page and equation numbers.
xlvii
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0 Introduction
0.1 Finite Sums
0.11 Progressions
0.111 Arithmetic progression.
n−1/summationdisplay
k=0(a+kr)=n
2[2a+(n−1)r]=n
2(a+l)[ l=a+(n−1)ris the last term]
0.112 Geometric progression.
n/summationdisplay
k=1aqk−1=a(qn−1)
q−1[q/negationslash=1 ]
0.113 Arithmetic-geometric progression.
n−1/summationdisplay
k=0(a+kr)qk=a−[a+(n−1)r]qn
1−q+rq/parenleftbig
1−qn−1/parenrightbig
(1−q)2
[q/negationslash=1,n > 1] JO (5)
0.1148n−1/summationdisplay
k=1k2xk=/parenleftbig
−n2+2n−1/parenrightbig
xn+2+/parenleftbig
2n2−2n−1/parenrightbig
xn+1−n2xn+x2+x
(1−x)3
0.12 Sums of powers of natural numbers
0.121n/summationdisplay
k=1kq=nq+1
q+1+nq
2+1
2/parenleftBigq
1/parenrightBig
B2nq−1+1
4/parenleftBigq
3/parenrightBig
B4nq−3+1
6/parenleftBigq
5/parenrightBig
B6nq−5+···
=nq+1
q+1+nq
2+qnq−1
12−q(q−1)(q−2)
720nq−3+q(q−1)(q−2)(q−3)(q−4)
30,240nq−5−···
[last term contains either norn2]CE 332
1.n/summationdisplay
k=1k=n(n+1 )
2CE 333
2.n/summationdisplay
k=1k2=n(n+ 1)(2 n+1 )
6CE 333
3.n/summationdisplay
k=1k3=/bracketleftbiggn(n+1 )
2/bracketrightbigg2
CE 333
1
2 Finite Sums 0.122
4.n/summationdisplay
k=1k4=1
30n(n+ 1)(2 n+ 1)(3 n2+3n−1) CE 333
5.n/summationdisplay
k=1k5=1
12n2(n+1 )2(2n2+2n−1) CE 333
6.n/summationdisplay
k=1k6=1
42n(n+ 1)(2 n+ 1)(3 n4+6n3−3n+1 ) CE 333
7.n/summationdisplay
k=1k7=1
24n2(n+1 )2(3n4+6n3−n2−4n+2 ) CE 333
0.122n/summationdisplay
k=1(2k−1)q=2q
q+1nq+1−1
2/parenleftBigq
1/parenrightBig
2q−1B2nq−1−1
4/parenleftBigq
3/parenrightBig
2q−3/parenleftbig
23−1/parenrightbig
B4nq−3−···
[last term contains either norn2.]
1.n/summationdisplay
k=1(2k−1) =n2
2.n/summationdisplay
k=1(2k−1)2=1
3n(4n2−1) JO (32a)
3.n/summationdisplay
k=1(2k−1)3=n2(2n2−1) JO (32b)
4.11n/summationdisplay
k=1(mk−1) =n
2[m(n+1 )−2]
5.10n/summationdisplay
k=1(mk−1)2=1
6n[m2(n+ 1)(2 n+1 )−6m(n+1 )+6 ]
6.10n/summationdisplay
k=1(mk−1)3=1
4n[m3n(n+1 )2−2m2(n+ 1)(2 n+1 )+6 m(n+1 )−4]
0.123n/summationdisplay
k=1k(k+1 )2=1
12n(n+1 ) (n+ 2)(3 n+5 )
0.124
1.q/summationdisplay
k=1k/parenleftbig
n2−k2/parenrightbig
=1
4q(q+1 )/parenleftbig
2n2−q2−q/parenrightbig
[q=1,2,...]
2.10n/summationdisplay
k=1k(k+1 )3=1
60n(n+1 )/parenleftbig
12n3+6 3n2+ 107 n+5 8/parenrightbig
0.125n/summationdisplay
k=1k!·k=(n+1 ) !−1 AD (188.1)
0.126n/summationdisplay
k=0(n+k)!
k!(n−k)!=/radicalbigge
πKn+1
2/parenleftbigg1
2/parenrightbigg
WA 94
0.142 Sums of the binomial coefficients 3
0.13 Sums of reciprocals of natural numbers
0.13111n/summationdisplay
k=11
k=C+l nn+1
2n−∞/summationdisplay
k=2Ak
n(n+1 )...(n+k−1), JO (59), AD (1876)
where
Ak=1
k/integraldisplay1
0x(1−x)(2−x)(3−x)···(k−1−x)dx
A2=1
12,A 3=1
12
A4=19
120,A 5=9
20,
0.1327n/summationdisplay
k=11
2k−1=1
2(C+l nn)+l n2+B2
8n2+/parenleftbig
23−1/parenrightbig
B4
64n4+... JO (71a)a
0.133n/summationdisplay
k=21
k2−1=3
4−2n+1
2n(n+1 )JO (184f)
0.14 Sums of products of reciprocals of natural numbers
1.n/summationdisplay
k=11
[p+(k−1)q](p+kq)=n
p(p+nq)GI III (64)a
2.n/summationdisplay
k=11
[p+(k−1)q](p+kq)[p+(k+1 )q]=n(2p+nq+q)
2p(p+q)(p+nq)[p+(n+1 )q]GI III (65)a
3.n/summationdisplay
k=11
[p+(k−1)q](p+kq)...[p+(k+l)q]
=1
(l+1 )q/braceleftbigg1
p(p+q)...(p+lq)−1
(p+nq)[p+(n+1 )q]...[p+(n+l)q]/bracerightbigg
AD (1856)a
4.n/summationdisplay
k=11
[1 + (k−1)q][1 + ( k−l)q+p]=1
p/bracketleftBiggn/summationdisplay
k=11
1+(k−1)q−n/summationdisplay
k=11
1+(k−1)q+p/bracketrightBigg
GI III (66)a
0.142n/summationdisplay
k=1k2+k−1
(k+2 ) !=1
2−n+1
(n+2 ) !JO (157)
0.15 Sums of the binomial coefficients
Notation :nis a natural number.
1.m/summationdisplay
k=0/parenleftbiggn+k
n/parenrightbigg
=/parenleftbiggn+m+1
n+1/parenrightbigg
KR 64 (70.1)
2. 1 +/parenleftBign
2/parenrightBig
+/parenleftBign
4/parenrightBig
+...=2n−1KR 62 (58.1)
4 Finite Sums 0.152
3./parenleftBign
1/parenrightBig
+/parenleftBign
3/parenrightBig
+/parenleftBign
5/parenrightBig
+...=2n−1KR 62 (58.1)
4.m/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig
=(−1)m/parenleftbiggn−1
m/parenrightbigg
[n≥1] KR 64 (70.2)
0.152
1./parenleftBign
0/parenrightBig
+/parenleftBign
3/parenrightBig
+/parenleftBign
6/parenrightBig
+...=1
3/parenleftBig
2n+ 2cosnπ
3/parenrightBig
KR 62 (59.1)
2./parenleftBign
1/parenrightBig
+/parenleftBign
4/parenrightBig
+/parenleftBign
7/parenrightBig
+...=1
3/parenleftbigg
2n+ 2cos(n−2)π
3/parenrightbigg
KR 62 (59.2)
3./parenleftBign
2/parenrightBig
+/parenleftBign
5/parenrightBig
+/parenleftBign
8/parenrightBig
+...=1
3/parenleftbigg
2n+ 2cos(n−4)π
3/parenrightbigg
KR 62 (59.3)
0.153
1./parenleftBign
0/parenrightBig
+/parenleftBign
4/parenrightBig
+/parenleftBign
8/parenrightBig
+...=1
2/parenleftBig
2n−1+2n
2cosnπ
4/parenrightBig
KR 63 (60.1)
2./parenleftBign
1/parenrightBig
+/parenleftBign
5/parenrightBig
+/parenleftBign
9/parenrightBig
+...=1
2/parenleftBig
2n−1+2n
2sinnπ
4/parenrightBig
KR 63 (60.2)
3./parenleftBign
2/parenrightBig
+/parenleftBign
6/parenrightBig
+/parenleftBign
10/parenrightBig
+...=1
2/parenleftBig
2n−1−2n
2cosnπ
4/parenrightBig
KR 63 (60.3)
4./parenleftBign
3/parenrightBig
+/parenleftBign
7/parenrightBig
+/parenleftBign
11/parenrightBig
+...=1
2/parenleftBig
2n−1−2n
2sinnπ
4/parenrightBig
KR 63 (60.4)
0.154
1.n/summationdisplay
k=0(k+1 )/parenleftBign
k/parenrightBig
=2n−1(n+2 ) [ n≥0] KR 63 (66.1)
2.n/summationdisplay
k=1(−1)k+1k/parenleftBign
k/parenrightBig
=0 [ n≥2] KR 63 (66.2)
3.N/summationdisplay
k=0(−1)k/parenleftbiggN
k/parenrightbigg
kn−1=0 [ N≥n≥1; 00≡1]
4.n/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig
kn=(−1)nn![ n≥0; 00≡1]
5.n/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig
(α+k)n=(−1)nn![ n≥0; 00≡1]
6.N/summationdisplay
k=0(−1)k/parenleftbiggN
k/parenrightbigg
(α+k)n−1=0 [ N≥n≥1,00≡1N, n∈N+]
0.155
1.n/summationdisplay
k=1(−1)k+1
k+1/parenleftBign
k/parenrightBig
=n
n+1KR 63 (67)
0.159 Sums of the binomial coefficients 5
2.n/summationdisplay
k=01
k+1/parenleftBign
k/parenrightBig
=2n+1−1
n+1KR 63 (68.1)
3.n/summationdisplay
k=0αk+1
k+1/parenleftBign
k/parenrightBig
=(α+1 )n+1−1
n+1KR 63 (68.2)
4.n/summationdisplay
k=1(−1)k+1
k/parenleftBign
k/parenrightBig
=n/summationdisplay
m=11
mKR 64 (69)
0.156
1.p/summationdisplay
k=0/parenleftBign
k/parenrightBig/parenleftbiggm
p−k/parenrightbigg
=/parenleftbiggn+m
p/parenrightbigg
[mis a natural number] KR 64 (71.1)
2.n−p/summationdisplay
k=0/parenleftBign
k/parenrightBig/parenleftbiggn
p+k/parenrightbigg
=(2n)!
(n−p)!(n+p)!KR 64 (71.2)
0.157
1.n/summationdisplay
k=0/parenleftBign
k/parenrightBig
2=/parenleftbigg2n
n/parenrightbigg
KR 64 (72.1)
2.2n/summationdisplay
k=0(−1)k/parenleftbigg2n
k/parenrightbigg
2=(−1)n/parenleftbigg2n
n/parenrightbigg
KR 64 (72.2)
3.2n+1/summationdisplay
k=0(−1)k/parenleftbigg2n+1
k/parenrightbigg
2=0 KR 64 (72.3)
4.n/summationdisplay
k=1k/parenleftBign
k/parenrightBig
2=(2n−1)!
[(n−1)!]2KR 64 (72.4)
0.15810
1.n/summationdisplay
k=1/bracketleftbigg
2k/parenleftbigg2n−k
n−k/parenrightbigg
−2k+1/parenleftbigg2n−k−1
n−k−1/parenrightbigg/bracketrightbigg
k=4n−/parenleftbigg2n
n/parenrightbigg
2.n/summationdisplay
k=1/bracketleftbigg
2k/parenleftbigg2n−k
n−k/parenrightbigg
−2k+1/parenleftbigg2n−k−1
n−k−1/parenrightbigg/bracketrightbigg
k2=4n−/parenleftbigg2n
n/parenrightbigg
3·4n
3.n/summationdisplay
k=1/bracketleftbigg
2k/parenleftbigg2n−k
n−k/parenrightbigg
−2k+1/parenleftbigg2n−k−1
n−k−1/parenrightbigg/bracketrightbigg
k3=( 6n+ 13)4n−18n/parenleftbigg2n
n/parenrightbigg
4.n/summationdisplay
k=1/bracketleftbigg
2k/parenleftbigg2n−k
n−k/parenrightbigg
−2k+1/parenleftbigg2n−k−1
n−k−1/parenrightbigg/bracketrightbigg
k4=( 3 2n2+ 104 n)/parenleftbigg2n
n/parenrightbigg
−(60n+ 75)4n
0.15910
1.n/summationdisplay
k=0/bracketleftbigg/parenleftbigg2n
n−k/parenrightbigg
−/parenleftbigg2n
n−k−1/parenrightbigg/bracketrightbigg
k=1
2/bracketleftbigg
4n−/parenleftbigg2n
n/parenrightbigg/bracketrightbigg
6 Numerical Series and Infinite Products 0.160
2.n/summationdisplay
k=0/bracketleftbigg/parenleftbigg2n
n−k/parenrightbigg
−/parenleftbigg2n
n−k−1/parenrightbigg/bracketrightbigg
k2=1
2/bracketleftbigg
(2n+1 )/parenleftbigg2n
n/parenrightbigg
−4n/bracketrightbigg
3.n/summationdisplay
k=0/bracketleftbigg/parenleftbigg2n
n−k/parenrightbigg
−/parenleftbigg2n
n−k−1/parenrightbigg/bracketrightbigg
k3=(3n+2 )
4·4n−1
2/parenleftbigg2n
n/parenrightbigg
(3n+1 )
0.16010
1.2n/summationdisplay
k=n+1/parenleftbigg2n
k/parenrightbigg
αk+1
2/parenleftbigg2n
n/parenrightbigg
αn+(1 +α)2n−1(1−α)
2n−1/summationdisplay
k=0/parenleftbigg2k
k/parenrightbigg/bracketleftbiggα
(1 +α)2/bracketrightbiggk
=1
2(1 +α)2n
2.n/summationdisplay
r=0(−1)r/parenleftBign
r/parenrightBigΓ(r+b)
Γ(r+a)=B(n+a−b,b)
Γ(a−b)
0.2 Numerical Series and Infinite Products
0.21 The convergence of numerical series
The series
0.211∞/summationdisplay
k=1uk=u1+u2+u3+...
is said to converge absolutely if the series
0.212∞/summationdisplay
k=1|uk|=|u1|+|u2|+|u3|+···,
composed of the absolute values of its terms converges. If the series 0.211 converges and the series 0.212
diverges, the series 0.211 is said to converge conditionally . Every absolutely convergent series converges.
0.22 Convergence tests
Suppose that
lim
k→∞|uk|1/k=q
Ifq<1, the series 0.211 converges absolutely. On the other hand, if q>1, the series 0.211 diverges.
(Cauchy)
0.222 Suppose that
lim
k→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingleu
k+1
uk/vextendsingle/vextendsingle/vextendsingle/vextendsingle=q
Here, if q<1, the series 0.211 converges absolutely. If q>1, the series 0.211 diverges. If/vextendsingle/vextendsingle/vextendsingle/vextendsingleu
k+1
uk/vextendsingle/vextendsingle/vextendsingle/vextendsingle
approaches 1 but remains greater than unity, then the series 0.211 diverges. (d’Alembert)
0.223 Suppose that
lim
k→∞k/braceleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleu
k
uk+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1/bracerightbigg
=q
Here, if q>1, the series 0.211 converges absolutely. If q<1, the series 0.211 diverges. (Raabe)
0.229 Convergence tests 7
0.224 Suppose that f(x) is a positive decreasing function and that
lim
k→∞ekf/parenleftbig
ek/parenrightbig
f(k)=q
for natural k.I fq<1, the series/summationtext∞
k=1f(k) converges. If q>1, this series diverges. (Ermakov)
0.225 Suppose that/vextendsingle/vextendsingle/vextendsingle/vextendsingleu
k
uk+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle=1+q
k+|vk|
kp,
where p>1a n dt h e |vk|are bounded, that is, the |vk|are all less than some M, which is independent
ofk. Here, if q>1, the series 0.211 converges absolutely. If q≤1, this series diverges. (Gauss)
0.226 Suppose that a function f(x) defined for x≥q≥1 is continuous, positive, and decreasing. Under
these conditions, the series
∞/summationdisplay
k=1f(k)
converges or diverges accordingly as the integral/integraldisplay∞
qf(x)dx
converges or diverges (the Cauchy integral test).
0.227 Suppose that all terms of a sequence u1,u2,...,u nare positive. In such a case, the series
1.∞/summationdisplay
k=1(−1)k+1uk=u1−u2+u3−...
is called an alternating series.
If the terms of an alternating series decrease monotonically in absolute value and approach zero,
that is, if
2. uk+1<ukand lim
k→∞uk=0,
the series 0.227 1 converges. Here, the remainder of the series is
3.∞/summationdisplay
k=n+1(−1)k−n+1uk=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
k=1(−1)k+1uk−n/summationdisplay
k=1(−1)k+1uk<un+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(Leibniz)
0.228 If the series
1.∞/summationdisplay
k=1vk=v1+v2+...+vk+...
converges and the numbers ukform a monotonic bounded sequence, that is, if |uk|<Mfor some
number Mand for all k,t h es e r i e s
2.∞/summationdisplay
k=1ukvk=u1v1+u2v2+...+ukvk+... FI II 354
converges. (Abel)
0.229 If the partial sums of the series 0.228 1 are bounded and if the numbers ukconstitute a monotonic
sequence that approaches zero, that is, if
8 Numerical Series and Infinite Products 0.231
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay
k=1vk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<M [n=1,2,...] and lim
k→∞uk=0, FI II 355
then the series 0.228 2 converges (Dirichlet).
0.23–0.24 Examples of numerical series
0.231 Progressions
1.∞/summationdisplay
k=0aqk=a
1−q[|q|<1]
2.∞/summationdisplay
k=0(a+kr)qk=a
1−q+rq
(1−q)2[|q|<1] (cf. 0.113 )
0.232
1.∞/summationdisplay
k=1(−1)k+11
k=l n2 ( c f . 1.511 )
2.∞/summationdisplay
k=1(−1)k+11
2k−1=1−2∞/summationdisplay
k=11
(4k−1)(4k+1 )=π
4
(cf.1.643 )
3.∗∞/summationdisplay
k=1ka
bk=1
(b−1)a+1a/summationdisplay
i=1⎡
⎣1
ba−ii/summationdisplay
j=0(−1)j(a+1 ) ! ( i−j)a
j!(a+1−j)!⎤
⎦
[a=1,2,3,..., b /negationslash=1 ]
0.233
1.∞/summationdisplay
k=11
kp=1+1
2p+1
3p+...=ζ(p)[ R e p>1] WH
2.∞/summationdisplay
k=1(−1)k+11
kp=( 1−21−p)ζ(p)[ R e p>0] WH
3.10∞/summationdisplay
k=11
k2n=22n−1π2n
(2n)!|B2n|,∞/summationdisplay
k=11
k2=π2
6FI II 721
4.∞/summationdisplay
k=1(−1)k+11
k2n=(22n−1−1)π2n
(2n)!|B2n| JO (165)
5.∞/summationdisplay
k=11
(2k−1)2n=(22n−1)π2n
2·(2n)!|B2n| JO (184b)
6.∞/summationdisplay
k=1(−1)k+1 1
(2k−1)2n+1=π2n+1
22n+2(2n)!|E2n| JO (184d)
0.236 Examples of numerical series 9
0.234
1.∞/summationdisplay
k=1(−1)k+11
k2=π2
12EU
2.∞/summationdisplay
k=11
(2k−1)2=π2
8EU
3.∞/summationdisplay
k=0(−1)k
(2k+1 )2=G FI II 482
4.∞/summationdisplay
k=1(−1)k+1
(2k−1)3=π3
32EU
5.∞/summationdisplay
k=11
(2k−1)4=π4
96EU
6.∞/summationdisplay
k=1(−1)k+1
(2k−1)5=5π5
1536EU
7.∞/summationdisplay
k=1(−1)k+1 k
(k+1 )2=π2
12−ln2
8.6∞/summationdisplay
k=11
k(2k+1 )=2−2ln2
9.∗∞/summationdisplay
n=1Γ/parenleftbig
n+1
2/parenrightbig
n2Γ(n)=√πln4
0.235 Sn=∞/summationdisplay
k=11
(4k2−1)n
S1=1
2,S 2=π2−8
16,S 3=32−3π2
64,S 4=π4+3 0π2−384
768
JO (186)
0.236
1.∞/summationdisplay
k=11
k(4k2−1)=2l n2 −1 BR 51a
2.∞/summationdisplay
k=11
k(9k2−1)=3
2(ln 3−1) BR 51a
3.∞/summationdisplay
k=11
k(36k2−1)=−3+3
2ln 3 + 2ln 2 BR 52, AD (6913.3)
4.∞/summationdisplay
k=1k
(4k2−1)2=1
8BR 52
5.∞/summationdisplay
k=11
k(4k2−1)2=3
2−2ln2 BR 52
10 Numerical Series and Infinite Products 0.237
6.∞/summationdisplay
k=112k2−1
k(4k2−1)2=2l n2 AD (6917.3), BR 52
7.6∞/summationdisplay
k=11
k(2k+1 )2=4−π2
4−2l n2
0.237
1.∞/summationdisplay
k=11
(2k−1)(2k+1 )=1
2AD (6917.2), BR 52
2.∞/summationdisplay
k=11
(4k−1)(4k+1 )=1
2−π
8
3.∞/summationdisplay
k=21
(k−1)(k+1 )=3
4[cf.0.133 ],
4.∞/summationdisplay/prime
k=1,k/negationslash=m1
(m+k)(m−k)=−3
4m2[mis an integer] AD (6916.1)
5.∞/summationdisplay/prime
k=1,k/negationslash=m(−1)k−1
(m−k)(m+k)=3
4m2[mis an even number] AD (6916.2)
0.238
1.∞/summationdisplay
k=11
(2k−1)2k(2k+1 )=l n2 −1
2GI III (93)
2.∞/summationdisplay
k=1(−1)k+1
(2k−1)2k(2k+1 )=1
2(1−ln2) GI III (94)a
3.∞/summationdisplay
k=01
(3k+ 1)(3 k+ 2)(3 k+ 3)(3 k+4 )=1
6−1
4ln3 +π
12√
3GI III (95)
0.239
1.11∞/summationdisplay
k=1(−1)k+11
3k−2=1
3/parenleftbiggπ√
3+l n2/parenrightbigg
GI III (85), BR∗161 (1)
2.7∞/summationdisplay
k=1(−1)k+11
3k−1=1
3/parenleftbiggπ√
3−ln 2/parenrightbigg
BR∗161 (1)
3.∞/summationdisplay
k=1(−1)k+11
4k−3=1
4√
2/bracketleftBig
π+2l n/parenleftBig√
2+1/parenrightBig/bracketrightBig
BR∗161 (1)
4.∞/summationdisplay
k=1(−1)[k+3
2]1
k=π
4+1
2ln2 GI III (87)
0.241 Examples of numerical series 11
5.∞/summationdisplay
k=1(−1)[k+3
2]1
2k−1=π
2√
2
6.∞/summationdisplay
k=1(−1)[k+5
3]1
2k−1=5π
12GI III (88)
7.∞/summationdisplay
k=11
(8k−1)(8k+1 )=1
2−π
16/parenleftBig√
2+1/parenrightBig
0.241
1.∞/summationdisplay
k=11
2kk=l n2 JO (172g)
2.∞/summationdisplay
k=11
2kk2=π2
12−1
2(ln 2)2JO (174)
3.11∞/summationdisplay
n=0/parenleftbigg2n
n/parenrightbigg
pn=1√1−4p/bracketleftbig
0≤p<1
4/bracketrightbig
4.10∞/summationdisplay
n=1pn
n2=π2
6−/integraldisplayp
1ln(1−x)
xdx [0≤p≤1]
5.10i/summationdisplay
j=1/bracketleftbigg
2j/parenleftbigg2i−j
i−j/parenrightbigg
−2j+1/parenleftbigg2i−(j+1 )
i−(j+1 )/parenrightbigg/bracketrightbigg
j=4i−/parenleftbigg2i
i/parenrightbigg
/bracketleftBig/parenleftBign
m/parenrightBig
=0,m < 0/bracketrightBig
6.10i/summationdisplay
j=1/bracketleftbigg
2j/parenleftbigg2i−j
i−j/parenrightbigg
−2j+1/parenleftbigg2i−(j+1 )
i−(j+1 )/parenrightbigg/bracketrightbigg
j2=4i/parenleftbigg2i
i/parenrightbigg
−3·4i
/bracketleftBig/parenleftBign
m/parenrightBig
=0,m < 0/bracketrightBig
7.10i/summationdisplay
j=1/bracketleftbigg
2j/parenleftbigg2i−j
i−j/parenrightbigg
−2j+1/parenleftbigg2i−(j+1 )
i−(j+1 )/parenrightbigg/bracketrightbigg
j3=( 6i+ 13)4i−18i/parenleftbigg2i
i/parenrightbigg
/bracketleftBig/parenleftBign
m/parenrightBig
=0,m < 0/bracketrightBig
8.10i/summationdisplay
j=1/bracketleftbigg
2j/parenleftbigg2i−j
i−j/parenrightbigg
−2j+1/parenleftbigg2i−(j+1 )
i−(j+1 )/parenrightbigg/bracketrightbigg
j4=/parenleftbig
32i2+ 104 i/parenrightbig/parenleftbigg2i
i/parenrightbigg
−(60i+ 75)4i
9.102n/summationdisplay
j=n+1/parenleftbigg2n
j/parenrightbigg
kj+1
2/parenleftbigg2n
n/parenrightbigg
kn+(1 +k)2n−1(1−k)
2n−1/summationdisplay
i=0/parenleftbigg2i
i/parenrightbigg/bracketleftbiggk
(1 +k)2/bracketrightbiggi
=1
2(1 +k)2n
10.10i/summationdisplay
k=0/parenleftbiggi+k
k/parenrightbigg
2i−k=4i
12 Numerical Series and Infinite Products 0.242
11.10i/summationdisplay
k=0/parenleftbiggi+k
h/parenrightbigg
i−kk=(i+1 ) 4i−(2i+1 )/parenleftbigg2i
i/parenrightbigg
12.10i/summationdisplay
k=0/parenleftbigg2i
k/parenrightbigg
=1
2/bracketleftbigg
4i+/parenleftbigg2i
i/parenrightbigg/bracketrightbigg
13.10i/summationdisplay
k=0/parenleftbigg2i
k/parenrightbigg
k=i
24i
14.10i/summationdisplay
k=0/parenleftbigg2i
k/parenrightbigg
k2=( 2i+1 )i4i−1−i2
2/parenleftbigg2i
i/parenrightbigg
0.242∞/summationdisplay
k=0(−1)k1
n2k=n2
n2+1[n>1]
0.243
1.∞/summationdisplay
k=11
[p+(k−1)q](p+kq)...[p+(k+l)q]=1
(l+1 )q1
p(p+q)...(p+lq)
(see also 0.141 3)
2.7∞/summationdisplay
k=1xk−1
[p+(k−1)q][p+(k−1)q+ 1][p+(k−1)q+2 ]...[p+(k−1)q+l]=1
l!/integraldisplayi
0tp−1(1−t)t
1−xtqdt
/bracketleftbig
p>0,x2<1/bracketrightbig
BR∗161 (2), AD (6.704)
3.∞/summationdisplay
k=01
(2k+1 )3/parenleftbigg1
xtanh/bracketleftbigg(2k+1 )πx
2/bracketrightbigg
+xtanh/bracketleftbigg(2k+1 )π
2x/bracketrightbigg/parenrightbigg
=π3
16
0.244
1.∞/summationdisplay
k=11
(k+p)(k+q)=1
q−p/integraldisplay1
0xp−xq
1−xdx [p>−1,q > −1,p/negationslash=q]GI III (90)
2.∞/summationdisplay
k=1(−1)k+1 1
p+(k−1)q=/integraldisplay1
0tp−1
1+tqdt [p>0,q > 0] BR∗161 (1)
3.10∞/summationdisplay
k=11
(k+p)(k+q)=1
q−pq/summationdisplay
m=p+11
m[q>p> −1,pandqintegers]
Summations of reciprocals of factorials
0.245
1.∞/summationdisplay
k=01
k!=e=2.71828 ...
2.11∞/summationdisplay
k=0(−1)k
k!=1
2e≈0.1839397 ...
0.249 Examples of numerical series 13
3.∞/summationdisplay
k=1k
(2k+1 ) !=1
e=0.36787 ...
4.∞/summationdisplay
k=1k
(k+1 ) !=1
5.∞/summationdisplay
k=01
(2k)!=1
2/parenleftbigg
e+1
e/parenrightbigg
=1.54308 ...
6.∞/summationdisplay
k=01
(2k+1 ) !=1
2/parenleftbigg
e−1
e/parenrightbigg
=1.17520 ...
7.∞/summationdisplay
k=0(−1)k
(2k)!=c o s1=0 .54030 ...
8.∞/summationdisplay
k=0(−1)k−1
(2k−1)!=s i n1=0 .84147 ...
0.246
1.∞/summationdisplay
k=01
(k!)2=I0(2) = 2 .27958530 ...
2.∞/summationdisplay
k=01
k!(k+1 ) !=I1(2) = 1 .590636855 ...
3.∞/summationdisplay
k=01
k!(k+n)!=In(2)
4.∞/summationdisplay
k=0(−1)k
(k!)2=J0(2) = 0 .22389078 ...
5.∞/summationdisplay
k=0(−1)k
k!(k+1 ) !=J1(2) = 0 .57672481 ...
6.∞/summationdisplay
k=0(−1)k
k!(k+n)!=Jn(2)
0.247∞/summationdisplay
k=1k!
(n+k−1)!=1
(n−2)·(n−1)!
0.248∞/summationdisplay
k=1kn
k!=Sn,
S1=e, S 2=2e, S 3=5e, S 4=1 5e
S5=5 2e, S 6= 203 e, S 7= 877 e, S 8= 4140 e
0.2497∞/summationdisplay
k=0(k+1 )3
k!=1 5e
14 Numerical Series and Infinite Products 0.250
0.25 Infinite products
0.250 Suppose that a sequence of numbers a1,a2,...,a k,...is given. If the limit lim
n→∞n/productdisplay
k=1(1 +ak)
exists, whether finite or infinite (but of definite sign), this limit is called the value of the infinite product
∞/productdisplay
k=1(1 +ak), and we write
1. lim
n→∞n/productdisplay
k=1(1 +ak)=∞/productdisplay
k=1(1 +ak)
If an infinite product has a finite nonzero value, it is said to converge. Otherwise, the infinite product is
said to diverge. We assume that no akis equal to −1. FI II 400
0.251 For the infinite product 0.250 1. to converge, it is necessary that lim
k→∞ak=0 . FI II 403
0.252 Ifak>0o rak<0 for all values of the index kstarting with some particular value, then, for the
product 0.250 1 to converge, it is necessary and sufficient that the series/summationtext∞
k=1akconverge.
0.253 The product∞/productdisplay
k=1(1 +ak) is said to converge absolutely if the product∞/productdisplay
k=1(1 +|ak|) converges.
FI II 403
0.254 Absolute convergence of an infinite product implies its convergence.
0.255 The product∞/productdisplay
k=1(1 +ak) converges absolutely if, and only if, the series∞/summationdisplay
k=1akconverges ab-
solutely. FI II 406
0.26 Examples of infinite products
0.261∞/productdisplay
k=1/parenleftbigg
1+(−1)k+1
2k−1/parenrightbigg
=√
2 EU
0.262
1.∞/productdisplay
k=2/parenleftbigg
1−1
k2/parenrightbigg
=1
2FI II 401
2.∞/productdisplay
k=1/parenleftbigg
1−1
(2k)2/parenrightbigg
=2
πFI II 401
3.∞/productdisplay
k=1/parenleftbigg
1−1
(2k+1 )2/parenrightbigg
=π
4FI II 401
0.263
1. e=2
1·/parenleftbigg4
3/parenrightbigg1/2/parenleftbigg6·8
5·7/parenrightbigg1/4/parenleftbigg10·12·14·16
9·11·13·15/parenrightbigg1/8
...
2.∗e=/parenleftbigg2
1/parenrightbigg1/2/parenleftbigg22
1·3/parenrightbigg1/3/parenleftbigg23·4
1·33/parenrightbigg1/4/parenleftbigg24·44
1·36·5/parenrightbigg1/5
···
0.303 Definitions and theorems 15
3.∗π
2=/parenleftbigg1
2/parenrightbigg1/2/parenleftbigg22
1·3/parenrightbigg1/4/parenleftbigg23·4
1·33/parenrightbigg1/8/parenleftbigg24·44
1·36·5/parenrightbigg1/16
···
where the nthfactor is the ( n+1 )throot of the product/producttextn
k=0(k+1 )(−1)k+1(n
k).
0.264
1. eC=∞/productdisplay
k=1k√e
1+1
kFI II 402
2.∗eC=/parenleftbigg2
1/parenrightbigg1/2/parenleftbigg22
1·3/parenrightbigg1/3/parenleftbigg23·4
1·33/parenrightbigg1/4/parenleftbigg24·44
1·36·5/parenrightbigg1/5
···
where the nthfactor is the ( n+1 )throot of the product/producttextn
k=0(k+1 )(−1)k+1(n
k). Here Cis the
Euler constant, denoted in other works by γ.
0.2652
π=/radicalbigg
1
2·/radicalBigg
1
2+1
2/radicalbigg
1
2·/radicaltp/radicalvertex/radicalvertex/radicalbt1
2+1
2/radicalBigg
1
2+1
2/radicalbigg
1
2... FI II 402
0.2668∞/productdisplay
k=0/parenleftBig
1+x2k/parenrightBig
=1
1−x[0<x< 1] FI II 401
0.3 Functional Series
0.30 Definitions and theorems
0.301 The series
1.∞/summationdisplay
k=1fk(x),
the terms of which are functions, is called a functional series . The set of values of the independent
variable xfor which the series 0.301 1 converges constitutes what is called the region of convergence of
that series.
0.302 A series that converges for all values of xin a region Mis said to converge uniformly in that
region if, for every ε≥0, there exists a number Nsuch that, for n>N , the inequality/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
k=n+1fk(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ε
holds for allxinM.
0.303 If the terms of the functional series 0.301 1 satisfy the inequalities:
|f
k(x)|<uk(k=1,2,3,...),
throughout the region M,w h e r et h e ukare the terms of some convergent numerical series
∞/summationdisplay
k=1uk=u1+u2+...+uk+...,
the series 0.301 1 converges uniformly in M. (Weierstrass)
16 Functional Series 0.304
0.304 Suppose that the series 0.301 1 converges uniformly in a region Mand that a set of functions
gk(x) constitutes (for each x) a monotonic sequence, and that these functions are uniformly bounded;
that is, suppose that a number Lexists such that the inequalities
1. |gn(x)|≤L
hold for all nandx. Then, the series
2.∞/summationdisplay
k=1fk(x)gk(x)
converges uniformly in the region M. (Abel) FI II 451
0.305 Suppose that the partial sums of the series 0.301 1 are uniformly bounded; that is, suppose that,
for some Land for all nandxinM, the inequalities/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay
k=1fk(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<L
hold. Suppose also that for each xthe functions g
n(x) constitute a monotonic sequence that approaches
zero uniformly in the region M. Then, the series 0.304 2 converges uniformly in the region M. (Dirichlet)
FI II 451
0.3066If the functions fk(x)( f o r k=1,2,3,...) are integrable on the interval [ a,b] and if the series
0.301 1 made up of these functions converges uniformly on that interval, this series may be integrated
termwise ;t h a ti s ,
/integraldisplayb
a/parenleftBigg∞/summationdisplay
k=1fk(x)/parenrightBigg
dx=∞/summationdisplay
k=1/integraldisplayb
afk(x)dx [a≤x≤b] FI II 459
0.307 Suppose that the functions fk(x)( f o r k=1,2,3,...) have continuous derivatives f/prime
k(x)o nt h e
interval [ a,b]. If the series 0.301 1 converges on this interval and if the series/summationtext∞
k=1f/prime
k(x)o ft h e s e
derivatives converges uniformly, the series 0.301 1 may be differentiated termwise; that is,/braceleftBigg∞/summationdisplay
k=1fk(x)/bracerightBigg/prime
=∞/summationdisplay
k=1f/prime
k(x) FI II 460
0.31 Power series
0.311 A functional series of the form
1.∞/summationdisplay
k=0ak(x−ξ)k=a0+a1(x−ξ)+a2(x−ξ)2+...
is called a power series . The following is true of any power series: if it is not everywhere convergent, the
region of convergence is a circle with its center at the point ξand a radius equal to R; at every interior
point of this circle, the power series 0.311 1 converges absolutely, and outside this circle, it diverges. This
circle is called the circle of convergence , and its radius is called the radius of convergence .I f t h e s e r i e s
converges at all points of the complex plane, we say that the radius of convergence is infinite ( R=+∞).
0.315 Power series 17
0.312 Power series may be integrated and differentiated termwise inside the circle of convergence; that
is,
/integraldisplayx
ξ/braceleftBigg∞/summationdisplay
k=0ak(x−ξ)k/bracerightBigg
dx=∞/summationdisplay
k=0ak
k+1(x−ξ)k+1,
d
dx/braceleftBigg∞/summationdisplay
k=0ak(x−ξ)k/bracerightBigg
=∞/summationdisplay
k=1kak(x−ξ)k−1.
The radius of convergence of a series that is obtained from termwise integration or differentiation of
another power series coincides with the radius of convergence of the original series.
Operations on power series
0.313 Division of power series.
∞/summationdisplay
k=0bkxk
∞/summationdisplay
k=0akxk=1
a0∞/summationdisplay
k=0ckxk,
where
cn+1
a0n/summationdisplay
k=1cn−kak−bn=0,
or
cn=(−1)n
an
0⎡
⎢⎢⎢⎢⎢⎢⎢⎣a
1b0−a0b1 a0 0··· 0
a2b0−a0b2 a1 a0 0
a3b0−a0b3 a2 a1 0
............
a
n−1b0−a0bn−1an−2an−3···a0
anb0−a0bn an−1an−2···a1⎤
⎥⎥⎥⎥⎥⎥⎥⎦AD (6360)
0.314 Power series raised to powers./parenleftBigg∞/summationdisplay
k=0akxk/parenrightBiggn
=∞/summationdisplay
k=0ckxk,
where
c0=an
0,c m=1
ma0m/summationdisplay
k=1(kn−m+k)akcm−kform≥1[ nis a natural number] AD (6361)
0.315 The substitution of one series into another.
∞/summationdisplay
k=1bkyk=∞/summationdisplay
k=1ckxky=∞/summationdisplay
k=1akxk;
c1=a1b1,c2=a2b1+a2
1b2,c3=a3b1+2a1a2b2+a3
1b3,
c4=a4b1+a2
2b2+2a1a3b2+3a2
1a2b3+a4
1b4, ...AD (6362)
18 Functional Series 0.316
0.316 Multiplication of power series
∞/summationdisplay
k=0akxk∞/summationdisplay
k=0bkxk=∞/summationdisplay
k=0ckxkcn=n/summationdisplay
k=0akbn−k FI II 372
Taylor series
0.317 If a function f(x) has derivatives of all orders throughout a neighborhood of a point ξ,t h e nw e
may write the series
1. f(ξ)+(x−ξ)
1!f/prime(ξ)+(x−ξ)2
2!f/prime/prime(ξ)+(x−ξ)3
3!f/prime/prime/prime(ξ)+...,
which is known as the Taylor series of the function f(x).
The Taylor series converges to the function f(x) if the remainder
2. Rn(x)=f(x)−f(ξ)−n/summationdisplay
k=1(x−ξ)k
k!f(k)(ξ)
approaches zero as n→∞.
The following are different forms for the remainder of a Taylor series:
3. Rn(x)=(x−ξ)n+1
(n+1 ) !f(n+1)(ξ+θ(x−ξ)) [0 <θ< 1] (Lagrange)
4. Rn(x)=(x−ξ)n+1
n!(1−θ)nf(n+1)(ξ+θ(x−ξ)) [0 <θ< 1] (Cauchy)
5. Rn(x)=ψ(x−ξ)−ψ(0)
ψ/prime[(x−ξ)(1−θ)](x−ξ)n(1−θ)n
n!f(n+1)(ξ+θ(x−ξ))
[0<θ< 1], (Schl¨omilch)
where ψ(x) is an arbitrary function satisfying the following two conditions: (1) It and its derivative
ψ/prime(x) are continuous in the interval (0 ,x−ξ); and (2) the derivative ψ/prime(x) does not change sign
in that interval. If we set ψ(x)=xp+1, we obtain the following form for the remainder:
Rn(x)=(x−ξ)n+1(1−θ)n−p−1
(p+1 )n!f(n+1)(ξ+θ(x−ξ)) [0 <p≤n;0<θ< 1](Rouch´ e)
6. Rn(x)=1
n!/integraldisplayx
ξf(n+1)(t)(x−t)ndt
0.318 Other forms in which a Taylor series may be written:
1.11f(a+x)=∞/summationdisplay
k=0xk
k!f(k)(a)=f(a)+x
1!f/prime(a)+x2
2!f/prime/prime(a)+...
2.∞/summationdisplay
k=0xk
k!f(k)(0) = f(0) +x
1!f/prime(0) +x2
2!f/prime/prime(0) + ... (Maclaurin series)
0.323 Fourier series 19
0.319 The Taylor series of functions of several variables:
f(x, y)=f(ξ,η)+(x−ξ)∂f(ξ,η)
∂x+(y−η)∂f(ξ,η)
∂y
+1
2!/braceleftbigg
(x−ξ)2∂2f(ξ,η)
∂x2+2 (x−ξ)(y−η)∂2f(ξ,η)
∂x∂y+(y−η)2∂2f(ξ,η)
∂y2/bracerightbigg
+...
0.32 Fourier series
0.320 Suppose that f(x)i saperiodic function of period 2 land that it is absolutely integrable (possibly
improperly) over the interval ( −l,l). The following trigonometric series is called the Fourier series of
f(x):
1.a0
2+∞/summationdisplay
k=1/parenleftbigg
akcoskπx
l+bksinkπx
l,/parenrightbigg
the coefficients of which (the Fourier coefficients) are given by the formulas
2. ak=1
l/integraldisplayl
−lf(t)coskπt
ldt=1
l/integraldisplayα+2l
αf(t)coskπt
ldt(k=0,1,2,...)
3.11bk=1
l/integraldisplayl
−lf(t)sinkπt
ldt=1
l/integraldisplayα+2l
αf(t)sinkπt
ldt(k=1,2,...)
Convergence tests
0.321 The Fourier series of a function f(x)a tap o i n t x0converges to the number
f(x0+0 )+ f(x0−0)
2,
if, for some h>0, the integral
/integraldisplayh
0|f(x0+t)+f(x0−t)−f(x0+0 )−f(x0−0)|
tdt
exists. Here, it is assumed that the function f(x) either is continuous at the point x0or has a discontinuity
of the first kind (a saltus ) at that point and that both one-sided limits f(x0+0 )a n d f(x0−0) exist.
(Dini) FI III 524
0.322 The Fourier series of a periodic function f(x) that satisfies the Dirichlet conditions on the interval
[a,b] converges at every point x0to the value1
2[f(x0+0 )+ f(x0−0)]. (Dirichlet)
We say that a function f(x) satisfies the Dirichlet conditions on the interval [ a,b] if it is bounded on
that interval and if the interval [ a,b] can be partitioned into a finite number of subintervals inside each
of which the function f(x) is continuous and monotonic.
0.323 The Fourier series of a function f(x)a tap o i n t x0converges to1
2[f(x0+0 )+ f(x0−0)] iff(x)i s
of bounded variation in some interval ( x0−h, x0+h) with center at x0. (Jordan–Dirichlet) FI III 528
The definition of a function of bounded variation. Suppose that a function f(x) is defined on some
interval [ a,b], where z<b. Let us partition this interval in an arbitrary manner into subintervals with
the dividing points
a=x0<x1<x2<...<x n−1<xn=b
and let us form the sum
20 Functional Series 0.324
n/summationdisplay
k=1|f(xk)−f(xk−1)|
Different partitions of the interval [ a,b] (that is, different choices of points of division xi) yield, generally
speaking, different sums. If the set of these sums is bounded above, we say that the function f(x)i sof
bounded variation on the interval [ a,b]. The least upper bound of these sums is called the total variation
of the function f(x) on the interval [ a,b].
0.324 Suppose that a function f(x) is piecewise-continuous on the interval [ a,b] and that in each interval
of continuity it has a piecewise-continuous derivative. Then, at every point x0of the interval [ a,b],the
Fourier series of the function f(x)c o n v e r g e st o1
2[f(x0+0 )+ f(x0−0)].
0.325 A function f(x) defined in the interval (0 ,l) can be expanded in a cosine series of the form
1.a0
2+∞/summationdisplay
k=1akcoskπx
l,
where
2. ak=2
l/integraldisplayl
0f(t)coskπt
ldt
0.326 A function f(x) defined in the interval (0 ,l) can be expanded in a sine series of the form
1.∞/summationdisplay
k=1bksinkπx
l,
where
2. bk=2
l/integraldisplayl
0f(t)sinkπt
ldt
The convergence tests for the series 0.325 1a n d0.326 1 are analogous to the convergence tests for the
series0.320 1( s e e0.321–0.324 ).
0.327 The Fourier coefficients akandbk(given by formulas 0.320 2a n d 0.320 3) of an absolutely
integrable function approach zero as k→∞.
If a function f(x) is square-integrable on the interval ( −l,l), the equation of closure is satisfied:
a2
0
2+∞/summationdisplay
k=1/parenleftbig
a2
k+b2
k/parenrightbig
=1
l/integraldisplayl
−lf2(x)dx (A. M. Lyapunov) FI III 705
0.328 Suppose that f(x)a n d ϕ(x) are two functions that are square-integrable on the interval ( −l,l)
and that ak,bkandαk,βkare their Fourier coefficients. For such functions, the generalized equation of
closure (Parseval’s equation) holds:
a0α0
2+∞/summationdisplay
k=1(akαk+bkβk)=1
l/integraldisplayl
−lf(x)ϕ(x)dx FI III 709
For examples of Fourier series, see 1.44and1.45.
0.411 Differentiation of a definite integral with respect to a parameter 21
0.33 Asymptotic series
0.330 Included in the collection of all divergent series is the broad class of series known as asymptotic
orsemiconvergent series. Despite the fact that these series diverge , the values of the functions that they
represent can be calculated with a high degree of accuracy if we take the sum of a suitable number of
terms of such series. In the case of alternating asymptotic series, we obtain greatest accuracy if we breakoff the series in question at whatever term is of lowest absolute value. In this case, the error (in absolutevalue) does not exceed the absolute value of the first of the discarded terms (cf. 0.227 3).
Asymptotic series have many properties that are analogous to the properties of convergent series, and,
for that reason, they play a significant role in analysis.
The asymptotic expansion of a function is denoted as follows:
f(z)∼
∞/summationdisplay
n=0Anz−n
This is the definition of an asymptotic expansion. The divergent series∞/summationdisplay
n=0An
znis called the asymptotic
expansion of a function f(z) in a given region of values of arg zif the expression Rn(z)=zn[f(z)−Sn(z)],
where Sn(z)=n/summationdisplay
k=0Ak
zk, satisfies the condition lim
|z|→∞Rn(z)=0f o rfi x e d n. FI II 820
A divergent series that represents the asymptotic expansion of some function is called an asymptotic
series.
0.331 Properties of asymptotic series
1. The operations of addition, subtraction, multiplication, and raising to a power can be performed
on asymptotic series just as on absolutely convergent series. The series obtained as a result ofthese operations will also be asymptotic.
2. One asymptotic series can be divided by another, provided that the first term A
0of the divisor
is not equal to zero. The series obtained as a result of division will also be asymptotic.
FI II 823-825
3. An asymptotic series can be integrated termwise, and the resultant series will also be asymptotic.
In contrast, differentiation of an asymptotic series is, in general, not permissible. FI II 824
4. A single asymptotic expansion can represent different functions. On the other hand, a given
function can be expanded in an asymptotic series in only one manner.
0.4 Certain Formulas from Differential Calculus
0.41 Differentiation of a definite integral with respect to a parameter
0.410d
da/integraldisplayϕ(a)
ψ(a)f(x, a)dx=f(ϕ(a),a)dϕ(a)
da−f(ψ(a),a)dψ(a)
da+/integraldisplayϕ(a)
ψ(a)d
daf(x, a)dx FI II 680
0.411 In particular,
1.d
da/integraldisplaya
bf(x)dx=f(a)
2.d
db/integraldisplaya
bf(x)dx=−f(b)
22 Certain Formulas from Differential Calculus 0.430
0.42 The nthderivative of a product (Leibniz’s rule)
Suppose that uandvaren-times-differentiable functions of x. Then,
dn(uv)
dxn=udnv
dxn+/parenleftBign
1/parenrightBigdu
dxdn−1v
dxn−1+/parenleftBign
2/parenrightBigd2u
dx2dn−2v
dxn−2+/parenleftBign
3/parenrightBigd3u
dx3dn−3v
dxn−3+···+vdnu
dxn
or, symbolically,
dn(uv)
dxn=(u+v)(n)FI I 272
0.43 The nthderivative of a composite function
0.430 Iff(x)=F(y)andy=ϕ(x), then
1.dn
dxnf(x)=U1
1!F/prime(y)+U2
2!F/prime/prime(y)+U3
3!F/prime/prime/prime(y)+...+Un
n!F(n)(y),
where
Uk=dn
dxnyk−k
1!ydn
dxnyk−1+k(k−1)
2!y2dn
dxnyk−2−...+(−1)k−1kyk−1dny
dxnAD (7361) GO
2.dn
dxnf(x)=/summationdisplay n!
i!j!h!...k!dmF
dym/parenleftbiggy/prime
1!/parenrightbiggi/parenleftbiggy/prime/prime
2!/parenrightbiggj/parenleftbiggy/prime/prime/prime
3!/parenrightbiggh
···/parenleftbiggy(l)
l!/parenrightbiggk
,
Here, the symbol/summationtextindicates summation over all solutions in non-negative integers of the equa-
tioni+2j+3h+...+lk=nandm=i+j+h+...+k.
0.431
1. ( −1)ndn
dxnF/parenleftbigg1
x/parenrightbigg
=1
x2nF(n)/parenleftbigg1
x/parenrightbigg
+n−1
x2n−1n
1!F(n−1)/parenleftbigg1
x/parenrightbigg
+(n−1)(n−2)
x2n−2n(n−1)
2!F(n−2)/parenleftbigg1
x/parenrightbigg
+...
AD (7362.1)
2. ( −1)ndn
dxnea
x=1
xnea
x⎡
⎣/parenleftBiga
x/parenrightBign
+(n−1)/parenleftBign
1/parenrightBig/parenleftBiga
x/parenrightBign−1
+(n−1)(n−2)/parenleftBign
2/parenrightBig/parenleftBiga
x/parenrightBign−2
+(n−1)(n−2)(n−3)/parenleftBign
3/parenrightBig/parenleftBiga
x/parenrightBign−3
+...⎤
⎦
AD (7362.2)
0.432
1.dn
dxnF/parenleftbig
x2/parenrightbig
=( 2x)nF(n)/parenleftbig
x2/parenrightbig
+n(n−1)
1!(2x)n−2F(n−1)/parenleftbig
x2/parenrightbig
+n(n−1)(n−2)(n−3)
2!(2x)n−4F(n−2)/parenleftbig
x2/parenrightbig
+
+n(n−1)(n−2)(n−3)(n−4)(n−5)
3!(2x)n−6F(n−3)/parenleftbig
x2/parenrightbig
+...
AD (7363.1)
0.440 Integration by substitution 23
2.dn
dxneax2=( 2ax)neax2⎡
⎣1+n(n−1)
1! (4ax2)+n(n−1)(n−2)(n−3)
2! (4ax2)2
+n(n−1)(n−2)(n−3)(n−4)(n−5)
3! (4ax2)3+···⎤
⎦
AD (7363.2)
3.dn
dxn/parenleftbig
1+ax2/parenrightbigp=p(p−1)(p−2)...(p−n+ 1)(2 ax)n
(1 +ax2)n−p
×/braceleftBigg
1+n(n−1)
1!(p−n+1 )1+ax2
4ax2+n(n−1)(n−2)(n−3)
2!(p−n+1 ) (p−n+2 )/parenleftbigg1+ax2
4ax2/parenrightbigg2
+.../bracerightBigg
,
AD (7363.3)
4.dm−1
dxm−1/parenleftbig
1−x2/parenrightbigm−1
2=(−1)m−1(2m−1)!!
msin (marccos x) AD (7363.4)
5. ( −1)n∂n
∂an/parenleftbigga
a2+b2/parenrightbigg
=n!/parenleftbigga
a2+b2/parenrightbiggn+1/summationdisplay
0≤2k≤n+1(−1)k/parenleftbiggn+1
2k/parenrightbigg/parenleftbiggb
a/parenrightbigg2k
(3.944.12)
6. ( −1)n∂n
∂an/parenleftbiggb
a2+b2/parenrightbigg
=n!/parenleftbigga
a2+b2/parenrightbiggn+1/summationdisplay
0≤2k≤n(−1)k/parenleftbiggn+1
2k+1/parenrightbigg/parenleftbiggb
a/parenrightbigg2k+1
(3.944.11)
0.433
1.dn
dxnF/parenleftbig√x/parenrightbig
=F(n)(√x)
(2√x)n−n(n−1)
1!F(n−1)(√x)
(2√x)n+1+(n+1 )n(n−1)(n−2)
2!F(n−2)(√x)
(2√x)n+2−...
AD (7364.1)
2.dn
dxn/parenleftbig
1+a√x/parenrightbig2n−1=(2n−1)!!
2na√x/parenleftbigg
a2−1
x/parenrightbiggn−1
AD (7364.2)
0.434dn
dxnyp=p/parenleftbiggn−p
n/parenrightbigg/braceleftBigg
−/parenleftBign
1/parenrightBig1
p−1yp−1dny
dxn+/parenleftBign
2/parenrightBig1
p−2yp−2dn/parenleftbig
y2/parenrightbig
dxn−.../bracerightBigg
AD (737.1)
0.435dn
dxnlny=/braceleftBigg/parenleftBign
1/parenrightBig1
1·ydny
dxn−/parenleftBign
2/parenrightBig1
2·y2dn/parenleftbig
y2/parenrightbig
dxn+dn/parenleftbig
y3/parenrightbig
dxnxn−.../bracerightBigg
AD (737.2)
0.44 Integration by substitution
0.44011Letf(g(x)) and g(x) be continuous in [ a,b]. Further, let g/prime(x) exist and be continuous there.
Then/integraldisplayb
af[g(x)]g/prime(x)dx=/integraldisplayg(b)
g(a)f(u)du
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1 Elementary Functions
1.1 Power of Binomials
1.11 Power series
1.110 (1 +x)q=1+ qx+q(q−1)
2!x2+···+q(q−1)...(q−k+1 )
k!xk+···=∞/summationdisplay
k=0/parenleftBigq
k/parenrightBig
xk
Ifqis neither a natural number nor zero, the series converges absolutely for |x|<1 and diverges for
|x|>1. For x= 1, the series converges for q>−1 and diverges for q≤−1. For x=1 ,t h es e r i e s
converges absolutely for q>0. For x=−1, it converges absolutely for q>0 and diverges for q<0. If
q=nis a natural number, the series 1.110 is reduced to the finite sum 1.111 . FI II 425
1.111 (a+x)n=n/summationdisplay
k=0/parenleftBign
k/parenrightBig
xkan−k
1.112
1. (1 + x)−1=1−x+x2−x3+···=∞/summationdisplay
k=1(−1)k−1xk−1
(see also 1.121 2)
2. (1 + x)−2=1−2x+3x2−4x3+···=∞/summationdisplay
k=1(−1)k−1kxk−1
3.11(1 +x)1/2=1+1
2x−1·1
2·4x2+1·1·3
2·4·6x3−1·1·3·5
2·4·6·8x4+...
4. (1 + x)−1/2=1−1
2x+1·3
2·4x2−1·3·5
2·4·6x3+...
1.113x
(1−x)2=∞/summationdisplay
k=1kxk/bracketleftbig
x2<1/bracketrightbig
1.114
1./parenleftbig
1+√
1+x/parenrightbigq=2q/bracketleftbigg
1+q
1!/parenleftBigx
4/parenrightBig
+q(q−3)
2!/parenleftBigx
4/parenrightBig2
+q(q−4)(q−5)
3!/parenleftBigx
4/parenrightBig3
+.../bracketrightbigg
/bracketleftbig
x2<1,qis a real number/bracketrightbig
AD (6351.1)
25
26 The Exponential Function 1.121
2./parenleftBig
x+/radicalbig
1+x2/parenrightBigq
=1+∞/summationdisplay
k=0q2/parenleftbig
q2−22/parenrightbig/parenleftbig
q2−42/parenrightbig
.../bracketleftbig
q2−(2k)2/bracketrightbig
x2k+2
(2k+2 ) !
+qx+q∞/summationdisplay
k=1/parenleftbig
q2−12/parenrightbig/parenleftbig
q2−32/parenrightbig
.../bracketleftbig
q2−(2k−1)2/bracketrightbig
(2k+1 ) !x2k+1
/bracketleftbig
x2<1,qi sar e a ln u m b e r/bracketrightbig
AD(6351.2)
1.12 Series of rational fractions
1.121
1.x
1−x=∞/summationdisplay
k=12k−1x2k−1
1+x2k−1=∞/summationdisplay
k=1x2k−1
1−x2k/bracketleftbig
x2<1/bracketrightbig
AD (6350.3)
2.1
x−1=∞/summationdisplay
k=12k−1
x2k−1+1/bracketleftbig
x2>1/bracketrightbig
AD (6350.3)
1.2 The Exponential Function
1.21 Series representation
1.211
1.11ex=∞/summationdisplay
k=0xk
k!
2. ax=∞/summationdisplay
k=0(xlna)k
k!
3. e−x2=∞/summationdisplay
k=0(−1)kx2k
k!
4.∗ex= lim
n→∞/parenleftBig
1+x
n/parenrightBign
1.212 ex(1 +x)=∞/summationdisplay
k=0xk(k+1 )
k!
1.213x
ex−1=1−x
2+∞/summationdisplay
k=1B2kx2k
(2k)![x<2π] FI II 520
1.214 eex=e/parenleftbigg
1+x+2x2
2!+5x3
3!+15x4
4!+.../parenrightbigg
AD (6460.3)
1.215
1. esinx=1+ x+x2
2!−3x4
4!−8x5
5!−3x6
6!+56x7
7!+... AD (6460.4)
2. ecosx=e/parenleftbigg
1−x2
2!+4x4
4!−31x6
6!+.../parenrightbigg
AD (6460.5)
1.232 Series of exponentials 27
3. etanx=1+ x+x2
2!+3x3
3!+9x4
4!+37x5
5!+... AD (6460.6)
1.216
1. earcsin x=1+ x+x2
2!+2x3
3!+5x4
4!+... AD (6460.7)
2. earctan x=1+ x+x2
2!−x3
3!−7x4
4!+... AD (6460.8)
1.217
1. πeπx+e−πx
eπx−e−πx=x∞/summationdisplay
k=−∞1
x2+k2(cf.1.421 3) AD (6707.1)
2.2π
eπx−e−πx=x∞/summationdisplay
k=−∞(−1)k
x2+k2(cf.1.422 3) AD (6707.2)
1.22 Functional relations
1.221
1. ax=exlna
2. alogax=a1
logxa=x
1.222
1. ex=c o s h x+s i n h x
2. eix=c o s x+isinx
1.223 eax−ebx=(a−b)xexp/bracketleftbigg1
2(a+b)x/bracketrightbigg∞/productdisplay
k=1/bracketleftbigg
1+(a−b)2x2
2k2π2/bracketrightbigg
MO 216
1.23 Series of exponentials
1.231∞/summationdisplay
k=0akx=1
1−ax[a>1a n d x<0o r0 <a< 1a n d x>0]
1.232
1. tanh x=1+2∞/summationdisplay
k=1(−1)ke−2kx[x>0]
2. sech x=2∞/summationdisplay
k=0(−1)ke−(2k+1)x[x>0]
3. cosech x=2∞/summationdisplay
k=0e−(2k+1)x[x>0]
4.∗sinx=e x p/bracketleftBigg
−∞/summationdisplay
n=1cos2nx
2n/bracketrightBigg
[0≤x≤π]
28 Trigonometric and Hyperbolic Functions 1.311
1.3–1.4 Trigonometric and Hyperbolic Functions
1.30 Introduction
The trigonometric and hyperbolic sines are related by the identities
sinhx=1
isin(ix),sinx=1
isinh(ix).
The trigonometric and hyperbolic cosines are related by the identities
coshx=c o s ( ix),cosx=c o s h ( ix).
Because of this duality, every relation involving trigonometric functions has its formal counterpart involv-
ing the corresponding hyperbolic functions, and vice versa. In many (though not all) cases, both pairs ofrelationships are meaningful.
The idea of matching the relationships is carried out in the list of formulas given below. However, not
all the meaningful “pairs” are included in the list.
1.31 The basic functional relations
1.311
1. sin x=1
2i/parenleftbig
eix−e−ix/parenrightbig
=−isinh(ix)
2. sinh x=1
2/parenleftbig
ex−e−x/parenrightbig
=−isin(ix)
3. cos x=1
2/parenleftbig
eix+e−ix/parenrightbig
=c o s h ( ix)
4. cosh x=1
2/parenleftbig
ex+e−x/parenrightbig
=c o s ( ix)
5. tan x=sinx
cosx=1
itanh(ix)
6. tanh x=sinhx
coshx=1
itan(ix)
7. cot x=cosx
sinx=1
tanx=icoth(ix)
8. coth x=coshx
sinhx=1
tanhx=icot(ix)
1.312
1. cos2x+s i n2x=1
1.314 The basic functional relations 29
2. cosh2x−sinh2x=1
1.313
1. sin ( x±y)=s i n xcosy±sinycosx
2. sinh ( x±y)=s i n h xcoshy±sinhycoshx
3. sin ( x±iy)=s i n xcoshy±isinhycosx
4. sinh ( x±iy)=s i n h xcosy±isinycoshx
5. cos( x±y) = cos xcosy∓sinxsiny
6. cosh ( x±y)=c o s h xcoshy±sinhxsinhy
7. cos( x±iy) = cos xcoshy∓isinxsinhy
8. cosh ( x±iy)=c o s h xcosy±isinhxsiny
9. tan ( x±y)=tanx±tany
1∓tanxtany
10. tanh ( x±y)=tanhx±tanhy
1±tanhxtanhy
11. tan ( x±iy)=tanx±itanhy
1∓itanxtanhy
12. tanh ( x±iy)=tanhx±itany
1±itanhxtany
1.314
1. sin x±siny=2s i n1
2(x±y)c o s1
2(x∓y)
2. sinh x±sinhy=2s i n h1
2(x±y)cosh1
2(x∓y)
3. cos x+c o s y= 2cos1
2(x+y)cos1
2(x−y)
4. cosh x+c o s h y=2c o s h1
2(x+y)cosh1
2(x−y)
5. cos x−cosy=2s i n1
2(x+y)sin1
2(y−x)
6. cosh x−coshy=2s i n h1
2(x+y)sin h1
2(x−y)
7. tan x±tany=sin (x±y)
cosxcosy
8. tanh x±tanhy=sinh (x±y)
coshxcoshy
9.∗sinx±cosy=±2s in/bracketleftbigg1
2(x+y)±π
4/bracketrightbigg
sin/bracketleftbigg1
2(x−y)±π
4/bracketrightbigg
=±2c os/bracketleftbigg1
2(x+y)∓π
4/bracketrightbigg
cos/bracketleftbigg1
2(x−y)∓π
4/bracketrightbigg
=2s i n/bracketleftbigg1
2(x±y)±π
4/bracketrightbigg
cos/bracketleftbigg1
2(x∓y)∓π
4/bracketrightbigg
30 Trigonometric and Hyperbolic Functions 1.315
10.∗asinx±bcosx=a/radicalBigg
1+/parenleftbiggb
a/parenrightbigg2
sin/bracketleftbigg
x±arctan/parenleftbiggb
a/parenrightbigg/bracketrightbigg
[a/negationslash=0 ]
11.∗±asinx+bcosx=b/radicalbigg
1+/parenleftBiga
b/parenrightBig2
cos/bracketleftBig
x∓arctan/parenleftBiga
b/parenrightBig/bracketrightBig
[b/negationslash=0 ]
12.∗asinx±bcosy=q/radicalBigg
1+/parenleftbiggr
q/parenrightbigg2
sin/bracketleftbigg1
2(x±y) + arctan/parenleftbiggr
q/parenrightbigg/bracketrightbigg
q=(a+b)cos/bracketleftbigg1
2(x∓y)/bracketrightbigg
,r=(a−b)sin/bracketleftbigg1
2(x∓y)/bracketrightbigg
[q/negationslash=0 ]
13.∗acosx+bcosy=t/radicalbigg
1+/parenleftBigs
t/parenrightBig2
cos/bracketleftbigg1
2(x∓y) + arctan/parenleftBigs
t/parenrightBig/bracketrightbigg
[t/negationslash=0 ]
=−s/radicalBigg
1+/parenleftbiggt
s/parenrightbigg2
cos/bracketleftbigg1
2(x∓y)−arctan/parenleftbiggt
s/parenrightbigg/bracketrightbigg
[s/negationslash=0 ]
s=(a−b)sin/bracketleftbigg1
2(x±y)/bracketrightbigg
,t=(a+b)cos/bracketleftbigg1
2(x±y)/bracketrightbigg
1.315
1. sin2x−sin2y=s i n ( x+y)sin(x−y) = cos2y−cos2x
2. sinh2x−sinh2y=s i n h ( x+y)sin h( x−y)=c o s h2x−cosh2y
3. cos2x−sin2y=c o s ( x+y)cos(x−y) = cos2y−sin2x
4. sinh2x+c o s h2y=c o s h ( x+y)cosh( x−y)=c o s h2x+s i n h2y
1.316
1. (cos x+isinx)n=c o s nx+isinnx [nis an integer]
2. (cosh x+s i n h x)n=s i n h nx+c o s h nx [nis an integer]
1.317
1. sinx
2=±/radicalbigg
1
2(1−cosx)
2. sinhx
2=±/radicalbigg
1
2(coshx−1)
3. cosx
2=±/radicalbigg
1
2(1 + cos x)
4. coshx
2=/radicalbigg
1
2(coshx+1 )
5. tanx
2=1−cosx
sinx=sinx
1 + cos x
1.321 Trigonometric and hyperbolic functions: expansion in multiple angles 31
6. tanhx
2=coshx−1
sinhx=sinhx
coshx+1
The signs in front of the radical in formulas 1.317 1,1.317 2, and 1.317 3a r et a k e ns oa st oa g r e e
with the signs of the left-hand members. The sign of the left hand members depends in turn on the valueofx.
1.32 The representation of powers of trigonometric and hyperbolic functions in terms
of functions of multiples of the argument (angle)
1.320
1. sin2nx=1
22n/braceleftBiggn−1/summationdisplay
k=0(−1)n−k2/parenleftbigg2n
k/parenrightbigg
cos2(n−k)x+/parenleftbigg2n
n/parenrightbigg/bracerightBigg
KR 56 (10, 2)
2. sinh2nx=(−1)n
22n/braceleftBiggn−1/summationdisplay
k=0(−1)n−k2/parenleftbigg2n
k/parenrightbigg
cosh 2( n−k)x+/parenleftbigg2n
n/parenrightbigg/bracerightBigg
3. sin2n−1x=1
22n−2n−1/summationdisplay
k=0(−1)n+k−1/parenleftbigg2n−1
k/parenrightbigg
sin(2n−2k−1)x KR 56 (10, 4)
4. sinh2n−1x=(−1)n−1
22n−2n−1/summationdisplay
k=0(−1)n+k−1/parenleftbigg2n−1
k/parenrightbigg
sinh(2 n−2k−1)x
5. cos2nx=1
22n/braceleftBiggn−1/summationdisplay
k=02/parenleftbigg2n
k/parenrightbigg
cos2(n−k)x+/parenleftbigg2n
n/parenrightbigg/bracerightBigg
KR 56 (10, 1)
6. cosh2nx=1
22n/braceleftBiggn−1/summationdisplay
k=02/parenleftbigg2n
k/parenrightbigg
cosh2( n−k)x+/parenleftbigg2n
n/parenrightbigg/bracerightBigg
7. cos2n−1x=1
22n−2n−1/summationdisplay
k=0/parenleftbigg2n−1
k/parenrightbigg
cos(2n−2k−1)x KR 56 (10, 3)
8. cosh2n−1x=1
22n−2n−1/summationdisplay
k=0/parenleftbigg2n−1
k/parenrightbigg
cosh(2 n−2k−1)x
Special cases
1.321
1. sin2x=1
2(−cos 2x+1 )
2. sin3x=1
4(−sin 3x+3s i n x)
3. sin4x=1
8(cos 4x−4c os2 x+3 )
4. sin5x=1
16(sin 5x−5s in3 x+1 0s i n x)
32 Trigonometric and Hyperbolic Functions 1.322
5. sin6x=1
32(−cos 6x+6c o s4 x−15cos2 x+ 10)
6. sin7x=1
64(−sin 7x+7s i n5 x−21 sin3 x+3 5s i n x)
1.322
1. sinh2x=1
2(cosh 2 x−1)
2. sinh3x=1
4(sinh 3 x−3s in h x)
3. sinh4x=1
8(cosh 4 x−4c o s h2 x+3 )
4. sinh5x=1
16(sinh 5 x−5s i n h3 x+1 0s i n h x)
5. sinh6x=1
32(cosh6 x−6c o s h4 x+1 5c o s h2 x+ 10)
6. sinh7x=1
64(sinh 7 x−7s i n h5 x+2 1s i n h3 x+3 5s i n h x)
1.323
1. cos2x=1
2(cos2x+1 )
2. cos3x=1
4(cos3x+ 3cos x)
3. cos4x=1
8(cos4x+4c o s2 x+3 )
4. cos5x=1
16(cos5x+5c o s3 x+1 0c o s x)
5. cos6x=1
32(cos6x+6c o s4 x+1 5c o s2 x+ 10)
6. cos7x=1
64(cos7x+7c o s5 x+2 1c o s3 x+3 5c o s x)
1.324
1. cosh2x=1
2(cosh 2 x+1 )
2. cosh3x=1
4(cosh 3 x+3c o s h x)
3. cosh4x=1
8(cosh 4 x+ 4 cosh 2 x+3 )
4. cosh5x=1
16(cosh 5 x+ 5 cosh3 x+1 0c o s h x)
5. cosh6x=1
32(cosh 6 x+ 6 cosh4 x+1 5c o s h2 x+ 10)
6. cosh7x=1
64(cosh 7 x+ 7 cosh5 x+2 1c o s h3 x+3 5c o s h x)
1.332 Trigonometric and hyperbolic functions: expansion in powers 33
1.33 The representation of trigonometric and hyperbolic functions of multiples of
the argument (angle) in terms of powers of these functions
1.331
1.7sinnx=ncosn−1xsinx−/parenleftBign
3/parenrightBig
cosn−3xsin3x+/parenleftBign
5/parenrightBig
cosn−5xsin5x−...;
=s i nx⎧
⎨
⎩2n−1cosn−1x−/parenleftbiggn−2
1/parenrightbigg
2n−3cosn−3x
+/parenleftbiggn−3
2/parenrightbigg
2n−5cosn−5x−/parenleftbiggn−4
3/parenrightbigg
2n−7cosn−7x+...⎫
⎬
⎭
AD (3.175)
2. sinh nx=x[(n+1)/2]/summationdisplay
k=1/parenleftbiggn
2k−1/parenrightbigg
sinh2k−2xcoshn−2k+1x
=s i n h x[(n−1)/2]/summationdisplay
k=0(−1)k/parenleftbiggn−k−1
k/parenrightbigg
2n−2k−1coshn−2k−1x
3. cos nx=c o snx−/parenleftBign
2/parenrightBig
cosn−2xsin2x+/parenleftBign
4/parenrightBig
cosn−4xsin4x−...;
=2n−1cosnx−n
12n−3cosn−2x+n
2/parenleftbiggn−3
1/parenrightbigg
2n−5cosn−4x
−n
3/parenleftbiggn−4
2/parenrightbigg
2n−7cosn−6x+...
AD (3.175)
4.3coshnx=[n/2]/summationdisplay
k=0/parenleftBign
2k/parenrightBig
sinh2kxcoshn−2kx
=2n−1coshnx+n[n/2]/summationdisplay
k=1(−1)k1
k/parenleftbiggn−k−1
k−1/parenrightbigg
2n−2k−1coshn−2kx
1.332
1. sin 2 nx=2ncosx/braceleftBigg
sinx−4n2−22
3!sin3x+/parenleftbig
4n2−22/parenrightbig/parenleftbig
4n2−42/parenrightbig
5!sin5x−.../bracerightBigg
AD (3.171)
=(−1)n−1cosx/braceleftbigg
22n−1sin2n−1x−2n−2
1!22n−3sin2n−3x
+(2n−3)(2n−4)
2!22n−5sin2n−5x
−(2n−4)(2n−5)(2n−6)
3!22n−7sin2n−7x+.../bracerightbigg
AD (3.173)
34 Trigonometric and Hyperbolic Functions 1.333
2. sin(2 n−1)x=( 2n−1)/braceleftbigg
sinx−(2n−1)2−12
3!sin3x
+/bracketleftbig
(2n−1)2−12/bracketrightbig/bracketleftbig
(2n−1)2−32/bracketrightbig
5!sin5x−.../bracerightBigg
AD (3.172)
=(−1)n−1/braceleftbigg
22n−2sin2n−1x−2n−1
1!22n−4sin2n−3x
+(2n−1)(2n−4)
2!22n−6sin2n−5x
−(2n−1)(2n−5)(2n−6)
3!22n−8sin2n−7x+.../bracerightbigg
AD (3.174)
3. cos2 nx=1−4n2
2!sin2x+4n2/parenleftbig
4n2−22/parenrightbig
4!sin4x−4n2/parenleftbig
4n2−2/parenrightbig/parenleftbig
4n2−42/parenrightbig
6!sin6x+...
AD (3.171)
=(−1)n/braceleftbigg
22n−1sin2nx−2n
1!22n−3sin2n−2x
+2n(2n−3)
2!22n−5sin2n−4x−2n(2n−4)(2n−5)
3!22n−7sin2n−6x+.../bracerightbigg
AD (3.173)a
4. cos(2 n−1)x=c o s x/braceleftbigg
1−(2n−1)2−12
2!sin2x
+/bracketleftbig
(2n−1)2−12/bracketrightbig/bracketleftbig
(2n−1)2−32/bracketrightbig
4!sin4x−.../bracerightBigg
AD (3.172)
=(−1)n−1cosx/braceleftbigg
22n−2sin2n−2x−2n−3
1!22n−4sin2n−4x
+(2n−4)(2n−5)
2!22n−6sin2n−6x
−(2n−5)(2n−6)(2n−7)
3!22n−8sin2n−8x+.../bracerightbigg
AD (3.174)
By using the formulas and values of 1.30, we can write formulas for sinh 2 nx,s i n h ( 2 n−1)x,c o s h2 nx,
and cosh(2 n−1)xthat are analogous to those of 1.332 , just as was done in the formulas in 1.331 .
Special cases
1.333
1. sin 2 x=2s i n xcosx
2. sin 3 x=3s i n x−4s in3x
3. sin 4 x=c o s x/parenleftbig
4s inx−8s in3x/parenrightbig
4. sin 5 x=5s i n x−20 sin3x+1 6s i n5x
5. sin 6 x=c o s x/parenleftbig
6s inx−32 sin3x+3 2s i n5x/parenrightbig
1.337 Trigonometric and hyperbolic functions: expansion in powers 35
6. sin 7 x=7s i n x−56 sin3x+ 112sin5x−64 sin7x
1.334
1. sinh 2 x=2s i n h xcoshx
2. sinh 3 x=3s i n h x+4s i n h3x
3.11sinh 4x=c o s h x/parenleftbig
4s in h x+8s i n h3x/parenrightbig
4. sinh 5 x=5s i n h x+2 0s i n h3x+1 6s i n h5x
5.11sinh 6x=c o s h x/parenleftbig
6s in h x+3 2s i n h3x+3 2s i n h5x/parenrightbig
6. sinh 7 x=7s i n h x+5 6s i n h3x+ 112 sinh5x+6 4s i n h7x
1.335
1. cos2 x= 2cos2x−1
2. cos3 x= 4cos3x−3c osx
3. cos4 x= 8cos4x−8c os2x+1
4. cos5 x=1 6c o s5x−20cos3x+ 5cos x
5. cos6 x=3 2c o s6x−48cos4x+1 8c o s2x−1
6. cos7 x=6 4c o s7x−112cos5x+5 6c o s3x−7c osx
1.336
1. cosh 2 x=2c o s h2x−1
2. cosh 3 x=4c o s h3x−3c os h x
3. cosh 4 x=8c o s h4x−8c os h2x+1
4. cosh 5 x=1 6c o s h5x−20 cosh3x+5c o s h x
5. cosh 6 x=3 2c o s h6x−48 cosh4x+1 8c o s h2x−1
6. cosh 7 x=6 4c o s h7x−112 cosh5x+5 6c o s h3x−7c os h x
1.337
1.∗cos 3x
cos3x=1−3t an2x
2.∗cos 4x
cos4x=1−6t an2x+t a n4x
3.∗cos 5x
cos5x=1−10tan2x+5t a n4x
4.∗cos 6x
cos6x=1−15tan2x+1 5t a n4x−tan6x
5.∗sin 3x
cos3x=3t a n x−tan3x
6.∗sin 4x
cos4x=4t a n x−4t an3x
36 Trigonometric and Hyperbolic Functions 1.341
7.∗sin 5x
cos5x=5t a n x−10tan3x+t a n5x
8.∗sin 6x
cos6x=6t a n x−20tan3x+6t a n5x
9.∗cos 3x
sin3x=c o t3x−3c otx
10.∗cos 4x
sin4x=c o t4x−6c ot2x+1
11.∗cos 5x
sin5x=c o t5x−10cot3x+ 5cot x
12.∗cos 6x
sin6x=c o t6x−15cot4x+1 5c o t2x−1
13.∗sin 3x
sin3x= 3cot2x−1
14.∗sin 4x
sin4x= 4cot3x−4c otx
15.∗sin 5x
sin5x= 5cot4x−10cot2x+1
16.∗sin 6x
sin6x= 6cot5x−20cot3x+ 6cot x
1.34 Certain sums of trigonometric and hyperbolic functions
1.341
1.n−1/summationdisplay
k=0sin(x+ky)=s i n/parenleftbigg
x+n−1
2y/parenrightbigg
sinny
2cosecy
2AD (361.8)
2.n−1/summationdisplay
k=0sinh(x+ky)=s i n h/parenleftbigg
x+n−1
2y/parenrightbigg
sinhny
21
sinhy
2
3.n−1/summationdisplay
k=0cos(x+ky) = cos/parenleftbigg
x+n−1
2y/parenrightbigg
sinny
2cosecy
2AD (361.9)
4.n−1/summationdisplay
k=0cosh(x+ky)=c o s h/parenleftbigg
x+n−1
2y/parenrightbigg
sinhny
21
sinhy
2
5.2n−1/summationdisplay
k=0(−1)kcos(x+ky)=s i n/parenleftbigg
x+2n−1
2y/parenrightbigg
sinnysecy
2JO (202)
6.n−1/summationdisplay
k=0(−1)ksin(x+ky)=s i n/parenleftbigg
x+n−1
2(y+π)/parenrightbigg
sinn(y+π)
2secy
2AD (202a)
1.351 Sums of powers of trigonometric functions of multiple angles 37
Special cases
1.342
1.n/summationdisplay
k=1sinkx=s i nn+1
2xsinnx
2cosecx
2AD (361.1)
2.10n/summationdisplay
k=0coskx=c o sn+1
2xsinnx
2cosecx
2+1
=c o snx
2sinn+1
2xcosecx
2=1
2/parenleftBigg
1+sin/parenleftbig
n+1
2/parenrightbig
x
sinx
2/parenrightBigg
AD (361.2)
3.n/summationdisplay
k=1sin(2k−1)x=s i n2nxcosecx AD (361.7)
4.n/summationdisplay
k=1cos(2k−1)x=1
2sin 2nxcosecx JO (207)
1.343
1.n/summationdisplay
k=1(−1)kcoskx=−1
2+(−1)ncos/parenleftbig2n+1
2x/parenrightbig
2c osx
2AD (361.11)
2.n/summationdisplay
k=1(−1)k+1sin(2k−1)x=(−1)n+1sin 2nx
2c osxAD (361.10)
3.n/summationdisplay
k=1cos(4k−3)x+n/summationdisplay
k=1sin(4k−1)x=s i n2 nx(cos2nx+s i n2 nx)(cosx+s i nx)cosec 2 x
JO (208)
1.344
1.n−1/summationdisplay
k=1sinπk
n=c o tπ
2nAD (361.19)
2.n−1/summationdisplay
k=1sin2πk2
n=√n
2/parenleftBig
1 + cosnπ
2−sinnπ
2/parenrightBig
AD (361.18)
3.n−1/summationdisplay
k=0cos2πk2
n=√n
2/parenleftBig
1 + cosnπ
2+s i nnπ
2/parenrightBig
AD (361.17)
1.35 Sums of powers of trigonometric functions of multiple angles
1.351
1.n/summationdisplay
k=1sin2kx=1
4[(2n+1 )s i n x−sin(2n+1 )x]c os e c x
=n
2−cos(n+1 )xsinnx
2s inx
AD (361.3)
38 Trigonometric and Hyperbolic Functions 1.352
2.n/summationdisplay
k=1cos2kx=n−1
2+1
2cosnxsin(n+1 )xcosecx
=n
2+cos(n+1 )xsinnx
2s inx
AD (361.4)a
3.n/summationdisplay
k=1sin3kx=3
4sinn+1
2xsinnx
2cosecx
2−1
4sin3(n+1 )x
2sin3nx
2cosec3x
2JO (210)
4.n/summationdisplay
k=1cos3kx=3
4cosn+1
2xsinnx
2cosecx
2+1
4cos3(n+1 )
2xsin3nx
2cosec3x
2JO (211)a
5.n/summationdisplay
k=1sin4kx=1
8[3n−4c os (n+1 )xsinnxcosecx+ cos 2( n+1 )xsin 2nxcosec 2 x] JO (212)
6.n/summationdisplay
k=1cos4kx=1
8[3n+4c o s ( n+1 )xsinnxcosecx+ cos2( n+1 )xsin 2nxcosec 2 x] JO (213)
1.352
1.11n−1/summationdisplay
k=1ksinkx=sinnx
4s in2x
2−ncos/parenleftbig2n−1
2x/parenrightbig
2s inx
2AD (361.5)
2.11n−1/summationdisplay
k=1kcoskx=nsin/parenleftbig2n−1
2x/parenrightbig
2s inx
2−1−cosnx
4s in2x
2AD (361.6)
1.353
1.n−1/summationdisplay
k=1pksinkx=psinx−pnsinnx+pn+1sin(n−1)x
1−2pcosx+p2AD (361.12)a
2.n−1/summationdisplay
k=1pksinhkx=psinhx−pnsinhnx+pn+1sinh(n−1)x
1−2pcoshx+p2
3.n−1/summationdisplay
k=0pkcoskx=1−pcosx−pncosnx+pn+1cos(n−1)x
1−2pcosx+p2AD (361.13)a¡
4.n−1/summationdisplay
k=0pkcoshkx=1−pcoshx−pncoshnx+pn+1cosh(n−1)x
1−2pcoshx+p2JO (396)
1.36 Sums of products of trigonometric functions of multiple angles
1.361
1.n/summationdisplay
k=1sinkxsin(k+1 )x=1
4[(n+1 )s i n2 x−sin 2(n+1 )x]c os e c x JO (214)
2.n/summationdisplay
k=1sinkxsin(k+2 )x=n
2cos2x−1
2cos(n+3 )xsinnxcosecx JO (216)
1.381 Sums leading to hyperbolic tangents and cotangents 39
3. 2n/summationdisplay
k=1sinkxcos(2k−1)y=s i n/parenleftbigg
ny+n+1
2x/parenrightbigg
sinn(x+2y)
2cosecx+2y
2
−sin/parenleftbigg
ny−n+1
2x/parenrightbigg
sinn(2y−x)
2cosec2y−x
2
JO (217)
1.362
1.n/summationdisplay
k=1/parenleftBig
2ksin2x
2k/parenrightBig2
=/parenleftBig
2nsinx
2n/parenrightBig2
−sin2x AD (361.15)
2.n/summationdisplay
k=1/parenleftbigg1
2ksecx
2k/parenrightbigg2
=c o s e c2x−/parenleftbigg1
2ncosecx
2n/parenrightbigg2
AD (361.14)
1.37 Sums of tangents of multiple angles
1.371
1.n/summationdisplay
k=01
2ktanx
2k=1
2ncotx
2n−2c ot2 x AD (361.16)
2.n/summationdisplay
k=01
22ktan2x
2k=22n+2−1
3·22n−1+ 4cot22x−1
22ncot2x
2nAD (361.20)
1.38 Sums leading to hyperbolic tangents and cotangents
1.381
1.n−1/summationdisplay
k=0tanh⎛
⎜⎜⎝x1
nsin2/parenleftbigg2k+1
4nπ/parenrightbigg⎞
⎟⎟⎠
1+tanh2x
tan2/parenleftbigg2k+1
4nπ/parenrightbigg=t a n h( 2 nx) JO (402)a
2.n−1/summationdisplay
k=1tanh⎛
⎜⎜⎝x1
nsin2/parenleftbiggkπ
2n/parenrightbigg⎞
⎟⎟⎠
1+tanh2x
tan2/parenleftbiggkπ
2n/parenrightbigg=c o t h( 2 nx)−1
2n(tanh x+c o t h x) JO (403)
40 Trigonometric and Hyperbolic Functions 1.382
3.n−1/summationdisplay
k=0tanh⎛
⎜⎜⎝x2
(2n+1 )s i n2/parenleftbigg2k+1
2(2n+1 )π/parenrightbigg⎞
⎟⎟⎠
1+tanh2x
tan2/parenleftbigg2k+1
2(2n+1 )π/parenrightbigg=t a n h( 2 n+1 )x−tanhx
2n+1JO (404)
4.n/summationdisplay
k=1tanh⎛
⎜⎜⎝x2
(2n+1 )s i n2/parenleftbiggkπ
2(2n+1 )/parenrightbigg⎞
⎟⎟⎠
1+tanh2x
tan2/parenleftbiggkπ
(2n+1 )/parenrightbigg=c o t h( 2 n+1 )x−cothx
2n+1JO (405)
1.382
1.n−1/summationdisplay
k=01⎛
⎜⎜⎝sin2/parenleftbigg2k+1
4nπ/parenrightbigg
sinhx+1
2tanh/parenleftBigx
2/parenrightBig⎞
⎟⎟⎠=2ntanh ( nx) JO (406)
2.n−1/summationdisplay
k=11⎛
⎜⎜⎝sin2/parenleftbiggkπ
2n/parenrightbigg
sinhx+1
2tanh/parenleftBigx
2/parenrightBig⎞
⎟⎟⎠=2ncoth (nx)−2c ot h x JO (407)
3.n−1/summationdisplay
k=01⎛
⎜⎜⎝sin2/parenleftbigg2k+1
2(2n+1 )π/parenrightbigg
sinhx+1
2tanh/parenleftBigx
2/parenrightBig⎞
⎟⎟⎠=( 2n+1 )t a n h/parenleftbigg(2n+1 )x
2/parenrightbigg
−tanhx
2JO (408)
4.n/summationdisplay
k=11⎛
⎜⎜⎝sin2/parenleftbiggkπ
2n+1/parenrightbigg
sinhx+1
2tanh/parenleftBigx
2/parenrightBig⎞
⎟⎟⎠=( 2n+1 )c o t h/parenleftbigg(2n+1 )x
2/parenrightbigg
−cothx
2JO (409)
1.395 Representing sines and cosines as finite products 41
1.39 The representation of cosines and sines of multiples of the angle as finite
products
1.391
1. sin nx=nsinxcosxn−2
2/productdisplay
k=1⎛
⎜⎝1−sin2x
sin2kπ
n⎞
⎟⎠ [nis even] JO (568)
2. cos nx=n
2/productdisplay
k=1⎛
⎜⎝1−sin2x
sin2(2k−1)π
2n⎞
⎟⎠ [nis even] JO (569)
3. sin nx=nsinxn−1
2/productdisplay
k=1⎛
⎜⎝1−sin2x
sin2kπ
n⎞
⎟⎠ [nis odd] JO (570)
4. cos nx=c o s xn−1
2/productdisplay
k=1⎛
⎜⎝1−sin2x
sin2(2k−1)π
2n⎞
⎟⎠ [nis odd] JO (571)a
1.392
1. sin nx=2n−1n−1/productdisplay
k=0sin/parenleftbigg
x+kπ
n/parenrightbigg
JO (548)
2. cos nx=2n−1n/productdisplay
k=1sin/parenleftbigg
x+2k−1
2nπ/parenrightbigg
JO (549)
1.393
1.n−1/productdisplay
k=0cos/parenleftbigg
x+2k
nπ/parenrightbigg
=1
2n−1cosnx [nodd]
=1
2n−1/bracketleftbig
(−1)n
2−cosnx/bracketrightbig
[neven]
JO (543)
2.11n−1/productdisplay
k=0sin/parenleftbigg
x+2k
nπ/parenrightbigg
=(−1)n−1
2
2n−1sinnx [nodd]
=(−1)n
2
2n−1(1−cosnx)[ neven]
JO (544)
1.394n−1/productdisplay
k=0/braceleftbigg
x2−2xycos/parenleftbigg
α+2kπ
n/parenrightbigg
+y2/bracerightbigg
=x2n−2xnyncosnα+y2nJO (573)
1.395
1. cos nx−cosny=2n−1n−1/productdisplay
k=0/braceleftbigg
cosx−cos/parenleftbigg
y+2kπ
n/parenrightbigg/bracerightbigg
JO (573)
42 Trigonometric and Hyperbolic Functions 1.396
2. cosh nx−cosny=2n−1n−1/productdisplay
k=0/braceleftbigg
coshx−cos/parenleftbigg
y+2kπ
n/parenrightbigg/bracerightbigg
JO (538)
1.396
1.n−1/productdisplay
k=1/parenleftbigg
x2−2xcoskπ
n+1/parenrightbigg
=x2n−1
x2−1KR 58 (28.1)
2.n/productdisplay
k=1/parenleftbigg
x2−2xcos2kπ
2n+1+1/parenrightbigg
=x2n+1−1
x−1KR 58 (28.2)
3.n/productdisplay
k=1/parenleftbigg
x2+2xcos2kπ
2n+1+1/parenrightbigg
=x2n+1−1
x+1KR 58 (28.3)
4.n−1/productdisplay
k=0/parenleftbigg
x2−2xcos(2k+1 )π
2n+1/parenrightbigg
=x2n+1 KR 58 (28.4)
1.41 The expansion of trigonometric and hyperbolic functions in power series
1.411
1. sin x=∞/summationdisplay
k=0(−1)kx2k+1
(2k+1 ) !
2. sinh x=∞/summationdisplay
k=0x2k+1
(2k+1 ) !
3. cos x=∞/summationdisplay
k=0(−1)kx2k
(2k)!
4. cosh x=∞/summationdisplay
k=0x2k
(2k)!
5. tan x=∞/summationdisplay
k=122k/parenleftbig
22k−1/parenrightbig
(2k)!|B2k|x2k−1/bracketleftbigg
x2<π2
4/bracketrightbigg
FI II 523
6.11tanhx=x−x3
3+2x5
15−17
315x7+···=∞/summationdisplay
k=122k/parenleftbig
22k−1/parenrightbig
(2k)!B2kx2k−1
/bracketleftbigg
x2<π2
4/bracketrightbigg
7. cot x=1
x−∞/summationdisplay
k=122k|B2k|
(2k)!x2k−1/bracketleftbig
x2<π2/bracketrightbig
FI II 523a
8. coth x=1
x+x
3−x3
45+2x5
945−···=1
x+∞/summationdisplay
k=122kB2k
(2k)!x2k−1
/bracketleftbig
x2<π2/bracketrightbig
FI II 522a
1.414 Trigonometric and hyperbolic functions: power series expansion 43
9. sec x=∞/summationdisplay
k=0|E2k|
(2k)!x2k/bracketleftbigg
x2<π2
4/bracketrightbigg
CE 330a
10. sech x=1−x2
2+5x4
24−61x6
720+···=1+∞/summationdisplay
k=1E2k
(2k)!x2k
/bracketleftbigg
x2<π2
4/bracketrightbigg
CE 330
11. cosec x=1
x+∞/summationdisplay
k=12/parenleftbig
22k−1−1/parenrightbig
|B2k|x2k−1
(2k)!/bracketleftbig
x2<π2/bracketrightbig
CE 329a
12. cosech x=1
x−1
6x+7x3
360−31x5
15120+···=1
x−∞/summationdisplay
k=12/parenleftbig
22k−1−1/parenrightbig
B2k
(2k)!x2k−1
/bracketleftbig
x2<π2/bracketrightbig
JO (418)
1.412
1. sin2x=∞/summationdisplay
k=1(−1)k+122k−1x2k
(2k)!JO (452)a
2. cos2x=1−∞/summationdisplay
k=1(−1)k+122k−1x2k
(2k)!JO (443)
3. sin3x=1
4∞/summationdisplay
k=1(−1)k+132k+1−3
(2k+1 ) !x2k+1JO (452a)a
4. cos3x=1
4∞/summationdisplay
k=0(−1)k/parenleftbig
32k+3/parenrightbig
x2k
(2k)!JO (443a)
1.413
1. sinh x=c o s e c x∞/summationdisplay
k=1(−1)k+122k−1x4k−2
(4k−1)!JO (508)
2. cosh x=s e c x+s e cx∞/summationdisplay
k=1(−1)k22kx4k
(4k)!JO (507)
3. sinh x=s e c x∞/summationdisplay
k=1(−1)[k/2]2k−1x2k−1
(2k−1)!JO (510)
4. cosh x=c o s e c x∞/summationdisplay
k=1(−1)[(k−1)/2]2k−1x2k−1
(2k−1)!JO (509)
1.414
1. cos/bracketleftBig
nln/parenleftBig
x+/radicalbig
1+x2/parenrightBig/bracketrightBig
=1−∞/summationdisplay
k=0(−1)k/parenleftbig
n2+02/parenrightbig/parenleftbig
n2+22/parenrightbig
.../bracketleftbig
n2+( 2k)2/bracketrightbig
(2k+2 ) !x2k+2
/bracketleftbig
x2<1/bracketrightbig
AD (6456.1)
44 Trigonometric and Hyperbolic Functions 1.421
2. sin/bracketleftBig
nln/parenleftBig
x+/radicalbig
1+x2/parenrightBig/bracketrightBig
=nx−n∞/summationdisplay
k=1(−1)k+1/parenleftbig
n2+12/parenrightbig/parenleftbig
n2+32/parenrightbig
.../bracketleftbig
n2+( 2k−1)2/bracketrightbig
x2k+1
(2k+1 ) !
/bracketleftbig
x2<1/bracketrightbig
AD (6456.2)
Power series for ln sin x,l nc o s x,a n dl n t a n xsee1.518 .
1.42 Expansion in series of simple fractions
1.421
1. tanπx
2=4x
π∞/summationdisplay
k=11
(2k−1)2−x2BR* (191), AD (6495.1)
2.10tanhπx
2=4x
π∞/summationdisplay
k=11
(2k−1)2+x2
3. cot πx=1
πx+2x
π∞/summationdisplay
k=11
x2−k2=1
πx+x
π∞/summationdisplay
k=−∞
k/negationslash=01
k(x−k)AD (6495.2), JO (450a)
4. coth πx=1
πx+2x
π∞/summationdisplay
k=11
x2+k2(cf.1.217 1)
5. tan2πx
2=x2∞/summationdisplay
k=12(2k−1)2−x2
(12−x2)2(32−x2)2...[(2k−1)2−x2]2JO (450)
1.422
1. secπx
2=4
π∞/summationdisplay
k=1(−1)k+1 2k−1
(2k−1)2−x2AD (6495.3)a
2. sec2πx
2=4
π2∞/summationdisplay
k=1/braceleftbigg1
(2k−1−x)2+1
(2k−1+x)2/bracerightbigg
JO (451)a
3. cosec πx=1
πx+2x
π∞/summationdisplay
k=1(−1)k
x2−k2(see also 1.217 2) AD (6495.4)a
4. cosec2πx=1
π2∞/summationdisplay
k=−∞1
(x−k)2=1
π2x2+2
π2∞/summationdisplay
k=1x2+k2
(x2−k2)2JO (446)
5.1+xcosecx
2x2=1
x2−∞/summationdisplay
k=1(−1)k+1
(x2−k2π2)JO (449)
6. cosec πx=2
π∞/summationdisplay
k=−∞(−1)k
x2−k2JO (450b)
1.423π2
4m2cosec2π
m+π
4mcotπ
m−1
2=∞/summationdisplay
k=11
(1−k2m2)2JO (477)
1.439 Representation in the form of an infinite product 45
1.43 Representation in the form of an infinite product
1.431
1. sin x=x∞/productdisplay
k=1/parenleftbigg
1−x2
k2π2/parenrightbigg
EU
2. sinh x=x∞/productdisplay
k=1/parenleftbigg
1+x2
k2π2/parenrightbigg
EU
3. cos x=∞/productdisplay
k=0/parenleftbigg
1−4x2
(2k+1 )2π2/parenrightbigg
EU
4. cosh x=∞/productdisplay
k=0/parenleftbigg
1+4x2
(2k+1 )2π2/parenrightbigg
EU
1.432
1.11cosx−cosy=2/parenleftbigg
1−x2
y2/parenrightbigg
sin2y
2∞/productdisplay
k=1/parenleftBigg
1−x2
(2kπ+y)2/parenrightBigg/parenleftbigg
1−x2
(2kπ−y)2/parenrightbigg
AD (653.2)
2. cosh x−cosy=2/parenleftbigg
1+x2
y2/parenrightbigg
sin2y
2∞/productdisplay
k=1/parenleftbigg
1+x2
(2kπ+y)2/parenrightbigg/parenleftbigg
1+x2
(2kπ−y)2/parenrightbigg
AD (653.1)
1.433 cosπx
4−sinπx
4=∞/productdisplay
k=1/bracketleftbigg
1+(−1)kx
2k−1/bracketrightbigg
BR* 189
1.434 cos2x=1
4(π+2x)2∞/productdisplay
k=1/bracketleftBigg
1−/parenleftbiggπ+2x
2kπ/parenrightbigg2/bracketrightBigg2
MO 216
1.435sinπ(x+a)
sinπa=x+a
a∞/productdisplay
k=1/parenleftbigg
1−x
k−a/parenrightbigg/parenleftbigg
1+x
k+a/parenrightbigg
MO 216
1.436 1−sin2πx
sin2πa=∞/productdisplay
k=−∞/bracketleftBigg
1−/parenleftbiggx
k−a/parenrightbigg2/bracketrightBigg
MO 216
1.437sin 3x
sinx=−∞/productdisplay
k=−∞/bracketleftBigg
1−/parenleftbigg2x
x+kπ/parenrightbigg2/bracketrightBigg
MO 216
1.438coshx−cosa
1−cosa=∞/productdisplay
k=−∞/bracketleftBigg
1+/parenleftbiggx
2kπ+a/parenrightbigg2/bracketrightBigg
MO 216
1.439
1. sin x=x∞/productdisplay
k=1cosx
2k[|x|<1] AD (615), MO 216
2.sinx
x=∞/productdisplay
k=1/bracketleftbigg
1−4
3sin2/parenleftBigx
3k/parenrightBig/bracketrightbigg
MO 216
46 Trigonometric and Hyperbolic Functions 1.441
1.44–1.45 Trigonometric (Fourier) series
1.441
1.∞/summationdisplay
k=1sinkx
k=π−x
2[0<x< 2π] FI III 539
2.∞/summationdisplay
k=1coskx
k=−1
2ln[2 (1 −cosx)] [0 <x< 2π] FI III 530a, AD (6814)
3.∞/summationdisplay
k=1(−1)k−1sinkx
k=x
2[−π<x<π ] FI III 542
4.∞/summationdisplay
k=1(−1)k−1coskx
k=l n/parenleftBig
2c osx
2/parenrightBig
[−π<x<π ] FI III 550
1.442
1.11∞/summationdisplay
k=1sin(2k−1)x
2k−1=π
4signx [−π<x<π ] FI III 541
2.∞/summationdisplay
k=1cos(2k−1)x
2k−1=1
2ln cotx
2[0<x<π ]
BR* 168, JO (266), GI III(195)
3.∞/summationdisplay
k=1(−1)k−1sin(2k−1)x
2k−1=1
2ln tan/parenleftBigπ
4+x
2/parenrightBig/bracketleftBig
−π
2<x<π
2/bracketrightBig
BR* 168, JO (268)a
4.10∞/summationdisplay
k=1(−1)k−1cos(2k−1)x
2k−1=π
4/bracketleftBig
−π
2<x<π
2/bracketrightBig
=−π
4/bracketleftbiggπ
2<x<3π
2/bracketrightbigg
BR* 168, JO (269)
1.443
1.8∞/summationdisplay
k=1coskπx
k2n=(−1)n−122n−1π2n
(2n)!2n/summationdisplay
k=0/parenleftbigg2n
k/parenrightbigg
B2n−kρk
=(−1)n−11
2(2π)2n
(2n)!B2n/parenleftBigx
2/parenrightBig
/bracketleftBig
0≤x≤2,ρ=x
2−/floorleftBigx
2/floorrightBig/bracketrightBig
CE 340, GE 71
2.∞/summationdisplay
k=1sinkπx
k2n+1=(−1)n−122nπ2n+1
(2n+1 ) !2n+1/summationdisplay
k=0/parenleftbigg2n+1
k/parenrightbigg
B2n−k+1ρk
=(−1)n−11
2(2π)2n+1
(2n+1 ) !B2n+1/parenleftBigx
2/parenrightBig
/bracketleftBig
0<x< 1;ρ=x
2−/floorleftBigx
2/floorrightBig/bracketrightBig
CE 340
1.445 Trigonometric (Fourier) series 47
3.∞/summationdisplay
k=1coskx
k2=π2
6−πx
2+x2
4[0≤x≤2π] FI III 547
4.∞/summationdisplay
k=1(−1)k−1coskx
k2=π2
12−x2
4[−π≤x≤π] FI III 544
5.∞/summationdisplay
k=1sinkx
k3=π2x
6−πx2
4+x3
12[0≤x≤2π]
6.∞/summationdisplay
k=1coskx
k4=π4
90−π2x2
12+πx3
12−x4
48[0≤x≤2π] AD (6617)
7.∞/summationdisplay
k=1sinkx
k5=π4x
90−π2x3
36+πx4
48−x5
240[0≤x≤2π] AD (6818)
1.444
1.∞/summationdisplay
k=1sin 2(k+1 )x
k(k+1 )=s i n2 x−(π−2x)sin2x−sinxcosxln/parenleftbig
4s in2x/parenrightbig
[0≤x≤π] BR* 168, GI III (190)
2.∞/summationdisplay
k=1cos2(k+1 )x
k(k+1 )=c o s2 x−/parenleftBigπ
2−x/parenrightBig
sin 2x+s i n2xln/parenleftbig
4s in2x/parenrightbig
[0≤x≤π] BR* 168
3.∞/summationdisplay
k=1(−1)ksin(k+1 )x
k(k+1 )=s i nx−x
2(1 + cos x)−sinxln/vextendsingle/vextendsingle/vextendsingle2c osx
2/vextendsingle/vextendsingle/vextendsingleMO 213
4.∞/summationdisplay
k=1(−1)kcos(k+1 )x
k(k+1 )=c o s x−x
2sinx−(1 + cos x)l n/vextendsingle/vextendsingle/vextendsingle2c osx
2/vextendsingle/vextendsingle/vextendsingleMO 213
5.∞/summationdisplay
k=0(−1)ksin(2k+1 )x
(2k+1 )2=π
4x/bracketleftBig
−π
2≤x≤π
2/bracketrightBig
=π
4(π−x)/bracketleftbiggπ
2≤x≤3
2π/bracketrightbigg
MO 213
6.6∞/summationdisplay
k=1cos(2k−1)x
(2k−1)2=π
4/parenleftBigπ
2−|x|/parenrightBig
[−π≤x≤π] FI III 546
7.∞/summationdisplay
k=1cos2kx
(2k−1)(2k+1 )=1
2−π
4sinx/bracketleftBig
0≤x≤π
2/bracketrightBig
JO (591)
1.445
1.∞/summationdisplay
k=1ksinkx
k2+α2=π
2sinhα(π−x)
sinhαπ[0<x< 2π] BR* 157, JO (411)
2.∞/summationdisplay
k=1coskx
k2+α2=π
2αcoshα(π−x)
sinhαπ−1
2α2[0≤x≤2π] BR* 257, JO (410)
48 Trigonometric and Hyperbolic Functions 1.446
3.∞/summationdisplay
k=1(−1)kcoskx
k2+α2=π
2αcoshαx
sinhαπ−1
2α2[−π≤x≤π] FI III 546
4.∞/summationdisplay
k=1(−1)k−1ksinkx
k2+α2=π
2sinhαx
sinhαπ[−π<x<π ] FI III, 546
5.∞/summationdisplay
k=1ksinkx
k2−α2=πsin{α[(2m+1 )π−x]}
2s inαπ/bracketleftBig
ifx=2mπ,t h e n/summationdisplay
···=0/bracketrightBig
[2mπ < x < (2m+2 )π, α not an integer] MO 213
6.∞/summationdisplay
k=1coskx
k2−α2=1
2α2−π
2cos[α{(2m+1 )π−x}]
αsinαπ
[2mπ≤x≤(2m+2 )π, α not an integer] MO 213
7.∞/summationdisplay
k=1(−1)kksinkx
k2−α2=πsin[α(2mπ−x)]
2s inαπ/bracketleftBig
ifx=( 2m+1 )π,t h e n/summationdisplay
···=0/bracketrightBig
,
[(2m−1)π<x< (2m+1 )π,αnot an integer] FI III 545a
8.∞/summationdisplay
k=1(−1)kcoskx
k2−α2=1
2α2−π
2cos[α(2mπ−x)]
αsinαπ
[(2m−1)π≤x≤(2m+1 )π,αnot an integer] FI III 545a
9.∗∞/summationdisplay
n=−∞einα
(n−β)2+γ2=π
γeiβ(α−2π)sinh(γα)+eiβαsinh [γ(2π−α)]
cosh(2 πγ)−cos(2πβ)
[0≤α≤2π]
1.446∞/summationdisplay
k=1(−1)k+1cos(2k+1 )x
(2k−1)(2k+ 1)(2 k+3 )=π
8cos2x−1
3cosx
/bracketleftBig
−π
2≤x≤π
2/bracketrightBig
BR* 256, GI III (189)
1.447
1.∞/summationdisplay
k=1pksinkx=psinx
1−2pcosx+p2
[|p|<1] FI II 559
2.∞/summationdisplay
k=0pkcoskx=1−pcosx
1−2pcosx+p2
[|p|<1] FI II 559
3. 1 + 2∞/summationdisplay
k=1pkcoskx=1−p2
1−2pcosx+p2
[|p|<1] FI II 559a, MO 213
1.449 Trigonometric (Fourier) series 49
1.448
1.∞/summationdisplay
k=1pksinkx
k=a r c t a npsinx
1−pcosx/bracketleftbig
0<x< 2π, p2≤1/bracketrightbig
FI II 559
2.∞/summationdisplay
k=1pkcoskx
k=−1
2ln/parenleftbig
1−2pcosx+p2/parenrightbig
/bracketleftbig
0<x< 2π, p2≤1/bracketrightbig
FI II 559
3.∞/summationdisplay
k=1p2k−1sin(2k−1)x
2k−1=1
2arctan2psinx
1−p2
/bracketleftbig
0<x< 2π, p2≤1/bracketrightbig
JO (594)
4.∞/summationdisplay
k=1p2k−1cos(2k−1)x
2k−1=1
4ln1+2pcosx+p2
1−2pcosx+p2
/bracketleftbig
0<x< 2π, p2≤1/bracketrightbig
JO (259)
5.∞/summationdisplay
k=1(−1)k−1p2k−1sin(2k−1)x
2k−1=1
4ln1+2psinx+p2
1−2psinx+p2
/bracketleftbig
0<x<π , p2≤1/bracketrightbig
JO (261)
6.∞/summationdisplay
k=1(−1)k−1p2k−1cos(2k−1)x
2k−1=1
2arctan2pcosx
1−p2
/bracketleftbig
0<x<π , p2≤1/bracketrightbig
JO (597)
1.449
1.∞/summationdisplay
k=1pksinkx
k!=epcosxsin (psinx)
/bracketleftbig
p2≤1/bracketrightbig
JO (486)
2.∞/summationdisplay
k=0pkcoskx
k!=epcosxcos(psinx)
/bracketleftbig
p2≤1/bracketrightbig
JO (485)
LetS(x)=−1
xcosx+1
xandC(x)=1
xsinx.
3.∗∞/summationdisplay
n=1n
n2−a2S(nx)=π
2[C(ax)−cot(πa)S(ax)] [0 <x< 2π, a /negationslash=0,±1,±2,...]
4.∗∞/summationdisplay
n=11
n2−a2C(nx)=1
2a2−π
2a[S(ax)−cot(πa)C(ax)]
[0≤x≤2π, a /negationslash=0,±1,±2,...]
5.∗∞/summationdisplay
n=1(−1)n−1n
n2−a2S(nx)=π
2cosec( πa)S(ax)[ −π<x<π , a /negationslash=0,±1,±2,...]
50 Trigonometric and Hyperbolic Functions 1.451
6.∗∞/summationdisplay
n=1(−1)n−1
n2−a2C(nx)=−1
2a2+π
2acosec( πa)C(ax)[ −π<x<π , a /negationslash=0,±1,±2,...]
7.∗∞/summationdisplay
n=12n−1
(2n−1)2−a2S(nx)=π
4/bracketleftBig
C(ax)+t a n/parenleftBigπa
2/parenrightBig
S(ax)/bracketrightBig
[0<x<π , a /negationslash=0,±1,±2,...]
8.∗∞/summationdisplay
n=11
(2n−1)2−a2C(nx)=−π
4a/bracketleftBig
S(ax)−tan/parenleftBigπa
2/parenrightBig
C(ax)/bracketrightBig
[0≤x≤π, a /negationslash=0,±1,±2,...]
9.∗∞/summationdisplay
n=1(−1)n−1
(2n−1)2−a2S(nx)=π
4asec/parenleftBigπa
2/parenrightBig
S(ax)/bracketleftBig
−π
2≤x≤π
2,a/negationslash=0,±1,±2,.../bracketrightBig
10.∗∞/summationdisplay
n=1(−1)n−1(2n−1)
(2n−1)2−a2C(nx)=π
4sec/parenleftBigπa
2/parenrightBig
C(ax)/bracketleftBig
−π
2≤x≤π
2,a/negationslash=0,±1,±2,.../bracketrightBig
Fourier expansions of hyperbolic functions
1.451
1. sinh x=c o s x∞/summationdisplay
k=0/parenleftbig
12+02/parenrightbig/parenleftbig
12+22/parenrightbig
.../bracketleftbig
12+( 2k)2/bracketrightbig
(2k+1 ) !sin2k+1x JO (504)
2. cosh x=c o s x+c o s x∞/summationdisplay
k=1/parenleftbig
12+12/parenrightbig/parenleftbig
12+32/parenrightbig
.../bracketleftbig
12+( 2k−1)2/bracketrightbig
(2k)!sin2kx JO (503)
1.452
1. sinh ( xcosθ)=s e c( xsinθ)∞/summationdisplay
k=0x2k+1cos(2k+1 )θ
(2k+1 ) !/bracketleftbig
x2<1/bracketrightbig
JO (391)
2. cosh ( xcosθ)=s e c( xsinθ)∞/summationdisplay
k=0x2kcos2kθ
(2k)!/bracketleftbig
x2<1/bracketrightbig
JO (390)
3. sinh ( xcosθ)=c o s e c( xsinθ)∞/summationdisplay
k=1x2ksin 2kθ
(2k)!/bracketleftbig
x2<1,xsinθ/negationslash=0/bracketrightbig
JO (393)
4. cosh ( xcosθ)=c os e c( xsinθ)∞/summationdisplay
k=0x2k+1sin(2k+1 )θ
(2k+1 ) !/bracketleftbig
x2<1,xsinθ/negationslash=0/bracketrightbig
JO (392)
1.480 Lobachevskiy’s “Angle of Parallelism” 51
1.46 Series of products of exponential and trigonometric functions
1.461
1.∞/summationdisplay
k=0e−ktsinkx=1
2sinx
cosht−cosx[t>0] MO 213
2. 1 + 2∞/summationdisplay
k=1e−ktcoskx=sinht
cosht−cosx[t>0] MO 213
1.4629∞/summationdisplay
k=1sinkxsinky
ke−2k|t|=1
4ln⎡
⎢⎣sin2x+y
2+s i n h2t
sin2x−y
2+s i n h2t⎤
⎥⎦ MO 214
1.463
1. excosϕcos(xsinϕ)=∞/summationdisplay
n=0xncosnϕ
n!/bracketleftbig
x2<1/bracketrightbig
AD (6476.1)
2. excosϕsin(xsinϕ)=∞/summationdisplay
n=1xnsinnϕ
n!/bracketleftbig
x2<1/bracketrightbig
AD (6476.2)
1.47 Series of hyperbolic functions
1.471
1.∞/summationdisplay
k=1sinhkx
k!=ecoshxsinh (sinh x). JO (395)
2.∞/summationdisplay
k=0coshkx
k!=ecoshxcosh (sinh x). JO (394)
3.∞/summationdisplay
k=01
(2k+1 )3/bracketleftbigg1
xtanh(2m+1 )πx
2+xtanh(2m+1 )π
2x/bracketrightbigg
=π3
16
1.472
1.∞/summationdisplay
k=1pksinhkx=psinhx
1−2pcoshx+p2/bracketleftbig
p2<1/bracketrightbig
JO (396)
2.∞/summationdisplay
k=0pkcoshkx=1−pcoshx
1−2pcoshx+p2/bracketleftbig
p2<1/bracketrightbig
JO (397)a
1.48 Lobachevskiy’s “Angle of Parallelism” Π(x)
1.480 Definition.
1. Π( x) = 2 arccot ex= 2arctan e−x[x≥0] LO III 297, LOI 120
52 Trigonometric and Hyperbolic Functions 1.481
2. Π( x)=π−Π(−x)[ x<0] LO III 183, LOI 193
1.481 Functional relations
1. sin Π( x)=1
coshxLO III 297
2. cosΠ( x)=t a n h x LO III 297
3. tan Π( x)=1
sinhxLO III 297
4. cotΠ( x)=s i n h x LO III 297
5. sin Π( x+y)=sin Π( x)sinΠ( y)
1 + cosΠ( x)cosΠ( y)LO III 297
6. cosΠ( x+y)=cosΠ( x) + cos Π( y)
1 + cosΠ( x)cosΠ( y)LO III 183
1.482 Connection with the Gudermannian.
gd(−x)=Π ( x)−π
2
(Definite) integral of the angle of parallelism: cf. 4.581 and4.561 .
1.49 The hyperbolic amplitude (the Gudermannian) gdx
1.490 Definition.
1. gd x=/integraldisplayx
0dt
cosht= 2arctan ex−π
2JA
2. x=/integraldisplaygdx
0dt
cost=l nt a n/parenleftbigggdx
2+π
4/parenrightbigg
JA
1.491 Functional relations.
1. cosh x= sec(gd x) AD (343.1), JA
2. sinh x=t a n ( g d x) AD (343.2), JA
3. ex= sec(gd x)+t a n ( g d x)=t a n/parenleftbiggπ
4+gdx
2/parenrightbigg
=1+s i n ( g d x)
cos(gd x)AD (343.5), JA
4. tanh x=s i n ( g d x) AD (343.3), JA
5. tanhx
2=t a n/parenleftbigg1
2gdx/parenrightbigg
AD (343.4), JA
6. arctan(tanh x)=1
2gd 2x AD (343.6a)
1.492 Ifγ=g dx,t h e n ix=g diγ JA
1.493 Series expansion.
1.gdx
2=∞/summationdisplay
k=0(−1)k
2k+1tanh2k+1x
2JA
1.513 Series representation 53
2.x
2=∞/summationdisplay
k=01
2k+1tan2k+1/parenleftbigg1
2gdx/parenrightbigg
JA
3. gd x=x−x3
6+x5
24−61x7
5040+··· JA
4. x=g dx+(gdx)3
6+(gdx)5
24+61(gd x)7
5040+.../bracketleftBig
gdx<π
2/bracketrightBig
JA
1.5 The Logarithm
1.51 Series representation
1.511 ln(1 + x)=x−1
2x2+1
3x3−1
4x4+···=∞/summationdisplay
k=1(−1)k+1xk
k
[−1<x≤1]
1.512
1. ln x=(x−1)−1
2(x−1)2+1
3(x−1)3−···=∞/summationdisplay
k=1(−1)k+1(x−1)k
k
[0<x≤2]
2. ln x=2/bracketleftBigg
x−1
x+1+1
3/parenleftbiggx−1
x+1/parenrightbigg3
+1
5/parenleftbiggx−1
x+1/parenrightbigg5
+.../bracketrightBigg
=2∞/summationdisplay
k=11
2k−1/parenleftbiggx−1
x+1/parenrightbigg2k−1
[0<x]
3. ln x=x−1
x+1
2/parenleftbiggx−1
x/parenrightbigg2
+1
3/parenleftbiggx−1
x/parenrightbigg3
+···=∞/summationdisplay
k=11
k/parenleftbiggx−1
x/parenrightbiggk
/bracketleftbig
x≥1
2/bracketrightbig
AD (644.6)
4.∗lnx= lim
/epsilon1→0/parenleftbiggx/epsilon1−1
/epsilon1/parenrightbigg
1.513
1. ln1+x
1−x=2∞/summationdisplay
k=11
2k−1x2k−1/bracketleftbig
x2<1/bracketrightbig
FI II 421
2. lnx+1
x−1=2∞/summationdisplay
k=11
(2k−1)x2k−1/bracketleftbig
x2>1/bracketrightbig
AD (644.9)
3. lnx
x−1=∞/summationdisplay
k=11
kxk[x≤−1o rx>1] JO (88a)
4. ln1
1−x=∞/summationdisplay
k=1xk
k[−1≤x<1] JO (88b)
5.1−x
xln1
1−x=1−∞/summationdisplay
k=1xk
k(k+1 )[−1≤x<1] JO (102)
54 The Logarithm 1.514
6.1
1−xln1
1−x=∞/summationdisplay
k=1xkk/summationdisplay
n=11
n/bracketleftbig
x2<1/bracketrightbig
JO (88e)
7.(1−x)2
2x3ln1
1−x=1
2x2−3
4x+∞/summationdisplay
k=1xk−1
k(k+1 ) (k+2 )[−1≤x<1] AD (6445.1)
1.514 ln/parenleftbig
1−2xcosϕ+x2/parenrightbig
=−2∞/summationdisplay
k=1coskϕ
kxk;l n/parenleftBig
x+/radicalbig
1+x2/parenrightBig
=a r c s i n h x
(see1.631, 1.641, 1.642, 1.646 )/bracketleftbig
x2≤1,xcosϕ/negationslash=1/bracketrightbig
M O9 8 ,F II I4 8 5
1.515
1.11ln/parenleftBig
1+/radicalbig
1+x2/parenrightBig
=l n2+1·1
2·2x2−1·1·3
2·4·4x4+1·1·3·5
2·4·6·6x6−...
=l n2 −∞/summationdisplay
k=1(−1)k(2k−1)!
22k(k!)2x2k
/bracketleftbig
x2≤1/bracketrightbig
JO (91)
2. ln/parenleftBig
1+/radicalbig
1+x2/parenrightBig
=l nx+1
x−1
2·3x3+1·3
2·4·5x5−...
=l nx+1
x+∞/summationdisplay
k=1(−1)k (2k−1)!
22k−1·k!(k−1)!(2k+1 )x2k+1
/bracketleftbig
x2≥1/bracketrightbig
AD (644.4)
3./radicalbig
1+x2ln/parenleftBig
x+/radicalbig
1+x2/parenrightBig
=x−∞/summationdisplay
k=1(−1)k22k−1(k−1)!k!
(2k+1 ) !x2k+1
/bracketleftbig
x2≤1/bracketrightbig
JO (93)
4.ln/parenleftbig
x+√
1+x2/parenrightbig
√
1+x2=∞/summationdisplay
k=0(−1)k22k(k!)2
(2k+1 ) !x2k+1/bracketleftbig
x2≤1/bracketrightbig
JO (94)
1.516
1.1
2{ln(1±x)}2=∞/summationdisplay
k=1(∓1)k+1xk+1
k+1k/summationdisplay
n=11
n/bracketleftbig
x2<1/bracketrightbig
J O( 8 6 ) ,J O( 8 5 )
2.1
6{ln(1 + x)}3=∞/summationdisplay
k=1(−1)k+1xk+2
k+2k/summationdisplay
n=11
n+1n/summationdisplay
m=11
m/bracketleftbig
x2<1/bracketrightbig
AD (644.14)
3. −ln(1 + x)·ln(1−x)=∞/summationdisplay
k=1x2k
k2k−1/summationdisplay
n=1(−1)n+1
n/bracketleftbig
x2<1/bracketrightbig
JO (87)
4.1
4x/braceleftbigg1+x√xln1+√x
1−√x+2l n ( 1 −x)/bracerightbigg
=1
2x+∞/summationdisplay
k=1xk−1
(2k−1)2k(2k+1 )
[0<x< 1] AD (6445.2)
1.521 Series of logarithms (cf. 1.431) 55
1.517
1.61
2x/braceleftbigg
1−ln(1 + x)−1−x√xarctan√x/bracerightbigg
=∞/summationdisplay
k=1(−1)k+1xk−1
(2k−1)2k(2k+1 )
[0<x≤1] AD (6445.3)
2.1
2arctan xln1+x
1−x=∞/summationdisplay
k=1x4k−2
2k−12k−1/summationdisplay
n=1(−1)n−1
2n−1/bracketleftbig
x2<1/bracketrightbig
BR* 163
3.1
2arctan xln/parenleftbig
1+x2/parenrightbig
=∞/summationdisplay
k=1(−1)k+1x2k+1
2k+12k/summationdisplay
n=11
n/bracketleftbig
x2≥1/bracketrightbig
AD (6455.3)
1.518
1. lnsin x=l nx−x2
6−x4
180−x6
2835−...
=l nx+∞/summationdisplay
k=1(−1)k22k−1B2kx2k
k(2k)!
[0<x<π ] AD (643.1)a
2.3lncos x=−x2
2−x4
12−x6
45−17x8
2520−...
=−∞/summationdisplay
k=122k−1/parenleftbig
22k−1/parenrightbig
|B2k|
k(2k)!x2k=−1
2∞/summationdisplay
k=1sin2kx
k/bracketleftbigg
x2<π2
4/bracketrightbigg
FI II 524
3. lntan x=l nx+x2
3+7
90x4+62
2835x6+127
18,900x8+...
=l nx+∞/summationdisplay
k=1(−1)k+1/parenleftbig
22k−1−1/parenrightbig
22kB2kx2k
k(2k)!/bracketleftBig
0<x<π
2/bracketrightBig
AD (643.3)a
1.52 Series of logarithms (cf. 1.431)
1.521
1.∞/summationdisplay
k=1ln/parenleftbigg
1−4x2
(2k−1)2π2/parenrightbigg
=l nc o s x/bracketleftBig
−π
2<x<π
2/bracketrightBig
2.∞/summationdisplay
k=1ln/parenleftbigg
1−x2
k2π2/parenrightbigg
=l ns i n x−lnx [0<x<π ]
56 The Inverse Trigonometric and Hyperbolic Functions 1.621
1.6 The Inverse Trigonometric and Hyperbolic Functions
1.61 The domain of definition
The principal values of the inverse trigonometric functions are defined by the inequalities:
1. −π
2≤arcsin x≤π
2;0≤arccos x≤π [−1≤x≤1] FI II 553
2. −π
2<arctan x<π
2;0<arccot x<π [−∞<x< +∞] FI II 552
1.62–1.63 Functional relations
1.621 The relationship between the inverse and the direct trigonometric functions.
1. arcsin(sin x)=x−2nπ/bracketleftBig
2nπ−π
2≤x≤2nπ+π
2/bracketrightBig
=−x+( 2n+1 )π/bracketleftBig
(2n+1 )π−π
2≤x≤(2n+1 )π+π
2/bracketrightBig
2. arccos(cos x)=x−2nπ [2nπ≤x≤(2n+1 )π]
=−x+2 (n+1 )π [(2n+1 )π≤x≤2(n+1 )π]
3. arctan(tan x)=x−nπ/bracketleftBig
nπ−π
2<x<n π +π
2/bracketrightBig
4. arccot(cot x)=x−nπ [nπ < x < (n+1 )π]
1.622 The relationship between the inverse trigonometric functions, the inverse hyperbolic functions,
and the logarithm.
1. arcsin z=1
iln/parenleftBig
iz+/radicalbig
1−z2/parenrightBig
=1
iarcsinh( iz)
2. arccos z=1
iln/parenleftBig
z+/radicalbig
z2−1/parenrightBig
=1
iarccosh z
3. arctan z=1
2iln1+iz
1−iz=1
iarctanh( iz)
4. arccot z=1
2ilniz−1
iz+1=iarccoth( iz)
5. arcsinh z=l n/parenleftBig
z+/radicalbig
z2+1/parenrightBig
=1
iarcsin( iz)
6. arccosh z=l n/parenleftBig
z+/radicalbig
z2−1/parenrightBig
=iarccos z
7. arctanh z=1
2ln1+z
1−z=1
iarctan( iz)
8. arccoth z=1
2lnz+1
z−1=1
iarccot( −iz)
1.624 Functional relations 57
Relations between different inverse trigonometric functions
1.623
1. arcsin x+ arccos x=π
2NV 43
2. arctan x+ arccot x=π
2NV 43
1.624
1. arcsin x= arccos/radicalbig
1−x2 [0≤x≤1] NV 47 (5)
=−arccos/radicalbig
1−x2[−1≤x≤0] NV 46 (2)
2. arcsin x=a r c t a nx√
1−x2/bracketleftbig
x2<1/bracketrightbig
3. arcsin x= arccot√
1−x2
x[0<x≤1]
= arccot√
1−x2
x−π[−1≤x<0] NV 49 (10)
4. arccos x=a r c s i n/radicalbig
1−x2 [0≤x≤1]
=π−arcsin/radicalbig
1−x2[−1≤x≤0] NV 48 (6)
5. arccos x=a r c t a n√
1−x2
x[0<x≤1]
=π+a r c t a n√
1−x2
x[−1≤x<0] NV 48 (8)
6. arccos x= arccotx√
1−x2[−1≤x<1] NV 46 (4)
7. arctan x=a r c s i nx√
1+x2NV 6 (3)
8. arctan x= arccos1√
1+x2[x≥0]
=−arccos1√
1+x2[x≤0] NV 48 (7)
9. arctan x= arccot1
x[x>0]
=−arccot1
x−π[x<0] NV 49 (9)
10.11arccot x=a r c s i n1√
1+x2[x>0]
=π−arcsin1√
1+x2[x<0] NV 49 (11)
11. arccot x= arccosx√
1+x2NV 46 (4)
58 The Inverse Trigonometric and Hyperbolic Functions 1.625
12. arccot x=a r c t a n1
x[x>0]
=π+a r c t a n1
x[x<0] NV 49 (12)
1.625
1. arcsin x+a r c s i n y=a r c s i n/parenleftBig
x/radicalbig
1−y2+y/radicalbig
1−x2/parenrightBig/bracketleftbig
xy≤0o rx2+y2≤1/bracketrightbig
=π−arcsin/parenleftBig
x/radicalbig
1−y2+y/radicalbig
1−x2/parenrightBig/bracketleftbig
x>0,y > 0a n d x2+y2>1/bracketrightbig
=−π−arcsin/parenleftBig
x/radicalbig
1−y2+y/radicalbig
1−x2/parenrightBig/bracketleftbig
x<0,y < 0a n d x2+y2>1/bracketrightbig
NV 54(1), GI I (880)
2. arcsin x+a r c s i n y= arccos/parenleftBig/radicalbig
1−x2/radicalbig
1−y2−xy/parenrightBig
[x≥0,y≥0]
=−arccos/parenleftBig/radicalbig
1−x2/radicalbig
1−y2−xy/parenrightBig
[x<0,y < 0] NV 55
3. arcsin x+a r c s i n y=a r c t a nx/radicalbig
1−y2+y√
1−x2
√
1−x2/radicalbig
1−y2−xy/bracketleftbig
xy≤0o rx2+y2<1/bracketrightbig
=a r c t a nx/radicalbig
1−y2+y√
1−x2
√
1−x2/radicalbig
1−y2−xy+π/bracketleftbig
x>0,y > 0a n d x2+y2>1/bracketrightbig
=a r c t a nx/radicalbig
1−y2+y√
1−x2
√
1−x2/radicalbig
1−y2−xy−π/bracketleftbig
x<0,y < 0a n d x2+y2>1/bracketrightbig
NV 56
4. arcsin x−arcsin y=a r c s i n/parenleftBig
x/radicalbig
1−y2−y/radicalbig
1−x2/parenrightBig/bracketleftbig
xy≥0o rx2+y2≤1/bracketrightbig
=π−arcsin/parenleftBig
x/radicalbig
1−y2−y/radicalbig
1−x2/parenrightBig/bracketleftbig
x>0,y < 0a n d x2+y2>1/bracketrightbig
=−π−arcsin/parenleftBig
x/radicalbig
1−y2−y/radicalbig
1−x2/parenrightBig/bracketleftbig
x<0,y > 0a n d x2+y2>1/bracketrightbig
NV 55(2)
5. arcsin x−arcsin y= arccos/parenleftBig
x/radicalbig
1−x2/radicalbig
1−y2+xy/parenrightBig
[xy > y ]
=−arccos/parenleftBig/radicalbig
1−x2/radicalbig
1−y2+xy/parenrightBig
[x<y] NV 56
6. arccos x+ arccos y= arccos/parenleftBig
xy−/radicalbig
1−x2/radicalbig
1−y2/parenrightBig
[x+y≥0]
=2π−arccos/parenleftBig
xy−/radicalbig
1−x2/radicalbig
1−y2/parenrightBig
[x+y<0] NV 57 (3)
7.11arccos x−arccos y=−arccos/parenleftBig
xy+/radicalbig
1−x2/radicalbig
1−y2/parenrightBig
[x≥y]
= arccos/parenleftBig
xy+/radicalbig
1−x2/radicalbig
1−y2/parenrightBig
[x<y] NV 57 (4)
1.627 Functional relations 59
8. arctan x+a r c t a n y=a r c t a nx+y
1−xy[xy <1]
=π+a r c t a nx+y
1−xy[x>0,x y > 1]
=−π+a r c t a nx+y
1−xy[x<0,x y > 1]
NV 59(5), GI I (879)
9. arctan x−arctan y=a r c t a nx−y
1+xy[xy >−1]
=π+a r c t a nx−y
1+xy[x>0,x y < −1]
=−π+a r c t a nx−y
1+xy[x<0,x y < −1]
NV 59(6)
1.626
1. 2 arcsin x=a r c s i n/parenleftBig
2x/radicalbig
1−x2/parenrightBig /bracketleftbigg
|x|≤1√
2/bracketrightbigg
=π−arcsin/parenleftBig
2x/radicalbig
1−x2/parenrightBig /bracketleftbigg1√
2<x≤1/bracketrightbigg
=−π−arcsin/parenleftBig
2x/radicalbig
1−x2/parenrightBig /bracketleftbigg
−1≤x<−1√
2/bracketrightbigg
NV 61 (7)
2. 2arccos x= arccos/parenleftbig
2x2−1/parenrightbig
[0≤x≤1]
=2π−arccos/parenleftbig
2x2−1/parenrightbig
[−1≤x<0] NV 61 (8)
3. 2arctan x=a r c t a n2x
1−x2[|x|<1]
=a r c t a n2x
1−x2+π [x>1]
=a r c t a n2x
1−x2−π [x<−1]
NV 61 (9)
1.627
1. arctan x+a r c t a n1
x=π
2[x>0]
=−π
2[x<0] GI I (878)
2. arctan x+a r c t a n1−x
1+x=π
4[x>−1]
=−3
4π[x<−1] NV 62, GI I (881)
60 The Inverse Trigonometric and Hyperbolic Functions 1.628
1.628
1. arcsin2x
1+x2=−π−2arc t an x [x≤−1]
= 2arctan x [−1≤x≤1]
=π−2arc t an x [x≥1]
NV 65
2. arccos1−x2
1+x2= 2arctan x [x≥0]
=−2arc t an x[x≤0] NV 66
1.6292x−1
2−1
πarctan/parenleftbigg
tan2x−1
2π/parenrightbigg
=E(x) GI (886)
1.631 Relations between the inverse hyperbolic functions.
1. arcsinh x= arccosh/radicalbig
x2+1=a r c t a n hx√
x2+1JA
2. arccosh x=a r c s i n h/radicalbig
x2−1 = arctanh√
x2−1
xJA
3. arctanh x=a r c s i n hx√
1−x2= arccosh1√
1−x2= arccoth1
xJA
4. arcsinh x±arcsinh y=a r c s i n h/parenleftBig
x/radicalbig
1+y2±y/radicalbig
1+x2/parenrightBig
JA
5. arccosh x±arccosh y= arccosh/parenleftBig
xy±/radicalbig
(x2−1)(y2−1)/parenrightBig
JA
6. arctanh x±arctanh y=a r c t a n hx±y
1±xyJA
1.64 Series representations
1.641
1. arcsin x=π
2−arccos x=x+1
2·3x3+1·3
2·4·5x5+1·3·5
2·4·6·7x7+...
=∞/summationdisplay
k=0(2k)!
22k(k!)2(2k+1 )x2k+1=xF/parenleftbigg1
2,1
2;3
2;x2/parenrightbigg
/bracketleftbig
x2≤1/bracketrightbig
FI II 479
2. arcsinh x=x−1
2·3x3+1·3
2·4·5x5−...;
=∞/summationdisplay
k=0(−1)k (2k)!
22k(k!)2(2k+1 )x2k+1
=xF/parenleftbig1
2,1
2;3
2;−x2/parenrightbig
/bracketleftbig
x2≤1/bracketrightbig
FI II 480
1.645 Series representations 61
1.642
1. arcsinh x=l n2 x+1
21
2x2−1·3
2·41
4x4+...
=l n2 x+∞/summationdisplay
k=1(−1)k+1(2k)!x−2k
22k(k!)22k[x≥1]
AD (6480.2)a
2. arccosh x=l n2 x−∞/summationdisplay
k=1(2k)!x−2k
22k(k!)22k[x≥1] AD (6480.3)a
1.643
1. arctan x=x−x3
3+x5
5−x7
7+...
=∞/summationdisplay
k=0(−1)kx2k+1
2k+1/bracketleftbig
x2≤1/bracketrightbig
FI II 479
2. arctanh x=x+x3
3+x5
5+···=∞/summationdisplay
k=0x2k+1
2k+1/bracketleftbig
x2<1/bracketrightbig
AD (6480.4)
1.644
1. arctan x=x√
1+x2∞/summationdisplay
k=0(2k)!
22k(k!)2(2k+1 )/parenleftbiggx2
1+x2/parenrightbiggk
=x√
1+x2F/parenleftbigg1
2,1
2;3
2;x2
1+x2/parenrightbigg/bracketleftbig
x2<∞/bracketrightbig
AD (641.3)
2. arctan x=π
2−1
x+1
3x3−1
5x5+1
7x7−···=π
2−∞/summationdisplay
k=0(−1)k 1
(2k+1 )x2k+1AD (641.4)
1.645
1. arcsec x=π
2−1
x−1
2·3x3−1·3
2·4·5x5−···=π
2−∞/summationdisplay
k=0(2k)!x−(2k+1)
(k!)222k(2k+1 )
=π
2−1
xF/parenleftbigg1
2,1
2;3
2;1
x2/parenrightbigg/bracketleftbig
x2>1/bracketrightbig
AD (641.5)
2. (arcsin x)2=∞/summationdisplay
k=022k(k!)2x2k+2
(2k+1 ) ! ( k+1 )/bracketleftbig
x2≤1/bracketrightbig
AD (642.2), GI III (152)a
3. (arcsin x)3=x3+3!
5!32/parenleftbigg
1+1
32/parenrightbigg
x5+3!
7!32·52/parenleftbigg
1+1
32+1
52/parenrightbigg
x7+...
/bracketleftbig
x2≤1/bracketrightbig
BR* 188, AD (642.2), GI III (153)a
62 The Inverse Trigonometric and Hyperbolic Functions 1.646
1.646
1. arcsinh1
x= arcosech x=∞/summationdisplay
k=0(−1)k(2k)!
22k(k!)2(2k+1 )x−2k−1
/bracketleftbig
x2≥1/bracketrightbig
AD (6480.5)
2. arccosh1
x=a r c s e c h x=l n2
x−∞/summationdisplay
k=1(2k)!
22k(k!)22kx2k[0<x≤1] AD (6480.6)
3. arcsinh1
x= arcosech x=l n2
x+∞/summationdisplay
k=1(−1)k+1(2k)!
22k(k!)22kx2k
[0<x≤1] AD (6480.7)a
4. arctanh1
x= arccoth x=∞/summationdisplay
k=0x−(2k+1)
2k+1/bracketleftbig
x2>1/bracketrightbig
AD (6480.8)
1.647
1.∞/summationdisplay
k=1tanh(2 k−1)(π/2)
(2k−1)4n+3=π4n+3
2⎛
⎝2n/summationdisplay
j=1(−1)j−1/parenleftbig
22j−1/parenrightbig/parenleftbig
24n−2j+4−1/parenrightbig
B∗
2j−1B∗
4n−2j+3
(2j)!(4n−2j+4 ) !
+(−1)n/parenleftbig
22n+2−1/parenrightbig2B∗
2n+12
[(2n+2 ) ! ]2⎞
⎠
n=0,1,2,...,
2.∞/summationdisplay
k=1(−1)k−1sech(2 k−1)(π/2)
(2k−1)4n+1=π4n+1
24n+3⎛
⎝2n−1/summationdisplay
j=1(−1)jB∗
2jB∗
4n−2j
(2j)!(4n−2j)!+2B∗
4n
(4n)!+(−1)nB∗
2n2
[(2n)]!2⎞
⎠,
n=1,2,...
(The summation term on the right is to be omitted for n= 1.) (See page xxxiii for the definition of B∗
r.)
2 Indefinite Integrals of Elementary
Functions
2.0 Introduction
2.00 General remarks
We omit the constant of integration in all the formulas of this chapter. Therefore, the equality sign (=)
means that the functions on the left and right of this symbol differ by a constant. For example (see20115), we write/integraldisplaydx
1+x2=a r c t a n x=−arctan x
although
arctan x=−arctan x+π
2.
When we integrate certain functions, we obtain the logarithm of the absolute value (for example,/integraltextdx√
1+x2=l n/vextendsingle/vextendsinglex+√
1+x2/vextendsingle/vextendsingle). In such formulas, the absolute-value bars in the argument of the logarithm
are omitted for simplicity in writing.
In certain cases, it is important to give the complete form of the primitive function. Such primitive
functions, written in the form of definite integrals, are given in Chapter 2 and in other chapters.
Closely related to these formulas are formulas in which the limits of integration and the integrand
depend on the same parameter.
A number of formulas lose their meaning for certain values of the constants (parameters) or for certain
relationships between these constants (for example, formula 2.028f o rn=−1 or formula 2.0215 for
a=b). These values of the constants and the relationships between them are for the most part completely
clear from the very structure of the right-hand member of the formula (the one not containing an integralsign). Therefore, throughout the chapter, we omit remarks to this effect. However, if the value of theintegral is given by means of some other formula for those values of the parameters for which the formula
in question loses meaning, we accompany this second formula with the appropriate explanation.
The letters x,y,t,...denote independent variables; f,g,ϕ,...denote functions of x,y,t,...;f
/prime,
g/prime,ϕ/prime,...,f/prime/prime,g/prime/prime,ϕ/prime/prime,...denote their first, second, etc., derivatives; a,b,m,p,...denote constants, by
which we generally mean arbitrary real numbers. If a particular formula is valid only for certain valuesof the constants (for example, only for positive numbers or only for integers), an appropriate remark ismade, provided the restriction that we make does not follow from the form of the formula itself. Thus,
in formulas 2.148 4a n d 2.424 6, we make no remark since it is clear from the form of these formulas
themselves that nmust be a natural number (that is, a positive integer).
63
64 Introduction
2.01 The basic integrals
1./integraldisplay
xndx=xn+1
n+1(n/negationslash=−1)
2./integraldisplaydx
x=l nx
3./integraldisplay
exdx=ex
4./integraldisplay
axdx=ax
lna
5./integraldisplay
sinxdx=−cosx
6.11/integraldisplay
cosxdx=s i nx
7./integraldisplaydx
sin2x=−cotx
8.11/integraldisplaydx
cos2x=t a n x9./integraldisplaysinx
cos2xdx=s e c x
10./integraldisplaycosx
sin2xdx=−cosecx
11./integraldisplay
tanxdx=−ln cos x
12./integraldisplay
cotxdx=l ns i n x
13./integraldisplaydx
sinx=l nt a nx
2
14./integraldisplaydx
cosx=l nt a n/parenleftBigπ
4+x
2/parenrightBig
=l n( s e c x+t a n x)
15./integraldisplaydx
1+x2=a r c t a n x=π
2−arccot x
16./integraldisplaydx
1−x2=a r c t a n h x=1
2ln1+x
1−x
17./integraldisplaydx√
1−x2=a r c s i n x=−arccos x
18./integraldisplaydx√
x2+1=a r c s i n h x=l n/parenleftBig
x+/radicalbig
x2+1/parenrightBig
19./integraldisplaydx√
x2−1= arccosh x=l n/parenleftBig
x+/radicalbig
x2−1/parenrightBig
20./integraldisplay
sinhxdx=c o s h x
21./integraldisplay
coshxdx=s i n h x
22.11/integraldisplaydx
sinh2x=−cothx
23./integraldisplaydx
cosh2x=t a n h x
24./integraldisplay
tanhxdx=l nc o s h x
25./integraldisplay
cothxdx=l ns i n h x
26./integraldisplaydx
sinhx=l nt a n hx
2
General formulas 65
2.02 General formulas
1./integraldisplay
af dx =a/integraldisplay
fd x
2./integraldisplay
[af±bϕ±cψ±...]dx=a/integraldisplay
fd x±b/integraldisplay
ϕdx±c/integraldisplay
ψd x±...
3.d
dx/integraldisplay
fd x=f
4./integraldisplay
f/primedx=f
5./integraldisplay
f/primeϕdx=fϕ−/integraldisplay
fϕ/primedx [integration by parts]
6./integraldisplay
f(n+1)ϕdx=ϕf(n)−ϕ/primef(n−1)+ϕ/prime/primef(n−2)−...+(−1)nϕ(n)f+(−1)n+1/integraldisplay
ϕ(n+1)fd x
7./integraldisplay
f(x)dx=/integraldisplay
f[ϕ(y)]ϕ/prime(y)dy [x=ϕ(y)] [change of variable]
8.11/integraldisplay
(f)nf/primedx=(f)n+1
n+1[n/negationslash=−1]
Forn=−1/integraldisplayf/primedx
f=l nf
9./integraldisplay
(af+b)nf/primedx=(af+b)n+1
a(n+1 )
10./integraldisplayf/primedx√af+b=2√af+b
a
11./integraldisplayf/primeϕ−ϕ/primef
ϕ2dx=f
ϕ
12./integraldisplayf/primeϕ−ϕ/primef
fϕdx=l nf
ϕ
13./integraldisplaydx
f(f±ϕ)=±/integraldisplaydx
fϕ∓/integraldisplaydx
ϕ(f±ϕ)
14./integraldisplayf/primedx/radicalbig
f2+a=l n/parenleftBig
f+/radicalbig
f2+a/parenrightBig
15./integraldisplayfd x
(f+a)(f+b)=a
a−b/integraldisplaydx
(f+a)−b
a−b/integraldisplaydx
(f+b)
Fora=b/integraldisplayfd x
(f+a)2=/integraldisplaydx
f+a−a/integraldisplaydx
(f+a)2
16./integraldisplayfd x
(f+ϕ)n=/integraldisplaydx
(f+ϕ)n−1−/integraldisplayϕdx
(f+ϕ)n
17./integraldisplayf/primedx
p2+q2f2=1
pqarctanqf
p
66 Rational Functions 2.101
18./integraldisplayf/primedx
q2f2−p2=1
2pqlnqf−p
qf+p
19./integraldisplayfd x
1−f=−x+/integraldisplaydx
1−f
20./integraldisplayf2dx
f2−a2=1
2/integraldisplayfd x
f−a+1
2/integraldisplayfd x
f+a
21./integraldisplayf/primedx/radicalbig
a2−f2=a r c s i nf
a
22./integraldisplayf/primedx
af2+bf=1
blnf
af+b
23./integraldisplayf/primedx
f/radicalbig
f2−a2=1
aarcsecf
a
24./integraldisplay(f/primeϕ−fϕ/prime)dx
f2+ϕ2=a r c t a nf
ϕ
25./integraldisplay(f/primeϕ−fϕ/prime)dx
f2−ϕ2=1
2lnf−ϕ
f+ϕ
2.1 Rational Functions
2.10 General integration rules
2.101 To integrate an arbitrary rational functionF(x)
f(x),w h e r e F(x)a n d f(x) are polynomials with
no common factors, we first need to separate out the integral part E(x) [where E(x) is a polynomial],
if there is an integral part, and then to integrate separately the integral part and the remainder; thus:/integraldisplayF(x)dx
f(x)=/integraldisplay
E(x)dx+/integraldisplayϕ(x)
f(x)dx.
Integration of the remainder, which is then a proper rational function (that is, one in which the degree
of the numerator is less than the degree of the denominator) is based on the decomposition of the fraction
into elementary fractions, the so-called partial fractions .
2.102 Ifa,b,c,...,mare roots of the equation f(x)=0a n di f α,β,γ,...,μare their corresponding
multiplicities, so that f(x)=(x−a)α(x−b)β...(x−m)μ,thenϕ(x)
f(x)can be decomposed into the following
partial fractions:
ϕ(x)
f(x)=Aα
(x−a)α+Aα−1
(x−a)α−1+...+A1
x−a+Bβ
(x−b)β+Bβ−1
(x−b)β−1+...+B1
x−b+...
+Mμ
(x−m)μ+Mμ−1
(x−m)μ−1+...+M1
x−m,
where the numerators of the individual fractions are determined by the following formulas:
Aα−k+1=ψ(k−1)
1(a)
(k−1)!,B β−k+1=ψ(k−1)
2(b)
(k−1)!, ..., M μ−k+1=ψ(k−1)
m(m)
(k−1)!,
ψ1(x)=ϕ(x)(x−a)α
f(x),ψ 2(x)=ϕ(x)(x−b)β
f(x), ..., ψ m(x)=ϕ(x)(x−m)μ
f(x)
2.104 General integration rules 67
TI 51a
Ifa,b,...,m are simple roots, that is, if α=β=...=μ=1, then
ϕ(x)
f(x)=A
x−a+B
x−b+···+M
x−m,
where
A=ϕ(a)
f/prime(a),B =ϕ(b)
f/prime(b), ..., M =ϕ(m)
f/prime(m).
If some of the roots of the equation f(x) = 0 are imaginary, we group together the fractions that represent
conjugate roots of the equation. Then, after certain manipulations, we represent the corresponding pairs
of fractions in the form of real fractions of the form
M1x+N1
x2+2Bx+C+M2x+N2
(x2+2Bx+C)2+...+Mpx+Np
(x2+2Bx+C)p.
2.103 Thus, the integration of a proper rational fractionϕ(x)
f(x)reduces to integrals of the form/integraldisplaygd x
(x−a)α
or/integraldisplayMx+N
(A+2Bx+Cx2)pdx. Fractions of the first form yield rational functions for α>1 and logarithms
forα= 1. Fractions of the second form yield rational functions and logarithms or arctangents:
1./integraldisplaygd x
(x−a)α=g/integraldisplayd(x−a)
(x−a)α=−g
(α−1)(x−a)α−1
2./integraldisplaygd x
x−a=g/integraldisplayd(x−a)
x−a=gln|x−a|
3./integraldisplayMx+N
(A+2Bx+Cx2)pdx=NB−MA+(NC−MB)x
2(p−1)(AC−B2)(A+2Bx+Cx2)p−1
+(2p−3)(NC−MB)
2(p−1)(AC−B2)/integraldisplaydx
(A+2Bx+Cx2)p−1
4./integraldisplaydx
A+2Bx+Cx2=1√
AC−B2arctanCx+B√
Ac−B2for/bracketleftbig
AC > B2/bracketrightbig
=1
2√
B2−ACln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleCx+B−√
B2−AC
Cx+B+√
B2−AC/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglefor/bracketleftbig
AC < B
2/bracketrightbig
5./integraldisplay(Mx+N)dx
A+2Bx+Cx2
=M
2Cln/vextendsingle/vextendsingleA+2Bx+Cx2/vextendsingle/vextendsingle+NC−MB
C√
AC−B2arctanCx+B√
AC−B2for/bracketleftbig
AC > B2/bracketrightbig
=M
2Cln/vextendsingle/vextendsingleA+2Bx+Cx2/vextendsingle/vextendsingle+NC−MB
2C√
B2−ACln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleCx+B−√
B2−AC
Cx+B+√
B2−AC/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglefor/bracketleftbig
AC < B 2/bracketrightbig
The Ostrogradskiy–Hermite method
2.104 By means of the Ostrogradskiy-Hermite method, we can find the rational part of/integraldisplayϕ(x)
f(x)dx
without finding the roots of the equation f(x) = 0 and without decomposing the integrand into partial
fractions:
68 Rational Functions 2.110
/integraldisplayϕ(x)
f(x)dx=M
D+/integraldisplayNd x
QFI II 49
Here, M,N,D,a n d Qare rational functions of x. Specifically, Dis the greatest common divisor of
the function f(x) and its derivative f/prime(x);Q=f(x)
D;Mis a polynomial of degree no higher than m−1,
where mis the degree of the polynomial D;Nis a polynomial of degree no higher than n−1, where
nis the degree of the polynomial Q. The coefficients of the polynomials MandNare determined by
equating the coefficients of like powers of xin the following identity:
ϕ(x)=M/primeQ−M(T−Q/prime)+ND
where T=f/prime(x)
DandM/primeandQ/primeare the derivatives of the polynomials MandQ.
2.11–2.13 Forms containing the binomial a+bxk
2.110 Reduction formulas for zk=a+bxkand an explicit expression for the general case.
1./integraldisplay
xnzm
kdx=xn+1zm
k
km+n+1+amk
km+n+1/integraldisplay
xnzm−1
kdx
=xn+1
m+1p/summationdisplay
s=0(ak)s(m+1 )m(m−1)...(m−s+1 )zm−s
k
[mk+n+ 1][(m−1)k+n+1 ]...[(m−s)k+n+1 ]
+(ak)p+1m(m−1)...(m−p+1 ) (m−p)
[mk+n+ 1][(m−1)k+n+1 ]...[(m−p)k+n+1 ]/integraldisplay
xnzm−p−1
k dx
LA 126(4)
2./integraldisplay
xnzm
kdx=−xn+1zm+1
k
ak(m+1 )+km+k+n+1
ak(m+1 )/integraldisplay
xnzm+1
kdx LA 126 (6)
3./integraldisplay
xnzm
kdx=xn+1zm
k
n+1−bkm
n+1/integraldisplay
xn+kzm−1
kdx
4./integraldisplay
xnzm
kdx=xn+1−kzm+1
k
bk(m+1 )−n+1−k
bk(m+1 )/integraldisplay
xn−kzm+1
kdx LA 125 (2)
5./integraldisplay
xnzm
kdx=xn+1−kzm+1
k
b(km+n+1 )−a(n+1−k)
b(km+n+1 )/integraldisplay
xn−kzm
kdx LA 126 (3)
6./integraldisplay
xnzm
kdx=xn+1zm+1
k
a(n+1 )−b(km+k+n+1 )
a(n+1 )/integraldisplay
xn+kzm
kdx LA 126 (5)
7.∗/integraldisplay
xn/parenleftbig
nxb+c/parenrightbigkdx=nk
bk/summationdisplay
i=0(−1)ik!Γ/parenleftbiga+1
b/parenrightbig/parenleftbig
nb+c
n/parenrightbigk−i
(k−i)! Γ/parenleftbiga+1
b+i+1/parenrightbigxa+1+ib
[a,b,k≥0 are all integers]
8.∗/integraldisplay
xnzm
kdx=bm
km/summationdisplay
i=0(−1)im!J!/parenleftbig
xk+a
b/parenrightbigm−ixk(J+i+1)
(m−i)!(J+i+1 ) !
J=n+1
k−1[ a,b,k,m,n real,k/negationslash=0,m≥0 an integer]
2.114 Forms containing the binomial a+bxk69
Forms containing the binomial z1=a+bx
2.111
1./integraldisplay
zm
1dx=zm+1
1
b(m+1 )
Form=−1/integraldisplaydx
z1=1
blnz1
2./integraldisplayxndx
zm
1=xn
zm−1
1(n+1−m)b−na
(n+1−m)b/integraldisplayxn−1dx
zm
1
Forn=m−1, we may use the formula
3.8/integraldisplayxm−1dx
zm
1=−xm−1
zm−1
1(m−1)b+1
b/integraldisplayxm−2dx
zm−1
1
Form=1/integraldisplayxndx
z1=xn
nb−axn−1
(n−1)b2+a2xn−2
(n−2)b3−...+(−1)n−1an−1x
1·bn+(−1)nan
bn+1lnz1
4./integraldisplayxndx
z2
1=n−1/summationdisplay
k=1(−1)k−1kak−1xn−k
(n−k)bk+1+(−1)n−1an
bn+1z1+(−1)n+1nan−1
bn+1lnz1
5./integraldisplayxdx
z1=x
b−a
b2lnz1
6./integraldisplayx2dx
z1=x2
2b−ax
b2+a2
b3lnz1
2.113
1./integraldisplaydx
z2
1=−1
bz1
2./integraldisplayxdx
z2
1=−x
bz1+1
b2lnz1=a
b2z1+1
b2lnz1
3./integraldisplayx2dx
z2
1=x
b2−a2
b3z1−2a
b3lnz1
2.114
1./integraldisplaydx
z3
1=−1
2bz2
1
2./integraldisplayxdx
z3
1=−/bracketleftBigx
b+a
2b2/bracketrightBig1
z2
1
3./integraldisplayx2dx
z3
1=/bracketleftbigg2ax
b2+3a2
2b3/bracketrightbigg1
z2
1+1
b3lnz1
4.6/integraldisplayx3dx
z3
1=/bracketleftbiggx3
b+2a
b2x2−2a2
b3x−5
2a3
b4/bracketrightbigg1
z2
1−3a
b4lnz1
70 Rational Functions 2.115
2.115
1./integraldisplaydx
z4
1=−1
3bz3
1
2./integraldisplayxdx
z4
1=−/bracketleftBigx
2b+a
6b2/bracketrightBig1
z3
1
3./integraldisplayx2dx
z4
1=−/bracketleftbiggx2
b+ax
b2+a2
3b3/bracketrightbigg1
z3
1
4./integraldisplayx3dx
z4
1=/bracketleftbigg3ax2
b2+9a2x
2b2+11a3
6b4/bracketrightbigg1
z3
1+1
b4lnz1
2.116
1./integraldisplaydx
z5
1=−1
4bz4
1
2./integraldisplayxdx
z5
1=−/bracketleftBigx
3b+a
12b2/bracketrightBig1
z4
1
3./integraldisplayx2dx
z5
1=−/bracketleftbiggx2
2b+ax
3b2+a2
12b3/bracketrightbigg1
z4
1
4./integraldisplayx3dx
z5
1=−/bracketleftbiggx3
b+3ax2
2b2+a2x
b3+a3
4b4/bracketrightbigg1
z4
1
2.117
1./integraldisplaydx
xnzm
1=−1
(n−1)axn−1zm−1
1+b(2−n−m)
a(n−1)/integraldisplaydx
xn−1zm
1
2./integraldisplaydx
zm
1=−1
(m−1)bzm−1
1
3./integraldisplaydx
xzm
1=1
zm−1
1a(m−1)+1
a/integraldisplaydx
xzm−1
1
4./integraldisplaydx
xnz1=n−1/summationdisplay
k=1(−1)kbk−1
(n−k)akxn−k+(−1)nbn−1
anlnz1
x
2.118
1./integraldisplaydx
xz1=−1
alnz1
x,
2./integraldisplaydx
x2z1=−1
ax+b
a2lnz1
x
3./integraldisplaydx
x3z1=−1
2ax2+b
a2x−b2
a3lnz1
x
2.119
1./integraldisplaydx
xz2
1=1
az1−1
a2lnz1
x
2.124 Forms containing the binomial a+bxk71
2./integraldisplaydx
x2z2
1=−/bracketleftbigg1
ax+2b
a2/bracketrightbigg1
z1+2b
a3lnz1
x
3./integraldisplaydx
x3z2
1=/bracketleftbigg
−1
2ax2+3b
2a2x+3b2
a3/bracketrightbigg1
z1−3b2
a4lnz1
x
2.121
1./integraldisplaydx
xz3
1=/bracketleftbigg3
2a+bx
a2/bracketrightbigg1
z2
1−1
a3lnz1
x
2./integraldisplaydx
x2z3
1=−/bracketleftbigg1
ax+9b
2a2+3b2x
a3/bracketrightbigg1
z2
1+3b
a4lnz1
x
3./integraldisplaydx
x3z3
1=/bracketleftbigg
−1
2ax2+2b
a2x+9b2
a3+6b3x
a4/bracketrightbigg1
z2
1−6b2
a5lnz1
x
2.122
1./integraldisplaydx
xz4
1=/bracketleftbigg11
6a+5bx
2a2+b2x2
a3/bracketrightbigg1
z3
1−1
a4lnz1
x
2./integraldisplaydx
x2z4
1=−/bracketleftbigg1
ax+22b
3a2+10b2x
a3+4b3x2
a4/bracketrightbigg1
z3
1+4b
a5lnz1
x
3./integraldisplaydx
x3z4
1=/bracketleftbigg
−1
2ax2+5b
2a2x+55b2
3a3+25b3x
a4+10b4x2
a5/bracketrightbigg1
z3
1−10b2
a6lnz1
x
2.123
1.11/integraldisplaydx
xz5
1=/bracketleftbigg25
12a+13bx
3a2+7b2x2
2a3+b3x3
a4/bracketrightbigg1
z4
1−1
a5lnz1
x
2./integraldisplaydx
x2z5
1=/bracketleftbigg
−1
ax−125b
12a2−65b2x
3a3−35b3x2
2a4−5b4x3
a5/bracketrightbigg1
z4
1+5b
a6lnz1
x
3./integraldisplaydx
x3z5
1=/bracketleftbigg
−1
2ax2+3b
a2x+125b2
4a3+65b3x
a4+105b4x2
2a5+15b5x3
a6/bracketrightbigg1
z4
1−15b2
a7lnz1
x
2.124 Forms containing the binomial z2=a+bx2.
1./integraldisplaydx
z2=1√
abarctan x/radicalbigg
b
aif [ab >0] (see also 2.141 2)
=1
2i√
ablna+xi√
ab
a−xi√
abif [ab <0] (see also 2.143 2a n d2.143 3)
2./integraldisplayxdx
zm
2=−1
2b(m−1)zm−1
2(see also 2.145 2,2.145 6, and 2.18)
72 Rational Functions 2.125
Forms containing the binomial z3=a+bx3
Notation :α=3/radicalbigga
b
2.125
1./integraldisplayxndx
zm
3=xn−2
zm−1
3(n+1−3m)b−(n−2)a
b(n+1−3m)/integraldisplayxn−3dx
zm
3
2./integraldisplayxndx
zm
3=xn+1
3a(m−1)zm−1
3−n+4−3m
3a(m−1)/integraldisplayxndx
zm−1
3LA 133 (1)
2.126
1./integraldisplaydx
z3=α
3a/braceleftBigg
1
2ln(x+α)2
x2−αx+α2+√
3arc t anx√
3
2α−x/bracerightBigg
=α
3a/braceleftbigg1
2ln(x+α)2
x2−αx+α2+√
3arc t an2x−α
α√
3/bracerightbigg
(see also 2.141 3a n d2.143 )
2./integraldisplayxdx
z3=−1
3bα/braceleftbigg1
2ln(x+α)2
x2−αx+α2−√
3arc t an2x−α
α√
3/bracerightbigg
(see also 2.145 3. and 2.145 7)
3./integraldisplayx2dx
z3=1
3bln/parenleftbig
1+x3α−3/parenrightbig
=1
3blnz3
4./integraldisplayx3dx
z3=x
b−a
b/integraldisplaydx
z3(see2.126 1)
5./integraldisplayx4dx
z3=x2
2b−a
b/integraldisplayxdx
z3(see2.126 2)
2.127
1./integraldisplaydx
z2
3=x
3az3+2
3a/integraldisplaydx
z3(see2.126 1)
2./integraldisplayxdx
z2
3=x2
3az3+1
3a/integraldisplayxdx
z3(see2.126 2)
3./integraldisplayx2dx
z2
3=−1
3bz3
4./integraldisplayx3dx
z2
3=−x
3bz3+1
3b/integraldisplaydx
z3(see2.126 1)
2.128
1./integraldisplaydx
xnzm
3=−1
(n−1)axn−1zm−1
3−b(3m+n−4)
a(n−1)/integraldisplaydx
xn−3zm
3
2./integraldisplaydx
xnzm
3=1
3a(m−1)xn−1zm−1
3+n+3m−4
3a(m−1)/integraldisplaydx
xnzm−1
3LA 133 (2)
2.133 Forms containing the binomial a+bxk73
2.129
1./integraldisplaydx
xz3=1
3alnx3
z3
2./integraldisplaydx
x2z3=−1
ax−b
a/integraldisplayxdx
z3(see2.126 2)
3./integraldisplaydx
x3z3=−1
2ax2−b
a/integraldisplaydx
z3(see2.126 1)
2.131
1./integraldisplaydx
xz2
3=1
3az3+1
3a2lnx3
z3
2./integraldisplaydx
x2z2
3=−/bracketleftbigg1
ax+4bx2
3a2/bracketrightbigg1
z3−4b
3a2/integraldisplayxdx
z3(see2.126 2)
3./integraldisplaydx
x3z2
3=−/bracketleftbigg1
2ax2+5bx
6a2/bracketrightbigg1
z3−5b
3a2/integraldisplaydx
z3(see2.126 1)
Forms containing the binomial z4=a+bx4
Notation :α=4/radicalbigga
bα/prime=4/radicalbigg
−a
b
2.132
1.8/integraldisplaydx
z4=α
4a√
2/braceleftBigg
lnx2+αx√
2+α2
x2−αx√
2+α2+ 2arctanαx√
2
α2−x2/bracerightBigg
forab >0 (see also 2.141 4)
=α/prime
4a/braceleftbigg
lnx+α/prime
x−α/prime+ 2arctanx
α/prime/bracerightbigg
forab <0 (see also 2.143 5)
2./integraldisplayxdx
z4=1
2√
abarctan x2/radicalbigg
b
aforab >0 (see also 2.145 4)
=1
4i√
ablna+x2i√
ab
a−x2i√
abforab <0 (see also 2.145 8)
3./integraldisplayx2dx
z4=1
4bα√
2/braceleftBigg
lnx2−αx√
2+α2
x2+αx√
2+α2+ 2arctanαx√
2
α2−x2/bracerightBigg
forab >0
=−1
4bα/prime/braceleftbigg
lnx+α/prime
x−α/prime−2arc t anx
α/prime/bracerightbigg
forab <0
4./integraldisplayx3dx
z4=1
4blnz4
2.133
1./integraldisplayxndx
zm
4=xn+1
4a(m−1)zm−1
4+4m−n−5
4a(m−1)/integraldisplayxndx
zm−1
4LA 134 (1)
2./integraldisplayxndx
zm
4=xn−3
zm−1
4(n+1−4m)b−(n−3)a
b(n+1−4m)/integraldisplayxn−4dx
zm
4
74 Rational Functions 2.134
2.134
1./integraldisplaydx
z2
4=x
4az4+3
4a/integraldisplaydx
z4(see2.132 1)
2./integraldisplayxdx
z2
4=x2
4az4+1
2a/integraldisplayxdx
z4(see2.132 2)
3./integraldisplayx2dx
z2
4=x3
4az4+1
4a/integraldisplayx2dx
z4(see2.132 3)
4./integraldisplayx3dx
z2
4=x4
4az4=−1
4bz4
2.135/integraldisplaydx
xnzm
4=−1
(n−1)axn−1zm−1
4−b(4m+n−5)
(n−1)a/integraldisplaydx
xn−4zm
4
Forn=1/integraldisplaydx
xzm
4=1
a/integraldisplaydx
xzm−1
4−b
a/integraldisplaydx
x−3zm
4
2.136
1./integraldisplaydx
xz4=lnx
a−lnz4
4a=1
4alnx4
z4
2./integraldisplaydx
x2z4=−1
ax−b
a/integraldisplayx2dx
z4(see2.132 3)
2.14 Forms containing the binomial 1±xn
2.141
1./integraldisplaydx
1+x=l n ( 1+ x)
2.11/integraldisplaydx
1+x2=a r c t a n x=−arctan/parenleftbigg1
x/parenrightbigg
(see also 2.124 1)
3./integraldisplaydx
1+x3=1
3ln1+x√
1−x+x2+1√
3arctanx√
3
2−x(see also 2.126 1)
4./integraldisplaydx
1+x4=1
4√
2ln1+x√
2+x2
1−x√
2+x2+1
2√
2arctanx√
2
1−x2
(see also 2.132 1)
2.142/integraldisplaydx
1+xn=−2
nn
2−1/summationdisplay
k=0Pkcos/parenleftbigg2k+1
nπ/parenrightbigg
+2
nn
2−1/summationdisplay
k=0Qksin/parenleftbigg2k+1
nπ/parenrightbigg
forna positive even number
TI (43)a
=1
nln(1 + x)−2
nn−3
2/summationdisplay
k=0Pkcos/parenleftbigg2k+1
nπ/parenrightbigg
+2
nn−3
2/summationdisplay
k=0Qksin/parenleftbigg2k+1
nπ/parenrightbigg
forna positive odd number
TI (45)
2.145 Forms containing the binomial 1±xn75
where
Pk=1
2ln/parenleftbigg
x2−2xcos/parenleftbigg2k+1
nπ/parenrightbigg
+1/parenrightbigg
Qk=a r c t a nxsin/parenleftbig2k+1
nπ/parenrightbig
1−xcos/parenleftbig2k+1
nπ/parenrightbig=a r c t a nx−cos/parenleftbig2k+1
nπ/parenrightbig
sin/parenleftbig2k+1
nπ/parenrightbig
2.143
1./integraldisplaydx
1−x=−ln(1−x)
2./integraldisplaydx
1−x2=1
2ln1+x
1−x=a r c t a n h x [−1<x< 1] (see also 2.141 1)
3./integraldisplaydx
x2−1=1
2lnx−1
x+1=−arccoth x [x>1,x < −1]
4./integraldisplaydx
1−x3=1
3ln√
1+x+x2
1−x+1√
3arctanx√
3
2+x(see also 2.126 1)
5./integraldisplaydx
1−x4=1
4ln1+x
1−x+1
2arctan x=1
2(arctanh x+a r c t a n x)
(see also 2.132 1)
2.144
1./integraldisplaydx
1−xn=1
nln1+x
1−x−2
nn
2−1/summationdisplay
k=1Pkcos2k
nπ+2
nn
2−1/summationdisplay
k=1Qksin2k
nπ
forna positive even number TI (47)
where Pk=1
2ln/parenleftbigg
x2+2xcos2k+1
nπ+1/parenrightbigg
,Q k=a r c t a nx+c o s2k+1
nπ
sin2k+1
nπ
2./integraldisplaydx
1−xn=−1
nln(1−x)+2
nn−3
2/summationdisplay
k=0Pkcos2k+1
nπ+2
nn−3
2/summationdisplay
k=0Qksin2k+1
nπ
forna positive odd number TI (49)
where Pk=1
2ln/parenleftbigg
x2−2xcos2k
nπ+1/parenrightbigg
,Q k=a r c t a nx−cos2k
nπ
sin2k
nπ
2.145
1./integraldisplayxdx
1+x=x−ln(1 + x)
2./integraldisplayxdx
1+x2=1
2ln/parenleftbig
1+x2/parenrightbig
3./integraldisplayxdx
1+x3=−1
6ln(1 +x)2
1−x+x2+1√
3arctan2x−1√
3(see also 2.126 2)
76 Rational Functions 2.146
4./integraldisplayxdx
1+x4=1
2arctan x2
5./integraldisplayxdx
1−x=−ln(1−x)−x
6./integraldisplayxdx
1−x2=−1
2ln/parenleftbig
1−x2/parenrightbig
7./integraldisplayxdx
1−x3=−1
6ln(1−x)2
1+x+x2−1√
3arctan2x+1√
3(see also 2.126 2)
8./integraldisplayxdx
1−x4=1
4ln1+x2
1−x2(see also 2.132 2)
2.146 Formandnnatural numbers.
1./integraldisplayxm−1dx
1+x2n=−1
2nn/summationdisplay
k=1cosmπ(2k−1)
2nln/braceleftbigg
1−2xcos2k−1
2nπ+x2/bracerightbigg
+1
nn/summationdisplay
k=1sinmπ(2k−1)
2narctanx−cos2k−1
2nπ
sin2k−1
2nπ
[m<2n] TI (44)a
2./integraldisplayxm−1dx
1+x2n+1=(−1)m+1ln(1 + x)
2n+1−1
2n+1n/summationdisplay
k=1cosmπ(2k−1)
2n+1ln/braceleftbigg
1−2xcos2k−1
2n+1π+x2/bracerightbigg
+2
2n+1n/summationdisplay
k=1sinmπ(2k−1)
2n+1arctanx−cos2k−1
2n+1π
sin2k−1
2n+1π
[m≤2n] TI (46)a
3.11/integraldisplayxm−1dx
1−x2n=1
2n/braceleftbig
(−1)m+1ln(1 + x)−ln(1−x)/bracerightbig
−1
2nn−1/summationdisplay
k=1coskmπ
nln/parenleftbigg
1−2xcoskπ
n+x2/parenrightbigg
+1
nn−1/summationdisplay
k=1sinkmπ
narctan/parenleftBigg
x−coskπ
n
sinkπ
n/parenrightBigg
[m<2n] TI (48)
4./integraldisplayxm−1dx
1−x2n+1=−1
2n+1ln(1−x)
+(−1)m+11
2n+1n/summationdisplay
k=1cosmπ(2k−1)
2n+1ln/parenleftbigg
1+2xcos2k−1
2n+1π+x2/parenrightbigg
+(−1)m+12
2n+1n/summationdisplay
k=1sinmπ(2k−1)
2n+1arctanx+c o s2k−1
2n+1π
sin2k−1
2n+1π
[m≤2n] TI (50)
2.147
1./integraldisplayxmdx
1−x2n=1
2/integraldisplayxmdx
1−xn+1
2/integraldisplayxmdx
1+xn
2./integraldisplayxmdx
(1 +x2)n=−1
2n−m−1·xm−1
(1 +x2)n−1+m−1
2n−m−1/integraldisplayxm−2dx
(1 +x2)n LA 139 (28)
2.149 Forms containing the binomial 1±xn77
3./integraldisplayxm
1+x2dx=xm−1
m−1−/integraldisplayxm−2
1+x2dx
4./integraldisplayxmdx
(1−x2)n=1
2n−m−1xm−1
(1−x2)n−1−m−1
2n−m−1/integraldisplayxm−2dx
(1−x2)n
=1
2n−2xm−1
(1−x2)n−1−m−1
2n−2/integraldisplayxm−2dx
(1−x2)n−1
LA 139 (33)
5./integraldisplayxmdx
1−x2=−xm−1
m−1+/integraldisplayxm−2dx
1−x2
2.148
1./integraldisplaydx
xm(1 +x2)n=−1
m−11
xm−1(1 +x2)n−1−2n+m−3
m−1/integraldisplaydx
xm−2(1 +x2)n LA 139 (29)
Form=1/integraldisplaydx
x(1 +x2)n=1
2n−21
(1 +x2)n−1+/integraldisplaydx
x(1 +x2)n−1LA 139 (31)
Form=1a n d n=1/integraldisplaydx
x(1 +x2)=l nx√
1+x2
2./integraldisplaydx
xm(1 +x2)=−1
(m−1)xm−1−/integraldisplaydx
xm−2(1 +x2)
3./integraldisplaydx
(1 +x2)n=1
2n−2x
(1 +x2)n−1+2n−3
2n−2/integraldisplaydx
(1 +x2)n−1FI II 40
4./integraldisplaydx
(1 +x2)n=x
2n−1n−1/summationdisplay
k=1(2n−1)(2n−3)(2n−5)···(2n−2k+1 )
2k(n−1)(n−2)...(n−k)(1+ x2)n−k+(2n−3)!!
2n−1(n−1)!arctan x
TI (91)
2.149
1./integraldisplaydx
xm(1−x2)n=−1
(m−1)xm−1(1−x2)n−1+2n+m−3
m−1/integraldisplaydx
xm−2(1−x2)n LA 139 (34)
Form=1/integraldisplaydx
x(1−x2)n=1
2(n−1)(1−x2)n−1+/integraldisplaydx
x(1−x2)n−1LA 139 (36)
Form=1a n d n=1/integraldisplaydx
x(1−x2)=l nx√
1−x2
2./integraldisplaydx
(1−x2)n=1
2n−2x
(1−x2)n−1+2n−3
2n−2/integraldisplaydx
(1−x2)n−1LA 139 (35)
3./integraldisplaydx
(1−x2)n=x
2n−1n−1/summationdisplay
k=1(2n−1)(2n−3)(2n−5)...(2n−2k+1 )
2k(n−1)(n−2)...(n−k)(1−x2)n−k+(2n−3)!!
2n·(n−1)!ln1+x
1−x
TI (91)
78 Rational Functions 2.151
2.15 Forms containing pairs of binomials: a+bxandα+βx
Notation :z=a+bx;t=α+βx;Δ = aβ−αb
2.151/integraldisplay
zntmdx=zn+1tm
(m+n+1 )b−mΔ
(m+n+1 )b/integraldisplay
zntm−1dx
2.152
1./integraldisplayz
tdx=bx
β+Δ
β2lnt
2./integraldisplayt
zdx=βx
b−Δ
b2lnz
2.153/integraldisplaytmdx
zn=1
(m−n+1 )btm
zn−1−mΔ
(m−n+1 )b/integraldisplaytm−1dx
zn
=1
(n−1)Δtm+1
zn−1−(m−n+2 )β
(n−1)Δ/integraldisplaytmdx
zn−1
=−1
(n−1)btm
zn−1+mβ
(n−1)b/integraldisplaytm−1
zn−1dx
2.154/integraldisplaydx
zt=1
Δlnt
z
2.155/integraldisplaydx
zntm=−1
(m−1)Δ1
tm−1zn−1−(m+n−2)b
(m−1)Δ/integraldisplaydx
tm−1zn
=1
(n−1)Δ1
tm−1zn−1+(m+n−2)β
(n−1)Δ/integraldisplaydx
tmzn−1
2.156/integraldisplayxdx
zt=1
Δ/parenleftbigga
blnz−α
βlnt/parenrightbigg
2.16 Forms containing the trinomial a+bxk+cx2k
2.160 Reduction formulas for Rk=a+bxk+cx2k.
1./integraldisplay
xm−1Rn
kdx=xmRn+1
k
ma−(m+k+nk)b
ma/integraldisplay
xm+k−1Rn
kdx−(m+2k+2kn)c
ma/integraldisplay
xm+2k−1Rn
kdx
2./integraldisplay
xm−1Rn
kdx=xmRn
k
m−bkn
m/integraldisplay
xm+k−1Rn−1
kdx−2ckn
m/integraldisplay
xm+2k−1Rn−1
kdx
3./integraldisplay
xm−1Rn
kdx=xm−2kRn+1
k
(m+2kn)c−(m−2k)a
(m+2kn)c/integraldisplay
xm−2k−1Rn
kdx−(m−k+kn)b
(m+2kn)c/integraldisplay
xm−k−1Rn
kdx
=xmRn
k
m+2kn+2kna
m+2kn/integraldisplay
xm−1Rn−1
kdx+bkn
m+2kn/integraldisplay
xm+k−1Rn−1
kdx
2.161 Forms containing the trinomial R2=a+bx2+cx4.
Notation :f=b
2−1
2/radicalbig
b2−4ac, g =b
2+1
2/radicalbig
b2−4ac,
h=/radicalbig
b2−4ac, q =4/radicalbigga
c,l=2a(n−1)/parenleftbig
b2−4ac/parenrightbig
,cosα=−b
2√ac
2.172 Forms containing a+bx+cx2and powers of x 79
1./integraldisplaydx
R2
=c
h/braceleftbigg/integraldisplaydx
cx2+f−/integraldisplaydx
cx2+g/bracerightbigg/bracketleftbig
h2>0/bracketrightbig
LA 146 (5)
=1
4cq3sinα/braceleftbigg
sinα
2lnx2+2qxcosα
2+q2
x2−2qxcosα
2+q2+ 2cosα
2arctanx2−q2
2qxsinα
2/bracerightbigg/bracketleftbig
h2<0/bracketrightbig
LA 146 (8)a
2./integraldisplayxdx
R2=1
2hlncx2+f
cx2+g/bracketleftbig
h2>0/bracketrightbig
LA 146 (6)
=1
2cq2sinαarctanx2−q2cosα
q2sinα/bracketleftbig
h2<0/bracketrightbig
LA 146 (9)a
3./integraldisplayx2dx
R2=g
h/integraldisplaydx
cx2+g−f
h/integraldisplaydx
cx2+f/bracketleftbig
h2>0/bracketrightbig
LA 146 (7)
4./integraldisplaydx
R2
2=bcx3+/parenleftbig
b2−2ac/parenrightbig
x
lR2+b2−6ac
l/integraldisplaydx
R2+bc
l/integraldisplayx2dx
R2
5./integraldisplaydx
Rn
2=bcx3+/parenleftbig
b2−2ac/parenrightbig
x
lR2
n−1+(4n−7)bc
l/integraldisplayx2dx
Rn−1
2+2(n−1)h2+2ac−b2
l/integraldisplaydx
Rn−1
2
[n>1] LA 146
6.9/integraldisplaydx
xmRn
2=−1
(m−1)axm−1Rn−1
2−(m+2n−3)b
(m−1)a/integraldisplaydx
xm−2Rn
2−(m+4n−5)bc
(m−1)a/integraldisplaydx
xm−4Rn
2
LA 147 (12)a
2.17 Forms containing the quadratic trinomial a+bx+cx2and powers of x
Notation :R=a+bx+cx2;Δ = 4 ac−b2
2.171
1./integraldisplay
xm+1Rndx=xmRn+1
c(m+2n+2 )−am
c(m+2n+2 )/integraldisplay
xm−1Rndx−b(m+n+1 )
c(m+2n+2 )/integraldisplay
xmRndx
TI (97)
2./integraldisplayRndx
xm+1=−Rn+1
amxm+b(n−m+1 )
am/integraldisplayRndx
xm+c(2n−m+2 )
am/integraldisplayRndx
xm−1LA 142(3), TI (96)a
3./integraldisplaydx
Rn+1=b+2cx
nΔRn+(4n−2)c
nΔ/integraldisplaydx
RnTI (94)a
4./integraldisplaydx
Rn+1=(2cx+b)
2n+1n−1/summationdisplay
k=02k(2n+ 1)(2 n−1)(2n−3)...(2n−2k+1 )ck
n(n−1)···(n−k)Δk+1Rn−k+2n(2n−1)!!cn
n!Δn/integraldisplaydx
R
TI (96)a
2.17211/integraldisplaydx
R=1√
−Δln√
−Δ−(b+2cx)
(b+2cx)+√
−Δ=−2√
−Δarctanhb+2cx√
−Δfor [Δ <0]
=−2
b+2cxfor [Δ = 0 ,bandcnon-zero]
=2√
Δarctanb+2cx√
Δfor [Δ >0]
80 Rational Functions 2.173
2.173
1./integraldisplaydx
R2=b+2cx
ΔR+2c
Δ/integraldisplaydx
R(see2.172 )
2./integraldisplaydx
R3=b+2cx
Δ/braceleftbigg1
2R2+3c
ΔR/bracerightbigg
+6c2
Δ2/integraldisplaydx
R(see2.172 )
2.174
1./integraldisplayxmdx
Rn=−xm−1
(2n−m−1)cRn−1−(n−m)b
(2n−m−1)c/integraldisplayxm−1dx
Rn+(m−1)a
(2n−m−1)c/integraldisplayxm−2dx
Rn
Form=2n−1, this formula is inapplicable. Instead, we may use
2./integraldisplayx2n−1dx
Rn=1
c/integraldisplayx2n−3dx
Rn−1−a
c/integraldisplayx2n−3dx
Rn−b
c/integraldisplayx2n−2dx
Rn
2.175
1./integraldisplayxdx
R=1
2clnR−b
2c/integraldisplaydx
R(see2.172 )
2./integraldisplayxdx
R2=−2a+bx
ΔR−b
Δ/integraldisplaydx
R(see2.172 )
3./integraldisplayxdx
R3=−2a+bx
2ΔR2−3b(b+2cx)
2Δ2R−3bc
Δ2/integraldisplaydx
R(see2.172 )
4./integraldisplayx2dx
R=x
c−b
2c2lnR+b2−2ac
2c2/integraldisplaydx
R(see2.172 )
5./integraldisplayx2dx
R2=ab+/parenleftbig
b2−2ac/parenrightbig
x
cΔR+2a
Δ/integraldisplaydx
R(see2.172 )
6./integraldisplayx2dx
R3=ab+/parenleftbig
b2−2ac/parenrightbig
x
2cΔR2+/parenleftbig
2ac+b2/parenrightbig
(b+2cx)
2cΔ2R+2ac+b2
Δ2/integraldisplaydx
R
(see2.172 )
7./integraldisplayx3dx
R=x2
2c−bx
c2+b2−ac
2c3lnR−b/parenleftbig
b2−3ac/parenrightbig
2c3/integraldisplaydx
R
(see2.172 )
8./integraldisplayx3dx
R2=1
2c2lnR+a/parenleftbig
2ac−b2/parenrightbig
+b/parenleftbig
3ac−b2/parenrightbig
x
c2ΔR−b/parenleftbig
6ac−b2/parenrightbig
2c2Δ/integraldisplaydx
R
(see2.172 )
9./integraldisplayx3dx
R3=−/parenleftbiggx2
c+abx
cΔ+2a2
cΔ/parenrightbigg1
2R2−3ab
2cΔ/integraldisplaydx
R2(see2.173 1)
2.176/integraldisplaydx
xmRn=−1
(m−1)axm−1Rn−1−b(m+n−2)
a(m−1)/integraldisplaydx
xm−1Rn−c(m+2n−3)
a(m−1)/integraldisplaydx
xm−2Rn
2.177
1./integraldisplaydx
xR=1
2alnx2
R−b
2a/integraldisplaydx
R(see2.172 )
2.177 Quadratic trinomials and binomials 81
2./integraldisplaydx
xR2=1
2a2lnx2
R+1
2aR/braceleftbigg
1−b(b+2cx)
Δ/bracerightbigg
−b
2a2/parenleftbigg
1+2ac
Δ/parenrightbigg/integraldisplaydx
R
(see2.172 )
3./integraldisplaydx
xR3=1
4aR2+1
2a2R+1
2a3lnx2
R−b
2a/integraldisplaydx
R3−b
2a2/integraldisplaydx
R2−b
2a3/integraldisplaydx
R
(see2.172 ,2.173 )
4./integraldisplaydx
x2R=−b
2a2lnx2
R−1
ax+b2−2ac
2a2/integraldisplaydx
R(see2.172 )
5./integraldisplaydx
x2R2=−b
a3lnx2
R−a+bx
a2xR+/parenleftbig
b2−3ac/parenrightbig
(b+2cx)
a2ΔR−1
Δ/parenleftbiggb4
a3−6b2c
a2+6c2
a/parenrightbigg/integraldisplaydx
R
(see2.172 )
6./integraldisplaydx
x2R3=−1
axR2−3b
a/integraldisplaydx
xR3−5c
a/integraldisplaydx
R3(see2.173 and2.177 3)
7./integraldisplaydx
x3R=−ac−b2
2a3lnx2
R+b
a2x−1
2ax2+b/parenleftbig
3ac−b2/parenrightbig
2a3/integraldisplaydx
R
(see2.172 )
8./integraldisplaydx
x3R2=/parenleftbigg
−1
2ax2+3b
2a2x/parenrightbigg1
R+/parenleftbigg3b2
a2−2c
a/parenrightbigg/integraldisplaydx
xR2+9bc
2a2/integraldisplaydx
R2
(see2.173 1a n d2.177 2)
9./integraldisplaydx
x3R3=/parenleftbigg−1
2ax2+2b
a2x/parenrightbigg1
R2+/parenleftbigg6b2
a2−3c
a/parenrightbigg/integraldisplaydx
xR3+10bc
a2/integraldisplaydx
R3
(see2.173 2a n d2.177 3)
2.18 Forms containing the quadratic trinomial a+bx+cx2and the binomial α+βx
Notation :R=a+bx+cx2;z=α+βx;A=aβ2−αbβ+cα2;
B=bβ−2cα;Δ = 4 ac−b2
1./integraldisplay
zmRndx=βzm−1Rn+1
(m+2n+1 )c−(m+n)B
(m+2n+1 )c/integraldisplay
zm−1Rndx−(m−1)A
(m+2n+1 )c/integraldisplay
zm−2Rndx
2./integraldisplayRndx
zm=−1
(m−2n−1)βRn
zm−1−2nA
(m−2n−1)β2/integraldisplayRn−1dx
zm
−nB
(m−2n−1)β2/integraldisplayRn−1dx
zm−1; LA 184 (4)a
=−β
(m−1)ARn+1
zm−1−(m−n−2)B
(m−1)A/integraldisplayRndx
zm−1−(m−2n−3)c
(m−1)A/integraldisplayRndx
zm−2LA 148 (5)
=−1
(m−1)βRn
zm−1+nB
(m−1)β2/integraldisplayRn−1dx
zm−1+2nc
(m−1)β2/integraldisplayRn−1dx
zm−2LA 418 (6)
82 Algebraic Functions 2.201
3./integraldisplayzmdx
Rn=β
(m−2n+1 )czm−1
Rn−1−(m−n)B
(m−2n+1 )c/integraldisplayzm−1dx
Rn−(m−1)A
(m−2n+1 )c/integraldisplayzm−2dx
Rn
LA 147 (1)
=b+2cx
(n−1)Δzm
Rn−1−2(m−2n+3 )c
(n−1)Δ/integraldisplayzmdx
Rn−1−Bm
(n−1)Δ/integraldisplayzm−1dx
Rn−1
LA 148 (3)
4.3/integraldisplaydx
zmRn=−β
(m−1)A1
zm−1Rn−1−(m+n−2)B
(m−1)A/integraldisplaydx
zm−1Rn−(m+2n−3)c
(m−1)A/integraldisplaydx
zm−2Rn
LA 148 (7)
=β
2(n−1)A1
zm−1Rn−1−B
2A/integraldisplaydx
zm−1Rn+(m+2n−3)β2
2(n−1)A/integraldisplaydx
zmRn−1
LA 148 (8)
Form=1a n d n=1/integraldisplaydx
zR=β
2Alnz2
R−B
2A/integraldisplaydx
R
ForA=0/integraldisplaydx
zmRn=−β
(m+n−1)B1
zmRn−1−(m+2n−2)c
(m+n−1)B/integraldisplaydx
zm−1RnLA 148 (9)
2.2 Algebraic Functions
2.20 Introduction
2.201 The integrals/integraldisplay
R/parenleftbigg
x,/parenleftbiggαx+β
γx+δ/parenrightbiggr
,/parenleftbiggαx+β
γx+δ/parenrightbiggs
,.../parenrightbigg
dx,w h e r e r ,s ,... are rational numbers,
can be reduced to integrals of rational functions by means of the substitution
αx+β
γx+δ=tm, FI II 57
where mis the common denominator of the fractions r ,s ,... .
2.202 Integrals of the form/integraldisplay
xm(a+bxn)pdx,∗where m,n,a n d pare rational numbers, can be
expressed in terms of elementary functions only in the following cases:
(a) When pis an integer; then, this integral takes the form of a sum of the integrals shown in 2.201 ;
(b) Whenm+1
nis an integer: by means of the substitution xn=z, this integral can be transformed
to the form1
n/integraldisplay
(a+bz)pzm+1
n−1dz, which we considered in 2.201 ;
(c) Whenm+1
n+pis an integer; by means of the same substitution xn=z,t h i si n t e g r a lc a nb e
reduced to an integral of the form1
n/integraldisplay/parenleftbigga+bz
z/parenrightbiggp
zm+1
n+p−1dz,c o n s i d e r e di n 2.201 ;
For reduction formulas for integrals of binomial differentials, see 2.110 .
∗Translator: The authors term such integrals “integrals of binomial differentials.”
2.214 Forms containing the binomial a+bxkand√x 83
2.21 Forms containing the binomial a+bxkand√x
Notation :z1=a+bx.
2.211/integraldisplaydx
z1√x=2√
abarctan/radicalbigg
bx
a[ab >0]
=1
i√
ablna−bx+2i√
xab
z1[ab <0]
2.212/integraldisplayxm√x
z1dx=2√xm/summationdisplay
k=0(−1)kakxm−k
(2m−2k+1 )bk+1+(−1)m+1am+1
bm+1/integraldisplaydx
z1√x
(see2.211 )
2.213
1./integraldisplay√xd x
z1=2√x
b−a
b/integraldisplaydx
z1√x(see2.211 )
2./integraldisplayx√xd x
z1=/parenleftBigx
3b−a
b2/parenrightBig
2√x+a2
b2/integraldisplaydx
z1√x(see2.211 )
3./integraldisplayx2√xdx
z1=/parenleftbiggx2
5b−xa
3b2+a2
b3/parenrightbigg
2√x−a3
b3/integraldisplaydx
z1√x(see2.211 )
4./integraldisplaydx
z2
1√x=√x
az1+1
2a/integraldisplaydx
z1√x(see2.211 )
5./integraldisplay√xd x
z2
1=−√x
bz1+1
2b/integraldisplaydx
z1√x(see2.211 )
6./integraldisplayx√xd x
z2
1=2x√x
bz1−3a
b/integraldisplay√xd x
z2
1(see2.213 5)
7./integraldisplayx2√xdx
z2
1=/parenleftbiggx2
3b−5ax
3b2/parenrightbigg2√x
z1+5a2
b2/integraldisplay√xdx
z2
1(see2.213 5)
8./integraldisplaydx
z3
1√x=/parenleftbigg1
2az2
1+3
4a2z1/parenrightbigg√x+3
8a2/integraldisplaydx
z1√x(see2.211 )
9./integraldisplay√xd x
z3
1=/parenleftbigg
−1
2bz2
1+1
4abz1/parenrightbigg√x+1
8ab/integraldisplaydx
z1√x(see2.211 )
10./integraldisplayx√xd x
z3
1=−2x√x
bz2
1+3a
b/integraldisplay√xdx
z3
1(see2.213 9)
11./integraldisplayx2√xdx
z3
1=/parenleftbiggx2
b+5ax
b2/parenrightbigg2√x
z2
1−15a2
b2/integraldisplay√xdx
z3
1(see2.213 9)
Notation :z2=a+bx2,α=4/radicalbigga
b,α/prime=4/radicalbigg
−a
b.
2.214/integraldisplaydx
z2√x=1
bα3√
2/bracketleftBigg
lnx+α√
2x+α2
√z2+a r c t a nα√
2x
α2−x/bracketrightBigg/bracketleftBiga
b>0/bracketrightBig
=1
2bα/prime3/parenleftbigg
lnα/prime−√x
α/prime+√x−2arc t an√x
α/prime/parenrightbigg/bracketleftBiga
b<0/bracketrightBig
84 Algebraic Functions 2.215
2.215/integraldisplay√xd x
z2=1
bα√
2/bracketleftBigg
−lnx+α√
2x+α2
√z2+a r c t a nα√
2x
α2−x/bracketrightBigg/bracketleftBiga
b>0/bracketrightBig
=1
2bα/prime/bracketleftbigg
lnα/prime−√x
α/prime+√x+ 2arctan√x
α/prime/bracketrightbigg /bracketleftBiga
b<0/bracketrightBig
2.216
1./integraldisplayx√xd x
z2=2√x
b−a
b/integraldisplaydx
z2√x(see2.214 )
2./integraldisplayx2√xdx
z2=2x√x
3b−a
b/integraldisplay√xdx
z2(see2.215 )
3./integraldisplaydx
z2
2√x=√x
2az2+3
4a/integraldisplaydx
z2√x(see2.214 )
4./integraldisplay√xd x
z2
2=x√x
2az2+1
4a/integraldisplay√xd x
z2(see2.215 )
5./integraldisplayx√xd x
z2
2=−√x
2bz2+1
4b/integraldisplaydx
z2√x(see2.214 )
6./integraldisplayx2√xdx
z2
2=−x√x
2bz2+3
4b/integraldisplay√xd x
z2(see2.215 )
7./integraldisplaydx
z3
2√x=/parenleftbigg1
4az2
2+7
16a2z2/parenrightbigg√x+21
32a2/integraldisplaydx
z2√x(see2.214 )
8./integraldisplay√xd x
z3
2=/parenleftbigg1
4az2
2+5
16a2z2/parenrightbigg
x√x+5
32a2/integraldisplay√xdx
z2(see2.215 )
9./integraldisplayx√xd x
z3
2=/parenleftbig
bx2−3a/parenrightbig√x
16abz2
2+3
32ab/integraldisplaydx
z2√x(see2.214 )
10./integraldisplayx2√xdx
z3
2=−2x√x
5bz2
2+3a
5b/integraldisplay√xdx
z3
2(see2.216 8)
2.22–2.23 Forms containingn/radicalbig
(a+bx)k
Notation :z=a+bx.
2.220/integraldisplay
xnl√
zlm+fdx=/braceleftBiggn/summationdisplay
k=0(−1)k/parenleftbign
k/parenrightbig
zn−kak
ln−lk+l(m+1 )+ f/bracerightBigg
ll√
zl(m+1)+ f
bn+1
The square root
2.221/integraldisplay
xn√
z2m−1dx=/braceleftBiggn/summationdisplay
k=0(−1)k/parenleftbign
k/parenrightbig
zn−kak
2n−2k+2m+1/bracerightBigg
2√
z2m+1
bn+1
2.222
1./integraldisplaydx√z=2
b√z
2.225 Forms containing the binomial a+bxkand√x 85
2./integraldisplayxdx√z=/parenleftbigg1
3z−a/parenrightbigg2√z
b2
3./integraldisplayx2dx√z=/parenleftbigg1
5z2−2
3az+a2/parenrightbigg2√z
b3
2.223
1./integraldisplaydx√
z3=−2
b√z
2./integraldisplayxdx√
z3=(z+a)2
b2√z
3./integraldisplayx2dx√
z3=/parenleftbiggz2
3−2az−a2/parenrightbigg2
b3√z
2.224
1./integraldisplayzmdx
xn√z=−zm√z
(n−1)axn−1+2m−2n+3
2(n−1)b
a/integraldisplayzmdx
xn−1√z
2./integraldisplayzmdx
xn√z=−zm√z⎧
⎨
⎩1
(n−1)axn−1
+n−2/summationdisplay
k=1(2m−2n+ 3)(2 m−2n+5 )...(2m−2n+2k+1 )
2k(n−1)(n−2)...(n−k−1)xn−k−1bk
ak+1⎫
⎬
⎭
+(2m−2n+ 3)(2 m−2n+5 )...(2m−3)(2m−1)
2n−1(n−1)!xbn−1
an−1/integraldisplayzmdx
x√z
Forn=1
3./integraldisplayzm
x√zdx=2zm
(2m−1)√z+a/integraldisplayzm−1
x√zdx
4./integraldisplayzm
x√zdx=m/summationdisplay
k=12am−kzk
(2k−1)√z+am/integraldisplaydx
x√z
5.6/integraldisplaydx
x√z=1√aln/vextendsingle/vextendsingle/vextendsingle/vextendsingle√
z−√a√z+√a/vextendsingle/vextendsingle/vextendsingle/vextendsingle[a>0]
=2
√−aarctan√z√−a[a<0]
2.225
1./integraldisplay√zd x
x=2√z+a/integraldisplaydx
x√z(see2.224 4)
2./integraldisplay√zd x
x2=−√z
x+b
2/integraldisplaydx
x√z(see2.224 4)
3./integraldisplay√zd x
x3=−√
z3
2ax2+b√z
4ax−b2
8a/integraldisplaydx
x√z(see2.224 4)
86 Algebraic Functions 2.226
2.226
1./integraldisplay√
z3dx
x=/parenleftBigz
3+a/parenrightBig
2√z+a2/integraldisplaydx
x√z(see2.224 4)
2./integraldisplay√
z3dx
x2=−√
z5
ax+3b
2a/integraldisplay√
z3dx
x(see2.226 1)
3./integraldisplay√
z3dx
x3=−/parenleftbigg1
2ax2+b
4a2x/parenrightbigg√
z5+3b2
8a2/integraldisplay√
z3dx
x
(see2.226 1)
2.227/integraldisplaydx
xzm√z=m−1/summationdisplay
k=02
(2k+1 )am−kzk√z+1
am/integraldisplaydx
x√z(see2.224 4)
2.228
1./integraldisplaydx
x2√z=−√z
ax−b
2a/integraldisplaydx
x√z(see2.224 4)
2./integraldisplaydx
x3√z=/parenleftbigg
−1
2ax2+3b
4a2x/parenrightbigg√z+3b2
8a2/integraldisplaydx
x√z(see2.224 4)
2.229
1./integraldisplaydx
x√
z3=2
a√z+1
a/integraldisplaydx
x√z(see2.224 4)
2./integraldisplaydx
x2√
z3=/parenleftbigg
−1
ax−3b
a2/parenrightbigg1√z−3b
2a2/integraldisplaydx
x√z(see2.224 4)
3./integraldisplaydx
x3√
z3=/parenleftbigg
−1
2ax2+5b
4a2x+15b2
4a3/parenrightbigg1√z+15b2
8a3/integraldisplaydx
x√z
(see2.224 4)
Cube root
2.231
1./integraldisplay
3√
z3m+1xndx=/braceleftBiggn/summationdisplay
k=0(−1)k/parenleftbign
k/parenrightbig
zn−kak
3n−3k+3 (m+1 )+1/bracerightBigg
33√
z3(m+1)+1
bn+1
2./integraldisplayxndx
3√
z3m+2=/braceleftBiggn/summationdisplay
k=0(−1)k/parenleftbign
k/parenrightbig
zn−kak
3n−3k−3(m−1)−2/bracerightBigg
3
bn+13√
z3(m−1)+2
3./integraldisplay
3√
z3m+2xndx=/braceleftBiggn/summationdisplay
k=0(−1)k/parenleftbign
k/parenrightbig
zn−kak
3n−3k+3 (m+1 )+2/bracerightBigg
33√
z3(m+1)+2
bn+1
4./integraldisplayxndx
3√
z3m+1=/braceleftBiggn/summationdisplay
k=0(−1)k/parenleftbign
k/parenrightbig
zn−kak
3n−3k−3(m−1)−1/bracerightBigg
3
bn+13√
z3(m−1)+1
2.236 Forms containing the binomial a+bxkand√x 87
5./integraldisplayzndx
xm3√
x2=−zn+1
3
(m−1)axm−1+3n−3m+4
3(m−1)b
a/integraldisplayzndx
xm−13√
z2
Form=1/integraldisplayzndx
x3√
z2=3zn
(3n−2)3√
z2+a/integraldisplayzn−1dx
x3√
z2
6./integraldisplaydx
xzn3√
z2=33√z
(3n−1)azn+1
a/integraldisplay3√zd x
xzn
2.232/integraldisplaydx
x3√
z2=1
3√
a2/braceleftBigg
3
2ln3√z−3√a
3√x−√
3arc t an√
33√z
3√z+23√a/bracerightBigg
2.233
1./integraldisplay3√zd x
x=33√z+a/integraldisplaydx
x3√
z2(see2.232 )
2./integraldisplay3√zd x
x2=−z3√z
ax+b
a3√z+b
3/integraldisplaydx
x3√
z2(see2.232 )
3./integraldisplay3√zd x
x3=/parenleftbigg
−1
2ax2+b
3a2x/parenrightbigg
z3√z−b2
3a23√z−b2
9a/integraldisplaydx
x3√
z2
(see2.232 )
4./integraldisplaydx
x23√
z2=−3√z
ax−2b
3a/integraldisplaydx
x3√
z2(see2.232 )
5./integraldisplaydx
x33√
z2=/bracketleftbigg
−1
2ax2+5b
6a2x/bracketrightbigg
3√z+5b2
9a2/integraldisplaydx
x3√
z2(see2.232 )
2.234
1./integraldisplayzndx
xm3√
z2=−zn3√
z2
(m−1)axm−1+3n−3m+5
3(m−1)b
a/integraldisplayzndx
xm−13√z
Form=1 :
2./integraldisplayzndx
x3√z=3zn
(3n−1)3√z+a/integraldisplayzn−1dx
x3√z
3./integraldisplaydx
xzn3√z=33√
z2
(3n−2)azn+1
a/integraldisplay3√
z2dx
xzn
2.235/integraldisplaydx
x3√z=1
3√
a2/braceleftBigg
3
2ln3√z−3√a
3√x+√
3arc t an√
33√z
3√z+23√a/bracerightBigg
2.236
1./integraldisplay3√
z2dx
x=3
23√
z2+a/integraldisplaydx
x3√z(see2.235 )
2./integraldisplay3√
z2dx
x2=−3√
z5
ax+b
a3√
z2+2b
3/integraldisplaydx
x3√z(see2.235 )
88 Algebraic Functions 2.241
3./integraldisplay3√
z2dx
x3=/bracketleftbigg
−1
2ax2+b
6a2x/bracketrightbigg
z5/3−b2
6a23√
z2−b2
9a/integraldisplaydx
x3√z
(see2.235 )
4./integraldisplaydx
x23√z=−3√
z2
ax−b
3a/integraldisplaydx
x3√z(see2.235 )
5./integraldisplaydx
x33√z=/bracketleftbigg
−1
2ax2+2b
3a2x/bracketrightbigg
3√z+2b2
9a2/integraldisplaydx
x3√z(see2.235 )
2.24 Forms containing√a+bxand the binomial α+βx
Notation :z=a+bx,t=α+βx, Δ = aβ−bα.
2.241
1./integraldisplayzmtndx√z=2
(2n+2m+1 )βtn+1zm−1√z+(2m−1)Δ
(2n+2m+1 )β/integraldisplayzm−1tndx√zLA 176 (1)
2./integraldisplaytnzmdx√z=2√
z2m+1n/summationdisplay
k=0/parenleftBign
k/parenrightBigαn−kβk
bk+1k/summationdisplay
p=0(−1)p/parenleftbiggk
p/parenrightbiggzk−pap
2k−2p+2m+1
2.242
1.11/integraldisplaytd x√z=2α√z
b+β/parenleftBigz
3−a/parenrightBig2√z
b2
2./integraldisplayt2dx√z=2α2√z
b+2αβ/parenleftBigz
3−a/parenrightBig2√z
b2+β2/parenleftbiggz2
5−2
3za+a2/parenrightbigg2√z
b3
3./integraldisplayt3dx√z=2α3√z
b+3α2β/parenleftBigz
3−α/parenrightBig2√z
b2+3αβ2/parenleftbiggz2
5−2
3za+a2/parenrightbigg2√z
b3
+βα/parenleftbiggz3
7−3z2a
5+za2−a3/parenrightbigg2√z
b4
4./integraldisplaytz dx√z=2α√
z3
3b+β/parenleftBigz
5−a
3/parenrightBig2√
z3
b2
5./integraldisplayt2zd x√z=2α2√
z3
3b+2αβ/parenleftBigz
5−a
3/parenrightBig2√
z3
b2+β2/parenleftbiggz2
7−2za
5+a2
3/parenrightbigg2√
z3
b3
6./integraldisplayt3zd x√z=2α3√
z3
3b+3α2β/parenleftBigz
5−a
3/parenrightBig2√
z3
b2+3αβ2/parenleftbiggz2
7−2za
5+a2
3/parenrightbigg2√
z3
b3
+β3/parenleftbiggz3
9−3z2a
7+3za2
5−a3
3/parenrightbigg2√
z3
b4
7./integraldisplaytz2dx√z=2α√
z5
5b+β/parenleftBigz
7−a
5/parenrightBig2√
z5
b2
8./integraldisplayt2z2dx√z=2α2√
z5
5b+2αβ/parenleftBigz
7−a
5/parenrightBig2√
z5
b2+β2/parenleftbiggz2
9−2za
7+a2
5/parenrightbigg2√
z5
b3
2.244 Forms with√
a+bxandα+βx 89
9./integraldisplayt3z2dx√z=2α3√
z5
5b+3α2β/parenleftBigz
7−a
5/parenrightBig2√
z5
b2+3αβ2/parenleftbiggz2
9−2za
7+a2
5/parenrightbigg2√
z5
b3
+β3/parenleftbiggz3
11−3z2a
9+3za2
7−a3
5/parenrightbigg2√
z5
b4
10./integraldisplaytz3dx√z=2α√
z7
7b+β/parenleftBigz
9−a
7/parenrightBig2√
z7
b2
11./integraldisplayt2z3dx√z=2α2√
z7
7b+2αβ/parenleftBigz
9−a
7/parenrightBig2√
z7
b2+β2/parenleftbiggz2
11−2za
9+a2
7/parenrightbigg2√
z7
b3
12./integraldisplayt3z3dx√z=2α3√
z7
7b+3α2β/parenleftBigz
9−a
7/parenrightBig2√
z7
b2+3αβ2/parenleftbiggz2
11−2za
9+a2
7/parenrightbigg2√
z7
b3
+β3/parenleftbiggz3
13−3z2a
11+3za2
9−a3
7/parenrightbigg2√
z7
b4
2.243
1./integraldisplaytndx
zm√z=2
(2m−1)Δtn+1
zm√z−(2n−2m+3 )β
(2m−1)Δ/integraldisplaytndx
zm−1√z
=−2
(2m−1)btn
zm√z+2nβ
(2m−1)b/integraldisplaytn−1dx
zm−1√z
LA 176 (2)
2./integraldisplaytndx
zm√z=2√
z2m−1n/summationdisplay
k=0/parenleftBign
k/parenrightBigan−kβk
bk+1k/summationdisplay
p=0(−1)p/parenleftbiggk
p/parenrightbiggzk−pap
2k−2p−2m+1
2.244
1./integraldisplaytd x
z√z=−2a
b√z+2β(z+a)
b2√z
2./integraldisplayt2dx
z√z=−2α2
b√z+4αβ(z+a)
b2√z+2β2/parenleftBig
z2
3−2za−a2/parenrightBig
b3√z
3./integraldisplayt3dx
z√z=−2α3
b√z+6α2β(z+a)
b2√z+6αβ2/parenleftBig
z2
3−2za−a2/parenrightBig
b3√z+2β3/parenleftBig
z3
5−z2a+3za2+a3/parenrightBig
b4√z
4./integraldisplaytd x
z2√z=−2a
3b√
z3−2β/parenleftbig
z−a
3/parenrightbig
b2√
z3
5./integraldisplayt2dx
z2√z=−2a2
3b√
z3−4αβ/parenleftbig
z−a
3/parenrightbig
b2√
z3+2β2/parenleftBig
z2+2az−a2
3/parenrightBig
b3√
z3
6./integraldisplayt3dx
z2√z=−2α3
3b√
z3−6α2β/parenleftbig
z−a
3/parenrightbig
b2√
z3+6αβ2/parenleftBig
z2+2za−a2
3/parenrightBig
b3√
z3+2β3/parenleftBig
z3
3−3z2a−3za2+a3
3/parenrightBig
b4√
z3
7./integraldisplaytd x
z3√z=−2α
5b√
z5−2β/parenleftbigz
3−a
5/parenrightbig
b2√
z5
90 Algebraic Functions 2.245
8./integraldisplayt2dx
z3√z=−2α2
5b√
z5−4αβ/parenleftbigz
3−a
5/parenrightbig
b2√
z5−2β2/parenleftBig
z2−2za
3+a2
5/parenrightBig
b3√
z5
9./integraldisplayt3dx
z3√z=−2α3
5b√
z5−6α2β/parenleftbigz
3−a
5/parenrightbig
b2√
z5−6αβ2/parenleftBig
z2−2za
3+a2
5/parenrightBig
b3√
z5
+2β3/parenleftBig
z3+3z2a−za2+a3
5/parenrightBig
b4√
z5
2.245
1./integraldisplayzmdx
tn√z=−2
(2n−2m−1)βzm−1
tn−1√z−(2m−1)Δ
(2n−2m−1)β/integraldisplayzm−1dx
tn√zLA 176 (3)
=−1
(n−1)βzm−1
tn−1√z+(2m−1)b
2(n−1)β/integraldisplayzm−1
tn−1√zdx
=−1
(n−1)Δzm
tn−1√z−(2n−2m−3)b
2(n−1)Δ/integraldisplayzmdx
tn−1√z
2./integraldisplayzmdz
tn√z=−zm√z⎡
⎣1
(n−1)Δ1
tn−1
+n−1/summationdisplay
k=2(2n−2m−3)(2n−2m−5)...(2n−2m−2k+1 )bk−1
2k−1(n−1)(n−2)...(n−k)Δk1
tn−k/bracerightBigg⎤
⎦
−(2n−2m−3)(2n−2m−5)...(−2m+3 ) (−2m+1 )bn−1
2n−1·(n−1)!Δn/integraldisplayzmdx
t√z
Forn=1
3./integraldisplayzmdx
t√z=2
(2m−1)βzm
√z+Δ
β/integraldisplayzm−1dx
t√z
4./integraldisplayzmdx
t√z=2m−1/summationdisplay
k=0Δk
(2m−2k−1)βk+1zm−k
√z+Δm
βm/integraldisplaydx
t√z
2.246/integraldisplaydx
t√z1√βΔlnβ√z−√βΔ
β√z+√βΔ[βΔ>0]
=2√−βΔarctanβ√z√−βΔ[βΔ<0]
=−2√z
bt[Δ = 0]
2.247/integraldisplaydx
tzm√z=2
zm−1√z+m/summationdisplay
k=1βk−1zk
Δk(2m−2k+1 )+βm
Δm/integraldisplaydx
t√z
(see2.246 )
2.248
1./integraldisplaydx
tz√z=2
Δ√z+β
Δ/integraldisplaydx
t√z(see2.246 )
2.248 Forms with√
a+bxandα+βx 91
2./integraldisplaydx
tz2√z=2
3Δz√z+2β
Δ2√z+β2
Δ2/integraldisplaydx
t√z(see2.246 )
3./integraldisplaydx
tz3√z=2
5Δz2√z+2β
3Δ2z√z+2β2
Δ3√z+β3
Δ3/integraldisplaydx
t√z
(see2.246 )
4./integraldisplaydx
t2√z=−√z
Δt−b
2Δ/integraldisplaydx
t√z(see2.246 )
5./integraldisplaydx
t2z√z=−1
Δt√z−3b
Δ2√z−3bβ
2Δ2/integraldisplaydx
t√z(see2.246 )
6./integraldisplaydx
t2z2√z=−1
Δtz2√z−5b
3Δ2z√z−5bβ
Δ3√z−5bβ2
2Δ3/integraldisplaydx
t√z
(see2.246 )
7./integraldisplaydx
t2z3√z=−1
Δtz2√z−7b
5Δ2z2√z−7bβ
3Δ3z√z−7bβ2
Δ4√z−7bβ3
2Δ4/integraldisplaydx
t√z
(see2.246 )
8./integraldisplaydx
t3√z=−√z
2Δt2+3b√z
4Δ2t+3b2
8Δ2/integraldisplaydx
t√z(see2.246 )
9./integraldisplaydx
t3z√z=−1
2Δt2√z+5b
4Δ2t√z+15b2
4Δ3√z+15b2β
8Δ3/integraldisplaydx
t√z
(see2.246 )
10./integraldisplaydx
t3z2√z=−1
2Δt2z√z+7b√z
4Δ2tz√z+35b2
12Δ2z√z+35b2β
4Δ4√z+35b2β2
8Δ4/integraldisplaydx
t√z
(see2.246 )
11./integraldisplaydx
t3z3√z=−1
2Δt2z2√z+9b
4Δ2tz2√z+63b2
20Δ3z2√z+21b2β
4Δ4z√z+63b2β2
4Δ5√z+63b2β3
8Δ5/integraldisplaydx
t√z
(see2.246 )
12./integraldisplayzd x
t√z=2√z
β+Δ
β/integraldisplaydx
t√z(see2.246 )
13./integraldisplayz2dx
t√z=2z√z
3β+2Δ√z
β2+Δ2
β2/integraldisplaydx
t√z(see2.246 )
14./integraldisplayz3dx
t√z=2z2√z
5β+2Δz√z
3β2+2Δ2√z
β3+Δ3
β3/integraldisplaydx
t√z(see2.246 )
15./integraldisplayzd x
t2√z=−z√z
Δt+b√z
βΔ+b
2β/integraldisplaydx
t√z(see2.246 )
16./integraldisplayz2dx
t2√z=−z2√z
Δt+bz√z
βΔ+3b√z
β2+3bΔ
2β2/integraldisplaydx
t√z(see2.246 )
92 Algebraic Functions 2.249
17./integraldisplayz3dx
t2√z=−z3√z
Δt+bz2√z
βΔ+5bz√z
3β2+5bΔ√z
β3+5Δ2b
2β3/integraldisplaydx
t√z
(see2.246 )
18.3/integraldisplayzd x
t3√z=−z√z
2Δt2+bz√z
4Δ2t−b2√z
4βΔ2+b2
8βΔ/integraldisplaydx
t√z(see2.246 )
19./integraldisplayz2dx
t3√z=−z2√z
2Δt2+bz2√z
4Δ2t+b2z√z
4βΔ2+3b2√z
4β2Δ+3b2
8β2/integraldisplaydx
t√z
(see2.246 )
20./integraldisplayz3dx
t3√z=−z3√z
2Δt2+3bz3√z
Δ2t+3b2z2√z
4βΔ2+5b2z√z
4β2Δ+15b2√z
4β3+15b2Δ
8β3/integraldisplaydx
t√z
(see2.246 )
2.249
1./integraldisplaydx
zmtn√z=2
(2m−1)Δ√z
tn−1zm+(2n+2m−3)β
(2m−1)Δ/integraldisplaydx
tnzm−1√zLA 177 (4)
=−1
(n−1)Δ√z
zmtn−1−(2n+2m−3)b
2(n−1)Δ/integraldisplaydx
tn−1zm√z
2./integraldisplaydx
zmtn√z=√z
zm⎡
⎣−1
(n−1)Δ1
tn−1
+n−1/summationdisplay
k=2(−1)k(2n+2m−3)(2n+2m−5)...(2n+2m−2k+1 )bk−1
2k−1(n−1)(n−2)...(n−k)Δk·1
tn−k⎤
⎦
+(−1)n−1(2n+2m−3)(2n+2m−5)...(−2m+3 ) (−2m+1 )bn−1
2n−1(n−1)!Δn−1/integraldisplaydx
tzm√z
Forn=1/integraldisplaydx
zmt√z=2
(2m−1)Δ1
zm−1√z+β
Δ/integraldisplaydx
tzm−1√z
2.25 Forms containing√a+bx+cx2
Integration techniques
2.251 It is possible to rationalize the integrand in integrals of the form/integraldisplay
R/parenleftBig
x,/radicalbig
a+bx+cx2/parenrightBig
dxby
using one or more of the following three substitutions, known as the “Euler substitutions”:
1.√
a+bx+cx2=xt±√afora>0;
2.√
a+bx+cx2=t±x√cforc>0;
3./radicalbig
c(x−x1)(x−x2)=t(x−x1)w h e n x1andx2are real roots of the equation a+bx+cx2=0.
2.252 Forms containing√
a+bx+cx2 93
2.252 Besides the Euler substitutions, there is also the following method of calculating integrals of
the form/integraldisplay
R/parenleftBig
x,/radicalbig
a+bx+cx2/parenrightBig
dx. By removing the irrational expressions in the denominator and
performing simple algebraic operations, we can reduce the integrand to the sum of some rational function
ofxand an expression of the formP1(x)
P2(x)√
a+bx+cx2,w h e r e P1(x)a n d P2(x) are both polynomials.
By separating the integral portion of the rational functionP1(x)
P2(x)from the remainder and decomposing
the latter into partial fractions, we can reduce the integral of these partial fractions to the sum of integrals,
each of which is in one of the following three forms:
1./integraldisplayP(x)dx√
a+bx+cx2,w h e r e P(x) is a polynomial of some degree r;
2./integraldisplaydx
(x+p)k√
a+bx+cx2;
3./integraldisplay(Mx+N)dx
(a+βx+x2)m/radicalbig
c(a1+b1x+x2),/parenleftbigg
a1=a
c,b1=b
c/parenrightbigg
.
In more detail:
1./integraldisplayP(x)dx√
a+bx+cx2=Q(x)/radicalbig
a+bx+cx2+λ/integraldisplaydx√
a+bx+cx2,w h e r e Q(x) is a polynomial of
degree ( r−1). Its coefficients, and also the number λ, can be calculated by the method of
undetermined coefficients from the identity
P(x)=Q/prime(x)/parenleftbig
a+bx+cx2/parenrightbig
+1
2Q(x)(b+2cx)+λ LI II 77
Integrals of the form/integraldisplayP(x)dx√
a+bx+cx2(where r≤3) can also be calculated by use of formulas
2.26.
2. Integrals of the form/integraldisplayP(x)dx
(x+p)k√
a+bx+cx2, where the degree nof the polynomial P(x)i s
lower than kcan, by means of the substitution t=1
x+p, be reduced to an integral of the form
/integraldisplayP(t)dt/radicalbig
a+βt+γt2. (See also 2.281 ).
3. Integrals of the form/integraldisplay(Mx+N)dx
(α+βx+x2)m/radicalbig
c(a1+b1x+x2)can be calculated by the following
procedure:
•Ifb1/negationslash=β, by using the substitution
x=a1−α
βb1+t−1
t+1/radicalBig
(a1−α)2−(αb1−a1β)(β−b1)
β−b1
94 Algebraic Functions 2.260
we can reduce this integral to an integral of the form/integraldisplayP(t)dt
(t2+p)m/radicalbig
c(t2+q),w h e r e P(t)
is a polynomial of degree no higher than 2 m−1. The integral/integraldisplayP(t)dt
(t2+p)m/radicalbig
t2+qcan be
reduced to the sum of integrals of the forms/integraldisplaytd t
(t2+p)k/radicalbig
t2+qand/integraldisplaydt
(t2+p)k/radicalbig
t2+q.
•Ifb1=β, we can reduce it to integrals of the form/integraldisplayP(t)dt
(t2+p)m/radicalbig
c(t2+q)by means of the
substitution t=x+b1
2.
The integral/integraldisplaytd t
(t2+p)k/radicalbig
c(t2+q)can be evaluated by means of the substitution t2+q=
u2.
The integral/integraldisplaydt
(t2+p)k/radicalbig
c(t2+q)can be evaluated by means of the substitutiont/radicalbig
t2+q=
υ(see also 2.283 ). FI II 78-82
2.26 Forms containing√a+bx+cx2and integral powers of x
Notation :R=a+bx+cx2,Δ = 4 ac−b2
For simplified formulas for the case b=0 ,s e e 2.27.
2.260
1./integraldisplay
xm√
R2n+1dx=xm−1√
R2n+3
(m+2n+2 )c−(2m+2n+1 )b
2(m+2n+2 )c/integraldisplay
xm−1√
R2n+1dx
−(m−1)a
(m+2n+2 )c/integraldisplay
xm−2√
R2n+1dx
TI (192)a
2./integraldisplay√
R2n+1dx=2cx+b
4(n+1 )c√
R2n+1+2n+1
8(n+1 )Δ
c/integraldisplay√
R2n−1dx TI (188)
3./integraldisplay√
R2n+1dx=(2cx+b)√
R
4(n+1 )c/braceleftBigg
Rn+n−1/summationdisplay
k=0(2n+ 1)(2 n−1)...(2n−2k+1 )
8k+1n(n−1)...(n−k)/parenleftbiggΔ
c/parenrightbiggk+1
Rn−k−1/bracerightBigg
+(2n+1 ) ! !
8n+1(n+1 ) !/parenleftbiggΔ
c/parenrightbiggn+1/integraldisplaydx√
R
TI (190)
2.26111Forn=−1/integraldisplaydx√
R=1√cln/parenleftBigg
2√
cR+2cx+b√
Δ/parenrightBigg
[c>0] TI (127)
=1√carcsinh/parenleftbigg2cx+b√
Δ/parenrightbigg
[c>0,Δ>0] DW
=1√cln(2cx+b) [c>0,Δ=0 ] DW
=−1√−carcsin/parenleftbigg2cx+b√
−Δ/parenrightbigg
[c<0,Δ<0] TI (128)
2.263 Forms containing√
a+bx+cx2andxn95
2.262
1./integraldisplay√
Rd x=(2cx+b)√
R
4c+Δ
8c/integraldisplaydx√
R(see2.261 )
2./integraldisplay
x√
Rd x=√
R3
3c−(2cx+b)b
8c2√
R−bΔ
16c2/integraldisplaydx√
R(see2.261 )
3./integraldisplay
x2√
Rd x=/parenleftbiggx
4c−5b
24c2/parenrightbigg√
R3+/parenleftbigg5b2
16c2−a
4c/parenrightbigg(2cx+b)√
R
4c+/parenleftbigg5b2
16c2−a
4c/parenrightbiggΔ
8c/integraldisplaydx√
R
(see2.261 )
4./integraldisplay
x3√
Rd x=/parenleftbiggx2
5c−7bx
40c2+7b2
48c3−2a
15c2/parenrightbigg√
R3−/parenleftbigg7b3
32c3−3ab
8c2/parenrightbigg(2cx+b)√
R
4c
−/parenleftbigg7b3
32c3−3ab
8c2/parenrightbiggΔ
8c/integraldisplaydx√
R
(see2.261 )
5./integraldisplay√
R3dx=/parenleftbiggR
8c+3Δ
64c2/parenrightbigg
(2cx+b)√
R+3Δ2
128c2/integraldisplaydx√
R
(see2.261 )
6./integraldisplay
x√
R3dx=√
R5
5c−(2cx+b)/parenleftbiggb
16c2√
R3+3Δb
128c3√
R/parenrightbigg
−3Δ2b
256c3/integraldisplaydx√
R
(see2.261 )
7./integraldisplay
x2√
R3dx=/parenleftbiggx
6c−7b
60c2/parenrightbigg√
R5+/parenleftbigg7b2
24c2−a
6c/parenrightbigg/parenleftbigg
2x+b
c/parenrightbigg/parenleftBigg√
R3
8+3Δ
64c√
R/parenrightBigg
+/parenleftbigg7b2
4c−a/parenrightbiggΔ2
256c3/integraldisplaydx√
R
(see2.261 )
8./integraldisplay
x3√
R3dx=/parenleftbiggx2
7c−3bx
28c2+3b2
40c3−2a
35c2/parenrightbigg√
R5
−/parenleftbigg3b3
16c3−ab
4c2/parenrightbigg/parenleftbigg
2x+b
c/parenrightbigg/parenleftBigg√
R3
8+3Δ
64c√
R/parenrightBigg
−/parenleftbigg3b2
4c−a/parenrightbigg3Δ2b
512c4/integraldisplaydx√
R
(see2.261 )
2.263
1./integraldisplayxmdx√
R2n+1=xm−1
(m−2n)c√
R2n−1−(2m−2n−1)b
2(m−2n)c/integraldisplayxm−1dx√
R2n+1−(m−1)a
(m−2n)c/integraldisplayxm−2dx√
R2n+1
TI (193)a
Form=2n
2./integraldisplayx2ndx√
R2n+1=−x2n−1
(2n−1)c√
R2n−1−b
2c/integraldisplayx2n−1
√
R2n+1dx+1
c/integraldisplayx2n−2
√
R2n−1dx TI (194)a
96 Algebraic Functions 2.264
3./integraldisplaydx√
R2n+1=2(2cx+b)
(2n−1)Δ√
R2n−1+8(n−1)c
(2n−1)Δ/integraldisplaydx√
R2n−1TI (189)
4./integraldisplaydx√
R2n+1=2(2cx+b)
(2n−1)Δ√
R2n−1/braceleftBigg
1+n−1/summationdisplay
k=18k(n−1)(n−2)...(n−k)
(2n−3)(2n−5)...(2n−2k−1)ck
ΔkRk/bracerightBigg
[n≥1]. TI (191)
2.264
1./integraldisplaydx√
R(see2.261 )
2./integraldisplayxdx√
R=√
R
c−b
2c/integraldisplaydx√
R(see2.261 )
3./integraldisplayx2dx√
R=/parenleftbiggx
2c−3b
4c2/parenrightbigg√
R+/parenleftbigg3b2
8c2−a
2c/parenrightbigg/integraldisplaydx√
R(see2.261 )
4./integraldisplayx3dx√
R=/parenleftbiggx2
3c−5bx
12c2+5b2
8c3−2a
3c2/parenrightbigg√
R−/parenleftbigg5b3
16c3−3ab
4c2/parenrightbigg/integraldisplaydx√
R
(see2.261 )
5./integraldisplaydx√
R3=2(2cx+b)
Δ√
R
6./integraldisplayxdx√
R3=−2(2a+bx)
Δ√
R
7./integraldisplayx2dx√
R3=−/parenleftbig
Δ−b2/parenrightbig
x−2ab
cΔ√
R+1
c/integraldisplaydx√
R(see2.261 )
8./integraldisplayx3dx√
R3=cΔx2+b/parenleftbig
10ac−3b2/parenrightbig
x+a/parenleftbig
8ac−3b2/parenrightbig
c2Δ√
R−3b
2c2/integraldisplaydx√
R
(see2.261 )
2.265/integraldisplay√
R2n+1
xmdx=−√
R2n+3
(m−1)axm−1+(2n−2m+5 )b
2(m−1)a/integraldisplay√
R2n+1
xm−1dx
+(2n−m+4 )c
(m−1)a/integraldisplay√
R2n+1
xm−2dx
TI (195)
Form=1/integraldisplay√
R2n+1
xdx=√
R2n+1
2n+1+b
2/integraldisplay√
R2n−1dx+a/integraldisplay√
R2n−1
xdx TI (198)
Fora=0
/integraldisplay/radicalBig
(bx+cx2)2n+1
xmdx=2/radicalBig
(bx+cx2)2n+3
(2n−2m+3 )bxm+2(m−2n−3)c
(2n−2m+3 )b/integraldisplay/radicalBig
(bx+cx2)2n+1
xm−1LA 169 (3)
Form=0s e e 2.260 2a n d2.260 3.
Forn=−1a n d m=1 :
2.267 Forms containing√
a+bx+cx2andxn97
2.2668/integraldisplaydx
x√
R=−1√aln2a+bx+2√
aR
x[a>0] TI (137)
=1√−aarcsin2a+bx
x√
b2−4ac[a<0,Δ<0] TI (138)
=1√−aarctan2a+bx
2√−a√
R[a<0] LA 178 (6)a
=−1√aarcsinh2a+bx
x√
Δ[a>0,Δ>0] DW
=−1√aarctanh2a+bx
2√a√
R[a>0]
=1√alnx
2a+bx[a>0,Δ=0 ]
=−2√
bx+cx2
bx[a=0,b/negationslash=0 ] LA 170 (16)
=1√aarccosh/parenleftbigg2a+bx
x√
−Δ/parenrightbigg
[a>0,Δ<0]
2.267
1./integraldisplay√
Rd x
x=√
R+a/integraldisplaydx
x√
R+b
2/integraldisplaydx√
R(see2.261 and2.266 )
2./integraldisplay√
Rd x
x2=−√
R
x+b
2/integraldisplaydx
x√
R+c/integraldisplaydx√
R(see2.261 and2.266 )
Fora=0/integraldisplay√
bx+cx2
x2dx=−2√
bx+cx2
x+c/integraldisplaydx√
bx+cx2(see2.261 )
3./integraldisplay√
Rd x
x3=−/parenleftbigg1
2x2+b
4ax/parenrightbigg√
R−/parenleftbiggb2
8a−c
2/parenrightbigg/integraldisplaydx
x√
R
(see2.266 )
Fora=0
/integraldisplay√
bx+cx2
x3dx=−2/radicalBig
(bx+cx2)3
3bx3
4./integraldisplay√
R3
xdx=√
R3
3+2bcx+b2+8ac
8c√
R+a2/integraldisplaydx
x√
R+b/parenleftbig
12ac−b2/parenrightbig
16c/integraldisplaydx√
R
(see2.261 and2.266 )
5./integraldisplay√
R3
x2dx=−√
R5
ax+cx+b
a√
R3+3
4(2cx+3b)√
R+3
2ab/integraldisplaydx
x√
R+3/parenleftbig
4ac+b2/parenrightbig
8/integraldisplaydx√
R
(see2.261 and2.266 )
Fora=0
/integraldisplay/radicalBig
(bx+cx2)3
x2=/radicalBig
(bx+cx2)3
2x+3b
4/radicalbig
bx+cx2+3b2
8/integraldisplaydx√
bx+cx2
(see2.261 )
98 Algebraic Functions 2.268
6./integraldisplay√
R3
x3dx=−/parenleftbigg1
2ax2+b
4a2x/parenrightbigg√
R5+bcx+2ac+b2
4a2√
R3+3/parenleftbig
bcx+2ac+b2/parenrightbig
4a√
R
+3
8/parenleftbig
4ac+b2/parenrightbig/integraldisplaydx
x√
R+3
2bc/integraldisplaydx√
R
(see2.261 and2.266 )
Fora=0
/integraldisplay/radicalBig
(bx+cx2)3
x3dx=/parenleftbigg
c−2b
x/parenrightbigg/radicalbig
bx+cx2+3bc
2/integraldisplaydx√
bx+cx2
(see2.261 )
2.268/integraldisplaydx
xm√
R2n+1=−1
(m−1)axm−1√
R2n−1
−(2n+2m−3)b
2(m−1)a/integraldisplaydx
xm−1√
R2n+1−(2n+m−2)c
(m−1)a/integraldisplaydx
xm−2√
R2n+1
TI (196)
Form=1/integraldisplaydx
x√
R2n+1=1
(2n−1)a√
R2n−1−b
2a/integraldisplaydx√
R2n+1+1
a/integraldisplaydx
x√
R2n−1TI (199)
Fora=0/integraldisplaydx
xm/radicalBig
(bx+cx2)2n+1=−2
(2n+2m−1)bxm/radicalBig
(bx+cx2)2n−1
−(4n+2m−2)c
(2n+2m−1)b/integraldisplaydx
xm−1/radicalBig
(bx+cx2)2n+1
(cf.2.265 )
2.269
1./integraldisplaydx
x√
R(see2.266 )
2./integraldisplaydx
x2√
R=−√
R
ax−b
2a/integraldisplaydx
x√
R(see2.266 )
Fora=0/integraldisplaydx
x2√
bx+cx2=2
3/parenleftbigg
−1
bx2+2c
b2x/parenrightbigg/radicalbig
bx+cx2
3./integraldisplaydx
x3√
R=/parenleftbigg
−1
2ax2+3b
4a2x/parenrightbigg√
R+/parenleftbigg3b2
8a2−c
2a/parenrightbigg/integraldisplaydx
x√
R
(see2.266 )
Fora=0/integraldisplaydx
x3√
bx+cx2=2
5/parenleftbigg
−1
bx3+4c
3b2x2−8c2
3b3x/parenrightbigg/radicalbig
bx+cx2
4./integraldisplaydx
x√
R3=−2/parenleftbig
bcx−2ac+b2/parenrightbig
aΔ√
R+1
a/integraldisplaydx
x√
R(see2.266 )
Fora=0
2.271 Forms containing√
a+cx2andxn99
/integraldisplaydx
x/radicalBig
(bx+cx2)3=2
3/parenleftbigg
−1
bx+4c
b2+8c2x
b3/parenrightbigg1√
bx+cx2
5.11/integraldisplaydx
x2√
R3=−A√
R−3b
2a2/integraldisplaydx
x√
R
where A=/parenleftBigg
−1
ax−b/parenleftbig
10ac−3b2/parenrightbig
a2Δ−c/parenleftbig
8ac−3b2/parenrightbig
x
a2Δ/parenrightBigg
(see2.266 )
Fora=0/integraldisplaydx
x2/radicalBig
(bx+cx2)3=2
5/parenleftbigg
−1
bx2+2c
b2x−8c2
b3−16c3x
b4/parenrightbigg1√
bx+cx2
6./integraldisplaydx
x3√
R3
=/parenleftBigg
−1
ax2+5b
2a2x−15b4−62acb2+2 4a2c2
2a3Δ−bc/parenleftbig
15b2−52ac/parenrightbig
x
2a3Δ/parenrightBigg
1
2√
R+15b2−12ac
8a3/integraldisplaydx
x√
R
(see2.266 )
Fora=0/integraldisplaydx
x3/radicalBig
(bx+cx2)3=2
7/parenleftbigg
−1
bx3+8c
5b2x2−16c2
5b3x+64c3
5b4+128c4x
5b5/parenrightbigg1√
bx+cx2
2.27 Forms containing√a+cx2and integral powers of x
Notation :u=√
a+cx2.
I1=1√cln/parenleftbig
x√c+u/parenrightbig
[c>0]
=1√−carcsin x/radicalbigg
−c
a[c<0a n d a>0]
I2=1
2√alnu−√a
u+√a[a>0a n d c>0]
=1
2√aln√a−u√a+u[a>0a n d c>0]
=1√−aarcsec x/radicalbigg
−c
a=1√−aarccos1
x/radicalbigg
−a
c[a<0a n d c>0]
2.271
1./integraldisplay
u5dx=1
6xu5+5
24axu3+5
16a2xu+5
16a3I1 DW
2./integraldisplay
u3dx=1
4xu3+3
8axu+3
8a2I1 DW
3./integraldisplay
ud x=1
2xu+1
2aI1 DW
4./integraldisplaydx
u=I1 DW
100 Algebraic Functions 2.272
5./integraldisplaydx
u3=1
ax
uDW
6./integraldisplaydx
u2n+1=1
ann−1/summationdisplay
k=0(−1)k
2k+1/parenleftbiggn−1
k/parenrightbiggckx2k+1
u2k+1
7./integraldisplayxdx
u2n+1=−1
(2n−1)cu2n−1DW
2.272
1./integraldisplay
x2u3dx=1
6xu5
c−1
24axu3
c−1
16a2xu
c−1
16a3
cI1 DW
2./integraldisplay
x2ud x=1
4xu3
c−1
8axu
c−1
8a2
cI1 DW
3./integraldisplayx2
udx=1
2xu
c−1
2a
cI1 DW
4./integraldisplayx2
u3dx=−x
cu+1
cI1 DW
5./integraldisplayx2
u5dx=1
3x3
au3DW
6./integraldisplayx2dx
u2n+1=1
an−1n−2/summationdisplay
k=0(−1)k
2k+3/parenleftbiggn−2
k/parenrightbiggckx2k+3
u2k+3
7./integraldisplayx3dx
u2n+1=−1
(2n−3)c2u2n−3+a
(2n−1)c2u2n−1DW
2.273
1./integraldisplay
x4u3dx=1
8x3u5
c−axu5
16c2+a2xu3
64c2+3a3xu
128c2+3a4
128c2I1 DW
2./integraldisplay
x4ud x=1
6x3u3
c−axu3
8c2+a2xu
16c2+a3
16c2I1 DW
3./integraldisplayx4
udx=1
4x3u
c−3
8axu
c2+3
8a2
c2I1 DW
4./integraldisplayx4
u3dx=1
2xu
c2+ax
c2u−3
2a
c2I1 DW
5./integraldisplayx4
u5dx=−x
c2u−1
3x3
cu3+1
c2I1 DW
6./integraldisplayx4
u7dx=1
5x5
au5DW
7./integraldisplayx4dx
u2n+1=1
an−2n−3/summationdisplay
k=0(−1)k
2k+5/parenleftbiggn−3
k/parenrightbiggckx2k+5
u2k+5
2.275 Forms containing√
a+cx2andxn101
8./integraldisplayx5dx
u2n+1=−1
(2n−5)c3u2n−5+2a
(2n−3)cu2n−3−a2
(2n−1)c3u2n−1DW
2.274
1./integraldisplay
x6u3dx=1
10x5u5
c−ax3u5
16c2+a2xu5
32c3−a3xu3
128c3−3a4xu
256c3−3
256a5
c3I1
2./integraldisplay
x6ud x=1
8x5u3
c−5
48ax3u3
c2+5a2xu3
64c3−5a3xu
128c3−5
128a4
c3I1
3./integraldisplayx6
udx=1
6x5u
c−5
24ax3u
c2+5
16a2xu
c3−5
16a3
c3I1 DW
4./integraldisplayx6
u3dx=1
4x5
cu−5
8ax3
c2u−15
8a2x
c3u+15
8a2
c3I1 DW
5./integraldisplayx6
u5dx=1
2x5
cu3+10
3ax3
c2u3+5
2a2x
c3u3−5
2a
c3I1 DW
6./integraldisplayx6
u7dx=−23
15x5
cu5−7
3ax3
c2u5−a2x
c3u5+1
c3I1 DW
7./integraldisplayx6
u9dx=1
7x7
au7DW
8./integraldisplayx6dx
u2n+1=1
an−3n−4/summationdisplay
k=0(−1)k
2k+7/parenleftbiggn−4
k/parenrightbiggckx2k+7
u2k+7
9./integraldisplayx7dx
u2n+1=−1
(2n−7)c4u2n−7+3a
(2n−5)c4u2n−5−3a2
(2n−3)c4u2n−3+a3
(2n−1)c4u2n−1DW
2.275
1./integraldisplayu5
xdx=u5
5+1
3au3+a2u+a3I2 DW
2./integraldisplayu3
xdx=u3
3+au+a2I2 DW
3./integraldisplayu
xdx=u+aI2 DW
4./integraldisplaydx
xu=I2 DW
5./integraldisplaydx
xu2n+1=1
anI2+n−1/summationdisplay
k=01
(2k+1 )an−ku2k+1
6./integraldisplayu5
x2dx=−u5
x+5
4cxu3+15
8acxu+15
8a2I1 DW
7./integraldisplayu3
x2dx=−u3
x+3
2cxu+3
2aI1 DW
8./integraldisplayu
x2dx=−u
x+cI1 DW
102 Algebraic Functions 2.276
9./integraldisplaydx
x2u2n+1=−1
an+1/braceleftBigg
u
x+n/summationdisplay
k=1(−1)k+1
2k−1/parenleftBign
k/parenrightBig
ck/parenleftBigx
u/parenrightBig2k−1/bracerightBigg
2.276
1./integraldisplayu5
x3dx=−u5
2x2+5
6cu3+5
2acu+5
2a2cI2 DW
2./integraldisplayu3
x3dx=−u3
2x2+3
2cu+3
2acI2 DW
3./integraldisplayu
x3dx=−u
2x2+c
2I2 DW
4./integraldisplaydx
x3u=−u
2ax2−c
2aI2 DW
5./integraldisplaydx
x3u3=−1
2ax2u−3c
2a2u−3c
2a2I2 DW
6./integraldisplaydx
x3u5=−1
2ax2u3−5
6c
a2u3−5
2c
a3u−5
2c
a3I2 DW
7./integraldisplayu5
x4dx=−au3
3x3−2acu
x+c2xu
2+5
2acI1 DW
8./integraldisplayu3
x4dx=−u3
3x3−cu
x+cI1 DW
9./integraldisplayu
x4dx=−u3
3ax3DW
10./integraldisplaydx
x4u2n+1=1
an+2/braceleftBigg
−u3
3x3+(n+1 )cu
x+n+1/summationdisplay
k=2(−1)k
2k−3/parenleftbiggn+1
k/parenrightbigg
ck/parenleftBigx
u/parenrightBig2k−3/bracerightBigg
2.277
1./integraldisplayu3
x5dx=−u3
4x4−3
8cu3
ax2+3
8c2u
a+3
8c2I2 DW
2./integraldisplayu
x5dx=−u
4x4−1
8cu
ax2−1
8c2
aI2 DW
3./integraldisplaydx
x5u=−u
4ax4+3
8cu
a2x2+3
8c2
a2I2 DW
4./integraldisplaydx
x5u3=−1
4ax4u+5
8c
a2x2u+15
8c2
a3u+15
8c2
a3I2
DW
2.278
1./integraldisplayu3
x6dx=−u5
5ax5DW
2./integraldisplayu
x6dx=−u3
5ax5+2
15cu3
a2x3DW
2.284 Forms containing√
a+bx+cx2and polynomials 103
3./integraldisplaydx
x6u=1
a3/parenleftbigg
−u5
5x5+2
3cu3
x3−c2u
x/parenrightbigg
DW
4./integraldisplaydx
x6u2n+1=1
an+3/braceleftBigg
−u5
5x5+1
3/parenleftbiggn+2
1/parenrightbiggcu3
x3−/parenleftbiggn+2
2/parenrightbiggc2u
x+n+2/summationdisplay
k=3(−1)k
2k−5/parenleftbiggn+2
k/parenrightbigg
ck/parenleftBigx
u/parenrightBig2k−5/bracerightBigg
2.28 Forms containing√a+bx+cx2and first- and second-degree polynomials
Notation :R=a+bx+cx2
See also 2.252
2.2813/integraldisplaydx
(x+p)n√
R=−/integraldisplaytn−1dt/radicalbig
c+(b−2pc)t+(a−bp+cp2)t2
/bracketleftbigg
t=1
x+p>0/bracketrightbigg
2.282
1.3/integraldisplay√
Rd x
x+p=c/integraldisplayxdx√
R+(b−cp)/integraldisplaydx√
R+/parenleftbig
a−bp+cp2/parenrightbig/integraldisplaydx
(x+p)√
R
[x+p>0]
2./integraldisplaydx
(x+p)(x+q)√
R=1
q−p/integraldisplaydx
(x+p)√
R+1
p−q/integraldisplaydx
(x+q)√
R
3./integraldisplay √
Rd x
(x+p)(x+q)=1
q−p/integraldisplay√
Rd x
x+p+1
p−q/integraldisplay√
Rd x
x+q
4./integraldisplay(x+p)√
Rd x
x+q=/integraldisplay√
Rd x+(p−q)/integraldisplay√
Rd x
x+q
5./integraldisplay(rx+s)dx
(x+p)(x+q)√
R=s−pr
q−p/integraldisplaydx
(x+p)√
R+s−qr
p−q/integraldisplaydx
(x+q)√
R
2.283/integraldisplay(Ax+B)dx
(p+R)n√
R=A
c/integraldisplaydu
(p+u2)n+2Bc−Ab
2c/integraldisplay/parenleftbig
1−cυ2/parenrightbign−1dυ/bracketleftbigg
p+a−b2
4c−cpυ2/bracketrightbiggn,
where u=√
Randυ=b+2cx
2c√
R.
2.284/integraldisplayAx+B
(p+R)√
Rdx=A
cI1+2Bc−Ab/radicalbig
c2p[b2−4(a+p)c]I2,
where
I1=1√parctan/radicalBigg
R
p[p>0]
=1
2√−pln√−p−√
R√−p+√
R[p<0]
104 Algebraic Functions 2.290
I2=a r c t a n/radicalbiggp
b2−4(a+p)cb+2cx√
R/bracketleftbig
p/braceleftbig
b2−4(a+p)c/bracerightbig
>0,p < 0/bracketrightbig
=−arctan/radicalbiggp
b2−4(a+p)cb+2cx√
R/bracketleftbig
p/braceleftbig
b2−4(a+p)c/bracerightbig
>0,p > 0/bracketrightbig
=1
2iln/radicalbig
4(a+p)c−b2√
R+√p(b+2cx)/radicalbig
4(a+p)c−b2√
R−√p(b+2cx)/bracketleftbig
p/braceleftbig
b2−4(a+p)c/bracerightbig
<0,p > 0/bracketrightbig
=1
2iln/radicalbig
b2−4(a+p)c√
R−√−p(b+2cx)/radicalbig
b2−4(a+p)c√
R+√−p(b+2cx)/bracketleftbig
p/braceleftbig
b2−4(a+p)c/bracerightbig
<0,p < 0/bracketrightbig
2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals
2.290 Integrals of the form/integraldisplay
R/parenleftBig
x,/radicalbig
P(x)/parenrightBig
dx,w h e r e P(x) is a third- or fourth-degree polynomial,
can, by means of algebraic transformations, be reduced to a sum of integrals expressed in terms of elemen-
tary functions and elliptic integrals (see 8.11). Since the substitutions that transform the given integral
into an elliptic integral in the normal Legendre form are different for different intervals of integration, the
corresponding formulas are given in the chapter on definite integrals (see 3.13,3.17).
2.291 Certain integrals of the form/integraltext
R/parenleftBig
x,/radicalbig
P(x)/parenrightBig
dx,w h e r e Pn(x) is a polynomial of not more than
fourth degree, can be reduced to integrals of the form/integraltext
R/parenleftBig
x,k/radicalbig
Pn(x)/parenrightBig
dxwithk≥2. Below are examples
of this procedure.
1./integraldisplaydx√
1−x6=−/integraldisplaydz√
3+3z2+z4/bracketleftbigg
x2=1
1+z2/bracketrightbigg
2./integraldisplaydx√
a+bx2+cx4+dx6=1
2/integraldisplaydz√
az+bz2+cz3+dz4
/bracketleftbig
x2=z/bracketrightbig
3./integraldisplay/parenleftbig
a+2bx+cx2+gx3/parenrightbig±1/3dx=3
2/integraldisplayz2A±1
3dz
B⎡
⎣a+2bx+cx2=z3,A=g/parenleftBigg
−b+/radicalbig
b2+(z3−a)c
c/parenrightBigg3
+z3,B =/radicalbig
b2+(z3−a)c⎤
⎦
4./integraldisplaydx√
a+bx+cx2+dx3+cx4+bx5+ax6
=−1√
2/integraldisplaydx/radicalbig
(z+1 )p−1√
2/integraldisplaydz/radicalbig
(z−1)p/bracketleftBig
x=z+/radicalbig
z2−1/bracketrightBig
=−1√
2/integraldisplayd/radicalbig
(z+1 )p+1√
2/integraldisplaydz/radicalbig
(z−1)p/bracketleftBig
x=z−/radicalbig
z2−1/bracketrightBig
where p=2a/parenleftbig
4z3−3z/parenrightbig
+2b/parenleftbig
2z2−1/parenrightbig
+2cz+d.
2.292 Integrals reducible to elliptic integrals 105
5./integraldisplaydx√
a+bx2+cx4+bx6+ax8=1
2/integraldisplaydy
√y/radicalbig
a+by+cy2+by3+ay4[x=√y]
=−1
2√
2/integraldisplaydz/radicalbig
(z+1 )p+1
2√
2/integraldisplaydz/radicalbig
(z−1)p/bracketleftBig
y=z+/radicalbig
z2−1/bracketrightBig
=1
2√
2/integraldisplaydz/radicalbig
(z+1 )p−1
2√
2/integraldisplaydz/radicalbig
(z−1)p/bracketleftBig
y=z−/radicalbig
z2−1/bracketrightBig
where p=2a/parenleftbig
2z2−1/parenrightbig
+2bz+c.
6./integraldisplaydx√
a+bx4+cx8=1
28/radicalbigga
c/integraldisplaydt√
t√
ab1t2+at4/bracketleftbigg
x=8/radicalbigga
c√
t/bracketrightbigg
;
=−1
2√
28/radicalbigga
c/braceleftBigg/integraldisplaydz/radicalbig
(z+1 )p−/integraldisplaydz/radicalbig
(z−1)p/bracerightBigg/bracketleftBig
t=z+/radicalbig
z2−1/bracketrightBig
=−1
2√
28/radicalbigga
c/braceleftBigg/integraldisplaydz/radicalbig
(z+1 )p+/integraldisplaydz/radicalbig
(z−1)p/bracerightBigg/bracketleftBig
t=z−/radicalbig
z2−1/bracketrightBig
where p=2a/parenleftbig
2z2−1/parenrightbig
+b1;b1=b/radicalbiga
c.
7./integraldisplayxdx
4√
a+bx2+cx4=2/integraldisplayz2dz√
A+Bz4/bracketleftbig
a+bx2+cx4=z4,A=b2−4ac, B =4c/bracketrightbig
8./integraldisplaydx
4√
a+2bx2+cx4=/integraldisplay/radicalbig
b2−a(c−z4)+b
(c−z4)/radicalbig
b2−a(c−z4)z2dz=/integraldisplay
R1/parenleftbig
z4/parenrightbig
z2dz+/integraldisplayR2/parenleftbig
z4/parenrightbig
z2dz/radicalbig
b2−a(c−z4),
where R1/parenleftbig
z4/parenrightbig
andR2/parenleftbig
z4/parenrightbig
are rational functions of z4anda+2bx2+cx4=x4z4.
2.292 In certain cases, integrals of the form/integraltext
R/parenleftBig
x,/radicalbig
P(x)/parenrightBig
dx,w h e r e P(x) is a third- or fourth-degree
polynomial, can be expressed in terms of elementary functions. Such integrals are called pseudo-elliptic
integrals.
Thus, if the relations
f1(x)=f1/parenleftbigg1
k2x/parenrightbigg
,f 2(x)=f2/parenleftbigg1−k2x
k2(1−x)/parenrightbigg
,f 3(x)=f3/parenleftbigg1−x
1−k2x/parenrightbigg
,
hold, then
1./integraldisplayf1(x)dx/radicalbig
x(1−x)(1−k2x)=/integraldisplay
R1(z)dz/bracketleftBig
z=/radicalbig
x(1−x)(1−k2x)/bracketrightBig
2./integraldisplayf2(x)dx/radicalbig
x(1−x)(1−k2x)=/integraldisplay
R2(z)dz/bracketleftBigg
z=/radicalbig
x(1−k2x)√1−x/bracketrightBigg
3./integraldisplayf3(x)dx/radicalbig
x(1−x)(1−k2x)=/integraldisplay
R3(z)dz/bracketleftBigg
z=/radicalbig
x(1−x)√
1−k2x/bracketrightBigg
where R1(z),R2(z), and R3(z) are rational functions of z.
106 The Exponential Function 2.311
2.3 The Exponential Function
2.31 Forms containing eax
2.311/integraldisplay
eaxdx=eax
a
2.312 axin the integrands should be replaced with exlna=ax
2.313
1./integraldisplaydx
a+bemx=1
am[mx−ln (a+bemx)] PE (410)
2./integraldisplaydx
1+ex=l nex
1+ex=x−ln(1 + ex) PE (409)
2.314/integraldisplaydx
aemx+be−mx=1
m√
abarctan/parenleftbigg
emx/radicalbigga
b/parenrightbigg
[ab >0] PE (411)
=1
2m√
−abln/vextendsingle/vextendsingle/vextendsingle/vextendsingleb+e
mx√
−ab
b−emx√
−ab/vextendsingle/vextendsingle/vextendsingle/vextendsingle[ab <0]
2.315/integraldisplaydx
√
a+bemx=1
m√aln√
a+bemx−√a√
a+bemx+√a[a>0]
=2
m√−aarctan√
a+bemx
√−a[a<0]
2.32 The exponential combined with rational functions of x
2.321
1./integraldisplay
xneaxdx=xneax
a−n
a/integraldisplay
xn−1eaxdx
2.11/integraldisplay
xneaxdx=eax/parenleftBiggn/summationdisplay
k=0(−1)kk!/parenleftbign
k/parenrightbig
ak+1xn−k/parenrightBigg
2.322
1./integraldisplay
xeaxdx=eax/parenleftbiggx
a−1
a2/parenrightbigg
2./integraldisplay
x2eaxdx=eax/parenleftbiggx2
a−2x
a2+2
a3/parenrightbigg
3./integraldisplay
x3eaxdx=eax/parenleftbiggx3
a−3x2
a2+6x
a3−6
a4/parenrightbigg
4.10/integraldisplay
x4eaxdx=eax/parenleftbiggx4
a−4x3
a2+12x2
a3−24x
a4+24
a5/parenrightbigg
2.323/integraldisplay
Pm(x)eaxdx=eax
am/summationdisplay
k=0(−1)kP(k)(x)
ak,
where Pm(x) is a polynomial in xof degree mandP(k)(x)i st h e kthderivative of Pm(x) with respect
tox.
2.325 Exponentials and rational functions of x 107
2.324
1./integraldisplayeaxdx
xm=1
m−1/bracketleftbigg
−eax
xm−1+a/integraldisplayeaxdx
xm−1/bracketrightbigg
2./integraldisplayeax
xndx=−eaxn−1/summationdisplay
k=1ak−1
(n−1)(n−2)...(n−k)xn−k+an−1
(n−1)!Ei(ax)
2.325
1./integraldisplayeax
xdx=E i (ax)
2./integraldisplayeax
x2dx=−eax
x+aEi(ax)
3./integraldisplayeax
x3dx=−eax
2x2−aeax
2x+a2
2Ei(ax)
4.∗/integraldisplayeax
x4dx=−eax
3x3−aeax
6x2−a2eax
6x+a3
6Ei(ax)
5.∗/integraldisplaye±axn
xmdx=1
m−1/bracketleftbigg
−e±axn
xm−1±na/integraldisplaye±axn
xm−ndx/bracketrightbigg
[m/negationslash=1 ]
6.∗/integraldisplayeaxn
xmdx=(−1)z+1azΓ(−z,−axn)
n
=(−1)z+1az
n/integraldisplay∞
−axne−t
tz+1dt
z=m−1
n,for Γ( α,x) see 8.350.2 [ n/negationslash=0 ]
7.∗/integraldisplayeaxn
xdx=Ei(axn)
n[a/negationslash=0,n/negationslash=0 ]
8.∗/integraldisplayeaxn
xmdx=−eaxn/summationtextz−1
k=0k!az−k−1
xn(k+1)
nz!+azEi(axn)
nz!
/bracketleftbigg
a/negationslash=0,z=m−1
n=1,2,..., m =2,3,.../bracketrightbigg
9.∗/integraldisplayeaxn
xmdx=−eaxn
nxn+aEi(axn)
n/bracketleftbigg
a/negationslash=0,z=m−1
n=1/bracketrightbigg
10.∗/integraldisplayeaxn
xmdx=−eaxn
2nx2n−aeaxn
2nxn+a2Ei(axn)
2n/bracketleftbigg
a/negationslash=0,z=m−1
n=2/bracketrightbigg
11.∗/integraldisplayeaxn
xmdx=−eaxn
3nx3n−eaxn
6nx2n−a2eaxn
6nxn+a3Ei(axn)
6n
/bracketleftbigg
a/negationslash=0,z=m−1
n=3/bracketrightbigg
12.∗/integraldisplayeax2
x2dx=−eax2
x+√aπerfi/parenleftbig√ax/parenrightbig
where erfi( z)=erf(iz)
i
108 The Exponential Function 2.326
13.∗/integraldisplay
e(ax2+2bx+c)dx=1
2/radicalbiggπ
aexp/parenleftbiggac−b2
a/parenrightbigg
erfi/parenleftbigg√ax+b√a/parenrightbigg
[a/negationslash=0 ]
2.326/integraldisplayxeaxdx
(1 +ax)2=eax
a2(1 +ax)[a/negationslash=0 ]
2.33
1.8/integraldisplay
e−(ax2+2bx+c)dx=1
2/radicalbiggπ
aexp/parenleftbiggb2−ac
a/parenrightbigg
erf/parenleftbigg√ax+b√a/parenrightbigg
[a/negationslash=0 ]
2.∗/integraldisplay
eax2dx=1
2/radicalbiggπ
aerfi/parenleftbig√ax/parenrightbig
where erfi( z)=erf(iz)
i[a/negationslash=0 ]
3.∗/integraldisplay
eax2+bx+cdx=1
2/radicalbiggπ
aexp/parenleftbiggac−b2
a/parenrightbigg
erfi/parenleftbigg√ax+b√a/parenrightbigg
where erfi( z)=erf(iz)
i[a/negationslash=0 ]
4.∗/integraldisplay
xme±axndx=±xm+1−n
na∓m+1−n
na/integraldisplay
xm−ne±axndx
[a/negationslash=0,n/negationslash=0 ]
5.∗/integraldisplay
xmeaxndx=eaxn
n/bracketleftBigg
(γ−1)!γ−1/summationdisplay
k=0(−1)k+1−γxnk
k!aγ−k/bracketrightBigg
/bracketleftbigg
a/negationslash=0,γ=m+1
n=1,2,.../bracketrightbigg
6.∗/integraldisplay
xmeaxndx=eaxn
na/bracketleftbigg
a/negationslash=0,γ=m+1
n=1/bracketrightbigg
7.∗/integraldisplay
xmeaxndx=eaxn
n/parenleftbiggxn
a−1
a2/parenrightbigg/bracketleftbigg
a/negationslash=0,γ=m+1
n=2/bracketrightbigg
8.∗/integraldisplay
xmeaxndx=eaxn
n/parenleftbiggx2n
a−2xn
a2+2
a3/parenrightbigg/bracketleftbigg
a/negationslash=0,γ=m+1
n=3/bracketrightbigg
9.∗/integraldisplay
xmeaxndx=eaxn
n/parenleftbiggx3n
a−3x2n
a2+6xn
a3−6
a4/parenrightbigg/bracketleftbigg
a/negationslash=0,γ=m+1
n=4/bracketrightbigg
10.∗/integraldisplay
xme−βxndx=−Γ(γ,βxn)
nβγfor Γ( α,x) see 8.350.2
=−1
nβγ/integraldisplay∞
βxntγ−1e−tdx/bracketleftbigg
γ=m+1
n,β/negationslash=0,n/negationslash=0/bracketrightbigg
11.∗/integraldisplay
xmexp (−βxn)dx=−(γ−1)!
nexp (−βxn)/bracketleftBiggγ−1/summationdisplay
k=0xnk
k!βγ−k/bracketrightBigg
/bracketleftbigg
γ=m+1
n=1,2,.../bracketrightbigg
2.33 Exponentials and rational functions of x 109
12.∗/integraldisplay
xmexp (−βxn)dx=−exp (−βxn)
nβ/bracketleftbigg
γ=m+1
n=1/bracketrightbigg
13.∗/integraldisplay
xmexp (−βxn)dx=−exp (−βxn)
n/parenleftbiggxn
β+1
β2/parenrightbigg/bracketleftbigg
γ=m+1
n=2/bracketrightbigg
14.∗/integraldisplay
xmexp (−βxn)dx=−exp (−βxn)
n/parenleftbiggx2n
β+2xn
β2+2
β3/parenrightbigg
/bracketleftbigg
γ=m+1
n=3/bracketrightbigg
15.∗/integraldisplay
xmexp (−βxn)dx=−exp (−βxn)
n/parenleftbiggx3n
β+3x2n
β2+6xn
β3+6
β4/parenrightbigg
/bracketleftbigg
γ=m+1
n=4/bracketrightbigg
16.∗/integraldisplay
e−βxndx=1
2/radicalbiggπ
βerf/parenleftBig/radicalbig
βx/parenrightBig
[β/negationslash=0 ]
17.∗/integraldisplayexp (−βxn)
xmdx=−βzΓ(−z,βxn)
n
=−βz
n/integraldisplay∞
βxne−t
tz+adt
z=m−1
n
18.∗/integraldisplayexp (−βxn)
xdx=Ei(−βxn)
n
19.∗/integraldisplayexp (−βxn)
xmdx=(−1)zexp (−βxn)
nz!z−1/summationdisplay
k=0(−1)k!βz−k−1
xn(k+1)+(−1)zβz
nz!Ei(−βxn)
/bracketleftbigg
z=m−1
n=1,2,..., m =2,3,.../bracketrightbigg
20.∗/integraldisplayexp (−βxn)
xmdx=−exp (−βxn)
nxn−βEi(−βxn)
n/bracketleftbigg
z=m−1
n=1/bracketrightbigg
21.∗/integraldisplayexp (−βxn)
xmdx=−exp (−βxn)
2nx2n+βexp (−βxn)
2nxn+β2Ei(−βxn)
2n
/bracketleftbigg
z=m−1
n=2/bracketrightbigg
22.∗/integraldisplayexp (−βxn)
xmdx=−exp (−βxn)
3nx3n+βexp (−βxn)
6nx2n−β2exp (−βxn)
6nxn−β3Ei(−βxn)
6n
/bracketleftbigg
z=m−1
n=3/bracketrightbigg
23.∗/integraldisplayexp/parenleftbig
−βx2/parenrightbig
x2dx=−exp/parenleftbig
−βx2/parenrightbig
x−/radicalbig
βπerf/parenleftBig/radicalbig
βx/parenrightBig
110 Hyperbolic Functions 2.411
2.4 Hyperbolic Functions
2.41–2.43 Powers of sinhx,coshx,tanhx, andcothx
2.411/integraldisplay
sinhpxcoshqxdx=sinhp+1xcoshq−1x
p+q+q−1
p+q/integraldisplay
sinhpxcoshq−2xdx
=sinhp−1xcoshq+1x
p+q−p−1
p+q/integraldisplay
sinhp−2xcoshqxdx
=sinhp−1xcoshq+1x
q+1−p−1
q+1/integraldisplay
sinhp−2xcoshq+2xdx
=sinhp+1xcoshq−1x
p+1−q−1
p+1/integraldisplay
sinhp+2xcoshq−2xdx
=sinhp+1xcoshq+1x
p+1−p+q+2
p+1/integraldisplay
sinhp+2xcoshqxdx
=−sinhp+1xcoshq+1x
q+1+p+q+2
q+1/integraldisplay
sinhpxcoshq+2xdx
2.412
1./integraldisplay
sinhpxcosh2nxdx=sinhp+1x
2n+p⎡
⎣cosh2n−1x
+n−1/summationdisplay
k=1(2n−1)(2n−3)...(2n−2k+1 )
(2n+p−2)(2n+p−4)...(2n+p−2k)cosh2n−2k−1x⎤
⎦
+(2n−1)!!
(2n+p)(2n+p−2)...(p+2 )/integraldisplay
sinhpxdx
This formula is applicable for arbitrary real p, except for the following negative even integers:
−2,−4,...,−2n.I fpis a natural number and n=0 ,w eh a v e
2./integraldisplay
sinh2mxdx=(−1)m/parenleftbigg2m
m/parenrightbiggx
22m+1
22m−1m−1/summationdisplay
k=0(−1)k/parenleftbigg2m
k/parenrightbiggsinh(2 m−2k)x
2m−2kTI (543)
3./integraldisplay
sinh2m+1xdx=1
22mm/summationdisplay
k=0(−1)k/parenleftbigg2m+1
k/parenrightbiggcosh(2 m−2k+1 )x
2m−2k+1; TI (544)
=(−1)nm/summationdisplay
k=0(−1)k/parenleftBigm
k/parenrightBigcosh2k+1x
2k+1GU (351) (5)
4./integraldisplay
sinhpxcosh2n+1xdx
=sinhp+1x
2n+p+1/braceleftBigg
cosh2nx+n/summationdisplay
k=12kn(n−1)...(n−k+1 )c o s h2n−2kx
(2n+p−1)(2n+p−3)...(2n+p−2k+1 )/bracerightBigg
This formula is applicable for arbitrary real p, except for the following negative odd integers: −1,
−3,...,−(2n+1 ) .
2.414 Powers of hyperbolic functions 111
2.413
1./integraldisplay
coshpxsinh2nxdx=coshp+1x
2n+p⎡
⎣sinh2n−1x
+n−1/summationdisplay
k=1(−1)k(2n−1)(2n−3)...(2n−2k+1 )s i n h2n−2k−1x
(2n+p−2)(2n+p−4)...(2n+p−2k)⎤
⎦
+(−1)n (2n−1)!!
(2n+p)(2n+p−2)...(p+2 )/integraldisplay
coshpxdx
This formula is applicable for arbitrary real p, except for the following negative even integers:
−2,−4,...,−2n.I fpis a natural number and n=0 ,w eh a v e
2./integraldisplay
cosh2mxdx=/parenleftbigg2m
m/parenrightbiggx
22m+1
22m−1m−1/summationdisplay
k=0/parenleftbigg2m
k/parenrightbiggsinh(2 m−2k)x
2m−2kTI (541)
3./integraldisplay
cosh2m+1xdx=1
22mm/summationdisplay
k=0/parenleftbigg2m+1
k/parenrightbiggsinh(2 m−2k+1 )x
2m−2k+1TI (542)
=m/summationdisplay
k=0/parenleftBigm
k/parenrightBigsinh2k+1x
2k+1GU (351) (8)
4./integraldisplay
coshpxsinh2n+1xdx=coshp+1x
2n+p+1⎡
⎣sinh2nx
+n/summationdisplay
k=1(−1)k2kn(n−1)...(n−k+1 )s i n h2n−2kx
(2n+p−1)(2n+p−3)...(2n+p−2k+1 )⎤
⎦
This formula is applicable for arbitrary real p, except for the following negative odd integers: −1,
−3,...,−(2n+1 ) .
2.414
1./integraldisplay
sinhaxdx =1
acoshax
2./integraldisplay
sinh2axdx =1
4asinh 2ax−x
2
3./integraldisplay
sinh3xdx=−3
4coshx+1
12cosh3 x=1
3cosh3x−coshx
4./integraldisplay
sinh4xdx=3
8x−1
4sinh 2x+1
32sinh 4x=3
8x−3
8sinhxcoshx+1
4sinh3xcoshx
5./integraldisplay
sinh5xdx=5
8coshx−5
48cosh 3 x+1
80cosh5 x
=4
5coshx+1
5sinh4xcoshx−4
15cosh3x
6./integraldisplay
sinh6xdx=−5
16x+15
64sinh 2x−3
64sinh 4x+1
192sinh 6x
=−5
16x+1
6sinh5xcoshx−5
24sinh3xcoshx+5
16sinhxcoshx
112 Hyperbolic Functions 2.415
7./integraldisplay
sinh7xdx=−35
64coshx+7
64cosh 3 x−7
320cosh 5 x+1
448cosh 7 x
=−24
35coshx+8
35cosh3x−6
35coshxsinh4x+1
7coshxsinh6x
8./integraldisplay
coshaxdx =1
asinhax
9./integraldisplay
cosh2axdx =x
2+1
4asinh 2ax
10./integraldisplay
cosh3xdx=3
4sinhx+1
12sinh 3x=s i n h x+1
3sinh3x
11./integraldisplay
cosh4xdx=3
8x+1
4sinh 2x+1
32sinh 4x=3
8x+3
8sinhxcoshx+1
4sinhxcosh3x
12./integraldisplay
cosh5xdx=5
8sinhx+5
48sinh 3x+1
80sinh 5x
=4
5sinhx+1
5cosh4xsinhx+4
15sinh3x
13./integraldisplay
cosh6xdx=5
16x+15
64sinh 2x+3
64sinh 4x+1
192sinh 6x
=5
16x+5
16sinhxcoshx+5
24sinhxcosh3x+1
6sinhxcosh5x
14./integraldisplay
cosh7xdx=35
64sinhx+7
64sinh 3x+7
320sinh 5x+1
448sinh 7x
=24
35sinhx+8
35sinh3x+6
35sinhxcosh4x+1
7sinhxcosh6x
2.415
1./integraldisplay
sinhaxcoshbxdx =cosh(a+b)x
2(a+b)+cosh(a−b)x
2(a−b)
2./integraldisplay
sinhaxcoshaxdx =1
4acosh 2 ax
3./integraldisplay
sinh2xcoshxdx=1
3sinh3x
4./integraldisplay
sinh3xcoshxdx=1
4sinh4x
5./integraldisplay
sinh4xcoshxdx=1
5sinh5x
6./integraldisplay
sinhxcosh2xdx=1
3cosh3x
7./integraldisplay
sinh2xcosh2xdx=−x
8+1
32sinh 4x
8./integraldisplay
sinh3xcosh2xdx=1
5/parenleftbig
sinh2x−2
3/parenrightbig
cosh3x
9./integraldisplay
sinh4xcosh2xdx=x
16−1
64sinh 2x−1
64sinh 4x+1
192sinh 6x
2.416 Powers of hyperbolic functions 113
10./integraldisplay
sinhxcosh3xdx=1
4cosh4x
11./integraldisplay
sinh2xcosh3xdx=1
5/parenleftbig
cosh2x+2
3/parenrightbig
sinh3x
12./integraldisplay
sinh3xcosh3xdx=−3
64cosh 2 x+1
192cosh6 x=1
48cosh32x−1
16cosh2 x
=sinh6x
6+sinh4x
4=cosh6x
6−cosh4x
4
13./integraldisplay
sinh4xcosh3xdx=1
7sinh3x/parenleftbig
cosh4x−3
5cosh2x−2
5/parenrightbig
=1
7/parenleftbig
cosh2x+2
5/parenrightbig
sinh5x
14./integraldisplay
sinhxcosh4xdx=1
5cosh5x
15./integraldisplay
sinh2xcosh4xdx=−x
16−1
64sinh 2x+1
64sinh 4x+1
192sinh 6x
16./integraldisplay
sinh3xcosh4xdx=1
7cosh3x/parenleftbig
sinh4x+3
5sinh2x−2
5/parenrightbig
=1
7/parenleftbig
sinh2x−2
5/parenrightbig
cosh5x
17./integraldisplay
sinh4xcosh4xdx=3x
128−1
128sinh 4x+1
1024sinh 8x
2.416
1.10/integraldisplaysinhpx
cosh2nxdx=sinhp+1x
2n−1⎡
⎣sech2n−1x
+n−1/summationdisplay
k=1(2n−p−2)(2n−p−4)...(2n−p−2k)
(2n−3)(2n−5)...(2n−2k−1)sech2n−2k−1x⎤
⎦
+(2n−p−2)(2n−p−4)...(−p+2 ) (−p)
(2n−1)!!/integraldisplay
sinhpxdx
This formula is applicable for arbitrary real p.F o r/integraltext
sinhpxdx,w h e r e pis a natural number, see
2.412 2a n d2.412 3. For n=0a n d pa negative integer, we have for this integral:
2./integraldisplaydx
sinh2mx=coshx
2m−1⎡
⎣−cosech2m−1x
+m−1/summationdisplay
k=1(−1)k−1·2k(m−1)(m−2)...(m−k)
(2m−3)(2m−5)...(2m−2k−1)cosech2m−2k−1x⎤
⎦
3./integraldisplaydx
sinh2m+1x=coshx
2m⎡
⎣−cosech2mx
+m−1/summationdisplay
k=1(−1)k−1·(2m−1)(2m−3)...(2m−2k+1 )
2k(m−1)(m−2)...(m−k)cosech2m−2kx⎤
⎦
+(−1)m(2m−1)!!
(2m)!!lntanhx
2
114 Hyperbolic Functions 2.417
2.417
1./integraldisplaysinhpx
cosh2n+1xdx=sinhp+1x
2n⎡
⎣sech2nx
+n−1/summationdisplay
k=1(2n−p−1)(2n−p−3)...(2n−p−2k+1 )
2k(n−1)(n−2)...(n−k)sech2n−2kx⎤
⎦
+(2n−p−1)(2n−p−3)...(3−p)(1−p)
2nn!/integraldisplaysinhpx
coshxdx
This formula is applicable for arbitrary real p.F o r n=0a n d pintegral, we have
2./integraldisplaysinh2m+1x
coshxdx=m/summationdisplay
k=1(−1)m+k
2ksinh2kx+(−1)mlncosh x
=m/summationdisplay
k=1(−1)m+k
2k/parenleftBigm
k/parenrightBig
cosh2kx+(−1)mlncosh x[m≥1]
3./integraldisplaysinh2mx
coshxdx=m/summationdisplay
k=1(−1)m+k
2k−1sinh2k−1x+(−1)marctan(sinh x)
[m≥1]
4./integraldisplaydx
sinh2m+1xcoshx=m/summationdisplay
k=1(−1)kcosech2m−2k+2x
2m−2k+2+(−1)mln tanh x
5./integraldisplaydx
sinh2mxcoshx=m/summationdisplay
k=1(−1)kcosech2m−2k+2x
2m−2k+1+(−1)marctansinh x
2.418
1./integraldisplaycoshpx
sinh2nxdx=−coshp+1x
2n−1⎡
⎣cosech2n−1x
+n−1/summationdisplay
k=1(−1)k(2n−p−2)(2n−p−4)...(2n−p−2k)
(2n−3)(2n−5)...(2n−2k−1)cosech2n−2k−1x⎤
⎦
+(−1)n(2n−p−2)(2n−p−4)...(−p+2 ) (−p)
(2n−1)!!/integraldisplay
coshpxdx
This formula is applicable for arbitrary real p. For the integral/integraltext
coshpxdx,w h e r e pis a natural
number, see 2.413 2a n d2.413 3. Ifpis a negative integer, we have for this integral:
2./integraldisplaydx
cosh2mx=sinhx
2m−1/braceleftBigg
sech2m−1x+m−1/summationdisplay
k=12k(m−1)(m−2)...(m−k)
(2m−3)(2m−5)...(2m−2k−1)sech2m−2k−1x/bracerightBigg
3./integraldisplaydx
cosh2m+1x=sinhx
2m/braceleftBigg
sech2mx+m−1/summationdisplay
k=1(2m−1)(2m−3)...(2m−2k+1 )
2k(m−1)(m−2)...(m−k)sech2m−2kx/bracerightBigg
+(2m−1)!!
(2m)!!arctansinh x
2.423 Powers of hyperbolic functions 115
2.419
1./integraldisplaycoshpx
sinh2n+1xdx=−coshp+1x
2n⎡
⎣cosech2nx
+n−1/summationdisplay
k=1(−1)k(2n−p−1)(2n−p−3)...(2n−p−2k+1 )
2k(n−1)(n−2)...(n−k)cosech2n−2kx⎤
⎦
+(−1)n(2n−p−1)(2n−p−3)...(3−p)(1−p)
2nn!/integraldisplaycoshpx
sinhxdx
This formula is applicable for arbitrary real p.F o r n=0a n d pan integer
2./integraldisplaycosh2mx
sinhxdx=m/summationdisplay
k=1cosh2k−1x
2k−1+l nt a n hx
2
3./integraldisplaycosh2m+1x
sinhxdx=m/summationdisplay
k=1cosh2kx
2k+l ns i n h x
=m/summationdisplay
k=1/parenleftBigm
k/parenrightBigsinh2kx
2k+l ns i n h x
4./integraldisplaydx
sinhxcosh2mx=m/summationdisplay
k=1sech2m−2k+1x
2m−2k+1+l nt a n hx
2
5./integraldisplaydx
sinhxcosh2m+1x=m/summationdisplay
k=1sech2m−2k+2x
2m−2k+2+l nt a n h x
2.421 In formulas 2.421 1a n d 2.421 2,s=1f o r modd and m<2n+ 1; in all other cases, s=0.
GI (351)(11, 13)
1.10/integraldisplaysinh2n+1x
coshmxdx=n/summationdisplay
k=0
k/negationslash=m−1
2(−1)n+k/parenleftBign
k/parenrightBigcosh2k−m+1x
2k−m+1+s(−1)n+m−1
2/parenleftbiggn
m−1
2/parenrightbigg
lncosh x
2./integraldisplaycosh2n+1x
sinhmxdx=n/summationdisplay
k=0
k/negationslash=m−1
2/parenleftBign
k/parenrightBigsinh2k−m+1x
2k−m+1+s/parenleftbiggn
m−1
2/parenrightbigg
lnsinh x
2.422
1./integraldisplaydx
sinh2mxcosh2nx=m+n−1/summationdisplay
k=0(−1)k+1
2m−2k−1/parenleftbiggm+n−1
k/parenrightbigg
tanh2k−2m+1x
2./integraldisplaydx
sinh2m+1xcosh2n+1x=m+n/summationdisplay
k=0
k/negationslash=m(−1)k+1
2m−2k/parenleftbiggm+n
k/parenrightbigg
tanh2k−2mx+(−1)m/parenleftbiggm+n
m/parenrightbigg
lntanh x
GI (351)(15)
2.423
1./integraldisplaydx
sinhx=l nt a n hx
2=1
2lncoshx−1
coshx+1
116 Hyperbolic Functions 2.423
2./integraldisplaydx
sinh2x=−cothx
3./integraldisplaydx
sinh3x=−coshx
2s in h2x−1
2lntanhx
2
4./integraldisplaydx
sinh4x=−coshx
3s in h3x+2
3cothx=−1
3coth3x+c o t h x
5./integraldisplaydx
sinh5x=−coshx
4s in h4x+3
8coshx
sinh2x+3
8lntanhx
2
6./integraldisplaydx
sinh6x=−coshx
5s in h5x+4
15coth3x−4
5cothx
=−1
5coth5x+2
3coth3x−cothx
7./integraldisplaydx
sinh7x=−coshx
6s in h2x/parenleftbigg1
sinh4x−5
4s in h2x+15
8/parenrightbigg
−5
16ln tanhx
2
8./integraldisplaydx
sinh8x=c o t h x−coth3x+3
5coth5x−1
7coth7x
9./integraldisplaydx
coshx=a r c t a n( s i n h x)
= arcsin(tanh x)
= 2arctan( ex)
=g dx
10./integraldisplaydx
cosh2x=t a n h x
11./integraldisplaydx
cosh3x=sinhx
2c os h2x+1
2arctan(sinh x)
12./integraldisplaydx
cosh4x=sinhx
3c os h3x+2
3tanhx
=−1
3tanh3x+t a n h x
13./integraldisplaydx
cosh5x=sinhx
4c os h4x+3
8sinhx
cosh2x+3
8arctan(sinh x)
14./integraldisplaydx
cosh6x=sinhx
5c os h5x−4
15tanh3x+4
5tanhx
=1
5tanh5x−2
3tanh3x+t a n h x
15./integraldisplaydx
cosh7x=sinhx
6c os h2x/parenleftbigg1
cosh4x+5
4c os h2x+15
8/parenrightbigg
+5
16arctan(sinh x)
16./integraldisplaydx
cosh8x=−1
7tanh7x+3
5tanh5x−tanh3x+t a n h x
17./integraldisplaysinhx
coshxdx=l nc o s h x
2.423 Powers of hyperbolic functions 117
18./integraldisplaysinh2x
coshxdx=s i n h x−arctan(sinh x)
19./integraldisplaysinh3x
coshxdx=1
2sinh2x−lncosh x
=1
2cosh2x−ln cosh x
20./integraldisplaysinh4x
coshxdx=1
3sinh3x−sinhx+a r c t a n( s i n h x)
21./integraldisplaysinhx
cosh2xdx=−1
coshx
22./integraldisplaysinh2x
cosh2xdx=x−tanhx
23./integraldisplaysinh3x
cosh2xdx=c o s h x+1
coshx
24./integraldisplaysinh4x
cosh2xdx=−3
2x+1
4sinh 2x+t a n h x
25./integraldisplaysinhx
cosh3xdx=−1
2c os h2x
=1
2tanh2x
26./integraldisplaysinh2x
cosh3xdx=−sinhx
2c os h2x+1
2arctan (sinh x)
27./integraldisplaysinh3x
cosh3xdx=−1
2tanh2x+l nc o s h x
=1
2c os h2x+l nc o s h x
28./integraldisplaysinh4x
cosh3xdx=sinhx
2c os h x+s i n h x−3
2arctan(sinh x)
29./integraldisplaysinhx
cosh4xdx=−1
3c os h3x
30./integraldisplaysinh2x
cosh4xdx=1
3tanh3x
31./integraldisplaysinh3x
cosh4xdx=−1
coshx+1
3c os h3x
32./integraldisplaysinh4x
cosh4xdx=−1
3tanh3x−tanhx+x
33./integraldisplaycoshx
sinhxdx=l ns i n h x
34./integraldisplaycosh2x
sinhxdx=c o s h x+l nt a n hx
2
118 Hyperbolic Functions 2.423
35./integraldisplaycosh3x
sinhxdx=1
2cosh2x+l ns i n h x
36./integraldisplaycosh4x
sinhxdx=1
3cosh3x+c o s h x+l nt a n hx
2
37./integraldisplaycoshx
sinh2xdx=−1
sinhx
38./integraldisplaycosh2x
sinh2xdx=x−cothx
39./integraldisplaycosh3x
sinh2xdx=s i n h x−1
sinhx
40./integraldisplaycosh4x
sinh2xdx=3
2x+1
4sinh 2x−cothx
41./integraldisplaycoshx
sinh3xdx=−1
2s in h2x
=−1
2coth2x
42./integraldisplaycosh2x
sinh3xdx=−coshx
2s in h2x+l nt a n hx
2
43./integraldisplaycosh3x
sinh3xdx=−1
2s in h2x+l ns i n h x
=−1
2coth2x+l ns i n h x
44./integraldisplaycosh4x
sinh3xdx=−coshx
2s in h2x+c o s h x+3
2ln tanhx
2
45./integraldisplaycoshx
sinh4xdx=−1
3s in h3x
46./integraldisplaycosh2x
sinh4xdx=−1
3coth3x
47./integraldisplaycosh3x
sinh4xdx=−1
sinhx−1
3s in h3x
48./integraldisplaycosh4x
sinh4xdx=−1
3coth3x−cothx+x
49./integraldisplaydx
sinhxcoshx=l nt a n h x
50./integraldisplaydx
sinhxcosh2x=1
coshx+l nt a n hx
2
51./integraldisplaydx
sinhxcosh3x=1
2c os h2x+l nt a n h x
=−1
2tanh2x+l nt a n h x
2.424 Powers of hyperbolic functions 119
52./integraldisplaydx
sinhxcosh4x=1
coshx+1
3c os h3x+l nt a n hx
2
53./integraldisplaydx
sinh2xcoshx=−1
sinhx−arctansinh x
54./integraldisplaydx
sinh2xcosh2x=−2c o t h2 x
55./integraldisplaydx
sinh2xcosh3x=−sinhx
2c os h2x−1
sinhx−3
2arctansinh x
56./integraldisplaydx
sinh2xcosh4x=1
3s in h xcosh3x−8
3coth 2 x
57./integraldisplaydx
sinh3xcoshx=−1
2s in h2x−lntanh x
=−1
2coth2x+l nc o t h x
58./integraldisplaydx
sinh3xcosh2x=−1
coshx−coshx
2s in h2x−3
2lntanhx
2
59./integraldisplaydx
sinh3xcosh3x=−2c o s h2 x
sinh22x−2lntan h x
=1
2tanh2x−1
2coth2x−2l nt a n h x
60./integraldisplaydx
sinh3xcosh4x=−2
coshx−1
3c os h2x−coshx
2s in h2x−5
2ln tanhx
2
61./integraldisplaydx
sinh4xcoshx=1
sinhx−1
3s in h3x+a r c t a ns i n h x
62./integraldisplaydx
sinh4xcosh2x=−1
3c os h xsinh3x+8
3coth2 x
63./integraldisplaydx
sinh4xcosh3x=2
sinhx−1
3s in h3x+sinhx
2c os h2x+5
2arctan sinh x
64./integraldisplaydx
sinh4xcosh4x= 8 coth 2 x−8
3coth32x
2.424
1./integraldisplay
tanhpxdx=−tanhp−1x
p−1+/integraldisplay
tanhp−2xdx [p/negationslash=1 ]
2./integraldisplay
tanh2n+1xdx=n/summationdisplay
k=1(−1)k−1
2k/parenleftBign
k/parenrightBig1
cosh2kx+l nc o s h x
=−n/summationdisplay
k=1tanh2n−2k+2x
2n−2k+2+l nc o s h x
3./integraldisplay
tanh2nxdx=−n/summationdisplay
k=1tanh2n−2k+1x
2n−2k+1+x GU (351)(12)
4./integraldisplay
cothpxdx=−cothp−1x
p−1+/integraldisplay
cothp−2xdx [p/negationslash=1 ]
120 Hyperbolic Functions 2.425
5./integraldisplay
coth2n+1xdx=−n/summationdisplay
k=11
2n/parenleftBign
k/parenrightBig1
sinh2kx+l ns i n h x
=−n/summationdisplay
k=1coth2n−2k+2x
2n−2k+2+l ns i n h x
6./integraldisplay
coth2nxdx=−n/summationdisplay
k=1coth2n−2k+1x
2n−2k+1+x GU (351)(14)
For formulas containing powers of tanh xand coth xequal to n=1 ,2 ,3 ,4 ,s e e 2.423 17,2.423 22,
2.423 27,2.423 32,2.423 33,2.423 38,2.423 43,2.423 48.
Powers of hyperbolic functions and hyperbolic functions of linear functions of the argument
2.425
1./integraldisplay
sinh(ax+b)sin h( cx+d)dx=1
2(a+c)sinh[(a+c)x+b+d]
−1
2(a−c)sinh[(a−c)x+b−d]
/bracketleftbig
a2/negationslash=c2/bracketrightbig
GU (352)(2a)
2./integraldisplay
sinh(ax+b)cosh( cx+d)dx=1
2(a+c)cosh[( a+c)x+b+d]
+1
2(a−c)cosh[( a−c)x+b−d]
/bracketleftbig
a2/negationslash=c2/bracketrightbig
GU (352)(2c)
3./integraldisplay
cosh(ax+b)cosh ( cx+d)dx=1
2(a+c)sinh[(a+c)x+b+d]
+1
2(a−c)sinh[(a−c)x+b−d]
/bracketleftbig
a2/negationslash=c2/bracketrightbig
GU (352)(2b)
When a=c:
4./integraldisplay
sinh(ax+b)sin h( ax+d)dx=−x
2cosh(b−d)+1
4asinh(2 ax+b+d) GU (352)(3a)
5./integraldisplay
sinh(ax+b)cosh( ax+d)dx=x
2sinh(b−d)+1
4acosh(2 ax+b+d) GU (352)(3c)
6./integraldisplay
cosh(ax+b)cosh ( ax+d)dx=x
2cosh(b−d)+1
4asinh(2 ax+b+d) GU (352)(3b)
2.426
1./integraldisplay
sinhaxsinhbxsinhcxdx=cosh(a+b+c)x
4(a+b+c)−cosh(−a+b+c)x
4(−a+b+c)
−cosh(a−b+c)x
4(a−b+c)−cosh(a+b−c)x
4(a+b−c)
GU (352)(4a)
2.428 Powers of hyperbolic functions 121
2./integraldisplay
sinhaxsinhbxcoshcxdx=sinh(a+b+c)x
4(a+b+c)−sinh(−a+b+c)x
4(−a+b+c)
−sinh(a−b+c)x
4(a−b+c)+sinh(a+b−c)x
4(a+b−c)
GU (352)(4b)
3./integraldisplay
sinhaxcoshbxcoshcxdx=cosh(a+b+c)x
4(a+b+c)−cosh(−a+b+c)x
4(−a+b+c)
+cosh(a−b+c)x
4(a−b+c)+cosh(a+b−c)x
4(a+b−c)
GU (352)(4c)
4./integraldisplay
coshaxcoshbxcoshcxdx=sinh(a+b+c)x
4(a+b+c)+sinh(−a+b+c)x
4(−a+b+c)
+sinh(a−b+c)x
4(a−b+c)+sinh(a+b−c)x
4(a+b−c)
GU (352)(4d)
2.427
1./integraldisplay
sinhpxsinhaxdx =1
p+a/braceleftbigg
sinhpxcoshax−p/integraldisplay
sinhp−1xcosh(a−1)xdx/bracerightbigg
2./integraldisplay
sinhpxsinh(2 n+1 )xdx=Γ(p+1 )
Γ/parenleftbigp+3
2+n/parenrightbig
×⎡
⎣n−1/summationdisplay
k=0Γ/parenleftbigp+1
2+n−2k/parenrightbig
22k+1Γ(p−2k+1 )sinhp−2kxcosh(2 n−2k+1 )x
−Γ/parenleftbigp−1
2+n−2k/parenrightbig
22k+2Γ(p−2k)sinhp−2k−1xsinh(2 n−2k)x⎤
⎦
+Γ/parenleftbigp+3
2−n/parenrightbig
22nΓ(p+1−2n)/integraldisplay
sinhp−2nxsinhxdx
[pis not a negative integer]
3./integraldisplay
sinhpxsinh 2nxdx =Γ(p+1 )
Γ/parenleftbigp
2+n+1/parenrightbig
×n−1/summationdisplay
k=0⎡
⎣Γ/parenleftbigp
2+n−2k/parenrightbig
22k+1Γ(p−2k+1 )sinhp−2kxcosh(2 n−2k)x
−Γ/parenleftbigp
2+n−2k−1/parenrightbig
22k+2Γ(p−2k)sinhp−2k−1xsinh(2 n−2k−1)x⎤
⎦
[pis not a negative integer] GU (352)(5)a
2.428
1./integraldisplay
sinhpxcoshaxdx =1
p+a/braceleftbigg
sinhpxsinhax−p/integraldisplay
sinhp−1xsinh(a−1)xdx/bracerightbigg
122 Hyperbolic Functions 2.429
2./integraldisplay
sinhpxcosh(2 n+1 )xdx=Γ(p+1 )
Γ/parenleftbigp+3
2+n/parenrightbig
×⎧
⎨
⎩⎡
⎣n−1/summationdisplay
k=0Γ/parenleftbigp+1
2+n−2k/parenrightbig
22k+1Γ(p−2k+1 )sinhp−2kxsinh(2 n−2k+1 )x
−Γ/parenleftbigp−1
2+n−2k/parenrightbig
22k+2Γ(p−2k)sinhp−2k−1xcosh(2 n−2k)x⎤
⎦
+Γ/parenleftbigp+3
2−n/parenrightbig
22nΓ(p+1−2n)/integraldisplay
sinhp−2nxcoshxdx⎫
⎬
⎭
[pis not a negative integer]
3./integraldisplay
sinhpxcosh 2 nxdx =Γ(p+1 )
Γ/parenleftbigp
2+n+1/parenrightbig
×⎧
⎨
⎩n−1/summationdisplay
k=0⎡
⎣Γ/parenleftbigp
2+n−2k/parenrightbig
22k+1Γ(p−2k+1 )sinhp−2kxsinh(2 n−2k)x
−Γ/parenleftbigp
2+n−2k−1/parenrightbig
22k+2Γ(p−2k)sinhp−2k−1xcosh(2 n−2k−1)x⎤
⎦
+Γ/parenleftbigp
2−n+1/parenrightbig
22nΓ(p+1−2n)/integraldisplay
sinhp−2nxdx⎫
⎬
⎭
[pis not a negative integer] GU (352)(6)a
2.429
1./integraldisplay
coshpxsinhaxdx =1
p+a/braceleftbigg
coshpxcoshax+p/integraldisplay
coshp−1xsinh(a−1)xdx/bracerightbigg
2./integraldisplay
coshpxsinh(2 n+1 )xdx=Γ(p+1 )
Γ/parenleftbigp+3
2+n/parenrightbig⎡
⎣n−1/summationdisplay
k=0Γ/parenleftbigp+1
2+n−k/parenrightbig
2k+1Γ(p−k+1 )coshp−kxcosh(2 n−k+1 )x
+Γ/parenleftbigp+3
2/parenrightbig
2nΓ(p−n+1 )/integraldisplay
coshp−nxsinh(n+1 )xdx⎤
⎦
[pis not a negative integer]
3./integraldisplay
coshpxsinh 2nxdx =Γ(p+1 )
Γ/parenleftbigp
2+n+1/parenrightbig⎡
⎣n−1/summationdisplay
k=0Γ/parenleftbigp
2+n−k/parenrightbig
2k+1Γ(p−k+1 )coshp−kxcosh(2 n−k)x
+Γ/parenleftbigp
2+1/parenrightbig
2nΓ(p−n+1 )/integraldisplay
coshp−nxsinhnxdx⎤
⎦
[pis not a negative integer] GU (352)(7)a
2.433 Powers of hyperbolic functions 123
2.431
1./integraldisplay
coshpxcoshaxdx =1
p+a/braceleftbigg
coshpxsinhax+p/integraldisplay
coshp−1xcosh(a−1)xdx/bracerightbigg
2./integraldisplay
coshpxcosh(2 n+1 )xdx=Γ(p+1 )
Γ/parenleftbigp+3
2+n/parenrightbig⎡
⎣n−1/summationdisplay
k=0Γ/parenleftbigp+1
2+n−k/parenrightbig
2k+1Γ(p−k+1 )coshp−kxsinh(2 n−k+1 )x
+Γ/parenleftbigp+3
2/parenrightbig
2nΓ(p−n+1 )/integraldisplay
coshp−nxcosh(n+1 )xdx⎤
⎦
[pis not a negative integer]
3./integraldisplay
coshpxcosh 2 nxdx =Γ(p+1 )
Γ/parenleftbigp
2+n+1/parenrightbig⎡
⎣n−1/summationdisplay
k=0Γ/parenleftbigp
2+n−k/parenrightbig
2k+1Γ(p−k+1 )coshp−kxsinh(2 n−k)x
+Γ/parenleftbigp
2+1/parenrightbig
2nΓ(p−n+1 )coshp−nxcoshnxdx⎤
⎦
[pis not a negative integer] GU (352)(8)a
2.432
1./integraldisplay
sinh(n+1 )xsinhn−1xdx=1
nsinhnxsinhnx
2./integraldisplay
sinh(n+1 )xcoshn−1xdx=1
ncoshnxcoshnx
3./integraldisplay
cosh(n+1 )xsinhn−1xdx=1
nsinhnxcoshnx
4./integraldisplay
cosh(n+1 )xcoshn−1xdx=1
ncoshnxsinhnx
2.433
1./integraldisplaysinh(2 n+1 )x
sinhxdx=2n−1/summationdisplay
k=0sinh(2 n−2k)x
2n−2k+x
2./integraldisplaysinh 2nx
sinhxdx=2n−1/summationdisplay
k=0sinh(2 n−2k−1)x
2n−2k−1GU (352)(5d)
3./integraldisplaycosh(2 n+1 )x
sinhxdx=2n−1/summationdisplay
k=0cosh(2 n−2k)x
2n−2k+l ns i n h x
4./integraldisplaycosh 2 nx
sinhxdx=2n−1/summationdisplay
k=0cosh(2 n−2k−1)x
2n−2k−1+l nt a n hx
2GU (352)(6d)
5./integraldisplaysinh(2 n+1 )x
coshxdx=2n−1/summationdisplay
k=0(−1)kcosh(2 n−2k)x
2n−2k+(−1)nlncosh x
124 Hyperbolic Functions 2.433
6./integraldisplaysinh 2nx
coshxdx=2n−1/summationdisplay
k=0(−1)kcosh(2 n−2k−1)x
2n−2k−1GU (352)(7d)
7./integraldisplaycosh(2 n+1 )x
coshxdx=2n−1/summationdisplay
k=0(−1)ksinh(2 n−2k)x
2n−2k+(−1)nx
8./integraldisplaycosh 2 nx
coshxdx=2n−1/summationdisplay
k=0(−1)ksinh(2 n−2k−1)x
2n−2k−1+(−1)narcsin(tanh x) GU (352)(8d)
9./integraldisplaysinh 2x
sinhnxdx=−2
(n−2)sinhn−2x
Forn=2 :
10./integraldisplaysinh 2x
sinh2xdx=2l ns i n h x
11./integraldisplaysinh 2xdx
coshnx=2
(2−n)coshn−2x
Forn=2 :
12./integraldisplaysinh 2x
cosh2xdx=2l nc o s h x
13./integraldisplaycosh 2 x
sinhxdx=2c o s h x+l nt a n hx
2
14./integraldisplaycosh 2 x
sinh2xdx=−cothx+2x
15./integraldisplaycosh 2 x
sinh3xdx=−coshx
2s in h2x+3
2ln tanhx
2
16./integraldisplaycosh 2 x
coshxdx=2s i n h x−arcsin(tanh x)
17./integraldisplaycosh 2 x
cosh2xdx=−tanhx+2x
18./integraldisplaycosh 2 x
cosh3xdx=−sinhx
2c os h2x+3
2arcsin(tanh x)
19./integraldisplaysinh 3x
sinhxdx=x+s i n h2 x
20./integraldisplaysinh 3x
sinh2xdx=3l nt a n hx
2+4c o s h x
21./integraldisplaysinh 3x
sinh3xdx=−3c ot h x+4x
22./integraldisplaysinh 3x
coshnxdx=4
(3−n)coshn−3x−1
(1−n)coshn−1x
Forn=1a n d n=3 :
23./integraldisplaysinh 3x
coshxdx=2s i n h2x−lncosh x
2.442 Rational functions of hyperbolic functions 125
24./integraldisplaysinh 3x
cosh3xdx=1
2c os h2x+4l nc o s h x
25./integraldisplaycosh 3 x
sinhnxdx=4
(3−n)sin hn−3x+1
(1−n)sin hn−1x
Forn=1a n d n=3 :
26./integraldisplaycosh 3 x
sinhxdx=2s i n h2x+l ns i n h x
27./integraldisplaycosh 3 x
sinh3xdx=−1
2s in h2x+4l ns i n h x
28./integraldisplaycosh 3 x
coshxdx=s i n h2 x−x
29./integraldisplaycosh 3 x
cosh2xdx=4s i n h x−3 arcsin(tanh x)
30./integraldisplaycosh 3 x
cosh3xdx=4x−3t an h x
2.44–2.45 Rational functions of hyperbolic functions
2.441
1./integraldisplayA+Bsinhx
(a+bsinhx)ndx=aB−bA
(n−1)(a2+b2)·coshx
(a+bsinhx)n−1
+1
(n−1)(a2+b2)/integraldisplay(n−1)(aA+bB)+(n−2)(aB−bA)sin h x
(a+bsinhx)n−1dx
Forn=1 :
2./integraldisplayA+Bsinhx
a+bsinhxdx=B
bx−aB−bA
b/integraldisplaydx
a+bsinhx(see2.441 3)
3./integraldisplaydx
a+bsinhx=1√
a2+b2lnatanhx
2−b+√
a2+b2
atanhx
2−b−√
a2+b2
=2√
a2+b2arctanhatanhx
2−b√
a2+b2
2.442
1./integraldisplayA+Bcoshx
(a+bsinhx)ndx=−B
(n−1)b(a+bsinhx)n−1+A/integraldisplaydx
(a+bsinhx)n
Forn=1 :
2./integraldisplayA+Bcoshx
a+bsinhxdx=B
bln (a+bsinhx)+A/integraldisplaydx
a+bsinhx
(see2.441 3)
126 Hyperbolic Functions 2.443
2.443
1./integraldisplayA+Bcoshx
(a+bcoshx)ndx=aB−bA
(n−1)(a2−b2)·sinhx
(a+bcoshx)n−1
+1
(n−1)(a2−b2)/integraldisplay(n−1)(aA−bB)+(n−2)(aB−bA)cosh x
(a+bcoshx)n−1dx
Forn=1 :
2./integraldisplayA+Bcoshx
a+bcoshxdx=B
bx−aB−bA
b/integraldisplaydx
a+bcoshx(see2.443 3)
3./integraldisplaydx
a+bcoshx=1√
b2−a2arcsinb+acoshx
a+bcoshx/bracketleftbig
b2>a2,x < 0/bracketrightbig
=−1√
b2−a2arcsinb+acoshx
a+bcoshx/bracketleftbig
b2>a2,x > 0/bracketrightbig
=1√
a2−b2lna+b+√
a2−b2tanhx
2
a+b−√
a2−b2tanhx
2/bracketleftbig
a2>b2/bracketrightbig
2.444
1./integraldisplaydx
cosha+c o s h x= cosech a/bracketleftbigg
ln coshx+a
2−lncoshx−a
2/bracketrightbigg
=2c o s e c h aarctanh/parenleftBig
tanhx
2tanha
2/parenrightBig
2.11/integraldisplaydx
cosa+c o s h x= 2 cosec aarctan/parenleftBig
tanhx
2tana
2/parenrightBig
2.445
1./integraldisplayBsinhx
(a+bcoshx)ndx=−B
(n−1)b(a+bcoshx)n−1[n/negationslash=1 ]
Forn=1 :
2./integraldisplayBsinhx
a+bcoshxdx=B
bln(a+bcoshx)( s e e 2.443 3)
In evaluating definite integrals by use of formulas 2.441–2.443 and2.445 , one may not take the integral
over points at which the integrand becomes infinite, that is, over the points
x=a r c s i n h/parenleftBig
−a
b/parenrightBig
in formulas 2.441 or2.442 or over the points
x= arccosh/parenleftBig
−a
b/parenrightBig
in formulas 2.443 or2.445 . Formulas 2.443 are not applicable for a2=b2. Instead, we may use the
following formulas in these cases:
2.449 Rational functions of hyperbolic functions 127
2.446
1./integraldisplayA+Bcoshx
(ε+c o s h x)ndx
=Bsinhx
(1−n)(ε+c o s h x)n+/parenleftbigg
εA+n
n−1B/parenrightbigg(n−1)!
(2n−1)!!sinhxn−1/summationdisplay
k=0(2n−2k−3)!!
(n−k−1)!
×εh
(ε+c o s h x)n−k
[ε=±1,n > 1]
Forn=1 :
2./integraldisplayA+Bcoshx
ε+c o s h xdx=Bx+(εA−B)coshx−ε
sinhx[ε=±1]
2.447
1./integraldisplaysinhxdx
acoshx+bsinhx=alncosh/parenleftbigg
x+a r c t a n hb
a/parenrightbigg
bx
a2−b2[a>|b|]
=bx−aln sinh/parenleftBig
x+a r c t a n ha
b/parenrightBig
b2−a2[b>|a|] MZ 215
Fora=b=1 :
2./integraldisplaysinhxdx
coshx+s i n h x=x
2+1
4e−2x
Fora=−b=1 :
3./integraldisplaysinhxdx
coshx−sinhx=−x
2+1
4e2xMZ 215
2.448
1./integraldisplaycoshxdx
acoshx+bsinhx=ax−bln cosh/parenleftbig
x+a r c t a n hb
a/parenrightbig
a2−b2[a>|b|]
=−ax+blnsinh/parenleftbig
x+a r c t a n ha
b/parenrightbig
b2−a2[b>|a|]
Fora=b=1 :
2./integraldisplaycoshxdx
coshx+s i n h x=x
2−1
4e−2x
Fora=−b=1 :
3./integraldisplaycoshxdx
coshx−sinhx=x
2+1
4e2xMZ 214, 215
2.449
1.6/integraldisplaydx
(acoshx+bsinhx)n=1/radicalbig
(a2−b2)n/integraldisplaydx
sinhn/parenleftbigg
x+a r c t a n hb
a/parenrightbigg[a>|b|]
=1/radicalbig
(b2−a2)n/integraldisplaydx
coshn/parenleftBig
x+a r c t a n ha
b/parenrightBig[b>|a|]
128 Hyperbolic Functions 2.451
Forn=1 :
2./integraldisplaydx
acoshx+bsinhx=1√
a2−b2arctan/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinh/parenleftbigg
x+a r c t a n hb
a/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle[a>|b|]
=1
√
b2−a2ln/vextendsingle/vextendsingle/vextendsingle/vextendsingletanhx+a r c t a n h a
b
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle[b>|a|]
Fora=b=1 :
3./integraldisplayax
coshx+s i n h x=−e−x=s i n h x−coshx
Fora=−b=1 :
4./integraldisplaydx
coshx−sinhx=ex=s i n h x+c o s h x MZ 214
2.451
1./integraldisplayA+Bcoshx+Csinhx
(a+bcoshx+csinhx)ndx
=Bc−Cb+(Ac−Ca)cosh x+(Ab−Ba)sin h x
(1−n)(a2−b2+c2)(a+bcoshx+csinhx)n−1+1
(n−1)(a2−b2+c2)
×/integraldisplay(n−1)(Aa−Bb+Cc)−(n−2)(Ab−Ba)cosh x−(n−2)(Ac−Ca)sin h x
(a+bcoshx+csinhx)n−1dx
/bracketleftbig
a2+c2/negationslash=b2/bracketrightbig
=Bc−Cb−Cacoshx−Basinhx
(n−1)a(a+bcoshx+csinhx)n+/bracketleftbiggA
a+n(Bb−Cc)
(n−1)a2/bracketrightbigg
(ccoshx+bsinhx)(n−1)!
(2n−1)!!
×n−1/summationdisplay
k=0(2n−2k−3)!!
(n−k−1)!ak1
(a+bcoshx+csinhx)n−k
/bracketleftbig
a2+c2=b2/bracketrightbig
2./integraldisplayA+Bcoshx+Csinhx
a+bcoshx+csinhxdx=Cb−Bc
b2−c2ln(a+bcoshx+csinhx)
+Bb−Cc
b2−c2x+/parenleftbigg
A−aBb−Cc
b2−c2/parenrightbigg/integraldisplaydx
a+bcoshx+csinhx/bracketleftbig
b2/negationslash=c2/bracketrightbig
(see2.451 4)
3./integraldisplayA+Bcoshx+Csinhx
a+bcoshx±bsinhxdx=C∓B
2a(coshx∓sinhx)+/bracketleftbiggA
a−(B∓C)b
2a2/bracketrightbigg
x
+/bracketleftbiggC±B
2b±A
a−(C∓B)b
2a2/bracketrightbigg
ln (a+bcoshx±bsinhx)
[ab/negationslash=0 ]
2.452 Rational functions of hyperbolic functions 129
4./integraldisplaydx
a+bcoshx+csinhx
=2√
b2−a2−c2arctan(b−a)tan hx
2+c√
b2−a2−c2/bracketleftbig
b2>a2+c2anda/negationslash=b/bracketrightbig
=1√
a2−b2+c2ln(a−b)tan hx
2−c+√
a2−b2+c2
(a−b)tan hx
2−c−√
a2−b2+c2/bracketleftbig
b2<a2+c2anda/negationslash=b/bracketrightbig
=1
cln/parenleftBig
a+ctanhx
2/parenrightBig
[a=bandc/negationslash=0 ]
=2
(a−b)tan hx
2+c/bracketleftbig
b2=a2+c2/bracketrightbig
GU (351)(18)
2.452
1./integraldisplayA+Bcoshx+Csinhx
(a1+b1coshx+c1sinhx)(a2+b2coshx+c2sinhx)dx
=A0lna1+b1coshx+c1sinhx
a2+b2coshx+c2sinhx+A1/integraldisplaydx
a1+b1coshx+c1sinhx+A2/integraldisplaydx
a2+b2coshx+c2sinhx
where GU (351)(19)
A0=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1b1c1
ABC
a2b2c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1b1
a2b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
1c1
b2c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
−/vextendsingle/vextendsingle/vextendsingle/vextendsinglec
1a1
c2a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2A1=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1 b1 c1 /vextendsingle/vextendsingle/vextendsingle/vextendsingleb
1c1
BC/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglec
1a1
CA/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1b1
AB/vextendsingle/vextendsingle/vextendsingle/vextendsingle
a
2 b2 c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1b1
a2b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
1c1
b2c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
−/vextendsingle/vextendsingle/vextendsingle/vextendsinglec
1a1
c2a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2,
A2=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1 b1 c1 /vextendsingle/vextendsingle/vextendsingle/vextendsingleCB
c
2b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleCA
c
2a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleBA
b
2a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle
a
2 b2 c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1b1
a2b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
1c1
b2c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
−/vextendsingle/vextendsingle/vextendsingle/vextendsinglec
1a1
c2a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2,/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1b1
a2b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
1c1
b2c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
/negationslash=/vextendsingle/vextendsingle/vextendsingle/vextendsinglec
1a1
c2a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/bracketrightBigg
.
2./integraldisplayAcosh2x+2Bsinhxcoshx+Csinh2x
acosh2x+2bsinhxcoshx+csinh2xdx
=1
4b2−(a+c)2⎧
⎨
⎩[4Bb−(A+C)(a+c)]x
+[(A+C)b−B(a+c)] ln/parenleftbig
acosh2x+2bsinhxcoshx+csinh2x/parenrightbig
+/bracketleftbig
2(A−C)b2−2Bb(a−c)+(Ca−Ac)(a+c)/bracketrightbig
f(x)⎫
⎬
⎭
130 Hyperbolic Functions 2.453
where GU (351)(24)
f(x)=1
2√
b2−aclnctanhx+b−√
b2−ac
ctanhx+b+√
b2−ac/bracketleftbig
b2>a c/bracketrightbig
=1√
ac−b2arctanctanhx+b√
ac−b2/bracketleftbig
b2<a c/bracketrightbig
=−1
ctanhx+b/bracketleftbig
b2=ac/bracketrightbig
2.453
1./integraldisplay(A+Bsinhx)dx
sinhx(a+bsinhx)=1
a/bracketleftbigg
Aln/vextendsingle/vextendsingle/vextendsingletanhx
2/vextendsingle/vextendsingle/vextendsingle+(aB−bA)/integraldisplaydx
a+bsinhx/bracketrightbigg
(see2.441 3)
2./integraldisplay(A+Bsinhx)dx
sinhx(a+bcoshx)=A
a2−b2/parenleftbigg
aln/vextendsingle/vextendsingle/vextendsingletanhx
2/vextendsingle/vextendsingle/vextendsingle+bln/vextendsingle/vextendsingle/vextendsingle/vextendsinglea+bcoshx
sinhx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg
+B/integraldisplaydx
a+bcoshx
(see2.443 3)
Fora2=b2=1 :
3./integraldisplay(A+Bsinhx)dx
sinhx(1 + cosh x)=A
2/parenleftbigg
ln/vextendsingle/vextendsingle/vextendsingletanhx
2/vextendsingle/vextendsingle/vextendsingle−1
2tanh2x
2/parenrightbigg
+Btanhx
2
4./integraldisplay(A+Bsinhx)dx
sinhx(1−coshx)=A
2/parenleftbigg
−ln/vextendsingle/vextendsingle/vextendsinglecothx
2/vextendsingle/vextendsingle/vextendsingle+1
2coth2x
2/parenrightbigg
+Bcothx
2
2.454
1./integraldisplay(A+Bsinhx)dx
coshx(a+bsinhx)=1
a2+b2/bracketleftbigg
(Aa+Bb)arctan(sin h x)+(Ab−Ba)ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglea+bsinhx
coshx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbigg
2./integraldisplay(A+Bcoshx)dx
sinhx(a+bsinhx)=1
a/parenleftbigg
Aln/vextendsingle/vextendsingle/vextendsingletanhx
2/vextendsingle/vextendsingle/vextendsingle+Bln/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinhx
a+bsinhx/vextendsingle/vextendsingle/vextendsingle/vextendsingle−Ab/integraldisplaydx
a+bsinhx/parenrightbigg
(see2.441 3)
2.455
1./integraldisplay(A+Bcoshx)dx
sinhx(a+bcoshx)=1
a2−b2/bracketleftbigg
(Aa+Bb)ln/vextendsingle/vextendsingle/vextendsingletanhx
2/vextendsingle/vextendsingle/vextendsingle+(Ab−Ba)ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglea+bcoshx
sinhx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbigg
Fora
2=b2=1 :
2./integraldisplay(A+Bcoshx)dx
sinhx(1 + cosh x)=A+B
2ln/vextendsingle/vextendsingle/vextendsingletanhx
2/vextendsingle/vextendsingle/vextendsingle−A−B
4tanh2x
2
3./integraldisplay(A+Bcoshx)dx
sinhx(1−coshx)=A+B
4coth2x
2−A−B
2ln cothx
2
2.456/integraldisplay(A+Bcoshx)dx
coshx(a+bsinhx)=A
a2+b2/bracketleftbigg
aarctan(sinh x)+bln/vextendsingle/vextendsingle/vextendsingle/vextendsinglea+bsinhx
coshx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbigg
+B/integraldisplaydx
a+bsinhx
(see2.441 3)
2.459 Rational functions of hyperbolic functions 131
2.457
1./integraldisplay(A+Bcoshx)dx
coshx(a+bcoshx)=1
a/bracketleftbigg
Aarctansinh x−(Ab−Ba)/integraldisplaydx
a+bcoshx/bracketrightbigg
(see2.443 3)
2.458
1./integraldisplaydx
a+bsinh2x
=1/radicalbig
a(b−a)arctan/parenleftBigg/radicalbigg
b
a−1t an h x/parenrightBigg/bracketleftbiggb
a>1/bracketrightbigg
=1/radicalbig
a(a−b)arctanh/parenleftBigg/radicalbigg
1−b
atanhx/parenrightBigg/bracketleftbigg
0<b
a<1o rb
a<0 and sinh2x<−a
b/bracketrightbigg
=1/radicalbig
a(a−b)arccoth/parenleftBigg/radicalbigg
1−b
atanhx/parenrightBigg/bracketleftbiggb
a<0 and sinh2x>−a
b/bracketrightbigg
MZ 195
2./integraldisplaydx
a+bcosh2x
=1/radicalbig
−a(a+b)arctan/parenleftBigg/radicalBigg
−/parenleftbigg
1+b
a/parenrightbigg
cothx/parenrightBigg/bracketleftbiggb
a<−1/bracketrightbigg
=1/radicalbig
a(a+b)arctanh/parenleftBigg/radicalbigg
1+b
acothx/parenrightBigg/bracketleftbigg
−1<b
a<0a n dc o s h2x>−a
b/bracketrightbigg
=1/radicalbig
a(a+b)arccoth/parenleftBigg/radicalbigg
1+b
acothx/parenrightBigg/bracketleftbiggb
a>0o r−1<b
a<0a n d c o s h2x<−a
b/bracketrightbigg
MZ 202
Fora2=b2=1 :
3./integraldisplaydx
1+s i n h2x=t a n h x
4./integraldisplaydx
1−sinh2x=1√
2arctanh/parenleftBig√
2t an h x/parenrightBig /bracketleftbig
sinh2x<1/bracketrightbig
=1√
2arccoth/parenleftBig√
2t an h x/parenrightBig /bracketleftbig
sinh2x>1/bracketrightbig
5./integraldisplaydx
1+c o s h2x=1√
2arccoth/parenleftBig√
2c ot h x/parenrightBig
6./integraldisplaydx
1−cosh2x=c o t h x
2.459
1./integraldisplaydx
/parenleftbig
a+bsinh2x/parenrightbig2=1
2a(b−a)/bracketleftbiggbsinhxcoshx
a+bsinh2x+(b−2a)/integraldisplaydx
a+bsinh2x/bracketrightbigg
(see2.458 1) MZ 196
132 Hyperbolic Functions 2.461
2./integraldisplaydx
/parenleftbig
a+bcosh2x/parenrightbig2=1
2a(a+b)/bracketleftbigg
−bsinhxcoshx
a+bcosh2x+( 2a+b)/integraldisplaydx
a+bcosh2x/bracketrightbigg
(see2.458 2) MZ 203
3./integraldisplaydx
/parenleftbig
a+bsinh2x/parenrightbig3=1
8pa3⎡
⎣/parenleftbigg
3−2
p2+3
p4/parenrightbigg
arctan( ptanhx)+/parenleftbigg
3−2
p2−3
p4/parenrightbiggptanhx
1+p2tanh2x
+/parenleftbigg
1+2
p2−1
p2tanh2x/parenrightbigg2ptanhx
/parenleftbig
1+p2tanh2x/parenrightbig2⎤
⎦
/bracketleftbigg
p2=b
a−1>0/bracketrightbigg
=1
8qa3⎡
⎣/parenleftbigg
3+2
q2+3
q4/parenrightbigg
arctanh ( qtanhx)+/parenleftbigg
3+2
q2−3
q4/parenrightbiggqtanhx
1−q2tanh2x
+/parenleftbigg
1−2
q2+1
q2tanh2x/parenrightbigg2qtanhx
/parenleftbig
1−q2tanh2x/parenrightbig2⎤
⎦
/bracketleftbigg
q2=1−b
a>0/bracketrightbigg
MZ 196
4./integraldisplaydx
/parenleftbig
a+bcosh2x/parenrightbig3=1
8pa3⎡
⎣/parenleftbigg
3−2
p2+3
p4/parenrightbigg
arctan( pcothx)+/parenleftbigg
3−2
p2−3
p4/parenrightbiggpcothx
1+p2coth2x
+/parenleftbigg
1+2
p2−1
p2coth2x/parenrightbigg2pcothx
/parenleftbig
1+p2coth2x/parenrightbig2⎤
⎦
/bracketleftbigg
p2=−1−b
a>0/bracketrightbigg
=1
8qa3⎡
⎣/parenleftbigg
3+2
q2+3
q4/parenrightbigg
ϕ(x)∗+/parenleftbigg
3+2
q2−3
q4/parenrightbiggqcothx
1−q2coth2x
+/parenleftbigg
1−2
q2+1
q2coth2x/parenrightbigg2qcothx
/parenleftbig
1−q2coth2x/parenrightbig2⎤
⎦
/bracketleftbigg
q2=1+b
a>0/bracketrightbigg
2.46 Algebraic functions of hyperbolic functions
2.461
1./integraldisplay√
tanhxdx=a r c t a n h√
tanhx−arctan√
tanhx MZ 221
∗In 2.459.4, ifb
a<0a n dc o s h2x>−a
b,t h e n ϕ(x) = arctanh ( qcothx). Ifb
a<0, but cosh2x<−a
b,o ri fb
a>0, then
ϕ(x) = arccoth ( qcothx).
2.462 Algebraic functions of hyperbolic functions 133
2./integraldisplay√
cothxd x= arccoth√
cothx−arctan√
cothx MZ 222
2.462
1./integraldisplaysinhxdx/radicalbig
a2+s i n h2x=a r c s i n hcoshx√
a2−1=l n/parenleftBig
coshx+/radicalbig
a2+s i n h2x/parenrightBig/bracketleftbig
a2>1/bracketrightbig
= arccoshcoshx√
1−a2=l n/parenleftBig
coshx+/radicalbig
a2+s i n h2x/parenrightBig/bracketleftbig
a2<1/bracketrightbig
=l nc o s h x/bracketleftbig
a2=1/bracketrightbig
2./integraldisplaysinhxdx/radicalbig
a2−sinh2x=a r c s i ncoshx√
a2+1/bracketleftbig
sinh2x<a2/bracketrightbig
3./integraldisplaysinhxdx/radicalbig
sinh2x−a2= arccoshcoshx√
a2+1=l n/parenleftBig
coshx+/radicalbig
sinh2x−a2/parenrightBig
/bracketleftbig
sinh2x>a2/bracketrightbig
MZ 199
4./integraldisplaycoshxdx/radicalbig
a2+s i n h2x=a r c s i n hsinhx
a=l n/parenleftBig
sinhx+/radicalbig
a2+s i n h2x/parenrightBig
5./integraldisplaycoshxdx/radicalbig
a2−sinh2x=a r c s i nsinhx
a/bracketleftbig
sinh2x<a2/bracketrightbig
6./integraldisplaycoshxdx/radicalbig
sinh2x−a2= arccoshsinhx
a=l n/parenleftBig
sinhx+/radicalbig
sinh2x−a2/parenrightBig
/bracketleftbig
sinh2x>a2/bracketrightbig
7./integraldisplaysinhxdx/radicalbig
a2+c o s h2x=a r c s i n hcoshx
a=l n/parenleftBig
coshx+/radicalbig
a2+c o s h2x/parenrightBig
8./integraldisplaysinhxdx/radicalbig
a2−cosh2x=a r c s i ncoshx
a/bracketleftbig
cosh2x<a2/bracketrightbig
9./integraldisplaysinhxdx/radicalbig
cosh2x−a2= arccoshcoshx
a=l n/parenleftBig
coshx+/radicalbig
cosh2x−a2/parenrightBig
/bracketleftbig
cosh2x>a2/bracketrightbig
MZ 215, 216
10./integraldisplaycoshxdx/radicalbig
a2+c o s h2x=a r c s i n hsinhx√
a2+1=l n/parenleftBig
sinhx+/radicalbig
a2+c o s h2x/parenrightBig
11./integraldisplaycoshxdx/radicalbig
a2−cosh2x=a r c s i nsinhx√
a2−1/bracketleftbig
cosh2x<a2/bracketrightbig
12./integraldisplaycoshxdx/radicalbig
cosh2x−a2= arccoshsinhx√
a2−1/bracketleftbig
a2>1/bracketrightbig
=l ns i n h x/bracketleftbig
a2=1/bracketrightbig
MZ 206
134 Hyperbolic Functions 2.463
13./integraldisplaycothxdx√
a+bsinhx=2√aarccoth/radicalbigg
1+b
asinhx [bsinhx>0,a > 0]
=2√aarctanh/radicalbigg
1+b
asinhx [bsinhx<0,a > 0]
=2√
−aarctanh/radicalBigg
−/parenleftbigg
1+b
asinhx/parenrightbigg
a<0
14./integraldisplaytanhxdx√
a+bcoshx=2√aarccoth/radicalbigg
1+b
acoshx [bcoshx>0,a > 0]
=2√aarctanh/radicalbigg
1+b
acoshx [bcoshx<0,a > 0]
=2√
−aarctanh/radicalBigg
−/parenleftbigg
1+b
acoshx/parenrightbigg
[a<0]
MZ 220, 221
2.463
1./integraldisplaysinhx√
a+bcoshx
p+qcoshxdx
=2/radicalBigg
aq−bp
qarccoth/radicalBigg
q(a+bcoshx)
aq−bp/bracketleftbigg
bcoshx>0,aq−bp
q>0/bracketrightbigg
=2/radicalBigg
aq−bp
qarctanh/radicalBigg
q(a+bcoshx)
aq−bp/bracketleftbigg
bcoshx<0,aq−bp
q>0/bracketrightbigg
=2/radicalBigg
bp−aq
qarctanh/radicalBigg
q(a+bcoshx)
bp−aq/bracketleftbiggaq−bp
q<0/bracketrightbigg
MZ 220
2./integraldisplaycoshx√
a+bsinhx
p+qsinhxdx
=2/radicalBigg
aq−bp
qarccoth/radicalBigg
q(a+bsinhx)
aq−bp/bracketleftbigg
bsinhx>0,aq−bp
q>0/bracketrightbigg
=2/radicalBigg
aq−bp
qarctanh/radicalBigg
q(a+bsinhx)
aq−bp/bracketleftbigg
bsinhx<0,aq−bp
q>0/bracketrightbigg
=2/radicalBigg
bp−aq
qarctanh/radicalBigg
q(a+bsinhx)
bp−aq/bracketleftbiggaq−bp
q<0/bracketrightbigg
MZ 221
2.464
1./integraldisplaydx/radicalbig
k2+k/prime2cosh2x=/integraldisplaydx/radicalbig
1+k/prime2sinh2x=F(arcsin(tanh x),k)
[x>0] BY (295.00)(295.10)
2./integraldisplaydx/radicalbig
cosh2x−k2=/integraldisplaydx/radicalbig
sinh2x+k/prime2=F/parenleftbigg
arcsin/parenleftbigg1
coshx/parenrightbigg
,k/parenrightbigg
[x>0] BY (295.40)(295.30)
2.464 Algebraic functions of hyperbolic functions 135
3./integraldisplaydx/radicalbig
1−k/prime2cosh2x=F/parenleftbigg
arcsin/parenleftbiggtanhx
k/parenrightbigg
,k/parenrightbigg/bracketleftbigg
0<x< arccosh1
k/prime/bracketrightbigg
BY (295.20)
Notation :I n2.464 4–2.464 8, we set α= arccos1−sinh 2ax
1+s i n h2 ax,r=1√
2[ax >0]
4./integraldisplaydx√
sinh 2ax=1
2aF(α,r) BY (296.50)
5./integraldisplay√
sinh 2axdx =1
2a[F(α,r)−2E(α,r)] +1
a/radicalBig
sinh 2ax/parenleftbig
1+s i n h22ax/parenrightbig
1+s i n h2 axBY (296.53)
6./integraldisplaycosh22axdx
(1 + sinh 2 ax)2√
sinh 2ax=1
2aE(α,r) BY (296.51)
7./integraldisplay(1−sinh 2ax)2dx
(1 + sinh 2 ax)2√
sinh 2ax=1
2a[2E(α,r)−F(α,r)] BY (296.55)
8./integraldisplay√
sinh 2axdx
(1 + sinh 2 ax)2=1
4a[F(α,r)−E(α,r)] BY (296.54)
Notation :I n2.464 9–2.464 15, we set α=a r c s i n/radicalbigg
cosh 2 ax−1
cosh2 ax,r=1√
2[x/negationslash=0 ] :
9./integraldisplaydx√
cosh 2 ax=1
a√
2F(α,r) BY (296.00)
10./integraldisplay√
cosh2 axdx =1
a√
2[F(α,r)−2E(α,r)] +sinh 2ax
a√
cosh 2 axBY (296.03)
11./integraldisplaydx√
cosh32ax=1
a√
2[2E(α,r)−F(α,r)] BY (296.04)
12./integraldisplaydx√
cosh52ax=1
3√
2aF(α,r)+tanh 2 ax
3a√
cosh2 axBY (296.04)
13./integraldisplaysinh22axdx√
cosh 2 ax=−√
2
3aF(α,r)+1
3asinh 2ax√
cosh 2 ax BY (296.07)
14./integraldisplaytanh22axdx√
cosh 2 ax=√
2
3aF(α,r)−tanh 2 ax
3a√
cosh 2 axBY (296.05)
15./integraldisplay √
cosh 2 axdx
p2+( 1−p2)c o s h2 ax=1
a√
2Π/parenleftbig
α,p2,r/parenrightbig
BY (296.02)
Notation :I n2.464 16–2.464 20, we set:
α= arccos√
a2+b2−a−bsinhx√
a2+b2+a+bsinhx,
r=/radicalBigg
a+√
a2+b2
2√
a2+b2/bracketleftBig
a>0,b > 0,x > −arcsinha
b/bracketrightBig
16./integraldisplaydx√
a+bsinhx=1
4√
a2+b2F(α,r) BY (298.00)
136 Hyperbolic Functions 2.464
17./integraldisplay√
a+bsinhxdx=4/radicalbig
a2+b2[F(α,r)−2E(α,r)] +2bcoshx√
a+bsinhx√
a2+b2+a+bsinhxBY (298.02)
18./integraldisplay√
a+bsinhx
cosh2xdx=4/radicalbig
a2+b2E(α,r)−√
a2+b2−a
24√
a2+b2F(α,r)
−a+√
a2+b2
b·√
a2+b2−a−bsinhx√
a2+b2+a+bsinhx·√
a+bsinhx
coshx
BY (298.03)
19./integraldisplaycosh2xdx
/bracketleftbig√
a2+b2+a+bsinhx/bracketrightbig2√
a+bsinhx=1
b24√
a2+b2E(α,r) BY (298.01)
20./integraldisplay √
a+bsinhxd x
/bracketleftbig√
a2+b2−a−bsinhx/bracketrightbig2=−1
4√
a2+b2/parenleftbig√
a2+b2−a/parenrightbigE(α,r)
+b√
a2+b2−a·coshx√
a+bsinhx
a2+b2−(a+bsinhx)2
BY (298.04)
Notation :I n2.464 21–2.464 31, we set α=a r c s i n/parenleftBig
tanhx
2/parenrightBig
,r=/radicalbigg
a−b
a+b[0<b<a ,x> 0]:
21./integraldisplaydx√
a+bcoshx=2√
a+bF(α,r) BY (297.25)
22./integraldisplay√
a+bcoshxd x=2√
a+b[F(α,r)−E(α,r)] + 2 tanhx
2√
a+bcoshx BY (297.29)
23./integraldisplaycoshxdx√
a+bcoshx=2√
a+bF(α,r)−2√
a+b
bE(α,r)+2
btanhx
2√
a+bcoshx BY (297.33)
24./integraldisplaytanh2x
2√
a+bcoshxdx=2√
a+b
a−b[F(α,r)−E(α,r)] BY (297.28)
25.11/integraldisplaytanh4x
2√
a+bcoshxdx=2√
a+b
3(a−b)2[(3a+b)F(α,r)−4aE(α,r)] +2
3(a−b)sinhx
2√
a+bcoshx
cosh3x
2
BY (297.28)
26./integraldisplaycoshx−1√
a+bcoshxdx=2
b/bracketleftBig/parenleftBig
tanhx
2/parenrightBig√
a+bcoshx−√
a+bE(α,r)/bracketrightBig
BY (297.31)
27./integraldisplay(coshx−1)2
√
a+bcoshxdx=4√
a+b
3b2[(a+3b)E(α,r)−bF(α,r)]
+4
3b2/bracketleftBig
bcosh2x
2−(a+3b)/bracketrightBig
tanhx
2√
a+bcoshx
BY (297.31)
28./integraldisplay√
a+bcoshx
coshx+1dx=√
a+bE(α,r) BY (297.26)
29./integraldisplaydx
(coshx+1 )√
a+bcoshx=√
a+b
a−bE(α,r)−2b
(a−b)√
a+bF(α,r) BY (297.30)
2.464 Algebraic functions of hyperbolic functions 137
30./integraldisplaydx
(coshx+1 )2√
a+bcoshx=1
3(a−b)2√
a+b/bracketleftbigg
b(5b−a)F(α,r)
+(a−3b)(a+b)E(α,r)/bracketrightbigg
+1
6(a−b)·sinhx
2
cosh3x
2√
a+bcoshx
297.30)
31./integraldisplay(1 + cosh x)dx
[1 +p2+( 1−p2)c o s h x]√
a+bcoshx=2√
a+bΠ/parenleftbig
α,p2,r/parenrightbig
BY (297.27)
Notation :I n2.464 32–2.464 40, we set:
α=a r c s i n/radicalbigg
a−bcoshx
a−b
r=/radicalbigg
a−b
a+b/bracketleftBig
0<b<a , 0<x< arccosha
b/bracketrightBig
32./integraldisplaydx√
a−bcoshx=2√
a+bF(α,r) BY (297.50)
33./integraldisplay√
a−bcoshxd x=2√
a+b[F(α,r)−E(α,r)] BY (297.54)
34./integraldisplaycoshxdx√
a−bcoshx=2√
a+b
bE(α,r)−2√
a+bF(α,r) BY (297.56)
35./integraldisplaycosh2xdx√
a−bcoshx=2(b−2a)
3b√
a+bF(α,r)+4a√
a+b
3b2E(α,r)+2
3bsinhx√
a−bcoshx BY (297.56)
36./integraldisplay(1 + cosh x)dx√
a−bcoshx=2√
a+b
bE(α,r) BY (297.51)
37./integraldisplaydx
coshx√
a−bcoshx=2b
a√
a+bΠ/parenleftbigg
α,a−b
a,r/parenrightbigg
BY (297.57)
38./integraldisplaydx
(1 + cosh x)√
a−bcoshx=1√
a+bE(α,r)−1
a+btanhx
2√
a−bcoshx BY (297.58)
39./integraldisplaydx
(1 + cosh x)2√
a−bcoshx=1
3/radicalbig
(a+b)3[(a+3b)E(α,r)−bF(α,r)]
−1
3(a+b)2tanhx
2√
a−bcoshx
coshx+1[2a+4b+(a+3b)cosh x]
BY (297.58)
40./integraldisplaydx
(a−b−ap2+bp2coshx)√
a−bcoshx=2
(a−b)√
a+bΠ/parenleftbig
α,p2,r/parenrightbig
BY (297.52)
Notation :I n2.464 41 –2.464 47, we set:
α=a r c s i n/radicalbigg
b(coshx−1)
bcoshx−a,
r=/radicalbigg
a+b
2b[0<a<b ,x> 0]
138 Hyperbolic Functions 2.464
41./integraldisplaydx√
bcosh−a=/radicalbigg
2
bF(α,r) BY (297.00)
42./integraldisplay√
bcoshx−ad x=(b−a)/radicalbigg
2
bF(α,r)−2√
2bE(α,r)+2bsinhx√
bcoshx−aBY (297.05)
43./integraldisplaydx/radicalBig
(bcoshx−a)3=1
b2−a2·/radicalbigg
2
b[2bE(α,r)−(b−a)F(α,r)] BY (297.06)
44./integraldisplaydx/radicalBig
(bcoshx−a)5=1
3(b2−a2)2/radicalbigg
2
b[(b−3a)(b−a)F(α,r)+8abE(α,r)]
+2b
3(b2−a2)·sinhx/radicalBig
(bcoshx−a)3
BY (297.06)
45./integraldisplaycoshxdx√
bcoshx−a=/radicalbigg
2
b[F(α,r)−2E(α,r)] +2s in h x√
bcoshx−aBY (297.03)
46./integraldisplay(coshx+1 )dx/radicalBig
(bcoshx−a)3=2
b−a/radicalbigg
2
bE(α,r) BY (297.01)
47./integraldisplay √
bcoshx−ad x
p2b−a+b(1−p2)cosh x=/radicalbigg
2
bΠ/parenleftbig
α,p2,r/parenrightbig
BY (297.02)
Notation :I n2.464 48–2.464 55, we set α=a r c s i n/radicalBigg
bcoshx−a
b(coshx−1)andr=/radicalbigg
2b
a+bfor
/bracketleftBig
0<b<a ,x> arccosha
b/bracketrightBig
:
48./integraldisplaydx√
bcoshx−a=2√
a+bF(α,r) BY (297.75)
49./integraldisplay√
bcoshx−ad x=−2√
a+bE(α,r)+2c o t hx
2√
bcoshx−a BY (297.79)
50./integraldisplaycoth2x
2dx√
bcoshx−a=2√
a+b
a−bE(α,r) BY (297.76)
51./integraldisplay√
bcoshx−a
coshx−1dx=√
a+b[F(α,r)−E(α,r)] BY (297.77)
52./integraldisplaydx
(coshx−1)√
bcoshx−a=√
a+b
a−bE(α,r)−1√
a+bF(α,r) BY (297.78)
53./integraldisplaydx
(coshx−1)2√
bcoshx−a=1
3(a−b)2√
a+b/bracketleftbigg
(a−2b)(a−b)F(α,r)
+( 3a−b)(a+b)E(α,r)/bracketrightbigg
+a+b
6b(a−b)·coshx
2
sinh3x
2√
bcoshx−a
BY (297.78)
54./integraldisplaydx
(coshx+1 )√
bcoshx−a=1√
a+b[F(α,r)−E(α,r)] +2√
bcoshx−a
(a+b)sin h xBY (297.80)
2.471 Hyperbolic functions and powers 139
55./integraldisplaydx
(coshx+1 )2√
bcoshx−a=1
3/radicalbig
(a+b)3/bracketleftbigg
(a+b)F(α,r)
−(a+3b)E(α,r)/bracketrightbigg
+√
bcoshx−a
3(a+b)sin h x/parenleftbigg
2a+3b
a+b−tanh2x
2/parenrightbigg
BY (297.80)
Notation :I n2.464 56–2.464 60, we set
α= arccos4√
b2−a2
√
asinhx+bcoshx,
r=1√
2/bracketleftbigg
0<a<b , −arcsinha√
b2−a2<x/bracketrightbigg
56./integraldisplaydx√
asinhx+bcoshx=4/radicalbigg
4
b2−a2F(α,r) BY (299.00)
57./integraldisplay√
asinhx+bcoshxd x=4/radicalbig
4(b2−a2)[F(α,r)−2E(α,r)] +2(acoshx+bsinhx)√
asinhx+bcoshx
BY (299.02)
58./integraldisplaydx/radicalBig
(asinhx+bcoshx)3=4/radicalBigg
4
(b2−a2)3[2E(α,r)−F(α,r)] BY (299.03)
59./integraldisplaydx/radicalBig
(asinhx+bcoshx)5=1
34/radicalBigg
4
(b2−a2)5F(α,r)+2
3(b2−a2)·acoshx+bsinhx/radicalBig
(asinhx+bcoshx)3
BY (299.03)
60./integraldisplay/parenleftbig√
b2−a2+asinhx+bcoshx/parenrightbig
dx/radicalBig
(asinhx+bcoshx)3=24/radicalbigg
4
b2−a2E(α,r) BY (299.01)
2.47 Combinations of hyperbolic functions and powers
2.471
1./integraldisplay
xrsinhpxcoshqxdx
=1
(p+q)2/bracketleftbigg
(p+q)xrsinhp−1xcoshq−1x
−rxr−1sinhpxcoshqx+r(r+1 )/integraldisplay
xr−2sinhpxcoshqxdx
+rp/integraldisplay
xr−1sinhp−1xcoshq−1xdx+(q−1)(p+q)/integraldisplay
xrsinhpxcoshq−2xdx/bracketrightbigg
=1
(p+q)2/bracketleftbigg
(p+q)xrsinhp−1xcoshq+1x
−rxr−1sinhpxcoshqx+r(r−1)/integraldisplay
xr−2sinhpxcoshqxdx
−rq/integraldisplay
xr−1sinhp−1xcoshq−1xdx−(p−1)(p+q)/integraldisplay
xrsinhp−2xcoshqxdx/bracketrightbigg
GU (353)(1)
140 Hyperbolic Functions 2.472
2./integraldisplay
xnsinh2mxdx=(−1)m/parenleftbigg2m
m/parenrightbiggxn+1
22m(n+1 )+1
22m−1m−1/summationdisplay
k=0(−1)k/parenleftbigg2m
k/parenrightbigg/integraldisplay
xncosh(2 m−2k)xdx
3./integraldisplay
xnsinh2m+1xdx=1
22mm/summationdisplay
k=0(−1)k/parenleftbigg2m+1
k/parenrightbigg/integraldisplay
xnsinh(2 m−2k+1 )xdx
4./integraldisplay
xncosh2mxdx=/parenleftbigg2m
m/parenrightbiggxn+1
22m(n+1 )+1
22m−1m−1/summationdisplay
k=0/parenleftbigg2m
k/parenrightbigg/integraldisplay
xncosh(2 m−2k)xdx
5./integraldisplay
xncosh2m+1xdx=1
22mm/summationdisplay
k=0/parenleftbigg2m+1
k/parenrightbigg/integraldisplay
xncosh(2 m−2k+1 )xdx
2.472
1./integraldisplay
xnsinhxdx=xncoshx−n/integraldisplay
xn−1coshxdx
=xncoshx−nxn−1sinhx+n(n−1)/integraldisplay
xn−2sinhxdx
2./integraldisplay
xncoshxdx=xnsinhx−n/integraldisplay
xn−1sinhxdx
=xnsinhx−nxn−1coshx+n(n−1)/integraldisplay
xn−2coshxdx
3./integraldisplay
x2nsinhxdx=( 2n)!/braceleftBiggn/summationdisplay
k=0x2k
(2k)!coshx−n/summationdisplay
k=1x2k−1
(2k−1)!sinhx/bracerightBigg
4./integraldisplay
x2n+1sinhxdx=( 2n+1 ) !n/summationdisplay
k=0/braceleftbiggx2k+1
(2k+1 ) !coshx−x2k
(2k)!sinhx/bracerightbigg
5.11/integraldisplay
x2ncoshxdx=( 2n)!/braceleftBiggn/summationdisplay
k=0x2k
(2k)!sinhx−n/summationdisplay
k=1x2k−1
(2k−1)!coshx/bracerightBigg
6./integraldisplay
x2n+1coshxdx=( 2n+1 ) !n/summationdisplay
k=0/braceleftbiggx2k+1
(2k+1 ) !sinhx−x2k
(2k)!coshx/bracerightbigg
7./integraldisplay
xsinhxdx=xcoshx−sinhx
8./integraldisplay
x2sinhxdx=/parenleftbig
x2+2/parenrightbig
coshx−2xsinhx
9./integraldisplay
xcoshxdx=xsinhx−coshx
10./integraldisplay
x2coshxdx=/parenleftbig
x2+2/parenrightbig
sinhx−2xcoshx
2.473 Notation :z1=a+bx
1./integraldisplay
z1sinhkxdx =1
kz1coshkx−b
k2sinhkx
2.474 Hyperbolic functions and powers 141
2./integraldisplay
z1coshkxdx =1
kz1sinhkx−b
k2coshkx
3./integraldisplay
z2
1sinhkxdx =1
k/parenleftbigg
z2
1+2b2
k2/parenrightbigg
coshkx−2bz1
k2sinhkx
4./integraldisplay
z2
1coshkxdx =1
k/parenleftbigg
z2
1+2b2
k2/parenrightbigg
sinhkx−2bz1
k2coshkx
5./integraldisplay
z3
1sinhkxdx =z1
k/parenleftbigg
z2
1+6b2
k2/parenrightbigg
coshkx−3b
k2/parenleftbigg
z2
1+2b2
k2/parenrightbigg
sinhkx
6./integraldisplay
z3
1coshkxdx =z1
k/parenleftbigg
z2
1+6b2
k2/parenrightbigg
sinhkx−3b
k2/parenleftbigg
z3
1+2b2
k2/parenrightbigg
coshkx
7./integraldisplay
z4
1sinhkxdx =1
k/parenleftbigg
z4
1+12b2
k2z2
1+24b4
k4/parenrightbigg
coshkx−4bz1
k2/parenleftbigg
z2
1+6b2
k2/parenrightbigg
sinhkx
8./integraldisplay
z4
1coshkxdx =1
k/parenleftbigg
z4
1+12b2
k2z2
1+24b4
k4/parenrightbigg
sinhkx−4bz1
k2/parenleftbigg
z2
1+6b2
k2/parenrightbigg
coshkx
9./integraldisplay
z5
1sinhkxdx =z1
k/parenleftbigg
z4
1+20b2
k2z2
1+ 120b4
k4/parenrightbigg
coshkx−5b
k2/parenleftbigg
z4
1+1 2b2
k2z2
1+2 4b4
k4/parenrightbigg
sinhkx
10./integraldisplay
z5
1coshkxdx =z1
k/parenleftbigg
z4
1+2 0b2
k2z2
1+ 120b4
k4/parenrightbigg
sinhkx−5b
k2/parenleftbigg
z4
1+1 2b2
k2z2
1+2 4b4
k4/parenrightbigg
coshkx
11./integraldisplay
z6
1sinhkxdx =1
k/parenleftbigg
z6
1+3 0b2
k2z4
1+ 360b4
k4z2
1+ 720b6
k6/parenrightbigg
coshkx
−6bz1
k2/parenleftbigg
z4
1+2 0b2
k2z2
1+ 120b4
k4/parenrightbigg
sinhkx
12./integraldisplay
z6
1coshkxdx =1
k/parenleftbigg
z6
1+3 0b2
k2z4
1+ 360b4
k4z2
1+ 720b6
k6/parenrightbigg
sinhkx
−6bz1
k2/parenleftbigg
z4
1+2 0b2
k2z1+ 120b4
k4/parenrightbigg
coshkx
2.474
1./integraldisplay
xnsinh2xdx=−xn+1
2(n+1 )+n!
4⌊n/2⌋/summationdisplay
k=0/braceleftbiggxn−2k
22k(n−2k)!sinh 2x−xn−2k−1
22k+1(n−2k−1)!cosh 2 x/bracerightbigg
GU (353)(2b)
2./integraldisplay
xncosh2xdx=xn+1
2(n+1 )+n!
4⌊n/2⌋/summationdisplay
k=0/braceleftbiggxn−2k
22k(n−2k)!sinh 2x−xn−2k−1
22k+1(n−2k−1)!cosh 2 x/bracerightbigg
GU (353)(3e)
3./integraldisplay
xsinh2xdx=1
4xsinh 2x−1
8cosh 2 x−x2
4
4./integraldisplay
x2sinh2xdx=1
4/parenleftbigg
x2+1
2/parenrightbigg
sinh 2x−x
4cosh2 x−x3
6MZ 257
142 Hyperbolic Functions 2.475
5./integraldisplay
xcosh2xdx=x
4sinh 2x−1
8cosh 2 x+x2
4
6./integraldisplay
x2cosh2xdx=1
4/parenleftbigg
x2+1
2/parenrightbigg
sinh 2x−x
4cosh 2 x+x3
6MZ 261
7./integraldisplay
xnsinh3xdx
=n!
4⌊n/2⌋/summationdisplay
k=0/braceleftbiggxn−2k
(n−2k)!/parenleftbiggcosh3 x
32k+1−3c os h x/parenrightbigg
−xn−2k−1
(n−2k−1)!/parenleftbiggsinh 3x
32k+2−3s in h x/parenrightbigg/bracerightbigg
GU (353)(2f)
8./integraldisplay
xncosh3xdx
=n!
4⌊n/2⌋/summationdisplay
k=0/braceleftbiggxn−2k
(n−2k)!/parenleftbiggsinh 3x
32k+1+3s i n h x/parenrightbigg
−xn−2k−1
(n−2k−1)!/parenleftbiggcosh3 x
32k+2+3c o s h x/parenrightbigg/bracerightbigg
GU (353)(3f)
9./integraldisplay
xsinh3xdx=3
4sinhx−1
36sinh 3x−3
4xcoshx−x
12cosh 3 x
10./integraldisplay
x2sinh3xdx=−/parenleftbigg3x2
4+3
2/parenrightbigg
coshx+/parenleftbiggx2
12+1
54/parenrightbigg
cosh 3 x+3x
2sinhx−x
18sinh 3x. MZ 257
11./integraldisplay
xcosh3xdx=−3
4coshx−1
36cosh3 x+3
4xsinhx+x
12sinh 3x
12./integraldisplay
x2cosh3xdx=/parenleftbigg3
4x2+3
2/parenrightbigg
sinhx+/parenleftbiggx2
12+1
54/parenrightbigg
sinh 3x−3
2xcoshx−x
18cosh 3 x MZ 262
2.475
1./integraldisplaysinhqx
xpdx=−(p−2)sinhqx+qxsinhq−1xcoshx
(p−1)(p−2)xp−1
+q(q−1)
(p−1)(p−2)/integraldisplaysinhq−2x
xp−2dx+q2
(p−1)(p−2)/integraldisplaysinhqx
xp−2dx [p>2]
GU (353)(6a)
2./integraldisplaycoshqx
xpdx=−(p−2)coshqx+qxcoshq−1xsinhx
(p−1)(p−2)xp−1
−q(q−1)
(p−1)(p−2)/integraldisplaycoshq−2x
xp−2dx+q2
(p−1)(p−2)/integraldisplaycoshqx
xp−2dx [p>2]
GU (353)(7a)
3./integraldisplaysinhx
x2ndx=−1
x(2n−1)!/braceleftBiggn−2/summationdisplay
k=0(2k+1 ) !
x2k+1coshx+n−1/summationdisplay
k=0(2k)!
x2ksinhx/bracerightBigg
+1
(2n−1)!chi(x)
GU (353)(6b)
4./integraldisplaysinhx
x2n+1dx=−1
x(2n)!/braceleftBiggn−1/summationdisplay
k=0(2k)!
x2kcoshx+n−1/summationdisplay
k=0(2k+1 ) !
x2k+1sinhx/bracerightBigg
+1
(2n)!shi(x) GU (353)(6b)
2.476 Hyperbolic functions and powers 143
5./integraldisplaycoshx
x2ndx=−1
x(2n−1)!/braceleftBiggn−2/summationdisplay
k=0(2k+1 ) !
x2k+1sinhx+n−1/summationdisplay
k=0(2k)!
x2kcoshx/bracerightBigg
+1
(2n−1)!shi(x)
GU (353)(7b)
6./integraldisplaycoshx
x2n+1dx=−1
(2n)!x/braceleftBiggn−1/summationdisplay
k=0(2k)!
x2ksinhx+n−1/summationdisplay
k=0(2k+1 ) !
x2k+1coshx/bracerightBigg
+1
(2n)!chi(x) GU (353)(7b)
7./integraldisplaysinh2mx
xdx=1
22m−1m−1/summationdisplay
k=0(−1)k/parenleftbigg2m
k/parenrightbigg
chi(2m−2k)x+(−1)m
22m/parenleftbigg2m
m/parenrightbigg
lnx GU (353)(6c)
8./integraldisplaysinh2m+1x
xdx=1
22mm/summationdisplay
k=0(−1)k/parenleftbigg2m+1
k/parenrightbigg
shi(2m−2k+1 )x GU (353)(6d)
9./integraldisplaycosh2mx
xdx=1
22m−1m−1/summationdisplay
k=0/parenleftbigg2m
k/parenrightbigg
chi(2m−2k)x+1
22m/parenleftbigg2m
m/parenrightbigg
lnx GU (353)(7c)
10./integraldisplaycosh2m+1x
xdx=1
22mm/summationdisplay
k=0/parenleftbigg2m+1
k/parenrightbigg
chi(2m−2k+1 )x GU (353)(7c)
11./integraldisplaysinh2mx
x2dx=(−1)m−1
22mx/parenleftbigg2m
m/parenrightbigg
+1
22m−1m−1/summationdisplay
k=0(−1)k+1/parenleftbigg2m
k/parenrightbigg/braceleftbiggcosh(2 m−2k)x
x−(2m−2k)sh i(2 m−2k)x/bracerightbigg
12./integraldisplaysinh2m+1x
x2dx=1
22mm/summationdisplay
k=0(−1)k+1/parenleftbigg2m+1
k/parenrightbigg
×/braceleftbiggsinh(2 m−2k+1 )x
x−(2m−2k+1 )c h i ( 2 m−2k+1 )x/bracerightbigg
13./integraldisplaycosh2mx
x2dx
=−1
22mx/parenleftbigg2m
m/parenrightbigg
−1
22m−1m−1/summationdisplay
k=0/parenleftbigg2m
k/parenrightbigg/braceleftbiggcosh(2 m−2k)x
x−(2m−2k)sh i(2 m−2k)x/bracerightbigg
14./integraldisplaycosh2m+1x
x2dx
=−1
22mm/summationdisplay
k=0/parenleftbigg2m+1
k/parenrightbigg/braceleftbiggcosh(2 m−2k+1 )x
x−(2m−2k+1 )s h i ( 2 m−2k+1 )x/bracerightbigg
2.476
1./integraldisplaysinhkx
a+bxdx=1
b/bracketleftbigg
coshka
bshi(u)−sinhka
bchi(u)/bracketrightbigg
=1
2b/bracketleftbigg
exp/parenleftbigg
−ka
b/parenrightbigg
Ei(u)−exp/parenleftbiggka
b/parenrightbigg
Ei(−u)/bracketrightbigg/bracketleftbigg
u=k
b(a+bx)/bracketrightbigg
144 Hyperbolic Functions 2.477
2./integraldisplaycoshkx
a+bxdx=1
b/bracketleftbigg
coshka
bchi(u)−sinhka
bshi(u)/bracketrightbigg
=1
2b/bracketleftbigg
exp/parenleftbigg
−ka
b/parenrightbigg
Ei(u) + exp/parenleftbiggka
b/parenrightbigg
Ei(−u)/bracketrightbigg/bracketleftbigg
u=k
b(a+bx)/bracketrightbigg
3./integraldisplaysinhkx
(a+bx)2dx=−1
b·sinhkx
a+bx+k
b/integraldisplaycoshkx
a+bxdx (see2.476 2)
4./integraldisplaycoshkx
(a+bx)2dx=−1
b·coshkx
a+bx+k
b/integraldisplaysinhkx
a+bxdx (see2.476 1)
5./integraldisplaysinhkx
(a+bx)3dx=−sinhkx
2b(a+bx)2−kcoshkx
2b2(a+bx)+k2
2b2/integraldisplaysinhkx
a+bxdx
(see2.476 1)
6./integraldisplaycoshkx
(a+bx)3dx=−coshkx
2b(a+bx)2−ksinhkx
2b2(a+bx)+k2
2b2/integraldisplaycoshkx
a+bxdx
(see2.476 2)
7./integraldisplaysinhkx
(a+bx)4dx=−sinhkx
3b(a+bx)3−kcoshkx
6b2(a+bx)2−k2sinhkx
6b3(a+bx)+k3
6b3/integraldisplaycoshkx
a+bxdx
(see2.476 2)
8./integraldisplaycoshkx
(a+bx)4dx=−coshkx
3b(a+bx)3−ksinhkx
6b2(a+bx)2−k2coshkx
6b3(a+bx)+k3
6b3/integraldisplaysinhkx
a+bxdx
(see2.476 1)
9./integraldisplaysinhkx
(a+bx)5dx=−sinhkx
4b(a+bx)4−kcoshkx
12b2(a+bx)3−k2sinhkx
24b3(a+bx)2
−k3coshkx
24b4(a+bx)+k4
24b4/integraldisplaysinhkx
a+bxdx
(see2.476 1)
10./integraldisplaycoshkx
(a+bx)5dx=−coshkx
4b(a+bx)4−ksinhkx
12b2(a+bx)3−k2coshkx
24b3(a+bx)2
−k3sinhkx
24b4(a+bx)+k4
24b4/integraldisplaycoshkx
a+bxdx
(see2.476 2)
11./integraldisplaysinhkx
(a+bx)6dx=−sinhkx
5b(a+bx)5−kcoshkx
20b2(a+bx)4−k2sinhkx
60b3(a+bx)3−k3coshkx
120b4(a+bx)2
−k4sinhkx
120b5(a+bx)+k5
120b5/integraldisplaycoshkx
a+bxdx
(see2.476 2)
12./integraldisplaycoshkx
(a+bx)6dx=−coshkx
5b(a+bx)5−ksinhkx
20b2(a+bx)4−k2coshkx
60b3(a+bx)3−k3sinhkx
120b4(a+bx)2
−k4coshkx
120b5(a+bx)+k5
120b5/integraldisplaysinhkx
a+bxdx
(see2.476 1)
2.477 Hyperbolic functions and powers 145
2.477
1./integraldisplayxpdx
sinhqx=−pxp−1sinhx−(q−2)xpcoshx
(q−1)(q−2)sinhq−1x+p(p−1)
(q−1)(q−2)/integraldisplayxp−2
sinhq−2xdx
−q−2
q−1/integraldisplayxpdx
sinhq−2x
[q>2] GU (353)(8a)
2./integraldisplayxpdx
coshqx=pxp−1coshx+(q−2)xpsinhx
(q−1)(q−2)coshq−1x−p(p−1)
(q−1)(q−2)/integraldisplayxp−2dx
coshq−2x
+q−2
q−1/integraldisplayxpdx
coshq−2x
[q>2] GU (353)(10a)
3./integraldisplayxn
sinhxdx=∞/summationdisplay
k=0/parenleftbig
2−22k/parenrightbig
B2k
(n+2k)(2k)!xn+2k[|x|<π , n> 0] GU(353)(8b)
4./integraldisplayxn
coshxdx=∞/summationdisplay
k=0E2kxn+2k+1
(n+2k+ 1)(2 k)!/bracketleftBig
|x|<π
2,n≥0/bracketrightBig
GU (353)(10b)
5./integraldisplaydx
xnsinhx=−[1 + (−1)n]2n−1−1
n!Bnlnx
+∞/summationdisplay
k=0
k/negationslash=n
22−22k
(2k−n)(2k)!B2kx2k−n
[|x|<π , n ≥1] GU (353)(9b)
6.11/integraldisplaydx
xncoshx=∞/summationdisplay
k=0
k/negationslash=n−1
2E2k
(2k−n+ 1)(2 k)!x2k−n+1+1
2[1 + (−1)n]+En−1
(n−1)!lnx
/bracketleftBig
|x|<π
2/bracketrightBig
GU (353)(11b)
7./integraldisplayxn
sinh2xdx=−xncothx+n∞/summationdisplay
k=022kB2k
(n+2k−1)(2k)!xn+2k−1
[n>1,|x|<π] GU (353)(8c)
8./integraldisplayxn
cosh2xdx=xntanhx−n∞/summationdisplay
k=122k/parenleftbig
22k−1/parenrightbig
B2k
(n+2k−1)(2k)!xn+2k−1
/bracketleftBig
n>1,|x|<π
2/bracketrightBig
GU (353)(10c)
9./integraldisplaydx
xnsinh2x=−cothx
xn−[1−(−1)n]2nn
(n+1 ) !Bn+1lnx
−n
xn+1∞/summationdisplay
k=0
k/negationslash=n+1
2B2k
(2k−n−1)(2k)!(2x)2k
[|x|<π] GU (353)(9c)
146 Hyperbolic Functions 2.477
10./integraldisplaydx
xncosh2x=tanhx
xn+[ 1−(−1)n]−2n/parenleftbig
2n+1−1/parenrightbig
n
(n+1 ) !Bn+1lnx
+n
xn+1∞/summationdisplay
k=1
k/negationslash=n+1
2/parenleftbig
22k−1/parenrightbig
B2k
(2k−n−1)(2k)!(2x)2k
/bracketleftBig
|x|<π
2/bracketrightBig
GU (353)(11c)
11./integraldisplayx
sinh2nxdx=n−1/summationdisplay
k=1(−1)k(2n−2)(2n−4)...(2n−2k+2 )
(2n−1)(2n−3)...(2n−2k+1 )
×/braceleftbiggxcoshx
sinh2n−2k+1x+1
(2n−2k)sin h2n−2kx/bracerightbigg
+(−1)n−1(2n−2)!!
(2n−1)!!/integraldisplayxdx
sinh2x
(see2.477 17) GU (353)(8e)
12./integraldisplayx
sinh2n−1xdx
=n−1/summationdisplay
k=1(−1)k(2n−3)(2n−5)...(2n−2k+1 )
(2n−2)(2n−4)...(2n−2k)
×/braceleftbiggxcoshx
sinh2n−2kx+1
(2n−2k−1)sinh2n−2k−1x/bracerightbigg
+(−1)n−1(2n−3)!!
(2n−2)!!/integraldisplayxdx
sinhx
(see2.477 15) GU (353)(8e)
13./integraldisplayx
cosh2nxdx=n−1/summationdisplay
k=1(2n−2)(2n−4)...(2n−2k+2 )
(2n−1)(2n−3)...(2n−2k+1 )
×/braceleftbiggxsinhx
cosh2n−2k+1x+1
(2n−2k)cosh2n−2kx/bracerightbigg
+(2n−2)!!
(2n−1)!!/integraldisplayxdx
cosh2x
(see2.477 18) GU (353)(10e)
14./integraldisplayx
cosh2n−1xdx=n−1/summationdisplay
k=1(2n−3)(2n−5)...(2n−2k+1 )
(2n−2)(2n−4)...(2n−2k)
×/braceleftbiggxsinhx
cosh2n−2kx+1
(2n−2k−1)cosh2n−2k−1x/bracerightbigg
+(2n−3)!!
(2n−2)!!/integraldisplayxdx
coshx
(see2.477 16) GU (353)(10e)
15./integraldisplayxdx
sinhx=∞/summationdisplay
k=02−22k
(2k+ 1)(2 k)!B2kx2k+1|x|<π GU (353)(8b)a
16./integraldisplayxdx
coshx=∞/summationdisplay
k=0E2kx2k+2
(2k+ 2)(2 k)!|x|<π
2GU (353)(10b)a
17./integraldisplayxdx
sinh2x=−xcothx+l ns i n h x MZ 257
18./integraldisplayxdx
cosh2x=xtanhx−ln cosh x MZ 262
19./integraldisplayxdx
sinh3x=−xcoshx
2s in h2x−1
2s in h x−1
2/integraldisplayxdx
sinhx(see2.477 15) MZ 257
2.478 Hyperbolic functions and powers 147
20./integraldisplayxdx
cosh3x=xsinhx
2c os h2x+1
2c os h x+1
2/integraldisplayxdx
coshx(see2.477 16) MZ 262
21./integraldisplayxdx
sinh4x=−xcoshx
3s in h3x−1
6s in h2x+2
3xcothx−2
3lnsinh x MZ 258
22./integraldisplayxdx
cosh4x=xsinhx
3c os h3x+1
6c os h2x+2
3xtanhx−2
3ln cosh x MZ 262
23./integraldisplayxdx
sinh5x=−xcoshx
4s in h4x−1
12 sinh3x+3xcoshx
8s in h2x+3
8s in h x+3
8/integraldisplayxdx
sinhx
(see2.477 15) MZ 258
24./integraldisplayxdx
cosh5x=xsinhx
4c os h4x+1
12 cosh3x+3xsinhx
8c os h2x+3
8c os h x+3
8/integraldisplayxdx
coshx
(see2.477 16) MZ 262
2.478
1./integraldisplayxncoshxdx
(a+bsinhx)m=−xn
(m−1)b(a+bsinhx)m−1+n
(m−1)b/integraldisplayxn−1dx
(a+bsinhx)m−1
[m/negationslash=1 ] MZ 263
2./integraldisplayxnsinhxdx
(a+bcoshx)m=−xn
(m−1)b(a+bcoshx)m−1+n
(m−1)b/integraldisplayxn−1dx
(a+bcoshx)m−1
[m/negationslash=1 ] MZ 263
3./integraldisplayxdx
1+c o s h x=xtanhx
2−2lncos hx
2
4./integraldisplayxdx
1−coshx=xcothx
2−2lns in hx
2
5./integraldisplayxsinhxdx
(1 + cosh x)2=−x
1+c o s h x+t a n hx
2
6./integraldisplayxsinhxdx
(1−coshx)2=x
1−coshx−cothx
2MZ 262-264
7./integraldisplayxdx
cosh 2 x−cos 2t=1
2s in2 t[L(u+t)−L(u−t)−2L(t)]
[u= arctan(tanh xcott),t/negationslash=±nπ]
LO III 402
8./integraldisplayxcoshxdx
cosh 2 x−cos 2t=1
2s int/bracketleftbigg
L/parenleftbiggu+t
2/parenrightbigg
−L/parenleftbiggu−t
2/parenrightbigg
+L/parenleftbigg
π−υ+t
2/parenrightbigg
+L/parenleftbiggυ−t
2/parenrightbigg
−2L/parenleftbiggt
2/parenrightbigg
−2L/parenleftbiggπ−t
2/parenrightbigg/bracketrightbigg
/bracketleftbigg
u= 2arctan/parenleftbigg
tanhx
2·cott
2/parenrightbigg
,υ= 2arctan/parenleftbigg
cothx
2·cott
2/parenrightbigg
;t/negationslash=±nπ/bracketrightbigg
LO III 403
148 Hyperbolic Functions 2.479
2.479
1./integraldisplay
xpsinh2mx
coshnxdx=m/summationdisplay
k=0(−1)m+k/parenleftBigm
k/parenrightBig/integraldisplayxpdx
coshn−2kx(see4.477 2)
2./integraldisplay
xpsinh2m+1x
coshnxdx=m/summationdisplay
k=0(−1)m+k/parenleftBigm
k/parenrightBig/integraldisplay
xpsinhx
coshn−2kxdx
[n>1] (see 2.479 3)
3./integraldisplay
xpsinhx
coshnxdx=−xp
(n−1)coshn−1x+p
n−1/integraldisplayxp−1dx
coshn−1x
[n>1] (see 2.477 2)GU (353)(12)
4./integraldisplay
xpcosh2mx
sinhnxdx=m/summationdisplay
k=0/parenleftBigm
k/parenrightBig/integraldisplayxpcoshx
sinhn−2kx(see2.477 1)
5./integraldisplay
xpcosh2m+1x
sinhnxdx=m/summationdisplay
k=0/parenleftBigm
k/parenrightBig/integraldisplayxpcoshx
sinhn−2kxdx (see2.479 6)
6./integraldisplay
xpcoshx
sinhnxdx=−xp
(n−1)sinhn−1x+p
n−1/integraldisplayxp−1dx
sinhn−1x
[n>1] (see 2.477 1)
GU (353)(13c)
7./integraldisplay
xptanhxdx=∞/summationdisplay
k=122k/parenleftbig
22k−1/parenrightbig
B2k
(2k+p)(2k)!xp+2k/bracketleftBig
p>−1,|x|<π
2/bracketrightBig
GU (353)(12d)
8./integraldisplay
xpcothxdx=∞/summationdisplay
k=022kB2k
(p+2k)(2k)!xp+2k[p≥+1,|x|<π] GU (353)(13d)
9./integraldisplayxcoshx
sinh2xdx=l nt a n hx
2−x
sinhx
10./integraldisplayxsinhx
cosh2xdx=−x
coshx+a r c t a n( s i n h x) MZ 263
2.48 Combinations of hyperbolic functions, exponentials, and powers
2.481
1./integraldisplay
eaxsinh(bx+c)dx=eax
a2−b2[asinh(bx+c)−bcosh(bx+c)]
/bracketleftbig
a2/negationslash=b2/bracketrightbig
2./integraldisplay
eaxcosh(bx+c)dx=eax
a2−b2[acosh(bx+c)−bsinh(bx+c)]
/bracketleftbig
a2/negationslash=b2/bracketrightbig
2.483 Hyperbolic functions, exponentials, and powers 149
Fora2=b2:
3./integraldisplay
eaxsinh(ax+c)dx=−1
2xe−c+1
4ae2ax+c
4./integraldisplay
e−axsinh(ax+c)dx=1
2xec+1
4ae−(2ax+c)
5./integraldisplay
eaxcosh(ax+c)dx=1
2xe−c+1
4ae2ax+c
6./integraldisplay
e−axcosh(ax+c)dx=1
2xec−1
4ae−(2ax+c)MZ 275-277
2.482
1./integraldisplay
xpeaxsinhbxdx =1
2/braceleftbigg/integraldisplay
xpe(a+b)xdx−/integraldisplay
xpe(a−b)xdx/bracerightbigg
/bracketleftbig
a2/negationslash=b2/bracketrightbig
2./integraldisplay
xpeaxcoshbxdx =1
2/braceleftbigg/integraldisplay
xpe(a+b)xdx+/integraldisplay
xpe(a−b)xdx/bracerightbigg
/bracketleftbig
a2/negationslash=b2/bracketrightbig
Fora2=b2:
3./integraldisplay
xpeaxsinhaxdx =1
2/integraldisplay
xpe2axdx−xp+1
2(p+1 )(see2.321 )
4./integraldisplay
xpe−axsinhaxdx =xp+1
2(p+1 )−1
2/integraldisplay
xpe−2axdx (see2.321 )
5./integraldisplay
xpeaxcoshaxdx =xp+1
2(p+1 )+1
2/integraldisplay
xpe2axdx (see2.321 ) MZ 276, 278
2.483
1./integraldisplay
xeaxsinhbxdx =eax
a2−b2/bracketleftbigg/parenleftbigg
ax−a2+b2
a2−b2/parenrightbigg
sinhbx−/parenleftbigg
bx−2ab
a2−b2/parenrightbigg
coshbx/bracketrightbigg
/bracketleftbig
a2/negationslash=b2/bracketrightbig
2./integraldisplay
xeaxcoshbxdx =eax
a2−b2/bracketleftbigg/parenleftbigg
ax−a2+b2
a2−b2/parenrightbigg
coshbx−/parenleftbigg
bx−2ab
a2−b2/parenrightbigg
sinhbx/bracketrightbigg
/bracketleftbig
a2/negationslash=b2/bracketrightbig
3./integraldisplay
x2eaxsinhbxdx=eax
a2−b2/braceleftBigg/bracketleftBigg
ax2−2/parenleftbig
a2+b2/parenrightbig
a2−b2x+2a/parenleftbig
a2+3b2/parenrightbig
(a2−b2)2/bracketrightBigg
sinhbx
−/bracketleftBigg
bx2−4ab
a2−b2x+2b/parenleftbig
3a2+b2/parenrightbig
(a2−b2)2/bracketrightBigg
coshx/bracerightBigg
/bracketleftbig
a2/negationslash=b2/bracketrightbig
150 Hyperbolic Functions 2.484
4./integraldisplay
x2eaxcoshbxdx=eax
a2−b2/braceleftBigg/bracketleftBigg
ax2−2/parenleftbig
a2+b2/parenrightbig
a2−b2x+2a/parenleftbig
a2+3b2/parenrightbig
(a2−b2)2/bracketrightBigg
coshbx
−/bracketleftBigg
bx2−4ab
a2−b2x+2b/parenleftbig
3a2+b2/parenrightbig
(a2−b2)2/bracketrightBigg
sinhx/bracerightBigg
/bracketleftbig
a2/negationslash=b2/bracketrightbig
Fora2=b2:
5./integraldisplay
xeaxsinhaxdx =e2ax
4a/parenleftbigg
x−1
2a/parenrightbigg
−x2
4
6./integraldisplay
xe−axsinhaxdx =e−2ax
4a/parenleftbigg
x+1
2a/parenrightbigg
+x2
4MZ 276, 278
7./integraldisplay
xeaxcoshaxdx =x2
4+e2ax
4a/parenleftbigg
x−1
2a/parenrightbigg
8./integraldisplay
xe−axcoshaxdx =x2
4−e−2ax
4a/parenleftbigg
x+1
2a/parenrightbigg
9./integraldisplay
x2eaxsinhaxdx =e2ax
4a/parenleftbigg
x2−x
a+1
2a2/parenrightbigg
−x3
6
10./integraldisplay
x2e−axsinhaxdx =e−2ax
4a/parenleftbigg
x2+x
a+1
2a2/parenrightbigg
+x3
6
11./integraldisplay
x2eaxcoshaxdx =x3
6+e2ax
4a/parenleftbigg
x2−x
a+1
2a2/parenrightbigg
2.484
1./integraldisplay
eaxsinhbxdx
x=1
2{Ei[(a+b)x]−Ei[(a−b)x]}/bracketleftbig
a2/negationslash=b2/bracketrightbig
2./integraldisplay
eaxcoshbxdx
x=1
2{Ei[(a+b)x]+E i [ ( a−b)x]}/bracketleftbig
a2/negationslash=b2/bracketrightbig
3./integraldisplay
eaxsinhbxdx
x2=−eaxsinhbx
2x+1
2{(a+b)Ei[(a+b)x]−(a−b)Ei[(a−b)x]}
/bracketleftbig
a2/negationslash=b2/bracketrightbig
4./integraldisplay
eaxcoshbxdx
x2=−eaxcoshbx
2x+1
2{(a+b)Ei[(a+b)x]+(a−b)Ei[(a−b)x]}
/bracketleftbig
a2/negationslash=b2/bracketrightbig
Fora2=b2:
5./integraldisplay
eaxsinhaxdx
x=1
2[Ei(2ax)−lnx]
6./integraldisplay
e−axsinhaxdx
x=1
2[lnx−Ei(−2ax)]
7./integraldisplay
eaxcoshaxdx
x=1
2[lnx+E i ( 2 ax)]
2.510 Powers of trigonometric functions 151
8./integraldisplay
eaxsinhaxdx
x2=−1
2x/parenleftbig
e2ax−1/parenrightbig
+aEi(2ax)
9./integraldisplay
e−axsinhaxdx
x2=−1
2x/parenleftbig
1−e−2ax/parenrightbig
+aEi(−2ax)
10./integraldisplay
eaxcoshaxdx
x2=−1
2x/parenleftbig
e2ax+1/parenrightbig
+aEi(2ax) MZ 276, 278
2.5–2.6 Trigonometric Functions
2.50 Introduction
2.501 Integrals of the form/integraldisplay
R(sinx,cosx)dxcan always be reduced to integrals of rational functions
by means of the substitution t=t a nx
2.
2.502 IfR(sinx,cosx) satisfies the relation
R(sinx,cosx)=−R(−sinx,cosx),
it is convenient to make the substitution t=c o s x.
2.503 If this function satisfies the relation
R(sinx,cosx)=−R(sinx,−cosx),
it is convenient to make the substitution t=s i nx.
2.504 If this function satisfies the relation
R(sinx,cosx)=R(−sinx,−cosx),
it is convenient to make the substitution t=t a n x.
2.51–2.52 Powers of trigonometric functions
2.510/integraldisplay
sinpxcosqxdx=−sinp−1xcosq+1x
q+1+p−1
q+1/integraldisplay
sinp−2xcosq+2xdx
=−sinp−1xcosq+1x
p+q+p−1
p+q/integraldisplay
sinp−2xcosqxdx
=sinp+1xcosq+1x
p+1+p+q+2
p+1/integraldisplay
sinp+2xcosqxdx
=sinp+1xcosq−1x
p+1+q−1
p+1/integraldisplay
sinp+2xcosq−2xdx
=sinp+1xcosq−1x
p+q+q−1
p+q/integraldisplay
sinpxcosq−2xdx
=−sinp+1xcosq+1x
q+1+p+q+2
q+1/integraldisplay
sinpxcosq+2xdx
=sinp−1xcosq−1x
p+q/braceleftbigg
sin2x−q−1
p+q−2/bracerightbigg
+(p−1)(q−1)
(p+q)(p+q−2)/integraldisplay
sinp−2xcosq−2xdx
FI II 89, TI 214
152 Trigonometric Functions 2.511
2.511
1./integraldisplay
sinpxcos2nxdx
=sinp+1x
2n+p/braceleftBigg
cos2n−1x+n−1/summationdisplay
k=1(2n−1)(2n−3)...(2n−2k+1 )c o s2n−2k−1x
(2n+p−2)(2n+p−4)...(2n+p−2k)/bracerightBigg
+(2n−1)!!
(2n+p)(2n+p−2)...(p+2 )/integraldisplay
sinpxdx
This formula is applicable for arbitrary real p, except for the following negative even integers:
−2,−4,...,−2n.I fpis a natural number and n=0 ,w eh a v e :
2./integraldisplay
sin2lxdx
=−cosx
2l/braceleftBigg
sin2l−1x+l−1/summationdisplay
k=1(2l−1)(2l−3)...(2l−2k+1 )
2k(l−1)(l−2)...(l−k)sin2l−2k−1x/bracerightBigg
+(2l−1)!!
2ll!x (see also 2.513 1)
TI (232)
3./integraldisplay
sin2l+1xdx=−cosx
2l+1/braceleftBigg
sin2lx+l−1/summationdisplay
k=02k+1l(l−1)...(l−k)
(2l−1)(2l−3)...(2l−2k−1)sin2l−2k−2x/bracerightBigg
(see also 2.513 2) TI (233)
4./integraldisplay
sinpxcos2n+1xdx=sinp+1x
2n+p+1/braceleftBigg
cos2nx+n/summationdisplay
k=12kn(n−1)...(n−k+1 )c o s2n−2kx
(2n+p−1)(2n+p−3)...(2n+p−2k+1 )/bracerightBigg
This formula is applicable for arbitrary real p, except for the negative odd integers: −1,−3,...,
−(2n+1 ) .
2.512
1./integraldisplay
cospxsin2nxdx
=−cosp+1x
2n+p/braceleftBigg
sin2n−1x+n−1/summationdisplay
k=1(2n−1)(2n−3)...(2n−2k+1 )s i n2n−2k−1x
(2n+p−2)(2n+p−4)...(2n+p−2k)/bracerightBigg
+(2n−1)!!
(2n+p)(2n+p−2)...(p+2 )/integraldisplay
cospxdx
This formula is applicable for arbitrary real p, except for the following negative even integers:
−2,−4,...,−2n.I fpis a natural number and n=0 ,w eh a v e
2./integraldisplay
cos2lxdx=sinx
2l/braceleftBigg
cos2l−1x+l−1/summationdisplay
k=1(2l−1)(2l−3)...(2l−2k+1 )
2k(l−1)(l−2)...(l−k)cos2l−2k−1x/bracerightBigg
+(2l−1)!!
2ll!x
(see also 2.513 3) TI (230)
2.513 Powers of trigonometric functions 153
3./integraldisplay
cos2l+1xdx=sinx
2l+1/braceleftBigg
cos2lx+l−1/summationdisplay
k=02k+1l(l−1)...(l−k)
(2l−1)(2l−3)...(2l−2k−1)cos2l−2k−2x/bracerightBigg
(see also 2.513 4) TI (231)
4./integraldisplay
cospxsin2n+1xdx
=−cosp+1x
2n+p+1/braceleftBigg
sin2nx+n/summationdisplay
k=12kn(n−1)...(n−k+1 )s i n2n−2kx
(2n+p−1)(2n+p−3)...(2n+p−2k+1 )/bracerightBigg
This formula is applicable for arbitrary real p, except for the following negative odd integers: −1,
−3,...,−(2n+1 ) .
2.513
1./integraldisplay
sin2nxdx=1
22n/parenleftbigg2n
n/parenrightbigg
x+(−1)n
22n−1n−1/summationdisplay
k=0(−1)k/parenleftbigg2n
k/parenrightbiggsin(2n−2k)x
2n−2k(see also 2.511 2)
TI (226)
2./integraldisplay
sin2n+1xdx=1
22n(−1)n+1n/summationdisplay
k=0(−1)k/parenleftbigg2n+1
k/parenrightbiggcos(2n+1−2k)x
2n+1−2k(see also 2.511 3)
TI (227)
3./integraldisplay
cos2nxdx=1
22n/parenleftbigg2n
n/parenrightbigg
x+1
22n−1n−1/summationdisplay
k=0/parenleftbigg2n
k/parenrightbiggsin(2n−2k)x
2n−2k
(see also 2.512 2) TI (224)
4./integraldisplay
cos2n+1xdx=1
22nn/summationdisplay
k=0/parenleftbigg2n+1
k/parenrightbiggsin(2n−2k+1 )x
2n−2k+1
(see also 2.512 3) TI (225)
5./integraldisplay
sin2xdx=−1
4sin 2x+1
2x=−1
2sinxcosx+1
2x
6./integraldisplay
sin3xdx=1
12cos 3x−3
4cosx=1
3cos3x−cosx
7./integraldisplay
sin4xdx=3x
8−sin 2x
4+sin 4x
32
=−3
8sinxcosx−1
4sin3xcosx+3
8x
8./integraldisplay
sin5xdx=−5
8cosx+5
48cos 3x−1
80cos5x
=−1
5sin4xcosx+4
15cos3x−4
5cosx
9./integraldisplay
sin6xdx=5
16x−15
64sin 2x+3
64sin 4x−1
192sin 6x
=−1
6sin5xcosx−5
24sin3xcosx−5
16sinxcosx+5
16x
154 Trigonometric Functions 2.513
10./integraldisplay
sin7xdx=−35
64cosx+7
64cos3x−7
320cos5x+1
448cos 7x
=−1
7sin6xcosx−6
35sin4xcosx+8
35cos3x−24
35cosx
11./integraldisplay
cos2xdx=1
4sin 2x+x
2=1
2sinxcosx+1
2x
12./integraldisplay
cos3xdx=1
12sin 3x+3
4sinx=s i nx−1
3sin3x
13./integraldisplay
cos4xdx=3
8x+1
4sin 2x+1
32sin 4x=3
8x+3
8sinxcosx+1
4sinxcos3x
14./integraldisplay
cos5xdx=5
8sinx+5
48sin 3x+1
80sin 5x=4
5sinx−4
15sin3x+1
5cos4xsinx
15./integraldisplay
cos6xdx=5
16x+15
64sin 2x+3
64sin 4x+1
192sin 6x
=5
16x+5
16sinxcosx+5
24sinxcos3x+1
6sinxcos5x
16./integraldisplay
cos7xdx=35
64sinx+7
64sin 3x+7
320sin 5x+1
448sin 7x
=24
35sinx−8
35sin3x+6
35sinxcos4x+1
7sinxcos6x
17./integraldisplay
sinxcos2xdx=−1
4/parenleftbigg1
3cos3x+c o s x/parenrightbigg
=−cos3x
3
18./integraldisplay
sinxcos3xdx=−cos4x
4
19./integraldisplay
sinxcos4xdx=−cos5x
5
20./integraldisplay
sin2xcosxdx=−1
4/parenleftbigg1
3sin 3x−sinx/parenrightbigg
=sin3x
3
21./integraldisplay
sin2xcos2xdx=−1
8/parenleftbigg1
4sin 4x−x/parenrightbigg
22./integraldisplay
sin2xcos3xdx=−1
16/parenleftbigg1
5sin 5x+1
3sin 3x−2s inx/parenrightbigg
=sin3x
5/parenleftbigg
cos2x+2
3/parenrightbigg
=sin3x
5/parenleftbigg5
3−sin2x/parenrightbigg
23./integraldisplay
sin2xcos4xdx=x
16+1
64sin 2x−1
64sin 4x−1
192sin 6x
24./integraldisplay
sin3xcosxdx=1
8/parenleftbigg1
4cos4x−cos2x/parenrightbigg
=sin4x
4
25./integraldisplay
sin3xcos2xdx=1
16/parenleftbigg1
5cos 5x−1
3cos3x−2c osx/parenrightbigg
=1
5cos5x−1
3cos3x
2.516 Powers of trigonometric functions 155
26./integraldisplay
sin3xcos3xdx=1
32/parenleftbigg1
6cos6x−3
2cos 2x/parenrightbigg
27./integraldisplay
sin3xcos4xdx=1
7cos3x/parenleftbigg
−2
5−3
5sin2x+s i n4x/parenrightbigg
28./integraldisplay
sin4xcosxdx=sin5x
5
29./integraldisplay
sin4xcos2xdx=1
16x−1
64sin2x−1
64sin4x+1
192sin 6x
30./integraldisplay
sin4xcos3xdx=1
7sin3x/parenleftbigg2
5+3
5cos2x−cos4x/parenrightbigg
31./integraldisplay
sin4xcos4xdx=3
128x−1
128sin 4x+1
1024sin 8x
2.514/integraldisplaysinpx
cos2nxdx
=sinp+1x
2n−1/braceleftBigg
sec2n−1x+n−1/summationdisplay
k=1(2n−p−2)(2n−p−4)...(2n−p−2k)
(2n−3)(2n−5)...(2n−2k−1)sec2n−2k−1x/bracerightBigg
+(2n−p−2)(2n−p−4)...(−p+2 ) (−p)
(2n−1)!!/integraldisplay
sinpxdx
This formula is applicable for arbitrary real p.F o r/integraltext
sinpxdx,w h e r e pis a natural number, see 2.511 2,
3a n d2.513 1, 2. If n=0a n d pis a negative integer, we have for this integral:
2.515
1./integraldisplaydx
sin2lx=−cosx
2l−1/braceleftBigg
cosec2l−1x+l−1/summationdisplay
k=12k(l−1)(l−2)...(l−k)
(2l−3)(2l−5)...(2l−2k−1)cosec2l−2k−1x/bracerightBigg
TI (242)
2./integraldisplaydx
sin2l+1x=−cosx
2l/braceleftBigg
cosec2lx+l−1/summationdisplay
k=1(2l−1)(2l−3)...(2l−2k+1 )
28k(l−1)(l−2)...(l−k)cosec2l−2kx/bracerightBigg
+(2l−1)!!
2ll!lntanx
2
TI (243)
2.516
1./integraldisplaysinpxdx
cos2n+1x
=sinp+1x
2n/braceleftBigg
sec2nx+n−1/summationdisplay
k=1(2n−p−1)(2n−p−3)···(2n−p−2k+1 )
2k(n−1)(n−2)···(n−k)sec2n−2kx/bracerightBigg
+(2n−p−1)(2n−p−3)···(3−p)(1−p)
2nn!/integraldisplaysinpx
cosxdx
This formula is applicable for arbitrary real p.F o r n=0a n d pa natural number, we have
2./integraldisplaysin2l+1xdx
cosx=−l/summationdisplay
k=1sin2kx
2k−ln cos x
156 Trigonometric Functions 2.517
3./integraldisplaysin2lxdx
cosx=−l/summationdisplay
k=1sin2k−1x
2k−1+l nt a n/parenleftBigπ
4+x
2/parenrightBig
2.517
1./integraldisplaydx
sin2m+1xcosx=−m/summationdisplay
k=11
(2m−2k+2 )s i n2m−2k+2x+l nt a n x
2./integraldisplaydx
sin2mxcosx=−m/summationdisplay
k=11
(2m−2k+1 )s i n2m−2k+1x+l nt a n/parenleftBigπ
4−x
2/parenrightBig
2.518
1./integraldisplaysinpx
cos2xdx=sinp−1x
cosx−(p−1)/integraldisplay
sinp−2xdx
2./integraldisplaycospxdx
sin2nx=cosp+1x
2n−1⎧
⎨
⎩cosec2n−1x
+n−1/summationdisplay
k=1(2n−p−2)(2n−p−4)...(2n−p−2k)
(2n−3)(2n−5)...(2n−2k−1)cosec2n−2k−1x⎫
⎬
⎭
+(2n−p−2)(2n−p−4)...(2−p)(−p)
(2n−1)!!/integraldisplay
cospxdx
This formula is applicable for arbitrary real p.F o r/integraltext
cospxdxwhere pis a natural number, see
2.512 2, 3 and 2.513 3, 4. If n=0a n d pis a negative integer, we have for this integral:
2.519
1./integraldisplaydx
cos2lx=sinx
2l−1/braceleftBigg
sec2l−1x+l−1/summationdisplay
k=12k(l−1)(l−2)...(l−k)
(2l−3)(2l−5)...(2l−2k−1)sec2l−2k−1x/bracerightBigg
TI (240)
2./integraldisplaydx
cos2l+1x=sinx
2l/braceleftBigg
sec2lx+l−1/summationdisplay
k=1(2l−1)(2l−3)...(2l−2k+1 )
2k(l−1)(l−2)...(l−k)sec2l−2kx/bracerightBigg
+(2l−1)!!
2ll!ln tan/parenleftBigπ
4+x
2/parenrightBig
TI (241)
2.521
1./integraldisplaycospxdx
sin2n+1x=−cosp+1x
2n⎧
⎨
⎩cosec2nx
+n−1/summationdisplay
k=1(2n−p−1)(2n−p−3)...(2n−p−2k+1 )
2k(n−1)(n−2)...(n−k)cosec2n−2kx⎫
⎬
⎭
+(2n−p−1)(2n−p−3)...(3−p)(1−p)
2n·n!/integraldisplaycospx
sinxdx
This formula is applicable for arbitrary real p.F o r n=0a n d pa natural number, we have
2.526 Powers of trigonometric functions 157
2./integraldisplaycos2l+1xdx
sinx=l/summationdisplay
k=1cos2kx
2k+l ns i n x
3./integraldisplaycos2lxdx
sinx=l/summationdisplay
k=1cos2k−1x
2k−1+l nt a nx
2
2.522
1./integraldisplaydx
sinxcos2m+1x=m/summationdisplay
k=11
(2m−2k+2 )c o s2m−2k+2x+l nt a n x
2./integraldisplaydx
sinxcos2mx=m/summationdisplay
k=11
(2m−2k+1 )c o s2m−2k+1x+l nt a nx
2GW (331)(15)
2.523/integraldisplaycosmx
sin2xdx=−cosm−1x
sinx−(m−1)/integraldisplay
cosm−2xdx
2.524 In formulas 2.524 1a n d2.524 2,s=1f o r modd and m<2n+ 1; in other cases, s=0 .
1./integraldisplaysin2n+1x
cosmxdx=n/summationdisplay
k=0
k/negationslash=m−1
2(−1)k+1/parenleftBign
k/parenrightBigcos2k−m+1x
2k−m+1+s(−1)m+1
2/parenleftbiggn
m−1
2/parenrightbigg
ln cos x
GU (331)(11d)
2./integraldisplaycos2n+1x
sinmxdx=n/summationdisplay
k=0
k/negationslash=m−1
2(−1)k/parenleftBign
k/parenrightBigsin2k−m+1x
2k−m+1+s(−1)m−1
2/parenleftbiggn
m−1
2/parenrightbigg
lnsinx
2.525
1./integraldisplaydx
sin2mxcos2nx=m+n−1/summationdisplay
k=0/parenleftbiggm+n−1
k/parenrightbiggtan2k−2m+1x
2k−2m+1TI (267)
2./integraldisplaydx
sin2m+1xcos2n+1x=m+n/summationdisplay
k=0/parenleftbiggm+n
k/parenrightbiggtan2k−2mx
2k−2m+/parenleftbiggm+n
m/parenrightbigg
ln tan x
TI (268), GU (331)(15f)
2.526
1./integraldisplaydx
sinx=l nt a nx
2
2./integraldisplaydx
sin2x=−cotx
3./integraldisplaydx
sin3x=−1
2cosx
sin2x+1
2ln tanx
2
4./integraldisplaydx
sin4x=−cosx
3s in3x−2
3cotx=−1
3cot3x−cotx
5./integraldisplaydx
sin5x=−cosx
4s in4x−3
8cosx
sin2x+3
8ln tanx
2
158 Trigonometric Functions 2.526
6./integraldisplaydx
sin6x=−cosx
5s in5x−4
15cot3x−4
5cotx
=−1
5cot5x−2
3cot3x−cotx
7./integraldisplaydx
sin7x=−cosx
6s in2x/parenleftbigg1
sin4x+5
4s in2x+15
8/parenrightbigg
+5
16ln tanx
2
8./integraldisplaydx
sin8x=−/parenleftbigg1
7cot7x+3
5cot5x+c o t3x+c o t x/parenrightbigg
9./integraldisplaydx
cosx=l nt a n/parenleftBigπ
4+x
2/parenrightBig
=l nc o t/parenleftBigπ
4−x
2/parenrightBig
=l n/radicalbigg
1+s i n x
1−sinx
10./integraldisplaydx
cos2x=t a n x
11./integraldisplaydx
cos3x=1
2sinx
cos2x+1
2lntan/parenleftBigπ
4+x
2/parenrightBig
12./integraldisplaydx
cos4x=sinx
3c os3x+2
3tanx=1
3tan3x+t a n x
13./integraldisplaydx
cos5x=sinx
4c os4x+3
8sinx
cos2x+3
8ln tan/parenleftBigx
2+π
4/parenrightBig
14./integraldisplaydx
cos6x=sinx
5c os5x+4
15tan3x+4
5tanx=1
5tan5x+2
3tan3x+t a n x
15./integraldisplaydx
cos7x=sinx
6c os6x+5s inx
24cos4x+5s inx
16cos2x+5
16lntan/parenleftBigx
2+π
4/parenrightBig
16./integraldisplaydx
cos8x=1
7tan7x+3
5tan5x+t a n3x+t a n x
17./integraldisplaysinx
cosxdx=−ln cos x
18./integraldisplaysin2x
cosxdx=−sinx+l nt a n/parenleftBigπ
4=x
2/parenrightBig
19./integraldisplaysin3x
cosxdx=−sin2x
2−lncos x=1
2cos2x−lncos x
20./integraldisplaysin4x
cosxdx=−1
3sin3x−sinx+l nt a n/parenleftBigx
2+π
4/parenrightBig
21./integraldisplaysin2xdx
cos2x=1
cosx
22./integraldisplaysin2xdx
cos2x=t a n x−x
23./integraldisplaysin3xdx
cos2x=c o s x+1
cosx
24./integraldisplaysin4xdx
cos2x=t a n x+1
2sinxcosx−3
2x
2.526 Powers of trigonometric functions 159
25./integraldisplaysinxdx
cos3x=1
2c os2x=1
2tan2x
26./integraldisplaysin2xdx
cos3x=sinx
2c os2x−1
2ln tan/parenleftBigπ
4+x
2/parenrightBig
27./integraldisplaysin3xdx
cos3x=1
2sinx
cos2x+l nc o s x
28./integraldisplaysin4xdx
cos3x=1
2sinx
cos2x+s i nx−3
2lntan/parenleftBigx
2+π
4/parenrightBig
29./integraldisplaysinxdx
cos4x=1
3c os3x
30./integraldisplaysin2xdx
cos4x=1
3tan3x
31./integraldisplaysin3xdx
cos4x=−1
cosx+1
3c os3x
32./integraldisplaysin4xdx
cos4x=1
3tan3x−tanx+x
33./integraldisplaycosxdx
sinx=l ns i n x
34./integraldisplaycos2xdx
sinx=c o s x+l nt a nx
2
35./integraldisplaycos3xdx
sinx=cos2x
2+l ns i n x
36./integraldisplaycos4xdx
sinx=1
3cos3x+c o s x+l nt a n/parenleftBigx
2/parenrightBig
37./integraldisplaycosx
sin2xdx=−1
sinx
38./integraldisplaycos2x
sin2xdx=−cotx−x
39./integraldisplaycos3x
sin2xdx=−sinx−1
sinx
40./integraldisplaycos4x
sin2xdx=−cotx−1
2sinxcosx−3
2x
41./integraldisplaycosx
sin3xdx=−1
2s in2x
42./integraldisplaycos2x
sin3xdx=−cosx
2s in2x−1
2lntanx
2
43./integraldisplaycos3x
sin3xdx=−1
2s in2x−ln sinx
44./integraldisplaycos4x
sin3xdx=−1
2cosx
sin2x−cosx−3
2lntanx
2
160 Trigonometric Functions 2.527
45./integraldisplaycosx
sin4xdx=−1
3s in3x
46./integraldisplaycos2x
sin4xdx=−1
3cot3x
47./integraldisplaycos3x
sin4xdx=1
sinx−1
3s in3x
48./integraldisplaycos4x
sin4xdx=−1
3cot3x+c o t x+x
49./integraldisplaydx
sinxcosx=l nt a n x
50./integraldisplaydx
sinxcos2x=1
cosx+l nt a nx
2
51./integraldisplaydx
sinxcos3x=1
2c os2x+l nt a n x
52./integraldisplaydx
sinxcos4x=1
cosx+1
3c os3x+l nt a nx
2
53./integraldisplaydx
sin2xcosx=l nt a n/parenleftBigπ
4+x
2/parenrightBig
−cosecx
54./integraldisplaydx
sin2xcos2x=−2c ot2 x
55./integraldisplaydx
sin2xcos3x=/parenleftbigg1
2c os2x−3
2/parenrightbigg1
sinx+3
2lntan/parenleftBigπ
4+x
2/parenrightBig
56./integraldisplaydx
sin2xcos4x=1
3s inxcos3x−8
3cot2x
57./integraldisplaydx
sin3xcosx=−1
2s in2x+l nt a n x
58./integraldisplaydx
sin3xcos2x=−1
cosx/parenleftbigg1
2s in2x−3
2/parenrightbigg
+3
2ln tanx
2
59./integraldisplaydx
sin3xcos3x=−2c os2 x
sin22x+2l nt a n x
60./integraldisplaydx
sin3xcos4x=2
cosx+1
3c os3x−cosx
2s in2x+5
2lntanx
2
61./integraldisplaydx
sin4xcosx=−1
sinx−1
3s in3x+l nt a n/parenleftBigx
2+π
4/parenrightBig
62./integraldisplaydx
sin4xcos2x=−1
3c osxsin3x−8
3cot2x
63./integraldisplaydx
sin4xcos3x=−2
sinx−1
3s in3x+sinx
2c os2x+5
2lntan/parenleftBigx
2+π
4/parenrightBig
64./integraldisplaydx
sin4xcos4x=−8c ot2 x−8
3cot32x
2.532 Sines and cosines of multiple angles and functions of the argument 161
2.527
1./integraldisplay
tanpxdx=tanp−1x
p−1−/integraldisplay
tanp−2xdx [p/negationslash=1 ]
2./integraldisplay
tan2n+1xdx=n/summationdisplay
k=1(−1)n+k/parenleftBign
k/parenrightBig1
2kcos2kx−(−1)nlncos x
=n/summationdisplay
k=1(−1)k−1tan2n−2k+2x
2n−2k+2−(−1)nln cos x
3./integraldisplay
tan2nxdx=n/summationdisplay
k=1(−1)k−1tan2n−2k+1x
2n−2k+1+(−1)nx GU (331)(12)
4./integraldisplay
cotpxdx=−cotp−1x
p−1−/integraldisplay
cotp−2xdx [p/negationslash=1 ]
5./integraldisplay
cot2n+1xdx=n/summationdisplay
k=1(−1)n+k+1/parenleftBign
k/parenrightBig1
2ksin2kx+(−1)nln sinx
=n/summationdisplay
k=1(−1)kcot2n−2k+2x
2n−2k+2+(−1)nln sinx
6./integraldisplay
cot2nxdx=n/summationdisplay
k=1(−1)kcot2n−2k+1x
2n−2k+1+(−1)nx GU (331)(14)
For special formulas for p=1 ,2 ,3 ,4 ,s e e 2.526 17,2.526 33,2.526 22,2.526 38,2.526 27,2.526 43,
2.526 32, and 2.526 48.
2.53–2.54 Sines and cosines of multiple angles and of linear and more complicated
functions of the argument
2.531
1./integraldisplay
sin(ax+b)dx=−1
acos(ax+b)
2./integraldisplay
cos(ax+b)dx=−1
asin(ax+b)
2.532
1./integraldisplay
sin(ax+b)sin (cx+d)dx=sin[(a−c)x+b−d]
2(a−c)−sin[(a+c)x+b+d]
2(a+c)
/bracketleftbig
a2/negationslash=c2/bracketrightbig
2.8/integraldisplay
sin(ax+b)cos(cx+d)dx=−cos[(a−c)x+b−d]
2(a−c)−cos[(a+c)x+b+d]
2(a+c)
/bracketleftbig
a2/negationslash=c2/bracketrightbig
162 Trigonometric Functions 2.533
3./integraldisplay
cos(ax+b)cos(cx+d)dx=sin[(a−c)x+b−d]
2(a−c)+sin[(a+c)x+b+d]
2(a+c)
/bracketleftbig
a2/negationslash=c2/bracketrightbig
Forc=a:
4./integraldisplay
sin(ax+b)sin (ax+d)dx=x
2cos(b−d)−sin(2ax+b+d)
4a
5./integraldisplay
sin(ax+b)cos(ax+d)dx=x
2sin(b−d)−cos(2ax+b+d)
4a
6./integraldisplay
cos(ax+b)cos(ax+d)dx=x
2cos(b−d)+sin(2ax+b+d)
4aGU (332)(3)
2.533
1.8/integraldisplay
sinaxcosbxdx =−cos(a+b)x
2(a+b)−cos(a−b)x
2(a−b)/bracketleftbig
a2/negationslash=b2/bracketrightbig
2.8/integraldisplay
sinaxsinbxsincxdx=−1
4⎧
⎨
⎩cos(a−b+c)x
a−b+c+cos(b+c−a)x
b+c−a
+cos(a+b−c)x
a+b−c−cos(a+b+c)x
a+b+c⎫
⎬
⎭
PE (376)
3./integraldisplay
sinaxcosbxcoscxdx=−1
4⎧
⎨
⎩cos(a+b+c)x
a+b+c−cos(b+c−a)x
b+c−a
+cos(a+b−c)x
a+b−c+cos(a+c−b)x
a+c−b⎫
⎬
⎭
PE (378)
4./integraldisplay
cosaxsinbxsincxdx=1
4⎧
⎨
⎩sin(a+b−c)x
a+b−c+sin(a+c−b)x
a+c−b
−sin(a+b+c)x
a+b+c−sin(b+c−a)x
b+c−a⎫
⎬
⎭
PE (379)
5./integraldisplay
cosaxcosbxcoscxdx=1
4⎧
⎨
⎩sin(a+b+c)x
a+b+c+sin(b+c−a)x
b+c−a
+sin(a+c−b)x
a+c−b+sin(a+b−c)x
a+b−c⎫
⎬
⎭
PE (377)
2.535 Sines and cosines of multiple angles and functions of the argument 163
2.534
1./integraldisplaycospx+isinpx
sinnxdx=−2/integraldisplayzp+n−1
1−z2ndz [z=c o s x+isinx] Pe (374)
2./integraldisplaycospx+isinpx
cosnxdx=−2i/integraldisplayzp+n−1
1−z2ndz [z=c o s x+isinx] Pe (373)
2.535
1./integraldisplay
sinpxsinaxdx =1
p+a/braceleftbigg
−sinpxcosax+p/integraldisplay
sinp−1xcos(a−1)xdx/bracerightbigg
GU (332)(5a)
2./integraldisplay
sinpxsin(2n+1 )xdx
=( 2n+1 )⎧
⎪⎪⎪⎨
⎪⎪⎪⎩/integraldisplay
sinp+1xdx+n/summationdisplay
k=1(−1)k/bracketleftbig
(2n+1 )2−12/bracketrightbig/bracketleftbig
(2n+1 )2−32/bracketrightbig
...
.../bracketleftbig
(2n+1 )2−(2k−1)2/bracketrightbig
(2k+1 ) !
×/integraldisplay
sin2k+p+1xdx⎫
⎬
⎭
TI (299)
=Γ(p+1 )
Γ/parenleftbiggp+3
2+n/parenrightbigg⎧
⎨
⎩n−1/summationdisplay
k=0⎡
⎣(−1)k−1Γ/parenleftbigp+1
2+n−2k/parenrightbig
22k+1Γ(p−2k+1 )sinp−2kxcos(2n−2k+1 )x
+(−1)kΓ/parenleftbigp−1
2+n−2k/parenrightbig
22k+2Γ(p−2k)sinp−2k−1xsin(2n−2k)x⎤
⎦
+(−1)nΓ/parenleftbigp+3
2−n/parenrightbig
22nΓ(p−2n+1 )/integraldisplay
sinp−2n+1xdx⎫
⎬
⎭
GU (332)(5c)
164 Trigonometric Functions 2.536
3./integraldisplay
sinpxsin 2nxdx =2n⎧
⎨
⎩sinp+2x
p+2
+n−1/summationdisplay
k=1(−1)k/parenleftbig
4n2−22/parenrightbig/parenleftbig
4n2−42/parenrightbig
.../bracketleftbig
4n2−(2k)2/bracketrightbig
(2k+ 1)!(2 k+p+2 )sin2k+p+2x⎫
⎬
⎭
TI (303)
=Γ(p+1 )
Γ/parenleftBigp
2+n+1/parenrightBig⎧
⎨
⎩n−1/summationdisplay
k=0(−1)k−1Γ/parenleftbigp
2+n−2k/parenrightbig
22k+1Γ(p−2k+1 )sinp−2kxcos(2n−2k)x
−(−1)kΓ/parenleftbigp
2+n−2k−1/parenrightbig
22k+2Γ(p−2k)sinp−2k−1xsin(2n−2k−1)x⎫
⎬
⎭
[pis not equal to −2,−4,...,−2n]
GU (332)(5c)
2.536
1./integraldisplay
sinpxcosaxdx =1
p+1/braceleftbigg
sinpxsinax−p/integraldisplay
sinp−1xsin(a−1)xdx/bracerightbigg
GU (332)(6a)
2./integraldisplay
sinpxcos(2n+1 )xdx
=sinp+1x
p+1+n/summationdisplay
k=1(−1)k/bracketleftbig
(2n+1 )2−12/bracketrightbig/bracketleftbig
(2n+1 )2−32/bracketrightbig
.../bracketleftbig
(2n+1 )2−(2k−1)2/bracketrightbig
(2k)!(2k+p+1 )
×sin2k+p+1x
TI (301)
=Γ(p+1 )
Γ/parenleftbigp+3
2+n/parenrightbig⎧
⎨
⎩n−1/summationdisplay
k=0⎡
⎣(−1)kΓ/parenleftbigp+1
2+n−2k/parenrightbig
22k+1Γ(p−2k+1 )sinp−2kxsin(2n−2k+1 )x
+(−1)kΓ/parenleftbigp−1
2+n−2k/parenrightbig
22k+2Γ(p−2k)sinp−2k−1xcos(2n−2k)x⎤
⎦
+(−1)nΓ/parenleftbigp+3
2−n/parenrightbig
22nΓ(p−2n+1 )/integraldisplay
sinp−2nxcosxdx⎫
⎬
⎭
[pis not equal to −3,−5,...,−(2n+1 ) ]
GU (332)(6c)
2.537 Sines and cosines of multiple angles and functions of the argument 165
3./integraldisplay
sinpxcos 2nxdx
=/integraldisplay
sinpxdx+n/summationdisplay
k=1(−1)k4n2·/parenleftbig
4n2−22/parenrightbig
.../bracketleftbig
4n2−(2k−2)2/bracketrightbig
(2k)!/integraldisplay
sin2k+pxdx
TI (300)
=Γ(p+1 )
Γ/parenleftbigp
2+n+1/parenrightbig⎧
⎨
⎩n−1/summationdisplay
k=0⎡
⎣(−1)kΓ/parenleftbigp
2+n−2k/parenrightbig
22k+1Γ(p−2k+1 )sinp−2kxsin(2n−2k)x
+(−1)kΓ/parenleftbigp
2+n−2k−1/parenrightbig
22k+2Γ(p−2k)sinp−2k−1xcos(2n−2k−1)x⎤
⎦
+(−1)nΓ/parenleftbigp
2−n+1/parenrightbig
22nΓ(p−2n+1 )/integraldisplay
sinp−2nxdx⎫
⎬
⎭
GU (332)(6c)
2.537
1./integraldisplay
cospxsinaxdx =1
p+a/braceleftbigg
−cospxcosax+p/integraldisplay
cosp−1xsin(a−1)xdx/bracerightbigg
GU (332)(7a)
2./integraldisplay
cospxsin(2n+1 )xdx
=(−1)n+1⎧
⎨
⎩cosp+1x
p+1
+n/summationdisplay
k=1(−1)k/bracketleftbig
(2n+1 )2−12/bracketrightbig/bracketleftbig
(2n+1 )2−32/bracketrightbig
.../bracketleftbig
(2n+1 )2−(2k−1)2/bracketrightbig
(2k)!(2k+p+1 )cos2k+p+1x⎫
⎬
⎭
TI (295)
=Γ(p+1 )
Γ/parenleftbiggp+3
2+n/parenrightbigg⎧
⎨
⎩−n−1/summationdisplay
k=0Γ/parenleftbigp+1
2+n−k/parenrightbig
22k+1Γ(p−2k+1 )cosp−kxcos(2n−k+1 )x
+Γ/parenleftbigp+3
2/parenrightbig
2nΓ(p−n+1 )/integraldisplay
cosp−nxsin(n+1 )xdx⎫
⎬
⎭
[pis not equal to −3,−5,...,−(2n+1 ) ]
GU (332)(7b)a
166 Trigonometric Functions 2.538
3./integraldisplay
cospxsin 2nxdx =(−1)n⎧
⎨
⎩cosp+2x
p+2
+n−1/summationdisplay
k=1(−1)k/parenleftbig
4n2−22/parenrightbig/parenleftbig
4n2−42/parenrightbig
.../bracketleftbig
4n2−(2k)2/bracketrightbig
(2k+ 1)!(2 k+p+2 )cos2k+p+2x⎫
⎬
⎭
TI (297)
=Γ(p+1 )
Γ/parenleftBigp
2+n+1/parenrightBig⎧
⎨
⎩−n−1/summationdisplay
k=0Γ/parenleftbigp
2+n−k/parenrightbig
2k+1Γ(p−k+1 )cosp−kxcos(2n−k)x
+Γ/parenleftbigp
2+1/parenrightbig
2nΓ(p−n+1 )/integraldisplay
cosp−nxsinnxdx⎫
⎬
⎭
[pis not equal to −2,−4,...,−2n]
GU (332)(7b)a
2.538
1./integraldisplay
cospxcosaxdx =1
p+a/braceleftbigg
cospxsinax+p/integraldisplay
cosp−1xcos(a−1)xdx/bracerightbigg
GU (332)(8a)
2./integraldisplay
cospxcos(2n+1 )xdx
=(−1)n(2n+1 )⎧
⎨
⎩/integraldisplay
cosp+1xdx
+n/summationdisplay
k=1(−1)k/bracketleftbig
(2n+1 )2−12/bracketrightbig/bracketleftbig
(2n+1 )2−32/bracketrightbig
.../bracketleftbig
(2n+1 )2−(2k−1)2/bracketrightbig
(2k+1 ) !
×/integraldisplay
cos2k+p+1xdx⎫
⎬
⎭
TI (293)
=Γ(p+1 )
Γ/parenleftbigp+3
2+n/parenrightbig⎧
⎨
⎩n−1/summationdisplay
k=0Γ/parenleftbigp+1
2+n−k/parenrightbig
2k+1Γ(p−k+1 )cosp−kxsin(2n−k+1 )x
+Γ/parenleftbigp+3
2/parenrightbig
2nΓ(p−n+1 )/integraldisplay
cosp−nxcos(n+1 )xdx⎫
⎬
⎭
GU (332)(8b)a
2.541 Sines and cosines of multiple angles and functions of the argument 167
3./integraldisplay
cospxcos 2nxdx
=(−1)n/braceleftBigg/integraldisplay
cospxdx+n/summationdisplay
k=1(−1)k4n2/bracketleftbig
4n2−22/bracketrightbig
.../bracketleftbig
4n2−(2k−2)2/bracketrightbig
(2k)!/integraldisplay
cos2k+pxdx/bracerightBigg
TI (294)
=Γ(p+1 )
Γ/parenleftbigp
2+n+1/parenrightbig⎧
⎨
⎩n−1/summationdisplay
k=0Γ/parenleftbigp
2+n−k/parenrightbig
2k+1Γ(p−k+1 )cosp−kxsin(2n−k)x
+Γ/parenleftbigp
2+1/parenrightbig
2nΓ(p−n+1 )/integraldisplay
cosp−nxcosnxdx⎫
⎬
⎭
GU (332)(8b)a
2.539
1./integraldisplaysin(2n+1 )x
sinxdx=2n/summationdisplay
k=1sin 2kx
2k+x
2./integraldisplaysin 2nx
sinxdx=2n/summationdisplay
k=1sin(2k−1)x
2k−1GU (332)(5e)
3./integraldisplaycos(2n+1 )x
sinxdx=2n/summationdisplay
k=1cos2kx
2k+l ns i n x
4./integraldisplaycos2nx
sinxdx=2n/summationdisplay
k=1cos(2k−1)x
2k−1+l nt a nx
2GI (332)(6e)
5./integraldisplaysin(2n+1 )x
cosxdx=2n/summationdisplay
k=1(−1)n−k+1cos2kx
2k+(−1)n+1ln cos x
6./integraldisplaysin 2nx
cosxdx=2n/summationdisplay
k=1(−1)n−k+1cos(2k−1)x
2k−1GU (332)(7d)
7./integraldisplaycos(2n+1 )x
cosxdx=2n/summationdisplay
k=1(−1)n−ksin 2kx
2k+(−1)nx
8./integraldisplaycos2nx
cosxdx=2n/summationdisplay
k=1(−1)n−ksin(2k−1)x
2k−1+(−1)nlntan/parenleftBigπ
4+x
2/parenrightBig
. GU (332)(8d)
2.541
1./integraldisplay
sin(n+1 )xsinn−1xdx=1
nsinnxsinnx BI (71)(1)a
2./integraldisplay
sin(n+1 )xcosn−1xdx=−1
ncosnxcosnx BI (71)(2)a
168 Trigonometric Functions 2.542
3./integraldisplay
cos(n+1 )xsinn−1xdx=1
nsinnxcosnx BI (71)(3)a
4./integraldisplay
cos(n+1 )xcosn−1xdx=1
ncosnxsinnx BI (71)(4)a
5./integraldisplay
sin/bracketleftBig
(n+1 )/parenleftBigπ
2−x/parenrightBig/bracketrightBig
sinn−1xdx=1
nsinnxcosn/parenleftBigπ
2−x/parenrightBig
BI (71)(5)a
6./integraldisplay
cos/bracketleftBig
(n+1 )/parenleftBigπ
2−x/parenrightBig/bracketrightBig
sinn−1xdx=−1
nsinnxsinn/parenleftBigπ
2−x/parenrightBig
BI (71)(6)a
2.542
1./integraldisplaysin 2x
sinnxdx=−2
(n−2)sinn−2x
Forn=2 :
2./integraldisplaysin 2x
sin2xdx=2l ns i n x
2.543
1./integraldisplaysin 2xdx
cosnx=2
(n−2)cosn−2x
Forn=2 :
2./integraldisplaysin 2x
cos2xdx=−2l nc o s x
2.544
1./integraldisplaycos2xdx
sinx= 2cos x+l nt a nx
2
2./integraldisplaycos2xdx
sin2x=−cotx−2x
3./integraldisplaycos2xdx
sin3x=−cosx
2s in2x−3
2ln tanx
2
4./integraldisplaycos2xdx
cosx=2s i n x−lntan/parenleftBigπ
4+x
2/parenrightBig
5./integraldisplaycos2xdx
cos2x=2x−tanx
6./integraldisplaycos2xdx
cos3x=−sinx
2c os2x+3
2lntan/parenleftBigπ
4+x
2/parenrightBig
7./integraldisplaysin 3xdx
sinx=x+s i n2 x
8./integraldisplaysin 3x
sin2xdx=3l nt a nx
2+ 4cos x
9./integraldisplaysin 3x
sin3xdx=−3c otx−4x
2.548 Sines and cosines of multiple angles and functions of the argument 169
2.545
1./integraldisplaysin 3x
cosnxdx=4
(n−3)cosn−3x−1
(n−1)cosn−1x
Forn=1a n d n=3 :
2./integraldisplaysin 3x
cosxdx=2s i n2x+l nc o s x
3./integraldisplaysin 3x
cos3xdx=−1
2c os2x−4lncos x
2.546
1./integraldisplaycos3x
sinnxdx=4
(n−3)sinn−3x−1
(n−1)sinn−1x
Forn=1a n d n=3 :
2./integraldisplaycos3x
sinxdx=−2s in2x+l ns i n x
3./integraldisplaycos3x
sin3xdx=−1
2s in2x−4l ns i n x
2.547
1./integraldisplaysinnx
cospxdx=2/integraldisplaysin(n−1)xdx
cosp−1x−/integraldisplaysin(n−2)xdx
cospx
2./integraldisplaycos3x
cosxdx=s i n2 x−x
3./integraldisplaycos3x
cos2xdx=4s i n x−3lntan/parenleftBigπ
4+x
2/parenrightBig
4./integraldisplaycos3x
cos3xdx=4x−3t anx
2.548
1./integraldisplaysinmxdx
sin(2n+1 )x=1
2n+12n/summationdisplay
k=0(−1)n+kcosm/bracketleftbigg2k+1
2(2n+1 )π/bracketrightbigg
lnsin/bracketleftbigg(k−n)π
2(2n+1 )+x
2/bracketrightbigg
sin/bracketleftbiggk+n+1
(2n+1 )π−x
2/bracketrightbigg
[ma natural number ≤2n] TI (378)
2./integraldisplaysin2mxdx
sin 2nx=(−1)n
2n/braceleftBigg
ln cos x+n−1/summationdisplay
k=1(−1)kcos2mkπ
2nln/parenleftbigg
cos2x−sin2kπ
2n/parenrightbigg/bracerightBigg
[ma natural number ≤n] TI (379)
3./integraldisplaysin2m+1x
sin 2nxdx=(−1)n
2n⎧
⎨
⎩ln tan/parenleftBigπ
4−x
2/parenrightBig
+n−1/summationdisplay
k=1(−1)kcos2m+1kπ
2nln/bracketleftbigg
tan/parenleftbiggn+k
4nπ−x
2/parenrightbigg
tan/parenleftbiggn−k
4nπ−x
2/parenrightbigg/bracketrightbigg⎫
⎬
⎭
[ma natural number <n]
170 Trigonometric Functions 2.549
4./integraldisplaysin2mxdx
cos(2n+1 )x=(−1)n+1
2n+1⎧
⎨
⎩lntan/parenleftBigπ
4−x
2/parenrightBig
+n/summationdisplay
k=1(−1)k
×cos2mkπ
2n+1ln/bracketleftbigg
tan/parenleftbigg2n+2k+1
4(2n+1 )π−x
2/parenrightbigg
tan/parenleftbigg2n−2k+1
2(2n+1 )π−x
2/parenrightbigg/bracketrightbigg⎫
⎬
⎭
[ma natural number ≤n] TI (381)
5./integraldisplaysin2m+1xdx
cos(2n+1 )x=(−1)n+1
2n+1/braceleftBigg
lncos x+n/summationdisplay
k=1(−1)kcos2m+1kπ
2n+1ln/parenleftbigg
cos2x−sin2kπ
2n+1/parenrightbigg/bracerightBigg
[ma natural number ≤n] TI (382)a
6./integraldisplaysinmxdx
cos 2nx=1
2n2n−1/summationdisplay
k=0(−1)n+kcosm/bracketleftbigg2k+1
4nπ/bracketrightbigg
lnsin/bracketleftbigg2k−2n+1
8nπ+x
2/bracketrightbigg
sin/bracketleftbigg2k+2n+1
8nπ−x
2/bracketrightbigg
[ma natural number <2n] TI (377)
7./integraldisplaycos2m+1xdx
sin(2n+1 )x=1
2n+1/braceleftBigg
ln sinx+n/summationdisplay
k=1(−1)kcos2m+1kπ
2n+1ln/parenleftbigg
sin2x−sin2kπ
2n+1/parenrightbigg/bracerightBigg
[ma natural number ≤n] TI (376)
8./integraldisplaycos2mxdx
sin(2n+1 )x=1
2n+1⎧
⎨
⎩lntanx
2
+n/summationdisplay
k=1(−1)kcos2mkπ
2n+1ln/bracketleftbigg
tan/parenleftbiggx
2+kπ
4n+2/parenrightbigg
tan/parenleftbiggx
2−kπ
4n+2/parenrightbigg/bracketrightbigg⎫
⎬
⎭
[ma natural number ≤n] TI (375)
9./integraldisplaycos2m+1x
sin 2nxdx=1
2n/braceleftBigg
lntanx
2+n−1/summationdisplay
k=1(−1)kcos2m+1kπ
2nln/bracketleftbigg
tan/parenleftbiggx
2+kπ
4/parenrightbigg
tan/parenleftbiggx
2−kπ
4n/parenrightbigg/bracketrightbigg/bracerightBigg
[ma natural number <n] TI (374)
10./integraldisplaycos2mx
sin2nxdx=1
2n/braceleftBigg
lnsinx+n−1/summationdisplay
k=1(−1)kcos2mkπ
2nln/parenleftbigg
sin2x−sin2kπ
2n/parenrightbigg/bracerightBigg
[ma natural number ≤n] TI (373)
11./integraldisplaycosmx
cosnxdx=1
nn−1/summationdisplay
k=0(−1)kcosm2k+1
2nπlnsin/bracketleftbigg2k+1
4nπ+x
2/bracketrightbigg
sin/bracketleftbigg2k+1
4nπ−x
2/bracketrightbigg
[mis a natural number ≤n]TI (372)
2.549
1./integraldisplay
sinx2dx=/radicalbiggπ
2S(x)
2.552 Rational functions of sine and cosine 171
2./integraldisplay
cosx2dx=/radicalbiggπ
2C(x)
3.11/integraldisplay
sin/parenleftbig
ax2+2bx+c/parenrightbig
dx=/radicalbiggπ
2a/braceleftbigg
cosac−b2
aS/parenleftbiggax+b√a/parenrightbigg
+s i nac−b2
aC/parenleftbiggax+b√a/parenrightbigg/bracerightbigg
[a>0]
4.11/integraldisplay
cos/parenleftbig
ax2+2bx+c/parenrightbig
dx=/radicalbiggπ
2a/braceleftbigg
cosac−b2
aC/parenleftbiggax+b√a/parenrightbigg
−sinac−b2
aS/parenleftbiggax+b√a/parenrightbigg/bracerightbigg
[a>0]
5./integraldisplay
sin lnxdx=x
2(sinln x−cosln x) PE (444)
6./integraldisplay
cosln xdx=x
2(sin ln x+ cosln x) PE (445)
2.55–2.56 Rational functions of the sine and cosine
2.551
1./integraldisplayA+Bsinx
(a+bsinx)ndx=1
(n−1)(a2−b2)⎡
⎣(Ab−aB)cosx
(a+bsinx)n−1
+/integraldisplay(Aa−Bb)(n−1) + ( aB−bA)(n−2)sinx
(a+bsinx)n−1dx⎤
⎦
TI (358)a
Forn=1 :
2./integraldisplayA+Bsinx
a+bsinxdx=B
bx+Ab−aB
b/integraldisplaydx
a+bsinx(see2.551 3) TI (342)
3./integraldisplaydx
a+bsinx=2√
a2−b2arctanatanx
2+b√
a2−b2/bracketleftbig
a2>b2/bracketrightbig
=1√
b2−a2lnatanx
2+b−√
b2−a2
atanx
2+b+/radicalbig
b2−a2/bracketleftbig
a2<b2/bracketrightbig
2.552
1./integraldisplayA+Bcosx
(a+bsinx)ndx=−B
(n−1)b(a+bsinx)n−1+A/integraldisplaydx
(a+bsinx)n
(see2.552 3) TI (361)
Forn=1 :
2./integraldisplayA+Bcosx
a+bsinxdx=B
bln(a+bsinx)+A/integraldisplaydx
a+bsinx
(see2.551 3) TI (344)
172 Trigonometric Functions 2.553
3./integraldisplaydx
(a+bsinx)n=1
(n−1)(a2−b2)⎡
⎣bcosx
(a+bsinx)n−1
+/integraldisplay(n−1)a−(n−2)bsinx
(a+bsinx)n−1dx⎤
⎦ (see2.551 1)
TI (359)
2.553
1./integraldisplayA+Bsinx
(a+bcosx)ndx=B
(n−1)b(a+bcosx)n−1+A/integraldisplaydx
(a+bcosx)n
(see2.554 3) TI (355)
Forn=1 :
2./integraldisplayA+Bsinx
a+bcosxdx=−B
bln(a+bcosx)+A/integraldisplaydx
a+bcosx
(see2.553 3∗) TI (343)
3.∗/integraldisplaydx
a+bcosx
=2√
a2−b2arctan/parenleftBigg
(a−b)tan/parenleftbigx
2/parenrightbig
√
a2−b2/parenrightBigg
/bracketleftbig
a2>b2/bracketrightbig
=2√
a2−b2ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(b−a)tan/parenleftbig
x
2/parenrightbig
+√
b2−ba
(b−a)tan/parenleftbigx
2/parenrightbig
−√
b2−ba/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbig
b
2>a2/bracketrightbig
=2√
b2−a2arctanh/parenleftBigg
(a−b)tan/parenleftbigx
2/parenrightbig
√
b2−a2/parenrightBigg/bracketleftBig
b2>a2,/vextendsingle/vextendsingle/vextendsingle(b−a)tan/parenleftBigx
2/parenrightBig/vextendsingle/vextendsingle/vextendsingle</radicalbig
b2−a2/bracketrightBig
=2√
b2−a2arccoth/parenleftBigg
(a−b)tan/parenleftbigx
2/parenrightbig
√
b2−a2/parenrightBigg/bracketleftBig
b2>a2,/vextendsingle/vextendsingle/vextendsingle(b−a)tan/parenleftBigx
2/parenrightBig/vextendsingle/vextendsingle/vextendsingle>/radicalbig
b2−a2/bracketrightBig
(compare with 2.551 3)
2.554
1./integraldisplayA+Bcosx
(a+bcosx)ndx=1
(n−1)(a2−b2)⎡
⎣(aB−Ab)sinx
(a+bcosx)n−1
+/integraldisplay(Aa−bB)(n−1) + ( n−2)(aB−bA)cosx
(a+bcosx)n−1dx⎤
⎦
TI (353)
2.557 Rational functions of sine and cosine 173
Forn=1 :
2./integraldisplayA+Bcosx
a+bcosxdx=B
bx+Ab−aB
b/integraldisplaydx
a+bcosx(see2.553 3) TI (341)
3./integraldisplaydx
(a+bcosx)n=−1
(n−1)(a2−b2)⎧
⎨
⎩bsinx
(a+bcosx)n−1
−/integraldisplay(n−1)a−(n−2)bcosx
(a+bcosx)n−1dx⎫
⎬
⎭(see2.554 1)
TI (354)
In integrating the functions in formulas 2.551 3a n d2.553 3, we may not take the integration over points
at which the integrand becomes infinite, that is, over the points x=a r c s i n/parenleftBig
−a
b/parenrightBig
in formula 2.551 3o r
over the points x= arccos/parenleftBig
−a
b/parenrightBig
in formula 2.553 3.
2.555 Formulas 2.551 3a n d2.553 3 are not applicable for a2=b2. Instead, we may use the following
formulas in these cases:
1./integraldisplayA+Bsinx
(1±sinx)ndx=−1
2n−1/braceleftBigg
2Bn−2/summationdisplay
k=0/parenleftbiggn−2
k/parenrightbiggtan2k+1/parenleftbigπ
4∓x
2/parenrightbig
2k+1
±(A∓B)n−1/summationdisplay
k=0/parenleftbiggn−1
k/parenrightbiggtan2k+1/parenleftbigπ
4∓x
2/parenrightbig
2k+1
TI (361)a
2./integraldisplayA+Bcosx
(1±cosx)ndx=1
2n−1/braceleftBigg
2Bn−2/summationdisplay
k=0/parenleftbiggn−2
k/parenrightbiggtan2k+1/bracketleftbigπ
4∓/parenleftbigπ
4−x
2/parenrightbig/bracketrightbig
2k+1
±(A∓B)n−1/summationdisplay
k=0/parenleftbiggn−1
k/parenrightbiggtan2k+1/bracketleftbigπ
4∓/parenleftbigπ
4−x
2/parenrightbig/bracketrightbig
2k+1/bracerightBigg
TI (356)
Forn=1:
3.11/integraldisplayA+Bsinx
1±sinxdx=±Bx+(B∓A)tan/parenleftBigπ
4∓x
2/parenrightBig
TI (250)
4./integraldisplayA+Bcosx
1±cosxdx=±Bx±(A∓B)tan/bracketleftBigπ
4∓/parenleftBigπ
4−x
2/parenrightBig/bracketrightBig
TI (248)
2.556
1./integraldisplay/parenleftbig
1−a2/parenrightbig
dx
1−2acosx+a2= 2arctan/parenleftbigg1+a
1−atanx
2/parenrightbigg
[0<a< 1,|x|<π] FI II 93
2./integraldisplay(1−acosx)dx
1−2acosx+a2=x
2+a r c t a n/parenleftbigg1+a
1−atanx
2/parenrightbigg
[0<a< 1,|x|<π] FI II 93
2.557
1./integraldisplaydx
(acosx+bsinx)n=1/radicalbig
(a2+b2)n/integraldisplaydx
sinn/parenleftBig
x+a r c t a na
b/parenrightBig
(see2.515 ) MZ 173a
174 Trigonometric Functions 2.558
2.6/integraldisplaysinxdx
asinx+bcosx=ax−blnsin/parenleftbig
x+a r c t a nb
a/parenrightbig
a2+b2
3./integraldisplaycosxdx
acosx+bsinx=ax+blnsin/parenleftbig
x+a r c t a na
b/parenrightbig
a2+b2MZ 174a
4./integraldisplaydx
acosx+bsinx=lntan/bracketleftbig1
2/parenleftbig
x+a r c t a na
b/parenrightbig/bracketrightbig
√
a2+b2
5./integraldisplaydx
(acosx+bsinx)2=−cot/parenleftbig
x+a r c t a na
b/parenrightbig
a2+b2=+1
a2+b2·asinx−bcosx
acosx+bsinxMZ 174a
2.558
1./integraldisplayA+Bcosx+Csinx
(a+bcosx+csinx)ndx
=(Bc−Cb)+(Ac−Ca)cosx−(Ab−Ba)sinx
(n−1)(a2−b2−c2)(a+bcosx+csinx)n−1+1
(n−1)(a2−b2−c2)
×/integraldisplay(n−1)(Aa−Bb−Cc)−(n−2)[(Ab−Ba)cosx−(Ac−Ca)sinx]
(a+bcosx+csinx)n−1dx
/bracketleftbig
n/negationslash=1,a2/negationslash=b2+c2/bracketrightbig
=Cb−Bc+Cacosx−Basinx
(n−1)a(a+bcosx+csinx)n+/parenleftbiggA
a+n(Bb+Cc)
(n−1)a2/parenrightbigg
(−ccosx+bsinx)
×(n−1)!
(2n−1)!!n−1/summationdisplay
k=0(2n−2k−3)!!
(n−k−1)!ak·1
(a+bcosx+csinx)n−k
/bracketleftbig
n/negationslash=1,a2=b2+c2/bracketrightbig
Forn=1:
2.11/integraldisplayA+Bcosx+Csinx
a+bcosx+csinxdx=Bc−Cb
b2+c2ln(a+bcosx+csinx)+Bb+Cc
b2+c2x
+/parenleftbigg
A−Bb+Cc
b2+c2a/parenrightbigg/integraldisplaydx
a+bcosx+csinx(see2.558 4)
GU (331)(18)
3./integraldisplaydx
(a+bcosx+csinx)n=/integraldisplayd(x−α)
[a+rcos(x−α)]n,
where b=rcosα,c=rsinα(see2.554 3)
4./integraldisplaydx
a+bcosx+csinx
=2√
a2−b2−c2arctan(a−b)tanx
2+c√
a2−b2−c2/bracketleftbig
a2>b2+c2/bracketrightbig
TI (253), FI II 94
=1√
b2+c2−a2ln(a−b)tanx
2+c−√
b2+c2−a2
(a−b)tanx
2+c+/radicalbig
b2+c2−a2/bracketleftbig
a2<b2+c2/bracketrightbig
TI (253)a
=1
cln/parenleftBig
a+c·tanx
2/parenrightBig
[a=b]
=−2
c+(a−b)tanx
2/bracketleftbig
a2=b2+c2/bracketrightbig
TI (253)a
2.561 Rational functions of sine and cosine 175
2.559
1./integraldisplaydx
[a(1 + cos x)+csinx]2=1
c3/bracketleftbiggc(asinx−ccosx)
a(1 + cos x)+csinx−aln/parenleftBig
a+ctanx
2/parenrightBig/bracketrightbigg
2./integraldisplayA+Bcosx+Csinx
(a1+b1cosx+c1sinx)(a2+b2cosx+c2sinx)dx
=A0lna1+b1cosx+c1sinx
a2+b2cosx+c+2s i n x+A1/integraldisplaydx
a1+b1cosx+c1sinx+A2/integraldisplaydx
a2+b2cosx+c2sinx
(see2.558 4 ) GU (331)(19)
where
A0=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleABC
a
1b1c1
a2b2c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1b1
a2b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
−/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
1c1
b2c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingle/vextendsinglec
1a1
c2a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2,A 1=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleBC
b
1c1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleAC
a
1c1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleBA
b
1a1/vextendsingle/vextendsingle/vextendsingle/vextendsingle
a
1 b1 c1
a2 b2 c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1b1
a2b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
−/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
1c1
b2c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingle/vextendsinglec
1a1
c2a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2,
A2=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleCB
c
2b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleCA
c
2a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleAB
a
2b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle
a
1 b1 c1
a2 b2 c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1b1
a2b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
−/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
1c1
b2c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingle/vextendsinglec
1a1
c2a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2,/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1b1
a2b2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingle/vextendsinglec
1a1
c2a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
/negationslash=/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
1c1
b2c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/bracketrightBigg
3./integraldisplayAcos2x+2Bsinxcosx+Csin2x
acos2x+2bsinxcosx+csin2xdx
=1
4b2+(a−c)2⎧
⎨
⎩[4Bb+(A−C)(a−c)]x+[ (A−C)b−B(a−c)]
×ln/parenleftbig
acos2x+2bsinxcosx+csin2x/parenrightbig
+/bracketleftbig
2(A+C)b2−2Bb(a+c)+(aC−Ac)(a−c)/bracketrightbig
f(x)⎫
⎬
⎭
where GU (331)(24)
f(x)=1
2√
b2−aclnctanx+b−√
b2−ac
ctanx+b+√
b2−ac/bracketleftbig
b2>a c/bracketrightbig
=1√
ac−b2arctanctanx+b√
ac−b2/bracketleftbig
b2<a c/bracketrightbig
=−1
ctanx+b/bracketleftbig
b2=ac/bracketrightbig
2.561
1./integraldisplay(A+Bsinx)dx
sinx(a+bsinx)=A
aln tanx
2+Ba−Ab
a/integraldisplaydx
a+bsinx
(see2.551 3) TI (348)
176 Trigonometric Functions 2.561
2./integraldisplay(A+Bsinx)dx
sinx(a+bcosx)=A
a2−b2/braceleftbigg
aln tanx
2+blna+bcosx
sinx/bracerightbigg
+B/integraldisplaydx
a+bcosx(see2.553 3)
TI (349)
Fora2=b2(= 1) :
3./integraldisplay(A+Bsinx)dx
sinx(a+bcosx)=A
2/braceleftbigg
lntanx
2+1
1 + cos x/bracerightbigg
+Btanx
2
4./integraldisplay(A+Bsinx)dx
sinx(1−cosx)=A
2/braceleftbigg
lntanx
2−1
1−cosx/bracerightbigg
−Bcotx
2
5./integraldisplay(A+Bsinx)dx
cosx(a+bsinx)=1
a2−b2/braceleftbigg
(Aa−Bb)lntan/parenleftBigπ
4+x
2/parenrightBig
−(Ab−aB)lna+bsinx
cosx/bracerightbigg
TI (346)
Fora2=b2(= 1):
6./integraldisplay(A+Bsinx)dx
cosx(1±sinx)=A±B
2ln tan/parenleftBigπ
4+x
2/parenrightBig
∓A∓B
2(1±sinx)
7./integraldisplay(A+Bsinx)dx
cosx(a+bcosx)=A
alntan/parenleftBigπ
4+x
2/parenrightBig
+B
alna+bcosx
cosx
−Ab
a/integraldisplaydx
a+bcosx(see2.553 3)
TI (351)a
8./integraldisplay(A+Bcosx)dx
sinx(a+bsinx)=A
alntanx
2−B
alna+bsinx
sinx−Ab
a/integraldisplaydx
a+bsinx
(see2.551 3) TI (352)
9./integraldisplay(A+Bcosx)dx
sinx(a+bcosx)=1
a2−b2/braceleftbigg
(Aa−Bb)lntanx
2+(Ab−Ba)lna+bcosx
sinx/bracerightbigg
TI (345)
Fora2=b2(= 1) :
10./integraldisplay(A+Bcosx)dx
sinx(1±cosx)=±A∓B
2(1±cosx)+A±B
2ln tanx
2
11./integraldisplay(A+Bcosx)dx
cosx(a+bsinx)=A
a2−b2/braceleftbigg
alntan/parenleftBigπ
4+x
2/parenrightBig
−blna+bsinx
cosx/bracerightbigg
+B/integraldisplaydx
a+bsinx
(see2.551 3) TI (350)
Fora2=b2(= 1):
12./integraldisplay(A+Bsinx)dx
cosx(1±sinx)=A±B
2ln tan/parenleftBigπ
4+x
2/parenrightBig
∓A∓B
2(1±sinx)
13./integraldisplay(A+Bcosx)dx
cosx(a+bcosx)=A
aln tan/parenleftBigπ
4+x
2/parenrightBig
+Ba−Ab
a/integraldisplaydx
a+bcosx
(see2.553 3) TI (347)
2.563 Rational functions of sine and cosine 177
2.562
1./integraldisplaydx
a+bsin2x=signa/radicalbig
a(a+b)arctan/parenleftBigg/radicalbigg
a+b
atanx/parenrightBigg/bracketleftbiggb
a>−1/bracketrightbigg
=signa/radicalbig
−a(a+b)arctanh/parenleftBigg/radicalbigg
−a+b
atanx/parenrightBigg/bracketleftbiggb
a<−1,sin2x<−a
b/bracketrightbigg
=signa/radicalbig
−a(a+b)arccoth/parenleftBigg/radicalbigg
−a+b
atanx/parenrightBigg/bracketleftbiggb
a<−1,sin2x>−a
b/bracketrightbigg
MZ 155
2./integraldisplaydx
a+bcos2x=−signa/radicalbig
a(a+b)arctan/parenleftBigg/radicalbigg
a+b
acotx/parenrightBigg/bracketleftbiggb
a>−1/bracketrightbigg
=−signa/radicalbig
−a(a+b)arctanh/parenleftBigg/radicalbigg
−a+b
acotx/parenrightBigg/bracketleftbiggb
a<−1,cos2x<−a
b/bracketrightbigg
=−signa/radicalbig
−a(a+b)arccoth/parenleftBigg/radicalbigg
−a+b
acotx/parenrightBigg/bracketleftbiggb
a<−1,cos2x>−a
b/bracketrightbigg
MZ 162
3./integraldisplaydx
1+s i n2x=1√
2arctan/parenleftBig√
2t anx/parenrightBig
4./integraldisplaydx
1−sin2x=t a n x
5./integraldisplaydx
1 + cos2x=−1√
2arctan/parenleftBig√
2c otx/parenrightBig
6./integraldisplaydx
1−cos2x=−cotx
2.563
1./integraldisplaydx
/parenleftbig
a+bsin2x/parenrightbig2=1
2a(a+b)/bracketleftbigg
(2a+b)/integraldisplaydx
a+bsin2x+bsinxcosx
a+bsin2x/bracketrightbigg
(see2.562 1) MZ 155
2./integraldisplaydx
(a+bcos2x)2=1
2a(a+b)/bracketleftbigg
(2a+b)/integraldisplaydx
a+bcos2x−bsinxcosx
a+bcos2x/bracketrightbigg
(see2.562 2) MZ 163
178 Trigonometric Functions 2.564
3./integraldisplaydx
/parenleftbig
a+bsin2x/parenrightbig3=1
8pa3⎡
⎣/parenleftbigg
3+2
p2+3
p4/parenrightbigg
arctan( ptanx)
+/parenleftbigg
3+2
p2−3
p4/parenrightbiggptanx
1+p2tan2x+/parenleftbigg
1−2
p2−1
p2tan2x/parenrightbigg2ptanx
/parenleftbig
1+p2tan2x/parenrightbig2⎤
⎦
/bracketleftbigg
p2=1+b
a>0/bracketrightbigg
=1
8qa3⎡
⎣/parenleftbigg
3−2
q2+3
q4/parenrightbigg
arctanh ( qtanx)
+/parenleftbigg
3−2
q2−3
q4/parenrightbiggqtanx
1−q2tan2x+/parenleftbigg
1+2
q2+1
q2tan2x/parenrightbigg2qtanx
/parenleftbig
1−q2tan2x/parenrightbig2⎤
⎦
/bracketleftbigg
q2=−1−b
a>0,sin2x<−a
b;f o r s i n2x>−a
b, change arctanh( qtanx) to arccoth( qtanx)/bracketrightbigg
MZ 156
4./integraldisplaydx
(a+bcos2x)3=−1
8pa3⎡
⎣/parenleftbigg
3+2
p2+3
p4/parenrightbigg
arctan( pcotx)
+/parenleftbigg
3+2
p2−3
p4/parenrightbiggpcotx
1+p2cot2x+/parenleftbigg
1−2
p2−1
p2cot2x/parenrightbigg2pcotx
/parenleftbig
1+p2cot2x/parenrightbig2⎤
⎦
/bracketleftbigg
p2=1+b
a>0/bracketrightbigg
=−1
8qa3⎡
⎣/parenleftbigg
3−2
q2+3
q4/parenrightbigg
arctanh( qcotx)
+/parenleftbigg
3−2
q2−3
q4/parenrightbiggqcotx
1−q2cot2x+/parenleftbigg
1+2
q2+1
q2cot2x/parenrightbigg2pcotx
/parenleftbig
1−q2cot2x/parenrightbig2⎤
⎦
/bracketleftbigg
q2=−1−b
a>0,cos2x<−a
b; for cos2x>−a
b, change arctanh( qcotx) to arccoth ( qcotx)/bracketrightbigg
MZ 163a
2.564
1./integraldisplaytanxdx
1+m2tan2x=ln/parenleftbig
cos2x+m2sin2x/parenrightbig
2(m2−1)LA 210 (10)
2./integraldisplaytanα−tanx
tanα+t a n xdx=s i n2 αln sin( x+α)−xcos 2α LA 210 (11)a
3./integraldisplaytanxdx
a+btanx=1
a2+b2{bx−aln (acosx+bsinx)} PE (335)
4./integraldisplaydx
a+btan2x=1
a−b/bracketleftBigg
x−/radicalbigg
b
aarctan/parenleftBigg/radicalbigg
b
atanx/parenrightBigg/bracketrightBigg
PE (334)
2.571 Integrals with√
a±bsinxor√
a±bcosx 179
2.57 Integrals containing√a±bsinxor√a±bcosx
Notation :
α=a r c s i n/radicalbigg
1−sinx
2,β =a r c s i n/radicalbigg
b(1−sinx)
a+b,
γ=a r c s i n/radicalbigg
b(1−cosx)
a+b,δ =a r c s i n/radicalBigg
(a+b)(1−cosx)
2(a−bcosx),r =/radicalbigg
2b
a+b
2.571
1./integraldisplaydx√
a+bsinx=−2√
a+bF(α,r)/bracketleftBig
a>b> 0,−π
2≤x<π
2/bracketrightBig
=−/radicalbigg
2
bF/parenleftbigg
β,1
r/parenrightbigg /bracketleftBig
0<|a|<b , −arcsina
b<x<π
2/bracketrightBig
BY (288.00, 288.50)
2./integraldisplaysinxdx√
a+bsinx
=2a
b√
a+bF(α,r)−2√
a+b
bE(α,r)/bracketleftBig
a>b> 0,−π
2≤x<π
2/bracketrightBig
BY (288.03)
=/radicalbigg
2
b/braceleftbigg
F/parenleftbigg
β,1
r/parenrightbigg
−2E/parenleftbigg
β,1
r/parenrightbigg/bracerightbigg/bracketleftBig
0<|a|<b , −arcsina
b<x<π
2/bracketrightBig
BY (288.54)
3./integraldisplaysin2xdx√
a+bsinx=4a√
a+b
3b2E(α,r)−2/parenleftbig
2a2+b2/parenrightbig
3b2√
a+bF(α,r)−2
3bcosx√
a+bsinx
/bracketleftBig
a>b> 0,−π
2≤x<π
2/bracketrightBig
=/radicalbigg
2
b/braceleftbigg4a
3bE/parenleftbigg
β,1
r/parenrightbigg
−2a+b
3bF/parenleftbigg
β,1
r/parenrightbigg/bracerightbigg
−2
3bcosx√
a+bsinx
/bracketleftBig
0<|a|<b , −arcsina
b<x<π
2/bracketrightBig
BY (288.03, 288.54)
4./integraldisplaydx√
a+bcosx=2√
a+bF/parenleftBigx
2,r/parenrightBig
[a>b> 0,0≤x≤π]
=/radicalbigg
2
bF/parenleftbigg
γ,1
r/parenrightbigg /bracketleftBig
b≥|a|>0,0≤x<arccos/parenleftBig
−a
b/parenrightBig/bracketrightBig
BY (289.00)
5./integraldisplaydx√
a−bcosx=2√
a+bF(δ, r)[ a>b> 0,0≤x≤π] BY (291.00)
180 Trigonometric Functions 2.572
6./integraldisplaycosxdx√
a+bcosx=2
b√
a+b/braceleftBig
(a+b)E/parenleftBigx
2,r/parenrightBig
−aF/parenleftBigx
2,r/parenrightBig/bracerightBig
[a>b> 0,0≤x≤π]
BY (289.03)
=/radicalbigg
2
b/braceleftbigg
2E/parenleftbigg
γ,1
r/parenrightbigg
−F/parenleftbigg
γ,1
r/parenrightbigg/bracerightbigg
/bracketleftBig
b>|a|>0,0≤x<arccos/parenleftBig
−a
b/parenrightBig/bracketrightBig
BY (290.04)
7.6/integraldisplaycosxdx√
a−bcosx=2
b√
a+b/braceleftbig
(b−a)Π/parenleftbig
δ, r2,r/parenrightbig
+aF(δ, r)/bracerightbig
[a>b> 0,0≤x≤π] BY (291.03)
8./integraldisplaycos2xdx√
a+bcosx=2
3b2√
a+b/braceleftBig/parenleftbig
2a2+b2/parenrightbig
F/parenleftBigx
2,r/parenrightBig
−2a(a+b)E/parenleftBigx
2,r/parenrightBig/bracerightBig
+2
3bsinx√
a+bcosx
[a>b> 0,0≤x≤π]
BY (289.03)
=1
3b/radicalbigg
2
b/braceleftbigg
(2a+b)F/parenleftbigg
γ,1
r/parenrightbigg
−4aE/parenleftbigg
γ,1
r/parenrightbigg/bracerightbigg
+2
3bsinx√
a+bcosx
/bracketleftBig
b≥|a|>0,0≤x<arccos/parenleftBig
−a
b/parenrightBig/bracketrightBig
BY (290.04)
9./integraldisplaycos2xdx√
a−bcosx=2
3b2√
a+b/braceleftbig/parenleftbig
2a2+b2/parenrightbig
F(δ, r)−2a(a+b)E(δ, r)/bracerightbig
+2
3bsinxa+bcosx√
a−bcosx[a>b> 0,]
BY (291.04)a
2.572/integraldisplaytan2xdx√
a+bsinx
=1√
a+bF(α,r)+a
(a−b)√
a+bE(α,r)
−b−asinx
(a2−b2)c o sx√
a+bsinx/bracketleftBig
0<b<a , −π
2<x<π
2/bracketrightBig
=/radicalbigg
2
b/braceleftbigg2a+b
2(a+b)F/parenleftbigg
β,1
r/parenrightbigg
+ab
a2−b2E/parenleftbigg
β,1
r/parenrightbigg/bracerightbigg
−b−asinx
(a2−b2)c o sx√
a+bsinx/bracketleftBig
0<|a|<b , −arcsina
b<x<π
2/bracketrightBig
BY(288.08, 288.58)
2.573
1./integraldisplay1−sinx
1+s i n x·dx√
a+bsinx=2
a−b/braceleftBig√
a+bE(α,r)/bracerightBig
−tan/parenleftBigπ
4−x
2/parenrightBig√
a+bsinx⎫
⎬
⎭
/bracketleftBig
0<b<a , −π
2≤x<π
2/bracketrightBig
BY (288.07)
2.575 Integrals with√
a±bsinxor√
a±bcosx 181
2./integraldisplay1−cosx
1 + cos xdx√
a+bcosx=2
a−btanx
2√
a+bcosx−2√
a+b
a−bE/parenleftBigx
2,r/parenrightBig
[a>b> 0,0≤x<π ] BY (289.07)
2.574
1./integraldisplaydx
(2−p2+p2sinx)√
a+bsinx=−1
a+bΠ/parenleftbig
α,p2,r/parenrightbig
/bracketleftBig
0<b<a , −π
2≤x<π
2/bracketrightBig
BY (288.02)
2./integraldisplaydx
(a+b−p2b+p2bsinx)√
a+bsinx=−1
a+b/radicalbigg
2
bΠ/parenleftbigg
β,p2,1
r/parenrightbigg
/bracketleftBig
0<|a|<b , −arcsina
b<x<π
2/bracketrightBig
BY (288.52)
3./integraldisplaydx
(2−p2+p2cosx)√
a+bcosx=1√
a+bΠ/parenleftBigx
2,p2,r/parenrightBig
[a>b> 0,0≤x<π ] BY (289.02)
4./integraldisplaydx
(a+b−p2b+p2bcosx)√
a+bcosx=√
2
(a+b)√
bΠ/parenleftbigg
γ,p2,1
r/parenrightbigg
/bracketleftBig
b≥|a|>0,0≤x<arccos/parenleftBig
−a
b/parenrightBig/bracketrightBig
BY (290.02)
2.575
1./integraldisplaydx/radicalBig
(a+bsinx)3=2bcosx
(a2−b2)√
a+bsinx−2
(a−b)√
a+bE(α,r)
/bracketleftBig
0<b<a , −π
2≤x<π
2/bracketrightBig
BY (288.05)
=/radicalbigg
2
b/braceleftbigg2b
b2−a2E/parenleftbigg
β,1
r/parenrightbigg
−1
a+bF/parenleftbigg
β,1
r/parenrightbigg/bracerightbigg
+2b
b2−a2·cosx√
a+bsinx/bracketleftBig
0<|a|<b , −arcsina
b<x<π
2/bracketrightBig
BY (288.56)
182 Trigonometric Functions 2.575
2./integraldisplaydx/radicalBig
(a+bsinx)5=2
3(a2−b2)2√
a+b/braceleftbig/parenleftbig
a2−b2/parenrightbig
F(α,r)−4a(a+b)E(α,r)/bracerightbig
+2b/parenleftbig
5a2−b2+4absinx/parenrightbig
3(a2−b2)2/radicalBig
(a+bsinx)3cosx
/bracketleftBig
0<b<a , −π
2≤x<π
2/bracketrightBig
BY (288.05)
=−1
3(a2−b2)2/radicalbigg
2
b/braceleftbigg
(3a−b)(a−b)F/parenleftbigg
β,1
r/parenrightbigg
+8abE/parenleftbigg
β,1
r/parenrightbigg/bracerightbigg
+2b/bracketleftbig
a2−b2+4a(a+bsinx)/bracketrightbig
3(a2−b2)2/radicalBig
(a+bsinx)3cosx
/bracketleftBig
0<|a|<b , −arcsina
b<x<π
2/bracketrightBig
BY (288.56)
3./integraldisplaydx/radicalBig
(a+bcosx)3=2
(a−b)√
a+bE/parenleftBigx
2,r/parenrightBig
−2b
a2−b2·sinx√
a+bcosx
[a>b> 0,0≤x≤π]
BY (289.05)
=1
a2−b2/radicalbigg
2
b/braceleftbigg
(a−b)F/parenleftbigg
γ,1
r/parenrightbigg
+2bE/parenleftbigg
γ,1
r/parenrightbigg/bracerightbigg
+2b
b2−a2·sinx√
a+bcosx/bracketleftBig
b≥|a|>0,0≤x<arccos/parenleftBig
−a
b/parenrightBig/bracketrightBig
BY (290.06)
4./integraldisplaydx/radicalBig
(a−bcosx)3=2
(a−b)√
a+bE(δ, r)[ a>b> 0,0≤x≤π] (291.01)
2.578 Integrals with√
a±bsinxor√
a±bcosx 183
5./integraldisplaydx/radicalBig
(a+bcosx)5=2√
a+b
3(a2−b2)2/braceleftBig
4aE/parenleftBigx
2,r/parenrightBig
−(a−b)F/parenleftBigx
2,r/parenrightBig/bracerightBig
−2b
3(a2−b2)2·5a2−b2+4abcosx/radicalBig
(a+bcosx)3sinx
[a>b> 0,0≤x≤π]
BY (289.05)
=1
3(a2−b2)2/radicalbigg
2
b/braceleftbigg
(a−b)(3a−b)F/parenleftbigg
γ,1
r/parenrightbigg
+8abE/parenleftbigg
γ,1
r/parenrightbigg/bracerightbigg
+2b/parenleftbig
5a2−b2+4abcosx/parenrightbig
sinx
3(ab−b2)2/radicalBig
(a+bcosx)3
/bracketleftBig
b≥|a|>0,0≤x<arccos/parenleftBig
−a
b/parenrightBig/bracketrightBig
BY (290.06)
2.576
1./integraldisplay√
a+bcosxdx=2√
a+bE/parenleftBigx
2,r/parenrightBig
[a>b> 0,0≤x≤π]
BY (289.01)
=/radicalbigg
2
b/braceleftbigg
(a−b)F/parenleftbigg
γ,1
r/parenrightbigg
+2bE/parenleftbigg
γ,1
r/parenrightbigg/bracerightbigg
/bracketleftBig
b≥|a|>0,0≤x<arccos/parenleftBig
−a
b/parenrightBig/bracketrightBig
BY (290.03)
2./integraldisplay√
a−bcosxdx=2√
a+bE(δ, r)−2bsinx√
a−bcosx[a>b> 0,0≤x≤π] BY (291.05)
2.577
1.3/integraldisplay√
a−bcosx
1+pcosxdx=2(a−b)
(1 +p)√
a+bΠ/parenleftbigg
δ,2ap
(a+b)(1 + p),r/parenrightbigg
[a>b> 0,0≤x≤π, p /negationslash=−1]
BY (291.02)
2.3/integraldisplay/radicalBigg
a−bcosx
1+pcosxdx=2(a−b)/radicalbig
(1 +p)(a+b)Π/parenleftBigg
δ,−r2,/radicalBigg
2(ap+b)
(1 +p)(a+b)/parenrightBigg
[a>b> 0,0≤x≤π, p /negationslash=−1]
2.578/integraldisplaytanxdx√
a+btan2x=1√
b−aarccos/parenleftbigg√
b−a√
bcosx/parenrightbigg
[b>a , b> 0] PE (333)
184 Trigonometric Functions 2.580
2.58–2.62 Integrals reducible to elliptic and pseudo-elliptic integrals
2.580
1./integraldisplaydϕ√a+bcosϕ+csinϕ=2/integraldisplaydψ/radicalbig
a−p+2pcos2ψ/bracketleftBig
ϕ=2ψ+α,tanα=c
b,p=/radicalbig
b2+c2/bracketrightBig
2./integraldisplaydϕ/radicalbig
a+bcosϕ+csinϕ+dcos2ϕ+esinϕcosϕ+fsin2ϕ=2/integraldisplaydx√
A+Bx+Cx2−Dx3+Ex4
/bracketleftBig
tanϕ
2=x, A=a+b+d, B=2c+2e,C=2a−2d+4f,D=2c−2e,E=a−b+d/bracketrightBig
Forms containing/radicalbig
1−k2sin2x
Notation :Δ=/radicalbig
1−k2sin2x,k/prime=√
1−k2
2.581
1./integraldisplay
sinmxcosnxΔrdx
=1
(m+n+r)k2⎧
⎨
⎩sinm−3xcosn+1xΔr+2+/bracketleftbig
(m+n−2) + ( m+r−1)k2/bracketrightbig
×/integraldisplay
sinm−2xcosnxΔrdx−(m−3)/integraldisplay
sinm−4xcosnxΔrdx⎫
⎬
⎭
=1
(m+n+r)k2⎧
⎨
⎩sinm+1xcosn−3xΔr+2+/bracketleftBig
(n+r−1)k2−(m+n−2)k/prime2/bracketrightBig
×/integraldisplay
sinmxcosn−2xΔrdx+(n−3)k/prime2/integraldisplay
sinmxcosn−4xΔrdx⎫
⎬
⎭
[m+n+r/negationslash=0 ]
Forr=−3a n d r=−5:
2./integraldisplaysinmxcosnx
Δ3dx=sinm−1xcosn−1x
k2Δ
−m−1
k2/integraldisplaysinm−2xcosnx
Δdx+n−1
k2/integraldisplaysinmxcosn−2x
Δdx
3./integraldisplaysinmxcosnx
Δ5dx=sinm−1xcosn−1x
3k2Δ3
−m−1
3k2/integraldisplaysinm−2xcosnx
Δ3dx+n−1
3k2/integraldisplaysinmxcosn−2x
Δ3dx
Form=1o r n=1 :
4./integraldisplay
sinxcosnxΔrdx=−cosn−1xΔr+2
(n+r+1 )k2−(n−1)k/prime2
(n+r+1 )k2/integraldisplay
cosn−2xsinxΔrdx
5./integraldisplay
sinmxcosxΔrdx=−sinm−1xΔr+2
(m+r+1 )k2+m−1
(m+r+1 )k2/integraldisplay
sinm−2xcosxΔrdx
Form=3o r n=3 :
2.583 Elliptic and pseudo-elliptic integrals 185
6./integraldisplay
sin3xcosnxΔrdx=(n+r+1 )k2cos2x−/bracketleftbig
(r+2 )k2+n+1/bracketrightbig
(n+r+1 ) (n+r+3 )k4cosn−1xΔr+2
−/bracketleftbig
(r+2 )k2+n+1/bracketrightbig
(n−1)k/prime2
(n+r+1 ) (n+r+3 )k4/integraldisplay
cosn−2xsinxΔrdx
7./integraldisplay
sinmxcos3xΔrdx
=(m+r+1 )k2sin2x−/bracketleftBig
(r+2 )k2−(m+1 )k/prime2/bracketrightBig
(m+r+1 ) (m+r+3 )k4
×sinm−1xnm−1xΔr+2+/bracketleftBig
(r+2 )k2−(m−1)k/prime2/bracketrightBig
(m−1)
(m+r+1 ) (m+r+3 )k4/integraldisplay
sinm−2xcosxΔrdx
2.582
1./integraldisplay
Δndx=n−1
n/parenleftbig
2−k2/parenrightbig/integraldisplay
Δn−2dx−n−2
n/parenleftbig
1−k2/parenrightbig/integraldisplay
Δn−4dx
+k2
nsinxcosx·Δn−2
LA (316)(1)a
2./integraldisplaydx
Δn+1=−k2sinxcosx
(n−1)k/prime2Δn−1+n−2
n−12−k2
k/prime2/integraldisplaydx
Δn−1−n−3
n−11
k/prime2/integraldisplaydx
Δn−3LA 317(8)a
3./integraldisplaysinnx
Δdx=sinn−3x
(n−1)k2cosx·Δ+n−2
n−11+k2
k2/integraldisplaysinn−2x
Δdx
−n−3
(n−1)k2/integraldisplaysinn−4x
Δdx
LA 316(1)a
4./integraldisplaycosnx
Δdx=cosn−3x
(n−1)k2sinx·Δ+n−2
n−12k2−1
k2/integraldisplaycosn−2x
Δdx
+n−3
n−1k/prime2
k2/integraldisplaycosn−4x
Δdx
LA 316(2)a
5./integraldisplaytannx
Δdx=tann−3x
(n−1)k/prime2Δ
cos2x−(n−2)/parenleftbig
2−k2/parenrightbig
(n−1)k/prime2/integraldisplaytann−2x
Δdx
−n−3
(n−1)k/prime2/integraldisplaytann−4x
Δdx
LA 317(3)
6./integraldisplaycotnx
Δdx=−cotn−1x
n−1Δ
cos2x−n−2
n−1/parenleftbig
2−k2/parenrightbig/integraldisplaycotn−2x
Δdx
−n−3
n−1k/prime2/integraldisplaycotn−4x
Δdx
LA 317(6)
2.583
1./integraldisplay
Δdx=E(x, k)
186 Trigonometric Functions 2.583
2./integraldisplay
Δsinxdx=−Δcos x
2−k/prime2
2kln(kcosx+Δ )
3./integraldisplay
Δcos xdx=Δsinx
2+1
2karcsin ( ksinx)
4./integraldisplay
Δsin2xdx=−Δ
3sinxcosx+k/prime2
3k2F(x, k)+2k2−1
3k2E(x, k)
5./integraldisplay
Δsinxcosxdx=−Δ3
3k2
6./integraldisplay
Δcos2xdx=Δ
3sinxcosx−k/prime2
3k2F(x, k)+k2+1
3k2E(x, k)
7./integraldisplay
Δsin3xdx=−2k2sin2x+3k2−1
8k2Δcos x+3k4−2k2−1
8k3ln (kcosx+Δ )
8./integraldisplay
Δsin2xcosxdx=2k2sin2x−1
8k2Δsinx+1
8k3arcsin( ksinx)
9./integraldisplay
Δsinxcos2xdx=−2k2cos2x+k/prime2
8k2Δcos x+k/prime4
8k3ln(kcosx+Δ )
10./integraldisplay
Δcos3xdx=2k2cos2x+2k2+1
8k2Δsinx+4k2−1
8k3arcsin ( ksinx)
11./integraldisplay
Δsin4xdx=−3k2sin2x+4k2−1
15k2Δsinxcosx
−2/parenleftbig
2k4−k2−1/parenrightbig
15k4F(x, k)+8k4−3k2−2
15k4E(x, k)
12./integraldisplay
Δsin3xcosxdx=3k4sin4x−k2sin2x−2
15k4Δ
13./integraldisplay
Δsin2xcos2xdx=−3k2cos2x−2k2+1
15k2Δsinxcosx
−k/prime2/parenleftBig
1+k/prime2/parenrightBig
15k4F(x, k)+2/parenleftbig
k4−k2+1/parenrightbig
15k4E(x, k)
14./integraldisplay
Δsinxcos3xdx=−3k4sin4x−k2/parenleftbig
5k2+1/parenrightbig
sin2x+5k2−2
15k4Δ
15./integraldisplay
Δcos4xdx=3k2cos2x+3k2+1
15k2Δsinxcosx
+2k/prime2/parenleftBig
k/prime2−2k2/parenrightBig
15k4F(x, k)+3k4+7k2−2
15k4E(x, k)
16./integraldisplay
Δsin5xdx=−8k4sin4x−2k2/parenleftbig
5k2−1/parenrightbig
sin2x−15k4+4k2+3
48k4Δcos x
+5k6−3k4−k2−1
16k5ln(kcosx+Δ )
17./integraldisplay
Δsin4xcosxdx=8k4sin4x−2k2sin2x−3
48k4Δsinx+1
16k5arcsin ( ksinx)
2.583 Elliptic and pseudo-elliptic integrals 187
18./integraldisplay
Δsin3xcos2xdx=8k4sin4x−2k2/parenleftbig
k2+1/parenrightbig
sin2x−3k4+2k2−3
48k4Δcos x
+k/prime4/parenleftbig
k2+1/parenrightbig
16k5ln (kcosx+Δ )
19./integraldisplay
Δsin2xcos3xdx=−8k4sin4x+2k2/parenleftbig
6k2+1/parenrightbig
sin2x−6k2+3
48k4Δsinx
+2k2−1
16k5arcsin ( ksinx)
20./integraldisplay
Δsinxcos4xdx=−8k4sin4x+2k2/parenleftbig
7k2+1/parenrightbig
sin2x−3k4−8k2+3
48k4Δcos x
−k/prime6
16k5ln(kcosx+Δ )
21./integraldisplay
Δcos5xdx=8k4sin4x−2k2/parenleftbig
12k2+1/parenrightbig
sin2x+2 4k4+1 2k2−3
48k4Δsinx
+8k4−4k2+1
16k5arcsin( ksinx)
22./integraldisplay
Δ3dx=2
3/parenleftBig
1+k/prime2/parenrightBig
E(x, k)−k/prime2
3F(x, F)+k2
3Δsinxcosx
23./integraldisplay
Δ3sinxdx=2k2sin2x+3k2−5
8Δcos x−3k/prime4
8kln(kcosx+Δ )
24./integraldisplay
Δ3cosxdx=−2k2sin2x+5
8Δsinx+3
8karcsin( ksinx)
25./integraldisplay
Δ3sin2xdx=3k2sin2x+4k2−6
15Δsinxcosx+k/prime2/parenleftbig
3−4k2/parenrightbig
15k2F(x, k)
−8k4−13k2+3
15k2E(x, k)
26./integraldisplay
Δ3sinxcosxdx=−Δ5
5k2
27./integraldisplay
Δ3cos2xdx=−3k2sin2x+k2+5
15Δsinxcosx−k/prime2/parenleftbig
k2+3/parenrightbig
15k2F(x, k)
−2k4−7k2−3
15k2E(x, k)
28./integraldisplay
Δ3sin3xdx=8k4sin4x+2k2/parenleftbig
5k2−7/parenrightbig
sin2x+1 5k4−22k2+3
48k2Δcos x
−5k6−9k4+3k2+1
16k3ln(kcosx+Δ )
29./integraldisplay
Δ3sin2xcosxdx=−8k4sin4x+1 4k2sin2x−3
48k2Δsinx
+1
16k3arcsin( ksinx)
188 Trigonometric Functions 2.583
30./integraldisplay
Δ3sinxcos2xdx=−8k4sin4x+2k2/parenleftbig
k2+7/parenrightbig
sin2x+3k4−8k2−3
48k2
×Δcos x+k/prime6
16k3ln (kcosx+Δ )
31./integraldisplay
Δ3cos3xdx=8k4sin4x−2k2/parenleftbig
6k2+7/parenrightbig
sin2x+3 0k2+3
48k2Δsinx
+6k2−1
16k3arcsin( ksinx)
32./integraldisplayΔdx
sinx=−1
2lnΔ + cos x
Δ−cosx+klnk(kcosx+Δ )
33./integraldisplayΔdx
cosx=k/prime
2lnΔ+k/primesinx
Δ−k/primesinx+karcsin( ksinx)
34./integraldisplayΔdx
sin2x=k/prime2F(x, k)−E(x, k)−Δcot x
35./integraldisplayΔdx
sinxcosx=1
2ln1−Δ
1+Δ+k/prime
2lnΔ+k/prime
Δ−k/prime
36./integraldisplayΔdx
cos2x=F(x, k)−E(x, k)+Δt a n x
37./integraldisplaysinx
cosxΔdx=/integraldisplay
Δtan xdx=−Δ+k/prime
2lnΔ+k/prime
Δ−k/prime
38./integraldisplaycosx
sinxΔdx=/integraldisplay
Δcot xdx=Δ+1
2ln1−Δ
1+Δ
39./integraldisplayΔdx
sin3x=−Δcos x
2s in2x+k/prime2
4lnΔ + cos x
Δ−cosx
40./integraldisplayΔdx
sin2xcosx=−Δ
sinx−1+k2
2k/primelnΔ−k/primesinx
Δ+k/primesinx
41./integraldisplayΔdx
sinxcos2x=Δ
cosx+1
2lnΔ + cos x
Δ−cosx
42./integraldisplayΔdx
cos3x=Δsinx
2c os2x+1
4k/primelnΔ+k/primesinx
Δ−k/primesinx
43./integraldisplayΔsinxdx
cos2x=Δ
cosx−kln (kcosx+Δ )
44./integraldisplayΔcos xdx
sin2x=−Δ
sinx−karcsin( ksinx)
45./integraldisplayΔsin2xdx
cosx=−Δsinx
2+2k2−1
2karcsin( ksinx)+k/prime
2lnΔ+k/primesinx
Δ−k/primesinx
46./integraldisplayΔcos2xdx
sinx=Δcos x
2+k2+1
2kln(kcosx+Δ )+1
2lnΔ + cos x
Δ−cosx
47./integraldisplayΔdx
sin4x=1
3/braceleftBig
−Δcot3x+/parenleftbig
k2−3/parenrightbig
Δcot x+2k/prime2F(x, k)+/parenleftbig
k2−2/parenrightbig
E(x, k)/bracerightBig
2.583 Elliptic and pseudo-elliptic integrals 189
48./integraldisplayΔdx
sin3xcosx=−Δ
2s in2x+k/prime
2lnΔ+k/prime
Δ−k/prime+k2−2
4ln1+Δ
1−Δ
49./integraldisplayΔdx
sin2xcos2x=/parenleftbigg1
k/prime2tanx−cotx/parenrightbigg
Δ+2 F(x, k)−1+k/prime2
k/prime2E(x, k)
50./integraldisplayΔdx
sinxcos3x=Δ
2c os2x−1
2ln1+Δ
1−Δ+2−k2
4k/primelnΔ+k/prime
Δ−k/prime
51./integraldisplayΔdx
cos4x=1
3k/prime2/braceleftBig/bracketleftBig
k/prime2tan2x−/parenleftbig
2k2−3/parenrightbig
tanx/bracketrightBig
Δ+2 k/prime2F(x, k)+/parenleftbig
k2−2/parenrightbig
E(x, k)/bracerightBig
52./integraldisplaysinx
cos3xΔdx=Δ
2c os2x+k2
4k/primelnΔ+k/prime
Δ−k/prime
53./integraldisplaycosx
sin3xΔdx=−Δ
2s in2x+k2
4ln1+Δ
1−Δ
54./integraldisplaysin2x
cos2xΔdx=/integraldisplay
tan2xΔdx=Δt a n x+F(x, k)−2E(x, k)
55./integraldisplaycos2x
sin2xΔdx=/integraldisplay
cot2xΔdx=−Δcot x+k/prime2F(x, k)−2E(x, k)
56./integraldisplaysin3x
cosxΔdx=−k2sin2x+3k2−1
3k2Δ+k/prime
2lnΔ+k/prime
Δ−k/prime
57./integraldisplaycos3x
sinxΔdx=−k2sin2x−3k2−1
3k2Δ+1
2ln1−Δ
1+Δ
58./integraldisplayΔdx
sin5x=/parenleftbig
k2−3/parenrightbig
sin2x+2
8s in4xcosxΔ+k/prime2/parenleftbig
k2+3/parenrightbig
16lnΔ + cos x
Δ−cosx
59./integraldisplayΔdx
sin4xcosx=−/parenleftbig
3−k2/parenrightbig
sin2x+1
3s in3xΔ−k/prime
2lnΔ−k/primesinx
Δ+k/primesinx
60./integraldisplayΔdx
sin3xcos2x=3s in2x−1
2s in2xcosxΔ+k2−3
4lnΔ−cosx
Δ + cos x
61./integraldisplayΔdx
sin2xcos3x=3s in2x−2
2s inxcos2xΔ−2k2−3
4k/primelnΔ+k/primesinx
Δ−k/primesinx
62./integraldisplayΔdx
sinxcos4x=/parenleftbig
2k2−3/parenrightbig
sin2x−3k2+4
3k/prime2cos3xΔ+1
2lnΔ + cos x
Δ−cosx
63./integraldisplayΔdx
cos5x=/parenleftbig
2k2−3/parenrightbig
sin2x−4k2+5
8k/prime2cos4xsinxΔ−4k2−3
16k/prime3lnΔ+k/primesinx
Δ−k/primesinx
64./integraldisplaysinx
cos4xΔdx=−/parenleftbig
2k2+1/parenrightbig
k2sin2x+3k4−k2+1
3k/prime2cos3xΔ
65./integraldisplaycosx
sin4xΔdx=−Δ3
3s in3x
66./integraldisplaysin2x
cos3xΔdx=sinx
2c os2xΔ+2k2−1
4k/primelnΔ+k/primesinx
Δ−k/primesinx−karcsin( ksinx)
190 Trigonometric Functions 2.584
67./integraldisplaycos2x
sin3xΔdx=−cosx
2s in2xΔ−k2+1
4lnΔ + cos x
Δ−cosx−kln(kcosx+Δ )
68./integraldisplaysin3x
cos2xΔdx=−sin2x−3
2c osxΔ−3k2−1
2kln(kcosx+Δ )
69./integraldisplaycos3x
sin2xΔdx=−sin2x+2
2s inxΔ−2k2+1
2karcsin( ksinx)
70./integraldisplaysin4x
cosxΔdx=−2k2sin2x+4k2−1
8k2Δsinx
+8k4−4k2−1
8k3arcsin( ksinx)+k/prime
2lnΔ+k/primesinx
Δ−k/primesinx
71./integraldisplaycos4x
sinxΔdx=−2k2sin2x+5k2+1
8k2Δcos x
+1
2lnΔ + cos x
Δ−cosx+3k4+6k2−1
8k3ln (kcosx+Δ )
2.584
1./integraldisplaydx
Δ=F(x, k)
2./integraldisplaysinxdx
Δ=1
2klnΔ−kcosx
Δ+kcosx=−1
kln(kcosx+Δ )
3./integraldisplaycosxdx
Δ=1
karcsin ( ksinx)=1
karctanksinx
Δ
4./integraldisplaysin2xdx
Δ=1
k2F(x, k)−1
k2E(x, k)
5./integraldisplaysinxcosxdx
Δ=−Δ
k2
6./integraldisplaycos2xdx
Δ=1
k2E(x, k)−k/prime2
k2F(x, k)
7./integraldisplaysin3xdx
Δ=cosxΔ
2k2−1+k2
2k3ln(kcosx+Δ )
8./integraldisplaysin2xcosxdx
Δ=−sinxΔ
2k2+arcsin( ksinx)
2k3
9./integraldisplaysinxcos2xdx
Δ=−cosxΔ
2k2+k/prime2
2k3ln(kcosx+Δ )
10./integraldisplaycos3xdx
Δ=sinxΔ
2k2+2k2−1
2k3arcsin ( ksinx)
11./integraldisplaysin4xdx
Δ=sinxcosxΔ
3k2+2+k2
3k4F(x, k)−2/parenleftbig
1+k2/parenrightbig
3k4E(x, k)
12./integraldisplaysin3xcosxdx
Δ=−1
3k4/parenleftbig
2+k2sin2x/parenrightbig
Δ
2.584 Elliptic and pseudo-elliptic integrals 191
13./integraldisplaysin2xcos2xdx
Δ=−sinxcosxΔ
3k2+2−k2
3k4E(x, k)+2k2−2
3k4F(x, k)
14./integraldisplaysinxcos3xdx
Δ=−1
3k4/parenleftBig
k2cos2x−2k/prime2/parenrightBig
Δ
15./integraldisplaycos4xdx
Δ=sinxcosxΔ
3k2+4k2−2
3k4E(x, k)+3k4−5k2+2
3k4F(x, k)
16./integraldisplaysin5xdx
Δ=2k2sin2x+3k2+3
8k4cosxΔ−3+2k2+3k4
8k5ln (kcosx+Δ )
17./integraldisplaysin4xcosxdx
Δ=−2k2sin2x+3
8k4sinxΔ+3
8k5arcsin( ksinx)
18./integraldisplaysin3xcosxdx
Δ=2k2cos2x−k2−3
8k4cosxΔ−k4+2k2−3
8k5ln (kcosx+Δ )
19./integraldisplaysin2xcos3xdx
Δ=−2k2cos2x+2k2−3
8k4sinxΔ+4k2−3
8k5arcsin( ksinx)
20./integraldisplaysinxcos4xdx
Δ=3−5k2+2k2sin2x
8k4cosxΔ−3k4−6k2+3
8k5ln (kcosx+Δ )
21./integraldisplaycos5xdx
Δ=2k2cos2x+6k2−3
8k4sinxΔ+8k4−8k2+3
8k5arcsin( ksinx)
22./integraldisplaysin6xdx
Δ=3k2sin2x+4k2+4
15k4sinxcosxΔ
+4k4+3k2+8
15k6F(x, k)−8k4+7k2+8
15k6E(x, k)
23./integraldisplaysin5xcosxdx
Δ=−3k4sin4x+4k2sin2x+8
15k6Δ
24./integraldisplaysin4xcosxdx
Δ=3k2cos2x−2k2−4
15k4sinxcosxΔ
+k4+7k2−8
15k6F(x, k)−2k4+3k2−8
15k6E(x, k)
25./integraldisplaysin3xcos3xdx
Δ=3k4sin4x−/parenleftbig
5k4−4k2/parenrightbig
sin2x−10k2+8
15k6Δ
26./integraldisplaysin2xcos4xdx
Δ=−3k2cos2x+3k2−4
15k4sinxcosxΔ
+9k4−17k2+8
15k6F(x, k)−3k4−13k2+8
15k6E(x, k)
27./integraldisplaysinxcos5xdx
Δ=−3k4cos4x+4k2k/prime2cos2x−8k4+1 6k2−8
15k6Δ
28./integraldisplaycos6xdx
Δ=3k2cos2x+8k2−4
15k4sinxcosxΔ
+15k6−34k4+2 7k2−8
15k6F(x, k)+23k4−23k2+8
15k6E(x, k)
192 Trigonometric Functions 2.584
29./integraldisplaysin7xdx
Δ=8k4sin4x+1 0k2/parenleftbig
k2+1/parenrightbig
sin2x+1 5k4+1 4k2+1 5
48k6cosxΔ
−/parenleftbig
5k4−2k2+5/parenrightbig/parenleftbig
k2+1/parenrightbig
16k7ln(kcosx+Δ )
30./integraldisplaysin6xcosxdx
Δ=−8k4sin4x+1 0k2sin2x+1 5
48k6sinxΔ+5
16k7arcsin( ksinx)
31./integraldisplaysin5xcos2xdx
Δ=−8k4sin4x+2k2/parenleftbig
k2−5/parenrightbig
sin2x+3k4+4k2−15
48k6cosxΔ
−k6+k4+3k2−5
16k7ln (kcosx+Δ )
32./integraldisplaysin4xcos3xdx
Δ=8k4sin4x−2k2/parenleftbig
6k2−5/parenrightbig
sin2x−18k2+1 5
48k6sinxΔ
+6k2−5
16k7arcsin( ksinx)
33./integraldisplaysin3xcos4xdx
Δ=8k4sin4x−2k2/parenleftbig
6k2−5/parenrightbig
sin2x+3k4−22k2+1 5
48k6cosxΔ
−k6+3k4−9k2+5
16k7ln (kcosx+Δ )
34./integraldisplaysin2xcos5xdx
Δ=−8k4sin4x+2k2/parenleftbig
12k2−5/parenrightbig
sin2x−24k4+3 6k2−15
48k6sinxΔ
+8k4−12k2+5
16k7arcsin ( ksinx)
35./integraldisplaysinxcos6xdx
Δ=−8k4sin4x+2k2/parenleftbig
13k2−5/parenrightbig
sin2x−33k4+4 0k2−15
48k6cosxΔ
+5k/prime6
16k7ln(kcosx+Δ )
36./integraldisplaycos7xdx
Δ=8k4sin4x−2k2/parenleftbig
18k2−5/parenrightbig
sin2x+7 2k4−54k2+1 5
48k6sinxΔ
+16k6−24k4+1 8k2−5
16k7arcsin ( ksinx)
37./integraldisplaydx
Δ3=1
k/prime2E(x, k)−k2
k/prime2sinxcosx
Δ
38./integraldisplaysinxdx
Δ3=−cosx
k/prime2Δ
39./integraldisplaycosxdx
Δ3=sinx
Δ
40.11/integraldisplaysin2xdx
Δ3=1
k/prime2k2E(x, k)−1
k2F(x, k)−1
k/prime2sinxcosx
Δ
41./integraldisplaysinxcosxdx
Δ3=1
k2Δ
42./integraldisplaycos2xdx
Δ3=1
k2F(x, k)−1
k2E(x, k)+sinxcosx
Δ
2.584 Elliptic and pseudo-elliptic integrals 193
43./integraldisplaysin3xdx
Δ3=−cosx
k2k/prime2Δ+1
k3ln (kcosx+Δ )
44./integraldisplaysin2xcosxdx
Δ3=sinx
k2Δ−1
k3arcsin( ksinx)
45./integraldisplaysinxcos2xdx
Δ3=cosx
k2Δ−1
k3ln (kcosx+Δ )
46./integraldisplaycos3xdx
Δ3=−k/prime2sinx
k2Δ+1
k3arcsin( ksinx)
47./integraldisplaysin4xdx
Δ3=k/prime2+1
k/prime2k4E(x, k)−2
k4F(x, k)−sinxcosx
k2k/prime2Δ
48./integraldisplaysin3xcosxdx
Δ3=2−k2sin2x
k4δ
49./integraldisplaysin2xcos2xdx
Δ3=2−k2
k4F(x, k)−2
k4E(x, k)+sinxcosx
k2Δ
50./integraldisplaysinxcos3xdx
Δ3=k2sin2x+k2−2
k4Δ
51./integraldisplaycos4xdx
Δ3=k/prime2+1
k4E(x, k)−2k/prime2
k4F(x, k)−k/prime2sinxcosx
k2Δ
52.9/integraldisplaysin5xdx
Δ3=k2k/prime2sin2x+k2−3
2k4k/prime2Δcosx+k2+3
2k5ln(kcosx+Δ )
53./integraldisplaysin4xcosxdx
Δ3=−k2sin2x+3
2k4Δsinx−3
2k5arcsin( ksinx)
54./integraldisplaysin3xcos2xdx
Δ=−k2sin2x+3
2k4Δcosx+k2−3
2k5ln (kcosx+Δ )
55./integraldisplaysin2xcos3xdx
Δ3=k2sin2x+2k2−3
2k4Δsinx−2k2−3
2k5arcsin( ksinx)
56./integraldisplaysinxcos4xdx
Δ3=k2sin2x+2k2−3
2k4Δcosx+3k/prime2
2k5ln (kcosx+Δ )
57./integraldisplaycos5xdx
Δ3=−k2sin2x+2k4−4k2+3
2k4Δsinx+4k2−3
2k5arcsin( ksinx)
58./integraldisplaydx
Δ5=−k2sinxcosx
3k/prime2Δ3−2k2/parenleftBig
k/prime2+1/parenrightBig
sinxcosx
3k/prime4Δ−1
3k/prime2F(x, k)
+2/parenleftBig
k/prime2+1/parenrightBig
3k/prime4E(x, k)
59./integraldisplaysinxdx
Δ5=2k2sin2x+k2−3
3k/prime4Δ3cosx
60./integraldisplaycosxdx
Δ5=−2k2sin2x+3
3Δ3sinx
194 Trigonometric Functions 2.584
61./integraldisplaysin2xdx
Δ5=k2+1
3k/prime4k2E(x, k)−1
3k/prime2k2F(x, k)
+k2/parenleftbig
k2+1/parenrightbig
sin2x−2
3k/prime4Δ3sinxcosx
62./integraldisplaysinxcosxdx
Δ5=1
3k2Δ3
63./integraldisplaycos2xdx
Δ5=1
3k2F(x, k)+2k2−1
3k2k/prime2E(x, k)+k2/parenleftbig
2k2−1/parenrightbig
sin2x−3k2+2
2k/prime2Δsinxcosx
64./integraldisplaysin3x
Δ5dx=/parenleftbig
3k2−1/parenrightbig
sin2x−2
3k/prime4Δ3cosx
65./integraldisplaysin2xcosx
Δ5dx=sin3x
3Δ3
66./integraldisplaysinxcos2x
Δ5dx=−cos3x
3k/prime2Δ3
67./integraldisplaycos3xdx
Δ5=−/parenleftbig
2k2+1/parenrightbig
sin2x+3
3Δ3sinx
68./integraldisplaydx
Δsinx=−1
2lnΔ + cos x
Δ−cosx
69./integraldisplaydx
Δcos x=−1
2k/primelnΔ−k/primesinx
Δ+k/primesinx
70./integraldisplaydx
Δsin2x=/integraldisplay1 + cot2x
Δdx=F(x, k)−E(x, k)−Δcot x
71./integraldisplaydx
Δsinxcosx=/integraldisplay
(tanx+c o t x)dx
Δ=1
2ln1−Δ
1+Δ+1
2k/primelnΔ+k/prime
Δ−k/prime
72./integraldisplaydx
Δcos2x=/integraldisplay/parenleftbig
1+t a n2x/parenrightbigdx
Δ=F(x, k)−1
k/prime2E(x, k)+1
k/prime2Δtan x
73./integraldisplaysinx
cosxdx
Δ=/integraldisplay
tanxdx
Δ=1
2k/primelnΔ+k/prime
Δ−k/prime
74./integraldisplaycosx
sinxdx
Δ=/integraldisplay
cotxdx
Δ=1
2ln1−Δ
1+Δ
75./integraldisplaydx
Δsin3x=−Δcos x
2s in2x−1+k2
4lnΔ + cos x
Δ−cosx
76./integraldisplaydx
Δsin2xcosx=−Δ
sinx−1
2k/primelnΔ−k/primesinx
Δ+k/primesinx
77./integraldisplaydx
Δsinxcos2x=Δ
k/prime2cosx+1
2lnΔ−cosx
Δ + cos x
78./integraldisplaydx
Δcos3x=Δsinx
2k/prime2cos2x+2k2−1
4k/prime3lnΔ−k/primesinx
Δ+k/primesinx
79./integraldisplaysinx
cos2xdx
Δ=Δ
k/prime2cosx
2.584 Elliptic and pseudo-elliptic integrals 195
80./integraldisplaycosx
sin2xdx
Δ=−Δ
sinx
81./integraldisplaysin2x
cosxdx
Δ=1
2k/primelnΔ+k/primesinx
Δ−k/primesinx−1
karcsin ( ksinx)
82./integraldisplaycos2x
sinxdx
Δ=1
2lnΔ + cos x
Δ−cosx+1
kln (kcosx+Δ )
83./integraldisplaydx
Δsin4x=1
3/braceleftbig
−Δcot3x−Δ/parenleftbig
2k2+3/parenrightbig
cotx+/parenleftbig
k2+2/parenrightbig
F(x, k)−2/parenleftbig
k2+1/parenrightbig
E(x, k)/bracerightbig
84./integraldisplaydx
Δsin3xcosx=/integraldisplay/parenleftbig
tanx+ 2cot x+c o t3x/parenrightbigdx
Δ
=−Δ
2s in2x+1
2k/primelnΔ+k/prime
Δ−k/prime−k2+2
4ln1+Δ
1−Δ
85./integraldisplaydx
Δsin2xcos2x=/integraldisplay/parenleftbig
tan2x+2+c o t2x/parenrightbigdx
Δ
=/parenleftbiggtanx
k/prime2−cotx/parenrightbigg
Δ+k2−2
k/prime2E(x, k)+2F(x, k)
86./integraldisplaydx
Δsinxcos3x=/integraldisplay/parenleftbig
cotx+2t a n x+t a n3x/parenrightbigdx
Δ
=−Δ
2k/prime2cos2x−1
2ln1+Δ
1−Δ+2−3k2
4k/prime3lnΔ+k/prime
Δ−k/prime
87./integraldisplaydx
Δcos4x=1
3k/prime2⎧
⎨
⎩Δtan3x−5k2−3
k/prime2Δtan x−/parenleftbig
3k2−2/parenrightbig
F(x, k)
+2/parenleftbig
2k2−1/parenrightbig
k/prime2E(x, k)⎫
⎬
⎭
88./integraldisplaysinx
cos3xdx
Δ=/integraldisplay
tanx/parenleftbig
1+t a n2x/parenrightbigdx
Δ=Δ
2k/prime2cos2x−k2
4k/prime3lnΔ+k/prime
Δ−k/prime
89./integraldisplaycosx
sin3xdx
Δ=−Δ
2s in2x−k2
4ln1+Δ
1−Δ
90./integraldisplaysin2x
cos2xdx
Δ=/integraldisplaytan2x
Δdx=Δ
k/prime2tanx−1
k/prime2E(x, k)
91./integraldisplaycos2x
sin2xdx
Δ=/integraldisplaycot2x
Δdx=−Δcot x−E(x, k)
92./integraldisplaysin3x
cosxdx
Δ=Δ
k2+1
2k/primelnΔ+k/prime
Δ−k/prime
93./integraldisplaycos3x
sinxdx
Δ=Δ
k2−1
2ln1+Δ
1−Δ
94./integraldisplaydx
Δsin5x=−/bracketleftbig
3/parenleftbig
1+k2/parenrightbig
sin2x+2/bracketrightbig
8s in2xΔcos x+3k4+2k2+3
16lnΔ + cos x
Δ−cosx
196 Trigonometric Functions 2.585
95./integraldisplaydx
Δsin4xcosx=−/parenleftbig
3+2k2/parenrightbig
sin2x+1
3s in3xΔ−1
2k/primelnΔ−k/primesinx
Δ+k/primesinx
96./integraldisplaydx
Δsin3xcos2x=/parenleftbig
3−k2/parenrightbig
sin2x−k/prime2
2k/prime2sin2xcosxΔ+k2+3
4lnΔ−cosx
Δ + cos x
97./integraldisplaydx
Δsin2xcos3x=/parenleftbig
3−2k2/parenrightbig
sin2x−2k/prime2
2k/prime2sinxcos2xΔ−4k2−3
4k/prime3lnΔ+k/primesinx
Δ−k/primesinx
98./integraldisplaydx
Δsinxcos4x=/parenleftbig
5k2−3/parenrightbig
sin2x−6k2+4
3k/prime4cos3xΔ−1
2lnΔ + cos x
Δ−cosx
99./integraldisplaydx
Δcos5x=3/parenleftbig
2k2−1/parenrightbig
sin2x−8k2+5
8k/prime4cos4xΔsinx+8k4−8k2+3
16k/prime5lnΔ+k/primesinx
Δ−k/primesinx
100./integraldisplaysinx
cos4xdx
Δ=−2k2cos2x−k/prime2
2k/prime4cos3xΔ
101./integraldisplaycosx
sin4xdx
Δ=−2k2sin2x+1
3s in3xΔ
102./integraldisplaysin2x
cos3xdx
Δ=Δsinx
2k/prime2cos2x−1
4k/prime3lnΔ+k/primesinx
Δ−k/primesinx
103./integraldisplaycos3x
sin3xdx
Δ=−Δcos x
2s in2x+k/prime2
4lnΔ + cos x
Δ−cosx
104./integraldisplaysin3x
cos2xdx
Δ=Δ
k/prime2cosx+1
kln(kcosx+Δ )
105./integraldisplaycos3x
sin2xdx
Δ=−Δ
sinx−1
karcsin( ksinx)
106./integraldisplaysin4x
cosxdx
Δ=Δsinx
2k2+1
2k/primelnΔ+k/primesinx
Δ−k/primesinx−2k2+1
2k3arcsin ( ksinx)
107./integraldisplaycos4x
sinxdx
Δ=Δcos x
2k2+1
2lnΔ + cos x
Δ−cosx+3k2−1
2k3ln(kcosx+Δ )
2.585
1./integraldisplay(a+s i nx)p+3dx
Δ
=1
(p+2 )k2/bracketleftbigg
(a+s i nx)pcosxΔ
+2(2p+3 )ak2/integraldisplay(a+s i nx)p+2dx
Δ+(p+1 )/parenleftbig
1+k2−6a2k2/parenrightbig/integraldisplay(a+s i nx)p+1dx
Δ
−a(2p+1 )/parenleftbig
1+k2−2a2k2/parenrightbig/integraldisplay(a+bsinx)pdx
Δ
−p/parenleftbig
1−a2/parenrightbig/parenleftbig
1−a2k2/parenrightbig/integraldisplay(a+s i nx)p−1dx
Δ/bracketrightBigg
/bracketleftbigg
p/negationslash=−2,a/negationslash=±1,a/negationslash=±1
k/bracketrightbigg
Forp=na natural number, this integral can be reduced to the following three integrals:
2.586 Elliptic and pseudo-elliptic integrals 197
2./integraldisplaya+s i nx
Δdx=aF(x, k)+1
2klnΔ−kcosx
Δ+kcosx
3./integraldisplay(a+s i nx)2
Δdx=1+k2a2
k2F(x, k)−1
k2E(x, k)+a
klnΔ−kcosx
Δ+kcosx
4.6/integraldisplaydx
(a+s i nx)Δ=1
aΠ/parenleftbigg
x,1
a2,k/parenrightbigg
−/integraldisplaysinxdx/parenleftbig
a2−sin2x/parenrightbig
Δ,
where
5./integraldisplaysinxdx/parenleftbig
a2−sin2x/parenrightbig
Δ=−1
2/radicalbig
(1−a2)(1−a2k2)ln√
1−a2Δ−√
1−k2a2cosx√
1−a2Δ+√
1−k2a2cosx
2.586
1./integraldisplaydx
(a+s i nx)nΔ=1
(n−1)(1−a2)( 1−a2k2)/bracketleftBigg
−cosxΔ
(a+s i nx)n−1
−(2n−3)/parenleftbig
1+k2−2a2k2/parenrightbig
a/integraldisplaydx
(a+s i nx)n−1Δ
−(n−2)/parenleftbig
6a2k2−k2−1/parenrightbig/integraldisplaydx
(a+s i nx)n−2Δ
−(10−4n)ak2/integraldisplaydx
(a+s i nx)n−3Δ−(n−3)k2/integraldisplaydx
(a+s i nx)n−4Δ/bracketrightBigg
/bracketleftbigg
n/negationslash=1,a/negationslash=±1,a/negationslash=±1
k/bracketrightbigg
This integral can be reduced to the integrals:
2./integraldisplaydx
(a+s i nx)2Δ=1
(1−a2)(1−a2k2)/bracketleftbigg
−cosxΔ
a+s i nx−a/parenleftbig
1+k2−2a2k2/parenrightbig/integraldisplaydx
(a+s i nx)Δ
−2ak2/integraldisplay(a+s i nx)dx
Δ+k2/integraldisplay(a+s i nx)2dx
Δ/bracketrightBigg
(see2.585 2, 3, 4)
3./integraldisplaydx
(a+s i nx)3Δ=1
2( 1−a2)(1−a2k2)⎡
⎣−cosxΔ
(a+s i nx)2−3a/parenleftbig
1+k2−2a2k2/parenrightbig/integraldisplaydx
(a+s i nx)2Δ
−/parenleftbig
6a2k2−k2−1/parenrightbig/integraldisplaydx
(a+s i nx)Δ+2ak2F(x, k)⎤
⎦
(see2.585 4a n d2.586 2)
Fora=±1, we have:
4./integraldisplaydx
(1±sinx)nΔ=1
(2n−1)k/prime2/bracketleftBigg
∓cosxΔ
(1±sinx)n+(n−1)/parenleftbig
1−5k2/parenrightbig/integraldisplaydx
(1±sinx)n−1Δ
+2 ( 2n−3)k2/integraldisplaydx
(1±sinx)n−2Δ−(n−2)k2/integraldisplaydx
(1±sinx)n−3Δ/bracketrightBigg
GU (241)(6a)
This integral can be reduced to the following integrals:
198 Trigonometric Functions 2.587
5./integraldisplaydx
(1±sinx)Δ=∓cosxΔ
k/prime2(1±sinx)+F(x, k)−1
k/prime2E(x, k) GU (241)(6c)
6./integraldisplaydx
(1±sinx)2Δ=1
3k/prime4/braceleftBigg
∓k/prime2cosxΔ
(1±sinx)2∓/parenleftbig
1−5k2/parenrightbig
cosxΔ
1±sinx
+/parenleftbig
1−3k2/parenrightbig
k/prime2F(x, k)−/parenleftbig
1−5k2/parenrightbig
E(x, k)/bracerightBig
GU (241)(6b)
Fora=±1
k,w eh a v e
7./integraldisplaydx
(1±ksinx)nΔ=1
(2n−1)k/prime2/bracketleftbigg
±kcosxΔ
(1±ksinx)n+(n−1)/parenleftbig
5−k2/parenrightbig/integraldisplaydx
(1±ksinx)n−1Δ
−2(2n−3)/integraldisplaydx
(1±ksinx)n−2Δ+(n−2)/integraldisplaydx
(1±ksinx)n−3Δ/bracketrightBigg
GU (241)(7a)
This integral can be reduced to the following integrals:
8./integraldisplaydx
(1±ksinx)Δ=±kcosxΔ
k/prime2(1±ksinx)+1
k/prime2E(x, k) GU (241)(7b)
9./integraldisplaydx
(1±ksinx)2Δ=1
3k/prime4/bracketleftBigg
±kk/prime2cosxΔ
(1±ksinx)2±k/parenleftbig
5−k2/parenrightbig
cosxΔ
1±ksinx
−2k/prime2F(x, k)+/parenleftbig
5−k2/parenrightbig
E(x, k)/bracketrightbigg
GU(241)(7c)
2.587
1./integraldisplay(b+c o s x)p+3dx
Δ1
(p+2 )k2/bracketleftBigg
(b+c o s x)psinxΔ+2 ( 2 p+3 )bk2/integraldisplay(b+c o s x)p+2dx
Δ
−(p+1 )/parenleftBig
k/prime2−k2+6b2k2/parenrightBig/integraldisplay(b+c o s x)p+1dx
Δ
+(2p+1 )b/parenleftBig
k/prime2−k2+b2k2/parenrightBig/integraldisplay(b+c o s x)pdx
Δ
+p/parenleftbig
1−b2/parenrightbig/parenleftBig
k/prime2+k2b2/parenrightBig/integraldisplay(b+c o s x)p−1dx
Δ/bracketrightBigg
/bracketleftbigg
p/negationslash=−2,b/negationslash=±1,b/negationslash=ik/prime
k/bracketrightbigg
Forp=na natural number, this integral can be reduced to the following three integrals:
2./integraldisplayb+c o s x
Δdx=bF(x, k)+1
karcsin( ksinx)
3./integraldisplay(b+c o s x)2
Δdx=b2k2−k/prime2
k2F(x, k)+1
k2E(x, k)+2b
karcsin( ksinx)
4./integraldisplaydx
(b+c o s x)Δ=b
b2−1Π/parenleftbigg
x,1
b2−1,k/parenrightbigg
+/integraldisplaycosxdx/parenleftbig
1−b2−sin2x/parenrightbig
Δ,
where
2.589 Elliptic and pseudo-elliptic integrals 199
5./integraldisplaycosxdx/parenleftbig
1−b2−sin2x/parenrightbig
Δ=1
2/radicalBig
(1−b2)/parenleftbig
k/prime2+k2b2/parenrightbigln√
1−b2Δ+k/radicalbig
k/prime2+k2b2sinx
√
1−b2Δ−k/radicalbig
k/prime2+k2b2sinx
2.588
1./integraldisplaydx
(b+c o s x)nΔ=1
(n−1)(1−b2)/parenleftbig
k/prime2+b2k2/parenrightbig/bracketleftBigg
−k/prime2sinxΔ
(b+c o s x)−1
−(2n−3)/parenleftbig
1−2k2+2b2k2/parenrightbig
b/integraldisplaydx
(b+c o s x)n−1Δ
−(n−2)/parenleftbig
2k2−1−6b2k2/parenrightbig/integraldisplaydx
(b+c o s x)n−2Δ
−(4n−10)bk2/integraldisplaydx
(b+c o s x)n−3Δ+(n−3)k2/integraldisplaydx
(b+c o s x)n−4Δ/bracketrightBigg
/bracketleftbigg
n/negationslash=1,b/negationslash=±1,b/negationslash=±ik/prime
k/bracketrightbigg
This integral can be reduced to the following integrals:
2./integraldisplaydx
(b+c o s x)2Δ=1
(1−b2)/parenleftbig
k/prime2+b2k2/parenrightbig⎡
⎣−k/prime2sinxΔ
b+c o s x−/parenleftbig
1−2k2+2b2k2/parenrightbig
b/integraldisplaydx
(b+c o s x)Δ
+2bk2/integraldisplayb+c o s x
Δdx−k2/integraldisplay(b+c o s x)2
Δdx⎤
⎦
(see2.587 2, 3, 4)
3./integraldisplaydx
(b+c o s x)3Δ=1
2( 1−b2)/parenleftbig
k/prime2+b2k2/parenrightbig⎡
⎣−k/prime2sinxΔ
(b+c o s x)2
−3b/parenleftbig
1−2k2+2k2b2/parenrightbig/integraldisplaydx
(b+c o s x)2Δ
−/parenleftbig
2k2−1−6b2k2/parenrightbig/integraldisplaydx
(b+c o s x)Δ−2bk2F(x, k)⎤
⎦
(see2.588 2a n d2.587 4)
2.589
1./integraldisplay(c+t a n x)p+3dx
Δ=1
(p+2 )k/prime2⎡
⎣(c+t a n x)pΔ
cos2x+2 ( 2n+3 )ck/prime2/integraldisplay(c+t a n x)p+2dx
Δ
−(p+1 )/parenleftBig
1+k/prime2+6c2k/prime2/parenrightBig/integraldisplay(c+t a n x)p+1dx
Δ
+(2p+1 )c/parenleftBig
1+k/prime2+2c2k/prime2/parenrightBig/integraldisplay(c+t a n x)pdx
Δ
−p/parenleftbig
1+c2/parenrightbig/parenleftBig
1+k/prime2c2/parenrightBig/integraldisplay(c+t a n x)p−1dx
Δ⎤
⎦
[p/negationslash=−2]
Forp=na natural number, this integral can be reduced to the following three integrals:
200 Trigonometric Functions 2.591
2./integraldisplayc+t a n x
Δdx=cF(x, k)+1
2k/primelnΔ+k/prime
Δ−k/prime
3./integraldisplay(c+t a n x)2
Δdx=1
k/prime2tanxΔ+c2F(x, k)−1
k/prime2E(x, k)+c
k/primelnΔ+k/prime
Δ−k/prime
4./integraldisplaydx
(c+t a n x)Δ=c
1+c2F(x, k)+1
c(1 +c2)Π/parenleftbigg
x,−1+c2
c2,k/parenrightbigg
−/integraldisplaysinxcosxdx/bracketleftbig
c2−(1 +c2)sin2x/bracketrightbig
Δ,
where
5./integraldisplaysinxcosxdx/bracketleftbig
c2−(1 +c2)s i n2x/bracketrightbig
Δ=1
2/radicalBig
(1 +c2)/parenleftbig
1+c2k/prime2/parenrightbigln/radicalbig
1+c2k/prime2+√
1+c2Δ/radicalbig
1+c2k/prime2−√
1+c2Δ
2.591
1./integraldisplaydx
(c+t a n x)nΔ=1
(n−1)(1 + c2)/parenleftbig
1+k/prime2c2/parenrightbig⎡
⎣−Δ
(c+t a n x)n−1cos2x
+(2n−3)c/parenleftBig
1+k/prime2+2c2k/prime2/parenrightBig/integraldisplaydx
(c+t a n x)n−1Δ
−(n−2)/parenleftBig
1+k/prime2+6c2k/prime2/parenrightBig/integraldisplaydx
(c+t a n x)n−2Δ
+( 4n−10)ck/prime2/integraldisplaydx
(c+t a n x)n−3Δ−(n−3)k/prime2/integraldisplaydx
(c+t a n x)n−4Δ⎤
⎦
This integral can be reduced to the integrals:
2./integraldisplaydx
(c+t a n x)2Δ=1
(1 +c2)/parenleftbig
1+k/prime2c2/parenrightbig⎡
⎣−Δ
(c+t a n x)c o s2x
+c/parenleftBig
1+k/prime2+2c2k/prime2/parenrightBig/integraldisplaydx
(c+t a n x)Δ
−2ck/prime2/integraldisplayc+t a n x
Δdx+k/prime2/integraldisplay(c+t a n x)2
Δdx⎤
⎦
(see2.589 2, 3, 4)
3./integraldisplaydx
(c+t a n x)3Δ=1
2( 1+ c2)/parenleftbig
1+k/prime2c2/parenrightbig⎡
⎣−Δ
(c+t a n x)2cos2x
+3c/parenleftBig
1+k/prime2+2c2k/prime2/parenrightBig/integraldisplaydx
(c+t a n x)2Δ
−/parenleftBig
1+k/prime2+6c2k/prime2/parenrightBig/integraldisplaydx
(c+t a n x)Δ+2ck/prime2F(x, k)⎤
⎦
(see2.591 2a n d2.589 4)
2.593 Elliptic and pseudo-elliptic integrals 201
2.592
1. Pn=/integraldisplay/parenleftbig
a+s i n2x/parenrightbign
Δdx
The recursion formula
Pn+1=1
(2n+3 )k2/braceleftbigg/parenleftbig
a+s i n2x/parenrightbignsinxcosxΔ+( 2 n+2 )/parenleftbig
1+k2+3ak2/parenrightbig
Pn+1
−(2n+1 )/bracketleftbig
1+2a/parenleftbig
1+k2/parenrightbig
+3a2k2/bracketrightbig
Pn+2na(1 +a)/parenleftbig
1+k2a/parenrightbig
Pn−1/bracerightbigg
reduces this integral (for nan integer) to the integrals:
2. P1 (see2.584 1a n d2.584 4)
3. P0 (see2.584 1)
4. P−1=/integraldisplaydx/parenleftbig
a+s i n2x/parenrightbig
Δ=1
aΠ/parenleftbigg
x,1
a,k/parenrightbigg
Fora=0
5./integraldisplaydx
sin2xΔ(see2.584 70) H (124)a
6. Tn=/integraldisplaydx/parenleftbig
h+gsin2x/parenrightbignΔ
can be calculated by means of the recursion formula:
Tn−3=1
(2n−5)k2/braceleftBigg
−g2sinxcosxΔ
/parenleftbig
h+gsin2x/parenrightbign−1+2 (n−2)/bracketleftbig
g/parenleftbig
1+k2/parenrightbig
+3hk2/bracketrightbig
Tn−2
−(2n−3)/bracketleftbig
g2+2hg/parenleftbig
1+k2/parenrightbig
+3h2k2/bracketrightbig
Tn−1+2 (n−1)h(g+h)/parenleftbig
g+hk2/parenrightbig
Tn/bracerightbigg
2.593
1. Qn=/integraldisplay/parenleftbig
b+c o s2x/parenrightbign
Δdx
The recursion formula
Qn+2=1
(2n+3 )k2⎧
⎨
⎩/parenleftbig
b+c o s2x/parenrightbignsinxsinxΔ−(2n+2 )/parenleftbig
1−2k2−3bk2/parenrightbig
Qn+1
+( 2n+1 )/bracketleftbig
k/prime2+2b/parenleftbig
k/prime2−k2/parenrightbig
−3b2k2/bracketrightbig
n−2nb(1−b)/parenleftbig
k/prime2−k2b/parenrightbig
Qn−1⎫
⎬
⎭
reduces this integral (for nan integer) to the integrals:
2. Q1 (see2.584 1a n d2.584 6)
3. Q0 (see2.584 1)
202 Trigonometric Functions 2.594
4. Q−1=/integraldisplaydx
(b+c o s2x)Δ=1
b+1Π/parenleftbigg
x,−1
b+1,k/parenrightbigg
Forb=0
5./integraldisplaydx
cos2xΔ(see2.584 72) H (123)
2.594
1. Rn=/integraldisplay/parenleftbig
c+t a n2x/parenrightbigndx
Δ
The recursion formula
Rn+2=1
(2n+3 )k/prime2⎧
⎨
⎩/parenleftbig
c+t a n2x/parenrightbigntanxΔ
cos2x−(2n+2 )/parenleftBig
1+k/prime2−3ck/prime2/parenrightBig
Rn+1
+( 2n−1)/bracketleftBig
1−2c/parenleftBig
1+k/prime2/parenrightBig
+3c2k/prime2/bracketrightBig
Rn+2nc(1−c)/parenleftBig
1−k/prime2c/parenrightBig
Rn−1⎫
⎬
⎭
reduces this integral (for nan integer) to the integrals:
2. R1 (see2.584 1a n d2.584 90)
3. R0 (see2.584 1)
4. R−1=/integraldisplaydx/parenleftbig
c+t a n2x/parenrightbig
Δ=1
c−1F(x, k)+1
c(1−c)Π/parenleftbigg
x,1−c
c,k/parenrightbigg
Forc=0 ,s e e 2.582 5.
2.595 Integrals of the type/integraldisplay
R/parenleftbigg
sinx,cosx,/radicalBig
1−p2sin2x/parenrightbigg
dxforp2>1.
Notation :α=a r c s i n( psinx).
Basic formulas
1./integraldisplaydx/radicalbig
1−p2sin2x=1
pF/parenleftbigg
α,1
p/parenrightbigg/bracketleftbig
p2>1/bracketrightbig
BY (283.00)
2./integraldisplay/radicalBig
1−p2sin2xd x=pE/parenleftbigg
α,1
p/parenrightbigg
−p2−1
pF/parenleftbigg
α,1
p/parenrightbigg
/bracketleftbig
p2>1/bracketrightbig
BY (283.03)
3./integraldisplaydx/parenleftbig
1−r2sin2x/parenrightbig/radicalbig
1−p2sin2x=1
pΠ/parenleftbigg
α,r2
p2,1
p/parenrightbigg/bracketleftbig
p2>1/bracketrightbig
BY (283.02)
To evaluate integrals of the form/integraldisplay
R/parenleftbigg
sinx,cosx,/radicalBig
1−p2sin2x/parenrightbigg
dxforp2>1, we may use formulas
2.583 and2.584 , making the following modifications in them. We replace
(1) kwithp;
(2) k/prime2with 1 −p2;
2.597 Elliptic and pseudo-elliptic integrals 203
(3) F(x, k) with1
pF/parenleftBig
α,1
p/parenrightBig
;
(4) E(x, k) with pE/parenleftbigg
α,1
p/parenrightbigg
−p2−1
pF/parenleftbigg
α,1
p/parenrightbigg
.
For example (see 2.584 15):
2.596
1.10/integraldisplaycos4xdx/radicalbig
1−p2sin2x=sinxcosx/radicalbig
1−p2sin2x
3p2+4p2−2
3p4⎡
⎣pE/parenleftbigg
α,1
p/parenrightbigg
−p2−1
pF/parenleftbigg
α,1
p/parenrightbigg⎤
⎦+2−5p2+3p4
3p4·1
pF/parenleftbigg
α,1
p/parenrightbigg
=sinxcosx/radicalbig
1−p2sin2x
3p2−p2−1
3p3F/parenleftbigg
α,1
p/parenrightbigg
+4p2−2
3p3E/parenleftbigg
α,1
p/parenrightbigg/bracketleftbig
p2>1/bracketrightbig
For example (see 2.583 36):
2./integraldisplay/radicalbig
1−p2sin2x
cos2xdx=t a n x/radicalBig
1−p2sin2x+1
pF/parenleftbigg
α,1
p/parenrightbigg
−/bracketleftbigg
pE/parenleftbigg
α,1
p/parenrightbigg
−p2−1
pF/parenleftbigg
α,1
p/parenrightbigg/bracketrightbigg
=p/bracketleftbigg
F/parenleftbigg
α,1
p/parenrightbigg
−E/parenleftbigg
α,1
p/parenrightbigg/bracketrightbigg
+t a n x/radicalBig
1−p2sin2x
/bracketleftbig
p2>1/bracketrightbig
For example (see 2.584 37):
3./integraldisplaydx/radicalBig/parenleftbig
1−p2sin2x/parenrightbig3=−1
p2−1/bracketleftbigg
pE/parenleftbigg
α,1
p/parenrightbigg
−p2−1
pF/parenleftbigg
α,1
p/parenrightbigg/bracketrightbigg
−p2
1−p2sinxcosx/radicalbig
1−p2sin2x
=p2
p2−1sinxcosx/radicalbig
1−p2sin2x+1
pF/parenleftbigg
α,1
p/parenrightbigg
−p
p2−1E/parenleftbigg
α,1
p/parenrightbigg
/bracketleftbig
p2>1/bracketrightbig
2.597 Integrals of the form/integraldisplay
R/parenleftbigg
sinx,cosx,/radicalBig
1+p2sin2x/parenrightbigg
dx
Notation :α=a r c s i n/parenleftBigg/radicalbig
1+p2sinx/radicalbig
1+p2sin2x/parenrightBigg
Basic formulas
1./integraldisplaydx/radicalbig
1+p2sin2x=1/radicalbig
1+p2F/parenleftBigg
α,p/radicalbig
1+p2/parenrightBigg
BY (282.00)
2./integraldisplay/radicalBig
1+p2sin2xd x=/radicalbig
1+p2E/parenleftBigg
α,p/radicalbig
1+p2/parenrightBigg
−p2sinxcosx/radicalbig
1+p2sin2xBY (282.03)
204 Trigonometric Functions 2.598
3./radicalbig
1+p2sin2xdx
1+(p2−r2p2−r2)s i n2x=1/radicalbig
1+p2Π/parenleftBigg
α,r2,p/radicalbig
1+p2/parenrightBigg
BY (282.02)
4./integraldisplaysinxdx/radicalbig
1+p2sin2x=−1
parcsin/parenleftBigg
pcosx/radicalbig
1+p2/parenrightBigg
5./integraldisplaycosxdx/radicalbig
1+p2sin2x=1
pln/parenleftbigg
psinx+/radicalBig
1+p2sin2x/parenrightbigg
6./integraldisplaydx
sinx/radicalbig
1+p2sin2x=1
2ln/radicalbig
1+p2sin2x−cosx/radicalbig
1+p2sin2x+c o s x
7./integraldisplaydx
cosx/radicalbig
1+p2sin2x=1
2/radicalbig
1+p2ln/radicalbig
1+p2sin2x+/radicalbig
1+p2sinx/radicalbig
1+p2sin2x−/radicalbig
1+p2sinx
8./integraldisplaytanxdx/radicalbig
1+p2sin2x=1
2/radicalbig
1+p2ln/radicalbig
1+p2sin2x+/radicalbig
1+p2
/radicalbig
1+p2sin2x−/radicalbig
1+p2
9./integraldisplaycotxdx/radicalbig
1+p2sin2x=1
2ln1−/radicalbig
1+p2sin2x
1+/radicalbig
1+p2sin2x
2.598 To calculate integrals of the form/integraltext
R/parenleftBig
sinx,cosx,/radicalbig
1+p2sin2x/parenrightBig
dx, we may use formulas
2.583 and2.584 , making the following modifications in them. We replace
(1) k2with−p2;
(2) k/prime2with 1 + p2;
(3) F(x, k) with1√
1+p2F/parenleftbigg
α,p√
1+p2/parenrightbigg
;
(4) E(x, k) with/radicalbig
1+p2E/parenleftbigg
α,p√
1+p2/parenrightbigg
−p2sinxcosx √
1+p2sin2x;
(5)1
kln(kcosx+ Δ) with1
parcsinpcosx√
1+p2;
(6)1
karcsin( ksinx) with1
pln/parenleftBig
psinx+/radicalbig
1+p2sin2x/parenrightBig
.
For example (see 2.584 90):
1./integraldisplaytan2xdx/radicalbig
1+p2sin2x=1
(1 +p2)⎡
⎣tanx/radicalBig
1+p2sin2x
−/radicalbig
1+p2E/parenleftBigg
α,p/radicalbig
1+p2/parenrightBigg
+p2sinxcosx/radicalbig
1+p2sin2x⎤
⎦
=−1/radicalbig
1+p2E/parenleftBigg
α,p/radicalbig
1+p2/parenrightBigg
+tanx/radicalbig
1+p2sin2x
For example (see 2.584 37):
2.611 Elliptic and pseudo-elliptic integrals 205
2./integraldisplaydx/radicalBig/parenleftbig
1+p2sin2x/parenrightbig3=1/radicalbig
1+p2E/parenleftBigg
α,p/radicalbig
1+p2/parenrightBigg
2.599 Integrals of the form/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
a2sin2x−1/parenrightBig
dx/bracketleftbig
a2>1/bracketrightbig
Notation :α=a r c s i n/parenleftBigg
acosx/radicalbig
a2−1/parenrightBigg
.
Basic formulas:
1./integraldisplaydx/radicalbig
a2sin2x−1=−1
aF/parenleftBigg
α,√
a2−1
a/parenrightBigg
/bracketleftbig
a2>1/bracketrightbig
BY (285.00)a
2./integraldisplay/radicalbig
a2sin2x−1dx=1
aF/parenleftBigg
α,√
a2−1
a/parenrightBigg
−aE/parenleftBigg
α,√
a2−1
a/parenrightBigg
/bracketleftbig
a2>1/bracketrightbig
BY (285.06)a
3./integraldisplaydx
/parenleftbig
1−r2sin2x/parenrightbig/radicalbig
a2sin2x−1=1
a(r2−1)Π/parenleftBigg
α,r2/parenleftbig
a2−1/parenrightbig
a2(r2−1),√
a2−1
a/parenrightBigg
/bracketleftbig
a2>1,r2>1/bracketrightbig
BY (285.02)a
4./integraldisplaysinxdx/radicalbig
a2sin2x−1=−α
a/bracketleftbig
a2>1/bracketrightbig
5./integraldisplaycosxdx/radicalbig
a2sin2x−1=1
aln/parenleftBig
asinx+/radicalbig
a2sin2x−1/parenrightBig/bracketleftbig
a2>1/bracketrightbig
6./integraldisplaydx
sinx/radicalbig
a2sin2x−1=−arctancosx/radicalbig
a2sin2x−1/bracketleftbig
a2>1/bracketrightbig
7./integraldisplaydx
cosx/radicalbig
a2sin2x−1=1
2√
a2−2ln√
a2−1s inx+/radicalbig
a2sin2x−1
√
a2−1s inx−/radicalbig
a2sin2x−1
/bracketleftbig
a2>1/bracketrightbig
8./integraldisplaytanxdx/radicalbig
a2sin2x−1=1
2√
a2−1ln√
a2−1+/radicalbig
a2sin2x−1
√
a2−1−/radicalbig
a2sin2x−1
/bracketleftbig
a2>1/bracketrightbig
9./integraldisplaycotxdx/radicalbig
a2sin2x−1=−arcsin/parenleftbigg1
asinx/parenrightbigg/bracketleftbig
a2>1/bracketrightbig
2.611 To calculate integrals of the type/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
a2sin2x−1/parenrightBig
dxfora2>1, we may use
formulas 2.583 and2.584 . In doing so, we should follow the procedure outlined below:
(1) In the right members of these formulas, the following functions should be replaced with integrals
equal to them:
206 Trigonometric Functions 2.611
F(x, k) should be replaced with/integraldisplaydx
Δ
E(x, k) should be replaced with/integraldisplay
Δdx
−1
kln (kcosx+ Δ) should be replaced with/integraldisplaysinxdx
Δ
1
karcsin( ksinx) should be replaced with/integraldisplaycosxdx
Δ
1
2lnΔ−cosx
Δ + cos xshould be replaced with/integraldisplaydx
Δsinx
1
2k/primelnΔ+k/primesinx
Δ−k/primesinxshould be replaced with/integraldisplaydx
Δcos x
1
2k/primelnΔ+k/prime
Δ−k/primeshould be replaced with/integraldisplaytanx
Δdx
1
2ln1−Δ
1+Δshould be replaced with/integraldisplaycotx
Δdx
(2) Then, on both sides of the equations, we should replace Δ with i/radicalbig
a2sin2x−1,kwithaandk/prime2
with 1 −a2.
(3) Both sides of the resulting equations should be multiplied by i, as a result of which only real
functions/parenleftbig
a2>1/parenrightbig
should appear on both sides of the equations.
(4) The integrals on the right sides of the equations should be replaced with their values found from
formulas 2.599 .
Examples:
1. We rewrite equation 2.584 4i nt h ef o r m
/integraldisplaysin2x
i/radicalbig
a2sin2x−1dx=1
a2/integraldisplaydx
i/radicalbig
a2sin2x−1−1
a2/integraldisplay
i/radicalbig
a2sin2x−1dx,
from which we get
/integraldisplaysin2xdx/radicalbig
a2sin2x−1=1
a2/braceleftBigg/integraldisplaydx/radicalbig
a2sin2x−1+/integraldisplay/radicalbig
a2sin2x−1dx/bracerightBigg
=−1
aE/parenleftBigg
α,√
a2−1
a/parenrightBigg
/bracketleftbig
a2>1/bracketrightbig
2. We rewrite equation 2.584 58 as follows:
/integraldisplaydx
i5/radicalBig/parenleftbig
a2sin2x−1/parenrightbig5=−2a4/parenleftbig
a2−2/parenrightbig
sin2x−/parenleftbig
3a2−5/parenrightbig
a2
3( 1−a2)2i3/radicalBig/parenleftbig
a2sin2x−1/parenrightbig3sinxcosx
−1
3(1−a2)/integraldisplaydx
i/radicalbig
a2sin2x−1−2a2−4
3( 1−a2)2/integraldisplay
i/radicalbig
a2sin2x−1dx
2.613 Elliptic and pseudo-elliptic integrals 207
from which we obtain/integraldisplaydx/radicalBig/parenleftbig
a2sin2x−1/parenrightbig5=2a4/parenleftbig
a2−2/parenrightbig
sin2x−/parenleftbig
3a2−5/parenrightbig
a2
3( 1−a2)2/radicalBig/parenleftbig
a2sin2x−1/parenrightbig3sinxcosx+1
3(1−a2)2a
×/braceleftBigg
/parenleftbig
a2−3/parenrightbig
F/parenleftBigg
α,√
a2−1
a/parenrightBigg
−2a2/parenleftbig
a2−2/parenrightbig
E/parenleftBigg
α,√
a2−1
a/parenrightBigg/bracerightBigg
/bracketleftbig
a2>1/bracketrightbig
3. We rewrite equation 2.584 71 in the form/integraldisplaydx
sinxcosxi/radicalbig
a2sin2x−1=/integraldisplaycotxdx
i/radicalbig
a2sin2x−1+/integraldisplaytanxdx
i/radicalbig
a2sin2x−1,
from which we obtain
/integraldisplaydx
sinxcosx/radicalbig
a2sin2x−1=1
2√
a2−1ln√
a2−1+/radicalbig
a2sin2x−1
√
a2−1−/radicalbig
a2sin2x−1−arcsin/parenleftbigg1
asinx/parenrightbigg
/bracketleftbig
a2>1/bracketrightbig
2.612 Integrals of the form/integraltext
R/parenleftbig
sinx,cosx,√
1−k2cos2x/parenrightbig
dx.
To find integrals of the form/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
1−k2cos2x/parenrightBig
dx, we make the substitution x=π
2−y,
which yields/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
1−k2cos2x/parenrightBig
dx=−/integraldisplay
R/parenleftbigg
cosy,siny,/radicalBig
1−k2sin2y/parenrightbigg
dy.
The integrals/integraldisplay
R/parenleftbigg
cosy,siny,/radicalBig
1−k2sin2y/parenrightbigg
dyare found from formulas 2.583 and2.584 .A s a
result of the use of these formulas (where it is assumed that the original integral can be reduced only to
integrals of the first and second Legendre forms), when we replace the functions F(x, k)a n dE(x, k) with
the corresponding integrals, we obtain an expression of the form
−g(cosy,siny)−A/integraldisplaydy/radicalbig
1−k2sin2y−B/integraldisplay/radicalBig
1−k2sin2ydy
Returning now to the original variable x, we obtain/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
1−k2cos2x/parenrightBig
dx=−g(sinx,cosx)−A/integraldisplaydx√
1−k2cos2x−B/integraldisplay/radicalbig
1−k2cos2xd x
The integrals appearing in this expression are found from the formulas
1./integraldisplaydx√
1−k2cos2x=F/parenleftbigg
arcsin/parenleftbiggsinx√
1−k2cos2x/parenrightbigg
,k/parenrightbigg
2./integraldisplay/radicalbig
1−k2cos2xdx=E/parenleftbigg
arcsin/parenleftbiggsinx√
1−k2cos2x/parenrightbigg
,k/parenrightbigg
−k2sinxcosx√
1−k2cos2x
2.613 Integrals of the form/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
1−p2cos2x/parenrightBig
dx [p>1].
To find integrals of the type/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
1−p2cos2x/parenrightBig
dx,w h e r e[ p>1], we proceed as in
section 2.612 . Here, we use the formulas
208 Trigonometric Functions 2.614
1./integraldisplaydx/radicalbig
1−p2cos2x=−1
pF/parenleftbigg
arcsin( pcosx),1
p/parenrightbigg
[p>1]
2./integraldisplay/radicalbig
1−p2cos2xdx=p2−1
pF/parenleftbigg
arcsin ( pcosx),1
p/parenrightbigg
−pE/parenleftbigg
arcsin ( pcosx),1
p/parenrightbigg
2.614 Integrals of the form/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
1+p2cos2x/parenrightBig
dx.
To find integrals of the type/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
1+p2cos2x/parenrightBig
dx, we need to make the substitution
x=π
2−y. This yields
/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
1+p2cos2x/parenrightBig
dx=−/integraldisplay
R/parenleftbigg
cosy,siny,/radicalBig
1+p2sin2y/parenrightbigg
dy.
To calculate the integrals −/integraldisplay
R/parenleftbigg
cosy,siny,/radicalBig
1+p2sin2y/parenrightbigg
dy, we need to use first what was said
in2.598 and2.612 and then, after returning to the variable x, the formulas
1./integraldisplaydx/radicalbig
1+p2cos2x=1/radicalbig
1+p2F/parenleftBigg
x,p/radicalbig
1+p2/parenrightBigg
2./integraldisplay/radicalbig
1+p2cos2xdx=/radicalbig
1+p2E/parenleftBigg
x,p/radicalbig
1+p2/parenrightBigg
2.615 Integrals of the form/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
a2cos2x−1/parenrightBig
dx [a>1].
To find integrals of the type/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
a2cos2x−1/parenrightBig
dx, we need to make the substitution
x=π
2−y. This yields
/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
a2cos2x−1/parenrightBig
dx=−/integraldisplay
R/parenleftbigg
cosy,siny,/radicalBig
a2sin2y−1/parenrightbigg
dy
To calculate the integrals −/integraltext
R/parenleftBig
cosy,siny,/radicalbig
a2sin2y−1/parenrightBig
dy, we use what was said in 2.611 and
then, after returning to the variable x, we use the formulas
1./integraldisplaydx√
a2cos2x−1=1
aF/parenleftBigg
arcsin/parenleftbiggasinx√
a2−1/parenrightbigg
,√
a2−1
a/parenrightBigg
[a>1]
2./integraldisplay/radicalbig
a2cos2x−1dx=aE/parenleftBigg
arcsin/parenleftbiggasinx√
a2−1/parenrightbigg
,√
a2−1
a/parenrightBigg
−1
aF/parenleftBigg
arcsin/parenleftbiggasinx√
a2−1/parenrightbigg
,√
a2−1
a/parenrightBigg
[a>1]
2.616 Elliptic and pseudo-elliptic integrals 209
2.61611Integrals of the form/integraldisplay
R/parenleftbigg
sinx,cosx,/radicalBig
1−p2sin2x,/radicalBig
1−q2sin2x/parenrightbigg
dx.
Notation :α=a r c s i n/parenleftBigg/radicalbig
1−p2sinx/radicalbig
1−p2sin2x/parenrightBigg
.
1./integraldisplaydx/radicalBig/parenleftbig
1−p2sin2x/parenrightbig/parenleftbig
1−q2sin2x/parenrightbig=1/radicalbig
1−p2F/parenleftBigg
α,/radicalBigg
q2−p2
1−p2/parenrightBigg
/bracketleftBig
0<p2<q2<1,0<x≤π
2/bracketrightBig
BY (284.00)
2./integraldisplaytan2xdx/radicalBig/parenleftbig
1−p2sin2x/parenrightbig/parenleftbig
1−q2sin2x/parenrightbig=tanx/radicalbig
1−q2sin2x
(1−q2)/radicalbig
1−p2sin2x
−1
(1−q2)/radicalbig
1−p2E/parenleftBigg
α,/radicalBigg
q2−p2
1−p2/parenrightBigg
/bracketleftBig
0<p2<q2<1,0<x≤π
2/bracketrightBig
BY (284.07)
3./integraldisplaytan4xdx/radicalBig/parenleftbig
1−p2sin2x/parenrightbig/parenleftbig
1−q2sin2x/parenrightbig
=1
3( 1−q2)2(1−p2)3
2×/bracketleftBigg
2/parenleftbig
2−p2−q2/parenrightbig
E/parenleftBigg
α,/radicalBigg
q2−p2
1−p2/parenrightBigg
−/parenleftbig
1−q2/parenrightbig
F/parenleftBigg
α,/radicalBigg
q2−p2
1−p2/parenrightBigg/bracketrightBigg
+2p2+q2−3+s i n2x/parenleftbig
4−3p2−2q2+p2q2/parenrightbig
3(1−p2)(1−q2)2sinx
cos2x/radicalBigg
1−q2sin2x
1−p2sin2x/bracketleftBig
0<p2<q2<1,0<x≤π
2/bracketrightBig
BY (284.07)
4./integraldisplaysin2xdx/radicalBig/parenleftbig
1−p2sin2x/parenrightbig/parenleftbig
1−q2sin2x/parenrightbig3
=/radicalbig
1−p2
(1−q2)(q2−p2)E/parenleftBigg
α,/radicalBigg
q2−p2
1−p2/parenrightBigg
−1
(q2−p2)/radicalbig
1−p2F/parenleftBigg
α,/radicalBigg
q2−p2
1−p2/parenrightBigg
−sinxcosx
(1−q2)/radicalBig/parenleftbig
1−p2sin2x/parenrightbig/parenleftbig
1−q2sin2x/parenrightbig
/bracketleftBig
0<p2<q2<1,0<x≤π
2/bracketrightBig
BY (284.06)
5./integraldisplaycos2xdx/radicalBig/parenleftbig
1−p2sin2x/parenrightbig3/parenleftbig
1−q2sin2x/parenrightbig
=/radicalbig
1−p2
q2−p2E/parenleftBigg
α,/radicalBigg
q2−p2
1−p2/parenrightBigg
−1−q2
(q2−p2)/radicalbig
1−p2F/parenleftBigg
α,/radicalBigg
q2−p2
1−p2/parenrightBigg
/bracketleftBig
0<p2<q2<1,0<x≤π
2/bracketrightBig
BY (284.05)
210 Trigonometric Functions 2.617
6./integraldisplaycos4xdx/radicalBig/parenleftbig
1−p2sin2x/parenrightbig5/parenleftbig
1−q2sin2x/parenrightbig
=/parenleftbig
1−p2/parenrightbig3
2
3(q2−p2)2⎡
⎣/parenleftbig
2+p2−3q2/parenrightbig/parenleftbig
1−q2/parenrightbig
(1−p2)2F/parenleftBigg
α,/radicalBigg
q2−p2
1−p2/parenrightBigg
+22q2−p2−1
1−p2E/parenleftBigg
α,/radicalBigg
q2−p2
1−p2/parenrightBigg⎤
⎦+/parenleftbig
1−p2/parenrightbig
sinxcosx/radicalbig
1−q2sin2x
3(q2−p2)/radicalBig/parenleftbig
1−p2sin2x/parenrightbig3
/bracketleftBig
0<p2<q2<1,0<x≤π
2/bracketrightBig
BY (284.05)
7./integraldisplaydx
1−p2sin2x/radicalBigg
1−q2sin2x
1−p2sin2x=1/radicalbig
1−p2E/parenleftBigg
α,/radicalBigg
q2−p2
1−p2/parenrightBigg
/bracketleftBig
0<p2<q2<1,0<x≤π
2/bracketrightBig
BY (284.01)
8./integraldisplay/radicaltp/radicalvertex/radicalvertex/radicalbt1−p2sin2x
/parenleftbig
1−q2sin2x/parenrightbig3dx=/radicalbig
1−p2
1−q2E/parenleftBigg
α,/radicalBigg
q2−p2
1−p2/parenrightBigg
−q2−p2
1−q2sinxcosx/radicalBig/parenleftbig
1−p2sin2x/parenrightbig/parenleftbig
1−q2sin2x/parenrightbig
/bracketleftBig
0<p2<q2<1,0<x≤π
2/bracketrightBig
.
BY (284.04)
9./integraldisplaydx
1+(p2r2−p2−r2)sin2x/radicalBigg
1−p2sin2x
1−q2sin2x=1/radicalbig
1−p2Π/parenleftBigg
α,r2,/radicalBigg
q2−p2
1−p2/parenrightBigg
/bracketleftBig
0<p2<q2<1,0<x≤π
2/bracketrightBig
.
BY (284.02)
2.617 Notation :α=a r c s i n/radicalBigg√
b2+c2−bsinx−ccosx
2√
b2+c2,r=/radicalBigg
2√
b2+c2
a+√
b2+c2.
1./integraldisplaydx√
a+bsinx+ccosx
=−2/radicalbig
a+√
b2+c2F(α,r)
/bracketleftbigg
0</radicalbig
b2+c2<a , arcsinb√
b2+c2−π≤x<arcsinb√
b2+c2/bracketrightbigg
BY (294.00)
=−√
2
4√
b2+c2F(α,r)
/bracketleftbigg
0<|a|</radicalbig
b2+c2,arcsinb√
b2+c2−arccos/parenleftbigg
−a√
b2+c2/parenrightbigg
≤x<arcsinb√
b2+c2/bracketrightbigg
BY (293.00)
2.618 Elliptic and pseudo-elliptic integrals 211
2./integraldisplaysinxdx√
a+bsinx+ccosx=−√
2b
4/radicalBig
(b2+c2)3{2E(α,r)−F(α,r)}+2c
b2+c2√
a+bsinx+ccosx
/bracketleftbigg
0<|a|</radicalbig
b2+c2,arcsinb√
b2+c2−arccos/parenleftbigg
−a√
b2+c2/parenrightbigg
≤x<arcsinb√
b2+c2/bracketrightbigg
BY (293.05)
3./integraldisplay(bcosx−csinx)dx√
a+bsinx+ccosx=2√
a+bsinx+ccosx
4./integraldisplay√
b2+c2+bsinx+ccosx√
a+bsinx+ccosxdx
=−2/radicalBig
a+/radicalbig
b2+c2E(α,r)+2/parenleftbig
a−√
b2+c2/parenrightbig
/radicalbig
a+√
b2+c2F(α,r)
/bracketleftbigg
0</radicalbig
b2+c2<a , arcsinb√
b2+c2−π≤x<arcsinb√
b2+c2/bracketrightbigg
BY (294.04)
=−2√
24/radicalbig
b2+c2E(α,r)
/bracketleftbigg
0<|a|</radicalbig
b2+c2,arcsinb√
b2+c2−arccos/parenleftbigg
−a√
b2+c2/parenrightbigg
≤x<arcsinb√
b2+c2/bracketrightbigg
BY (293.01)
5./integraldisplay√
a+bsinx+ccosxd x
=−2/radicalBig
a+/radicalbig
b2+c2E(α,r)
/bracketleftbigg
0</radicalbig
b2+c2<a , arcsinb√
b2+c2−π≤x<arcsinb√
b2+c2/bracketrightbigg
BY (294.01)
=−2√
24/radicalbig
b2+c2E(α,r)+√
2/parenleftbig√
b2+c2−a/parenrightbig
4√
b2+c2F(α,r)
/bracketleftbigg
0<|a|</radicalbig
b2+c2,arcsinb√
b2+c2−arccos/parenleftbigg−a√
b2+c2/parenrightbigg
≤x<arcsinb√
b2+c2/bracketrightbigg
BY (293.03)
2.618 Integrals of the form/integraldisplay
R/parenleftBig
sinax,cosax,√
cos2ax/parenrightBig
dx=1
a/integraldisplay
R/parenleftBig
sint,cost,/radicalbig
1−2s in2t/parenrightBig
dt
where the substitution t=axhas been used.
Notation :α=a r c s i n/parenleftbig√
2s inax/parenrightbig
The integrals/integraldisplay
R/parenleftBig
sinax,cosax,√
cos 2ax/parenrightBig
dxare special cases of the integrals 2.595 .f o r ( p=2 ) .
We give some formulas:
1./integraldisplaydx√
cos2ax=1
a√
2F/parenleftbigg
α,1√
2/parenrightbigg/bracketleftBig
0<a x≤π
4/bracketrightBig
2./integraldisplaycos2ax√
cos2axdx=1
a√
2E/parenleftbigg
α,1√
2/parenrightbigg /bracketleftBig
0<a x≤π
4/bracketrightBig
212 Trigonometric Functions 2.619
3./integraldisplaydx
cos2ax√
cos2ax=√
2
aE/parenleftbigg
α,1√
2/parenrightbigg
−tanx
a√
cos2ax
/bracketleftBig
0<a x≤π
4/bracketrightBig
4./integraldisplaydx
cos4ax√
cos2ax=2√
2
aE/parenleftbigg
α,1√
2/parenrightbigg
−√
2
3aF/parenleftbigg
α,1√
2/parenrightbigg
−/parenleftbig
6c os2ax+1/parenrightbig
sinax
3acos3ax√
cos 2ax
/bracketleftBig
0<x≤π
4/bracketrightBig
5./integraldisplaytan2axdx√
cos2ax=√
2
aE/parenleftbigg
α,1√
2/parenrightbigg
−1
a√
2F/parenleftbigg
α,1√
2/parenrightbigg
−1
atanax√
cos2ax
/bracketleftBig
0<x≤π
2/bracketrightBig
6./integraldisplaytan4axdx√
cos2ax=1
3a√
2F/parenleftbigg
α,1√
2/parenrightbigg
−sinax
3acos3ax√
cos2ax
/bracketleftBig
0<a x≤π
4/bracketrightBig
7./integraldisplaydx/parenleftbig
1−2r2sin2ax/parenrightbig√
cos 2ax=1
a√
2Π/parenleftbigg
α,r2,1√
2/parenrightbigg/bracketleftBig
0<a x≤π
4/bracketrightBig
8./integraldisplaydx√
cos32ax=1
a√
2F/parenleftbigg
α,1√
2/parenrightbigg
−√
2
aE/parenleftbigg
α,1√
2/parenrightbigg
+sin 2ax
a√
cos2ax
/bracketleftBig
0<a x≤π
4/bracketrightBig
9./integraldisplaysin2axdx√
cos32ax=sin 2ax
2a√
cos2ax−1
a√
2E/parenleftbigg
α,1√
2/parenrightbigg/bracketleftBig
0<a x≤π
4/bracketrightBig
10./integraldisplaydx√
cos52ax=1
3a√
2F/parenleftbigg
α,1√
2/parenrightbigg
+sin 2ax
3a√
cos32ax/bracketleftBig
0<a x≤π
4/bracketrightBig
11./integraldisplay√
cos2axdx =√
2
aE/parenleftbigg
α,1√
2/parenrightbigg
−1
a√
2F/parenleftbigg
α,1√
2/parenrightbigg
/bracketleftBig
0<a x≤π
4/bracketrightBig
12./integraldisplay√
cos2ax
cos2axdx=√
2
a/braceleftbigg
F/parenleftbigg
α,1√
2/parenrightbigg
−E/parenleftbigg
α,1√
2/parenrightbigg/bracerightbigg
+1
atanax√
cos2ax
/bracketleftBig
0<x≤π
4/bracketrightBig
2.619 Integrals of the form/integraldisplay
R/parenleftbig
sinax,cosax,√
−cos 2ax/parenrightbig
dx=1
a/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
2s in2x−1/parenrightBig
dx
Notation :α=a r c s i n/parenleftbig√
2c osax/parenrightbig
The integrals/integraldisplay
R/parenleftBig
sinx,cosx,/radicalbig
2s in2x−1/parenrightBig
dxare special cases of the integrals 2.599 and2.611
for (a= 2). We give some formulas:
2.621 Elliptic and pseudo-elliptic integrals 213
1./integraldisplaydx√−cos 2ax=−1
a√
2F/parenleftbigg
α,1√
2/parenrightbigg
2./integraldisplaycos2axdx√−cos 2ax=1
a√
2/bracketleftbigg
E/parenleftbigg
α,1√
2/parenrightbigg
−F/parenleftbigg
α,1√
2/parenrightbigg/bracketrightbigg
3./integraldisplaycos4axdx√−cos 2ax=1
3a√
2/bracketleftbigg
3F/parenleftbigg
α,1√
2/parenrightbigg
−5
2E/parenleftbigg
α,1√
2/parenrightbigg/bracketrightbigg
−1
12asin2ax√
−cos 2ax
4./integraldisplaydx
sin2ax√−cos 2ax=1
acotax√
−cos 2ax−√
2
aE/parenleftbigg
α,1√
2/parenrightbigg
5./integraldisplaydx
sin4ax√−cos 2ax=2
3a√
2/bracketleftbigg
F/parenleftbigg
α,1√
2/parenrightbigg
−6E/parenleftbigg
α,1√
2/parenrightbigg/bracketrightbigg
+1
3acosax
sin3ax/parenleftbig
6s in2ax+1/parenrightbig√
−cos 2ax
6./integraldisplaycot2axdx√−cos 2ax=1
a√
2/bracketleftbigg
F/parenleftbigg
α,1√
2/parenrightbigg
−2E/parenleftbigg
α,1√
2/parenrightbigg/bracketrightbigg
+1
acotax√
−cos 2ax
7./integraldisplaydx
(1−2r2cos2ax)√−cos 2ax=−1
a√
2Π/parenleftbigg
α,r2,1√
2/parenrightbigg
8./integraldisplaydx√
−cos32ax=1
a√
2/bracketleftbigg
F/parenleftbigg
α,1√
2/parenrightbigg
−2E/parenleftbigg
α,1√
2/parenrightbigg/bracketrightbigg
+sin2ax
a√−cos 2ax
9./integraldisplaycos2axdx√
−cos32ax=sin 2ax
2a√−cos 2ax−1
a√
2E/parenleftbigg
α,1√
2/parenrightbigg
10./integraldisplaydx√
−cos52ax=−1
3a√
2F/parenleftbigg
α,1√
2/parenrightbigg
−sin2ax
3a√
−cos32ax
11./integraldisplay√
−cos 2axdx =1
a√
2/bracketleftbigg
F/parenleftbigg
α,1√
2/parenrightbigg
−2E/parenleftbigg
α,1√
2/parenrightbigg/bracketrightbigg
2.621 Integrals of the form/integraldisplay
R/parenleftBig
sinax,cosax,√
sin 2ax/parenrightBig
dx.
Notation :α=a r c s i n/radicalbigg
2s inax
1+s i n ax+c o s ax.
1./integraldisplaydx√
sin 2ax=√
2
aF/parenleftbigg
α,1√
2/parenrightbigg
BY (287.50)
2./integraldisplaysinaxdx√
sin 2ax=√
2
a⎡
⎣1+i
2Π/parenleftbigg
α,1+i
2,1√
2/parenrightbigg
+1−i
2Π/parenleftbigg
α,1−i
2,1√
2/parenrightbigg
+F/parenleftbigg
α,1√
2/parenrightbigg
−2E/parenleftbigg
α,1√
2/parenrightbigg⎤
⎦
BY (287.57)
3./integraldisplaysinaxdx
(1 + sin ax+c o s ax)√
sin 2ax=√
2
a/bracketleftbigg
F/parenleftbigg
α,1√
2/parenrightbigg
−E/parenleftbigg
α,1√
2/parenrightbigg/bracketrightbigg
BY (287.54)
214 Trigonometric Functions 2.631
4./integraldisplaysinaxdx
(1−sinax+c o s ax)√
sin 2ax=√
2
a/braceleftbigg√
tanax−E/parenleftbigg
α,1√
2/parenrightbigg/bracerightbigg
/bracketleftBig
ax/negationslash=π
2/bracketrightBig
BY (287.55)
5./integraldisplay(1 + cos ax)dx
(1 + sin ax+c o s ax)√
sin 2ax=√
2
aE/parenleftbigg
α,1√
2/parenrightbigg
BY (287.51)
6./integraldisplay(1 + cos ax)dx
(1−sinax+c o s ax)√
sin 2ax=√
2
a/braceleftbigg
F/parenleftbigg
α,1√
2/parenrightbigg
−E/parenleftbigg
α,1√
2/parenrightbigg
+√
tanax/bracerightbigg
/bracketleftBig
ax/negationslash=π
2/bracketrightBig
BY (287.56)
7./integraldisplay(1−sinax+c o s ax)dx
(1 + sin ax+c o s ax)√
sin 2ax=√
2
a/braceleftbigg
2E/parenleftbigg
α,1√
2/parenrightbigg
−F/parenleftbigg
α,1√
2/parenrightbigg/bracerightbigg
BY (287.53)
8./integraldisplay(1 + sin ax+c o s ax)dx
[1 + cos ax+( 1−2r2)s i nax]√
sin 2ax=√
2
aΠ/parenleftbigg
α,r2,1√
2/parenrightbigg
. BY (287.52)
2.63–2.65 Products of trigonometric functions and powers
2.631
1./integraldisplay
xrsinpxcosqxdx=1
(p+q)2⎡
⎣(p+q)xrsinp+1xcosq−1x
+rxr−1sinpxcosqx−r(r−1)/integraldisplay
xr−2sinpxcosqxdx
−rp/integraldisplay
xr−1sinp−1xcosq−1xdx+(q−1)(p+q)/integraldisplay
xrsinpxcosq−2xdx⎤
⎦
=1
(p+q)2⎡
⎣−(p+q)xrsinp−1xcosq+1x
+rxr−1sinpxcosqx−r(r−1)/integraldisplay
xr−2sinpxcosqxdx
+rq/integraldisplay
xr−1sinp−1xcosq−1xdx+(p−1)(p+q)/integraldisplay
xrsinp−2xcosqxdx⎤
⎦
GU (331)(1)
2./integraldisplay
xmsinnxdx=xm−1sinn−1x
n2{msinx−nxcosx}
+n−1
n/integraldisplay
xmsinn−2xdx−m(m−1)
n2/integraldisplay
xm−2sinnxdx
3./integraldisplay
xmcosnxdx=xm−1cosn−1x
n2{mcosx+nxsinx}
+n−1
n/integraldisplay
xmcosn−2xdx−m(m−1)
n2/integraldisplay
xm−2cosnxdx
2.633 Trigonometric functions and powers 215
4./integraldisplay
xnsin2mxdx=/parenleftbigg2m
m/parenrightbiggxn+1
22m(n+1 )
+(−1)m
22m−1m−1/summationdisplay
k=0(−1)k/parenleftbigg2m
k/parenrightbigg/integraldisplay
xncos(2m−2k)xdx
(see2.633 2) TI 333
5./integraldisplay
xnsin2m+1xdx=(−1)m
22mm/summationdisplay
k=0(−1)k/parenleftbigg2m+1
k/parenrightbigg/integraldisplay
xnsin(2m−2k+1 )xdx
(see2.633 1) TI 333
6./integraldisplay
xncos2mxdx=/parenleftbigg2m
m/parenrightbiggxn+1
22m(n+1 )
+1
22m−1m−1/summationdisplay
k=0/parenleftbigg2m
k/parenrightbigg/integraldisplay
xncos(2m−2k)xdx
(see2.633 2) TI 333
7./integraldisplay
xncos2m+1xdx=1
22mm/summationdisplay
k=0/parenleftbigg2m+1
k/parenrightbigg/integraldisplay
xncos(2m−2k+1 )xdx
(see2.633 2) TI 333
2.632
1./integraldisplay
xμ−1sinβxdx =i
2(iβ)−μγ(μ, iβx )−i
2(−iβ)−μγ(μ,−iβx)
[Reμ>−1,x > 0] ET I 317(2)
2./integraldisplay
xμ−1sinaxdx =−1
2aμ/braceleftbigg
exp/bracketleftbiggπi
2(μ−1)/bracketrightbigg
Γ(μ,−iax) + exp/bracketleftbiggπi
2(1−μ)/bracketrightbigg
Γ(μ, iax)/bracerightbigg
[Reμ<1,a > 0,x > 0]ET I 317(3)
3./integraldisplay
xμ−1cosβxdx =1
2/braceleftBig
(iβ)−μγ(μ, iβx )+(−iβ)−μγ(μ,−iβx)/bracerightBig
[Reμ>0,x > 0] ET I 319(22)
4./integraldisplay
xμ−1cosaxdx =−1
2aμ/braceleftBig
exp/parenleftBig
iμπ
2/parenrightBig
Γ(μ,−iax) + exp/parenleftBig
−iμπ
2/parenrightBig
Γ(μ, iax)/bracerightBig
ET I 319(23)
2.633
1./integraldisplay
xnsinaxdx =−n/summationdisplay
k=0k!/parenleftBign
k/parenrightBigxn−k
ak+1cos/parenleftbigg
ax+1
2kπ/parenrightbigg
TI (487)
2.8/integraldisplay
xncosaxdx =n/summationdisplay
k=0k!/parenleftBign
k/parenrightBigxn−k
ak+1sin/parenleftbigg
ax+1
2kπ/parenrightbigg
TI (486)
3./integraldisplay
x2nsinxdx=( 2n)!/braceleftBiggn/summationdisplay
k=0(−1)k+1x2n−2k
(2n−2k)!cosx+n−1/summationdisplay
k=0(−1)kx2n−2k−1
(2n−2k−1)!sinx/bracerightBigg
216 Trigonometric Functions 2.634
4./integraldisplay
x2n+1sinxdx=( 2n+1 ) !/braceleftBiggn/summationdisplay
k=0(−1)k+1x2n−2k+1
(2n−2k+1 ) !cosx+n/summationdisplay
k=0(−1)kx2n−2k
(2n−2k)!sinx/bracerightBigg
5./integraldisplay
x2ncosxdx=( 2n)!/braceleftBiggn/summationdisplay
k=0(−1)kx2n−2k
(2n−2k)!sinx+n−1/summationdisplay
k=0(−1)kx2n−2k−1
(2n−2k−1)!cosx/bracerightBigg
6./integraldisplay
x2n+1cosxdx=( 2n+1 ) !/braceleftBiggn/summationdisplay
k=0(−1)kx2n−2k+1
(2n−2k+1 ) !sinx+n/summationdisplay
k=0x2n−2k
(2n−2k)!cosx/bracerightBigg
2.634
1./integraldisplay
Pn(x)sinmxdx =−cosmx
m⌊n/2⌋/summationdisplay
k=0(−1)kP(2k)
n(x)
m2k+sinmx
m⌊(n+1)/2⌋/summationdisplay
k=1(−1)k−1P(2k−1)
n (x)
m2k−1
2./integraldisplay
Pn(x)cosmxdx =sinmx
m⌊n/2⌋/summationdisplay
k=0(−1)kP(2k)
n(x)
m2k+cosmx
m⌊(n+1)/2⌋/summationdisplay
k=1(−1)k−1P(2k−1)
n (x)
m2k−1
In formulas 2.634 ,Pn(x)i sa n y nth-degree polynomial, and P(k)
n(x)i si t s kthderivative with respect to x.
2.635 Notation :z1=a+bx.
1./integraldisplay
z1sinkxdx =−1
kz1coskx+b
k2sinkx
2./integraldisplay
z1coskxdx =1
kz1sinkx+b
k2coskx
3./integraldisplay
z2
1sinkxdx =1
k/parenleftbigg2b2
k2−z2
1/parenrightbigg
coskx+2bz1
k2sinkx
4./integraldisplay
z2
1coskxdx =1
k/parenleftbigg
z2
1−2b2
k2/parenrightbigg
sinkx+2bz1
k2coskx
5./integraldisplay
z3
1sinkxdx =z1
k/parenleftbigg6b2
k2−z2
1/parenrightbigg
coskx+3b
k2/parenleftbigg
z2
1−2b2
k2/parenrightbigg
sinkx
6./integraldisplay
z3
1coskxdx =z1
k/parenleftbigg
z2
1−6b2
k2/parenrightbigg
sinkx+3b
k2/parenleftbigg
z2
1−2b2
k2/parenrightbigg
coskx
7./integraldisplay
z4
1sinkxdx =−1
k/parenleftbigg
z4
1−12b2
k2z2
1+24b4
k4/parenrightbigg
coskx+4bz1
k2/parenleftbigg
z2
1−6b2
k2/parenrightbigg
sinkx
8./integraldisplay
z4
1coskxdx =1
k/parenleftbigg
z4
1−12b2
k2z2
1+24b4
k4/parenrightbigg
sinkx+4bz1
k2/parenleftbigg
z2
1−6b2
k2/parenrightbigg
coskx
9./integraldisplay
z5
1sinkxdx =5b
k2/parenleftbigg
z4
1−12b2
k2z2
1+24b4
k4/parenrightbigg
sinkx−z1
k/parenleftbigg
z4
1−20b2
k2z2
1+120b4
k4/parenrightbigg
coskx
10./integraldisplay
z5
1coskxdx =5b
k2/parenleftbigg
z4
1−12b2
k2z2
1+24b4
k4/parenrightbigg
coskx+z1
k/parenleftbigg
z4
1−20b2
k2z2
1+120b4
k4/parenrightbigg
sinkx
2.637 Trigonometric functions and powers 217
11./integraldisplay
z6
1sinkxdx =6bz1
k2/parenleftbigg
z4
1−20b2
k2z2
1+120b4
k4/parenrightbigg
sinkx
−1
k/parenleftbigg
z6
1−30b2
k2z4
1+360b4
k4z2
1−720b6
k6/parenrightbigg
coskx
12./integraldisplay
z6
1coskxdx =6bz1
k2/parenleftbigg
z4
1−20b2
k2z2
1+120b4
k4/parenrightbigg
coskx
+1
k/parenleftbigg
z6
1−30b2
k2z4
1+360b4
k4z2
1−720b6
k6/parenrightbigg
sinkx
2.636
1./integraldisplay
xnsin2xdx=xn+1
2(n+1 )
+n!
4⎧
⎨
⎩⌊n/2⌋/summationdisplay
k=0(−1)k+1xn−2k
22k(n−2k)!sin 2x+⌊(n−1)/2⌋/summationdisplay
k=0(−1)k+1xn−2k−1
22k+1(n−2k−1)!cos2x⎫
⎬
⎭
GU (333)(2e)
2./integraldisplay
xncos2xdx=xn+1
2(n+1 )
−n!
4⎧
⎨
⎩⌊n/2⌋/summationdisplay
k=0(−1)k+1xn−2k
22k(n−2k)!sin 2x+⌊(n−1)/2⌋/summationdisplay
k=0(−1)k+1xn−2k−1
22k+1(n−2k−1)!cos2x⎫
⎬
⎭
GU (333)(3e)
3./integraldisplay
xsin2xdx=x2
4−x
4sin 2x−1
8cos2x
4./integraldisplay
x2sin2xdx=x3
6−x
4cos2x−1
4/parenleftbigg
x2−1
2/parenrightbigg
sin 2x MZ 241
5./integraldisplay
xcos2xdx=x2
4+x
4sin 2x+1
8cos2x
6./integraldisplay
x2cos2xdx=x3
6+x
4cos 2x+1
4/parenleftbigg
x2−1
2/parenrightbigg
sin2x MZ 245
2.637
1.11/integraldisplay
xnsin3xdx=n!
4⎧
⎨
⎩⌊n/2⌋/summationdisplay
k=0(−1)kxn−2k
(n−2k)!/parenleftbiggcos3x
32k+1−3c osx/parenrightbigg
−⌊(n−1)/2⌋/summationdisplay
k=0(−1)kxn−2k−1
(n−2k−1)!/parenleftbiggsin 3x
32k+2−3s inx/parenrightbigg⎫
⎬
⎭
GU(333)(2f)
218 Trigonometric Functions 2.638
2./integraldisplay
xncos3xdx=n!
4⎧
⎨
⎩⌊n/2⌋/summationdisplay
k=0(−1)kxn−2k
(n−2k)!/parenleftbiggsin 3x
32k+1+3s i n x/parenrightbigg
+[(n−1)/2]/summationdisplay
k=0(−1)kxn−2k−1
(n−2k−1)!/parenleftbiggcos 3x
32k+2+ 3cos x/parenrightbigg⎫
⎬
⎭
GU(333)(3f)
3./integraldisplay
xsin3xdx=3
4sinx−1
36sin 3x−3
4xcosx+x
12cos 3x
4./integraldisplay
x2sin3xdx=−/parenleftbigg3
4x2+3
2/parenrightbigg
cosx+/parenleftbiggx2
12+1
54/parenrightbigg
cos3x+3
2xsinx−x
18sin 3x MZ 241
5./integraldisplay
xcos3xdx=3
4cosx+1
36cos3x+3
4xsinx+x
12sin 3x
6./integraldisplay
x2cos3xdx=/parenleftbigg3
4x2−3
2/parenrightbigg
sinx+/parenleftbiggx2
12−1
54/parenrightbigg
sin 3x+3
2xcosx+x
18cos 3x MZ 245, 246
2.638
1./integraldisplaysinqx
xpdx=−sinq−1x[(p−2)sinx+qxcosx]
(p−1)(p−2)xp−1
−q2
(p−1)(p−2)/integraldisplaysinqxdx
xp−2+q(q−1)
(p−1)(p−2)/integraldisplaysinq−2xdx
xp−2
[p/negationslash=1,p/negationslash=2 ] TI (496)
2./integraldisplaycosqx
xpdx=−cosq−1x[(p−2)cos x−qxsinx]
(p−1)(p−2)xp−1
−q2
(p−1)(p−2)/integraldisplaycosqxdx
xp−2+q(q−1)
(p−1)(p−2)/integraldisplaycosq−2xdx
xp−2
[p/negationslash=1,p/negationslash=2 ] TI (495)
3.6/integraldisplaysinxdx
xp=−sinx
(p−1)xp−1+1
p−1/integraldisplaycosxdx
xp−1
=−sinx
(p−1)xp−1−cosx
(p−1)(p−2)xp−2−1
(p−1)(p−2)/integraldisplaysinxdx
xp−2
(p>2) TI (492)
4.6/integraldisplaycosxdx
xp=−cosx
(p−1)xp−1−1
p−1/integraldisplaysinxdx
xp−1
=−cosx
(p−1)xp−1+sinx
(p−1)(p−2)xp−2−1
(p−1)(p−2)/integraldisplaycosxdx
xp−2
(p>2) TI (491)
2.641 Trigonometric functions and powers 219
2.639
1./integraldisplaysinxdx
x2n=(−1)n+1
x(2n−1)!⎧
⎨
⎩n−2/summationdisplay
k=0(−1)k(2k+1 ) !
x2k+1cosx
+n−1/summationdisplay
k=0(−1)k+1(2k)!
x2ksinx⎫
⎬
⎭+(−1)n+1
(2n−1)!ci(x)
GU (333)(6b)a
2./integraldisplaysinx
x2n+1dx=(−1)n+1
x(2n)!⎧
⎨
⎩n−1/summationdisplay
k=0(−1)k+1(2k)!
x2kcosx
+n−1/summationdisplay
k=0(−1)k+1(2k+1 ) !
x2k+1sinx⎫
⎬
⎭+(−1)n
(2n)!si(x)
GU (333)(6b)a
3./integraldisplaycosxdx
x2ndx=(−1)n+1
x(2n−1)!⎧
⎨
⎩n−1/summationdisplay
k=0(−1)k+1(2k)!
x2kcosx
−n−2/summationdisplay
k=0(−1)k(2k+1 ) !
x2k+1sinx⎫
⎬
⎭+(−1)n
(2n−1)!si(x)
GU (333)(7b)
4./integraldisplaycosxdx
x2n+1=(−1)n+1
x(2n)!⎧
⎨
⎩n−1/summationdisplay
k=0(−1)k+1(2k+1 ) !
x2k+1cosx
−n−1/summationdisplay
k=0(−1)k+1(2k)!
x2ksinx⎫
⎬
⎭+(−1)n
(2n)!ci(x)
GU (333)(7b)
2.641
1./integraldisplaysinkx
a+bxdx=1
b/bracketleftbigg
coska
bsi(u)−sinka
bci(u)/bracketrightbigg/bracketleftbigg
u=k
b(a+bx)/bracketrightbigg
2./integraldisplaycoskx
a+bxdx=1
b/bracketleftbigg
coska
bci(u)+s i nka
bsi(u)/bracketrightbigg/bracketleftbigg
u=k
b(a+bx)/bracketrightbigg
3./integraldisplaysinkx
(a+bx)2dx=−1
bsinkx
a+bx+k
b/integraldisplaycoskx
a+bxdx (see2.641 2)
4./integraldisplaycoskx
(a+bx)2dx=−1
bcoskx
a+bx−k
b/integraldisplaysinkx
a+bxdx (see2.641 1)
5./integraldisplaysinkx
(a+bx)3dx=−sinkx
2b(a+bx)2−kcoskx
2b2(a+bx)−k2
2b2/integraldisplaysinkx
a+bxdx
(see2.641 1)
220 Trigonometric Functions 2.642
6./integraldisplaycoskx
(a+bx)3dx=−coskx
2b(a+bx)2+ksinkx
2b2(a+bx)−k2
2b2/integraldisplaycoskx
a+bxdx
(see2.641 2)
7./integraldisplaysinkx
(a+bx)4dx=−sinkx
3b(a+bx)3−kcoskx
6b2(a+bx)2
+k2sinkx
6b2(a+bx)−k3
6b3/integraldisplaycoskx
a+bxdx
(see2.641 2)
8./integraldisplaycoskx
(a+bx)4dx=−coskx
3b(a+bx)3+ksinkx
6b2(a+bx)2+k2coskx
6b3(a+bx)+k3
6b3/integraldisplaysinkx
a+bxdx
(see2.641 1)
9./integraldisplaysinkx
(a+bx)5dx=−sinkx
4b(a+bx)4−kcoskx
12b2(a+bx)3
+k2sinkx
24b3(a+bx)2+k3coskx
24b4(a+bx)k4
24b4/integraldisplaysinkx
a+bxdx
(see2.641 1)
10./integraldisplaycoskx
(a+bx)5dx=−coskx
4b(a+bx)4+ksinkx
12b2(a+bx)3
+k2coskx
24b3(a+bx)2−k3sinkx
24b4(a+bx)+k4
24b4/integraldisplaycoskx
a+bxdx
(see2.641 2)
11./integraldisplaysinkx
(a+bx)6dx=−sinkx
5b(a+bx)5−kcoskx
20b2(a+bx)4+k2sinkx
60b3(a+bx)3+k3coskx
120b4(a+bx)2
−k4sinkx
120b5(a+bx)+k5
120b5/integraldisplaycoskx
a+bxdx
(see2.641 2)
12./integraldisplaycoskx
(a+bx)6dx=−coskx
5b(a+bx)5+ksinkx
20b2(a+bx)4+k2coskx
60b3(a+bx)3
−k3sinkx
120b4(a+bx)2−k4coskx
120b5(a+bx)−k5
120b5/integraldisplaysinkx
a+bxdx
(see2.641 1)
2.642
1./integraldisplaysin2mx
xdx=/parenleftbigg2m
m/parenrightbigglnx
22m+(−1)m
22m−1m−1/summationdisplay
k=0(−1)k/parenleftbigg2m
k/parenrightbigg
ci[(2m−2k)x]
2./integraldisplaysin2m+1x
xdx=(−1)m
22mm/summationdisplay
k=0(−1)k/parenleftbigg2m+1
k/parenrightbigg
si[(2m−2k+1 )x]
3./integraldisplaycos2mx
xdx=/parenleftbigg2m
m/parenrightbigglnx
22m+1
22m−1m−1/summationdisplay
k=0/parenleftbigg2m
k/parenrightbigg
ci[(2m−2k)x]
4./integraldisplaycos2m+1x
xdx=1
22mm/summationdisplay
k=0/parenleftbigg2m+1
k/parenrightbigg
ci[(2m−2k+1 )x]
2.643 Trigonometric functions and powers 221
5./integraldisplaysin2mx
x2dx=−/parenleftbigg2m
m/parenrightbigg1
22mx
+(−1)m
22m−1m−1/summationdisplay
k=0(−1)k+1/parenleftbigg2m
k/parenrightbigg/braceleftbiggcos(2m−2k)x
x+( 2m−2k)si[(2m−2k)x]/bracerightbigg
6./integraldisplaysin2m+1x
x2dx=(−1)m
22mm/summationdisplay
k=0(−1)k+1/parenleftbigg2m+1
k/parenrightbigg
×/braceleftbiggsin(2m−2k+1 )x
x−(2m−2k+ 1)ci[(2 m−2k+1 )x]/bracerightbigg
7./integraldisplaycos2mx
x2dx=−/parenleftbigg2m
m/parenrightbigg1
22mx
−1
22m−1m−1/summationdisplay
k=0/parenleftbigg2m
k/parenrightbigg/braceleftbiggcos(2m−2k)x
x+( 2m−2k)si[(2m−2k)x]/bracerightbigg
8./integraldisplaycos2m+1x
x2=−1
22mm/summationdisplay
k=0/parenleftbigg2m+1
k/parenrightbigg⎧
⎨
⎩cos(2m−2k+1 )x
x
+( 2m−2k+1 )s i [ ( 2 m−2k+1 )x]⎫
⎬
⎭
2.643
1./integraldisplayxpdx
sinqx=−xp−1[psinx+(q−2)xcosx]
(q−1)(q−2)sinq−1x+q−2
q−1/integraldisplayxpdx
sinq−2x+p(p−1)
(q−1)(q−2)/integraldisplayxp−2dx
sinq−2x
2./integraldisplayxpdx
cosqx=−xp−1[pcosx−(q−2)xsinx]
(q−1)(q−2)cosq−1x
+q−2
q−1/integraldisplayxpdx
cosq−2x+p(p−1)
(q−1)(q−2)/integraldisplayxp−2dx
cosq−2x
3.4/integraldisplayxn
sinxdx=xn
n+∞/summationdisplay
k=1(−1)k+12/parenleftbig
22k−1−1/parenrightbig
(n+2k)(2k)!B2kxn+2k
[|x|<π , n> 0] TU (333)(8b)
4./integraldisplaydx
xnsinx=−1
nxn−[1 + (−1)n](−1)n
22n−1−1
n!Bnlnx−∞/summationdisplay
k=1
k/negationslash=n
2(−1)k2/parenleftbig
22n−1/parenrightbig
(2k−n)·(2k)!B2kx2k−n
[n>1,|x|>π] GU (333)(9b)
5.8/integraldisplayxndx
cosx=∞/summationdisplay
k=0|E2k|xn+2k+1
(n+2k+ 1)(2 k)!/bracketleftBig
|x|<π
2,n > 0/bracketrightBig
GU (333)(10b)
6./integraldisplaydx
xncosx=1
2[1−(−1)n]|En−1|
(n−1)!lnx+∞/summationdisplay
k=0
k/negationslash=n−1
2|E2k|x2k−n+1
(2k−n+1 )·(2k)!
/bracketleftBig
|x|<π
2/bracketrightBig
GU (333)(11b)
222 Trigonometric Functions 2.644
7./integraldisplayxndx
sin2x=−xncotx+n
n−1xn−1+n∞/summationdisplay
k=1(−1)k22kxn+2k−1
(n+2k−1)(2k)!B2k
[|x|<π , n> 1] GU (333)(8c)
8./integraldisplaydx
xnsin2x=−cotx
xn+n
(n+1 )xn+1−[1−(−1)n](−1)n+1
22nn
(n+1 ) !Bn+1lnx
−n
2n+1∞/summationdisplay
k=1
k/negationslash=n+1
2(−1)k(2x)2k
(2k−n−1)(2k)!B2k
[|x|<π] GU (333)(9c)
9./integraldisplayxndx
cos2x=xntanx+n∞/summationdisplay
k=1(−1)k22k/parenleftbig
22k−1/parenrightbig
xn+2k−1
(n+2k−1)·(2k)!B2k
/bracketleftBig
n>1,|x|<π
2/bracketrightBig
GU (333)(10c)
10./integraldisplaydx
xncos2x=tanx
xn−[1−(−1)n](−1)n+1
22nn
(n+1 ) !/parenleftbig
2n+1−1/parenrightbig
Bn+1lnx
−n
xn+1∞/summationdisplay
k=1
k/negationslash=n+1
2(−1)k/parenleftbig
22k−1/parenrightbig
(2x)2k
(2k−n−1)(2k)!B2k
/bracketleftBig
|x|<π
2/bracketrightBig
GU (333)(11c)
2.644
1./integraldisplayxdx
sin2nx=−n−1/summationdisplay
k=0(2n−2)(2n−4)...(2n−2k+2 )
(2n−1)(2n−3)...(2n−2k+3 )sinx+( 2n−2k)xcosx
(2n−2k+ 1)(2 n−2k)sin2n−2k+1x
+2n−1(n−1)!
(2n−1)!!(ln sin x−xcotx)
2./integraldisplayxdx
sin2n+1x=−n−1/summationdisplay
k=0(2n−1)(2n−3)...(2n−2k+1 )
2n(2n−2)...(2n−2k+2 )sinx+( 2n−2k−1)xcosx
(2n−2k)(2n−2k−1)sin2n−2kx
+(2n−1)!!
2nn!/integraldisplayxdx
sinx
(see2.644 5)
3./integraldisplayxdx
cos2nx=n−1/summationdisplay
k=0(2n−2)(2n−4)...(2n−2k+2 )
(2n−1)(2n−3)...(2n−2k+3 )(2n−2k)xsinx−cosx
(2n−2k+ 1)(2 n−2k)cos2n−2k+1x
+2n−1(n−1)!
(2n−1)!!(xtanx+l nc o s x)
4./integraldisplayxdx
cos2n+1x=n−1/summationdisplay
k=0(2n−1)(2n−3)...(2n−2k+1 )
2n(2n−2)...(2n−2k+2 )(2n−2k+1 )xsinx−cosx
(2n−2k)(2n−2k−1)cos2n−2kx
+(2n−1)!!
2nn!/integraldisplayxdx
cosx
(see2.644 6)
2.645 Trigonometric functions and powers 223
5./integraldisplayxdx
sinx=x+∞/summationdisplay
k=1(−1)k+12/parenleftbig
22k−1−1/parenrightbig
(2k+1 ) !B2kx2k+1
6./integraldisplayxdx
cosx=∞/summationdisplay
k=0|E2k|x2k+2
(2k+ 2)(2 k)!
7./integraldisplayxdx
sin2x=−xcotx+l ns i n x
8./integraldisplayxdx
cos2x=xtanx+l nc o s x
9./integraldisplayxdx
sin3x=−sinx+xcosx
2s in2x+1
2/integraldisplayx
sinxdx (see2.644 5)
10./integraldisplayxdx
cos3x=xsinx−cosx
2c os2x+1
2/integraldisplayxdx
cosx(see2.644 6)
11./integraldisplayxdx
sin4x=−xcosx
3s in3x−1
6s in2x−2
3xcotx+2
3ln(sin x)
12./integraldisplayxdx
cos4x=xsinx
3c os3x−1
6c os2x+2
3xtanx−2
3ln (cos x)
13./integraldisplayxdx
sin5x=−xcosx
4s in4x−1
12 sin3x−3xcosx
8s in2x−3
8s inx+3
8/integraldisplayxdx
sinx
(see2.644 5)
14./integraldisplayxdx
cos5x=xsinx
4c os4x−1
12cos3x+3xsinx
8c os2x−3
8c osx+3
8/integraldisplayxdx
cosx
(see2.644 6)
2.645
1./integraldisplay
xpsin2mx
cosnxdx=m/summationdisplay
k=0(−1)k/parenleftBigm
k/parenrightBig/integraldisplayxpdx
cosn−2kx(see2.643 2)
2./integraldisplay
xpsin2m+1x
cosnxdx=m/summationdisplay
k=0(−1)k/parenleftBigm
k/parenrightBig/integraldisplayxpsinx
cosn−2kxdx (see2.645 3)
3./integraldisplay
xpsinxdx
cosnx=xp
(n−1)cosn−1x−p
n−1/integraldisplayxp−1
cosn−1xdx
[n>1] (see 2.643 2)GU (333)(12)
4./integraldisplay
xpcos2mx
sinnxdx=m/summationdisplay
k=0(−1)k/parenleftBigm
k/parenrightBig/integraldisplayxpdx
sinn−2kx(see2.643 1)
5./integraldisplay
xpcos2m+1x
sinnxdx=m/summationdisplay
k=0(−1)k/parenleftBigm
k/parenrightBig/integraldisplayxpcosx
sinn−2kxdx (see2.645 6)
6./integraldisplay
xpcosx
sinnx=−xp
(n−1)sinn−1x+p
n−1/integraldisplayxp−1dx
sinn−1x[n>1] (see 2.643 1)GU (333)(13)
224 Trigonometric Functions 2.646
7./integraldisplayxcosx
sin2xdx=−x
sinx+l nt a nx
2
8./integraldisplayxsinx
cos2xdx=x
cosx−lntan/parenleftBigx
2+π
4/parenrightBig
2.646
1./integraldisplay
xptanxdx=∞/summationdisplay
k=1(−1)k+122k/parenleftbig
22k−1−1/parenrightbig
(p+2k)·(2k)!B2kxp+2k
/bracketleftBig
p≥−1,|x|<π
2/bracketrightBig
GU (333)(12d)
2./integraldisplay
xpcotxdx=∞/summationdisplay
k=0(−1)k22kB2k
(p+2k)(2k)!xp+2k[p≥1,|x|<π] GU (333)(13d)
3./integraldisplay
xptan2xdx=xtanx+l nc o s x−x2
2
4./integraldisplay
xcot2xdx=−xcotx+l ns i n x−x2
2
2.647
1./integraldisplayxncosxdx
(a+bsinx)m=−xn
(m−1)b(a+bsinx)m−1+n
(m−1)b/integraldisplayxn−1dx
(a+bsinx)m−1
[m/negationslash=1 ] MZ 247
2./integraldisplayxnsinxdx
(a+bcosx)m=xn
(m−1)b(a+bcosx)m−1−n
(m−1)b/integraldisplayxn−1dx
(a+bcosx)m−1
[m/negationslash=1 ] MZ 247
3./integraldisplayxdx
1+s i n x=−xtan/parenleftBigπ
4−x
2/parenrightBig
+ 2lncos/parenleftBigπ
4−x
2/parenrightBig
PE (329)
4./integraldisplayxdx
1−sinx=xcot/parenleftBigπ
4−x
2/parenrightBig
+2l ns i n/parenleftBigπ
4−x
2/parenrightBig
PE (330)
5./integraldisplayxdx
1 + cos x=xtanx
2+ 2ln cosx
2PE (331)
6./integraldisplayxdx
1−cosx=−xcotx
2+ 2lncosx
2PE (332)
7./integraldisplayxcosx
(1 + sin x)2dx=−x
1+s i n x+t a n/parenleftBigx
2−π
4/parenrightBig
8./integraldisplayxcosx
(1−sinx)2dx=x
1−sinx+t a n/parenleftBigx
2+π
4/parenrightBig
9./integraldisplayxsinx
(1 + cos x)2dx=x
1 + cos x−tanx
2
10./integraldisplayxsinx
(1−cosx)2dx=−x
1−cosx−cotx
2MZ 247a
2.654 Trigonometric functions and powers 225
2.648
1./integraldisplayx+s i nx
1 + cos xdx=xtanx
2
2./integraldisplayx−sinx
1−cosxdx=−xcotx
2GU (333)(16)
2.649/integraldisplayx2dx
[(ax−b)sinx+(a+bx)cosx]2=xsinx+c o s x
b[(ax−b)sinx+(a+bx)cosx]GU (333)(17)
2.651/integraldisplaydx
[a+(ax+b)tanx]2=tanx
a[a+(ax+b)tanx]GU (333)(18)
2.652/integraldisplayxdx
cos(x+t)cos(x−t)= cosec2 t/braceleftbigg
xlncos(x−t)
cos(x+t)−L(x+t)+L(x−t)/bracerightbigg
/bracketleftBig
t/negationslash=nπ;|x|</vextendsingle/vextendsingle/vextendsingleπ
2−|t0|/vextendsingle/vextendsingle/vextendsingle/bracketrightBig
,
where t
0is the value of the argument t, which is reduced by multiples of the argument πto lie in the
interval/parenleftbig
−π
2,π
2/parenrightbig
. LO III 288
2.653
1./integraldisplaysinx√xdx=√
2πS/parenleftbig√x/parenrightbig
(cf.8.251 21)
2./integraldisplaycosx√xdx=√
2πC/parenleftbig√x/parenrightbig
(cf.8.251 3)
2.654 Notation :Δ=/radicalbig
1−k2sin2x,k/prime=√
1−k2:
1./integraldisplayxsinxcosx
Δdx=−xΔ
k2+1
k2E(x, k)
2./integraldisplayxsin3xcosx
Δdx=−k/prime2
9k4F(x, k)+2k2+5
9k4E(x, k)−1
9k4/bracketleftbig
3/parenleftbig
3−Δ2/parenrightbig
x+k2sinxcosx/bracketrightbig
Δ
3./integraldisplayxsinxcos3x
Δdx=−k/prime2
9k4F(x, k)+7k2−5
9k4E(x, k)−1
9k4/bracketleftBig
3/parenleftBig
Δ2−3k/prime2/parenrightBig
x−k2sinxcosx/bracketrightBig
Δ
4./integraldisplayxsinxdx
Δ3dx=−xcosx
k/prime2Δ+1
kk/prime2arcsin ( ksinx)
5./integraldisplayxcosxdx
Δ3=xsinx
Δ+1
kln(kcosx+Δ )
6./integraldisplayxsinxcosxdx
Δ3=x
k2Δ−1
k2F(x, k)
7./integraldisplayxsin3xcosxdx
Δ3=x2−k2sin2x
k4Δ−1
k4[E(x, k)+F(x, k)]
8./integraldisplayxsinxcos3xdx
Δ3=xk2sin2x+k2−2
k4Δ+k/prime2
k4F(x, k)+1
k4E(x, k)
226 Trigonometric Functions 2.655
2.655 Integrals containing sin x2and cos x2
In integrals containing sin x2and cos x2, it is expedient to make the substitution x2=u.
1./integraldisplay
xpsinx2dx=−xp−1
2cosx2+p−1
2/integraldisplay
xp−2cosx2dx
2./integraldisplay
xpcosx2dx=xp−1
2sinx2−p−1
2/integraldisplay
xp−2sinx2dx
3./integraldisplay
xnsinx2dx=(n−1)!!⎧
⎨
⎩r/summationdisplay
k=1(−1)k/bracketleftbiggxn−4k+3cosx2
22k−1(n−4k+3 ) ! !−xn−4k+1sinx2
22k(n−4k+1 ) ! !/bracketrightbigg
+(−1)r
22r(n−4r−1)!!/integraldisplay
xn−4rsinx2dx⎫
⎬
⎭
/bracketleftBig
r=/floorleftBign
4/floorrightBig/bracketrightBig
GU (336)(4a)
4./integraldisplay
xncosx2dx=(n−1)!!⎧
⎨
⎩r/summationdisplay
k=1(−1)k−1/bracketleftbiggxn−4k+3sinx2
22k−1(n−4k+3 ) ! !+xn−4k+1cosx2
22k(n−4k+1 ) ! !/bracketrightbigg
+(−1)r
22r(n−4r−1)!!/integraldisplay
xn−4rcosx2dx⎫
⎬
⎭
/bracketleftBig
r=/floorleftBign
4/floorrightBig/bracketrightBig
GU (336)(5a)
5./integraldisplay
xsinx2dx=−cos2x
2
6./integraldisplay
xcosx2dx=−sin2x
2
7./integraldisplay
x2sinx2dx=−x
2cosx2+1
2/radicalbiggπ
2C(x)
8./integraldisplay
x2cosx2dx=x
2sinx2−1
2/radicalbiggπ
2S(x)
9./integraldisplay
x3sinx2dx=−x2
2cosx2+1
2sinx2
10./integraldisplay
x3cosx2dx=x2
2sinx2+1
2cosx2
2.661 Trigonometric functions and exponentials 227
2.66 Combinations of trigonometric functions and exponentials
2.661/integraldisplay
eaxsinpxcosqxdx=1
a2+(p+q)2/braceleftbigg
eaxsinpxcosq−1x[acosx+(p+q)sinx]
−pa/integraldisplay
eaxsinp−1xcosq−1xdx+(q−1)(p+q)/integraldisplay
eaxsinpxcosq−2xdx/bracerightbigg
TI (523)
=1
a2+(p+q)2/braceleftbigg
eaxsinp−1xcosqx[asinx−(p+q)cosx]
+qa/integraldisplay
eaxsinp−1xcosq−1xdx+(p−1)(p+q)/integraldisplay
eaxsinp−2xcosqxdx/bracerightbigg
TI (524)
=1
a2+(p+q)2/braceleftbigg
eaxsinp−1xcosq−1x/bracketleftbig
asinxcosx+qsin2x−pcos2x/bracketrightbig
+q(q−1)/integraldisplay
eaxsinpxcosq−2xdx+p(p−1)/integraldisplay
eaxsinp−2xcosqxdx/bracerightbigg
TI (525)
=1
a2+(p+q)2/braceleftbigg
eaxsinp−1xcosq−1x/parenleftbig
asinxcosx+qsin2x−pcos2x/parenrightbig
+q(q−1)/integraldisplay
eaxsinp−2xcosq−2xdx
−(q−p)(p+q−1)/integraldisplay
eaxsinp−2xcosqxdx/bracerightbigg
TI (526)
=1
a2+(p+q)2/bracketleftbigg
eaxsinp−1xcosq−1x/parenleftbig
asinxcosx+qsin2x−pcos2x/parenrightbig
+p(p−1)/integraldisplay
eaxsinp−2xcosq−2xdx
+(q−p)(p+q−1)/integraldisplay
eaxsinpxcosq−2xdx/bracketrightbigg
GU (334)(1a)
Forp=mandq=neven integers, the integral/integraldisplay
eaxsinmxcosnxdxcan be reduced by means of
these formulas to the integral/integraldisplay
eaxdx. However, when only mor only nis even, they can be reduced to
228 Trigonometric Functions 2.662
integrals of the form/integraldisplay
eaxcosnxdxor/integraldisplay
eaxsinmxdx, respectively.
2.662
1./integraldisplay
eaxsinnbxdx=1
a2+n2b2/bracketleftbigg
(asinbx−nbcosbx)eaxsinn−1bx
+n(n−1)b2/integraldisplay
eaxsinn−2bxdx/bracketrightbigg
2./integraldisplay
eaxcosnbxdx=1
a2+n2b2/bracketleftbigg
(acosbx+nbsinbx)eaxcosn−1bx
+n(n−1)b2/integraldisplay
eaxcosn−2bxdx/bracketrightbigg
3./integraldisplay
eaxsin2mbxdx
=m−1/summationdisplay
k=0(2m)!b2keaxsin2m−2k−1bx
(2m−2k)! [a2+( 2m)2b2][a2+( 2m−2)2b2]···[a2+( 2m−2k)2b2]
×[asinbx−(2m−2k)bcosbx]+(2m)!b2meax
[a2+( 2m)2b2][a2+( 2m−2)2b2]···[a2+4b2]a
=/parenleftbigg2m
m/parenrightbiggeax
22ma+eax
22m−1m/summationdisplay
k=1(−1)k/parenleftbigg2m
m−k/parenrightbigg1
a2+4b2k2(acos2bkx+2bksin 2bkx)
4./integraldisplay
eaxsin2m+1bxdx
=m/summationdisplay
k=0(2m+1 ) !b2keaxsin2m−2kbx[asinbx−(2m−2k+1 )bcosbx]
(2m−2k+1 ) ![ a2+( 2m+1 )2b2][a2+( 2m−1)2b2]···[a2+( 2m−2k+1 )2b2]
=eax
22mm/summationdisplay
k=0(−1)k
a2+( 2k+1 )2b2/parenleftbigg2m+1
m−k/parenrightbigg
[asin(2k+1 )bx−(2k+1 )bcos(2k+1 )bx]
5.8/integraldisplay
eaxcos2mbxdx=m−1/summationdisplay
k=0(2m)!b2keaxcos2m−2k−1bx[acosbx+( 2m−2k)bsinbx]
(2m−2k)! [a2+( 2m)2b2][a2+( 2m−2)2b2]···[a2+( 2m−2k)2b2]
+(2m)!b2meax
[a2+( 2m)2b2][a2+( 2m−2)2b2]···[a2+4b2]a
=/parenleftbigg2m
m/parenrightbiggeax
22ma+eax
22m−1m/summationdisplay
k=1/parenleftbigg2m
m−k/parenrightbigg1
a2+4b2k2[acos2kbx+2kbsin 2kbx]
6./integraldisplay
eaxcos2m+1bxdx
=m/summationdisplay
k=0(2m+1 ) !b2keaxcos2m−2kbx
(2m−2k+1 ) ![ a2+( 2m−1)2b2]···[a2+( 2m−2k+1 )2b2]
=eax
22mm/summationdisplay
k=0/parenleftbigg2m+1
m−k/parenrightbigg1
a2+( 2k+1 )2b2[acos(2k+1 )bx+( 2k+1 )bsin(2k+1 )bx]
2.663
1./integraldisplay
eaxsinbxdx =eax(asinbx−bcosbx)
a2+b2
2.666 Trigonometric functions and exponentials 229
2./integraldisplay
eaxsin2bxdx=eaxsinbx(asinbx−2bcosbx)
4b2+a2+2b2eax
(4b2+a2)a
=eax
2a−eax
a2+4b2/parenleftBiga
2cos2bx+bsin2bx/parenrightBig
3./integraldisplay
eaxcosbxdx =eax(acosbx+bsinbx)
a2+b2
4./integraldisplay
eaxcos2bxdx=eaxcosbx(acosbx+2bsinbx)
4b2+a2+2b2eax
(4b2+a2)a
=eax
2a+eax
a2+4b2/parenleftBiga
2cos2bx+bsin 2bx/parenrightBig
2.664
1./integraldisplay
eaxsinbxcoscxdx=eax
2/bracketleftbiggasin(b+c)x−(b+c)cos(b+c)x
a2+(b+c)2
+asin(b−c)x−(b−c)cos(b−c)x
a2+(b−c)2/bracketrightbigg
GU (334)(6b)
2./integraldisplay
eaxsin2bxcoscxdx=eax
4/bracketleftbigg
2acoscx+csincx
a2+c2−acos(2b+c)x+( 2b+c)sin(2 b+c)x
a2+( 2b+c)2
−acos(2b−c)x+( 2b−c)sin (2 b−c)x
a2+( 2b−c)2/bracketrightbigg
GU (334)(6c)
3./integraldisplay
eaxsinbxcos2cxdx=eax
4/bracketleftbigg
2asinbx−bcosbx
a2+b2+asin(b+2c)x−(b+2c)cos(b+2c)x
a2+(b+2c)2
+asin(b−2c)x−(b−2c)cos (b−2c)x
a2+(b−2c)2/bracketrightbigg
GU (334)(6d)
2.665
1./integraldisplayeaxdx
sinpbx=−eax[asinbx+(p−2)bcosbx]
(p−1)(p−2)b2sinp−1bx+a2+(p−2)2b2
(p−1)(p−2)b2/integraldisplayeaxdx
sinp−2bxTI (530)a
2./integraldisplayeaxdx
cospbx=−eax[acosbx−(p−2)bsinbx]
(p−1)(p−2)b2cosp−1bx+a2+(p−2)2b2
(p−1)(p−2)b2/integraldisplayeaxdx
cosp−2bxTI (529)a
By successive applications of formulas 2.665 forpa natural number, we obtain integrals of the form/integraldisplayeaxdx
sinbx,/integraldisplayeaxdx
sin2bx,/integraldisplayeaxdx
cosbx,/integraldisplayeaxdx
cos2bx, which are not expressible in terms of a finite combination
of elementary functions.
2.666
1./integraldisplay
eaxtanpxdx=eax
p−1tanp−1x−a
p−1/integraldisplay
eaxtanp−1xdx−/integraldisplay
eaxtanp−2xdx TI (527)
2./integraldisplay
eaxcotpxdx=−eaxcotp−1x
p−1+a
p−1/integraldisplay
eaxcotp−1xdx−/integraldisplay
eaxcotp−2xdx TI (528)
3./integraldisplay
eaxtanxdx=eaxtanx
a−1
a/integraldisplayeaxdx
cos2x(see remark following 2.665 )
230 Trigonometric Functions 2.667
4./integraldisplay
eaxtan2xdx=eax
a(atanx−1)−a/integraldisplay
eaxtanxdx (see2.666 3) TI 355
5./integraldisplay
eaxcotxdx=eaxcotx
a+1
a/integraldisplayeaxdx
sin2x(see remark following 2.665 )
6./integraldisplay
eaxcot2xdx=−eax
a(acotx+1 )+ a/integraldisplay
eaxcotxdx
(see2.666 5)
Integrals of type/integraldisplay
R(x, eax,sinbx,coscx)dx
Notation :s i nt=−b√
a2+b2;c o s t=a√
a2+b2.
2.667
1./integraldisplay
xpeaxsinbxdx=xpeax
a2+b2(asinbx−bcosbx)−p
a2+b2/integraldisplay
xp−1eax(asinbx−bcosbx)dx
=xpeax
√
a2+b2sin(bx+t)−p√
a2+b2/integraldisplay
xp−1eaxsin(bx+t)dx
2./integraldisplay
xpeaxcosbxdx=xpeax
a2+b2(acosbx+bsinbx)−p
a2+b2/integraldisplay
xp−1eax(acosbx+bsinbx)dx
=xpeax
√
a2+b2cos(bx+t)−p√
a2+b2/integraldisplay
xp−1eaxcos(bx+t)dx
3./integraldisplay
xneaxsinbxdx =eaxn+1/summationdisplay
k=1(−1)k+1n!xn−k+1
(n−k+1 ) !( a2+b2)k/2sin(bx+kt)
4./integraldisplay
xneaxcosbxdx =eaxn+1/summationdisplay
k=1(−1)k+1n!xn−k+1
(n−k+1 ) !( a2+b2)k/2cos(bx+kt)
5./integraldisplay
xeaxsinbxdx =eax
a2+b2/bracketleftbigg/parenleftbigg
ax−a2−b2
a2+b2/parenrightbigg
sinbx−/parenleftbigg
bx−2ab
a2+b2/parenrightbigg
cosbx/bracketrightbigg
6./integraldisplay
xeaxcosbxdx =eax
a2+b2/bracketleftbigg/parenleftbigg
ax−a2−b2
a2+b2/parenrightbigg
cosbx+/parenleftbigg
bx−2ab
a2+b2/parenrightbigg
sinbx/bracketrightbigg
7./integraldisplay
x2eaxsinbxdx=eax
a2+b2⎧
⎨
⎩/bracketleftBigg
ax2−2/parenleftbig
a2−b2/parenrightbig
a2+b2x+2a/parenleftbig
a2−3b2/parenrightbig
(a2+b2)2/bracketrightBigg
sinbx
−/bracketleftBigg
bx2−4ab
a2+b2x+2b/parenleftbig
3a2−b2/parenrightbig
(a2+b2)2/bracketrightBigg
cosbx⎫
⎬
⎭
2.672 Trigonometric and hyperbolic functions 231
8./integraldisplay
x2eaxcosbxdx=eax
a2+b2⎧
⎨
⎩/bracketleftBigg
ax2−2/parenleftbig
a2−b2/parenrightbig
a2+b2x+2a/parenleftbig
a2−3b2/parenrightbig
(a2+b2)2/bracketrightBigg
cosbx
+/bracketleftBigg
bx2−4ab
a2+b2x+2b/parenleftbig
3a2−b2/parenrightbig
(a2+b2)2/bracketrightBigg
sinbx⎫
⎬
⎭
GU (335), MZ 274-275
2.67 Combinations of trigonometric and hyperbolic functions
2.671
1./integraldisplay
sinh(ax+b)sin(cx+d)dx=a
a2+c2cosh(ax+b)sin(cx+d)
−c
a2+c2sinh(ax+b)cos(cx+d)
2./integraldisplay
sinh(ax+b)cos(cx+d)dx=a
a2+c2cosh(ax+b)cos(cx+d)
+c
a2+c2sinh(ax+b)sin(cx+d)
3./integraldisplay
cosh(ax+b)sin (cx+d)dx=a
a2+c2sinh(ax+b)sin(cx+d)
−c
a2+c2cosh(ax+b)cos(cx+d)
4./integraldisplay
cosh(ax+b)cos(cx+d)dx=a
a2+c2sinh(ax+b)cos(cx+d)
+c
a2+c2cosh(ax+b)sin(cx+d)
GU (354)(1)
2.672
1./integraldisplay
sinhxsinxdx=1
2(coshxsinx−sinhxcosx)
2./integraldisplay
sinhxcosxdx=1
2(coshxcosx+s i n h xsinx)
3./integraldisplay
coshxsinxdx=1
2(sinhxsinx−coshxcosx)
4./integraldisplay
coshxcosxdx=1
2(sinhxcosx+c o s h xsinx)
232 Trigonometric Functions 2.673
2.673
1./integraldisplay
sinh2m(ax+b)sin2n(cx+d)dx
=(−1)m
22m+2n/parenleftbigg2m
m/parenrightbigg/parenleftbigg2n
n/parenrightbigg
x+(−1)m+n
22m+2n−1/parenleftbigg2m
m/parenrightbiggn−1/summationdisplay
k=0(−1)k
(2n−2k)c/parenleftbigg2n
k/parenrightbigg
sin[(2n−2k)(cx+d)]
+(−1)n
22m+2n−2m−1/summationdisplay
j=0n−1/summationdisplay
k=0(−1)j+k/parenleftBig
2m
j/parenrightBig/parenleftbig2n
k/parenrightbig
(2m−2j)2a2+( 2n−2k)2c2
×{(2m−2j)asinh[(2 m−2j)(ax+b)] cos[(2 n−2k)(cx+d)]}
+(2−2k)ccosh[(2 m−2j)(ax+b)] sin[(2 n−2k)(cx+d)]
GU (354)(3a)
2./integraldisplay
sinh2m(ax+b)sin2n−1(cx+d)dx
=(−1)m+n
22m+2n−2/parenleftbigg2m
m/parenrightbiggn−1/summationdisplay
k=0(−1)k
(2n−2k−1)c/parenleftbigg2n−1
k/parenrightbigg
cos[(2 n−2k−1)(cx+d)]
+(−1)n−1
22m+2n−3m−1/summationdisplay
j=0n−1/summationdisplay
k=0(−1)j+k/parenleftBig
2m
j/parenrightBig/parenleftbig2n−1
k/parenrightbig
(2m−2j)2a2+( 2n−2k−1)2c2
×{(2m−2j)asinh[(2 m−2j)(ax+b)] sin[(2 n−2k−1)(cx+d)]}
−(2n−2k−1)ccosh[(2 m−2j)(ax+b)] cos[(2 n−2k−1)(cx+d)]
GU (354)(3b)
3./integraldisplay
sinh2m−1(ax+b)sin2n(cx+d)dx
=/parenleftbig2n
n/parenrightbig
22m+2n−2m−1/summationdisplay
j=0(−1)j/parenleftBig
2m−1
j/parenrightBig
(2m−2j−1)acosh[(2 m−2j−1)(ax+d)]
+(−1)n
22m+2n−3m−1/summationdisplay
j=0n−1/summationdisplay
k=0(−1)j+k/parenleftBig
2m−1
j/parenrightBig/parenleftbig2n
k/parenrightbig
(2m−2j−1)2a2+( 2n−2k)2c2
×{(2m−2j−1)acosh[(2 m−2j−1)(ax+b)] cos[(2 n−2k)(cx+d)]}
+(2n−2k)csinh[(2 m−2j−1)(ax+b)] sin[(2 n−2k)(cx+d)]
GU (354)(3c)
2.673 Trigonometric and hyperbolic functions 233
4./integraldisplay
sinh2m−1(ax+b)sin2n−1(cx+d)dx
=(−1)n−1
22m−2n−4m−1/summationdisplay
j=0n−1/summationdisplay
k=0(−1)j+k/parenleftBig
2m−1
j/parenrightBig/parenleftbig2n−1
k/parenrightbig
(2m−2j−1)2a2+( 2n−2k−1)2c2
×{(2m−2j−1)acosh[(2 m−2j−1)(ax+b)] sin[(2 n−2k−1)(cx+d)]}
−(2n−2k−1)csinh[(2 m−2j−1)(ax+b)] cos[(2 n−2k−1)(cx+d)]
GU (354)(3d)
5./integraldisplay
sinh2m(ax+b)cos2n(cx+d)dx
=(−1)m
22m+2n/parenleftbigg2m
m/parenrightbigg/parenleftbigg2n
n/parenrightbigg
x+/parenleftbig2n
n/parenrightbig
22m+2n−1m−1/summationdisplay
j=0(−1)j/parenleftBig
2m
j/parenrightBig
(2m−2j)asinh[(2 m−2j)(ax+b)]
+(−1)m/parenleftbig2m
m/parenrightbig
22m+2n−1n−1/summationdisplay
k=0/parenleftbig2n
k/parenrightbig
(2n−2k)csin[(2n−2k)(cx+d)]
+1
22m+2n−2m−1/summationdisplay
j=0n−1/summationdisplay
k=0(−1)j/parenleftBig
2m
j/parenrightBig/parenleftbig2n
k/parenrightbig
(2m−2j)2a2+( 2n−2k)2c2
×{(2m−2j)asinh[(2 m−2j)(ax+b)] cos[(2 n−2k)(cx+d)]}
+(2−2k)ccosh[(2 m−2j)(ax+b)] sin[(2 n−2k)(cx+d)]
GU (354)(4a)
6./integraldisplay
sinh2m(ax+b)cos2n−1(cx+d)dx
=(−1)m/parenleftbig2m
m/parenrightbig
22m+2n−2n−1/summationdisplay
k=0/parenleftbig2n−1
k/parenrightbig
(2n−2k−1)csin[(2n−2k−2)(cx+d)]
+1
22m+2n−3m−1/summationdisplay
j=0n−1/summationdisplay
k=0(−1)j/parenleftBig
2m
j/parenrightBig/parenleftbig2−1
k/parenrightbig
(2m−2j)2a2+( 2n−2k−1)2c2
×{(2m−2j)asinh[(2 m−2j)(ax+b)] cos[(2 n−2k−1)(cx+d)]}
+(2n−2k−1)ccosh[(2 m−2j)(ax+b)] sin[(2 n−2k−1)(cx+d)]
GU (354)(4a)
234 Trigonometric Functions 2.673
7./integraldisplay
sinh2m−1(ax+b)cos2n(cx+d)dx
=/parenleftbig2n
n/parenrightbig
22m+2n−2m−1/summationdisplay
j=0(−1)j/parenleftBig
2m−1
j/parenrightBig
(2m−2j−1)acosh[(2 m−2j−1)(ax+d)]
+1
22m−2n−3m−1/summationdisplay
j=0n−1/summationdisplay
k=0(−1)j/parenleftBig
2m
j/parenrightBig/parenleftbig2n
k/parenrightbig
(2m−2j−1)2a2+( 2n−2k)2c2
×{(2m−2j−1)acosh[(2 m−2j−1)(ax+b)] cos[(2 n−2k)(cx+d)]}
+(2n−2k)csinh[(2 m−2j−1)(ax+b)] sin[(2 n−2k)(cx+d)]
GU (354)(4b)
8./integraldisplay
sinh2m−1(ax+b)cos2n−1(cx+d)dx
=1
22m+2n−4m−1/summationdisplay
j=0n−1/summationdisplay
k=0(−1)j/parenleftBig
2m−1
j/parenrightBig/parenleftbig2n−1
k/parenrightbig
(2m−2j−1)2a2+( 2n−2k−1)2c2
×{(2m−2j−1)acosh[(2 m−2j−1)(ax+b)] cos[(2 n−2k−1)(cx+d)]}
+(2n−2k−1)csinh[(2 m−2j−1)(ax+b)] sin[(2 n−2k−1)(cx+d)]
GU (354)(4b)
9./integraldisplay
cosh2m(ax+b)sin2n(cx+d)dx
=/parenleftbig2m
m/parenrightbig/parenleftbig2n
n/parenrightbig
22m+2nx+(−1)n/parenleftbig2m
m/parenrightbig
22m+2n−1m−1/summationdisplay
k=0(−1)k/parenleftbig2n
k/parenrightbig
(2n−2k)csin[(2n−2k)(cx+d)]
+/parenleftbig2n
n/parenrightbig
22m+2n−1m−1/summationdisplay
j=0/parenleftBig
2m
j/parenrightBig
(2m−2j)asinh[(2 m−2j)(ax+b)]
+(−1)n
22m+2n−2m−1/summationdisplay
j=0n−1/summationdisplay
k=0(−1)k/parenleftBig
2m
j/parenrightBig/parenleftbig2n
k/parenrightbig
(2m−2j)2a2+( 2n−2k)2c2
×{(2m−2j)asinh[(2 m−2j)(ax+b)] cos[(2 n−2k)(cx+d)]}
+(2n−2k)ccosh[(2 m−2j)(ax+b)] sin[(2 n−2k)(cx+d)]
GU (354)(5a)
2.673 Trigonometric and hyperbolic functions 235
10./integraldisplay
cosh2m−1(ax+b)sin2n(cx+d)dx
=/parenleftbig2n
n/parenrightbig
22m+2n−2m−1/summationdisplay
j=0/parenleftBig
2m−1
j/parenrightBig
(2m−2j−1)asinh[(2 m−2j−1)(ax+b)]
+(−1)n
22m+2n−3m−1/summationdisplay
j=0n−1/summationdisplay
k=0(−1)k/parenleftBig
2m−1
j/parenrightBig/parenleftbig2n
k/parenrightbig
(2m−2j−1)2a2+( 2n−2k)2c2
×{(2m−2j−1)asinh[(2 m−2j−1)(ax+b)] cos[(2 n−2k)(cx+d)]}
+(2n−2k)ccosh[(2 m−2j−1)(ax+b)] sin[(2 n−2k)(cx+d)]
GU (354)(5a)
11./integraldisplay
cosh2m(ax+b)sin2n−1(cx+d)dx
=(−1)n−1/parenleftbig2m
m/parenrightbig
22m+2n−2n−1/summationdisplay
k=0(−1)k+1/parenleftbig2n−1
k/parenrightbig
(2n−2k−1)ccos[(2 n−2k−1)(cx+d)]
+(−1)n−1
22m+2n−3m−1/summationdisplay
j=0n−1/summationdisplay
k=0(−1)k/parenleftBig
2m
j/parenrightBig/parenleftbig2n−1
k/parenrightbig
(2m−2j)2a2+( 2n−2k−1)2c2
×{(2m−2j)asinh[(2 m−2j)(ax+b)] sin[(2 n−2k−1)(cx+d)]}
−(2n−2k−1)ccosh[(2 m−2j)(ax+b)] cos[(2 n−2k−1)(cx+d)]
GU (354)(5b)
12./integraldisplay
cosh2m−1(ax+b)sin2n−1(cx+d)dx
=(−1)n−1
22m+2n−4m−1/summationdisplay
j=0n−1/summationdisplay
k=0(−1)k/parenleftBig
2m−1
j/parenrightBig/parenleftbig2n−1
k/parenrightbig
(2m−2j−1)2a2+( 2n−2k−1)2c2
×{(2m−2j−1)asinh[(2 m−2j−1)(ax+b)] sin[(2 n−2k−1)(cx+d)]}
−(2n−2k−1)ccosh[(2 m−2j−1)(ax+b)] cos[(2 n−2k−1)(cx+d)]
GU (354)(5b)
236 Trigonometric Functions 2.673
13./integraldisplay
cosh2m(ax+b)cos2n(cx+d)dx
=/parenleftbig2m
m/parenrightbig/parenleftbig2n
n/parenrightbig
22m+2nx+/parenleftbig2m
m/parenrightbig
22m+2n−1n−1/summationdisplay
k=0/parenleftbig2
k/parenrightbig
(2n−2k)csin[(2n−2k)(cx+d)]
+/parenleftbig2n
n/parenrightbig
22m+2n−1m−1/summationdisplay
j=0/parenleftBig
2m
j/parenrightBig
(2m−2j)asinh[(2 m−2j)(ax+b)]
+1
22m+2n−2m−1/summationdisplay
j=0n−1/summationdisplay
k=0/parenleftBig
2m
j/parenrightBig/parenleftbig2n
k/parenrightbig
(2m−2j)2a2+( 2n−2k)2c2
×{(2m−2j)asinh[(2 m−2j)(ax+b)] cos[(2 n−2k)(cx+d)]}
+(2n−2k)ccosh[(2 m−2j)(ax+b)] sin[(2 n−2k)(cx+d)]
GU (354)(6)
14./integraldisplay
cosh2m−1(ax+b)cos2n(cx+d)dx
=/parenleftbig2n
n/parenrightbig
22m+2n−2m−1/summationdisplay
j=0/parenleftBig
2m−1
j/parenrightBig
(2m−2j−1)asinh[(2 m−2j−1)(ax+b)]
+1
22m+2n−3m−1/summationdisplay
j=0n−1/summationdisplay
k=0/parenleftBig
2m−1
j/parenrightBig/parenleftbig2n
k/parenrightbig
(2m−2j−1)2a2+( 2n−2k)2c2
×{(2m−2j−1)asinh[(2 m−2j−1)(ax+b)] cos[(2 n−2k)(cx+d)]}
+(2n−2k)ccosh[(2 m−2j−1)(ax+b)] sin[(2 n−2k)(cx+d)]
GU (354)(6)
15./integraldisplay
cosh2m(ax+b)cos2n−1(cx+d)dx
=/parenleftbig2m
m/parenrightbig
22m+2n−2n−1/summationdisplay
k=0/parenleftbig2n−1
k/parenrightbig
(2n−2k−1)csin[(2n−2k−1)(cx+d)]
+1
22m+2n−3m−1/summationdisplay
j=0n−1/summationdisplay
k=0/parenleftBig
2m
j/parenrightBig/parenleftbig2n−1
k/parenrightbig
(2m−2j)2a2+( 2n−2k−1)2c2
×{(2m−2j)asinh[(2 m−2j)(ax+b)] cos[(2 n−2k−1)(cx+d)]}
+(2n−2k−1)ccosh[(2 m−2j)(ax+b)] sin[(2 n−2k−1)(cx+d)]
GU (354)(6)
2.711 The logarithm 237
16./integraldisplay
cosh2m−1(ax+b)cos2n−1(cx+d)dx
=1
22m+2n−4m−1/summationdisplay
j=0n−1/summationdisplay
k=0/parenleftBig
2m−1
j/parenrightBig/parenleftbig2n−1
k/parenrightbig
(2m−2j−1)2a2+( 2n−2k−1)2c2
×{(2m−2j−1)asinh[(2 m−2j−1)(ax+b)] cos[(2 n−2k−1)(cx+d)]}
+(2n−2k−1)ccosh[(2 m−2j−1)(ax+b)] sin[(2 n−2k−1)(cx+d)]
GU (354)(6)
2.674
1./integraldisplay
eaxsinhbxsincxdx=e(a+b)x
2[ (a+b)2+c2][(a+b)sincx−ccoscx]
−e(a−b)x
2[ (a−b)2+c2][(a−b)sincx−ccoscx]
2./integraldisplay
eaxsinhbxcoscxdx=e(a+b)x
2[(a+b)2+c2][(a+b)coscx+csincx]
−e(a−b)x
2[(a−b)2+c2][(a−b)coscx+csincx]
3./integraldisplay
eaxcoshbxsincxdx=e(a+b)x
2[(a+b)2+c2][(a+b)sincx−ccoscx]
+e(a−b)x
2[(a−b)2+c2][(a−b)sincx−ccoscx]
4./integraldisplay
eaxcoshbxcoscxdx=e(a+b)x
2[ (a+b)2+c2][(a+b)coscx+csincx]
+e(a−b)x
2[(a−b)2+c2][(a−b)coscx+csincx]
MZ 379
2.7 Logarithms and Inverse-Hyperbolic Functions
2.71 The logarithm
2.711/integraldisplay
lnmxdx=xlnmx−m/integraldisplay
lnm−1xdx
=x
m+1m/summationdisplay
k=0(−1)k(m+1 )m(m−1)···(m−k+1 )l nm−kx
(m>0) TI (603)
238 Logarithms and Inverse-Hyperbolic Functions 2.721
2.72–2.73 Combinations of logarithms and algebraic functions
2.721
1./integraldisplay
xnlnmxdx=xn+1lnmx
n+1−m
n+1/integraldisplay
xnlnm−1xdx (see2.722 )
Forn=−1
2./integraldisplaylnmxdx
x=lnm+1x
m+1
Forn=−1a n d m=−1
3./integraldisplaydx
xlnx=l n( l n x)
2.722/integraldisplay
xnlnmxdx=xn+1
m+1m/summationdisplay
k=0(−1)k(m+1 )m(m−1)···(m−k+1 )lnm−kx
(n+1 )k+1TI (604)
2.723
1./integraldisplay
xnlnxdx=xn+1/bracketleftbigglnx
n+1−1
(n+1 )2/bracketrightbigg
TI 375
2./integraldisplay
xnln2xdx=xn+1/bracketleftbiggln2x
n+1−2lnx
(n+1 )2+2
(n+1 )3/bracketrightbigg
TI 375
3./integraldisplay
xnln3xdx=xn+1/bracketleftbiggln3x
n+1−3ln2x
(n+1 )2+6lnx
(n+1 )3−6
(n+1 )4/bracketrightbigg
2.724
1./integraldisplayxndx
(lnx)m=−xn+1
(m−1)(lnx)m−1+n+1
m−1/integraldisplayxndx
(lnx)m−1
Form=1
2./integraldisplayxndx
lnx=l i/parenleftbig
xn+1/parenrightbig
2.725
1./integraldisplay
(a+bx)mlnxdx=1
(m+1 )b/bracketleftbigg
(a+bx)m+1lnx−/integraldisplay(a+bx)m+1dx
x/bracketrightbigg
TI 374
2./integraldisplay
(a+bx)mlnxdx=1
(m+1 )b/bracketleftbig
(a+bx)m+1−am+1/bracketrightbig
lnx−m/summationdisplay
k=0/parenleftbigm
k/parenrightbig
am−kbkxk+1
(k+1 )2
Form=−1, see2.727 2.
2.726
1./integraldisplay
(a+bx)lnxdx=/bracketleftbigg(a+bx)2
2b−a2
2b/bracketrightbigg
lnx−/parenleftbigg
ax+1
4bx2/parenrightbigg
2./integraldisplay
(a+bx)2lnxdx=1
3b/bracketleftbig
(a+bx)3−a3/bracketrightbig
lnx−/parenleftbigg
a2x+abx2
2+b2x3
9/parenrightbigg
2.731 Combinations of logarithms and algebraic functions 239
3./integraldisplay
(a+bx)3lnxdx=1
4b/bracketleftbig
(a+bx)4−a4/bracketrightbig
lnx−/parenleftbigg
a3x+3
4a2bx2+1
3ab2x3+1
16b3x4/parenrightbigg
2.727
1.8/integraldisplaylnxdx
(a+bx)m=1
b(m−1)/bracketleftbigg
−lnx
(a+bx)m−1+/integraldisplaydx
x(a+bx)m−1/bracketrightbigg
TI 376
Form=1
2.8/integraldisplaylnxdx
a+bx=1
blnxln(a+bx)−1
b/integraldisplayln(a+bx)dx
x(see2.728 2)
3./integraldisplaylnxdx
(a+bx)2=−lnx
b(a+bx)+1
ablnx
a+bx
4./integraldisplaylnxdx
(a+bx)3=−lnx
2b(a+bx)2+1
2ab(a+bx)+1
2a2blnx
a+bx
5./integraldisplaylnxdx√
a+bx=2
b/braceleftbigg
(lnx−2)√
a+bx−2√aln/bracketleftbigg(a+bx)1/2−a1/2
x1/2/bracketrightbigg/bracerightbigg
[a>0]
=2
b/braceleftBigg
(lnx−2)√
a+bx+2√
−aarctan/radicalbigg
a+bx
−a/bracerightBigg
[a<0]
2.728
1./integraldisplay
xmln(a+bx)dx=1
m+1/bracketleftbigg
xm+1ln(a+bx)−b/integraldisplayxm+1dx
a+bx/bracketrightbigg
2.9/integraldisplayln(a+bx)
x=l nalnx+bx
aΦ/parenleftbigg
−bx
a,2,1/parenrightbigg
[a>0]
2.729
1./integraldisplay
xmln(a+bx)dx=1
m+1/bracketleftbigg
xm+1−(−a)m+1
bm+1/bracketrightbigg
ln(a+bx)+1
m+1m+1/summationdisplay
k=1(−1)kxm−k+2ak−1
(m−k+2 )bk−1
2./integraldisplay
xln(a+bx)dx=1
2/bracketleftbigg
x2−a2
b2/bracketrightbigg
ln(a+bx)−1
2/bracketleftbiggx2
2−ax
b/bracketrightbigg
3./integraldisplay
x2ln(a+bx)dx=1
3/bracketleftbigg
x3+a3
b3/bracketrightbigg
ln(a+bx)−1
3/bracketleftbiggx3
3−ax2
2b+a2x
b2/bracketrightbigg
4./integraldisplay
x3ln(a+bx)dx=1
4/bracketleftbigg
x4−a4
b4/bracketrightbigg
ln(a+bx)−1
4/bracketleftbiggx4
4−ax3
3b+a2x2
2b2−a3x
b3/bracketrightbigg
2.731/integraldisplay
x2nln/parenleftbig
x2+a2/parenrightbig
dx=1
2n+1⎧
⎨
⎩x2n+1ln/parenleftbig
x2+a2/parenrightbig
+(−1)n2a2n+1arctanx
a
−2n/summationdisplay
k=0(−1)n−k
2k+1a2n−2kx2k+1⎫
⎬
⎭
240 Logarithms and Inverse-Hyperbolic Functions 2.732
2.7327/integraldisplay
x2n+1ln/parenleftbig
x2+a2/parenrightbig
dx=1
2n+2⎧
⎨
⎩/parenleftbig
x2n+2+(−1)na2n+2/parenrightbig
ln/parenleftbig
x2+a2/parenrightbig
+n+1/summationdisplay
k=1(−1)n−k
ka2n−2k+2x2k⎫
⎬
⎭
2.733
1./integraldisplay
ln/parenleftbig
x2+a2/parenrightbig
dx=xln/parenleftbig
x2+a2/parenrightbig
−2x+2aarctanx
aDW
2./integraldisplay
xln/parenleftbig
x2+a2/parenrightbig
dx=1
2/bracketleftbig/parenleftbig
x2+a2/parenrightbig
ln/parenleftbig
x2+a2/parenrightbig
−x2/bracketrightbig
DW
3./integraldisplay
x2ln/parenleftbig
x2+a2/parenrightbig
dx=1
3/bracketleftbigg
x3ln/parenleftbig
x2+a2/parenrightbig
−2
3x3+2a2x−2a3arctanx
a/bracketrightbigg
DW
4./integraldisplay
x3ln/parenleftbig
x2+a2/parenrightbig
dx=1
4/bracketleftbigg/parenleftbig
x4−a4/parenrightbig
ln/parenleftbig
x2+a2/parenrightbig
−x4
2+a2x2/bracketrightbigg
DW
5./integraldisplay
x4ln/parenleftbig
x2+a2/parenrightbig
dx=1
5/bracketleftbigg
x5ln/parenleftbig
x2+a2/parenrightbig
−2
5x5+2
3a2x3−2a4x+2a5arctanx
a/bracketrightbigg
DW
2.734/integraldisplay
x2nln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingledx
=1
2n+1/braceleftBigg
x2n+1ln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingle+a2n+1ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglex+a
x−a/vextendsingle/vextendsingle/vextendsingle/vextendsingle−2n/summationdisplay
k=01
2k+1a2n−2kx2k+1/bracerightBigg
2.735/integraldisplay
x2n+1ln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingledx=1
2n+2/braceleftBigg
/parenleftbig
x2n+2−a2n+2/parenrightbig
ln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingle−n+1/summationdisplay
k=11
ka2n−2k+2x2k/bracerightBigg
2.736
1./integraldisplay
ln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingledx=xln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingle−2x+aln/vextendsingle/vextendsingle/vextendsingle/vextendsinglex+a
x−a/vextendsingle/vextendsingle/vextendsingle/vextendsingle
DW
2./integraldisplay
xln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingledx=1
2/braceleftbig/parenleftbig
x2−a2/parenrightbig
ln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingle−x2/bracerightbig
DW
3./integraldisplay
x2ln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingledx=1
3/braceleftbigg
x3ln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingle−2
3x3−2a2x+a3ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglex+a
x−a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracerightbigg
DW
4./integraldisplay
x3ln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingledx=1
4/braceleftbigg/parenleftbig
x4−a4/parenrightbig
ln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingle−x4
2−a2x2/bracerightbigg
DW
5./integraldisplay
x4ln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingledx=1
5/braceleftbigg
x5ln/vextendsingle/vextendsinglex2−a2/vextendsingle/vextendsingle−2
5x5−2
3a2x3−2a4x+a5ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglex+a
x−a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracerightbigg
DW
2.74 Inverse hyperbolic functions
2.741
1./integraldisplay
arcsinhx
adx=xarcsinhx
a−/radicalbig
x2+a2DW
2.813 Arcsines and arccosines 241
2./integraldisplay
arccoshx
adx=xarccoshx
a−/radicalbig
x2−a2/bracketleftBig
arccoshx
a>0/bracketrightBig
DW
=xarccoshx
a+/radicalbig
x2−a2/bracketleftBig
arccoshx
a<0/bracketrightBig
DW
3./integraldisplay
arctanhx
adx=xarctanhx
a+a
2ln/parenleftbig
a2−x2/parenrightbig
DW
4./integraldisplay
arccothx
adx=xarccothx
a+a
2ln/parenleftbig
x2−a2/parenrightbig
DW
2.742
1./integraldisplay
xarcsinhx
adx=/parenleftbiggx2
2+a2
4/parenrightbigg
arcsinhx
a−x
4/radicalbig
x2+a2DW
2./integraldisplay
xarccoshx
adx=/parenleftbiggx2
2−a2
4/parenrightbigg
arccoshx
a−x
4/radicalbig
x2−a2/bracketleftBig
arccoshx
a>0/bracketrightBig
=/parenleftbiggx2
2−a2
4/parenrightbigg
arccoshx
a+x
4/radicalbig
x2−a2/bracketleftBig
arccoshx
a<0/bracketrightBig
DW
2.8 Inverse Trigonometric Functions
2.81 Arcsines and arccosines
2.811/integraldisplay/parenleftBig
arcsinx
a/parenrightBign
dx=x⌊n/2⌋/summationdisplay
k=0(−1)k/parenleftBign
2k/parenrightBig
·(2k)!/parenleftBig
arcsinx
a/parenrightBign−2k
+/radicalbig
a2−x2⌊(n+1)/2⌋/summationdisplay
k=1(−1)k−1/parenleftbiggn
2k−1/parenrightbigg
·(2k−1)!/parenleftBig
arcsinx
a/parenrightBign−2k+1
2.812/integraldisplay/parenleftBig
arccosx
a/parenrightBign
dx=x⌊n/2⌋/summationdisplay
k=0(−1)k/parenleftBign
2k/parenrightBig
·(2k)!/parenleftBig
arccosx
a/parenrightBign−2k
+/radicalbig
a2−x2⌊(n+1)/2⌋/summationdisplay
k=1(−1)k/parenleftbiggn
2k−1/parenrightbigg
·(2k−1)!/parenleftBig
arccosx
a/parenrightBign−2k+1
2.813
1.11/integraldisplay
arcsinx
adx= sign( a)/bracketleftbigg
xarcsinx
|a|+/radicalbig
a2−x2/bracketrightbigg
2.9/integraldisplay/parenleftBig
arcsinx
a/parenrightBig2
dx=x/parenleftbigg
arcsinx
|a|/parenrightbigg2
+2/radicalbig
a2−x2arcsinx
|a|−2x
3./integraldisplay/parenleftBig
arcsinx
a/parenrightBig3
dx= sign( a)⎡
⎣x/parenleftbigg
arcsinx
|a|/parenrightbigg3
+3/radicalbig
a2−x2/parenleftbigg
arcsinx
|a|/parenrightbigg2
−6xarcsinx
|a|−6/radicalbig
a2−x2⎤
⎦
242 Inverse Trigonometric Functions 2.814
2.814
1./integraldisplay
arccosx
adx=xarccosx
a−/radicalbig
a2−x2
2./integraldisplay/parenleftBig
arccosx
a/parenrightBig2
dx=x/parenleftBig
arccosx
a/parenrightBig2
−2/radicalbig
a2−x2arccosx
a−2x
3./integraldisplay/parenleftBig
arccosx
a/parenrightBig3
dx=x/parenleftBig
arccosx
a/parenrightBig3
−3/radicalbig
a2−x2/parenleftBig
arccosx
a/parenrightBig2
−6xarccosx
a+6/radicalbig
a2−x2
2.82 The arcsecant, the arccosecant, the arctangent, and the arccotangent
2.821
1./integraldisplay
arccosecx
adx=/integraldisplay
arcsina
xdx=xarcsinx
2+aln/parenleftBig
x+/radicalbig
x2−a2/parenrightBig/bracketleftBig
0<arcsina
x<π
2/bracketrightBig
=xarcsina
x−aln/parenleftBig
x+/radicalbig
x2−a2/parenrightBig/bracketleftBig
−π
2<arcsina
x<0/bracketrightBig
DW
2./integraldisplay
arcsecx
adx=/integraldisplay
arccosa
xdx=xarccosa
x−aln/parenleftBig
x+/radicalbig
x2−a2/parenrightBig/bracketleftBig
0<arccosa
x<π
2/bracketrightBig
=xarccosa
x−aln/parenleftBig
x+/radicalbig
x2−a2/parenrightBig/bracketleftBig
−π
2<arccosa
x<0/bracketrightBig
DW
2.822
1.8/integraldisplay
arctanx
adx=xarctanx
a−a
2ln/parenleftbig
a2+x2/parenrightbig
DW
2./integraldisplay
arccotx
adx=xarccotx
a−a
2ln/parenleftbig
a2+x2/parenrightbig
DW
3.9/integraldisplay
xarctanx
adx=1
2/parenleftbig
x2+a2/parenrightbig
arctanx
a−ax
2
4.9/integraldisplay
xarccotx
adx=ax
2+πx2
4−1
2/parenleftbig
x2+a2/parenrightbig
arctanx
a
5.9/integraldisplay
x2arctanx
adx=1
3x3arctanx
a+1
6a3ln/parenleftbig
x2+a2/parenrightbig
−ax2
6
6.9/integraldisplay
x2arccotx
adx=−1
3x3arctanx
a−1
6a3ln/parenleftbig
x2+a2/parenrightbig
+πx3
6+ax2
6
2.83 Combinations of arcsine or arccosine and algebraic functions
2.831/integraldisplay
xnarcsinx
adx=xn+1
n+1arcsinx
a−1
n+1/integraldisplayxn+1dx√
a2−x2(see2.263 1,2.264,2.27)
2.832/integraldisplay
xnarccosx
adx=xn+1
n+1arccosx
a+1
n+1/integraldisplayxn+1dx√
a2−x2(see2.263 1,2.264,2.27)
1. For n=−1, these integrals (that is,/integraldisplayarcsin x
xdxand/integraldisplayarccos x
xdx)c a n n o tb ee x p r e s s e da sa
finite combination of elementary functions.
2.838 Arcsine or arccosine and algebraic functions 243
2./integraldisplayarccos x
xdx=−π
2ln1
x−/integraldisplayarcsin x
xdx
2.8339
1./integraldisplay
xarcsinx
adx= sign( a)/bracketleftbigg/parenleftbiggx2
2−a2
4/parenrightbigg
arcsinx
|a|+x
4/radicalbig
a2−x2/bracketrightbigg
2./integraldisplay
xarccosx
adx=πx2
4−sign(a)/bracketleftbigg1
4/parenleftbig
2x2−a2/parenrightbig
arcsinx
|a|+x
4/radicalbig
a2−x2/bracketrightbigg
3./integraldisplay
x2arcsinx
adx= sign( a)/bracketleftbiggx3
3arcsinx
|a|+1
9/parenleftbig
x2+2a2/parenrightbig/radicalbig
a2−x2/bracketrightbigg
4./integraldisplay
x2arccosx
adx=πx3
6−sign(a)/bracketleftbiggx3
3arcsinx
|a|+1
9/parenleftbig
x2+2a2/parenrightbig/radicalbig
a2−x2/bracketrightbigg
5./integraldisplay
x3arcsinx
adx= sign( a)/bracketleftbigg/parenleftbiggx4
4−3a4
32/parenrightbigg
arcsinx
|a|+1
32x/parenleftbig
2x2+3a2/parenrightbig/radicalbig
a2−x2/bracketrightbigg
6./integraldisplay
x3arccosx
adx=πx4
8−sign(a)/bracketleftBigg/parenleftbig
8x4−3a4/parenrightbig
32arcsinx
|a|+1
32x/parenleftbig
2x2+3a2/parenrightbig/radicalbig
a2−x2/bracketrightBigg
2.834
1./integraldisplay1
x2arcsinx
adx=−1
xarcsinx
a−1
alna+√
a2−x2
x
2./integraldisplay1
x2arccosx
adx=−1
xarccosx
a−1
alna+√
a2−x2
x
2.835/integraldisplayarcsin x
(a+bx)2dx=−arcsin x
b(a+bx)−2
b√
a2−b2arctan/radicalBigg
(a−b)(1−x)
(a+b)(1 + x)/bracketleftbig
a2>b2/bracketrightbig
=−arcsin x
b(a+bx)−1
b√
b2−a2ln/radicalbig
(a+b)(1 + x)+/radicalbig
(b−a)(1−x)/radicalbig
(a+b)(1 + x)−/radicalbig
(b−a)(1−x)/bracketleftbig
a2<b2/bracketrightbig
2.8368/integraldisplayxarcsin x
(1 +cx2)2dx=−arcsin x
2c(1 +cx2)+1
2c√c+1arctan√c+1x√
1−x2[c>−1]
=−arcsin x
2c(1 +cx2)+1
4c/radicalbig
−(c+1 )ln√
1−x2+x/radicalbig
−(c+1 )√
1−x2−x/radicalbig
−(c+1 )[c<−1]
2.837
1./integraldisplayxarcsin x√
1−x2dx=x−/radicalbig
1−x2arcsin x
2./integraldisplayxarcsin x√
1−x2dx=x2
4−x
2/radicalbig
1−x2arcsin x+1
4(arcsin x)2
3./integraldisplayx3arcsin x√
1−x2dx=x3
9+2x
3−1
3/parenleftbig
x2+2/parenrightbig/radicalbig
1−x2arcsin x
2.838
1./integraldisplayarcsin x/radicalBig
(1−x2)3dx=xarcsin x√
1−x2+1
2ln/parenleftbig
1−x2/parenrightbig
244 Inverse Trigonometric Functions 2.841
2./integraldisplayxarcsin x/radicalBig
(1−x2)3dx=arcsin x√
1−x2+1
2ln1−x
1+x
2.84 Combinations of the arcsecant and arccosecant with powers of x
2.841
1./integraldisplay
xarcsecx
adx=/integraldisplay
arccosa
xdx=1
2/braceleftBig
x2arccosa
x−a/radicalbig
x2−a2/bracerightBig/bracketleftBig
0<arccosa
x<π
2/bracketrightBig
=1
2/braceleftBig
x2arccosa
x+a/radicalbig
x2−a2/bracerightBig/bracketleftBigπ
2<arccosa
x<π/bracketrightBig
DW
2./integraldisplay
x2arcsecx
adx=/integraldisplay
arccosa
xdx=1
3/braceleftbigg
x3arccosa
x−a
2x/radicalbig
x2−a2−a3
2ln/parenleftBig
x+/radicalbig
x2−a2/parenrightBig/bracerightbigg
/bracketleftBig
0<arccosa
x<π
2/bracketrightBig
=1
3/braceleftbigg
x3arccosa
x+a
2x/radicalbig
x2−a2+a3
2ln/parenleftBig
x+/radicalbig
x2−z2/parenrightBig/bracerightbigg
/bracketleftBigπ
2<arccosa
x<π/bracketrightBig
DW
3./integraldisplay
xarccosecx
adx=/integraldisplay
arcsina
xdx=1
2/braceleftBig
x2arcsina
x+a/radicalbig
x2−a2/bracerightBig/bracketleftBig
0<arcsina
x<π
2/bracketrightBig
=1
2/braceleftBig
x2arcsina
x−a/radicalbig
x2−a2/bracerightBig/bracketleftBig
−π
2<arcsina
x<0/bracketrightBig
DW
2.85 Combinations of the arctangent and arccotangent with algebraic functions
2.851/integraldisplay
xnarctanx
adx=xn+1
n+1arctanx
a−a
n+1/integraldisplayxn+1dx
a2+x2
2.852
1./integraldisplay
xnarccotx
adx=xn+1
n+1arccotx
a+a
n+1/integraldisplayxn+1dx
a2+x2
Forn=−1
2./integraldisplayarctan x
xdxcannot be expressed as a finite combination of elementary functions.
3./integraldisplayarccot x
xdx=π
2lnx−/integraldisplayarctan x
xdx
2.853
1./integraldisplay
xarctanx
adx=1
2/parenleftbig
x2+a2/parenrightbig
arctanx
a−ax
2
2./integraldisplay
xarccotx
adx=1
2/parenleftbig
x2+a2/parenrightbig
arccotx
a+ax
2
2.859 Arctangent and arccotangent with algebraic functions 245
3.9/integraldisplay
x2arctanx
adx=x3
3arctanx
a+a3
6ln/parenleftbig
x2+a2/parenrightbig
−ax2
6
4.9/integraldisplay
x2arccotx
adx=−x3
3arctanx
a−a3
6ln/parenleftbig
x2+a2/parenrightbig
+πx3
6+ax2
6
2.854/integraldisplay1
x2arctanx
adx=−1
xarctanx
a−1
2alna2+x2
x2
2.855/integraldisplayarctan x
(α+βx)2dx=1
α2+β2/braceleftbigg
lnα+βx√
1+x2−β−αx
α+βxarctan x/bracerightbigg
2.856
1./integraldisplayxarctan x
1+x2dx=1
2arctan xln/parenleftbig
1+x2/parenrightbig
−1
2/integraldisplayln/parenleftbig
1+x2/parenrightbig
dx
1+x2TI (689)
2./integraldisplayx2arctan x
1+x2dx=xarctan x−1
2ln/parenleftbig
1+x2/parenrightbig
−1
2(arctan x)2TI (405)
3./integraldisplayx3arctan x
1+x2dx=−1
2x+1
2/parenleftbig
1+x2/parenrightbig
arctan x−/integraldisplayxarctan x
1+x2dx
(see2.8511 )
4./integraldisplayx4arctan x
1+x2dx=−1
6x2+2
3ln/parenleftbig
1+x2/parenrightbig
+/parenleftbiggx3
3−x/parenrightbigg
arctan x+1
2(arctan x)2
2.857/integraldisplayarctan xdx
(1 +x2)n+1=/bracketleftBiggn/summationdisplay
k=1(2n−2k)!!(2n−1)!!
(2n)!!(2n−2k+1 ) ! !x
(1 +x2)n−k+1+1
2(2n−1)!!
(2)!!arctan x/bracketrightBigg
arctan x
+1
2n/summationdisplay
k=1(2n−1)!!(2n−2k)!!
(2n)!!(2n−2k+ 1)!!( n−k+1 )1
(1 +x2)n−k+1
2.858/integraldisplayxarctan x√
1−x2dx=−/radicalbig
1−x2arctan x+√
2arc t anx√
2√
1−x2−arcsin x
2.859/integraldisplayarctan x/radicalBig
(a+bx2)3dx=xarctan x
a√
a+bx2−1
a√
b−aarctan/radicalbigg
a+bx2
b−a[a<b]
=xarctan x
a√
a+bx2+1
2a√
a−bln√
a+bx2−√
a−b√
a+bx2+√
a−b[a>b]
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3–4 Definite Integrals of Elementary
Functions
3.0 Introduction
3.01 Theorems of a general nature
3.011 Suppose that f(x) is integrable†over the largest of the intervals ( p, q),(p, r),(r, q). Then (de-
pending on the relative positions of the points p,q,a n dr) it is also integrable over the other two intervals,
and we have/integraldisplayq
pf(x)dx=/integraldisplayr
pf(x)dx+/integraldisplayq
rf(x)dx. FI II 126
3.012 The first mean-value theorem . Suppose (1) that f(x) is continuous and that g(x) is integrable
over the interval ( p, q), (2) that m≤f(x)≤M, and (3) that g(x) does not change sign anywhere in the
interval ( p, q). Then, there exists at least one point ξ(with p≤ξ≤q) such that/integraldisplayq
pf(x)g(x)dx=f(ξ)/integraldisplayq
pg(x)dx. FI II 132
3.013 The second mean-value theorem. Iff(x) is monotonic and non-negative throughout the interval
(p, q), where p<q, and if g(x) is integrable over that interval, then there exists at least one point ξ(with
p≤ξ≤q) such that
1./integraldisplayq
pf(x)g(x)dx=f(p)/integraldisplayξ
pg(x)dx
Under the conditions of Theorem 3.013 1, iff(x) is nondecreasing, then
2./integraldisplayq
pf(x)g(x)dx=f(q)/integraldisplayq
ξg(x)dx [p≤ξ≤q].
Iff(x) is monotonic in the interval ( p, q), where p<q, and if g(x) is integrable over that interval, then
∗We omit the definition of definite and multiple integrals since they are widely known and can easily be found in any
textbook on the subject. Here we give only certain theorems of a general nature which provide estimates, or which reduce
the given integral to a simpler one.
†A function f(x)i ss a i dt ob ei n t e g r a b l eo v e rt h ei n t e r v a l( p, q), if the integral/integraldisplayq
pf(x)dxexists. Here, we usually mean
the existence of the integral in the sense of Riemann. When it is a matter of the existence of the integral in the sense of
Stieltjes or Lebesgue, etc., we shall speak of integrability in the sense of Stieltjes or Lebesgue.
247
248 Introduction 3.020
3./integraldisplayq
pf(x)g(x)dx=f(p)/integraldisplayξ
pg(x)dx+f(q)/integraldisplayq
ξg(x)dx [p≤ξ≤q],
or
4./integraldisplayq
pf(x)g(x)dx=A/integraldisplayξ
pg(x)dx+B/integraldisplayq
ξg(x)dx [p≤ξ≤q],
where AandBare any two numbers satisfying the conditions
A≥f(p+0 ) a n d B≤f(q−0) [if fdecreases] ,
A≤f(p+0 ) a n d B≥f(q−0) [if fincreases] .
In particular,
5./integraldisplayq
pf(x)g(x)dx=f(p+0 )/integraldisplayξ
pg(x)dx+f(q−0)/integraldisplayq
ξg(x)dx FI II 138
3.02 Change of variable in a definite integral
3.020/integraldisplayβ
αf(x)dx=/integraldisplayψ
ϕf[g(t)]g/prime(t)dt;x=g(t).
This formula is valid under the following conditions:
1. f(x) is continuous on some interval A≤x≤Bcontaining the original limits of integration α
andβ.
2. The equalities α=g(ϕ)a n d β=g(ψ) hold.
3. g(t) and its derivative g/prime(t) are continuous on the interval ϕ≤t≤ψ.
4. As tvaries from ϕtoψ, the function g(t) always varies in the same direction from g(ϕ)=αto
g(ψ)=β.∗
3.021 The integral/integraldisplayβ
αf(x)dxcan be transformed into another integral with given limits ϕandψby
means of the linear substitution
x=β−α
ψ−ϕt+αψ−βϕ
ψ−ϕ:
1./integraldisplayβ
αf(x)dx=β−α
ψ−ϕ/integraldisplayψ
ϕf/parenleftbiggβ−α
ψ−ϕt+αψ−βϕ
ψ−ϕ/parenrightbigg
dt
In particular, for ϕ=0a n d ψ=1 ,
∗If this last condition is not satisfied, the interval ϕ≤t≤ψshould be partitioned into subintervals throughout each of
which the condition is satisfied:
/integraldisplayβ
αf(x)dx=/integraldisplayϕ1
ϕf[g(t)]g/prime(t)dt+/integraldisplayϕ2
ϕ1f[g(t)]g/prime(t)dt+···+/integraldisplayψ
ϕn−1f[g(t)]g/prime(t)dt.
3.033 General formulas 249
2./integraldisplayβ
αf(x)dx=(β−α)/integraldisplay1
0f((β−α)t+α)dt
Forϕ=0a n d ψ=∞,
3./integraldisplayβ
αf(x)dx=(β−α)/integraldisplay∞
0f/parenleftbiggα+βt
1+t/parenrightbiggdt
(1 +t)2
3.022 The following formulas also hold:
1./integraldisplayβ
αf(x)dx=/integraldisplayβ
αf(α+β−x)dx
2./integraldisplayβ
0f(x)dx=/integraldisplayβ
0f(β−x)dx
3./integraldisplayα
−αf(x)dx=/integraldisplayα
−αf(−x)dx
3.03 General formulas
3.031
1. Suppose that a function f(x) is integrable over the interval ( −p, p) and satisfies the relation
f(−x)=f(x) on that interval. (A function satisfying the latter condition is called an even
function.) Then,
/integraldisplayp
−pf(x)dx=2/integraldisplayp
0f(x)dx. FI II 159
2. Suppose that f(x) is a function that is integrable on the interval ( −p, p) and satisfies the relation
f(−x)=−f(x) on that interval. (A function satisfying the latter condition is called an odd
function). Then,
/integraldisplayp
−pf(x)dx=0. FI II 159
3.032
1./integraldisplayπ
2
0f(sinx)dx=/integraldisplayπ
2
0f(cosx)dx,
where f(x) is a function that is integrable on the interval (0 ,1). FI II 159
2./integraldisplay2π
0f(pcosx+qsinx)dx=2/integraldisplayπ
0f/parenleftBig/radicalbig
p2+q2cosx/parenrightBig
dx,
where f(x) is integrable on the interval/parenleftBig
−/radicalbig
p2+q2,/radicalbig
p2+q2/parenrightBig
. FI II 160
3./integraldisplayπ
2
0f(sin 2x)cosxdx=/integraldisplayπ
2
0f/parenleftbig
cos2x/parenrightbig
cosxdx,
where f(x) is integrable on the interval (0 ,1). FI II 161
3.033
1. If f(x+π)=f(x)a n d f(−x)=f(x), then
250 Introduction 3.034
/integraldisplay∞
0f(x)sinx
xdx=/integraldisplayπ
2
0f(x)dx LO V 277(3)
2. If f(x+π)=−f(x)a n d f(−x)=f(x), then
/integraldisplay∞
0f(x)sinx
xdx=/integraldisplayπ
2
0f(x)cosxdx LO V 279(4)
In formulas 3.033 , it is assumed that the integrals in the left members of the formulas exist.
3.034/integraldisplay∞
0f(px)−f(qx)
xdx=[f(0)−f(+∞)] lnq
p,
iff(x) is continuous for x≥0 and if there exists a finite limit f(+∞) = lim
x→+∞f(x). FI II 633
3.035
1./integraldisplayπ
0f/parenleftbig
α+exi/parenrightbig
+f/parenleftbig
α+e−xi/parenrightbig
1+2pcosx+p2dx=2π
1−p2f(α+p)[|p|<1] LA 230(16)
2./integraldisplayπ
01−pcosx
1−2pcosx+p2/braceleftbig
f/parenleftbig
α+exi/parenrightbig
+f/parenleftbig
α+e−xi/parenrightbig/bracerightbig
dx=π{f(α+p)+f(α)}
[|p|<1] BE 169
3./integraldisplayπ
0f/parenleftbig
α+e−xi/parenrightbig
−f/parenleftbig
α+exi/parenrightbig
1−2pcosx+p2sinxdx=π
π{f(α+p)−f(α)}
[|p|<1] BE 169
In formulas 3.035 , it is assumed that the function fis analytic in the closed unit circle with its center
at the point α.
3.036
1.11/integraldisplayπ
0f/parenleftbiggsin2x
1+2pcosx+p2/parenrightbigg
dx=/integraldisplayπ
0f/parenleftbig
sin2x/parenrightbig
dx/bracketleftbig
p2<1/bracketrightbig
=/integraldisplayπ
0f/parenleftbiggsin2x
p2/parenrightbigg
dx/bracketleftbig
p2≥1/bracketrightbig
LA 228(6)
2./integraldisplayπ
0F(n)(cosx)sin2nxdx=( 2n−1)!!/integraldisplayπ
0F(cosx)c o snxdx B1 7 4
3.037 Iffis analytic in the circle of radius rand if
f[r(cosx+isinx)] =f1(r, x)+if2(r, x),
then
1./integraldisplay∞
0f1(r, x)
p2+x2dx=π
2pf/parenleftbig
re−p/parenrightbig
LA 230(19)
2./integraldisplay∞
0f2(r, x)xdx
p2+x2=π
2/bracketleftbig
f/parenleftbig
re−p/parenrightbig
−f(0)/bracketrightbig
LA 230(20)
3./integraldisplay∞
0f2(r, x)
xdx=π
2[f(r)−f(0)] LA 230(21)
3.045 Improper integrals 251
4./integraldisplay∞
0f2(r, x)
x(p2+x2)dx=π
2p2/bracketleftbig
f(r)−f/parenleftbig
re−p/parenrightbig/bracketrightbig
LA 230(22)
3.038/integraldisplay∞
−∞xdx√
1+x2F/parenleftBig
qx+p/radicalbig
1+x2/parenrightBig
=/integraldisplay∞
−∞F(pcoshx+qsinhx)sin h xdx
=2q/integraldisplay∞
0F/prime/parenleftBig
signp·/radicalbig
p2−q2coshx/parenrightBig
sinh2xdx
[IfFis a function with a continuous derivative in the interval ( −∞,∞), all these integrals converge.]
3.04 Improper integrals
3.041 Suppose that a function f(x)i sd e fi n e do na ni n t e r v a l( p,+∞) and that it is integrable over an
arbitrary finite subinterval of the form ( p, P). Then, by definition
/integraldisplay+∞
pf(x)dx= lim
P→+∞/integraldisplayP
pf(x)dx,
if this limit exists. If it does exist, we say that the integral/integraldisplay+∞
pf(x)dxexists or that it converges.
Otherwise, we say that the integral diverges.
3.042 Suppose that a function f(x) is bounded and integrable in an arbitrary interval ( p, q−η)( f o r
0<η<q −p) but is unbounded in every interval ( q−η,q) to the left of the point q. The point qis then
called a singular point . Then, by definition,/integraldisplayq
pf(x)dx= lim
η→0/integraldisplayq−η
pf(x)dx,
if this limit exists. In this case, we say that the integral/integraldisplayq
pf(x)dxexists or that it converges .
3.043 If not only the integral of f(x) but also the integral of |f(x)|exists, we say that the integral of
f(x)c o n v e r g e s absolutely .
3.044 The integral/integraldisplay+∞
pf(x)dxconverges absolutely if there exists a number α>1 such that the limit
lim
x→+∞{xα|f(x)|}
exists. On the other hand, if
lim
x→+∞{x|f(x)|}=L>0,
the integral/integraldisplay+∞
p|f(x)|dxdiverges.
3.045 Suppose that the upper limit qof the integral/integraldisplayq
pf(x)dxis a singular point. Then, this integral
converges absolutely if there exists a number α<1 such that the limit
lim
x→q[(q−x)α|f(x)|]
exists. On the other hand, if
lim
x→q[(q−x)|f(x)|]=L>0,
the integral/integraldisplayq
pf(x)dxdiverges.
252 Introduction 3.046
3.046 Suppose that the functions f(x)a n d g(x) are defined on the interval ( p,+∞), that f(x)i s
integrable over every finite interval of the form ( p, P), that the integral
/integraldisplayP
pf(x)dx
is a bounded function of P,t h a t g(x) is monotonic, and that g(x)→0a sx→+∞. Then, the integral
/integraldisplay+∞
pf(x)g(x)dx
converges. FI II 577
3.05 The principal values of improper integrals
3.051 Suppose that a function f(x) has a singular point rsomewhere inside the interval ( p, q), that
f(x) is defined at r,a n dt h a t f(x) is integrable over every portion of this interval that does not contain
the point r. Then, by definition/integraldisplayq
pf(x)dx= lim
η→0
η/prime→0/braceleftbigg/integraldisplayr−η
pf(x)dx+/integraldisplayq
r+η/primef(x)dx/bracerightbigg
.
Here, the limit must exist for independent modes of approach of ηandη/primeto zero. If this limit does not
exist but the limit
lim
η→0/braceleftbigg/integraldisplayr−η
pf(x)dx+/integraldisplayq
r+ηf(x)dx/bracerightbigg
does exist, we say that this latter limit is the principal value of the improper integral/integraltextq
pf(x)dx,a n dw e
say that the integral/integraldisplayq
pf(x)dxexists in the sense of principal values. FI II 603
3.052 Suppose that the function f(x) is continuous over the interval ( p, q) and vanishes at only one
point rinside this interval. Suppose that the first derivative f/prime(x) exists in a neighborhood of the point
r. Suppose that f/prime(r)/negationslash= 0 and that the second derivative f/prime/prime(r) exists at the point ritself. Then,/integraldisplayq
pdx
f(x)FI II 605
diverges, but exists in the sense of principal values.
3.053 A divergent integral of a positive function cannot exist in the sense of principal values.
3.054 Suppose that the function f(x) has no singular points in the interval ( −∞,+∞). Then, by
definition/integraldisplay+∞
−∞f(x)dx= lim
P→−∞
Q→+∞/integraldisplayQ
Pf(x)dx.
Here, the limit must exist for independent approach of PandQto±∞. If this limit does not exist but
the limit
lim
P→+∞/integraldisplay+P
−Pf(x)dx
does exist, this last limit is called the principal value of the improper integral/integraldisplay+∞
−∞f(x)dx. FI II 607
3.055 The principal value of an improper integral of an even function exists only when this integral
converges (in the ordinary sense). FI II 607
3.112 Rational functions 253
3.1–3.2 Power and Algebraic Functions
3.11 Rational functions
1./integraldisplay∞
−∞p+qx
r2+2rxcosλ+x2dx=π
rsinλ(p−qrcosλ) (principal value)
(see also 3.194 8a n d3.252 1a n d2 ) BI (22)(14)
3.11211Integrals of the form/integraldisplay∞
−∞gn(x)dx
hn(x)hn(−x),w h e r e
gn(x)=b0x2n−2+b1x2n−4+···bn−1,
hn(x)=a0xn+a1xn−1+···an
[All roots of hn(x) lie in the upper half-plane.]
1./integraldisplay∞
−∞gn(x)dx
hn(x)hn(−x)=πi
a0Mn
Δn, JE
where
Δn=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1a3a5 0
a0a2a4 0
0a1a3 0
......
000 an/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,M
n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
0b1b2···bn−1
a0a2a4 0
0a1a3 0
......
000 an/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
2./integraldisplay
∞
−∞g1(x)dx
h1(x)h1(−x)=πib0
a0a1JE
3.8/integraldisplay∞
−∞g2(x)dx
h2(x)h2(−x)=πi−b0+a0b1
a2
a0a1
4.11/integraldisplay∞
−∞g3(x)dx
h3(x)h3(−x)=πi−a2b0+a0b1−a0a1b2
a3
a0(a0a3−a1a2)JE
5./integraldisplay∞
−∞g4(x)dx
h4(x)h4(−x)=πib0(−a1a4+a2a3)−a0a3b1+a0a1b2+a0b3
a4(a0a3−a1a2)
a0(a0a2
3+a2
1a4−a1a2a3)JE
6./integraldisplay∞
−∞g5(x)dx
h5(x)h5(−x)=πiM5
a0Δ5,
where
M5=b0/parenleftbig
−a0a4a5+a1a2
4+a2
2a5−a2a3a4/parenrightbig
+a0b1(−a2a5+a3a4)
+a0b2(a0a5−a1a4)+a0b3(−a0a3+a1a2)+a0b4
a5/parenleftbig
−a0a1a5+a0a2
3+a2
1a4−a1a2a3/parenrightbig
,
Δ5=a2
0a25−2a0a1a4a5−a0a2a3a5+a0a2
3a4+a2
1a24+a1a2
2a5−a1a2a3a4 JE
254 Power and Algebraic Functions 3.121
3.12 Products of rational functions and expressions that can be reduced to square
roots of first- and second-degree polynomials
3.121
1./integraldisplay1
01
1−2xcosλ+x2dx√x= 2 cosec λ∞/summationdisplay
k=1sinkλ
2k−1BI (10)(17)
2./integraldisplay1
01
q−pxdx/radicalbig
x(1−x)=π/radicalbig
q(q−p)[0<p<q ] BI (10)(9)
3./integraldisplay1
0dx
1−2rx+r2/radicalbigg
1∓x
1±x=±π
4r∓1
r1∓r
1±rarctan1+r
1−rLI (14)(5, 16)
3.13–3.17 Expressions that can be reduced to square roots of third- and fourth-
degree polynomials and their products with rational functions
Notation :I n3.131 –3.137 we set: α=a r c s i n/radicalbigg
a−c
a−u,β=a r c s i n/radicalbigg
c−u
b−u,
γ=a r c s i n/radicalbigg
u−c
b−c,δ =a r c s i n/radicalBigg
(a−c)(b−u)
(b−c)(a−u),
κ=a r c s i n/radicalBigg
(a−c)(u−b)
(a−b)(u−c),λ =a r c s i n/radicalbigg
a−u
a−b,
μ=a r c s i n/radicalbigg
u−a
u−b,ν =a r c s i n/radicalbigg
a−c
u−c,p =/radicalbigg
a−b
a−c,q =/radicalbigg
b−c
a−c.
3.131
1./integraldisplayu
−∞dx/radicalbig
(a−x)(b−x)(c−x)=2√a−cF(α,p)[ a>b>c ≥u] BY (231.00)
2./integraldisplayc
udx/radicalbig
(a−x)(b−x)(c−x)=2√a−cF(β,p)[ a>b>c>u ] BY (232.00)
3./integraldisplayu
cdx/radicalbig
(a−x)(b−x)(x−c)=2√a−cF(γ,q)[ a>b≥u>c] BY (233.00)
4./integraldisplayb
udx/radicalbig
(a−x)(b−x)(x−c)=2√a−cF(δ, q)[ a>b>u ≥c] BY (234.00)
5./integraldisplayu
bdx/radicalbig
(a−x)(x−b)(x−c)=2√a−cF(κ, p)[ a≥u>b>c ] BY (235.00)
6./integraldisplaya
udx/radicalbig
(a−x)(x−b)(x−c)=2√a−cF(λ,p)[ a>u ≥b>c] BY (236.00)
7./integraldisplayu
adx/radicalbig
(x−a)(x−b)(x−c)=2√a−cF(μ, q)[ u>a>b>c ] BY (237.00)
3.133 Square roots of polynomials 255
8./integraldisplay∞
udx/radicalbig
(x−a)(x−b)(x−c)=2√a−cF(ν,q)[ u≥a>b>c ] BY (238.00)
3.132
1./integraldisplayc
uxdx/radicalbig
(a−x)(b−x)(c−x)=2√a−c[cF(β,p)+(a−c)E(β,p)]−2/radicalbigg
(a−u)(c−u)
b−u
[a>b>c>u ] BY (232.19)
2./integraldisplayu
cxdx/radicalbig
(a−x)(b−x)(x−c)=2a√a−cF(γ,q)−2√
a−cE(γ,q)
[a>b≥u>c] BY (233.17)
3./integraldisplayb
uxdx/radicalbig
(a−x)(b−x)(x−c)=2√a−c/bracketleftbig
(b−a)Π/parenleftbig
δ, q2,q/parenrightbig
+aF(δ, q)/bracketrightbig
[a>b>u ≥c] BY (234.16)
4./integraldisplayu
bxdx/radicalbig
(a−x)(x−b)(x−c)=2√a−c/bracketleftbig
(b−c)Π/parenleftbig
κ, p2,p/parenrightbig
+cF(κ, p)/bracketrightbig
[a≥u>b>c ] BY (235.16)
5./integraldisplaya
uxdx/radicalbig
(a−x)(x−b)(x−c)=2c√a−cF(λ,p)+2√
a−cE(λ,p)
[a>u ≥b>c] BY (236.16)
6./integraldisplayu
axdx/radicalbig
(x−a)(x−b)(x−c)=2
b√a−c/bracketleftbig
a(a−b)Π(μ,1,q)+b2F(μ, q)/bracketrightbig
[u>a>b>c ] BY (237.16)
3.133
1./integraldisplayu
−∞dx/radicalbig
(a−x)3(b−x)(c−x)=2
(a−b)√a−c[F(α,p)−E(α,p)]
[a>b>c ≥u] BY (231.08)
2./integraldisplayc
udx/radicalbig
(a−x)3(b−x)(c−x)=2
(a−b)√a−c[F(β,p)−E(β,p)] +2
a−c/radicalbiggc−u
(a−u)(b−u)
[a>b>c>u ] BY (232.13)
3./integraldisplayu
cdx/radicalbig
(a−x)3(b−x)(x−c)=2
(a−b)√a−cE(γ,q)−2
(a−b)(a−c)/radicalbigg
(b−u)(u−c)
a−u
[a>b≥u>c] BY (233.09)
4./integraldisplayb
udx/radicalbig
(a−x)3(b−x)(x−c)=2
(a−b)√a−cE(δ, q)[a>b>u ≥c] BY (234.05)
5./integraldisplayu
bdx/radicalbig
(a−x)3(x−b)(x−c)=2
(a−b)√a−c[F(κ, p)−E(κ, p)] +2
a−b/radicalBigg
u−b
(a−u)(u−c)
[a>u>b>c ] BY (235.04)
256 Power and Algebraic Functions 3.133
6./integraldisplay∞
udx/radicalbig
(x−a)3(x−b)(x−c)=2
(b−a)√a−cE(ν,q)+2
a−b/radicalBigg
u−b
(u−a)(u−c)
[u>a>b>c ] BY (238.05)
7./integraldisplayu
−∞dx/radicalbig
(a−x)(b−x)3(c−x)=2√a−c
(a−b)(b−c)E(α,p)−2
(a−b)√a−cF(α,p)
−2
b−c/radicalbiggc−u
(a−u)(b−u)
[a>b>c ≥u] BY (231.09)
8./integraldisplayc
udx/radicalbig
(a−x)(b−x)3(c−x)=2√a−c
(a−b)(b−c)E(β,p)−2
(a−b)√a−cF(β,p)
[a>b>c>u ] BY (232.14)
9./integraldisplayu
cdx/radicalbig
(a−x)(b−x)3(x−c)=2
(b−c)√a−cF(γ,q)−2√a−c
(a−b)(b−c)E(γ,q)
+2
(a−b)(b−c)/radicalbigg
(a−u)(u−c)
b−u
[a>b>u>c ] BY (233.10)
10./integraldisplaya
udx/radicalbig
(a−x)(x−b)3(x−c)=2
(a−b)√a−cF(λ,p)−2√a−c
(a−b)(b−c)E(λ,p)
+2
(a−b)(b−c)/radicalbigg
(a−u)(u−c)
u−b
[a>u>b>c ] BY (236.09)
11./integraldisplayu
adx/radicalbig
(x−a)(x−b)3(x−c)=2√a−c
(a−b)(b−c)E(μ, q)−2
(b−c)√a−cF(μ, q)
[u>a>b>c ] BY (237.12)
12./integraldisplay∞
udx/radicalbig
(x−a)(x−b)3(x−c)=2√a−c
(a−b)(b−c)E(ν,q)−2
(b−c)√a−cF(ν,q)
−2
a−b/radicalbiggu−a
(u−b)(u−c)
[u≥a>b>c ] BY (238.04)
13./integraldisplayu
−∞dx/radicalbig
(a−x)(b−x)(c−x)3=2
(c−b)√a−cE(α,p)+2
b−c/radicalBigg
b−u
(a−u)(c−u)
[a>b>c>u ] BY (231.10)
14./integraldisplayb
udx/radicalbig
(a−x)(b−x)(x−c)3=2
(b−c)√a−c[F(δ, q)−E(δ, q)] +2
b−c/radicalBigg
b−u
(a−u)(u−c)
[a>b>u>c ] BY (234.04)
15./integraldisplayu
bdx/radicalbig
(a−x)(x−b)(x−c)3=2
(b−c)√a−cE(κ, p)
[a≥u>b>c ] BY (235.01)
3.134 Square roots of polynomials 257
16./integraldisplaya
udx/radicalbig
(a−x)(x−b)(x−c)3=2
(b−c)√a−cE(λ,p)−2
(b−c)(a−c)/radicalbigg
(a−u)(u−b)
u−c
[a>u ≥b>c] BY (236.10)
17./integraldisplayu
adx/radicalbig
(x−a)(x−b)(x−c)3=2
(b−c)√a−c[F(μ, q)−E(μ, q)] +2
a−c/radicalbiggu−a
(u−b)(u−c)
[u>a>b>c ] BY (237.13)
18./integraldisplay∞
udx/radicalbig
(x−a)(x−b)(x−c)3=2
(b−c)√a−c[F(ν,q)−E(ν,q)]
[u≥a>b>c ] BY (238.03)
3.134
1./integraldisplayu
−∞dx/radicalbig
(a−x)5(b−x)(c−x)
=2
3(a−b)2/radicalbig
(a−c)3[(3a−b−2c)F(α,p)−2(2a−b−c)E(α,p)]
+2
3(a−c)(a−b)/radicalBigg
(c−u)(b−u)
(a−u)3
[a>b>c ≥u] BY (231.08)
2./integraldisplayc
udx/radicalbig
(a−x)5(b−x)(c−x)=2
3(a−b)2/radicalbig
(a−c)3[(3a−b−2c)F(β,p)−2(2a−b−c)E(β,p)]
+2/bracketleftbig
4a2−3ab−2ac+bc−u(3a−2b−c)/bracketrightbig
3(a−b)(a−c)2/radicalbiggc−u
(a−u)3(b−u)
[a>b>c>u ] BY (232.13)
3./integraldisplayu
cdx/radicalbig
(a−x)5(b−x)(x−c)=2
3(a−b)3/radicalbig
(a−c)3[2(2a−b−c)E(γ,q)−(a−b)F(γ,q)]
−2/bracketleftbig
5a2−3ab−3ac+bc−2u(2a−b−c)/bracketrightbig
3(a−b)2(a−c)2/radicalBigg
(b−u)(u−c)
(a−u)3
[a>b≥u>c] BY (233.09)
4./integraldisplayb
udx/radicalbig
(a−x)5(b−x)(x−c)=2
3(a−b)2/radicalbig
(a−c)3[2(2a−b−c)E(δ, q)−(a−b)F(δ, q)]
−2
3(a−b)(a−c)/radicalBigg
(b−u)(u−c)
(a−u)3
[a>b>u ≥c] BY (234.05)
5./integraldisplayu
bdx/radicalbig
(a−x)5(x−b)(x−c)
=2
3(a−b)2/radicalbig
(a−c)3[(3a−b−2c)F(κ, p)−2(2a−b−c)E(κ, p)]
+2/bracketleftbig
4a2−2ab−3ac+bc−u(3a−b−2c)/bracketrightbig
3(a−b)2(a−c)/radicalBigg
u−b
(a−u)3(u−c)
[a>u>b>c ] BY (235.04)
258 Power and Algebraic Functions 3.134
6./integraldisplay∞
udx/radicalbig
(x−a)5(x−b)(x−c)=2
3(a−b)2/radicalbig
(a−c)3[2(2a−b−c)E(ν,q)−(a−b)F(ν,q)]
+2/bracketleftbig
4a2−2ab−3ac+bc+u(b+2c−3a)/bracketrightbig
3(a−b)2(a−c)/radicalBigg
u−b
(u−a)3(u−c)
[u>a>b>c ] BY (238.05)
7./integraldisplayu
−∞dx/radicalbig
(a−x)(b−x)5(c−x)=2
3(a−b)2(b−c)2√a−c
×[2(a−c)(a+c−2b)E(α,p)+(b−c)(3b−a−2c)F(α,p)]
−2/bracketleftbig
3ab−ac+2bc−4b2−u(2a−3b+c)/bracketrightbig
3(a−b)(b−c)2/radicalbiggc−u
(a−u)(b−u)3
[a>b>c ≥u] BY (231.09)
8./integraldisplayc
udx/radicalbig
(a−x)(b−x)5(c−x)=2
3(a−b)2(b−c)2√a−c
×[(b−c)(3b−a−2c)F(β,p)+2 ( a−c)(a−2b+c)E(β,p)]
+2
3(a−b)(b−c)/radicalBigg
(a−u)(c−u)
(b−u)3
[a>b>c>u ] BY (232.14)
9./integraldisplayu
cdx/radicalbig
(a−x)(b−x)5(x−c)=2
3(a−b)2(b−c)2√a−c
×[(a−b)(2a−3b+c)F(γ,q)+2 ( a−c)(2b−a−c)E(γ,q)]
+2/bracketleftbig
3ab+3bc−ac−5b2−2u(a−2b+c)/bracketrightbig
3(a−b)2(b−c)2/radicalBigg
(a−u)(u−c)
(b−u)3
[a>b>u>c ] BY (233.10)
10./integraldisplaya
udx/radicalbig
(a−x)(x−b)5(x−c)=2
3(a−b)2(b−c)2√a−c
×[(b−c)(3b−2c−a)F(λ,p)+2 ( a−c)(a+c−2b)E(λ,p)]
+2/bracketleftbig
3ab+3bc−ac−5b2+2u(2b−a−c)/bracketrightbig
3(a−b)2(b−c)2/radicalBigg
(a−u)(u−c)
(u−b)3
[a>u>b>c ] BY (236.09)
11./integraldisplayu
adx/radicalbig
(x−a)(x−b)5(x−c)=2
3(a−b)2(b−c)2√a−c
×[(a−b)(2a+c−3b)F(μ, q)+2 ( a−c)(2b−a−c)E(μ, q)]
+2
3(a−b)(b−c)/radicalBigg
(u−a)(u−c)
(u−b)3
[u>a>b>c ] BY (237.12)
3.134 Square roots of polynomials 259
12./integraldisplay∞
udx/radicalbig
(x−a)(x−b)5(x−c)=2
3(a−b)2(b−c)2√a−c
×[(a−b)(2a+c−3b)F(ν,q)+2 ( a−c)(2b−c−a)E(ν,q)]
−2/bracketleftbig
3bc+2ab−ac−4b2+u(3b−a−2c)/bracketrightbig
3(a−b)2(b−c)/radicalbiggu−a
(u−b)3(u−c)
[u≥a>b>c ] BY (238.04)
13./integraldisplayu
−∞dx/radicalbig
(a−x)(b−x)(c−x)5=2
3(b−c)2/radicalbig
(a−c)3[2(a+b−2c)E(α,p)−(b−c)F(α,p)]
+2/bracketleftbig
ab−3ac−2bc+4c2+u(2a+b−3c)/bracketrightbig
3(a−c)(b−c)2/radicalBigg
b−u
(a−u)(c−u)3
[a>b>c>u ] By (231.10)
14./integraldisplayb
udx/radicalbig
(a−x)(b−x)(x−c)5=2
3(b−c)2/radicalbig
(a−c)3[(2a+b−3c)F(δ, q)−2(a+b−2c)E(δ, q)]
+2/bracketleftbig
ab−3ac−2bc+4c2+u(2a+b−3c)/bracketrightbig
3(b−c)2(a−c)/radicalBigg
b−u
(a−u)(u−c)3
[a>b>u>c ] BY (234.04)
15./integraldisplayu
bdx/radicalbig
(a−x)(x−b)(x−c)5=2
3(b−c)2/radicalbig
(a−c)3[2(a+b−2c)E(κ, p)−(b−c)F(κ, p)]
+2
3(a−c)(b−c)/radicalBigg
(a−u)(u−b)
(u−c)3
[a≥u>b>c ] BY (235.20)
16./integraldisplaya
udx/radicalbig
(a−x)(x−b)(x−c)5=2
3(b−c)2/radicalbig
(a−c)3[2(a+b−2c)E(λ,p)−(b−c)F(λ,p)]
−2/bracketleftbig
ab−3ac−3bc+5c2+2u(a+b−2c)/bracketrightbig
3(b−c)2(a−c)2/radicalBigg
(a−u)(u−b)
(u−c)3
[a>u ≥b>c] BY (236.10)
17./integraldisplayu
adx/radicalbig
(x−a)(x−b)(x−c)5=2
3(b−c)2/radicalbig
(a−c)3[(2a+b−3c)F(μ, q)−2(a+b−2c)E(μ, q)]
+2/bracketleftbig
4c2−ab−2ac−bc+u(3a+2b−5c)/bracketrightbig
3(b−c)(a−c)2/radicalbiggu−a
(u−b)(u−c)3
[u>a>b>c ] BY (237.13)
18./integraldisplay∞
udx/radicalbig
(x−a)(x−b)(x−c)5=2
3(b−c)2/radicalbig
(a−c)3[(2a+b−3c)F(ν,q)−2(a+b−2c)E(ν,q)]
+2
3(a−c)(b−c)/radicalBigg
(u−a)(u−b)
(u−c)3
[u≥a>b>c ] BY (238.03)
260 Power and Algebraic Functions 3.135
3.135
1.6/integraldisplayu
−∞dx/radicalbig
(a−x)(b−x)3(c−x)3=2
(a−b)(b−c)2√a−c[(b−c)F(α,p)−(2a−b−c)E(α,p)]
+2(b+c−2u)
(b−c)2/radicalbig
(a−u)(b−u)(c−u)
[a>b>c>u ] BY (231.13)
2./integraldisplaya
udx/radicalbig
(a−x)(x−b)3(x−c)3=2
(a−b)(b−c)2√a−c[(b−c)F(λ,p)−2(2a−b−c)E(λ,p)]
+2(a−b−c+u)
(a−b)(b−c)(a−c)/radicalbigga−u
(u−b)(u−c)
[a>u>b>c ] BY (236.15)
3./integraldisplayu
adx/radicalbig
(x−a)(x−b)3(x−c)3=2
(a−b)(b−c)2√a−c[(2a−b−c)E(μ, q)−2(a−b)F(μ, q)]
+2
(a−c)(b−c)/radicalbiggu−a
(u−b)(u−c)
[u>a>b>c ] BY (236.14)
4./integraldisplay∞
udx/radicalbig
(x−a)(x−b)3(x−c)3=2
(a−b)(b−c)2√a−c[(2a−b−c)E(ν,q)−2(a−b)F(ν,q)]
−2
(a−b)(b−c)/radicalbiggu−a
(u−b)(u−c)
[u≥a>b>c ] BY (238.13)
5./integraldisplayu
−∞dx/radicalbig
(a−x)3(b−x)(c−x)3=2
(a−b)(b−c)/radicalbig
(a−c)3[(2b−a−c)E(α,p)−(b−c)F(α,p)]
+2
(b−c)(a−c)/radicalBigg
b−u
(a−u)(c−u)
[a>b>c>u ] BY(231.12)
6./integraldisplayb
udx/radicalbig
(a−x)3(b−x)(x−c)3=2
(b−c)(a−b)/radicalbig
(a−c)3[(a−b)F(δ, q)+( 2 b−a−c)E(δ, q)]
+2
(b−c)(a−c)/radicalBigg
b−u
(a−u)(u−c)
[a>b>u>c ] BY (234.03)
7./integraldisplayu
bdx/radicalbig
(a−x)3(x−b)(x−c)3=2
(a−b)(b−c)/radicalbig
(a−c)3[(b−c)F(κ, p)−(2b−a−c)E(κ, p)]
+2
(a−b)(a−c)/radicalBigg
u−b
(a−u)(u−c)
[a>u>b>c ] BY (235.15)
3.136 Square roots of polynomials 261
8./integraldisplay∞
udx/radicalbig
(x−a)3(x−b)(x−c)3=2
(a−b)(b−c)/radicalbig
(a−c)3[(a+c−2b)E(ν,q)−(a−b)F(ν,q)]
+2
(a−b)(a−c)/radicalBigg
u−b
(u−a)(u−c)
[u>a>b>c ] BY (238.14)
9./integraldisplayu
−∞dx/radicalbig
(a−x)3(b−x)3(c−x)=2
(b−c)(a−b)2√a−c[(a+b−2c)E(α,p)−2(b−c)F(α,p)]
−2
(a−b)(b−c)/radicalbiggc−u
(a−u)(b−u)
[a>b>c ≥u] BY (231.11)
10./integraldisplayc
udx/radicalbig
(a−x)3(b−x)3(c−x)=2
(a−b)2(b−c)√a−c[(a+b−2c)E(β,p)−2(b−c)F(β,p)]
+2
(a−b)(a−c)/radicalbiggc−u
(a−u)(b−u)
[a>b>c>u ] BY (232.15)
11./integraldisplayu
cdx/radicalbig
(a−x)3(b−x)3(x−c)=2
(a−b)2(b−c)√a−c[(a−b)F(γ,q)−(a+b−2c)E(γ,q)]
+2/bracketleftbig
a2+b2−ac−bc−u(a+b−2c)/bracketrightbig
(a−b)2(b−c)(a−c)/radicalbiggu−c
(a−u)(b−u)
[a>b>u>c ] BY (233.11)
12./integraldisplay∞
udx/radicalbig
(x−a)3(x−b)3(x−c)=2
(a−b)2(b−c)√a−c[(a−b)F(ν,q)−(a+b−2c)E(ν,q)]
+2u−a−b
(a−b)2/radicalbig
(u−a)(u−b)(u−c)
[u>a>b>c ] BY (238.15)
3.136
1./integraldisplayu
−∞dx/radicalbig
(a−x)3(b−x)3(c−x)3
=2
(a−b)2(b−c)2/radicalbig
(a−c)3
×/bracketleftbig
(b−c)(a+b−2c)F(α,p)−2/parenleftbig
c2+a2+b2−ab−ac−bc/parenrightbig
E(α,p)/bracketrightbig
+2[c(a−c)+b(a−b)−u(2a−c−b)]
(a−b)(a−c)(b−c)2/radicalbig
(a−u)(b−u)(c−u)
[a>b>c>u ] BY (231.14)
262 Power and Algebraic Functions 3.137
2./integraldisplay∞
udx/radicalbig
(x−a)3(x−b)3(x−c)3
=2
(a−b)2(b−c)2/radicalbig
(a−c)3
×/bracketleftbig
(a−b)(2a−b−c)F(ν,q)−2/parenleftbig
a2+b2+c2−ab−ac−bc/parenrightbig
E(ν,q)/bracketrightbig
+2[u(a+b−2c)−a(a−c)−b(b−c)]
(a−b)2(a−c)(b−c)/radicalbig
(u−a)(u−b)(u−c)
[u>a>b>c ] BY (238.16)
3.137
1.6/integraldisplayu
−∞dx
(r−x)/radicalbig
(a−x)(b−x)(c−x)=2
(a−r)√a−c/bracketleftbigg
Π/parenleftbigg
α,a−r
a−c,p/parenrightbigg
−F(α,p)/bracketrightbigg
[a>b>c ≥u] BY (231.15)
2./integraldisplayc
udx
(r−x)/radicalbig
(a−x)(b−x)(c−x)=2(c−b)
(r−b)(r−c)√a−c
×Π/parenleftbigg
β,r−b
r−c,p/parenrightbigg
+2
(r−b)√a−cF(β,p)
[a>b>c>u , r /negationslash=0 ] BY (232.17)
3./integraldisplayu
cdx
(r−x)/radicalbig
(a−x)(b−x)(x−c)=2
(r−c)√a−cΠ/parenleftbigg
γ,b−c
r−c,q/parenrightbigg
[a>b≥u>c , r /negationslash=c] BY (233.02)
4./integraldisplayb
udx
(r−x)/radicalbig
(a−x)(b−x)(x−c)=2
(r−a)(r−b)√a−c
×/bracketleftbigg
(b−a)Π/parenleftbigg
δ, q2r−a
r−b,q/parenrightbigg
+(r−b)F(δ, q)/bracketrightbigg
[a>b>u ≥c, r/negationslash=b] BY (234.18)
5./integraldisplayu
bdx
(x−r)/radicalbig
(a−x)(x−b)(x−c)=2
(c−r)(b−r)√a−c
×/bracketleftbigg
(c−b)Π/parenleftbigg
κ, p2c−r
b−r,p/parenrightbigg
+(b−r)F(κ, p)/bracketrightbigg
[a≥u>b>c , r /negationslash=b] BY (235.17)
6.8/integraldisplaya
udx
(x−r)/radicalbig
(a−x)(x−b)(x−c)=2
(a−r)√a−cΠ/parenleftbigg
λ,a−b
a−r,p/parenrightbigg
[a>u ≥b>c , r /negationslash=a] BY (236.02)
7./integraldisplayu
adx
(x−r)/radicalbig
(x−a)(x−b)(x−c)=2
(b−r)(a−r)√a−c
×/bracketleftbigg
(b−a)Π/parenleftbigg
μ,b−r
a−b,q/parenrightbigg
+(a−p)F(μ, q)/bracketrightbigg
[u>a>b>c , r /negationslash=a] BY (237.17)
3.139 Square roots of polynomials 263
8./integraldisplay∞
udx
(x−r)/radicalbig
(x−a)(x−b)(x−c)=2
(r−c)√a−c/bracketleftbigg
Π/parenleftbigg
ν,r−c
a−c,q/parenrightbigg
−F(ν,q)/bracketrightbigg
[u≥a>b>c ] BY (238.06)
3.138
1./integraldisplayu
0dx/radicalbig
x(1−x)(1−k2x)=2F/parenleftbig
arcsin√u,k/parenrightbig
[0<u< 1] PE (532), JA
2./integraldisplay1
udx/radicalBig
x(1−x)/parenleftbig
k/prime2+k2x/parenrightbig=2F/parenleftbig
arccos√u,k/parenrightbig
[0<u< 1] PE(533)
3./integraldisplay1
udx/radicalBig
x(1−x)/parenleftbig
x−k/prime2/parenrightbig=2F/parenleftbigg
arcsin√1−u
k,k/parenrightbigg
[0<u< 1] PE (534)
4./integraldisplayu
0dx/radicalBig
x(1 +x)/parenleftbig
1+k/prime2x/parenrightbig=2F/parenleftbig
arctan√u,k/parenrightbig
[0<u< 1] PE (535)
5./integraldisplayu
0dx/radicalBig
x/bracketleftbig
1+x2+2/parenleftbig
k/prime2−k2/parenrightbig
x/bracketrightbig=F/parenleftbig
2arc t an√u,k/parenrightbig
[0<u< 1] JA
6./integraldisplay1
udx/radicalBig
x/bracketleftbig
k/prime2(1 +x2)+2( 1+ k2)x/bracketrightbig=F/parenleftBigπ
2−2arc t an√u,k/parenrightBig
[0<u< 1] JA
7./integraldisplayu
adx/radicalbig
(x−α)[(x−m)2+n2]=1√pF/parenleftbigg
2arc t an/radicalbiggu−α
p,/radicalbiggp+m−α
2p/parenrightbigg
[α<u ],
8./integraldisplaya
udx/radicalbig
(α−x)[(x−m)2+n2]=1√pF/parenleftbigg
2 arccot/radicalbiggα−u
p,/radicalbiggp−m+α
2p/parenrightbigg
[u<α ],
where p=/radicalbig
(m−α)2+n2.
3.139 Notation α= arccos1−√
3−u
1+√
3−u,β = arccos√
3−1+u√
3+1−u,
γ= arccos√
3+1−u√
3−1+u,δ = arccosu−1−√
3
u−1+√
3.
1./integraldisplayu
−∞dx√
1−x3=1
4√
3F(α,sin 75◦) H 66 (285)
2./integraldisplay1
udx√
1−x3=1
4√
3F(β,sin 75◦) H 65 (284)
264 Power and Algebraic Functions 3.139
3./integraldisplayu
1dx√
x3−1=1
4√
3F(γ,sin15◦) H 65 (283)
4./integraldisplay∞
udx√
x3−1=1
4√
3F(δ,sin 15◦) H 65 (282)
5./integraldisplay1
0dx√
1−x3=1
2π√
33√
2/braceleftbigg
Γ/parenleftbigg1
3/parenrightbigg/bracerightbigg3
MO 9
6./integraldisplay1
0xdx√
1−x3=1
π√
3
3√
4/braceleftbigg
Γ/parenleftbigg2
3/parenrightbigg/bracerightbigg3
MO 9
7./integraldisplay1
u/radicalbig
1−x3dx=1
5/braceleftBig
4√
27F(β,sin 75◦)−2u/radicalbig
1−u3/bracerightBig
BY (244.01)
8./integraldisplay1
uxdx√
1−x3=/parenleftBig
3−1
4−31
4/parenrightBig
F(β,sin 75◦)+24√
3E(β,sin 75◦)−2√
1−u3
√
3+1−uBY (244.05)
9./integraldisplay1
uxmdx√
1−x3=2um−2√
1−u3
2m−1+2(m−2)
2m−1/integraldisplay1
uxm−3dx√
1−x3BY (244.07)
10./integraldisplayu
1xdx√
x3−1=/parenleftBig
3−1
4+31
4/parenrightBig
F(γ,sin15◦)−24√
3E(γ,sin15◦)+2√
u3−1√
3−1+uBY (240.05)
11./integraldisplayu
−∞dx
(1−x)√
1−x3=1
4√
27[F(α,sin 75◦)−2E(α,sin 75◦)] +2√
3√
1+u+u2
/parenleftbig
1+√
3−u/parenrightbig√1−u
[u/negationslash=1 ] BY (246.06)
12./integraldisplay∞
udx
(x−1)√
x3−1=1
4√
27[F(δ,sin 15◦)−2E(δ,sin 15◦)] +2√
3√
1+u+u2
/parenleftbig
u−1+√
3/parenrightbig√u−1
[u/negationslash=1 ] BY (242.03)
13./integraldisplayu
−∞(1−x)dx
/parenleftbig
1+√
3−x/parenrightbig2√
1−x3=2−√
3
4√
27[F(α,sin75◦)−E(α,sin75◦)] BY (246.07)
14./integraldisplay1
u(1−x)dx
/parenleftbig
1+√
3−x/parenrightbig2√
1−x3=2−√
3
4√
27[F(β,sin 75◦)−E(β,sin 75◦)] BY (244.04)
15./integraldisplayu
1(x−1)dx
/parenleftbig
1+√
3−x/parenrightbig2√
x3−1=2/parenleftbig√
3−2/parenrightbig
√
3√
u3−1
u2−2u−2−2−√
3
4√
27E(γ,sin 15◦) BY (240.08)
16./integraldisplay∞
u(x−1)dx
/parenleftbig
1+√
3−x/parenrightbig2√
x3−1=2/parenleftbig
2−√
3/parenrightbig
√
3√
u3−1
u2−2u−2−2−√
3
4√
27E(δ,sin 15◦) BY (242.07)
17./integraldisplayu
−∞(1−x)dx
/parenleftbig
1−√
3−x/parenrightbig2√
1−x3=2+√
3
4√
27/bracketleftBigg
24√
3√
1−u3
u2−2u−2−E(α,sin 75◦)/bracketrightBigg
BY (246.08)
18./integraldisplayu
1(x−1)dx
/parenleftbig
1−√
3−x/parenrightbig2√
x3−1=2+√
3
4√
27[F(γ,sin15◦)−E(γ,sin15◦)] BY (240.04)
3.141 Square roots of polynomials 265
19./integraldisplay∞
u(x−1)dx
/parenleftbig
1−√
3−x/parenrightbig2√
x3−1=2+√
3
4√
27[F(δ,sin 15◦)−E(δ,sin 15◦)] BY (242.05)
20./integraldisplayu
−∞/parenleftbig
x2+x+1/parenrightbig
dx
/parenleftbig
1+√
3−x/parenrightbig2√
1−x3=1
4√
3E(α,sin 75◦) BY (246.01)
21./integraldisplay1
u/parenleftbig
x2+x+1/parenrightbig
dx
/parenleftbig
x−1+√
3/parenrightbig2√
1−x3=1
4√
3E(β,sin 75◦) BY (244.02)
22./integraldisplayu
1/parenleftbig
x2+x+1/parenrightbig
dx
/parenleftbig√
3+x−1/parenrightbig2√
x3−1=1
4√
3E(γ,sin15◦) BY (240.01)
23./integraldisplay∞
u/parenleftbig
x2+x+1/parenrightbig
dx
/parenleftbig
x−1+√
3/parenrightbig2√
x3−1=1
4√
3E(δ,sin 15◦) BY (242.01)
24./integraldisplayu
1(x−1)dx
(x2+x+1 )√
x3−1=4
4√
27E(γ,sin 15◦)−2+√
3
4√
27F(γ,sin15◦)
−2−√
3√
32(u−1)/parenleftbig√
3+1−u/parenrightbig
/parenleftbig√
3−1+u/parenrightbig√
u3−1
BY (240.09)
25./integraldisplayu
−∞/parenleftbig
1+√
3−x/parenrightbig2dx/bracketleftBig/parenleftbig
1+√
3−x/parenrightbig2−4√
3p2(1−x)/bracketrightBig√
1−x3=1
4√
3Π/parenleftbig
α,p2,sin 75◦/parenrightbig
BY (246.02)
26./integraldisplay1
u/parenleftbig
1+√
3−x/parenrightbig2dx/bracketleftBig/parenleftbig
1+√
3−x/parenrightbig2−4√
3p2(1−x)/bracketrightBig√
1−x3=1
4√
3Π/parenleftbig
β,p2,sin 75◦/parenrightbig
BY (244.03)
27./integraldisplayu
1/parenleftbig
1−√
3−x/parenrightbig2dx/bracketleftBig/parenleftbig
1−√
3−x/parenrightbig2−4√
3p2(x−1)/bracketrightBig√
x3−1=1
4√
3Π/parenleftbig
γ,p2,sin 15◦/parenrightbig
BY (240.02)
28./integraldisplay∞
u/parenleftbig
1−√
3−x/parenrightbig2dx/bracketleftBig/parenleftbig
1−√
3−x/parenrightbig2−4√
3p2(x−1)/bracketrightBig√
x3−1=1
4√
3Π/parenleftbig
δ, p2,sin15◦/parenrightbig
BY (242.02)
3.141 Notation :I n3.141 and3.142 we set:
α=a r c s i n/radicalbigg
a−c
a−u,β =a r c s i n/radicalbigg
c−u
b−u,γ =a r c s i n/radicalbigg
u−c
b−c
δ=a r c s i n/radicalBigg
(a−c)(b−u)
(b−c)(a−u),κ =a r c s i n/radicalBigg
(a−c)(u−b)
(a−b)(u−c),λ =a r c s i n/radicalbigg
a−u
a−b,
μ=a r c s i n/radicalbigg
u−a
u−b,ν =a r c s i n/radicalbigg
a−c
u−c,p =/radicalbigg
a−b
a−c,q =/radicalbigg
b−c
a−c.
266 Power and Algebraic Functions 3.141
1./integraldisplayc
u/radicalbigga−x
(b−x)(c−x)dx=2√
a−c[F(β,p)−E(β,p)] + 2/radicalbigg
(a−u)(c−u)
b−u
[a>b>c>u ] BY (232.06)
2./integraldisplayu
c/radicalbigga−x
(b−x)(x−c)dx=2√
a−cE(γ,q)[ a>b≥u>c] BY (233.01)
3./integraldisplayb
u/radicalbigga−x
(b−x)(x−c)dx=2√
a−cE(δ, q)−2/radicalbigg
(b−u)(u−c)
a−u
[a>b>u ≥c] BY (234.06)
4./integraldisplayu
b/radicalbigga−x
(x−b)(x−c)dx=2√
a−c[F(κ, p)−E(κ, p)] + 2/radicalbigg
(a−u)(u−b)
u−c
[a≥u>b>c ] BY (235.07)
5./integraldisplaya
u/radicalbigga−x
(x−b)(x−c)dx=2√
a−c[F(λ,p)−E(λ,p)]
[a>u ≥b>c] BY (236.04)
6./integraldisplayu
a/radicalbiggx−a
(x−b)(x−c)dx=−2√
a−cE(μ, q)+2/radicalbigg
(u−a)(u−c)
u−b
[u>a>b>c ] BY (237.03)
7./integraldisplayc
u/radicalBigg
b−x
(a−x)(c−x)dx=2(b−c)√a−cF(β,p)−2√
a−cE(β,p)+2/radicalbigg
(a−u)(c−u)
b−u
[a>b>c>u ] BY (232.07)
8./integraldisplayu
c/radicalBigg
b−x
(a−x)(x−c)dx=2√
a−cE(γ,q)−2(a−b)√a−cF(γ,q)
[a>b≥u>c] BY (233.04)
9./integraldisplayb
u/radicalBigg
b−x
(a−x)(x−c)dx=2√
a−cE(δ, q)−2(a−b)√a−cF(δ, q)−2/radicalbigg
(b−u)(u−c)
a−u
[a>b>u ≥c] BY (234.07)
10./integraldisplayu
b/radicalBigg
x−b
(a−x)(x−c)dx=2√
a−cE(κ, p)−2(b−c)√a−cF(κ, p)−2/radicalbigg
(a−u)(u−b)
u−c
[a≥u>b>c ] BY (235.06)
11./integraldisplaya
u/radicalBigg
x−b
(a−x)(x−c)dx=2√
a−cE(λ,p)−2(b−c)√a−cF(λ,p)
[a>u ≥b>c] BY (236.03)
12./integraldisplayu
a/radicalBigg
x−b
(x−a)(x−c)dx=2(a−b)√a−cF(μ, q)−2√
a−cE(μ, q)+2/radicalbigg
(u−a)(u−c)
u−b
[u>a>b>c ] BY (237.04)
3.141 Square roots of polynomials 267
13./integraldisplayc
u/radicalbiggc−x
(a−x)(b−x)dx=−2√
a−cE(β,p)+2/radicalbigg
(a−u)(c−u)
b−u
[a>b>c>u ] BY (232.08)
14./integraldisplayu
c/radicalbiggx−c
(a−x)(b−x)dx=2√
a−c[F(γ,q)−E(γ,q)]
[a>b≥u>c] BY (233.03)
15./integraldisplayb
u/radicalbiggx−c
(a−x)(b−x)dx=2√
a−c[F(δ, q)−E(δ, q)] + 2/radicalbigg
(b−u)(u−c)
a−u
[a>b>u ≥c] BY (234.08)
16./integraldisplayu
b/radicalbiggx−c
(a−x)(x−b)dx=2√
a−cE(κ, p)−2/radicalbigg
(a−u)(u−b)
u−c
[a≥u>b>c ] BY (235.07)
17./integraldisplaya
u/radicalbiggx−c
(a−x)(x−b)dx=2√
a−cE(λ,p)[ a>u ≥b>c] BY (236.01)
18./integraldisplayu
a/radicalbiggx−c
(x−a)(x−b)dx=2√
a−c[F(μ, q)−E(μ, q)] + 2/radicalbigg
(u−a)(u−c)
u−b
[u>a>b>c ] BY (237.05)
19./integraldisplayc
u/radicalbigg
(b−x)(c−x)
a−xdx=2
3√
a−c[(2a−b−c)E(β,p)−(b−c)F(β,p)]
+2
3(2b−2a+c−u)/radicalbigg
(a−u)(c−u)
b−u
[a>b>c>u ] BY (232.11)
20./integraldisplayu
c/radicalbigg
(x−c)(b−x)
a−xdx=2
3√
a−c[(2a−b−c)E(γ,q)−2(a−b)F(γ,q)]
−2
3/radicalbig
(a−u)(b−u)(u−c)
[a>b≥u>c] BY (233.06)
21.11/integraldisplayb
u/radicalbigg
(x−c)(b−x)
a−xdx=2
3√
a−c[2(b−a)F(δ, q)+( 2 a−b−c)E(δ, q)]
+2
3(b+c−a−u)/radicalbigg
(b−u)(u−c)
a−u
[a>b>u ≥c] BY (234.11)
22./integraldisplayu
b/radicalbigg
(x−b)(x−c)
a−xdx=2
3√
a−c[(2a−b−c)E(κ, p)−(b−c)F(κ, p)]
+2
3(b+2c−2a−u)/radicalbigg
(a−u)(u−b)
u−c
[a≥u>b>c ] BY (235.10)
268 Power and Algebraic Functions 3.141
23.11/integraldisplaya
u/radicalbigg
(x−b)(x−c)
a−xdx=2
3√
a−c[(2a−b−c)E(λ,p)−(b−c)F(λ,p)]
+2
3/radicalbig
(a−u)(u−b)(u−c)
[a>u ≥b>c] BY (236.07)
24./integraldisplayu
a/radicalbigg
(x−b)(x−c)
x−adx=2
3√
a−c[2(a−b)F(μ, q)+(b+c−2a)E(μ, q)]
+2
3(u+2a−2b−c)/radicalbigg
(u−a)(u−b)
u−c
[u>a>b>c ] BY (237.08)
25./integraldisplayc
u/radicalbigg
(a−x)(c−x)
b−xdx=2
3√
a−c[(2b−a−c)E(β,p)−(b−c)F(β,p)]
+2
3(a+c−b−u)/radicalbigg
(a−u)(c−u)
b−u
[a>b>c>u ] BY (232.10)
26./integraldisplayu
c/radicalbigg
(a−x)(x−c)
b−xdx=2
3√
a−c[(2b−a−c)E(γ,q)+(a−b)F(γ,q)]
−2
3/radicalbig
(a−u)(b−u)(u−c)
[a>b≥u>c] BY (233.05)
27./integraldisplayb
u/radicalbigg
(a−x)(x−c)
b−xdx=2
3√
a−c[(a−b)F(δ, q)+( 2 b−a−c)E(δ, q)]
+2
3(2a+c−2b−u)/radicalbigg
(b−u)(u−c)
a−u
[a>b>u ≥c] BY (234.10)
28./integraldisplayu
b/radicalbigg
(a−x)(x−c)
x−bdx=2
3√
a−c[(b−c)F(κ, p)+(a+c−2b)E(κ, p)]
+2
3(2b−a−2c+u)/radicalbigg
(a−u)(u−b)
u−c
[a≥u>b>c ] BY (235.11)
29./integraldisplaya
u/radicalbigg
(a−x)(x−c)
x−bdx=2
3√
a−c[(a+c−2b)E(λ,p)+(b−c)F(λ,p)]
−2
3/radicalbig
(a−u)(u−b)(u−c)
[a>u ≥b>c] BY (236.06)
30.11/integraldisplayu
a/radicalbigg
(x−a)(x−c)
x−bdx=2
3√
a−c[(a+c−2b)E(μ, q)−(a−b)F(μ, q)]
+2
3(u+b−a−c)/radicalbigg
(u−a)(u−c)
u−b
[u>a>b>c ] BY (237.06)
3.142 Square roots of polynomials 269
31./integraldisplayc
u/radicalbigg
(a−x)(b−x)
c−xdx=2
3√
a−c[2(b−c)F(β,p)+( 2 c−a−b)E(β,p)]
+2
3(a+2b−2c−u)/radicalbigg
(a−u)(c−u)
b−u
[a>b>c>u ] BY (232.09)
32./integraldisplayu
c/radicalbigg
(a−x)(b−x)
x−cdx=2
3√
a−c[(a+b−2c)E(γ,q)−(a−b)F(γ,q)]
+2
3/radicalbig
(a−u)(b−u)(u−c)
[a>b≥u>c] BY (233.07)
33./integraldisplayb
u/radicalbigg
(a−x)(b−x)
x−cdx=2
3√
a−c[(a+b−2c)E(δ, q)−(a−b)F(δ, q)]
+2
3(2c−2a−b+u)/radicalbigg
(b−u)(u−c)
a−u
[a>b>u ≥c] BY (234.09)
34./integraldisplayu
b/radicalbigg
(a−x)(x−b)
x−cdx=2
3√
a−c[(a+b−2c)E(κ, p)−2(b−c)F(κ, p)]
+2
3(u+c−a−b)/radicalbigg
(a−u)(u−b)
u−c
[a≥u>b>c ] BY (235.09)
35./integraldisplaya
u/radicalbigg
(a−x)(x−b)
x−cdx=2
3√
a−c[(a+b−2c)E(λ,p)−2(b−c)F(λ,p)]
−2
3/radicalbig
(a−u)(u−b)(u−c)
[a>u ≥b>c] BY (236.05)
36./integraldisplayu
a/radicalbigg
(x−a)(x−b)
x−cdx=2
3√
a−c[(a+b−2c)E(μ, q)−(a−b)F(μ, q)]
+2
3(u+2c−a−2b)/radicalbigg
(u−a)(u−c)
u−b
[u>a>b>c ] BY (237.07)
3.142
1./integraldisplayu
−∞/radicalbigga−x
(b−x)(c−x)3dx=2√a−cF(α,p)−2√a−c
b−cE(α,p)+2(a−c)
b−c/radicalBigg
b−u
(a−u)(c−u)
[a>b>c>u ] BY (231.05)
2./integraldisplayb
u/radicalbigga−x
(b−x)(x−c)3dx=2a−b
(b−c)√a−cF(δ, q)−2√a−c
b−cE(δ, q)
+2a−c
b−c/radicalBigg
b−u
(a−u)(u−c)
[a>b>u>c ] BY (234.13)
3./integraldisplayu
b/radicalbigga−x
(x−b)(x−c)3dx=2√a−c
b−cE(κ, p)−2√a−cF(κ, p)
[a≥u>b>c ] BY (235.12)
270 Power and Algebraic Functions 3.142
4./integraldisplaya
u/radicalbigga−x
(x−b)(x−c)3dx=2√a−c
b−cE(λ,p)−2√a−cF(λ,p)−2
b−c/radicalbigg
(a−u)(u−b)
u−c
[a>u ≥b>c] BY (236.12)
5./integraldisplayu
a/radicalbiggx−a
(x−b)(x−c)3dx=2√a−c
b−cE(μ, q)−2(a−b)
(b−c)√a−cF(μ, q)−2/radicalbiggu−a
(u−b)(u−c)
[u>a>b>c ] BY (237.10)
6./integraldisplay∞
u/radicalbiggx−a
(x−b)(x−c)3dx=2√a−c
b−cE(ν,q)−2(a−b)
(b−c)√a−cF(ν,q)
[u≥a>b>c ] BY (238.09)
7./integraldisplayu
−∞/radicalbigga−x
(b−x)3(c−x)dx=2√a−c
b−cE(α,p)−2a−b
b−c/radicalbiggc−u
(a−u)(b−u)
[a>b>c ≥u] BY (231.03)
8./integraldisplayc
u/radicalbigga−x
(b−x)3(c−x)dx=2√a−c
b−cE(β,p)[ a>b>c>u ] BY (232.01)
9./integraldisplayu
c/radicalbigga−x
(b−x)3(x−c)dx=2√a−c
b−c[F(γ,q)−E(γ,q)] +2
b−c/radicalbigg
(a−u)(u−c)
b−u
[a>b>u>c ] BY (233.15)
10./integraldisplaya
u/radicalbigga−x
(x−b)3(x−c)dx=2√a−c
c−bE(λ,p)+2
b−c/radicalbigg
(a−u)(u−c)
u−b
[a>u>b>c ] BY (236.11)
11./integraldisplayu
a/radicalbiggx−a
(x−b)3(x−c)dx=2√a−c
b−c[F(μ, q)−E(μ, q)]
[u>a>b>c ] BY (237.09)
12./integraldisplay∞
u/radicalbiggx−a
(x−b)3(x−c)dx=2√a−c
b−c[F(ν,q)−E(ν,q)] + 2/radicalbiggu−a
(u−b)(u−c)
[u≥a>b>c ] BY (238.10)
13./integraldisplayu
−∞/radicalBigg
b−x
(a−x)3(c−x)dx=2√a−cE(α,p)[ a>b>c ≥u] BY (231.01)
14./integraldisplayc
u/radicalBigg
b−x
(a−x)3(c−x)dx=2√a−cE(β,p)−2(a−b)
a−c/radicalbiggc−u
(a−u)(b−u)
[a>b>c>u ] BY (232.05)
15./integraldisplayu
c/radicalBigg
b−x
(a−x)3(x−c)dx=2√a−c[F(γ,q)−E(γ,q)] +2
a−c/radicalbigg
(b−u)(u−c)
a−u
[a>b≥u>c] BY (233.13)
16./integraldisplayb
u/radicalBigg
b−x
(a−x)3(x−c)dx=2√a−c[F(δ, q)−E(δ, q)] [a>b>u ≥c] BY (234.15)
3.142 Square roots of polynomials 271
17./integraldisplayu
b/radicalBigg
x−b
(a−x)3(x−c)dx=−2√a−cE(κ, p)+2/radicalBigg
u−b
(a−u)(u−c)
[a>u>b>c ] BY (235.08)
18./integraldisplay∞
u/radicalBigg
x−b
(x−a)3(x−c)dx=2√a−c[F(ν,q)−E(ν,q)] + 2/radicalBigg
u−b
(u−a)(u−c)
[u>a>b>c ] BY (238.07)
19./integraldisplayu
−∞/radicalBigg
b−x
(a−x)(c−x)3dx=2√a−c[F(α,p)−E(α,p)] + 2/radicalBigg
b−u
(a−u)(c−u)
[a>b>c>u ] BY (231.04)
20./integraldisplayb
u/radicalBigg
b−x
(a−x)(x−c)3dx=−2√a−cE(δ, q)+2/radicalBigg
b−u
(a−u)(u−c)
[a>b>u>c ] BY (234.14)
21./integraldisplayu
b/radicalBigg
x−b
(a−x)(x−c)3dx=2√a−c[F(κ, p)−E(κ, p)]
[a≥u>b>c ] BY (235.03)
22./integraldisplaya
u/radicalBigg
x−b
(a−x)(x−c)3dx=2√a−c[F(λ,p)−E(λ,p)] +2
a−c/radicalbigg
(a−u)(u−b)
u−c
[a>u ≥b>c] BY (236.14)
23./integraldisplayu
a/radicalBigg
x−b
(x−a)(x−c)3dx=2√a−cE(μ, q)−2b−c
a−c/radicalbiggu−a
(u−b)(u−c)
[u>a>b>c ] BY (237.11)
24./integraldisplay∞
u/radicalBigg
x−b
(x−a)(x−c)3dx=2√a−cE(ν,q)[ u≥a>b>c ] BY (238.01)
25./integraldisplayu
−∞/radicalbiggc−x
(a−x)3(b−x)dx=2√a−c
a−bE(α,p)−2(b−c)
(a−b)√a−cF(α,p)
[a>b>c ≥u] BY (231.07)
26./integraldisplayc
u/radicalbiggc−x
(a−x)3(b−x)dx=2√a−c
a−bE(β,p)−2(b−c)
(a−b)√a−cF(β,p)−2/radicalbiggc−u
(a−u)(b−u)
[a>b>c>u ] BY (232.03)
27./integraldisplayu
c/radicalbiggx−c
(a−x)3(b−x)dx=2√a−c
a−bE(γ,q)−2√a−cF(γ,q)−2
a−b/radicalbigg
(b−u)(u−c)
a−u
[a>b≥u>c] BY (233.14)
28./integraldisplayb
u/radicalbiggx−c
(a−x)3(b−x)dx=2√a−c
a−bE(δ, q)−2√a−cF(δ, q)
[a>b>u ≥c] BY (234.20)
272 Power and Algebraic Functions 3.143
29./integraldisplayu
b/radicalbiggx−c
(a−x)3(x−b)dx=2(b−c)
(a−b)√a−cF(κ, p)−2√a−c
a−bE(κ, p)
+2a−c
a−b/radicalBigg
u−b
(a−u)(u−c)
[a>u>b>c ] BY (235.13)
30./integraldisplay∞
u/radicalbiggx−c
(x−a)3(x−b)dx=2√a−cF(ν,q)−2√a−c
a−bE(ν,q)+2(a−c)
a−b/radicalBigg
u−b
(u−a)(u−c)
[u>a>b>c ] BY (238.08)
31./integraldisplayu
−∞/radicalbiggc−x
(a−x)(b−x)3dx=2√a−c
a−b[F(α,p)−E(α,p)] + 2/radicalbiggc−u
(a−u)(b−u)
[a>b>c ≥u] BY (231.06)
32./integraldisplayc
u/radicalbiggc−x
(a−x)(b−x)3dx=2√a−c
a−b[F(β,p)−E(β,p)]
[a>b>c>u ] BY (232.04)
33./integraldisplayu
c/radicalbiggx−c
(a−x)(b−x)3dx=−2√a−c
a−bE(γ,q)+2
a−b/radicalbigg
(a−u)(u−c)
b−u
[a>b>u>c ] BY (233.16)
34./integraldisplaya
u/radicalbiggx−c
(a−x)(x−b)3dx=2√a−c
a−b[F(λ,p)−E(λ,p)] +2
a−b/radicalbigg
(a−u)(u−c)
u−b
[a>u>b>c ] BY (236.13)
35./integraldisplayu
a/radicalbiggx−c
(x−a)(x−b)3dx=2√a−c
a−bE(μ, q)[ u>a>b>c ] BY (237.01)
36./integraldisplay∞
u/radicalbiggx−c
(x−a)(x−b)3dx=2√a−c
a−bE(ν,q)−2b−c
a−b/radicalbiggu−a
(u−b)(u−c)
[u≥a>b>c ] BY (238.11)
3.143
1.6/integraldisplay1
udx√
1+x4=1
2F/parenleftBigg
arctan/parenleftbig
1+√
2/parenrightbig
(1−u)
(1 +u),24√
2/parenleftBig√
2−1/parenrightBig/parenrightBigg
H 66 (286)
2./integraldisplay∞
udx√
1+x4=1
2F/parenleftBigg
arccosu2−1
u2+1,√
2
2/parenrightBigg
H 66 (287)
3.144 Notation :α=a r c s i n1√
u2−u+1.
1./integraldisplay∞
udx/radicalbig
x(x−1)(x2−x+1 )=F/parenleftBigg
α,√
3
2/parenrightBigg
[u≥1] BY (261.50)
3.144 Square roots of polynomials 273
2./integraldisplay∞
udx/radicalbig
x3(x−1)3(x2−x+1 )=2(2u−1)/radicalbig
u(u−1)(u2−u+1 )−4E/parenleftBigg
α,√
3
2/parenrightBigg
[u>1] BY (261.54)
3./integraldisplay∞
u(2x−1)2dx/radicalbig
x3(x−1)3(x2−x+1 )=4/bracketleftBigg
F/parenleftBigg
α,√
3
2/parenrightBigg
−E/parenleftBigg
α,√
3
2/parenrightBigg
+2u−1
2/radicalbig
u(u−1)(u2−u−1)/bracketrightBigg
[u>1] BY (261.56)
4./integraldisplay∞
udx/radicalBig
x(x−1)(x2−x+1 )3=4
3/bracketleftBigg
F/parenleftBigg
α,√
3
2/parenrightBigg
−E/parenleftBigg
α,√
3
2/parenrightBigg/bracketrightBigg
[u≥1] BY (261.52)
5./integraldisplay∞
u(2x−1)2dx/radicalBig
x(x−1)(x2−x+1 )3=4E/parenleftBigg
α,√
3
2/parenrightBigg
[u>1] BY (261.51)
6./integraldisplay∞
u/radicalBigg
x(x−1)
(x2−x+1 )3dx=4
3E/parenleftBigg
α,√
3
2/parenrightBigg
−1
3F/parenleftBigg
α,√
3
2/parenrightBigg
[u>1] BY (261.53)
7./integraldisplay∞
udx
(2x−1)2/radicalbigg
x(x−1)
x2−x+1=1
3/bracketleftBigg
F/parenleftBigg
α,√
3
2/parenrightBigg
−E/parenleftBigg
α,√
3
2/parenrightBigg/bracketrightBigg
+1
2(2u−1)/radicalbigg
u(u−1)
u2−u+1
[u>1] BY (261.57)
8./integraldisplay∞
udx
(2x−1)2/radicalBigg
x2−x+1
x(x−1)=E/parenleftBigg
α,√
3
2/parenrightBigg
−3
2(2u−1)/radicalbigg
u(u−1)
u2−u+1
[u>1] BY (261.58)
9./integraldisplay∞
udx
(2x−1)2/radicalbig
x(x−1)(x2−x+1 )=4
3E/parenleftBigg
α,√
3
2/parenrightBigg
−1
3F/parenleftBigg
α,√
3
2/parenrightBigg
−2
2u−1/radicalbigg
u(u−1)
u2−u+1
[u>1] BY (261.55)
10./integraldisplay∞
udx/radicalbig
x5(x−1)5(x2−x+1 )=40
3E/parenleftBigg
α,√
3
2/parenrightBigg
−4
3F/parenleftBigg
α,√
3
2/parenrightBigg
−2(2u−1)/parenleftbig
9u2−9u−1/parenrightbig
3/radicalbig
u3(u−1)3(u2−u+1 )
[u>1] BY (261.54)
11./integraldisplay∞
udx/radicalBig
x(x−1)(x2−x+1 )5=44
27F/parenleftBigg
α,√
3
2/parenrightBigg
−56
27E/parenleftBigg
α,√
3
2/parenrightBigg
+2(2u−1)/radicalbig
u(u−1)
9/radicalBig
(u2−u+1 )3
[u>1] BY (261.52)
274 Power and Algebraic Functions 3.145
12./integraldisplay∞
udx
(2x−1)4/radicalbig
x(x−1)(x2−x+1 )=16
27E/parenleftBigg
α,√
3
2/parenrightBigg
−1
27F/parenleftBigg
α,√
3
2/parenrightBigg
−8/parenleftbig
5u2−5u+2/parenrightbig
9(2u−1)3/radicalbigg
u(u−1)
u2−u+1
[u>1] BY (261.55)
3.145
1./integraldisplayu
αdx/radicalbig
(x−α)(x−β)[(x−m)2+n2]=1√pqF/parenleftBigg
2arc t an/radicalBigg
q(u−α)
p(u−β),1
2/radicalBigg
(p+q)2+(α−β)2
pq/parenrightBigg
[β<α<u ]
2./integraldisplayu
βdx/radicalbig
(α−x)(x−β)[(x−m)2+n2]
=1√pqF/parenleftBigg
2 arccot/radicalBigg
q(α−u)
p(u−β),1
2/radicalBigg
−(p−q)2+(α−β)2
pq/parenrightBigg
[β<u<α ]
3./integraldisplayβ
udx/radicalbig
(x−α)(x−β)[(x−m)2+n2]=1√pqF/parenleftBigg
2arc t an/radicalBigg
q(β−u)
p(α−u),1
2/radicalBigg
(p+q)2+(α−β)2
pq/parenrightBigg
[u<β<α ]
where ( m−α)2+n2=p2,a n d( m−β)2+n2=q2.∗
4. Set
(m1−m)2+(n1+n)2=p2,(m1−m)2+(n1−n)2=p2
1,
cotα=/radicalBigg
(p+p1)2−4n2
4n2−(p−p1)2;
then/integraldisplayu
m−ntanαdx/radicalbigg
[(x−m)2+n2]/bracketleftBig
(xm1)2+n2
1/bracketrightBig=2
p+p1F/parenleftbigg
α+a r c t a nu−m
n,2√pp1
p+p1/parenrightbigg
[m−ntanα<u<m +ncotα]
3.146
1./integraldisplay1
01
1+x4dx√
1−x4=π
8+1
4√
2K/parenleftBigg√
2
2/parenrightBigg
BI (13)(6)
2./integraldisplay1
0x2
1+x4dx√
1−x4=π
8BI (13)(7)
∗Formulas 3.145 are not valid for α+β=2m. In this case, we make the substitution x−m=z, which leads to one of
the formulas in 3.152 .
3.147 Square roots of polynomials 275
3./integraldisplay1
0x4
1+x4dx√
1−x4=−π
8+1
4√
2K/parenleftBigg√
2
2/parenrightBigg
BI (13)(8)
3.147 Notation :I n3.147 –3.151 we set: α=a r c s i n/radicalBigg
(a−c)(d−u)
(a−d)(c−u),
β=a r c s i n/radicalBigg
(a−c)(u−d)
(c−d)(a−u),γ =a r c s i n/radicalBigg
(b−d)(c−u)
(c−d)(b−u),
δ=a r c s i n/radicalBigg
(b−d)(u−c)
(b−c)(u−d),κ =a r c s i n/radicalBigg
(a−c)(b−u)
(b−c)(a−u),
λ=a r c s i n/radicalBigg
(a−c)(u−b)
(a−b)(u−c),μ =a r c s i n/radicalBigg
(b−d)(a−u)
(a−b)(u−d),
ν=a r c s i n/radicalBigg
(b−d)(u−a)
(a−d)(u−b),q =/radicalBigg
(b−c)(a−d)
(a−c)(b−d),r =/radicalBigg
(a−b)(c−d)
(a−c)(b−d).
1./integraldisplayd
udx/radicalbig
(a−x)(b−x)(c−x)(d−x)=2/radicalbig
(a−c)(b−d)F(α,q)
[a>b>c>d>u ] BY (251.00)
2./integraldisplayu
ddx/radicalbig
(a−x)(b−x)(c−x)(x−d)=2/radicalbig
(a−c)(b−d)F(β,r)
[a>b>c ≥u>d] BY (254.00)
3./integraldisplayc
udx/radicalbig
(a−x)(b−x)(c−x)(x−d)=2/radicalbig
(a−c)(b−d)F(γ,r)
[a>b>c>u ≥d] BY (253.00)
4./integraldisplayu
cdx/radicalbig
(a−x)(b−x)(x−c)(x−d)=2/radicalbig
(a−c)(b−d)F(δ, q)
[a>b≥u>c>d ] BY (254.00)
5./integraldisplayb
udx/radicalbig
(a−x)(b−x)(x−c)(x−d)=2/radicalbig
(a−c)(b−d)F(κ, q)
[a>b>u ≥c>d] BY (255.00)
6./integraldisplayu
bdx/radicalbig
(a−x)(x−b)(x−c)(x−d)=2/radicalbig
(a−c)(b−d)F(λ,r)
[a≥u>b>c>d ] BY (256.00)
7.11/integraldisplaya
udx/radicalbig
(a−x)(x−b)(x−c)(x−d)=2/radicalbig
(a−c)(b−d)F(μ, r)
[a>u ≥b>c>d ] BY (257.00)
276 Power and Algebraic Functions 3.148
8./integraldisplayu
adx/radicalbig
(x−a)(x−b)(x−c)(x−d)=2/radicalbig
(a−c)(b−d)F(ν,q)
[u>a>b>c>d ] BY (258.00)
3.148
1.8/integraldisplayd
uxdx/radicalbig
(a−x)(b−x)(c−x)(d−x)=2/radicalbig
(a−c)(b−d)/braceleftbigg
(d−c)Π/parenleftbigg
α,a−d
a−c,q/parenrightbigg
+cF(α,q)/bracerightbigg
[a>b>c>d>u ] BY (251.03)
2./integraldisplayu
dxdx/radicalbig
(a−x)(b−x)(c−x)(x−d)=2/radicalbig
(a−c)(b−d)/braceleftbigg
(d−a)Π/parenleftbigg
β,d−c
a−c,r/parenrightbigg
+aF(β,r)/bracerightbigg
[a>b>c ≥u>d] BY (252.11)
3./integraldisplayc
uxdx/radicalbig
(a−x)(b−x)(c−x)(x−d)=2/radicalbig
(a−c)(b−d)/braceleftbigg
(c−b)Π/parenleftbigg
γ,c−d
b−d,r/parenrightbigg
+bF(γ,r)/bracerightbigg
[a>b>c>u ≥d] BY (253.11)
4./integraldisplayu
cxdx/radicalbig
(a−x)(b−x)(x−c)(x−d)=2/radicalbig
(a−c)(b−d)/braceleftbigg
(c−d)Π/parenleftbigg
δ,b−c
b−d,q/parenrightbigg
+dF(δ, q)/bracerightbigg
[a>b≥u>c>d ] BY (254.10)
5./integraldisplayb
uxdx/radicalbig
(a−x)(b−x)(x−c)(x−d)=2/radicalbig
(a−c)(b−d)/braceleftbigg
(b−a)Π/parenleftbigg
κ,b−c
a−c,q/parenrightbigg
+aF(κ, q)/bracerightbigg
[a>b>u ≥c>d] BY (255.17)
6.8/integraldisplayu
bxdx/radicalbig
(a−x)(x−b)(x−c)(x−d)=2/radicalbig
(a−c)(b−d)/braceleftbigg
(b−c)Π/parenleftbigg
λ,a−b
a−c,r/parenrightbigg
+cF(λ,r)/bracerightbigg
[a≥u>b>c>d ] BY (256.11)
7./integraldisplaya
uxdx/radicalbig
(a−x)(x−b)(x−c)(x−d)=2/radicalbig
(a−c)(b−d)/braceleftbigg
(a−d)Π/parenleftbigg
μ,b−a
b−d,r/parenrightbigg
+dF(μ, r)/bracerightbigg
[a>u ≥b>c>d ] BY (257.11)
8./integraldisplayu
axdx/radicalbig
(x−a)(x−b)(x−c)(x−d)=2/radicalbig
(a−c)(b−d)/braceleftbigg
(a−b)Π/parenleftbigg
ν,a−d
b−d,q/parenrightbigg
+bF(ν,q)/bracerightbigg
[u>a>b>c>d ] BY (258.11)
3.149
1./integraldisplayd
udx
x/radicalbig
(a−x)(b−x)(c−x)(d−x)
=2
cd/radicalbig
(a−c)(b−d)/braceleftbigg
(c−d)Π/parenleftbigg
α,c(a−d)
d(a−c),q/parenrightbigg
+dF(α,q)/bracerightbigg
[a>b>c>d>u ] BY (251.04)
3.149 Square roots of polynomials 277
2./integraldisplayu
ddx
x/radicalbig
(a−x)(b−x)(c−x)(x−d)
=2
ad/radicalbig
(a−c)(b−d)/braceleftbigg
(a−d)Π/parenleftbigg
β,a(d−c)
d(a−c),r/parenrightbigg
+dF(β,r)/bracerightbigg
[a>b>c ≥u>d] BY (252.12)
3./integraldisplayc
udx
x/radicalbig
(a−x)(b−x)(c−x)(x−d)
=2
bc/radicalbig
(a−c)(b−d)/braceleftbigg
(b−c)Π/parenleftbigg
γ,b(c−d)
c(b−d),r/parenrightbigg
+cF(γ,r)/bracerightbigg
[a>b>c>u ≥d] BY (253.12)
4./integraldisplayu
cdx
x/radicalbig
(a−x)(b−x)(x−c)(x−d)
=2
cd/radicalbig
(a−c)(b−d)/braceleftbigg
(d−c)Π/parenleftbigg
δ,d(b−c)
c(b−d),q/parenrightbigg
+cF(δ, q)/bracerightbigg
[a>b≥u>c>d ] BY (254.11)
5./integraldisplayb
udx
x/radicalbig
(a−x)(b−x)(x−c)(x−d)
=2
ab/radicalbig
(a−c)(b−d)×/braceleftbigg
(a−b)Π/parenleftbigg
κ,a(b−c)
b(a−c),q/parenrightbigg
+bF(κ, q)/bracerightbigg
[a>b>u ≥c>d] BY (255.18)
6./integraldisplayu
bdx
x/radicalbig
(a−x)(x−b)(x−c)(x−d)
=2
bc/radicalbig
(a−c)(b−d)×/braceleftbigg
(c−b)Π/parenleftbigg
λ,c(a−b)
b(a−c),r/parenrightbigg
+bF(λ,r)/bracerightbigg
[a≥u>b>c>d ] BY (256.12)
7./integraldisplaya
udx
x/radicalbig
(a−x)(x−b)(x−c)(x−d)
=2
ad/radicalbig
(a−c)(b−d)×/braceleftbigg
(d−a)Π/parenleftbigg
μ,d(b−a)
a(b−d),r/parenrightbigg
+aF(μ, r)/bracerightbigg
[a>u ≥b>c>d ] BY (257.12)
8./integraldisplayu
adx
x/radicalbig
(x−a)(x−b)(x−c)(x−d)
=2
ab/radicalbig
(a−c)(b−d)/braceleftbigg
(b−a)Π/parenleftbigg
ν,b(a−d)
a(b−d),q/parenrightbigg
+aF(ν,q)/bracerightbigg
[u>a>b>c>d ] BY (258.12)
278 Power and Algebraic Functions 3.151
3.151
1./integraldisplayd
udx
(p−x)/radicalbig
(a−x)(b−x)(c−x)(d−x)
=2
(p−c)(p−d)/radicalbig
(a−c)(b−d)
×/bracketleftbigg
(d−c)Π/parenleftbigg
α,(a−d)(p−c)
(a−c)(p−d),q/parenrightbigg
+(p−d)F(α,q)/bracketrightbigg
[a>b>c>d>u , p /negationslash=d]BY (251.39)
2./integraldisplayu
ddx
(p−x)/radicalbig
(a−x)(b−x)(c−x)(x−d)
=2
(p−a)(p−d)/radicalbig
(a−c)(b−d)
×/bracketleftbigg
(d−a)Π/parenleftbigg
β,(d−c)(p−a)
(a−c)(p−d),r/parenrightbigg
+(p−d)F(β,r)/bracketrightbigg
[a>b>c ≥u>d , p /negationslash=d]BY (252.39)
3./integraldisplayc
udx
(p−x)/radicalbig
(a−x)(b−x)(c−x)(x−d)=2
(p−b)(p−c)/radicalbig
(a−c)(b−d)
×/bracketleftbigg
(c−b)Π/parenleftbigg
γ,(c−d)(p−b)
(b−d)(p−c),r/parenrightbigg
+(p−c)F(γ,r)/bracketrightbigg
[a>b>c>u ≥d, p /negationslash=c]BY (253.39)
4./integraldisplayu
cdx
(p−x)/radicalbig
(a−x)(b−x)(x−c)(x−d)=2
(p−c)(p−d)/radicalbig
(a−c)(b−d)
×/bracketleftbigg
(c−d)Π/parenleftbigg
δ,(b−c)(p−d)
(b−d)(p−c),q/parenrightbigg
+(p−c)F(δ, q)/bracketrightbigg
[a>b≥u>c>d , p /negationslash=c]BY (254.39)
5./integraldisplayb
udx
(p−x)/radicalbig
(a−x)(b−x)(x−c)(x−d)
=2
(p−a)(p−b)/radicalbig
(a−c)(b−d)
×/bracketleftbigg
(b−a)Π/parenleftbigg
κ,(b−c)(p−a)
(a−c)(p−b),q/parenrightbigg
+(p−b)F(κ, q)/bracketrightbigg
[a>b>u ≥c>d , p /negationslash=b]BY (255.38)
6./integraldisplayu
bdx
(x−p)/radicalbig
(a−x)(x−b)(x−c)(x−d)
=2
(b−p)(p−c)/radicalbig
(a−c)(b−d)
×/bracketleftbigg
(b−c)Π/parenleftbigg
λ,(a−b)(p−c)
(a−c)(p−b),r/parenrightbigg
+(p−b)F(λ,r)/bracketrightbigg
[a≥u>b>c>d , p /negationslash=b]BY (256.39)
3.152 Square roots of polynomials 279
7./integraldisplaya
udx
(p−x)/radicalbig
(a−x)(x−b)(x−c)(x−d)
=2
(p−a)(p−d)/radicalbig
(a−c)(b−d)
×/bracketleftbigg
(a−d)Π/parenleftbigg
μ,(b−a)(p−d)
(b−d)(p−a),r/parenrightbigg
+(p−a)F(μ, r)/bracketrightbigg
[a>u ≥b>c>d , p /negationslash=a]BY (257.39)
8./integraldisplayu
adx
(p−x)/radicalbig
(x−a)(x−b)(x−c)(x−d)
=2
(p−a)(p−b)/radicalbig
(a−c)(b−d)
×/bracketleftbigg
(a−b)Π/parenleftbigg
ν,(a−d)(p−b)
(b−d)(p−a),q/parenrightbigg
+(p−a)F(ν,q)/bracketrightbigg
[u>a>b>c>d , p /negationslash=a]BY (258.39)
3.152 Notation :I n3.152 –3.163 we set: α=a r c t a nu
b,β = arccotu
a
γ=a r c s i nu
b/radicalbigg
a2+b2
a2+u2,δ = arccosu
b,ε = arccosb
u,ξ =a r c s i n/radicalbigg
a2+b2
a2+u2,
η=a r c s i nu
b,ζ =a r c s i na
b/radicalbigg
b2−u2
a2−u2,κ =a r c s i na
u/radicalbigg
u2−b2
a2−b2,
λ=a r c s i n/radicalbigg
a2−u2
a2−b2,μ =a r c s i n/radicalbigg
u2−a2
u2−b2,ν =a r c s i na
u,q =√
a2−b2
a,
r=b√
a2+b2,s =a√
a2+b2,t =b
a.
1./integraldisplayu
0dx/radicalbig
(x2+a2)(x2+b2)=1
aF(α,q)[ a>b> 0] H 62(258), BY (221.00)
2./integraldisplay∞
udx/radicalbig
(x2+a2)(x2+b2)=1
aF(β,q)[ a>b> 0] H 63 (259), BY (222.00)
3./integraldisplayu
0dx/radicalbig
(x2+a2)(b2−x2)=1√
a2+b2F(γ,r)[ b≥u>0] H 63 (260)
4./integraldisplayb
udx/radicalbig
(x2+a2)(b2−x2)=1√
a2+b2F(δ, r)[ b>u≥0] H 63 (261), BY (213.00)
5./integraldisplayu
bdx/radicalbig
(x2+a2)(x2−b2)=1√
a2+b2F(ε, s)[ u>b> 0] H 63 (262), BY (211.00)
6./integraldisplay∞
udx/radicalbig
(x2+a2)(x2−b2)=1√
a2+b2F(ξ,s)[ u>b> 0] H 63 (263), BY (212.00)
280 Power and Algebraic Functions 3.153
7./integraldisplayu
0dx/radicalbig
(a2−x2)(b2−x2)=1
aF(η,t)[ a>b≥u>0]H 63 (264), BY (219.00)
8./integraldisplayb
udx/radicalbig
(a2−x2)(b2−x2)=1
aF(ζ,t)[ a>b>u ≥0]H 63 (265), BY (220.00)
9./integraldisplayu
bdx/radicalbig
(a2−x2)(x2−b2)=1
aF(κ, q)[ a≥u>b> 0]H 63 (266), BY (217.00)
10./integraldisplaya
udx/radicalbig
(a2−x2)(x2−b2)=1
aF(λ,q)[ a>u ≥b>0]H 63 (257), BY (218.00)
11./integraldisplayu
adx/radicalbig
(x2−a2)(x2−b2)=1
aF(μ, t)[ u>a>b> 0]H 63 (268), BY (216.00)
12./integraldisplay∞
udx/radicalbig
(x2−a2)(x2−b2)=1
aF(ν,t)[ u≥a>b> 0] H 64(269), BY (215.00)
3.153
1./integraldisplayu
0x2dx/radicalbig
(x2+a2)(x2+b2)=u/radicalbigg
a2+u2
b2+u2−aE(α,q)[ u>0,a > b ] BY (221.09)
2./integraldisplayu
0x2dx/radicalbig
(a2+x2)(b2−x2)=/radicalbig
a2+b2E(γ,r)−a2
√
a2+b2F(γ,r)−u/radicalbigg
b2−u2
a2+u2
[b≥u>0] BY (214.05)
3./integraldisplayb
ux2dx/radicalbig
(a2+x2)(b2−x2)=/radicalbig
a2+b2E(δ, r)−a2
√
a2+b2F(δ, r)
[b>u≥0] BY (213.06)
4./integraldisplayu
bx2dx/radicalbig
(a2+x2)(x2−b2)=b2
√
a2+b2F(ε, s)−/radicalbig
a2+b2E(ε, s)+1
u/radicalbig
(u2+a2)(u2−b2)
[u>b> 0] BY (211.09)
5./integraldisplayu
0x2dx/radicalbig
(a2−x2)(b2−x2)=a{F(η,t)−E(η,t)} [a>b≥u>0] BY (219.05)
6./integraldisplayb
ux2dx/radicalbig
(a2−x2)(b2−x2)=a{F(ζ,t)−E(ζ,t)}+u/radicalbigg
b2−u2
a2−u2
[a>b>u ≥0] BY (220.06)
7./integraldisplayu
bx2dx/radicalbig
(a2−x2)(x2−b2)=aE(κ, q)−1
u/radicalbig
(a2−u2)(u2−b2)
[a≥u>b> 0] BY (217.05)
8./integraldisplaya
ux2dx/radicalbig
(a2−x2)(x2−b2)=aE(λ,q)[ a>u ≥b>0] BY (218.06)
3.154 Square roots of polynomials 281
9.6/integraldisplayu
ax2dx/radicalbig
(x2−a2)(x2−b2)=a{F(μ, t)−E(μ, t)}+u/radicalbigg
u2−a2
u2−b2
[u>a>b> 0] BY (216.06)
10./integraldisplay1
0x2dx/radicalbig
(1 +x2)(1+ k2x2)=1
k2/braceleftBigg/radicalbigg
1+k2
2−E/parenleftBigπ
4,/radicalbig
1−k2/parenrightBig/bracerightBigg
BI (14)(9)
3.154
1./integraldisplayu
0x4dx/radicalbig
(x2+a2)(x2+b2)=a
3/braceleftbig
2/parenleftbig
a2+b2/parenrightbig
E(α,q)−b2F(α,q)/bracerightbig
+u
3/parenleftbig
u2−2a2−b2/parenrightbig/radicalbigg
a2+u2
b2+u2
[a>b , u> 0] BY (221.09)
2./integraldisplayu
0x4dx/radicalbig
(a2+x2)(b2−x2)=1
3√
a2+b2/braceleftbig/parenleftbig
2a2−b2/parenrightbig
a2F(γ,r)−2/parenleftbig
a4−b4/parenrightbig
E(γ,r)/bracerightbig
−u
3/parenleftbig
2b2−a2+u2/parenrightbig/radicalbigg
b2−u2
a2+u2
[a≥u>0] BY (214.05)
3./integraldisplayb
ux4dx/radicalbig
(a2+x2)(b2−x2)=1
3√
a2+b2/braceleftbig/parenleftbig
2a2−b2/parenrightbig
a2F(δ, r)−2/parenleftbig
a4−b4/parenrightbig
E(δ, r)/bracerightbig
+u
3/radicalbig
(a2+u2)(b2−u2)
[b>u≥0] BY (213.06)
4./integraldisplayu
bx4dx/radicalbig
(a2+x2)(x2−b2)=1
3√
a2+b2/braceleftbig/parenleftbig
2b2−a2/parenrightbig
b2F(ε, s)+2/parenleftbig
a4−b4/parenrightbig
E(ε, s)/bracerightbig
+2b2−2a2+u2
3u/radicalbig
(u2+a2)(u2−b2)
[u>b> 0] BY (211.09)
5./integraldisplayu
0x4dx/radicalbig
(a2−x2)(b2−x2)=a
3/braceleftbig/parenleftbig
2a2+b2/parenrightbig
F(η,t)−2/parenleftbig
a2+b2/parenrightbig
E(η,t)/bracerightbig
+u
3/radicalbig
(a2−u2)(b2−u2)
[a>b≥u>0] BY (219.05)
6./integraldisplayb
ux4dx/radicalbig
(a2−x2)(b2−x2)=a
3/braceleftbig/parenleftbig
2a2+b2/parenrightbig
F(ζ,t)−2/parenleftbig
a2+b2/parenrightbig
E(ζ,t)/bracerightbig
+u
3/parenleftbig
u2+a2+2b2/parenrightbig/radicalbigg
b2−u2
a2−u2
[a>b>u ≥0] BY (220.06)
7./integraldisplayu
bx4dx/radicalbig
(a2−x2)(x2−b2)=a
3/braceleftbig
2/parenleftbig
a2+b2/parenrightbig
E(κ, q)−b2F(κ, q)/bracerightbig
−u2+2a2+2b2
3u/radicalbig
(a2−u2)(u2−b2)
[a≥u>b> 0] BY (217.05)
8./integraldisplaya
ux4dx/radicalbig
(a2−x2)(x2−b2)=a
3/braceleftbig
2/parenleftbig
a2+b2/parenrightbig
E(λ,q)−b2F(λ,q)/bracerightbig
+u
3/radicalbig
(a2−u2)(u2−b2)
[a>u ≥b>0] BY (218.06)
282 Power and Algebraic Functions 3.155
9./integraldisplayu
ax4dx/radicalbig
(x2−a2)(x2−b2)=a
3/braceleftbig/parenleftbig
2a2+b2/parenrightbig
F(μ, t)−2/parenleftbig
a2+b2/parenrightbig
E(μ, t)/bracerightbig
+u
3/parenleftbig
u2+2a2+b2/parenrightbig/radicalbigg
u2−a2
u2−b2
[u>a>b> 0] BY (216.06)
3.155
1./integraldisplaya
u/radicalbig
(a2−x2)(x2−b2)dx=a
3/braceleftbig/parenleftbig
a2+b2/parenrightbig
E(λ,q)−2b2F(λ,q)/bracerightbig
−u
3/radicalbig
(a2−u2)(u2−b2)
[a>u ≥b>0] BY (218.11)
2./integraldisplayu
a/radicalbig
(x2−a2)(x2−b2)dx=a
3/braceleftbig/parenleftbig
a2+b2/parenrightbig
E(μ, t)−/parenleftbig
a2−b2/parenrightbig
F(μ, t)/bracerightbig
+u
3/parenleftbig
u2−a2−2b2/parenrightbig/radicalbigg
u2−a2
u2−b2
[u>a>b> 0] BY (216.10)
3./integraldisplayu
0/radicalbig
(x2+a2)(x2+b2)dx=a
3/braceleftbig
2b2F(α,q)−/parenleftbig
a2+b2/parenrightbig
E(α,q)/bracerightbig
+u
3/parenleftbig
u2+a2+2b2/parenrightbig/radicalbigg
a2+u2
b2+u2
[a>b , u> 0] BY (221.08)
4./integraldisplayu
0/radicalbig
(a2+x2)(b2−x2)dx=1
3/radicalbig
a2+b2/braceleftbig
a2F(γ,r)−/parenleftbig
a2−b2/parenrightbig
E(γ,r)/bracerightbig
+u
3/parenleftbig
u2+2a2−b2/parenrightbig/radicalbigg
b2−u2
a2+u2
[a≥u>0] BY (214.12)
5.9/integraldisplayb
u/radicalbig
(a2+x2)(b2−x2)dx=1
3/radicalbig
a2+b2/braceleftbig
a2F(δ, r)+/parenleftbig
b2−a2/parenrightbig
E(δ, r)/bracerightbig
+u
3/radicalbig
(a2+u2)(b2−u2)
[b>u≥0] BY (213.13)
6./integraldisplayu
b/radicalbig
(a2+x2)(x2−b2)dx=1
3/radicalbig
a2+b2/braceleftbig/parenleftbig
b2−a2/parenrightbig
E(ε, s)−b2F(ε, s)/bracerightbig
+u2+a2−b2
3u/radicalbig
(a2+u2)(u2−b2)
[u>b> 0] BY (211.08)
7./integraldisplayu
0/radicalbig
(a2−x2)(b2−x2)dx=a
3/braceleftbig/parenleftbig
a2+b2/parenrightbig
E(η,t)−/parenleftbig
a2−b2/parenrightbig
F(η,t)/bracerightbig
+u
3/radicalbig
(a2−u2)(b2−u2)
[a>b≥u>0] BY (219.11)
8./integraldisplayb
u/radicalbig
(a2−x2)(b2−x2)dx=a
3/braceleftbig/parenleftbig
a2+b2/parenrightbig
E(ζ,t)−/parenleftbig
a2−b2/parenrightbig
F(ζ,t)/bracerightbig
+u
3/parenleftbig
u2−2a2−b2/parenrightbig/radicalbigg
b2−u2
a2−u2
[a>b>u ≥0] BY (220.05)
3.156 Square roots of polynomials 283
9./integraldisplayu
b/radicalbig
(a2−x2)(x2−b2)dx=a
3/braceleftbig/parenleftbig
a2+b2/parenrightbig
E(κ, q)−2b2F(κ, q)/bracerightbig
+u2−a2−b2
3u/radicalbig
(a2−u2)(u2−b2)
[a≥u>b> 0] BY (217.09)
3.156
1.6/integraldisplay∞
udx
x2/radicalbig
(x2+a2)(x2+b2)=1
ub2/radicalbigg
b2+u2
a2+u2−1
ab2E(β,q)
[a≥b, u > 0] BY (222.04)
2./integraldisplayb
udx
x2/radicalbig
(x2+a2)(b2−x2)=1
a2b2√
a2+b2/braceleftbig
a2F(δ, r)−/parenleftbig
a2+b2/parenrightbig
E(δ, r)/bracerightbig
+1
a2b2u/radicalbig
(a2+u2)(b2−u2)
[b>u> 0] BY (213.09)
3./integraldisplayu
bdx
x2/radicalbig
(x2+a2)(x2−b2)=1
a2b2√
a2+b2/braceleftbig/parenleftbig
a2+b2/parenrightbig
E(ε, s)−b2F(ε, s)/bracerightbig
[u>b> 0] BY (211.11)
4./integraldisplay∞
udx
x2/radicalbig
(x2+a2)(x2−b2)=1
a2b2√
a2+b2/braceleftbig/parenleftbig
a2+b2/parenrightbig
E(ξ,s)−b2F(ξ,s)/bracerightbig
−1
b2u/radicalbigg
u2−b2
a2+u2
[u≥b>0] BY (212.06)
5./integraldisplayb
udx
x2/radicalbig
(a2−x2)(b2−x2)=1
ab2{F(ζ,t)−E(ζ,t)}+1
b2u/radicalbigg
b2−u2
a2−u2
[a>b>u> 0] BY (220.09)
6./integraldisplayu
bdx
x2/radicalbig
(a2−x2)(x2−b2)=1
ab2E(κ, q)[ a≥u>b> 0] BY (217.01)
7./integraldisplaya
udx
x2/radicalbig
(a2−x2)(x2−b2)=1
ab2E(λ,q)−1
a2b2u/radicalbig
(a2−u2)(u2−b2)
[a>u ≥b>0] BY (218.12)
8./integraldisplayu
adx
x2/radicalbig
(x2−a2)(x2−b2)=1
ab2{F(μ, t)−E(μ, t)}+1
a2u/radicalbigg
u2−a2
u2−b2
[u>a>b> 0] BY (216.09)
9./integraldisplay∞
udx
x2/radicalbig
(x2−a2)(x2−b2)=1
ab2{F(ν,t)−E(ν,t)}
[u≥a>b> 0] BY (215.07)
284 Power and Algebraic Functions 3.157
3.157
1./integraldisplayu
0dx
(p−x2)/radicalbig
(x2+a2)(x2+b2)=1
a(p+b2)/braceleftbiggb2
pΠ/parenleftbigg
α,p+b2
p,q/parenrightbigg
+F(α,q)/bracerightbigg
[p/negationslash=0 ] BY (221.13)
2./integraldisplay∞
udx
(p−x2)/radicalbig
(x2+a2)(x2+b2)=−1
a(a2+p)/braceleftbigg
Π/parenleftbigg
β,a2+p
a2,q/parenrightbigg
−F(β,q)/bracerightbigg
BY (222.11)
3./integraldisplayu
0dx
(p−x2)/radicalbig
(a2+x2)(b2−x2)=1
p(p+a2)√
a2+b2/braceleftBigg
a2Π/parenleftBigg
γ,b2/parenleftbig
p+a2/parenrightbig
p(a2+b2),r/parenrightBigg
+pF(γ,r)/bracerightBigg
[b≥u>0,p/negationslash=0 ] BY (214.13)a
4./integraldisplayb
udx
(p−x2)/radicalbig
(a2+x2)(b2−x2)=1
(p−b2)√
a2+b2Π/parenleftbigg
δ,b2
b2−p,r/parenrightbigg
/bracketleftbig
b>u≥0,p/negationslash=b2/bracketrightbig
BY (213.02)
5./integraldisplayu
bdx
(p−x2)/radicalbig
(a2+x2)(x2−b2)=1
p(p−b2)√
a2+b2/braceleftbigg
b2Π/parenleftbigg
ε,p
p−b2,s/parenrightbigg
+/parenleftbig
p−b2/parenrightbig
F(ε, s)/bracerightbigg
/bracketleftbig
u>b> 0,p/negationslash=b2/bracketrightbig
BY (211.14)
6./integraldisplay∞
udx
(x2−p)/radicalbig
(a2+x2)(x2−b2)=1
(a2+p)√
a2+b2/braceleftbigg
Π/parenleftbigg
ξ,a2+p
a2+b2,s/parenrightbigg
−F(ξ,s)/bracerightbigg
[u≥b>0] BY (212.12)
7./integraldisplayu
0dx
(p−x2)/radicalbig
(a2−x2)(b2−x2)=1
apΠ/parenleftbigg
η,b2
p,t/parenrightbigg
[a>b≥u>0;p/negationslash=b] BY (219.02)
8./integraldisplayb
udx
(p−x2)/radicalbig
(a2−x2)(b2−x2)=1
a(p−a2)(p−b2)
×/braceleftBigg
/parenleftbig
b2−a2/parenrightbig
Π/parenleftBigg
ζ,b2/parenleftbig
p−a2/parenrightbig
a2(p−b2),t/parenrightBigg
+/parenleftbig
p−b2/parenrightbig
F(ζ,t)/bracerightBigg
/bracketleftbig
a>b>u ≥0;p/negationslash=b2/bracketrightbig
BY (220.13)
9./integraldisplayu
bdx
(p−x2)/radicalbig
(a2−x2)(x2−b2)=1
ap(p−b2)/braceleftBigg
b2Π/parenleftBigg
κ,p/parenleftbig
a2−b2/parenrightbig
a2(p−b2),q/parenrightBigg
+/parenleftbig
p−b2/parenrightbig
F(κ, q)/bracerightBigg
/bracketleftbig
a≥u>b> 0;p/negationslash=b2/bracketrightbig
BY (217.12)
10./integraldisplaya
udx
(x2−p)/radicalbig
(a2−x2)(x2−b2)=1
a(a2−p)Π/parenleftbigg
λ,a2−b2
a2−p,q/parenrightbigg
/bracketleftbig
a>u ≥b>0;p/negationslash=a2/bracketrightbig
BY (218.02)
11./integraldisplayu
adx
(p−x2)/radicalbig
(x2−a2)(x2−b2)
=1
a(p−a2)(p−b2)/braceleftbigg/parenleftbig
a2−b2/parenrightbig
Π/parenleftbigg
μ,p−b2
p−a2,t/parenrightbigg
+/parenleftbig
p−a2/parenrightbig
F(μ, t)/bracerightbigg
/bracketleftbig
u>a>b> 0;p/negationslash=a2,p/negationslash=b2/bracketrightbig
BY (216.12)
3.158 Square roots of polynomials 285
12./integraldisplay∞
udx
(x2−p)/radicalbig
(x2−a2)(x2−b2)=1
ap/braceleftBig
Π/parenleftBig
ν,p
a2,t/parenrightBig
−F(ν,t)/bracerightBig
[u≥a>b> 0;p/negationslash=0 ] BY (215.12)
3.158
1./integraldisplayu
0dx/radicalBig
(x2+a2)(x2+b2)3=1
ab2(a2−b2)/braceleftbig
a2E(α,q)−b2F(α,q)/bracerightbig
[a>b;u>0] BY (221.05)
2./integraldisplay∞
udx/radicalBig
(x2+a2)(x2+b2)3=1
ab2(a2−b2)/braceleftbig
a2E(β,q)−b2F(β,q)/bracerightbig
−u
b2/radicalbig
(a2+u2)(b2+u2)
[a>b , u ≥0] BY (222.05)
3./integraldisplayu
0dx/radicalBig
(x2+a2)3(x2+b2)=1
a(a2−b2){F(α,q)−E(α,q)}+u
a2/radicalbig
(u2+a2)(u2+b2)
[a>b;u>0] BY (221.06)
4./integraldisplay∞
udx/radicalBig
(a2+x2)3(x2+b2)=1
a(a2−b2){F(β,q)−E(β,q)}
[a>b , u ≥0] BY (222.03)
5./integraldisplayu
0dx/radicalBig
(a2+x2)3(b2−x2)=1
a2√
a2+b2E(γ,r)[ b≥u>0] BY (214.01)a
6./integraldisplayb
udx/radicalBig
(a2+x2)3(b2−x2)=1
a2√
a2+b2E(δ, r)−u
a2(a2+b2)/radicalbigg
b2−u2
a2+u2
[b>u≥0] BY (213.08)
7./integraldisplayu
bdx/radicalBig
(a2+x2)3(x2−b2)=1
a2√
a2+b2{F(ε, s)−E(ε, s)}+1
(a2+b2)u/radicalbigg
u2−b2
u2+a2
[u>b> 0] BY (211.05)
8./integraldisplay∞
udx/radicalBig
(a2+x2)3(x2−b2)=1
a2√
a2+b2{F(ξ,s)−E(ξ,s)}
[u≥b>0] BY (212.03)
9./integraldisplayu
0dx/radicalBig
(a2+x2)(b2−x2)3=1
b2√
a2+b2{F(γ,r)−E(γ,r)}+u
b2/radicalbig
(a2+u2)(b2−u2)
[b>u> 0] BY (214.10)
10./integraldisplay∞
udx/radicalBig
(a2+x2)(x2−b2)3=u
b2/radicalbig
(a2+u2)(u2−b2)−1
b2√
a2+b2E(ξ,s)
[u≥b>0] BY (212.04)
286 Power and Algebraic Functions 3.159
11./integraldisplayu
0dx/radicalBig
(a2−x2)3(b2−x2)=1
a2(a2−b2)/braceleftBigg
aE(η,t)−u/radicalbigg
b2−u2
a2−u2/bracerightBigg
[a>b≥u>0] BY (219.07)
12./integraldisplayb
udx/radicalBig
(a2−x2)3(b2−x2)=1
a(a2−b2)E(ζ,t)[ a>b>u ≥0] BY (220.10)
13./integraldisplayu
bdx/radicalBig
(a2−x2)3(x2−b2)=1
a(a2−b2)/braceleftBigg
F(κ, q)−E(κ, q)+a
u/radicalbigg
u2−b2
a2−u2/bracerightBigg
[a>u>b> 0] BY (217.10)
14./integraldisplay∞
udx/radicalBig
(x2−a2)3(x2−b2)=1
a(b2−a2)/braceleftBigg
E(ν,t)−a
u/radicalbigg
u2−b2
u2−a2/bracerightBigg
[u>a>b> 0] BY (215.04)
15./integraldisplayu
0dx/radicalBig
(a2−x2)(b2−x2)3=1
ab2F(η,t)−1
b2(a2−b2)/braceleftBigg
aE(η,t)−u/radicalbigg
a2−u2
b2−u2/bracerightBigg
[a>b>u> 0] BY (219.06)
16./integraldisplaya
udx/radicalBig
(a2−x2)(x2−b2)3=1
ab2(a2−b2)/braceleftBigg
b2F(λ,q)−a2E(λ,q)+au/radicalbigg
a2−u2
u2−b2/bracerightBigg
[a>u>b> 0] BY (218.04)
17./integraldisplayu
adx/radicalBig
(x2−a2)(x2−b2)3=a
b2(a2−b2)E(μ, t)−1
ab2F(μ, t)
[u>a>b> 0] BY (216.11)
18./integraldisplay∞
udx/radicalBig
(x2−a2)(x2−b2)3=1
b2(a2−b2)/braceleftBigg
aE(ν,t)−b2
u/radicalbigg
u2−a2
u2−b2/bracerightBigg
−1
ab2F(ν,t)
[u≥a>b> 0] BY (215.06)
3.159
1./integraldisplayu
0x2dx/radicalBig
(x2+a2)(x2+b2)3=a
a2−b2{F(α,q)−E(α,q)}
[a>b , u> 0] BY (221.12)
2./integraldisplay∞
ux2dx/radicalBig
(x2+a2)(x2+b2)3=a
a2−b2{F(β,q)−E(β,q)}+u/radicalbig
(a2+u2)(b2+u2)
[a>b , u ≥0] BY (222.10)
3.159 Square roots of polynomials 287
3./integraldisplayu
0x2dx/radicalBig
(x2+a2)3(x2+b2)=1
a(a2−b2)/braceleftbig
a2E(α,q)−b2F(α,q)/bracerightbig
−u/radicalbig
(a2+u2)(b2+u2)
[a>b , u> 0] BY (221.11)
4./integraldisplay∞
ux2dx/radicalBig
(x2+a2)3(x2+b2)=1
a(a2−b2)/braceleftbig
a2E(β,q)−b2F(β,q)/bracerightbig
[a>b , u ≥0] BY (222.07)
5./integraldisplayu
0x2dx/radicalBig
(a2+x2)3(b2−x2)=1√
a2+b2{F(γ,r)−E(γ,r)}
[b≥u>0] BY (214.04)
6./integraldisplayb
ux2dx/radicalBig
(a2+x2)3(b2−x2)=1√
a2+b2{F(δ, r)−E(δ, r)}+u
a2+b2/radicalbigg
b2−u2
a2+u2
[b>u≥0] BY (213.07)
7./integraldisplayu
bx2dx/radicalBig
(a2+x2)3(x2−b2)=1√
a2+b2E(ε, s)−a2
u(a2+b2)/radicalbigg
u2−b2
u2+a2
[u>b> 0] BY (211.13)
8./integraldisplay∞
ux2dx/radicalBig
(a2+x2)3(x2−b2)=1√
a2+b2E(ξ,s)[ u≥b>0] BY (212.01)
9./integraldisplayu
0x2dx/radicalBig
(a2+x2)(b2−x2)3=u/radicalbig
(a2+u2)(b2−u2)−1√
a2+b2E(γ,r)
[b>u> 0] BY (214.07)
10./integraldisplay∞
ux2dx/radicalBig
(a2+x2)(x2−b2)3=1√
a2+b2{F(ξ,s)−E(ξ,s)}+u/radicalbig
(a2+u2)(u2−b2)
[u>b> 0] BY (212.10)
11./integraldisplayu
0x2dx/radicalBig
(a2−x2)3(b2−x2)=1
a2−b2/braceleftBigg
aE(η,t)−u/radicalbigg
b2−u2
a2−u2/bracerightBigg
−1
aF(η,t)
[a>b≥u>0] BY (219.04)
12./integraldisplayb
ux2dx/radicalBig
(a2−x2)3(b2−x2)=a
a2−b2E(ζ,t)−1
aF(ζ,t)
[a>b>u ≥0] BY (220.08)
13./integraldisplayu
bx2dx/radicalBig
(a2−x2)3(x2−b2)=1
a(a2−b2)/braceleftBigg
b2F(κ, q)−a2E(κ, q)+a3
u/radicalbigg
u2−b2
a2−u2/bracerightBigg
[a>u>b> 0] BY (217.06)
288 Power and Algebraic Functions 3.161
14./integraldisplay∞
ux2dx/radicalBig
(x2−a2)3(x2−b2)=a
a2−b2/braceleftBigg
a
u/radicalbigg
u2−b2
u2−a2−E(ν,t)/bracerightBigg
+1
aF(ν,t)
[u>a>b> 0] BY (215.09)
15./integraldisplayu
0x2dx/radicalBig
(a2−x2)(b2−x2)3=1
a2−b2/braceleftBigg
u/radicalbigg
a2−u2
b2−u2−aE(η,t)/bracerightBigg
[a>b>u> 0] BY (219.12)
16./integraldisplaya
ux2dx/radicalBig
(a2−x2)(x2−b2)3=1
a2−b2/braceleftBigg
aF(λ,q)−aE(λ,q)+u/radicalbigg
a2−u2
u2−b2/bracerightBigg
[a>u>b> 0] BY (218.07)
17./integraldisplayu
ax2dx/radicalBig
(x2−a2)(x2−b2)3=a
a2−b2E(μ, t)[ u>a>b> 0] BY (216.01)
18./integraldisplay∞
ux2dx/radicalBig
(x2−a2)(x2−b2)3=1
a2−b2/braceleftBigg
aE(ν,t)−b2
u/radicalbigg
u2−a2
u2−b2/bracerightBigg
[u≥a>b> 0] BY (215.11)
3.161
1./integraldisplay∞
udx
x4/radicalbig
(x2+a2)(x2+b2)=1
3a3b4/braceleftbig
2/parenleftbig
a2+b2/parenrightbig
E(β,q)−b2F(β,q)/bracerightbig
+a2b2−u2/parenleftbig
2a2+b2/parenrightbig
3a2b4u3
[a>b , u> 0] BY (222.04)
2./integraldisplayb
udx
x4/radicalbig
(x2+a2)(b2−x2)=1
3a4b4√
a2+b2/braceleftbig
a2/parenleftbig
2a2−b2/parenrightbig
F(δ, r)−2/parenleftbig
a4−b4/parenrightbig
E(δ, r)/bracerightbig
+a2b2+2u2/parenleftbig
a2−b2/parenrightbig
3a4b4u3/radicalbig
(b2−u2)(a2+u2)
[b>u> 0] BY (213.09)
3./integraldisplayu
bdx
x4/radicalbig
(x2+a2)(x2−b2)=2b2−a2
3a4b2√
a2+b2F(ε, s)+2
3/parenleftbig
a2−b2/parenrightbig√
a2+b2
a4b4E(ε, s)
+1
3a2b2u3/radicalbig
(u2+a2)(u2−b2)
[u>b> 0] BY (211.11)
4./integraldisplay∞
udx
x4/radicalbig
(x2+a2)(x2−b2)=1
3a4b4√
a2+b2/braceleftbig
2/parenleftbig
a4−b4/parenrightbig
E(ξ,s)+b2/parenleftbig
2b2−a2/parenrightbig
F(ξ,s)/bracerightbig
−a2b2+u2/parenleftbig
2a2−b2/parenrightbig
3a2b4u3/radicalbigg
u2−b2
u2+a2
[u≥b>0] BY (212.06)
3.162 Square roots of polynomials 289
5./integraldisplayb
udx
x4/radicalbig
(a2−x2)(b2−x2)=1
3a3b4⎧
⎨
⎩/braceleftbig/parenleftbig
2a2+b2/parenrightbig
F(ζ,t)−2/parenleftbig
a2+b2/parenrightbig
E(ζ,t)/bracerightbig
+/bracketleftbig/parenleftbig
2a2+b2/parenrightbig
u2+a2b2/bracketrightbig
a
u3/radicalbigg
b2−u2
a2−u2⎫
⎬
⎭
[a>b>u> 0] BY (220.09)
6./integraldisplayu
bdx
x4/radicalbig
(a2−x2)(x2−b2)=1
3a3b4/braceleftbig
2/parenleftbig
a2+b2/parenrightbig
E(κ, q)−b2F(κ, q)/bracerightbig
+1
3a2b2u3/radicalbig
(a2−u2)(u2−b2)
[a≥u>b> 0] BY (217.14)
7./integraldisplaya
udx
x4/radicalbig
(a2−x2)(x2−b2)=1
3a3b4⎧
⎨
⎩2/parenleftbig
a2+b2/parenrightbig
E(λ,q)−b2F(λ,q)
−2/parenleftbig
a2+b2/parenrightbig
u2+a2b2
au3/radicalbig
(a2−u2)(u2−b2)⎫
⎬
⎭
[a>u ≥b>0] BY (218.12)
8./integraldisplayu
adx
x4/radicalbig
(x2−a2)(x2−b2)
=1
3a3b4⎧
⎨
⎩/braceleftbig/parenleftbig
2a2+b2/parenrightbig
F(μ, t)−2/parenleftbig
a2+b2/parenrightbig
E(μ, t)/bracerightbig
[u>a>b> 0]
+/bracketleftbig/parenleftbig
a2+2b2/parenrightbig
u2+a2b2/bracketrightbig
b2
au3/radicalbigg
u2−a2
u2−b2⎫
⎬
⎭
BY (216.09)
9./integraldisplay∞
udx
x4/radicalbig
(x2−a2)(x2−b2)=1
3a3b4⎧
⎨
⎩/parenleftbig
2a2+b2/parenrightbig
F(ν,t)−2/parenleftbig
a2+b2/parenrightbig
E(ν,t)
+ab2
u3/radicalbig
(u2−a2)(u2−b2)⎫
⎬
⎭
[u≥a>b> 0] BY (215.07)
3.162
1./integraldisplayu
0dx/radicalBig
(x2+a2)5(x2+b2)=1
3a3(a2−b2)2/braceleftbig/parenleftbig
3a2−b2/parenrightbig
F(α,q)−2/parenleftbig
2a2−b2/parenrightbig
E(α,q)/bracerightbig
+u/bracketleftbig
a2/parenleftbig
4a2−3b2/parenrightbig
+u2/parenleftbig
3a2−2b2/parenrightbig/bracketrightbig
3a4(a2−b2)/radicalBig
(u2+a2)3(u2+b2)
[a>b , u> 0] BY (221.06)
290 Power and Algebraic Functions 3.162
2./integraldisplay∞
udx/radicalBig
(x2+a2)5(x2+b2)=1
3a3(a2−b2)2/braceleftbig/parenleftbig
3a2−b2/parenrightbig
F(β,q)−2/parenleftbig
2a2−b2/parenrightbig
E(β,q)/bracerightbig
+u
3a2(a2−b2)/radicalBigg
u2+b2
(a2+u2)3
[a>b , u ≥0] BY (222.03)
3./integraldisplayu
0dx/radicalBig
(x2+a2)(x2+b2)5=3b2−a2
3ab2(a2−b2)2F(α,q)+a/parenleftbig
2a2−4b2/parenrightbig
3b4(a2−b2)2E(α,q)
+u
3b2(a2−b2)/radicalBigg
u2+a2
(u2+b2)3
[a>b , u> 0] BY (221.05)
4./integraldisplay∞
udx/radicalBig
(x2+a2)(x2+b2)5=1
3ab4(a2−b2)2/braceleftbig
2a2/parenleftbig
a2−2b2/parenrightbig
E(β,q)+b2/parenleftbig
3b2−a2/parenrightbig
F(β,q)/bracerightbig
−u/bracketleftbig
b2/parenleftbig
3a2−4b2/parenrightbig
+u2/parenleftbig
2a2−3b2/parenrightbig/bracketrightbig
3b4(a2−b2)/radicalBig
(u2+a2)(u2+b2)3
[a>b , u ≥0] BY (222.05)
5./integraldisplayu
0dx/radicalBig
(a2+x2)5(b2−x2)=1
3a4/radicalBig
(a2+b2)3/braceleftbig
2/parenleftbig
b2+2a2/parenrightbig
E(γ,r)−a2F(γ,r)/bracerightbig
+u
3a2(a2+b2)/radicalBigg
b2−u2
(a2+u2)3
[b≥u>0] BY (214.15)
6./integraldisplayb
udx/radicalBig
(a2+x2)5(b2−x2)=1
3a4/radicalBig
(a2+b2)3/braceleftbig/parenleftbig
4a2+2b2/parenrightbig
E(δ, r)−a2F(δ, r)/bracerightbig
−u/bracketleftbig
a2/parenleftbig
5a2+3b2/parenrightbig
+u2/parenleftbig
4a2+2b2/parenrightbig/bracketrightbig
3a4(a2+b2)2/radicalBigg
b2−u2
(a2+u2)3
[b>u> 0] BY (213.08)
7./integraldisplayu
bdx/radicalBig
(a2+x3)5(x2−b2)=1
3a4/radicalBig
(a2+b2)3/braceleftbig/parenleftbig
3a2+2b2/parenrightbig
F(ε, s)−/parenleftbig
4a2+2b2/parenrightbig
E(ε, s)/bracerightbig
+/parenleftbig
3a2+b2/parenrightbig
u2+2/parenleftbig
2a2+b2/parenrightbig
a2
3a2(a2+b2)2u/radicalBigg
u2−b2
(u2+a2)3
[u>b> 0] BY (211.05)
8./integraldisplay∞
udx/radicalBig
(a2+x2)5(x2−b2)=1
3a4/radicalBig
(a2+b2)3/braceleftbig/parenleftbig
3a2+2b2/parenrightbig
F(ξ,s)−/parenleftbig
4a2+2b2/parenrightbig
E(ξ,s)/bracerightbig
+u
3a2(a2+b2)/radicalBigg
u2−b2
(a2+u2)3
[u>b> 0] BY (212.03)
3.162 Square roots of polynomials 291
9./integraldisplayu
0dx/radicalBig
(a2+x2)(b2−x2)5=1
3b4/radicalBig
(a2+b2)3/braceleftbig/parenleftbig
2a2+3b2/parenrightbig
F(γ,r)−/parenleftbig
2a2+4b2/parenrightbig
E(γ,r)/bracerightbig
+u/bracketleftbig/parenleftbig
3a3+4b2/parenrightbig
b2−/parenleftbig
2a2+3b2/parenrightbig
u2/bracketrightbig
3b4(a2+b2)/radicalBig
(a2+u2)(b2−u2)3
[b>u> 0] BY (214.10)
10./integraldisplay∞
udx/radicalBig
(a2+x2)(x2−b2)5=1
3b4/radicalBig
(a2+b2)3/braceleftbig/parenleftbig
2a2+4b2/parenrightbig
E(ξ,s)−b2F(ξ,s)/bracerightbig
+u/bracketleftbig/parenleftbig
3a2+4b2/parenrightbig
b2−/parenleftbig
2a2+3b2/parenrightbig
u2/bracketrightbig
3b4(a2+b2)/radicalBig
(a2+u2)(u2−b2)3
[u>b> 0] BY (212.04)
11./integraldisplayu
0dx/radicalBig
(a2−x2)(b2−x2)5=2a2−3b2
3ab4(a2−b2)F(η,t)+2a/parenleftbig
2b2−a2/parenrightbig
3b4(a2−b2)2E(η,t)
+u/bracketleftbig/parenleftbig
3a2−5b2/parenrightbig
b2−2/parenleftbig
a2−2b2/parenrightbig
u2/bracketrightbig
3b4(a2−b2)2(b2−u2)/radicalbigg
a2−u2
b2−u2
[a>b>a> 0] BY (219.06)
12./integraldisplaya
udx/radicalBig
(a2−x2)(x2−b2)5=3b2−a2
3ab2(a2−b2)2F(λ,q)+2a/parenleftbig
a2−2b2/parenrightbig
3b4(a2−b2)2E(λ,q)
+u/bracketleftbig
2/parenleftbig
2b2−a2/parenrightbig
u2+/parenleftbig
3a2−5b2/parenrightbig
b2/bracketrightbig
3b4(a2−b2)2(u2−b2)/radicalbigg
a2−u2
u2−b2
[a>u>b> 0] BY (218.04)
13./integraldisplayu
adx/radicalBig
(x2−a2)(x2−b2)5=2a2−3b2
3ab4(a2−b2)F(μ, t)+2a/parenleftbig
2b2−a2/parenrightbig
3b4(a2−b2)2E(μ, t)
+u
3b2(a2−b2)(u2−b2)/radicalbigg
u2−a2
u2−b2
[u>a>b> 0] BY (216.11)
14./integraldisplay∞
udx/radicalBig
(x2−a2)(x2−b2)5=/parenleftbig
4b2−2a2/parenrightbig
a
3b4(a2−b2)2E(ν,t)+2a2−3b2
3ab4(a2−b2)F(ν,t)
−/parenleftbig
3b2−a2/parenrightbig
u2−/parenleftbig
4b2−2a2/parenrightbig
b2
3b2u(a2−b2)2(u2−b2)/radicalbigg
u2−a2
u2−b2
[u≥a>b> 0] BY (215.06)
15./integraldisplayu
0dx/radicalBig
(a2−x2)5(b2−x2)=1
3a3(a2−b2)2/braceleftbig/parenleftbig
4a2−2b2/parenrightbig
E(η,t)−/parenleftbig
a2−b2/parenrightbig
F(η,t)
−u/bracketleftbig/parenleftbig
5a2−3b2/parenrightbig
a2−/parenleftbig
4a2−2b2/parenrightbig
u2/bracketrightbig
a(a2−u2)/radicalbigg
b2−u2
a2−u2/bracerightBigg
[a>b≥u>0] BY (219.07)
292 Power and Algebraic Functions 3.163
16./integraldisplayb
udx/radicalBig
(a2−x2)5(b2−x2)=2/parenleftbig
2a2−b2/parenrightbig
3a3(a2−b2)2E(ζ,r)−1
3a3(a2−b2)F(ζ,t)
+u
3a2(a2−b2)(a2−u2)/radicalbigg
b2−u2
a2−u2
[a>b>u ≥0] BY (220.10)
17./integraldisplayu
bdx/radicalBig
(a2−x2)5(x2−b2)=1
3a3(a2−b2)2/braceleftbig/parenleftbig
3a2−b2/parenrightbig
F(κ, q)−/parenleftbig
4a2−2b2/parenrightbig
E(κ, q)/bracerightbig
+2/parenleftbig
2a2−b2/parenrightbig
a2+/parenleftbig
b2−3a2/parenrightbig
u2
3a2u(a2−b2)2(a2−u2)/radicalbigg
u2−b2
a2−u2,
[a>u>b> 0] BY (217.10)
18./integraldisplay∞
udx/radicalBig
(x2−a2)5(x2−b2)=1
3a3(a2−b2)2/braceleftbig/parenleftbig
4a2−2b2/parenrightbig
E(ν,t)−/parenleftbig
a2−b2/parenrightbig
F(ν,t)/bracerightbig
+/parenleftbig
4a2−2b2/parenrightbig
a2+/parenleftbig
b2−3a2/parenrightbig
u2
3a2u(a2−b2)2(u2−a2)/radicalbigg
u2−b2
u2−a2
[u>a>b> 0] BY (215.04)
3.163
1./integraldisplayu
0dx/radicalBig
(x2+a2)3(x2+b2)3=1
ab2(a2−b2)2/braceleftbig/parenleftbig
a2+b2/parenrightbig
E(α,q)−2b2F(α,q)/bracerightbig
−u
a2(a2−b2)/radicalbig
(a2+u2)(b2+u2)
[a>b , u> 0] BY (221.07)
2./integraldisplay∞
udx/radicalBig
(x2+a2)3(x2+b2)3=1
ab2(a2−b2)2/braceleftbig/parenleftbig
a2+b2/parenrightbig
E(β,q)−2b2F(β,q)/bracerightbig
−u
b2(a2−b2)/radicalbig
(a2+u2)(b2+u2)
[a>b , u ≥0] BY (222.12)
3./integraldisplayu
0dx/radicalBig
(x2+a2)3(b3−x2)3=1
a2b2/radicalBig
(a2+b2)3/braceleftbig
a2F(γ,r)−/parenleftbig
a2−b2/parenrightbig
E(γ,r)/bracerightbig
+u
b2(a2+b2)/radicalbig
(a2+u2)(b2−u2)
[b>u> 0] BY (214.15)
4./integraldisplay∞
udx/radicalBig
(x2+a2)3(x2−b2)3=b2−a2
a2b2/radicalBig
(a2+b2)3E(ξ,s)−1
a2/radicalBig
(a2+b2)3F(ξ,s)
+u
b2(a2+b2)/radicalbig
(u2+a2)(u2−b2)
[u>b> 0] BY (212.05)
3.165 Square roots of polynomials 293
5./integraldisplayu
0dx/radicalBig
(a2−x2)3(b2−x2)3=1
ab2(a2−b2)F(η,t)−a2+b2
ab2(a2−b2)2E(η,t)
+/bracketleftbig
a4+b4−/parenleftbig
a2+b2/parenrightbig
u2/bracketrightbig
u
a2b2(a2−b2)2/radicalbig
(a2−u2)(b2−u2)
[a>b>u> 0] BY (279.08)
6./integraldisplay∞
udx/radicalBig
(x2−a2)3(x2−b2)3=1
ab2(a2−b2)F(ν,t)−a2+b2
ab2(a2−b2)2E(ν,t)
+1
u(a2−b2)/radicalbig
(u2−a2)(u2−b2)
[u>a>b> 0] BY (215.10)
3.164 Notation :α= arccosu2−ρρ
u2+ρρ,r =1
2/radicalBigg
−(ρ−ρ)2
ρρ.
1./integraldisplay∞
udx/radicalbig
(x2+ρ2)(x2+ρ2)=1√ρρF(α,r) BY (225.00)
2./integraldisplay∞
ux2dx
(x2−ρρ)2/radicalbig
(x2+ρ2)(x2+ρ2)=2u/radicalbig
(u2+ρ2)(u2+ρ2)
(ρ+ρ)2(u4−ρ2ρ2)−1
(ρ+ρ)2√ρρE(α,r)
BY (225.03)
3./integraldisplay∞
ux2dx
(x2+ρρ)2/radicalbig
(x2+ρ2)(x2+ρ2)=−1
(ρ−ρ)2√ρρ[F(α,r)−E(α,r)] BY (225.07)
4./integraldisplay∞
ux2dx/radicalBig
(x2+ρ2)3(x2+ρ2)3=−4√ρρ
(ρ2−ρ2)2E(α,r)+1
(ρ−ρ)2√ρρF(α,r)
−2u/parenleftbig
u2−ρρ/parenrightbig
(ρ+ρ)2(u2+ρρ)/radicalbig
(u2+ρ2)(u2+ρ2)
BY (225.05)
5./integraldisplay∞
u/parenleftbig
x2−ρρ/parenrightbig2dx/radicalBig
(x2+ρ2)3(x2+ρ2)3=−4√ρρ
(ρ−ρ)2[F(α,r)−E(α,r)]
+2u/parenleftbig
u2−ρρ/parenrightbig
(u2+ρρ)/radicalbig
(u2+ρ2)(u2+ρ2)
BY (225.06)
6./integraldisplay∞
u/radicalbig
(x2+ρ2)(x2+ρ2)
(x2+ρρ)2dx=1√ρρE(α,r) BY(225.01)
7./integraldisplay∞
u/parenleftbig
x2−/rho1/rho1/parenrightbig2dx
(x2+/rho1/rho1)2/radicalbig
(x2+/rho12)(x2+/rho12)=−4√/rho1/rho1
(/rho1−/rho1)2E(α,r)+(/rho1+/rho1)2
(/rho1−/rho1)2√/rho1/rho1F(α,r) BY (225.08)
8./integraldisplay∞
u/parenleftbig
x2+/rho1/rho1/parenrightbig2dx/bracketleftBig
(x2+/rho1/rho1)2−4p2/rho1/rho1x2/bracketrightBig/radicalbig
(x2+/rho12)(x2+/rho12)=1√/rho1/rho1Π/parenleftbig
α,p2,r/parenrightbig
BY (225.02)
294 Power and Algebraic Functions 3.165
3.165 Notation :α= arccosu2−a2
u2+a2,r=√
a2−b2
a√
2.
1./integraldisplaya
udx√
x4+2b2x2+a4=√
2
a√
2+√
a2+b2
×F⎡
⎣arctan/parenleftBigg
a√
2+√
a2−b2
√
a2+b2a−u
a+u/parenrightBigg
,2/radicalBig
a/radicalbig
2(a2−b2)
a√
2+√
a2−b2⎤
⎦
[a>b , a>u ≥0] BY (264.00)
2./integraldisplay∞
udx√
x4+2b2x2+a4=1
2aF(α,r)/bracketleftbig
a2>b2>−∞,a2>0,u≥0/bracketrightbig
BY (263.00, 266.00)
3./integraldisplay∞
udx
x2√
x4+2b2x2+a4=1
2a3[F(α,r)−2E(α,r)] +√
u4+2b2u2+a4
a2u(u2+a2)
[a>b> 0,u > 0] BY (263.06)
4./integraldisplay∞
ux2dx
(x2+a2)2√
x4+2b2x2+a4=1
4a(a2−b2)[F(α,r)−E(α,r)]
/bracketleftbig
a2>b2>−∞,a2>0,u≥0/bracketrightbig
BY (263.03, 266.05)
5./integraldisplay∞
ux2dx
(x2−a2)2√
x4+2b2x2+a4=u√
u4+2b2u2+a4
2(a2+b2)(u4−a4)−1
4a(a2+b2)E(α,r)
/bracketleftbig
a2>b2>−∞,u2>a2>0/bracketrightbig
BY (263.05, 266.02)
6./integraldisplay∞
ux2dx/radicalBig
(x4+2b2x2+a4)3=a
2(a4−b4)E(α,r)−1
4a(a2−b2)F(α,r)
−u/parenleftbig
u2−a2/parenrightbig
2(a2+b2)(u2+a2)√
u4+2b2u2+a4/bracketleftbig
a2>b2>−∞,a2>0,u≥0/bracketrightbig
BY (263.08, 266.03)
7./integraldisplay∞
u/parenleftbig
x2−a2/parenrightbig2dx/radicalBig
(x4+2b2x2+a4)3=a
a2−b2[F(α,r)−E(α,r)] +u2−a2
u2+a2u√
u4+2b2u2+a4
/bracketleftbig/vextendsingle/vextendsingleb2/vextendsingle/vextendsingle<a2,u≥0/bracketrightbig
BY (266.08)
8./integraldisplay∞
u/parenleftbig
x2+a2/parenrightbig2dx/radicalBig
(x2+2b2x2+a4)3=a
a2+b2E(α,r)−a2−b2
a2+b2·u2−a2
u2+a2·u√
u4+2b2u2+a4
/bracketleftbig/vextendsingle/vextendsingleb2/vextendsingle/vextendsingle<a2,u≥0/bracketrightbig
BY (266.06)a
9./integraldisplay∞
u/parenleftbig
x2−a2/parenrightbig2dx
(x2+a2)2√
x4+2b2x2+a4=a
a2−b2E(α,r)−a2+b2
2a(a2−b2)F(α,r)
/bracketleftbig
a2>b2>−∞,a2>0,u≥0/bracketrightbig
BY (263.04, 266.07)
3.166 Square roots of polynomials 295
10./integraldisplay∞
u√
x4+2b2x2+a4
(x2+a2)2dx=1
2aE(α,r)/bracketleftbig
a2>b2>−∞,a2>0,u≥0/bracketrightbig
BY (263.01, 266.01)
11./integraldisplay∞
u√
x4+2b2x2+a4
(x2−a2)2dx=1
2a[F(α,r)−E(α,r)] +u
u4−a4/radicalbig
u4+2b2u2+a4
[a>b> 0,u > a ] BY (263)
12./integraldisplay∞
u/parenleftbig
x2+a2/parenrightbig2dx/bracketleftBig
(x2+a2)2−4a2p2x2/bracketrightBig√
x4+2b2x2+a4=1
2aΠ/parenleftbig
α,p2,r/parenrightbig
[a>b> 0,u≥0] BY (263.02)
3.166 Notation :α= arccosu2−1
u2+1,β=a r c t a n/braceleftbigg/parenleftBig
1+√
2/parenrightBig1−u
1+u/bracerightbigg
,
γ= arccos u, δ = arccos1
u,ε= arccos1−u2
1+u2,
r=√
2
2,q=2/radicalBig
3√
2−4=24√
2/parenleftBig√
2−1/parenrightBig
≈0.985171
1./integraldisplay∞
udx√
x4+1=1
2F(α,r)[ u≥0] H (287), BY (263.50)
2./integraldisplay∞
udx
x2√
x4+1=1
2[F(α,r)−2E(α,r)] +√
u4+1
u(u2+1 )
[u>0] BY (263.57)
3./integraldisplay∞
ux2dx
(x4+1 )√
x4+1=1
2E(α,r)−1
4F(α,r)−u/parenleftbig
u2−1/parenrightbig
2(u2+1 )√
u4+1
[u≥0] BY (263.59)
4./integraldisplay∞
ux2dx
(x2+1 )2√
x4+1=1
4[F(α,r)−E(α,r)] [ u≥0] BY (263.53)
5./integraldisplay∞
ux2dx
(x2−1)2√
x4+1=u√
u4+1
2(u4−1)−1
4E(α,r)[ u>1] BY (263.55)
6./integraldisplay∞
u√
x4+1
(x2−1)2dx=1
2[F(α,r)−E(α,r)] +u√
u4+1
u4−1
[u>1] BY (263.58)
7./integraldisplay∞
u/parenleftbig
x2−1/parenrightbig2dx
(x2+1 )2√
x4+1=E(α,r)−1
2F(α,r)[ u≥0] BY (263.54)
8./integraldisplay∞
u√
x4+1dx
(x2+1 )2=1
2E(α,r)[ u≥0] BY (263.51)
296 Power and Algebraic Functions 3.166
9./integraldisplay∞
u/parenleftbig
x2+1/parenrightbig2dx/bracketleftBig
(x2+1 )2−4p2x2/bracketrightBig√
x4+1=1
2Π/parenleftbig
α,p2,r/parenrightbig
[u≥0] BY (263.52)
10./integraldisplayu
0dx√
x4+1=1
2F(ε, r) H 66(288)
11./integraldisplay1
udx√
x4+1=/parenleftBig
2−√
2/parenrightBig
F(β,q)[ 0 ≤u<1] BY (264.50)
12./integraldisplay1
u/parenleftbig
x2+x√
2+1/parenrightbig
dx/parenleftbig
x2−x√
2+1/parenrightbig√
x4+1=/parenleftBig
2+√
2/parenrightBig
E(β,q)[ 0 ≤u<1] BY (264.51)
13./integraldisplay1
u(1−x)2dx/parenleftbig
x2−x√
2+1/parenrightbig√
x4+1=1√
2[F(β,q)−E(β,q)]
[0≤u<1] BY (264.55)
14./integraldisplay1
u(1 +x)2dx/parenleftbig
x2−x√
2+1/parenrightbig√
x4+1=3√
2+4
2E(β,q)−3√
2−4
2F(β,q)
[0≤u<1] BY (264.56)
15./integraldisplay1
udx√
1−x4=1√
2F(γ,r)[ u<1] H 66 (290), BY (259.75)
16./integraldisplay1
0dx√
1−x4=1
4√
2π/braceleftbigg
Γ/parenleftbigg1
4/parenrightbigg/bracerightbigg2
17./integraldisplayu
1dx√
x4−1=1√
2F(δ, r)[ u>1] H 66 (289), BY (260.75)
18.8/integraldisplay1
ux2dx√
1−x4=√
2E(γ,r)−1√
2F(γ,r)[ u<1]
=1√
2π/braceleftbigg
Γ/parenleftbigg3
4/parenrightbigg/bracerightbigg2
[u=0 ]
BY (259.76)
19./integraldisplayu
1x2dx√
x4−1=1√
2F(δ, r)−√
2E(δ, r)+1
u/radicalbig
u4−1[u>1] BY (260.77)
20./integraldisplay1
ux4dx√
1−x4=1
3√
2F(γ,r)+u
3/radicalbig
1−u4 [u<1] BY (259.76)
21.3/integraldisplayu
1x4dx√
x4−1=1
3√
2F(δ, r)+1
3u/radicalbig
u4−1[ u>1] BY (260.77)
22./integraldisplayu
0dx/radicalbig
x(1 +x3)=1
4√
3F/parenleftBigg
arccos1+/parenleftbig
1−√
3/parenrightbig
u
1+/parenleftbig
1+√
3/parenrightbig
u,/radicalbig
2+√
3
2/parenrightBigg
[u>0] BY (260.50)
23./integraldisplayu
0dx/radicalbig
x(1−x3)=1
4√
3F/parenleftBigg
arccos1−/parenleftbig
1+√
3/parenrightbig
u
1+/parenleftbig√
3−1/parenrightbig
u,/radicalbig
2−√
3
2/parenrightBigg
[1≥u>0] BY (259.50)
3.167 Square roots of polynomials 297
3.167 Notation :I n3.167 and3.168 we set: α=a r c s i n/radicalBigg
(a−c)(d−u)
(a−d)(c−u),
β=a r c s i n/radicalBigg
(a−c)(u−d)
(c−d)(a−u),γ =a r c s i n/radicalBigg
(b−d)(c−u)
(c−d)(b−u),
δ=a r c s i n/radicalBigg
(b−d)(u−c)
(b−c)(u−d),κ =a r c s i n/radicalBigg
(a−c)(b−u)
(b−c)(a−u),
λ=a r c s i n/radicalBigg
(a−c)(u−b)
(a−b)(u−c),μ =a r c s i n/radicalBigg
(b−d)(a−u)
(a−b)(u−d),
ν=a r c s i n/radicalBigg
(b−d)(u−a)
(a−d)(u−b),q =/radicalBigg
(b−c)(a−d)
(a−c)(b−d),r =/radicalBigg
(a−b)(c−d)
(a−c)(b−d).
1./integraldisplayd
u/radicalBigg
d−x
(a−x)(b−x)(c−x)dx=2(c−d)/radicalbig
(a−c)(b−d)/braceleftbigg
Π/parenleftbigg
α,a−d
a−c,q/parenrightbigg
−F(α,q)/bracerightbigg
[a>b>c>d>u ] BY (251.05)
2./integraldisplayu
d/radicalBigg
x−d
(a−x)(b−x)(c−x)dx=2(d−a)/radicalbig
(a−c)(b−d)/braceleftbigg
Π/parenleftbigg
β,d−c
a−c,r/parenrightbigg
−F(β,r)/bracerightbigg
[a>b>c ≥u>d] BY (252.14)
3./integraldisplayc
u/radicalBigg
x−d
(a−x)(b−x)(c−x)dx=2/radicalbig
(a−c)(b−d)/braceleftbigg
(c−b)Π/parenleftbigg
γ,c−d
b−d,r/parenrightbigg
+(b−d)F(γ,r)/bracerightbigg
[a>b>c>u ≥d] BY (253.14)
4./integraldisplayu
c/radicalBigg
x−d
(a−x)(b−x)(x−c)dx=2(c−d)/radicalbig
(a−c)(b−d)Π/parenleftbigg
δ,b−c
b−d,q/parenrightbigg
[a>b≥u>c>d ] BY (254.02)
5./integraldisplayb
u/radicalBigg
x−d
(a−x)(b−x)(x−c)dx=2/radicalbig
(a−c)(b−d)/braceleftbigg
(b−a)Π/parenleftbigg
κ,b−c
a−c,q/parenrightbigg
+(a−d)F(κ, q)/bracerightbigg
[a>b>u ≥c>d] BY (255.20)
6./integraldisplayu
b/radicalBigg
x−d
(a−x)(x−b)(x−c)dx=2/radicalbig
(a−c)(b−d)/braceleftbigg
(b−c)Π/parenleftbigg
λ,a−b
a−c,r/parenrightbigg
+(c−d)F(λ,r)/bracerightbigg
[a≥u>b>c>d ] BY (256.13)
7./integraldisplaya
u/radicalBigg
x−d
(a−x)(x−b)(x−c)dx=2(a−d)/radicalbig
(a−c)(b−d)Π/parenleftbigg
μ,b−a
b−d,r/parenrightbigg
[a>u ≥b>c>d ] BY (257.02)
298 Power and Algebraic Functions 3.167
8./integraldisplayu
a/radicalBigg
x−d
(x−a)(x−b)(x−c)dx=2/radicalbig
(a−c)(b−d)/braceleftbigg
(a−b)Π/parenleftbigg
ν,a−d
b−d,q/parenrightbigg
+(b−d)F(ν,q)/bracerightbigg
[u>a>b>c>d ] BY (258.14)
9./integraldisplayd
u/radicalbiggc−x
(a−x)(b−x)(d−x)dx=2(c−d)/radicalbig
(a−c)(b−d)Π/parenleftbigg
α,a−d
a−c,q/parenrightbigg
[a>b>c>d>u ] BY (251.02)
10./integraldisplayu
d/radicalbiggc−x
(a−x)(b−x)(x−d)dx=2/radicalbig
(a−c)(b−d)/bracketleftbigg
(a−d)Π/parenleftbigg
β,d−c
a−c,r/parenrightbigg
−(a−c)F(β,r)/bracketrightbigg
[a>b>c ≥u>d] BY (252.13)
11./integraldisplayc
u/radicalbiggc−x
(a−x)(b−x)(x−d)dx=2(b−c)/radicalbig
(a−c)(b−d)/bracketleftbigg
Π/parenleftbigg
γ,c−d
b−d,r/parenrightbigg
−F(γ,r)/bracketrightbigg
[a>b>c>u ≥d] BY (253.13)
12./integraldisplayu
c/radicalbiggx−c
(a−x)(b−x)(x−d)dx=2(c−d)/radicalbig
(a−c)(b−d)/bracketleftbigg
Π/parenleftbigg
δ,b−c
b−d,q/parenrightbigg
−F(δ, q)/bracketrightbigg
[a>b≥u>c>d ] BY (254.12)
13./integraldisplayb
u/radicalbiggx−c
(a−x)(b−x)(x−d)dx=2/radicalbig
(a−c)(b−d)/bracketleftbigg
(b−a)Π/parenleftbigg
κ,b−c
a−c,q/parenrightbigg
+(a−c)F(κ, q)/bracketrightbigg
[a>b>u ≥c>d] BY (259.19)
14./integraldisplayu
b/radicalbiggx−c
(a−x)(x−b)(x−d)dx=2(b−c)/radicalbig
(a−c)(b−d)Π/parenleftbigg
λ,a−b
a−c,r/parenrightbigg
[a≥u>b>c>d ] BY (256.02)
15./integraldisplaya
u/radicalbiggx−c
(a−x)(x−b)(x−d)dx=2/radicalbig
(a−c)(b−d)/bracketleftbigg
(a−d)Π/parenleftbigg
μ,b−a
b−d,r/parenrightbigg
+(d−c)F(μ, r)/bracketrightbigg
[a>u ≥b>c>d ] BY (257.13)
16./integraldisplayu
a/radicalbiggx−c
(x−a)(x−b)(x−d)dx=2/radicalbig
(a−c)(b−d)/bracketleftbigg
(a−b)Π/parenleftbigg
ν,a−d
b−d,q/parenrightbigg
+(b−c)F(ν,q)/bracketrightbigg
[u>a>b>c>d ] BY (258.13)
17./integraldisplayd
u/radicalBigg
b−x
(a−x)(c−x)(d−x)dx=2/radicalbig
(a−c)(b−d)/bracketleftbigg
(c−d)Π/parenleftbigg
α,a−d
a−c,q/parenrightbigg
+(b−c)F(α,q)/bracketrightbigg
[a>b>c>d>u ] BY (251.07)
18./integraldisplayu
d/radicalBigg
b−x
(a−x)(c−x)(x−d)dx=2/radicalbig
(a−c)(b−d)/bracketleftbigg
(a−d)Π/parenleftbigg
β,d−c
a−c,r/parenrightbigg
−(a−b)F(β,r)/bracketrightbigg
[a>b>c ≥u>d] BY (252.15)
3.167 Square roots of polynomials 299
19./integraldisplayc
u/radicalBigg
b−x
(a−x)(c−x)(x−d)dx=2(b−c)/radicalbig
(a−c)(b−d)Π/parenleftbigg
γ,c−d
b−d,r/parenrightbigg
[a>b>c>u ≥d] BY (253.02)
20./integraldisplayu
c/radicalBigg
b−x
(a−x)(x−c)(x−d)dx=2/radicalbig
(a−c)(b−d)/bracketleftbigg
(d−c)Π/parenleftbigg
δ,b−c
b−d,q/parenrightbigg
+(b−d)F(δ, q)/bracketrightbigg
[a>b≥u>c>d ] BY (254.14)
21./integraldisplayb
u/radicalBigg
b−x
(a−x)(x−c)(x−d)dx=2(a−b)/radicalbig
(a−c)(b−d)/bracketleftbigg
Π/parenleftbigg
κ,b−c
a−c,q/parenrightbigg
−F(κ, q)/bracketrightbigg
[a>b>u ≥c>d] BY (255.21)
22./integraldisplayu
b/radicalBigg
x−b
(a−x)(x−c)(x−d)dx=2(b−c)/radicalbig
(a−c)(b−d)/bracketleftbigg
Π/parenleftbigg
λ,a−b
a−c,r/parenrightbigg
−F(λ,r)/bracketrightbigg
[a≥u>b>c>d ] BY (256.15)
23.8/integraldisplaya
u/radicalBigg
x−b
(a−x)(x−c)(x−d)dx=2/radicalbig
(a−c)(b−d)/bracketleftbigg
(a−d)Π/parenleftbigg
μ,b−a
b−d,r/parenrightbigg
−(b−d)F(μ, r)/bracketrightbigg
[a>u ≥b>c>d ] BY (257.15)
24./integraldisplayu
a/radicalBigg
x−b
(x−a)(x−c)(x−d)dx=2(a−b)/radicalbig
(a−c)(b−d)Π/parenleftbigg
ν,a−d
b−d,q/parenrightbigg
[u>a>b>c>d ] BY (258.02)
25./integraldisplayd
u/radicalbigga−x
(b−x)(c−x)(d−x)dx=2/radicalbig
(a−c)(b−d)/bracketleftbigg
(c−d)Π/parenleftbigg
α,a−d
a−c,q/parenrightbigg
+(a−c)F(α,q)/bracketrightbigg
[a>b>c>d>u ] BY (251.06)
26./integraldisplayu
d/radicalbigga−x
(b−x)(c−x)(x−d)dx=2(a−d)/radicalbig
(a−c)(b−d)Π/parenleftbigg
β,d−c
a−c,r/parenrightbigg
[a>b>c ≥u>d] BY (252.02)
27./integraldisplayc
u/radicalbigga−x
(b−x)(c−x)(x−d)dx=2/radicalbig
(a−c)(b−d)/bracketleftbigg
(b−c)Π/parenleftbigg
γ,c−d
b−d,r/parenrightbigg
+(a−b)F(γ,r)/bracketrightbigg
[a>b>c>u ≥d] BY (253.15)
28./integraldisplayu
c/radicalbigga−x
(b−x)(x−c)(x−d)dx=2/radicalbig
(a−c)(b−d)/bracketleftbigg
(d−c)Π/parenleftbigg
δ,b−c
b−d,q/parenrightbigg
+(a−d)F(δ, q)/bracketrightbigg
[a>b≥u>c>d ] BY (254.13)
29./integraldisplayb
u/radicalbigga−x
(b−x)(x−c)(x−d)dx=2(a−b)/radicalbig
(a−c)(b−d)Π/parenleftbigg
κ,b−c
a−c,q/parenrightbigg
[a>b>u ≥c>d] BY (255.02)
300 Power and Algebraic Functions 3.168
30./integraldisplayu
b/radicalbigga−x
(x−b)(x−c)(x−d)dx=2/radicalbig
(a−c)(b−d)/bracketleftbigg
(c−b)Π/parenleftbigg
λ,a−b
a−c,r/parenrightbigg
+(a−c)F(λ,r)/bracketrightbigg
[a≥u>b>c>d ] BY (256.14)
31./integraldisplaya
u/radicalbigga−x
(x−b)(x−c)(x−d)dx=2(d−a)/radicalbig
(a−c)(b−d)/bracketleftbigg
Π/parenleftbigg
μ,b−a
b−d,r/parenrightbigg
−F(μ, r)/bracketrightbigg
[a>u ≥b>c>d ] BY (257.14)
32./integraldisplayu
a/radicalbiggx−a
(x−b)(x−c)(x−d)dx=2(a−b)/radicalbig
(a−c)(b−d)/bracketleftbigg
Π/parenleftbigg
ν,a−d
b−d,q/parenrightbigg
−F(ν,q)/bracketrightbigg
[u>a>b>c>d ] BY (258.15)
3.168
1./integraldisplayc
u/radicalbiggc−x
(a−x)(b−x)(x−d)3dx=2
d−a/bracketleftBigg/radicalbigg
a−c
b−dE(γ,r)−/radicalBigg
(a−u)(c−u)
(b−u)(u−d)/bracketrightBigg
[a>b>c>u>d ] BY (253.06)
2./integraldisplayu
c/radicalbiggx−c
(a−x)(b−x)(x−d)3dx=2
a−d/radicalbigg
a−c
b−d[F(δ, q)−E(δ, q)]
[a>b≥u>c>d ] BY (254.04)
3./integraldisplayb
u/radicalbiggx−c
(a−x)(b−x)(x−d)3dx=2
a−d/radicalbigg
a−c
b−d[F(κ, q)−E(κ, q)] +2
b−d/radicalBigg
(b−u)(u−c)
(a−u)(u−d)
[a>b>u ≥c>d] BY (255.09)
4./integraldisplayu
b/radicalbiggx−c
(a−x)(x−b)(x−d)3dx=2
a−d/bracketleftBigg/radicalbigg
a−c
b−dE(λ,r)−c−d
b−d/radicalBigg
(a−u)(u−b)
(u−c)(u−d)/bracketrightBigg
[a≥u>b>c>d ] BY (256.06)
5./integraldisplaya
u/radicalbiggx−c
(a−x)(x−b)(x−d)3dx=2
a−d/radicalbigg
a−c
b−dE(μ, r)
[a>u ≥b>c>d ] BY (257.01)
6./integraldisplayu
a/radicalbiggx−c
(x−a)(x−b)(x−d)3dx=2
a−d/radicalbigg
a−c
b−d[F(ν,q)−E(ν,q)]
+2
a−d/radicalBigg
(u−a)(u−c)
(u−b)(u−d)
[u>a>b>c>d ] BY (258.10)
7./integraldisplayc
u/radicalBigg
b−x
(a−x)(c−x)(x−d)3dx=2
(a−d)(c−d)/radicalbig
(a−c)(b−d)
×[(b−c)(a−d)F(γ,r)−(a−c)(b−d)E(γ,r)]
+2(b−d)
(a−d)(c−d)/radicalBigg
(a−u)(c−u)
(b−u)(u−d)
[a>b>c>u>d ] BY (253.03)
3.168 Square roots of polynomials 301
8./integraldisplayu
c/radicalBigg
b−x
(a−x)(x−c)(x−d)3dx=2
(a−d)(c−d)/radicalbig
(a−c)(b−d)
×[(a−c)(b−d)E(δ, q)−(a−b)(c−d)F(δ, q)]
[a>b≥u>c>d ] BY (254.15)
9./integraldisplayb
u/radicalBigg
b−x
(a−x)(x−c)(x−d)3dx=2
(a−d)(c−d)/radicalbig
(a−c)(b−d)
×[(a−c)(b−d)E(κ, q)−(a−b)(c−d)F(κ, q)]
−2
c−d/radicalBigg
(b−u)(u−c)
(a−u)(u−d)
[a>b>u ≥c>d] BY (255.06)
10./integraldisplayu
b/radicalBigg
x−b
(a−x)(x−c)(x−d)3dx=2
(a−d)(c−d)/radicalbig
(a−c)(b−d)
×[(a−c)(b−d)E(λ,r)−(a−d)(b−c)F(λ,r)]
−2
a−d/radicalBigg
(a−u)(u−b)
(u−c)(u−d)
[a≥u>b>c>d ] BY (256.03)
11./integraldisplaya
u/radicalBigg
x−b
(a−x)(x−c)(x−d)3dx=2/radicalbig
(a−c)(b−d)
(a−d)(c−d)E(μ, r)
−2(b−c)
(c−d)/radicalbig
(a−c)(b−d)F(μ, r)
[a>u ≥b>c>d ] BY (257.09)
12./integraldisplayu
a/radicalBigg
x−b
(x−a)(x−c)(x−d)3dx
=2(b−d)
(a−d)(c−d)/radicalBigg
(u−a)(u−c)
(u−b)(u−d)+2(a−b)
(a−d)/radicalbig
(a−c)(b−d)F(ν,q)
+2/radicalbig
(a−c)(b−d)
(a−d)(c−d)E(ν,q)
[u>a>b>c>d ] BY (258.09)
13./integraldisplayc
u/radicalbigga−x
(b−x)(c−x)(x−d)3dx=2
c−d/radicalbigg
a−c
b−d[F(γ,r)−E(γ,r)] +2
c−d/radicalBigg
(a−u)(c−u)
(b−u)(u−d)
[a>b>c>u>d ] BY (253.04)
14./integraldisplayu
c/radicalbigga−x
(b−x)(x−c)(x−d)3dx=2
c−d/radicalbigg
a−c
b−dE(δ, q)
[a>b≥u>c>d ] BY (254.01)
302 Power and Algebraic Functions 3.168
15./integraldisplayb
u/radicalbigga−x
(b−x)(x−c)(x−d)3dx=2
c−d/radicalbigg
a−c
b−dE(κ, q)−2(a−d)
(b−d)(c−d)/radicalBigg
(b−u)(u−c)
(a−u)(u−d)
[a>b>u ≥c>d] BY (255.08)
16./integraldisplayu
b/radicalbigga−x
(x−b)(x−c)(x−d)3dx=2
c−d/radicalbigg
a−c
b−d[F(λ,r)−E(λ,r)] +2
b−d/radicalBigg
(a−u)(u−b)
(u−c)(u−d)
[a≥u>b>c>d ] BY (256.05)
17./integraldisplaya
u/radicalbigga−x
(x−b)(x−c)(x−d)3dx=2
c−d/radicalbigg
a−c
b−d[F(μ, r)−E(μ, r)]
[a>u ≥b>c>d ] BY (257.06)
18./integraldisplayu
a/radicalbiggx−a
(x−b)(x−c)(x−d)3dx=−2
c−d/radicalbigg
a−c
b−dE(ν,q)+2
c−d/radicalBigg
(u−a)(u−c)
(u−b)(u−d)
[u>a>b>c>d ] BY (258.05)
19./integraldisplayd
u/radicalBigg
d−x
(a−x)(b−x)(c−x)3dx=2
b−c/radicalbigg
b−d
a−c[F(α,q)−E(α,q)]
[a>b>c>d>u ] BY (251.01)
20./integraldisplayu
d/radicalBigg
x−d
(a−x)(b−x)(c−x)3dx=−2
b−c/radicalbigg
b−d
a−cE(β,r)+2
b−c/radicalBigg
(b−u)(u−d)
(a−u)(c−u)
[a>b>c ≥u>d] BY (252.06)
21./integraldisplayb
u/radicalBigg
x−d
(a−x)(b−x)(x−c)3dx=2
b−c/radicalbigg
b−d
a−c[F(κ, q)−E(κ, q)] +2
b−c/radicalBigg
(b−u)(u−d)
(a−u)(u−c)
[a>b>u>c>d ] BY (255.05)
22./integraldisplayu
b/radicalBigg
x−d
(a−x)(x−b)(x−c)3dx=2
b−c/radicalbigg
b−d
a−cE(λ,r)
[a≥u>b>c>d ] BY (256.01)
23./integraldisplaya
u/radicalBigg
x−d
(a−x)(x−b)(x−c)3dx=2
b−c/radicalbigg
b−d
a−cE(μ, r)−2(c−d)
(a−c)(b−c)/radicalBigg
(a−u)(u−b)
(u−c)(u−d)
[a>u ≥b>c>d ] BY (257.06)
24./integraldisplayu
a/radicalBigg
x−d
(x−a)(x−b)(x−c)3dx=2
b−c/radicalbigg
b−d
a−c[F(ν,q)−E(ν,q)] +2
a−c/radicalBigg
(u−a)(u−d)
(u−b)(u−c)
[u>a>b>c>d ] BY (258.06)
25./integraldisplaya
u/radicalBigg
b−x
(a−x)(c−x)3(d−x)dx=2
c−d/radicalbigg
b−d
a−cE(α,q)
[a>b>c>d>u ] BY (251.01)
3.168 Square roots of polynomials 303
26./integraldisplayu
d/radicalBigg
b−x
(a−x)(c−x)3(x−d)dx=2
c−d/radicalbigg
b−d
a−c[F(β,r)−E(β,r)] +2
c−d/radicalBigg
(b−u)(u−d)
(a−u)(c−u)
[a>b>c>u>d ] BY (252.03)
27./integraldisplayb
u/radicalBigg
b−x
(a−x)(x−c)3(x−d)dx=2
d−c/radicalbigg
b−d
a−cE(κ, q)+2
c−d/radicalBigg
(b−u)(u−d)
(a−u)(u−c)
[a>b>u>c>d ] BY (255.03)
28./integraldisplayu
b/radicalBigg
x−b
(a−x)(x−c)3(x−d)dx=2
c−d/radicalbigg
b−d
a−c[F(λ,r)−E(λ,r)]
[a≥u>b>c>d ] BY (256.08)
29./integraldisplaya
u/radicalBigg
x−b
(a−x)(x−c)3(x−d)dx=2
c−d/radicalbigg
b−d
a−c[F(μ, r)−E(μ, r)] +2
a−c/radicalBigg
(a−u)(u−b)
(u−c)(u−d)
[a>u ≥b>c>d ] BY (257.03)
30./integraldisplayu
a/radicalBigg
x−b
(x−a)(x−c)3(x−d)dx=2
c−d/radicalbigg
b−d
a−cE(ν,q)−2(b−c)
(a−c)(c−d)/radicalBigg
(u−a)(u−d)
(u−b)(u−c)
[u>a>b>c>d ] BY (258.03)
31./integraldisplayd
u/radicalbigga−x
(b−x)(c−x)3(d−x)dx=2/radicalbig
(a−c)(b−d)
(b−c)(c−d)E(α,q)−a−b
b−c2/radicalbig
(a−c)(b−d)F(α,q)
[a>b>c>d>u ] BY (251.08)
32./integraldisplayu
d/radicalbigga−x
(b−x)(c−x)3(x−d)dx=2(a−d)
(c−d)/radicalbig
(a−c)(b−d)F(β,r)−2/radicalbig
(a−c)(b−d)
(b−c)(c−d)E(β,r)
+2a−c
(b−c)(c−d)/radicalBigg
(b−u)(u−d)
(a−u)(c−u)
[a>b>c>u>d ] BY (252.04)
33./integraldisplayb
u/radicalbigga−x
(b−x)(x−c)3(x−d)dx=2(a−b)
(b−c)/radicalbig
(a−c)(b−c)F(κ, q)−2/radicalBigg
(a−c)(b−d)
(b−c)(c−d)E(κ, q)
+2(a−c)
(b−c)(c−d)/radicalBigg
(b−u)(u−d)
(a−u)(u−c)
[a>b>u>c>d ] BY (255.04)
34./integraldisplayu
b/radicalbigga−x
(x−b)(x−c)3(x−d)dx=2/radicalbig
(a−c)(b−d)
(b−c)(c−d)E(λ,r)−2(a−d)
(c−d)/radicalbig
(a−c)(b−d)F(λ,r)
[a≥u>b>c>d ] BY (256.09)
35./integraldisplaya
u/radicalbigga−x
(x−b)(x−c)3(x−d)dx=2/radicalbig
(a−c)(b−d)
(b−c)(c−d)E(μ, r)−2(a−d)
(c−d)/radicalbig
(a−c)(b−d)F(μ, r)
−2
b−c/radicalBigg
(a−u)(u−b)
(u−c)(u−d)
[a>u ≥b>c>d ] BY (257.04)
304 Power and Algebraic Functions 3.168
36./integraldisplayu
a/radicalbiggx−a
(x−b)(x−c)3(x−d)dx=2/radicalbig
(a−c)(b−d)
(b−c)(c−d)E(ν,q)−2(a−b)
(b−c)/radicalbig
(a−c)(b−d)F(ν,q)
−2
c−d/radicalBigg
(u−a)(u−d)
(u−b)(u−c)
[u>a>b>c>d ] BY (258.04)
37./integraldisplayd
u/radicalBigg
d−x
(a−x)(b−x)3(c−x)dx=2/radicalbig
(a−c)(b−d)
(a−b)(b−c)E(α,q)−2(c−d)
(b−c)/radicalbig
(a−c)(b−d)F(α,q)
−2
a−b/radicalBigg
(a−u)(d−u)
(b−u)(c−u)
[a>b>c>d>u ] BY (251.11)
38./integraldisplayu
d/radicalBigg
x−d
(a−x)(b−x)3(c−x)dx=2/radicalbig
(a−c)(b−d)
(a−b)(b−c)E(β,r)−2(a−d)
(a−b)/radicalbig
(a−c)(b−d)F(β,r)
+2
b−c/radicalBigg
(c−u)(u−d)
(a−u)(b−u)
[a>b>c ≥u>d] BY (252.07)
39./integraldisplayc
u/radicalBigg
x−d
(a−x)(b−x)3(c−x)dx=2/radicalbig
(a−c)(b−d)
(a−b)(b−c)E(γ,r)−2(a−d)
(a−b)/radicalbig
(a−c)(b−d)F(γ,r)
[a>b>c>u ≥d] BY (253.07)
40./integraldisplayu
c/radicalBigg
x−d
(a−x)(b−x)3(x−c)dx=2(c−d)
(b−c)/radicalbig
(a−c)(b−d)F(δ, q)−2/radicalbig
(a−c)(b−d)
(a−b)(b−c)E(δ, q)
+2(b−d)
(a−b)(b−c)/radicalBigg
(a−u)(u−c)
(b−u)(u−d)
[a>b>u>c>d ] BY (254.05)
41./integraldisplaya
u/radicalBigg
x−d
(a−x)(x−b)3(x−c)dx=2(a−d)
(a−b)/radicalbig
(a−c)(b−d)F(μ, r)−2/radicalbig
(a−c)(b−d)
(a−b)(b−c)E(μ, r)
+2(b−d)
(a−b)(b−c)/radicalBigg
(a−u)(u−c)
(u−b)(u−d)
[a>u>b>c>d ] BY (257.07)
42./integraldisplayu
a/radicalBigg
x−d
(x−a)(x−b)3(x−c)dx=2/radicalbig
(a−c)(b−d)
(a−b)(b−c)E(ν,q)−2(c−d)
(b−c)/radicalbig
(a−c)(b−d)F(ν,q)
[u>a>b>c>d ] BY (258.07)
43./integraldisplayd
u/radicalbiggc−x
(a−x)(b−x)3(d−x)dx=2
a−b/radicalbigg
a−c
b−dE(α,q)−2(b−c)
(a−b)(b−d)/radicalBigg
(a−u)(d−u)
(b−u)(c−u)
[a>b>c>d>u ]
3.168 Square roots of polynomials 305
44./integraldisplayu
d/radicalbiggc−x
(a−x)(b−x)3(x−d)dx=2
a−b/radicalbigg
a−c
b−d[F(β,r)−E(β,r)] +2
b−d/radicalBigg
(c−u)(u−d)
(a−u)(b−u)
[a>b>c ≥u>d] BY (252.10)
45./integraldisplayc
u/radicalbiggc−x
(a−x)(b−x)3(x−d)dx=2
a−b/radicalbigg
a−c
b−d[F(γ,r)−E(γ,r)]
[a>b>c>u ≥d] BY (254.08)
46./integraldisplayu
c/radicalbiggx−c
(a−x)(b−x)3(x−d)dx=2
b−a/radicalbigg
a−c
b−dE(δ, q)+2
a−b/radicalBigg
(a−u)(u−c)
(b−u)(u−d)
[a>b≥u>c>d ] BY (254.08)
47./integraldisplaya
u/radicalbiggx−c
(a−x)(x−b)3(x−d)dx=2
a−b/radicalbigg
a−c
b−d[F(μ, r)−E(μ, r)] +2
a−b/radicalBigg
(a−u)(u−c)
(u−b)(u−d)
[a>u ≥b>c>d ] BY (257.10)
48./integraldisplayu
a/radicalbiggx−c
(x−a)(x−b)3(x−d)dx=2
a−b/radicalbigg
a−c
b−dE(ν,q)
[u>a>b>c>d ] BY (258.01)
49./integraldisplayd
u/radicalbigga−x
(b−x)3(c−x)(d−x)dx=2
b−c/radicalbigg
a−c
b−d[F(α,q)−E(α,q)] +2
b−d/radicalBigg
(a−u)(d−u)
(b−u)(c−u)
[a>b>c>d>u ] BY (251.12)
50./integraldisplayu
d/radicalbigga−x
(b−x)3(c−x)(x−d)dx=2
b−c/radicalbigg
a−c
b−dE(β,r)−2(a−b)
(b−c)(b−d)/radicalBigg
(u−d)(c−u)
(a−u)(b−u)
[a>b>c ≥u>d] BY (252.09)
51./integraldisplayc
u/radicalbigga−x
(b−x)3(c−x)(x−d)dx=2
b−c/radicalbigg
a−c
b−dE(γ,r)
[a>b>c>u ≥d] BY (253.01)
52./integraldisplayu
c/radicalbigga−x
(b−x)3(x−c)(x−d)dx=2
b−c/radicalbigg
a−c
b−d[F(δ, q)−E(δ, q)] +2
b−c/radicalBigg
(a−u)(u−c)
(b−u)(u−d)
[a>b>u>c>d ] BY (254.06)
53./integraldisplaya
u/radicalbigga−x
(x−b)3(x−c)(x−d)dx=2
c−b/radicalbigg
a−c
b−dE(μ, r)+2
b−c/radicalBigg
(a−u)(u−c)
(u−b)(u−d)
[a>u>b>c>d ] BY (257.08)
54./integraldisplayu
a/radicalbiggx−a
(x−b)3(x−c)(x−d)dx=2
b−c/radicalbigg
a−c
b−d[F(ν,q)−E(ν,q)]
[u>a>b>c>d ] BY (258.08)
306 Power and Algebraic Functions 3.168
55./integraldisplayd
u/radicalBigg
d−x
(a−x)3(b−x)(c−x)dx=2
b−a/radicalbigg
b−d
a−cE(α,q)+2
a−b/radicalBigg
(b−u)(d−u)
(a−u)(c−u)
[a>b>c>d>u ] BY (251.09)
56./integraldisplayu
d/radicalBigg
x−d
(a−x)3(b−x)(c−x)dx=2
a−b/radicalbigg
b−d
a−c[F(β,q)−E(β,q)]
[a>b>c ≥u>d] BY (252.05)
57./integraldisplayc
u/radicalBigg
x−d
(a−x)3(b−x)(c−x)dx=2
a−b/radicalbigg
b−d
a−c[F(γ,r)−E(γ,r)] +2
a−c/radicalBigg
(c−u)(u−d)
(a−u)(b−u)
[a>b>c>u ≥d] BY (253.05)
58./integraldisplayu
c/radicalBigg
x−d
(a−x)3(b−x)(x−c)dx=2
a−b/radicalbigg
b−d
a−cE(δ, q)−2(a−d)
(a−b)(a−c)/radicalBigg
(b−u)(u−c)
(a−u)(u−d)
[a>b≥u>c>d ] BY (254.03)
59./integraldisplayb
u/radicalBigg
x−d
(a−x)3(b−x)(x−c)dx=2
a−b/radicalbigg
b−d
a−cE(κ, q)
[a>b>u ≥c>d] BY (255.01)
60./integraldisplayu
b/radicalBigg
x−d
(a−x)3(x−b)(x−c)dx=2
a−b/radicalbigg
b−d
a−c[F(λ,r)−E(λ,r)] +2
a−b/radicalBigg
(u−b)(u−d)
(a−u)(u−c)
[a>u>b>c>d ] BY (256.10)
61./integraldisplayd
u/radicalbiggc−x
(a−x)3(b−x)(d−x)dx=2(c−d)
(a−d)/radicalbig
(a−c)(b−d)F(α,q)−2/radicalbig
(a−c)(b−d)
(a−b)(a−d)E(α,q)
+2(a−c)
(a−b)(a−d)/radicalBigg
(b−u)(d−u)
(a−u)(c−u)
[a>b>c>d>u ] BY (251.15)
62./integraldisplayu
d/radicalbiggc−x
(a−x)3(b−x)(x−d)dx=2/radicalbig
(a−c)(b−d)
(a−b)(a−d)E(β,r)−2(b−c)
(a−b)/radicalbig
(a−c)(b−d)F(β,r)
[a>b>c ≥u>d] BY (252.08)
63./integraldisplayc
u/radicalbiggc−x
(a−x)3(b−x)(x−d)dx=2/radicalbig
(a−c)(b−d)
(a−b)(a−d)E(γ,r)−2(b−c)
(a−b)/radicalbig
(a−c)(b−d)F(γ,r)
−2
a−d/radicalBigg
(c−u)(u−d)
(a−u)(b−u)
[a>b>c>u ≥d] BY (253.10)
64./integraldisplayu
c/radicalbiggx−c
(a−x)3(b−x)(x−d)dx=2/radicalbig
(a−c)(b−d)
(a−b)(a−d)E(δ, q)−2(c−d)
(a−d)/radicalbig
(a−c)(b−d)F(δ, q)
−2
a−b/radicalBigg
(b−u)(u−c)
(a−u)(u−d)
[a>b≥u>c>d ] BY (254.09)
3.169 Square roots of polynomials 307
65./integraldisplayb
u/radicalbiggx−c
(a−x)3(b−x)(x−d)dx=2/radicalbig
(a−c)(b−d)
(a−b)(a−d)E(κ, q)−2(c−d)
(a−d)/radicalbig
(a−c)(b−d)F(κ, q)
[a>b>u ≥c>d] BY (255.10)
66./integraldisplayu
b/radicalbiggx−c
(a−x)3(x−b)(x−d)dx=2(b−c)
(a−b)/radicalbig
(a−c)(b−d)F(λ,r)−2/radicalbig
(a−c)(b−d)
(a−b)(a−d)E(λ,r)
+2(a−c)
(a−b)(a−d)/radicalBigg
(u−b)(u−d)
(a−u)(u−c)
[a>u>b>c>d ] BY (256.07)
67./integraldisplayd
u/radicalBigg
b−x
(a−x)3(c−x)(d−x)dx=2
a−d/radicalbigg
b−d
a−c[F(α,q)−E(α,q)] +2
a−d/radicalBigg
(b−u)(d−u)
(a−u)(c−u)
[a>b>c>d>u ] BY (251.13)
68./integraldisplayu
d/radicalBigg
b−x
(a−x)3(c−x)(x−d)dx=2
a−d/radicalbigg
b−d
a−cE(β,r)
[a>b>c ≥u>d] BY (252.01)
69./integraldisplayc
u/radicalBigg
b−x
(a−x)3(c−x)(x−d)dx=2
a−d/radicalbigg
b−d
a−cE(γ,r)−2(a−b)
(a−c)(a−d)/radicalBigg
(c−u)(u−d)
(a−u)(b−u)
[a>b>c>u ≥d] BY (253.08)
70./integraldisplayu
c/radicalBigg
b−x
(a−x)3(x−c)(x−d)dx=2
a−d/radicalbigg
b−d
a−c[F(δ, q)−E(δ, q)] +2
a−c/radicalBigg
(b−u)(u−c)
(a−u)(u−d)
[a>b≥u>c>d ] BY (254.07)
71./integraldisplayb
u/radicalBigg
b−x
(a−x)3(x−c)(x−d)dx=2
a−d/radicalbigg
b−d
a−c[F(κ, q)−E(κ, q)]
[a>b>u ≥c>d] BY (255.07)
72./integraldisplayu
b/radicalBigg
x−b
(a−x)3(x−c)(x−d)dx=−2
a−d/radicalbigg
b−d
a−cE(λ,r)+2
a−d/radicalBigg
(u−b)(u−d)
(a−u)(u−c)
[a≥u>b>c>d ] BY (256.04)
3.169 Notation :I n3.169 –3.172 , we set: α=a r c t a nu
b,β=a r c t a na
u,
γ=a r c s i nu
b/radicalbigg
a2+b2
a2+u2,δ = arccosu
b,ε = arccosb
u,ξ =a r c s i n/radicalbigg
a2+b2
a2+u2,
η=a r c s i nu
b,ζ =a r c s i na
b/radicalbigg
b2−u2
a2−u2,κ =a r c s i na
u/radicalbigg
u2−b2
a2−b2,
λ=a r c s i n/radicalbigg
a2−u2
a2−b2,μ =a r c s i n/radicalbigg
u2−a2
u2−b2,ν =a r c s i na
u,q =√
a2−b2
a,
r=b√
a2+b2,s =a√
a2+b2t=b
a.
308 Power and Algebraic Functions 3.169
1./integraldisplayu
0/radicalbigg
x2+a2
x2+b2dx=a{F(α,q)−E(α,q)}+u/radicalbigg
a2+u2
b2+u2
[a>b , u> 0] BY (221.03)
2.6/integraldisplayu
0/radicalbigg
x2+b2
x2+a2dx=b2
aF(α,q)−aE(α,q)+u/radicalbigg
a2+u2
b2+u2
[a>b , u> 0] BY (221.04)
3./integraldisplayu
0/radicalbigg
x2+a2
b2−x2dx=/radicalbig
a2+b2E(γ,r)−u/radicalbigg
b2−u2
a2+u2[b≥u>0] BY (214.11)
4./integraldisplayb
u/radicalbigg
a2+x2
b2−x2dx=/radicalbig
a2+b2E(δ, r)[ b>u≥0] BY (213.01), ZH 64 (273)
5./integraldisplayu
b/radicalbigg
a2+x2
x2−b2dx=/radicalbig
a2+b2{F(ε, s)−E(ε, s)}+1
u/radicalbig
(u2+a2)(u2−b2)
[u>b> 0] BY (211.03)
6./integraldisplayu
0/radicalbigg
b2−x2
a2+x2dx=/radicalbig
a2+b2{F(γ,r)−E(γ,r)}+u/radicalbigg
b2−u2
a2+u2
[b≥u>0] BY (214.03)
7./integraldisplayb
u/radicalbigg
b2−x2
a2+x2dx=/radicalbig
a2+b2{F(δ, r)−E(δ, r)} [b>u≥0] BY (213.03)
8./integraldisplayu
b/radicalbigg
x2−b2
a2+x2dx=1
u/radicalbig
(a2+u2)(u2−b2)−/radicalbig
a2+b2E(ε, s)
[u>b> 0] BY (211.04)
9./integraldisplayu
0/radicalbigg
b2−x2
a2−x2dx=aE(η,t)−a2−b2
aF(η,t)[ a>b≥u>0] BY (219.03)
10./integraldisplayb
u/radicalbigg
b2−x2
a2−x2dx=aE(ζ,t)−a2−b2
aF(ζ,t)−u/radicalbigg
b2−u2
a2−u2
[a>b>u ≥0] BY (220.04)
11./integraldisplayu
b/radicalbigg
x2−b2
a2−x2dx=aE(κ, q)−b2
aF(κ, q)−1
u/radicalbig
(a2−u2)(u2−b2)
[a≥u>b> 0] BY (217.04)
12./integraldisplaya
u/radicalbigg
x2−b2
a2−x2dx=aE(λ,q)−b2
aF(λ,q)[ a>u ≥b>0] BY (218.03)
13./integraldisplayu
a/radicalbigg
x2−b2
x2−a2dx=a2−b2
aF(μ, t)−aE(μ, t)+μ/radicalbigg
u2−a2
u2−b2
[u>a>b> 0] BY (216.03)
14./integraldisplayu
0/radicalbigg
a2−x2
b2−x2dx=aE(η,t)[ a>b≥u>0]H 64 (276), BY (219.01)
3.171 Square roots of polynomials 309
15./integraldisplayb
u/radicalbigg
a2−x2
b2−x2dx=a/braceleftBigg
E(ζ,t)−u
a/radicalbigg
b2−u2
a2−u2/bracerightBigg
[a>b>u ≥0] BY (220.03)
16./integraldisplayu
b/radicalbigg
a2−x2
x2−b2dx=a{F(κ, q)−E(κ, q)}+1
u/radicalbig
(a2−u2)(u2−b2)
[a≥u>b> 0] BY (217.03)
17./integraldisplaya
u/radicalbigg
a2−x2
x2−b2dx=a{F(λ,q)−E(λ,q)} [a>u ≥b>0] BY (218.09)
18./integraldisplayu
a/radicalbigg
x2−a2
x2−b2dx=u/radicalbigg
u2−a2
u2−b2−aE(μ, t)[ u>a>b> 0] BY (216.04)
3.171
1./integraldisplayu
bdx
x2/radicalbigg
a2+x2
x2−b2=√
a2+b2
b2E(ε, s)[ u>b> 0] BY (211.01), ZH 64 (274)
2./integraldisplay∞
udx
x2/radicalbigg
a2+x2
x2−b2=√
a2+b2
b2E(ξ,s)−a2
b2u/radicalbigg
u2−b2
a2+u2
[u≥b>0] BY (212.09)
3./integraldisplayb
udx
x2/radicalbigg
a2−x2
b2−x2=a2−b2
ab2F(ζ,t)−a
b2E(ζ,t)+a2
b2u/radicalbigg
b2−u2
a2−u2
[a>b>u> 0] BY (220.12)
4./integraldisplayu
bdx
x2/radicalbigg
a2−x2
x2−b2=a
b2E(κ, q)−1
aF(κ, q)[ a≥u>b> 0] BY (217.11)
5./integraldisplaya
udx
x2/radicalbigg
a2−x2
x2−b2=a
b2E(λ,q)−1
af(λ,q)−/radicalbig
(a2−u2)(u2−b2)
b2u
[a>u ≥b>0] BY (218.10)
6./integraldisplayu
adx
x2/radicalbigg
x2−a2
x2−b2=a
b2E(μ, t)−a2−b2
ab2F(μ, t)−1
u/radicalbigg
u2−a2
u2−b2
[u>a>b> 0] BY (216.08)
7./integraldisplay∞
udx
x2/radicalbigg
x2+a2
x2+b2=1
aF(β,q)−a
b2E(β,q)+a2
b2u/radicalbigg
b2+u2
a2+u2
[a>b , u> 0] BY (222.08)
8./integraldisplay∞
udx
x2/radicalbigg
x2+b2
x2+a2=1
a{F(β,q)−E(β,q)}+1
u/radicalbigg
b2+u2
a2+u2
[a>b , u> 0] BY (222.09)
9./integraldisplayb
udx
x2/radicalbigg
b2−x2
a2+x2=/radicalbig
(b2−u2)(a2+u2)
a2u−√
a2+b2
a2E(δ, r)
[b>u> 0] BY (213.10)
310 Power and Algebraic Functions 3.172
10./integraldisplayu
bdx
x2/radicalbigg
x2−b2
a2+x2=√
a2+b2
a2{F(ε, s)−E(ε, s)} [a>b> 0] BY (211.07)
11./integraldisplay∞
udx
x2/radicalbigg
x2−b2
a2+x2=√
a2+b2
a2{F(ξ,s)−E(ξ,s)}+1
u/radicalbigg
u2−b2
a2+u2
[u≥b>0] BY (212.11)
12./integraldisplayb
udx
x2/radicalbigg
a2+x2
b2−x2=√
a2+b2
b2{F(δ, r)−E(δ, r)}+/radicalbig
(b2−u2)(a2+u2)
b2u
[b>u> 0] BY (213.05)
13./integraldisplay∞
udx
x2/radicalbigg
x2−a2
x2−b2=a
b2E(ν,t)−a2−b2
ab2F(ν,t)[ u≥a>b> 0] BY (215.08)
14./integraldisplayb
udx
x2/radicalbigg
b2−x2
a2−x2=1
u/radicalbigg
b2−u2
a2−u2−1
aE(ζ,t)[ a>b>u> 0] BY (220.11)
15./integraldisplayu
bdx
x2/radicalbigg
x2−b2
a2−x2=1
a{F(κ, q)−E(κ, q)} [a≥u>b> 0] BY (217.08)
16./integraldisplaya
udx
x2/radicalbigg
x2−b2
u2−x2=1
a{F(λ,q)−E(λ,q)}+/radicalbig
(a2−u2)(u2−b2)
a2u
[a>u ≥b>0] BY (218.08)
17./integraldisplayu
adx
x2/radicalbigg
x2−b2
x2−a2=1
aE(μ, t)−1
u/radicalbigg
u2−a2
u2−b2[u>a>b> 0] BY (216.07)
18./integraldisplay∞
udx
x2/radicalbigg
x2−b2
x2−a2=1
aE(ν,t)[ u≥a>b> 0]
BY (215.01), ZH 65 (281)
3.172
1./integraldisplayu
0/radicalBigg
x2+b2
(x2+a2)3dx=1
aE(α,q)−a2−b2
a2u/radicalbig
(a2+u2)(b2+u2)
[a>b , u> 0] BY (221.10)
2./integraldisplay∞
u/radicalBigg
x2+b2
(x2+a2)3dx=1
aE(β,q)[ a>b , u ≥0] H 64 (271)
3./integraldisplayu
0/radicalBigg
x2+a2
(x2+b2)3dx=a
b2E(α,q)[ a>b , u> 0] H 64 (270)
4./integraldisplay∞
u/radicalBigg
x2+a2
(x2+b2)3dx=a
b2E(β,q)−a2−b2
b2u/radicalbig
(a2+u2)(b2+u2)
[a>b , u ≥0] BY (222.06)
5./integraldisplayu
0/radicalBigg
b2−x2
(a2+x2)3dx=√
a2+b2
a2E(γ,r)−1√
a2+b2F(γ,r)
[b≥u>0] BY (214.08)
3.172 Square roots of polynomials 311
6./integraldisplayb
u/radicalBigg
b2−x2
(a2+x2)3dx=√
a2+b2
a2E(δ, r)−1√
a2+b2F(δ, r)−u
a2/radicalbigg
b2−u2
a2+u2
[b>u≥0] BY (213.04)
7./integraldisplayu
b/radicalBigg
x2−b2
(a2+x2)3dx=√
a2+b2
a2E(ε, s)−b2
a2√
a2+b2F(ε, s)−1
u/radicalbigg
u2−b2
u2+a2
[u>b> 0] BY (211.06)
8./integraldisplay∞
u/radicalBigg
x2−b2
(a2+x2)3dx=√
a2+b2
a2E(ξ,s)−b2
a2√
a2+b2F(ξ,s)
[u≥b>0] BY (212.08)
9./integraldisplayu
0/radicalBigg
x2+a2
(b2−x2)3dx=a2
b2√
a2+b2F(γ,r)−√
a2+b2
b2E(γ,r)+/parenleftbig
a2+b2/parenrightbig
u
b2/radicalbig
(a2+u2)(b2−u2)
[b>u> 0] BY (214.09)
10./integraldisplay∞
u/radicalBigg
x2+a2
(x2−b2)3dx=1√
a2+b2F(ξ,s)−√
a2+b2
b2E(ξ,s)+/parenleftbig
a2+b2/parenrightbig
u
b2/radicalbig
(a2+u2)(u2−b2)
[u>b> 0] BY (212.07)
11./integraldisplayu
0/radicalBigg
b2−x2
(a2−x2)3dx=1
a/braceleftBigg
F(η,t)−E(η,t)+u
a/radicalbigg
b2−u2
a2−u2/bracerightBigg
[a>b≥u>0] BY (219.09)
12./integraldisplayb
u/radicalBigg
b2−x2
(a2−x2)3dx=1
a{F(ζ,t)−E(ζ,t)} [a>b>u ≥0] BY (220.07)
13./integraldisplayu
b/radicalBigg
x2−b2
(a2−x2)3dx=1
u/radicalbigg
u2−b2
a2−u2−1
aE(κ, q)[ a>u>b> 0] BY (217.07)
14./integraldisplay∞
u/radicalBigg
x2−b2
(x2−a2)3dx=1
a[F(ν,t)−E(ν,t)] +1
u/radicalbigg
u2−b2
u2−a2
[u>a>b> 0] BY (215.05)
15./integraldisplayu
0/radicalBigg
a2−x2
(b2−x2)3dx=a
b2[F(η,t)−E(η,t)] +u
b2/radicalbigg
a2−u2
b2−u2
[a>b>u> 0] BY (219.10)
16./integraldisplaya
u/radicalBigg
a2−x2
(x2−b2)3dx=u
b2/radicalbigg
a2−u2
u2−b2−a
b2E(λ,q)[ a>u>b> 0] BY (218.05)
17./integraldisplayu
a/radicalBigg
x2−a2
(x2−b2)3dx=a
b2[F(μ, t)−E(μ, t)] [ u>a>b> 0] BY (216.05)
312 Power and Algebraic Functions 3.173
18./integraldisplay∞
u/radicalBigg
x2−a2
(x2−b2)3dx=a
b2[F(ν,t)−E(ν,t)] +1
u/radicalbigg
u2−a2
u2−b2
[u≥a>b> 0] BY (215.03)
3.173
1./integraldisplay1
udx
x2/radicalbigg
x2+1
1−x2=√
2/bracketleftBigg
F/parenleftBigg
arccos u,√
2
2/parenrightBigg
−E/parenleftBigg
arccos u,√
2
2/parenrightBigg/bracketrightBigg
+√
1−u4
u
[u<1] BY (259.77)
2./integraldisplayu
1dx
x2/radicalbigg
x2+1
x2−1=√
2E/parenleftBigg
arccos1
u,√
2
2/parenrightBigg
[u>1] BY (260.76)
3.174 Notation :I n3.174 and3.175 ,w et a k e : α= arccos1+/parenleftbig
1−√
3/parenrightbig
u
1+/parenleftbig
1+√
3/parenrightbig
u,
β= arccos1−/parenleftbig
1+√
3/parenrightbig
u
1+/parenleftbig√
3−1/parenrightbig
u,p =/radicalbig
2+√
3
2,q =/radicalbig
2−√
3
2.
1./integraldisplayu
0dx
/bracketleftbig
1+/parenleftbig
1+√
3/parenrightbig
x/bracketrightbig2/radicalBigg
1−x+x2
x(1 +x)=1
4√
3E(α,p)[ u>0] BY (260.51)
2./integraldisplayu
0dx
/bracketleftbig
1+/parenleftbig√
3−1/parenrightbig
x/bracketrightbig2/radicalBigg
1+x+x2
x(1−x)=1
4√
3E(β,q)[ 1 ≥u>0] BY (259.51)
3./integraldisplayu
0dx
1−x+x2/radicalbigg
x(1 +x)
1−x+x21
4√
27E(α,p)+−2−√
3
4√
27F(α,p)−2/parenleftbig
2+√
3/parenrightbig
√
31+/parenleftbig
1−√
3/parenrightbig
u
1+/parenleftbig
1+√
3/parenrightbig
u
×/radicalbigg
u(1 +u)
1−u+u2
[u>0] BY (260.54)
4./integraldisplayu
0dx
1+x+x2/radicalbigg
x(1−x)
1+x+x24
4√
27E(β,q)−2+√
3
4√
27F(β,q)−2/parenleftbig
2−√
3/parenrightbig
√
31−/parenleftbig
1+√
3/parenrightbig
u
1+/parenleftbig√
3−1/parenrightbig
u
×/radicalbigg
u(1−u)
1+u+u2
[1≥u>0] BY (259.55)
3.175
1./integraldisplayu
0dx
1+x/radicalbiggx
1+x3=1
4√
27[F(α,p)−2E(α,p)] +2√
3/radicalbig
u(1−u+u2)√1+u/bracketleftbig
1+/parenleftbig
1+√
3/parenrightbig
u/bracketrightbig
[u>0] BY (260.55)
2./integraldisplayu
0dx
1−x/radicalbiggx
1−x3=1
4√
27[F(β,q)−2E(β,q)] +2√
3/radicalbig
u(1 +u+u2)√1−u/bracketleftbig
1+/parenleftbig√
3−1/parenrightbig
u/bracketrightbig
[0<u< 1] BY (259.52)
3.183 Fourth roots of polynomials 313
3.18 Expressions that can be reduced to fourth roots of second-degree polynomials
and their products with rational functions
3.181
1./integraldisplayu
bdx
4/radicalbig
(a−x)(x−b)=√
a−b/braceleftBigg
2/bracketleftBigg
E/parenleftbigg1√
2/parenrightbigg
+E/parenleftBigg
arccos4/radicalBigg
4(a−u)(u−b)
(a−b)2,1√
2/parenrightBigg/bracketrightBigg
−/bracketleftBigg
K/parenleftbigg1√
2/parenrightbigg
+F/parenleftBigg
arccos4/radicalBigg
4(a−u)(u−b)
(a−b)2,1√
2/parenrightBigg/bracketrightBigg/bracerightBigg
[a≥u>b] BY (271.05)
2./integraldisplayu
adx
4/radicalbig
(x−a)(x−b)/radicalbigg
a−b
2F/bracketleftBigg/parenleftBigg
arccosa−b−2/radicalbig
(u−a)(u−b)
a−b+2/radicalbig
(u−a)(u−b),1√
2/parenrightBigg
−2E/parenleftBigg
arccosa−b−2/radicalbig
(u−a)(u−b)
a−b+2/radicalbig
(u−a)(u−b),1√
2/parenrightBigg/bracketrightBigg
+2(2u−a−b)4/radicalbig
(u−a)(u−b)
a−b+2/radicalbig
(u−a)(u−b)
[u>a>b ] BY (272.05)
3.182
1./integraldisplayu
bdx
4/radicalbig
[(a−x)(x−b)]3=2√
a−b/bracketleftBigg
K/parenleftbigg1√
2/parenrightbigg
+F/parenleftBigg
arccos/radicalBigg
4(a−u)(u−b)
(a−b)2,1√
2/parenrightBigg/bracketrightBigg
[a≥u>b] BY (271.01)
2./integraldisplayu
adx
4/radicalbig
[(x−a)(x−b)]3=√
2√
a−bF/parenleftBigg
arccosa−b−2/radicalbig
(u−a)(u−b)
a−b+2/radicalbig
(u−a)(u−b),1√
2/parenrightBigg
[u>a>b ] BY (272.00)
3.183 Notation :I n3.183 –3.186 we set:
α= arccos1
4√
u2+1,β = arccos4/radicalbig
1−u2,γ = arccos1−√
u2−1
1+√
u2−1.
1./integraldisplayu
0dx
4√
x2+1=√
2/bracketleftbigg
F/parenleftbigg
α,1√
2/parenrightbigg
−2E/parenleftbigg
α,1√
2/parenrightbigg/bracketrightbigg
+2u
4√
u2+1
[u>0] BY (273.55)
2./integraldisplayu
0dx
4√
1−x2=√
2/bracketleftbigg
2E/parenleftbigg
β,1√
2/parenrightbigg
−F/parenleftbigg
β,1√
2/parenrightbigg/bracketrightbigg
[0<u≤1] BY (271.55)
3./integraldisplayu
1dx
4√
x2−1=F/parenleftbigg
γ,1√
2/parenrightbigg
−2E/parenleftbigg
γ,1√
2/parenrightbigg
+2u4√
u2−1
1+√
u2−1
[u>1] BY (272.55)
314 Power and Algebraic Functions 3.184
3.184
1./integraldisplayu
0x2dx
4√
1−x2=2√
2
5/bracketleftbigg
2E/parenleftbigg
β,1√
2/parenrightbigg
−F/parenleftbigg
β,1√
2/parenrightbigg/bracketrightbigg
−2u
54/radicalBig
(1−u2)3
[0<u≤1] BY (271.59)
2./integraldisplayu
1dx
x24√
x2−1=E/parenleftbigg
γ,1√
2/parenrightbigg
−1
2F/parenleftbigg
γ,1√
2/parenrightbigg
−1−√
u2−1
1+√
u2−1·√
u2−1
u
[u>1] BY (272.54)
3.185
1./integraldisplayu
0dx
4/radicalBig
(x2+1 )3=√
2F/parenleftbigg
α,1√
2/parenrightbigg
[u>0] BY (273.50)
2./integraldisplayu
0dx
4/radicalBig
(1−x2)3=√
2F/parenleftbigg
β,1√
2/parenrightbigg
[0<u≤1] BY (271.51)
3./integraldisplayu
1dx
4/radicalBig
(x2−1)3=F/parenleftbigg
γ,1√
2/parenrightbigg
[u>1] BY (272.50)
4./integraldisplayu
0x2dx
4/radicalBig
(1−x2)3=2√
2
3F/parenleftbigg
β,1√
2/parenrightbigg
−2
3u4/radicalbig
1−u2 [0<u≤1] BY (271.54)
5./integraldisplayu
0dx
4/radicalBig
(x2+1 )5=2√
2E/parenleftbigg
α,1√
2/parenrightbigg
−√
2F/parenleftbigg
α,1√
2/parenrightbigg
[u>0] BY (273.54)
6./integraldisplayu
0x2dx
4/radicalBig
(x2+1 )5=2√
2/bracketleftbigg
F/parenleftbigg
α,1√
2/parenrightbigg
−2E/parenleftbigg
α,1√
2/parenrightbigg/bracketrightbigg
+2u
4√
u2+1
[u>0] BY (273.56)
7./integraldisplayu
0x2dx
4/radicalBig
(x2+1 )7=1
3√
2F/parenleftbigg
α,1√
2/parenrightbigg
−u
64/radicalBig
(u2+1 )3[u>0] BY (273.53)
3.186
1./integraldisplayu
01+√
x2+1
(x2+1 )4√
x2+1dx=2√
2E/parenleftbigg
α,1√
2/parenrightbigg
[u>0] BY (273.51)
2./integraldisplayu
0dx/parenleftbig
1+√
1−x2/parenrightbig4√
1−x2=√
2/bracketleftbigg
F/parenleftbigg
β,1√
2/parenrightbigg
−E/parenleftbigg
β,1√
2/parenrightbigg/bracketrightbigg
+u4√
1−u2
1+√
1−u2
[0<u≤1] BY (271.58)
3./integraldisplayu
1dx/parenleftbig
x2+2√
x2−1/parenrightbig4√
x2−1=1
2/bracketleftbigg
F/parenleftbigg
γ,1√
2/parenrightbigg
−E/parenleftbigg
γ,1√
2/parenrightbigg/bracketrightbigg
[u>1] BY (272.53)
3.194 Powers of xand binomials 315
4./integraldisplayu
01−√
1−x2
1+√
1−x2·dx
4/radicalBig
(1−x2)3=√
2/bracketleftbigg
2E/parenleftbigg
β,1√
2/parenrightbigg
−F/parenleftbigg
β,1√
2/parenrightbigg/bracketrightbigg
−2u4√
1−u2
1+√
1−u2
[0<u≤1] BY (271.57)
5./integraldisplayu
1x2dx
/parenleftbig
x2+2√
x2−1/parenrightbig4/radicalBig
(x2−1)3=E/parenleftbigg
γ,1√
2/parenrightbigg
[u>1] BY (272.51)
3.19–3.23 Combinations of powers of xand powers of binomials of the form (α+βx)
3.191
1./integraldisplayu
0xν−1(u−x)μ−1dx=uμ+ν−1B(μ, ν)[ R e μ>0,Reν>0] ET II 185(7)
2./integraldisplay∞
ux−ν(x−u)μ−1dx=uμ−νB(ν−μ, μ)[ R e ν>Reμ>0] ET II 201(6)
3./integraldisplay1
0xν−1(1−x)μ−1dx=/integraldisplay1
0xμ−1(1−x)ν−1dx=B (μ, ν)
[Reμ>0,Reν>0] FI II 774(1)
3.192
1./integraldisplay1
0xpdx
(1−x)p=pπcosecpπ/bracketleftbig
p2<1/bracketrightbig
BI (3)(4)
2./integraldisplay1
0xpdx
(1−x)p+1=−πcosecpπ [−1<p< 0] BI (3)(5)
3./integraldisplay1
0(1−x)p
xp+1dx=−πcosecpπ [−1<p< 0] BI (4)(6)
4./integraldisplay∞
1(x−1)p−1
2dx
x=πsecpπ/bracketleftbig
−1
2<p<1
2/bracketrightbig
BI (23)(7)
3.193/integraldisplayn
0xν−1(n−x)ndx=n!nν+n
ν(ν+1 ) (ν+2 )...(ν+n)[Reν>0] EH I 2
3.194
1./integraldisplayu
0xμ−1dx
(1 +βx)ν=uμ
μ2F1(ν,μ;1+μ;−βu)[ |arg(1 + βu)|<π , Reμ>0]
ET I 310(20)
2.6/integraldisplay∞
uxμ−1dx
(1 +βx)ν=uμ−ν
βν(ν−μ)2F1/parenleftbigg
ν,ν−μ;ν−μ+1 ; −1
βu/parenrightbigg
[Reν>Reμ] ET I 310(21)
3./integraldisplay∞
0xμ−1dx
(1 +βx)ν=β−μB(μ, ν−μ)[ |argβ|<π , Reν>Reμ>0]
FI II 775a, ET I 310(19)
316 Power and Algebraic Functions 3.195
4.11/integraldisplay∞
0xμ−1dx
(1 +βx)n+1=(−1)nπ
βμ/parenleftbiggμ−1
n/parenrightbigg
cosec( μπ)[ |argβ|<π , 0<Reμ<n +1 ]
ET I 308(6)
5./integraldisplayu
0xμ−1dx
1+βx=uμ
μ2F1(1,μ;1+μ;−βu)[ |arg(1 + uβ)|<π , Reμ>0]
ET I 308(5)
6./integraldisplay∞
0xμ−1dx
(1 +βx)2=(1−μ)π
βμcosecμπ [0<Reμ<2] BI (16)(4)
7./integraldisplay∞
0xmdx
(a+bx)n+1
2=2m+1m!(2n−2m−3)!!
(2n−1)!!am−n+1
2
bm+1
/bracketleftbig
m<n −1
2,a > 0,b > 0/bracketrightbig
BI (21)(2)
8./integraldisplay1
0xn−1dx
(1 +x)m=2−n∞/summationdisplay
k=0/parenleftbiggm−n−1
k/parenrightbigg(−2)−k
n+kBI (3)(1)
3.19511/integraldisplay∞
0(1 +x)p−1
(a+x)p+1dx=1−a−p
p(a−1)[p/negationslash=0,a > 0,a/negationslash=1 ]
=lna
a−1[p=0,a > 0,a/negationslash=1 ]
=1 [ a=1 ]
LI (19)(6)
3.196
1./integraldisplayu
0(x+β)ν(u−x)μ−1dx=βνuμ
μ2F1/parenleftbigg
1,−ν;1+μ;−u
β/parenrightbigg
/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleargu
β/vextendsingle/vextendsingle/vextendsingle/vextendsingle<π/bracketrightbigg
ET II 185(8)
2./integraldisplay∞
u(x+β)−ν(x−u)μ−1dx=(u+β)μ−νB(ν−μ, μ)
/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleargu
β/vextendsingle/vextendsingle/vextendsingle/vextendsingle<π , Reν>Reμ>0/bracketrightbigg
ET II 201(7)
3./integraldisplayb
a(x−a)μ−1(b−x)ν−1dx=(b−a)μ+ν−1B(μ, ν)[ b>a , Reμ>0,Reν>0]
EH I 10(13)
4./integraldisplay∞
1dx
(a−bx)(x−1)ν=−π
bcosecνπ/parenleftbiggb
b−a/parenrightbiggν
[a<b , b> 0,0<ν< 1] LI (23)(5)
5./integraldisplay1
−∞dx
(a−bx)(1−x)ν=π
bcosecνπ/parenleftbiggb
a−b/parenrightbiggν
[a>b> 0,0<ν< 1] LI (24)(10)
3.197 Powers of xand binomials 317
3.197
1./integraldisplay∞
0xν−1(β+x)−μ(x+γ)−/rho1dx=β−μγν−/rho1B(ν,μ−ν+/rho1)2F1/parenleftbigg
μ, ν;μ+/rho1;1−γ
β/parenrightbigg
[|argβ|<π , |argγ|<π , Reν>0,Reμ>Re(ν−/rho1)]ET II 233(9)
2.11/integraldisplay∞
ux−λ(x+β)ν(x−u)μ−1dx=u−λ(β+u)μ+νB(λ−μ−ν,μ)2F1/parenleftbigg
λ,μ;λ−μ;−β
u/parenrightbigg
/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleargu
β/vextendsingle/vextendsingle/vextendsingle/vextendsingle<πor/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ
u/vextendsingle/vextendsingle/vextendsingle/vextendsingle<1,0<Reμ<Re(λ−ν)/bracketrightbigg
ET II 201(8)
3./integraldisplay1
0xλ−1(1−x)μ−1(1−βx)−νdx=B (λ,μ)2F1(ν,λ;λ+μ;β)
[Reλ>0,Reμ>0,|β|<1] WH
4./integraldisplay1
0xμ−1(1−x)ν−1(1 +ax)−μ−νdx=( 1+ a)−μB(μ, ν)
[Reμ>0,Reν>0,a > −1]
BI(5)4, EH I 10(11)
5./integraldisplay∞
0xλ−1(1 +x)ν(1 +αx)μdx=B (λ,−μ−ν−λ)2F1(−μ, λ;−μ−ν;1−α)
[|argα|<π , −Re(μ+ν)>Reλ>0]
EH I 60(12), ET I 310(23)
6./integraldisplay∞
1xλ−ν(x−1)ν−μ−1(αx−1)−λdx=α−λB(μ, ν−μ)2F1/parenleftbig
ν,μ;λ;α−1/parenrightbig
[1 + Re ν>Reλ>Reμ,|arg(α−1)|<π]EH I 115(6)
7./integraldisplay∞
0xμ−1
2(x+a)−μ(x+b)−μdx=√π/parenleftBig√a+√
b/parenrightBig1−2μΓ/parenleftbig
μ−1
2/parenrightbig
Γ(μ)
[Reμ>0] BI 19(5)
8./integraldisplayu
0xν−1(x+α)λ(u−x)μ−1dx=αλuμ+ν−1B(μ, ν)2F1/parenleftBig
−λ,ν;μ+ν;−u
α/parenrightBig
/bracketleftBig/vextendsingle/vextendsingle/vextendsinglearg/parenleftBigu
α/parenrightBig/vextendsingle/vextendsingle/vextendsingle<π , Reμ>0,Reν>0/bracketrightBig
ET II 186(9)
9./integraldisplay∞
0xλ−1(1 +x)−μ+ν(x+β)−νdx=B (μ−λ,λ)2F1(ν,μ−λ;μ;1−β)
[Reμ>Reλ>0] EH I 205
10./integraldisplay1
0xq−1dx
(1−x)q(1 +px)=π
(1 +p)qcosecqπ [0<q< 1,p > −1] BI (5)(1)
11./integraldisplay1
0xp−1
2dx
(1−x)p(1 +qx)p=2Γ/parenleftbig
p+1
2/parenrightbig
Γ(1−p)√πcos2p(arctan√q)sin/bracketleftbig
(2p−1)arctan/parenleftbig√q/parenrightbig/bracketrightbig
(2p−1)sin/bracketleftbig
arctan/parenleftbig√q/parenrightbig/bracketrightbig
/bracketleftbig
−1
2<p< 1,q > 0/bracketrightbig
BI (11)(1)
318 Power and Algebraic Functions 3.198
12./integraldisplay1
0xp−1
2dx
(1−x)p(1−qx)p=Γ/parenleftbig
p+1
2/parenrightbig
Γ(1−p)√π/parenleftbig
1−√q/parenrightbig1−2p−/parenleftbig
1+√q/parenrightbig1−2p
(2p−1)√q
/bracketleftbig
−1
2<p< 1,0<q< 1/bracketrightbig
BI (11)(2)
3.198/integraldisplay1
0xμ−1(1−x)ν−1[ax+b(1−x)+c]−(μ+ν)dx=(a+c)−μ(b+c)−νB(μ, ν)
[a≥0,b≥0,c > 0,Reμ>0,Reν>0]FI II 787
3.199/integraldisplayb
a(x−a)μ−1(b−x)ν−1(x−c)−μ−νdx=(b−a)μ+ν−1(b−c)−μ(a−c)−νB(μ, ν)
[Reμ>0,Reν>0,c < a < b ]
EH I 10(14)
3.211/integraldisplay1
0xλ−1(1−x)μ−1(1−ux)−/rho1(1−vx)−σdx=B (μ, λ)F1((λ, /rho1, σ, λ +μ;u,v))
[Reλ>0,Reμ>0] EH I 231(5)
3.212/integraldisplay∞
0/bracketleftbig
(1 +ax)−p+( 1+ bx)−p/bracketrightbig
xq−1dx=2 (ab)−q
2B(q,p−q)cos/braceleftbigg
qarccos/bracketleftbigga+b
2√
ab/bracketrightbigg/bracerightbigg
[p>q> 0] BI (19)(9)
3.213/integraldisplay∞
0/bracketleftbig
(1 +ax)−p−(1 +bx)−p/bracketrightbig
xq−1dx=−2i(ab)−q
2B(q,p−q)sin/braceleftbigg
qarccos/bracketleftbigga+b
2√
ab/bracketrightbigg/bracerightbigg
[p>q> 0] BI (19)(10)
3.214/integraldisplay1
0/bracketleftbig
(1 +x)μ−1(1−x)ν−1+( 1+ x)ν−1(1−x)μ−1/bracketrightbig
dx=2μ+ν−1B(μ, ν)
[Reμ>0,Reν>0]
LI(1)(15), EH I 10(10)
3.215/integraldisplay1
0/braceleftbig
aμxμ−1(1−ax)ν−1+( 1−a)νxν−1[1−(1−a)x]μ−1/bracerightbig
dx=B (μ, ν)
[Reμ>0,Reν>0,|a|<1]
BI (1)(16)
3.216
1./integraldisplay1
0xμ−1+xν−1
(1 +x)μ+νdx=B (μ, ν)[ R e μ>0,Reν>0] FI II 775
2./integraldisplay∞
1xμ−1+xν−1
(1 +x)μ+νdx=B (μ, ν)[ R e μ>0,Reν>0] FI II 775
3.217/integraldisplay∞
0/braceleftbiggbpxp−1
(1 +bx)p−(1 +bx)p−1
bp−1xp/bracerightbigg
dx=πcotpπ [0<p< 1,b > 0] BI(18)(13)
3.218/integraldisplay∞
0x2p−1−(a+x)2p−1
(a+x)pxpdx=πcotpπ [p<1] (cf. 3.217 ) BI (18)(7)
3.219/integraldisplay∞
0/braceleftbiggxν
(x+1 )ν+1−xμ
(x+1 )μ+1/bracerightbigg
dx=ψ(μ+1 )−ψ(ν+1 )
[Reμ>−1,Reν>−1] BI (19)(13)
3.221
1./integraldisplay∞
a(x−a)p−1
x−bdx=π(a−b)p−1cosecpπ [a>b , 0<p< 1] LI (24)(8)
3.226 Powers of xand binomials 319
2./integraldisplaya
−∞(a−x)p−1
x−bdx=−π(b−a)p−1cosecpπ [a<b , 0<p< 1] LI (24)(8)
3.222
1./integraldisplay1
0xμ−1dx
1+x=β(μ)[ R e μ>0] WH
2./integraldisplay∞
0xμ−1dx
x+a=πcosec( μπ)aμ−1fora>0 FI II 718, FI II 737
=−πcot(μπ)(−a)μ−1fora<0 BI(18)(2), ET II 249(28)
[0<Reμ<1]
3.223
1./integraldisplay∞
0xμ−1dx
(β+x)(γ+x)=π
γ−β/parenleftbig
βμ−1−γμ−1/parenrightbig
cosec( μπ)
[|argβ|<π , |argγ|<π , 0<Reμ<2]ET I 309(7)
2./integraldisplay∞
0xμ−1dx
(β+x)(α−x)=π
α+β/bracketleftbig
βμ−1cosec( μπ)+αμ−1cot(μπ)/bracketrightbig
[|argβ|<π , α> 0,0<Reμ<2]
ET I 309(8)
3./integraldisplay∞
0xμ−1dx
(a−x)(b−x)=πcot(μπ)aμ−1−bμ−1
b−a[a>b> 0,0<Reμ<2]ET I 309(9)
3.224/integraldisplay∞
0(x+β)xμ−1dx
(x+γ)(x+δ)=πcosec( μπ)/braceleftbiggγ−β
γ−δγμ−1+δ−β
δ−γδμ−1/bracerightbigg
[|argγ|<π , |argδ|<π , 0<Reμ<1]ET I 309(10)
3.225
1./integraldisplay∞
1(x−1)p−1
x2dx=( 1−p)πcosecpπ [−1<p< 1] BI (23)(8)
2./integraldisplay∞
1(x−1)1−p
x3dx=1
2p(1−p)πcosecpπ [0<p< 1] BI (23)(1)
3./integraldisplay∞
0xpdx
(1 +x)3=π
2p(1−p)cosec pπ [−1<p< 2] BI (16)(5)
3.226
1./integraldisplay1
0xndx√1−x=2(2n)!!
(2n+1 ) ! !BI (8)(1)
2./integraldisplay1
0xn−1
2dx√1−x=(2n−1)!!
(2n)!!π. BI (8)(2)
320 Power and Algebraic Functions 3.227
3.227
1./integraldisplay∞
0xν−1(β+x)1−μ
γ+xdx=β1−μγν−1B(ν,μ−ν)2F1/parenleftbigg
μ−1,ν;μ;1−γ
β/parenrightbigg
[|argβ|<π , |argγ|<π , 0<Reν<Reμ]ET II 217(9)
2./integraldisplay∞
0x−/rho1(β−x)−σ
γ+xdx=πγ−/rho1(β−γ)−σcosec( /rho1π)I1−γ/β(σ, /rho1)
[|argβ|<π , |argγ|<π , −Reσ<Re/rho1<1]ET II 217(10)
3.228
1./integraldisplayb
a(x−a)ν(b−x)−ν
x−cdx=πcosec( νπ)/bracketleftbigg
1−/parenleftbigga−c
b−c/parenrightbiggν/bracketrightbigg
forc<a
=πcosec( νπ)/bracketleftbigg
1−cos(νπ)/parenleftbiggc−a
b−c/parenrightbiggν/bracketrightbigg
fora<c<b
=πcosec( νπ)/bracketleftbigg
1−/parenleftbiggc−a
c−b/parenrightbiggν/bracketrightbigg
forc>b
[|Reν|<1] ET II 250(31)
2./integraldisplayb
a(x−a)ν−1(b−x)−ν
x−cdx=πcosec( νπ)
b−c/vextendsingle/vextendsingle/vextendsingle/vextendsinglea−c
b−c/vextendsingle/vextendsingle/vextendsingle/vextendsingleν−1
forc<a orc>b;
=−π(c−a)ν−1
(b−c)νcot(νπ)f o r a<c<b
[0<Reν<1] ET II 250(32)
3./integraldisplayb
a(x−a)ν−1(b−x)μ−1
x−cdx
=(b−a)μ+ν−1
b−cB(μ, ν)2F1/parenleftbigg
1,μ;μ+ν;b−a
b−c/parenrightbigg
forc<a orc>b;
=π(c−a)ν−1(b−c)μ−1cotμπ−(b−a)μ+ν−2B(μ−1,ν)
×2F1/parenleftbigg
2−μ−ν,1;2−μ;b−c
b−a/parenrightbigg
fora<c<b
[Reμ>0,Reν>0,μ+ν/negationslash=1,μ/negationslash=1,2,...]ET II 250(33)
4./integraldisplay1
0(1−x)ν−1x−ν
a−bxdx=π(a−b)ν−1
aνcosec( νπ)[ 0 <Reν<1,0<b<a ] BI (5)(8)
5./integraldisplay∞
0xν−1(x+a)1−μ
x−cdx=a1−μ(−c)ν−1B(μ−ν,ν)2F1/parenleftBig
μ−1,ν;μ;1+c
a/parenrightBig
forc<0;
=πcν−1(a+c)1−μcot[(μ−ν)π]−a1−μ−ν
a+cB(μ−ν−1,ν)
×2F1/parenleftbigg
2−μ,1;2−μ+ν;a
a+c/parenrightbigg
forc>0
[a>0,0<Reν<Reμ]ET II 251(34)
3.237 Powers of xand binomials 321
6./integraldisplay∞
0xν−1(γ+x)−n
x+βdx=π
sinπνβν−1
(γ−β)n⎡
⎣1−/parenleftbiggγ
β/parenrightbiggν−1n−1/summationdisplay
j=0(1−ν)j
j!/parenleftbiggγ−β
γ/parenrightbiggj⎤
⎦
[|argβ|<π , |argγ|<π , 0<Reν<n]AS 256 (6.1.22)
3.229/integraldisplay1
0xμ−1dx
(1−x)μ(1 +ax)(1 + bx)=πcosecμπ
a−b/bracketleftbigga
(1 +a)μ−b
(1 +b)μ/bracketrightbigg
[0<Reμ<1] BI (5)(7)
3.231
1./integraldisplay1
0xp−1−x−p
1−xdx=πcotpπ/bracketleftbig
p2<1/bracketrightbig
BI (4)(4)
2.11/integraldisplay1
0xp−1+x−p
1+xdx=πcosecpπ/bracketleftbig
p2<1/bracketrightbig
BI (4)(1)
3./integraldisplay1
0xp−x−p
x−1dx=1
p−πcotpπ/bracketleftbig
p2<1/bracketrightbig
BI (4)(3)
4./integraldisplay1
0xp−x−p
1+xdx=1
p−πcosecpπ/bracketleftbig
p2<1/bracketrightbig
BI (4)(2)
5./integraldisplay1
0xμ−1−xν−1
1−xdx=ψ(ν)−ψ(μ)[ R e μ>0,Reν>0]
FI II 815, BI(4)(5)
6./integraldisplay∞
0xp−1−xq−1
1−xdx=π(cotpπ−cotqπ)[ p>0,q > 0] FI II 718
3.232/integraldisplay∞
0(c+ax)−μ−(c+bx)−μ
xdx=c−μlnb
a[Reμ>−1;a>0;b>0;c>0]
BI (18)(14)
3.233/integraldisplay∞
0/braceleftbigg1
1+x−(1 +x)−ν/bracerightbiggdx
x=ψ(ν)+C [Reν>0] EH I 17, WH
3.234
1.11/integraldisplay1
0/parenleftbiggxq−1
1−ax−x−q
a−x/parenrightbigg
dx=πa−qcotqπ [0<q< 1,a > 0] BI (5)(11)
2./integraldisplay1
0/parenleftbiggxq−1
1+ax+x−q
a+x/parenrightbigg
dx=πa−qcosecqπ [0<q< 1,a > 0] BI (5)(10)
3.235/integraldisplay∞
0(1 +x)μ−1
(1 +x)νdx
x=ψ(ν)−ψ(ν−μ)[ R e ν>Reμ>0] BI (18)(5)
3.23610/integraldisplay1
0xμ
2dx
[(1−x)(1−a2x)]μ+1
2=(1−a)−μ−(1 +a)−μ
2aμ√πΓ/parenleftBig
1+μ
2/parenrightBig
Γ/parenleftbigg1−μ
2/parenrightbigg
[−2<μ< 1,|a|<1] BI (12)(32)
3.237∞/summationdisplay
n=0(−1)n+1/integraldisplayn+1
ndx
x+u=l nu/bracketleftbig
Γ/parenleftbigu
2/parenrightbig/bracketrightbig2
2/bracketleftbigg
Γ/parenleftbiggu+1
2/parenrightbigg/bracketrightbigg2[|argu|<π] ET II 216(1)
322 Power and Algebraic Functions 3.238
3.238
1./integraldisplay∞
−∞|x|ν−1
x−udx=−πcotνπ
2|u|ν−1signu [0<Reν<1ureal,u/negationslash=0 ]
ET II 249(29)
2./integraldisplay∞
−∞|x|ν−1
x−usignxdx=πtanνπ
2|u|ν−1[0<Reν<1ureal,u/negationslash=0 ]
ET II 249(30)
3./integraldisplayb
a(b−x)μ−1(x−a)ν−1
|x−u|μ+νdx=(b−a)μ+ν−1
|a−u|μ|b−u|νΓ(μ)Γ(ν)
Γ(μ+ν)
[Reμ>0,Reν>0,0<u<a<b and 0 <a<b<u ]MO 7
3.24–3.27 Powers of x, of binomials of the form α+βxpand of polynomials in x
3.241
1./integraldisplay1
0xμ−1dx
1+xp=1
pβ/parenleftbiggμ
p/parenrightbigg
[Reμ>0,p > 0] WH, BI (2)(13)
2./integraldisplay∞
0xμ−1dx
1+xν=π
νcosecμπ
ν=1
νB/parenleftbiggμ
ν,ν−μ
ν/parenrightbigg
[Reν>Reμ>0]
ET I 309(15)a, BI (17)(10)
3.11PV/integraldisplay∞
0xp−1dx
1−xq=π
qcotpπ
q[p<q] BI (17)(11)
4.11/integraldisplay∞
0xμ−1dx
(p+qxν)n+1=1
νpn+1/parenleftbiggp
q/parenrightbiggμ/νΓ/parenleftbigμ
ν/parenrightbig
Γ/parenleftbig
1+n−μ
ν/parenrightbig
Γ(1 + n)/bracketleftBig
0<μ
ν<n+1,p/negationslash=0,q/negationslash=0/bracketrightBig
BI (17)(22)a
5./integraldisplay∞
0xp−1dx
(1 +xq)2=(p−q)π
q2cosec(p−q)π
q[p<2q] BI (17)(18)
6.10G(x)=/integraldisplayb
asign/bracketleftbiggx
c−/parenleftbiggb−u
b−a/parenrightbiggp/bracketrightbigg
du=(b−a)F/bracketleftbigg/parenleftBigx
c/parenrightBig1/p/bracketrightbigg
where
F(x)=/integraldisplay1
0sign(x−t)dt=⎧
⎪⎨
⎪⎩−1 x≤0
2x−10<x< 1
1 x≥1
3.242
1./integraldisplay∞
−∞x2mdx
x4n+2x2ncost+1=π
nsin/bracketleftbigg(2n−2m−1)
2nt/bracketrightbigg
cosectcosec(2m+1 )π
2n
/bracketleftbig
m<n , t2<π2/bracketrightbig
FI II 642
3.247 Powers of xand binomials and polynomials 323
2.11/integraldisplay∞
0/bracketleftbiggx2
x4+2ax2+1/bracketrightbiggc/parenleftbiggx2+1
xb+1/parenrightbiggdx
x2=2−1/2−c(1 +a)1/2−cB/parenleftbigg
c−1
2,1
2/parenrightbigg
3.24311/integraldisplay∞
0xμ−1dx
(1 +x2ν)(1+ x3ν)
=π
48ν/bracketleftbigg
8 cosec(2 ρ) + 12 cosec(3 ρ)−8c os e c/parenleftbigg
2ρ−4π
3/parenrightbigg
+ 8 cosec/parenleftbigg
2ρ−2π
3/parenrightbigg
−3c os e c/parenleftBig
ρ−π
6/parenrightBig
cosec/parenleftBig
ρ+π
6/parenrightBig
sec(ρ)/bracketrightBig
where ρ=μπ
6ν,[0<Reμ<5R eν]ET I 312(34)
3.244
1./integraldisplay1
0xp−1+xq−p−1
1+xqdx=π
qcosecpπ
q[q>p> 0] BI (2)(14)
2./integraldisplay1
0xp−1−xq−p−1
1−xqdx=π
qcotpπ
q[q>p> 0] BI (2)(16)
3./integraldisplay1
0xν−1−xμ−1
1−xνdx=1
ν/bracketleftBig
C+ψ/parenleftBigμ
ν/parenrightBig/bracketrightBig
[Reμ>Reν>0] BI (2)(17)
4./integraldisplay∞
−∞x2m−x2n
1−x2ldx=π
l/bracketleftbigg
cot/parenleftbigg2m+1
2lπ/parenrightbigg
−cot/parenleftbigg2n+1
2lπ/parenrightbigg/bracketrightbigg
[m<l , n<l ] FI II 640
3.245/integraldisplay∞
0/bracketleftbig
xν−μ−xν(1 +x)−μ/bracketrightbig
dx=ν
ν−μ+1B(ν,μ−ν)
[Reμ>Reν>0] BI (16)(13)
3.246/integraldisplay∞
01−xq
1−xrxp−1dx=π
rsinqπ
rcosecpπ
rcosec(p+q)π
r
[p+q<r , p> 0]
ET I 331(33), BI (17)(12)
Integrals of the form/integraldisplay
f/parenleftbig
xp±x−p,xq±x−q,.../parenrightbigdx
xcan be transformed by the substitution x=et
orx=e−t. For example, instead of/integraldisplay1
0/parenleftbig
x1+p+x1−p/parenrightbig−1dx, we should seek to evaluate/integraldisplay∞
0sechpxdx
and, instead of/integraldisplay1
0xn−m−1+xn+m−1
1+2xncosa+x2ndx, we should seek to evaluate/integraldisplay∞
0coshmx(coshnx−cosa)−1dx
(see3.514 2).
3.247
1.11/integraldisplay1
0xα−1(1−x)n−1
1−ξxbdx=(n−1)!∞/summationdisplay
k=0ξk
(α+kb)(α+kb+1 )...(α+kb+n−1)
[b>0,|ξ|<1] AD (6704)
2./integraldisplay∞
0(1−xp)xν−1
1−xnpdx=π
npsin/parenleftBigπ
n/parenrightBig
cosec(p+ν)π
npcosecπν
np
[0<Reν<(n−1)p] ET I 311(33)
324 Power and Algebraic Functions 3.248
3.248
1./integraldisplay∞
0xμ−1dx√1+xν=1
νB/parenleftbiggμ
ν,1
2−μ
ν/parenrightbigg
[Reν>Re 2μ>0] BI (21)(9)
2./integraldisplay1
0x2n+1dx√
1−x2=(2n)!!
(2n+1 ) ! !BI (8)(14)
3./integraldisplay1
0x2ndx√
1−x2=(2n−1)!!
(2n)!!π
2BI (8)(13)
4.3/integraldisplay∞
−∞dx
(1 +x2)√
4+3x2=π
3
6.∗/integraldisplay∞
−∞dx
/parenleftbig
1+x2/parenrightbig2/radicalbig
b+ax2=⎧
⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩2
√
b−aarctan/parenleftBigg/radicalbigg
b
a−1/parenrightBigg
ifa<b
2√aifa=b
1√
a−bln/parenleftbigg√a+√
a−b√a−√
a−b/parenrightbigg
ifa>b
3.249
1.0/integraldisplay∞
0dx
(x2+a2)n=(2n−3)!!
2·(2n−2)!!π
a2n−1FI II 743
2.9/integraldisplaya
0/parenleftbig
a2−x2/parenrightbign−1
2dx=a2n(2n−1)!!
2(2n)!!π. FI II 156
3./integraldisplay1
−1/parenleftbig
1−x2/parenrightbigndx
(a−x)n+1=2n+1Qn(a) EH II 181(31)
4./integraldisplay1
0xμdx
1+x2=1
2β/parenleftbiggμ+1
2/parenrightbigg
[Reμ>−1] BI (2)(7)
5./integraldisplay1
0/parenleftbig
1−x2/parenrightbigμ−1dx=22μ−2B(μ, μ)=1
2B/parenleftbig1
2,μ/parenrightbig
[Reμ>0] FI II 784
6./integraldisplay1
0/parenleftbig
1−√x/parenrightbigp−1dx=2
p(p+1 )[p>0] BI (7)(7)
7./integraldisplay1
0(1−xμ)−1
νdx=1
μB/parenleftbigg1
μ,1−1
ν/parenrightbigg
[Reμ>0,|ν|>1]
8.11/integraldisplay∞
−∞/parenleftbigg
1+x2
n−1/parenrightbigg−n/2
dx=/radicalbig
π(n−1)
Γ/parenleftbign
2/parenrightbigΓ/parenleftbiggn−1
2/parenrightbigg
[n>1]
3.251
1./integraldisplay1
0xμ−1/parenleftbig
1−xλ/parenrightbigν−1dx=1
λB/parenleftBigμ
λ,ν/parenrightBig
[Reμ>0,Reν>0,λ > 0]
FI II 787
2./integraldisplay∞
0xμ−1/parenleftbig
1+x2/parenrightbigν−1dx=1
2B/parenleftBigμ
2,1−ν−μ
2/parenrightBig/bracketleftbig
Reμ>0,Re/parenleftbig
ν+1
2μ/parenrightbig
<1/bracketrightbig
3.252 Powers of xand binomials and polynomials 325
3./integraldisplay∞
1xμ−1(xp−1)ν−1dx=1
pB/parenleftbigg
1−ν−μ
p,ν/parenrightbigg
[p>0,Reν>0,Reμ<p−pReν]
ET I 311(32)
4./integraldisplay∞
0x2mdx
(ax2+c)n=(2m−1)!!(2n−2m−3)!!π
2·(2n−2)!!amcn−m−1√ac[a>0,c > 0,n > m +1 ]
GU (141)(8a)
5./integraldisplay∞
0x2m+1dx
(ax2+c)n=m!(n−m−2)!
2(n−1)!am+1cn−m−1[ac >0,n > m +1≥1]GU (141)(8b)
6./integraldisplay∞
0xμ+1
(1 +x2)2dx=μπ
4s inμπ
2[−2<Reμ<2] WH
7./integraldisplay1
0xμdx
(1 +x2)2=−1
4+μ−1
4β/parenleftbiggμ−1
2/parenrightbigg
[Reμ>1] LI (3)(11)
8./integraldisplay1
0xq+p−1(1−xq)−p
qdx=pπ
q2cosecpπ
q[q>p] BI (9)(22)
9./integraldisplay1
0xq
p−1(1−xq)−1
pdx=π
qcosecπ
p[p>1,q > 0] BI (9)(23)a
10./integraldisplay1
0xp−1(1−xq)−p
qdx=π
qcosecpπ
q[q>p> 0] BI (9)(20)
11./integraldisplay∞
0xμ−1(1 +βxp)−νdx=1
pβ−μ
pB/parenleftbiggμ
p,ν−μ
p/parenrightbigg
[|argβ|<π , p> 0,0<Reμ<p Reν]BI (17)(20), EH I 10(16)
3.252
1./integraldisplay∞
0dx
(ax2+2bx+c)n=(−1)n−1
(n−1)!∂n−1
∂cn−1/bracketleftbigg1√
ac−b2arccotb√
ac−b2/bracketrightbigg
/bracketleftbig
a>0,a c > b2/bracketrightbig
GW (131)(4)
2./integraldisplay∞
−∞dx
(ax2+2bx+c)n=(2n−3)!!πan−1
(2n−2)!!(ac−b2)n−1
2/bracketleftbig
a>0,a c > b2/bracketrightbig
GW (131)(5)
3./integraldisplay∞
0dx
(ax2+2bx+c)n+3
2=(−2)n
(2n+1 ) ! !∂n
∂cn/braceleftbigg1√c(√ac+b)/bracerightbigg
/bracketleftbig
a≥0,c > 0,b > −√ac/bracketrightbig
GW (213)(4)
326 Power and Algebraic Functions 3.252
4./integraldisplay∞
0xdx
(ax2+2bx+c)n
=(−1)n
(n−1)!∂n−2
∂cn−2/braceleftBigg
1
2(ac−b2)−b
2(ac−b2)3
2arccotb√
ac−b2/bracerightBigg
forac > b2;
=(−1)n
(n−1)!∂n−2
∂cn−2/braceleftBigg
1
2(ac−b2)+b
4(b2−ac)3
2lnb+√
b2−ac
b−√
b2−ac/bracerightBigg
forb2>a c> 0;
=an−2
2(n−1)(2n−1)b2n−2forac=b2
[a>0,b > 0,n≥2] GW (141)(5)
5./integraldisplay∞
−∞xdx
(ax2+2bx+c)n=−(2n−3)!!πban−2
(2n−2)!!(ac−b2)(2n−1)
2/bracketleftbig
ac > b2,a > 0,n≥2/bracketrightbig
GW (141)(6)
6./integraldisplay∞
−∞xmdx
(ax2+2bx+c)n=(−1)mπan−m−1bm
(2n−2)!!(ac−b2)n−1
2
×[m/2]/summationdisplay
k=0/parenleftBigm
2k/parenrightBig
(2k−1)!!(2n−2k−3)!!/parenleftbiggac−b2
b2/parenrightbiggk
/bracketleftbig
ac > b2,0≤m≤2n−2/bracketrightbig
GW (141)(17)
7./integraldisplay∞
0xndx
(ax2+2bx+c)n+3
2=n!
(2n+1 ) ! !√c(√ac+b)n+1
/bracketleftbig
a≥0,c > 0,b > −√ac/bracketrightbig
GW (213)(5a)
8./integraldisplay∞
0xn+1dx
(ax2+2bx+c)n+3
2=n!
(2n+1 ) ! !√a(√ac+b)n+1
/bracketleftbig
a>0,c≥0,b > −√ac/bracketrightbig
GW (213)(5b)
9./integraldisplay∞
0xn+1
2dx
(ax2+2bx+c)n+1=(2n−1)!!π
22n+1
2(b+√ac)n+1
2n!√a/bracketleftbig
a>0,c > 0,b+√ac >0/bracketrightbig
LI (21)(19)
10.6/integraldisplay∞
0xμ−1dx
(1 + 2 xcost+x2)ν=2ν−1
2(sint)1
2−νtΓ/parenleftbigg
ν+1
2/parenrightbigg
B(μ,2ν−μ)P1
2−ν
μ−ν−1
2(cost)
[0<t<π , 0<Reμ<Re 2ν]
ET I 310(22)
3.255 Powers of xand binomials and polynomials 327
11./integraldisplay∞
0/parenleftbig
1+2βx+x2/parenrightbigμ−1
2x−ν−1dx=2−μ/parenleftbig
β2−1/parenrightbigμ
2Γ(1−μ)B(ν−2μ+1,−ν)Pμ
ν−μ(β)
[Reν<0,Re(2μ−ν)<1,|arg (β±1)|<π]
EH I 160(33)
=−πcosecνπC1
2−μ
ν(β)
/bracketleftbig
−2<Re/parenleftbig1
2−μ/parenrightbig
<Reν<0,|arg (β±1)|<π/bracketrightbig
EH I 178(24)
12./integraldisplay∞
0xμ−1dx
x2+2axcost+a2=−πaμ−2cosectcosec( μπ)sin[(μ−1)t]
[a>0,0<|t|<π , 0<Reμ<2]
FI II 738, BI(20)(3)
13./integraldisplay∞
0xμ−1dx
(x2+2axcost+a2)2=πaμ−4
2cosecμπcosec3t
×{(μ−1)sintcos[(μ−2)t]−sin[(μ−1)t]}
[a>0,0<|t|<π , 0<Reμ<4]LI(20)(8)a, ET I 309(13)
14./integraldisplay∞
0xμ−1dx√
1+2xcost+x2=πcosec( μπ)Pμ−1(cost)[ −π<t<π , 0<Reμ<1]
ET I 310(17)
3.253/integraldisplay1
−1(1 +x)2μ−1(1−x)2ν−1
(1 +x2)μ+νdx=2μ+ν−2B(μ, ν)[ R e μ>0,Reν>0] FI II 787
3.254
1./integraldisplayu
0xλ−1(u−x)μ−1/parenleftbig
x2+β2/parenrightbigνdx
=β2νuλ+μ−1B(λ,μ)3F2/parenleftbigg
−ν,λ
2,λ+1
2;λ+μ
2,λ+μ+1
2;−u2
β2/parenrightbigg
/bracketleftbigg
Re/parenleftbiggu
β/parenrightbigg
>0,λ > 0,Reμ>0/bracketrightbigg
ET II 186(10)
2.6/integraldisplay∞
u/parenleftbig
x−λ(x−u)μ−1/parenleftbig
x2+β2/parenrightbig/parenrightbigνdx
=uμ−λ+2νΓ(μ)Γ(λ−μ−2ν)
Γ(λ−2ν)
×3F2/parenleftbigg
−ν,λ−μ
2−ν,1+λ−μ
2−ν;λ
2−ν,1+λ
2−ν;−β2
u2/parenrightbigg
/bracketleftbigg
|u|>|β|and Re/parenleftbiggβ
u/parenrightbigg
>0,0<Reμ<Re(λ−2ν)/bracketrightbigg
ET II 202(9)
3.255/integraldisplay1
0xμ+1
2(1−x)μ−1
2
(c+2bx−ax2)μ+1dx=√π
/braceleftBig
a+/parenleftbig√
c+2b−a+√c/parenrightbig2/bracerightBigμ+1
2√
c+2b−aΓ/parenleftbig
μ+1
2/parenrightbig
Γ(μ+1 )
/bracketleftbigg
a+/parenleftBig√
c+2b−a+√c/parenrightBig2
>0,c+2b−a>0,Reμ>−1
2/bracketrightbigg
BI (14)(2)
328 Power and Algebraic Functions 3.256
3.256
1./integraldisplay1
0xp−1+xq−1
(1−x2)p+q
2dx=1
2cos/parenleftbiggq−p
4π/parenrightbigg
sec/parenleftbiggq+p
4π/parenrightbigg
B/parenleftBigp
2,q
2/parenrightBig
[p>0,q > 0,p+q<2] BI (8)(25)
2./integraldisplay1
0xp−1−xq−1
(1−x2)p+q
2dx=1
2sin/parenleftbiggq−p
4π/parenrightbigg
cosec/parenleftbiggq+p
4π/parenrightbigg
B/parenleftBigp
2,q
2/parenrightBig
[p>0,q > 0,p+q<2] BI (8)(26)
3.2579/integraldisplay∞
0/bracketleftBigg/parenleftbigg
ax+b
x/parenrightbigg2
+c/bracketrightBigg−p−1
dx
=√πΓ/parenleftbig
p+1
2/parenrightbig
2acp+1
2Γ(p+1 )/bracketleftbig
a>0,b < 0,c > 0,p > −1
2/bracketrightbig
BI (20)(4)
=1
2B/parenleftbig
p+1
2,1
2/parenrightbig
a(4ab+x)p+1
2/bracketleftbig
a>0,b > 0,c > −4ab, p > −1
2/bracketrightbig
3.258
1./integraldisplay∞
b/parenleftBig
x−/radicalbig
x2−a2/parenrightBign
dx=a2
2(n−1)/parenleftBig
b−/radicalbig
b2−a2/parenrightBign−1
−1
2(n+1 )/parenleftBig
b−/radicalbig
b2−a2/parenrightBign+1
[0<a≤b, n ≥2] GW (215)(5)
2./integraldisplay∞
b/parenleftBig/radicalbig
x2+1−x/parenrightBign
dx=/parenleftbig√
b2+1−b/parenrightbign−1
2(n−1)+/parenleftbig√
b2+1−b/parenrightbign+1
2(n+1 )
[n≥2] GW (214)(7)
3./integraldisplay∞
0/parenleftBig/radicalbig
x2+a2−x/parenrightBign
dx=nan+1
n2−1[n≥2] GW (214)(6a)
4./integraldisplay∞
0dx/parenleftbig
x+√
x2+a2/parenrightbign=n
an−1(n2−1)[n≥2] GW (214)(5a)
5./integraldisplay∞
0xm/parenleftBig/radicalbig
x2+a2−x/parenrightBign
dx=n·m!am+n+1
(n−m−1)(n−m+1 )...(m+n+1 )
[a>0,0≤m≤n−2] GW (214)(6)
6./integraldisplay∞
0xmdx/parenleftbig
x+√
x2+a2/parenrightbign=n·m!
(n−m−1)(n−m+1 )...(m+n+1 )an−m−1
[a>0,0≤m≤n−2] GW (214)(5)
7./integraldisplay∞
a(x−a)m/parenleftBig
x−/radicalbig
x2−a2/parenrightBign
dx=n·(n−m−2)!(2m+1 ) !am+n+1
2m(n+m+1 ) !
[a>0,n≥m+2 ] GH (215)(6)
3.264 Powers of xand binomials and polynomials 329
3.259
1.6/integraldisplay1
0xp−1(1−x)n−1(1 +bxm)ldx=(n−1)!∞/summationdisplay
k=0/parenleftbiggl
k/parenrightbiggbkΓ(p+km)
Γ(p+n+km)
[|b|<1 unless l=0,1,2,...;p, n, p +ml > 0]BI (1)(14)
2.11/integraldisplayu
0xν−1(u−x)μ−1(xm+βm)λdx
=βmλuμ+ν−1B(μ, ν)
×m+1Fm/parenleftbigg
−λ,ν
m,ν+1
m,...,ν+m−1
m;μ+ν
m,μ+ν+1
m,...,μ+ν+m−1
m;−um
βm/parenrightbigg
/bracketleftbigg
Reμ>0,Reν>0,/vextendsingle/vextendsingle/vextendsingle/vextendsinglearg/parenleftbiggu
β/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle<π
m/bracketrightbigg
ET II 186(11)
3.11/integraldisplay∞
0xλ−1(1 +αxp)−μ(1 +βxp)−νdx=1
pα−λ/pB/parenleftbiggλ
p,μ+ν−λ
p/parenrightbigg
2F1/parenleftbigg
ν,λ
p;μ+ν;1−β
α/parenrightbigg
[|argα|<π , |argβ|<π , p> 0,0<Reλ<2R e(μ+ν)]ET I 312(35)
3.261
1.11PV/integraldisplay1
0(1−xcost)xμ−1dx
1−2xcost+x2=∞/summationdisplay
k=0coskt
μ+k[Reμ>0,t/negationslash=2nπ] BI (6)(9)
2./integraldisplay1
0(xν+x−ν)dx
1+2xcost+x2=πsinνt
sintsinνπ/bracketleftbig
ν2<1,t/negationslash=( 2n+1 )π/bracketrightbig
BI (6)(8)
3./integraldisplay1
0/parenleftbig
x1+p+x1−p/parenrightbig
dx
(1 + 2 xcost+x2)2=π(psintcospt−costsinpt)
2s in3tsinpπ
/bracketleftbig
p2<1,t/negationslash=( 2n+1 )π/bracketrightbig
BI (6)(18)
4./integraldisplay1
0xμ−1
1+2axcost+a2x2·dx
(1−x)μ=πcosectcosecμπ
(1 + 2 acost+a2)μ
2sin/parenleftbigg
t−μarctanasint
1+acost/parenrightbigg
[a>0,0<Reμ<1] BI (6)(21)
3.262/integraldisplay∞
0x−pdx
1+x3=π
3cosec(1−p)π
3[−2<p< 1] LI (18)(3)
3.263/integraldisplay∞
0xνdx
(x+γ)(x2+β2)=π
2(β2+γ2)/bracketleftBig
γβν−1secνπ
2+βνcosecνπ
2−2γνcosec( νπ)/bracketrightBig
[Reβ>0,|argγ|<π , −1<Reν<2,ν/negationslash=0 ] ET II 216(7)
3.264
1./integraldisplay∞
0xp−1dx
(a2+x2)(b2−x2)=π
2ap−2+bp−2cospπ
2
a2+b2cosecpπ
2
[0<p< 4,a > 0,b > 0]
BI (19)(14)
330 Power and Algebraic Functions 3.265
2./integraldisplay∞
0xμ−1dx
(β+x2)(γ+x2)=π
2γμ
2−1−βμ
2−1
β−γcosecμπ
2
=π
2(γ−β)/parenleftbigg1√β−1√γ/parenrightbigg/bracketleftbig
μ=1
2/bracketrightbig
[|argβ|<π , |argγ|<π , 0<Reμ<4]ET I 309(4)
3./integraldisplay∞
0dx
(b+x2)(a+b+x2)2=π
2/parenleftBigg
1
a2b1/2−1
2a(a+b)3/2−1
a2(a+b)1/2/parenrightBigg
MC
4./integraldisplay∞
0dx
(b+x2)(a+b+x2)3=π
4/parenleftBigg
2
a3b1/2−3
4a(a+b)5/2−1
a2(a+b)3/2−2
a3(a+b)1/2/parenrightBigg
5./integraldisplay∞
0dx
(b+x2)(a+b+x2)4=π
4⎛
⎝2
a4b1/2−5
8a(a+b)7/2−3
4a2(a+b)5/2
−1
a3(a+b)3/2−2
a4(a+b)1/2⎞
⎠
6./integraldisplay∞
0dx
(b+x2)(a+b+x2)n
=π
21
anb1/2−1
2a(a+b)n−1/2B/parenleftbigg
n−1
2,1
2/parenrightbigg
2F1/parenleftbigg
1−n,1;3
2−n;a+b
a/parenrightbigg
AS 263 (6.6.3.2)
=π
21
anb1/2−π
2an(a+b)n−1/2n−1/summationdisplay
j=0/parenleftbig1
2/parenrightbig
j
j!/parenleftbigga
a+b/parenrightbiggj
[n>0,a+b>0]
7./integraldisplay∞
0x2dx
(x2+α2)(x2+β2)(x2+γ2)=π
2α(β2−γ2)/bracketleftbiggβ
β+α−γ
γ+α/bracketrightbigg
=π
2(α+β)(α+γ)(β+γ)
3.265/integraldisplay1
01−xμ−1
1−xdx=ψ(μ)+C [Reμ>0] FI II 796, WH, ET I 16(13)
=ψ(1−μ)+C−πcot(μπ)[ R e μ>0] EH I 16(15)a
3.266/integraldisplay∞
0(xν−aν)dx
(x−a)(β+x)=π
a+β/braceleftbigg
βνcosec( νπ)−aνcot(νπ)−aν
πlnβ
a/bracerightbigg
[|argβ|<π , |Reν|<1,ν/negationslash=0 ]
ET II 216(8)
3.267
1./integraldisplay1
0x3ndx
3√
1−x3=2π
3√
3Γ/parenleftbig
n+1
3/parenrightbig
Γ/parenleftbig1
3/parenrightbig
Γ(n+1 )BI (9)(6)
2./integraldisplay1
0x3n−1dx
3√
1−x3=(n−1)! Γ/parenleftbig2
3/parenrightbig
3Γ/parenleftbig
n+2
3/parenrightbig BI (9)(7)
3.272 Powers of xand binomials and polynomials 331
3.∗/integraldisplay1
0x3n−2dx
3√
1−x3=Γ/parenleftbig
n−1
3/parenrightbig
Γ/parenleftbig2
3/parenrightbig
3Γ/parenleftbig
n+1
3/parenrightbig
3.268
1./integraldisplay1
0/parenleftbigg1
1−x−pxp−1
1−xp/parenrightbigg
dx=l np BI (5)(14)
2./integraldisplay1
01−xμ
1−xxν−1dx=ψ(μ+ν)−ψ(ν)[ R e ν>0,Reμ>0] BI (2)(3)
3./integraldisplay1
0/bracketleftbiggn
1−x−xμ−1
1−n√x/bracketrightbigg
dx=nC+n/summationdisplay
k=1ψ/parenleftbigg
μ+n−k
n/parenrightbigg
[Reμ>0] BI (13)(10)
3.269
1./integraldisplay1
0xp−x−p
1−x2xdx=π
2cotpπ
2−1
p/bracketleftbig
p2<1/bracketrightbig
BI (4)(12)
2./integraldisplay1
0xp−x−p
1+x2xdx=1
p−π
2cosecpπ
2/bracketleftbig
p2<1/bracketrightbig
BI (4)(8)
3./integraldisplay1
0xμ−xν
1−x2dx=1
2ψ/parenleftbiggν+1
2/parenrightbigg
−1
2ψ/parenleftbiggμ+1
2/parenrightbigg
[Reμ>−1,Reν>−1] BI (2)(9)
3.271
1./integraldisplay∞
0xp−xq
x−1dx
x+a=π
1+a/parenleftbiggap−cospπ
sinpπ−aq−cosqπ
sinqπ/parenrightbigg
/bracketleftbig
p2<1,q2<1,a > 0/bracketrightbig
BI (19)(2)
2./integraldisplay∞
0xp−ap
x−axp−1
x−1dx=π
a−1/braceleftbigga2p−1
sin(2pπ)−1
πaplna/bracerightbigg
/bracketleftbigg
p2<1
4/bracketrightbigg
BI (19)(3)
3./integraldisplay∞
0xp−ap
x−ax−p−1
x−1dx=π
a−1/braceleftbigg
2(ap−1)cot pπ−1
π(ap+1 )l n a/bracerightbigg
/bracketleftbig
p2<1/bracketrightbig
BI (18)(9)
4./integraldisplay∞
0xp−ap
x−a1−x−p
1−xxqdx=π
a−1/braceleftbiggap+q−1
sin[(p+q)π]+ap−aq
sin[(q−p)π]/bracerightbiggsinpπ
sinqπ/bracketleftbig
(p+q)2<1,(p−q)2<1/bracketrightbig
BI (19)(4)
5./integraldisplay∞
0/parenleftbiggxp−x−p
1−x/parenrightbigg2
dx=2( 1 −2pπcot 2pπ)/bracketleftbig
0<p2<1
4/bracketrightbig
BI (16)(3)
3.272
1./integraldisplay1
0xn−1+xn−1
2−2x2n−1
1−xdx=2l n2 BI (8)(8)
332 Power and Algebraic Functions 3.273
2./integraldisplay1
0xn−1+xn−2
3+xn−1
3−3x3n−1
1−xdx=3l n3 BI (8)(9)
3.273
1./integraldisplay1
0sint−anxnsin[(n+1 )t]+an+1xn+1sinnt
1−2axcost+a2x2(1−x)p−1dx=Γ (p)n/summationdisplay
k=1(k−1)!ak−1sinkt
Γ(p+k)
[p>0] BI (6)(13)
2./integraldisplay1
0cost−ax−anxncos[(n+1 )t]+an+1xn+1cosnt
1−2axcost+a2x2(1−x)p−1dx=Γ (p)n/summationdisplay
k=1(k−1)!ak−1coskt
Γ(p+k)
[p>0] BI (6)(14)
3./integraldisplay1
0xsint−xnsin[(n+1 )t]+xn+1sinnt
1−2xcost+x2dx=n/summationdisplay
k=1sinkt
k+1BI (6)(12)
4./integraldisplay1
01−xcost−xn+1cos[(n+1 )t]+xn+2cosnt
1−2xcost+x2dx=n/summationdisplay
k=0coskt
k+1BI (6)(11)
3.274
1./integraldisplay∞
0xμ−1(1−x)
1−xndx=π
nsinπ
ncosecμπ
ncosec(μ+1 )π
n
[0<Reμ<n −1] BI (20)(13)
2./integraldisplay1
01−xn
(1 +x)n+1dx
1−x=1
2n+1n/summationdisplay
k=12k
kBI (5)(3)
3./integraldisplay∞
0xq−1
xp−x−pdx
x=π
2ptanqπ
2p[p>q] BI (18)(6)
3.275
1./integraldisplay1
0/parenleftbiggxn−1
1−x1/p−pxnp−1
1−x/parenrightbigg
dx=plnp [p>0] BI (13)(9)
2./integraldisplay1
0/parenleftbiggnxn−1
1−xn−xmn−1
1−x/parenrightbigg
dx=C+1
nn/summationdisplay
k=1ψ/parenleftbigg
m+n−k
n/parenrightbigg
BI (5)(13)
3./integraldisplay1
0/parenleftbiggxp−1
1−x−qxpq−1
1−xq/parenrightbigg
dx=l nq [q>0] BI (5)(12)
4./integraldisplay∞
0/parenleftbigg1
1+x2n−1
1+x2m/parenrightbiggdx
x=0. BI (18)(17)
3.276
1.10/integraldisplay∞
0/bracketleftBigg/parenleftbigg
ax+b
x/parenrightbigg2
+c/bracketrightBigg−p−1
dx
x2=1
2|b|B/parenleftbig
p+1
2,1
2/parenrightbig
(2a(b+|b|)+c)p+1
2
/bracketleftbig
a>0,c > −4ac, p > −1
2/bracketrightbig
3.278 Powers of xand binomials and polynomials 333
2.10/integraldisplay∞
0/parenleftbigg
a+b
x2/parenrightbigg/bracketleftBigg/parenleftbigg
ax+b
x/parenrightbigg2
+c/bracketrightBigg−p−1
dx=B/parenleftbig
p+1
2,1
2/parenrightbig
(4ab+c)p+1
2
/bracketleftbig
a>0,b > 0,c > −4ac, p > −1
2/bracketrightbig
3.277
1.11/integraldisplay∞
0xμ−1/bracketleftbig√
1+x2+β/bracketrightbigν
√
1+x2dx=2μ
2−1/parenleftbig
β2−1/parenrightbigν
2+μ
4Γ/parenleftBigμ
2/parenrightBig
Γ(1−μ−ν)Pν+μ
2μ
2−1(β)
[Reβ>−1,0<Reμ<1−Reν]
ET I 310(25)
2./integraldisplay∞
0xμ−1/bracketleftBig/radicalbig
β2+x2+x/bracketrightBigν
/radicalbig
β2+x2dx=βμ+ν−1
2μB/parenleftbigg
μ,1−μ−ν
2/parenrightbigg
[Reβ>0,0<Reμ<1−Reν]
ET I 311(28)
3./integraldisplay∞
0xμ−1/bracketleftbig
cost±isint√
1+x2/bracketrightbigν
√
1+x2dx=2μ−1
2sin1−μ
2tΓ/parenleftbigμ
2/parenrightbig
Γ(1−μ−ν)
Γ(−ν)
×/bracketleftbigg
π−1
2Qμ+1
2
−μ+1
2−ν(cost)∓i
2π1
2P−μ+1
2−ν
μ−1
2(cost)/bracketrightbigg
[Reμ>0] ET I 311 (27)
4./integraldisplay∞
0xμ−1/bracketleftBig/radicalbig
(β2−1)(x2+1 )+ β/bracketrightBigν
√
x2+1dx
=2μ−1
2√πe−1
2iπ(μ−1)Γ/parenleftbigμ
2/parenrightbig
Γ(1−μ−ν)
Γ(−ν)/parenleftbig
β2−1/parenrightbig1−μ
4Qμ−1
2
−μ+1
2−ν(β)
[Reβ>1,Reν<0,Reμ<1−Reν]ET I 311(26)
5./integraldisplay∞
u(x−u)μ−1/parenleftbig√x+1−√x−1/parenrightbig2ν
√
x2−1dx=2ν+1
2√πe(μ−1
2)πi/parenleftbig
u2−1/parenrightbig2μ−1
4Q1
2−μ
ν−1
2(u)
[|arg(u−1)|<π , 0<Reμ<1+R e ν]ET II 202(10)
6./integraldisplay∞
1xμ−1/bracketleftBig/parenleftbig
x−√
x2−1/parenrightbigν+/parenleftbig
x−√
x2−1/parenrightbig−ν/bracketrightBig
√
x2−1dx=2−μB/parenleftbigg1−μ+ν
2,1−μ−ν
2/parenrightbigg
[Reμ<1+R e ν] ET I 311(29)
7./integraldisplayu
0(u−x)μ−1/bracketleftBig/parenleftbig√x+2+√x/parenrightbig2ν+/parenleftbig√x+2−√x/parenrightbig2ν/bracketrightBig
/radicalbig
x(x+2 )dx=22μ+1
2/radicalBig
π[u(u+2 ) ]μ−1
2P1
2−μ
ν−1
2(u+
1)
[|argu|<π , Reμ>0] ET II 186(12)
3.2788
1./integraldisplay∞
0/parenleftbiggxp
1+x2p/parenrightbiggqdx
1−x2=0 [ pq >1]
334 Exponential Functions 3.310
3.3–3.4 Exponential Functions
3.31 Exponential functions
3.31011/integraldisplay∞
0e−pxdx=1
p[Rep>0]
3.311
1./integraldisplay∞
0dx
1+epx=ln 2
pLO III 284a
2./integraldisplay∞
0e−μx
1+e−xdx=β(μ)[ R e μ>0] EH I 20(3), ET I 144(7)
3.11/integraldisplay∞
−∞e−px
1+e−qxdx=π
|q|cosecpπ
q
[q>p> 0o r0 >p>q ] (cf. 3.241 2)BI (28)(7)
4./integraldisplay∞
0e−qxdx
1−ae−px=∞/summationdisplay
k=0ak
q+kp[0<a< 1] BI (27)(7)
5./integraldisplay∞
01−eνx
ex−1dx=ψ(ν)+C+πcot(πν)[ R e ν<1] (cf. 3.265 )EH I 16(16)
6./integraldisplay∞
0e−x−e−νx
1−e−xdx=ψ(ν)+C [Reν>0] WH, EH I 16(14)
7./integraldisplay∞
0e−μx−e−νx
1−e−xdx=ψ(ν)−ψ(μ)[ R e μ>0,Reν>0] (cf. 3.231 5)
BI (27)(8)
8./integraldisplay∞
−∞e−μxdx
b−e−x=πbμ−1cot(μπ)[ b>0,0<Reμ<1] ET I 120(14)a
9./integraldisplay∞
−∞e−μxdx
b+e−x=πbμ−1cosec( μπ)[ |argb|<π , 0<Reμ<1]
ET I 120(15)a
10.11/integraldisplay∞
0e−px−e−qx
1−e−(p+q)xdx=π
p+qcotpπ
p+q[p>0,q > 0] GW (311)(16c)
11./integraldisplay∞
0epx−eqx
erx−esxdx=1
r−s/bracketleftbigg
ψ/parenleftbiggr−q
r−s/parenrightbigg
−ψ/parenleftbiggr−p
r−s/parenrightbigg/bracketrightbigg
[r>s ,r>p ,r>q ] GW (311)(16)
12./integraldisplay∞
0ax−bx
cx−dxdx=1
lnc
d/bracketleftbigg
ψ/parenleftbigglnc
b
lnc
d/parenrightbigg
−ψ/parenleftbigglnc
a
lnc
d/parenrightbigg/bracketrightbigg
[c>a> 0,b > 0,d > 0]
GW (311)(16a)
13.∗/integraldisplay∞
0e−px+e−qx
1+e−(p+q)xdx=π
p+qcosec/parenleftbiggπp
p+q/parenrightbigg
3.317 Exponential functions 335
3.312
1./integraldisplay∞
0/parenleftBig
1−e−x
β/parenrightBigν−1
e−μxdx=βB(βμ,ν)[ R e β>0,Reν>0,Reμ>0]
LI(25)(13), EH I 11(24)
2./integraldisplay∞
0/parenleftbig
1−e−x/parenrightbig−1/parenleftbig
1−e−αx/parenrightbig/parenleftbig
1−e−βx/parenrightbig
e−pxdx=ψ(p+α)+ψ(p+β)−ψ(p+α+β)−ψ(p)
[Rep>0,Rep>−Reα,Rep>−Reβ,Rep>−Re(α+β)]ET I 145(15)
3.11/integraldisplay∞
0/parenleftbig
1−e−x/parenrightbigν−1/parenleftbig
1−βe−x/parenrightbig−/rho1e−μxdx=B (μ, ν)2F1(/rho1, μ;μ+ν;β)
[Reμ>0,Reν>0,|arg(1−β)|<π]EH I 116(15)
3.313
1.7PV/integraldisplay∞
−∞e−μxdx
1−e−x=πcotπμ [0<Reμ<1]
2.7/integraldisplay∞
−∞e−μxdx
(1 +e−x)ν=B (μ, ν−μ)[ 0 <Reμ<Reν]
3.314/integraldisplay∞
−∞e−μxdx/parenleftbig
eβ/γ+e−x/γ/parenrightbigν=γexp/bracketleftbigg
β/parenleftbigg
μ−ν
γ/parenrightbigg/bracketrightbigg
B(γμ,ν−γμ)
/bracketleftbigg
Re/parenleftbiggν
γ/parenrightbigg
>Reμ>0,|Imβ|<πReγ/bracketrightbigg
ET I 120(21)
3.315
1./integraldisplay∞
−∞e−μxdx
(eβ+e−x)ν(eγ+e−x)/rho1=e x p [ γ(μ−/rho1)−βν]B(μ, ν+/rho1−μ)2F1/parenleftbig
ν,μ;ν+/rho1;1−eν−β/parenrightbig
[|Imβ|<π , |Imγ|<π , 0<Reμ<Re(ν+/rho1)]ET I 121(22)
2./integraldisplay∞
−∞e−μxdx
(β+e−x)(γ+e−x)=π/parenleftbig
βμ−1−γμ−1/parenrightbig
γ−βcosec( μπ)
[|argβ|<π , |argγ|<π , β /negationslash=γ,0<Reμ<2]ET I 120(18)
3.316/integraldisplay∞
−∞(1 +e−x)ν−1
(1 +e−x)μdx=ψ(μ)−ψ(μ−ν)[ R e μ>Reν>0] (cf. 3.235 )
BI (28)(8)
3.317
1./integraldisplay∞
−∞/parenleftbigg1
1+e−x−1
(1 +e−x)μ/parenrightbigg
dx=C+ψ(μ)[ R e μ>0] (cf. 3.233 ) BI (28)(10)
2./integraldisplay∞
−∞/parenleftbigg1
(1 +e−x)ν−1
(1 +e−x)μ/parenrightbigg
dx=ψ(μ)−ψ(ν)[ R e μ>0,Reν>0] (cf. 3.219 )
BI (28)(11)
336 Exponential Functions 3.318
3.318
1./integraldisplay∞
0/bracketleftbig
β+√
1−e−x/bracketrightbig−ν+/bracketleftbig
β−√
1−e−x/bracketrightbig−ν
√
1−e−xe−μxdx
=2μ+1e(μ−ν)πi/parenleftbig
β2−1/parenrightbig(μ−ν)/2Γ(μ)Qν−μ
μ−1(β)
Γ(ν)
[Reμ>0] ET I 145(18)
2.7/integraldisplay∞
u1√
1−e−2x/parenleftBig
e−u/radicalbig
1−e−2x−e−x/radicalbig
1−e−2u/parenrightBigν
e−μxdx
=2−1
2(μ+ν)√πe−u
2(μ+ν)Γ(μ)Γ(ν+1 )P−1
2(μ+ν)
−1
2(μ−ν)/parenleftbig√
1−e−2u/parenrightbig
Γ[(μ+ν+1 )/2]
[u>0,Reμ>0,Reν>−1]ET I 145(19)
3.32–3.34 Exponentials of more complicated arguments
3.321
1.11√π
2Φ(u)=√π
2erf(u)=/integraldisplayu
0e−x2dx=∞/summationdisplay
k=0(−1)ku2k+1
k!(2k+1 )
=e−u2∞/summationdisplay
k=02ku2k+1
(2k+1 ) ! !
(cf.8.25) AD 6.700
2./integraldisplayu
0e−q2x2dx=√π
2qΦ(qu)[ q>0]
3./integraldisplay∞
0e−q2x2dx=√π
2q[q>0] FI II 624
4.∗/integraldisplayu
0xe−q2x2dx=1
2q2/bracketleftBig
1−e−q2u2/bracketrightBig
5.∗/integraldisplayu
0x2e−q2x2dx=1
2q3/bracketleftbigg√π
2Φ(qu)−que−q2u2/bracketrightbigg
6.∗/integraldisplayu
0x3e−q2x2dx=1
2q4/bracketleftBig
1−/parenleftbig
1+q2u2/parenrightbig
e−q2u2/bracketrightBig
7.∗/integraldisplayu
0x4e−q2x2dx=1
2q5/bracketleftbigg3√π
4Φ(qu)−/parenleftbigg3
2+q2u2/parenrightbigg
que−q2u2/bracketrightbigg
3.322
1.11/integraldisplay∞
uexp/parenleftbigg
−x2
4β−γx/parenrightbigg
dx=/radicalbig
πβeβγ2/bracketleftbigg
1−Φ/parenleftbigg
γ/radicalbig
β+u
2√β/parenrightbigg/bracketrightbigg
[Reβ>0] ET I 146(21)
2./integraldisplay∞
0exp/parenleftbigg
−x2
4β−γx/parenrightbigg
dx=/radicalbig
πβexp/parenleftbig
βγ2/parenrightbig/bracketleftBig
1−Φ/parenleftBig
γ/radicalbig
β/parenrightBig/bracketrightBig
[Reβ>0] NT 27(1)a
3.326 Exponentials of more complicated arguments 337
3.11PV/integraldisplay∞
0e±iλx2dx=1
2/radicalbiggπ
λe±πi/4[λ>0] PBM 343 (2.3.15(2))
3.323
1.11/integraldisplay∞
1exp/parenleftbig
−qx−x2/parenrightbig
dx=√π
2eq2/4/bracketleftbigg
1−Φ/parenleftbigg
1+1
2q/parenrightbigg/bracketrightbigg
BI (29)(4)
2.10/integraldisplay∞
−∞exp/parenleftbig
−p2x2±qx/parenrightbig
dx=e x p/parenleftbiggq2
4p2/parenrightbigg√π
p/bracketleftbig
Rep2>0/bracketrightbig
BI (28)(1)
3.11/integraldisplay∞
0exp/parenleftbig
−β2x4−2γ2x2/parenrightbig
dx=2−3
2γ
βeγ4
2β2K1
4/parenleftbiggγ4
2β2/parenrightbigg
/bracketleftBig
|argβ|<π
4,|argγ|<π
4/bracketrightBig
ET I 147(34)a
3.324
1./integraldisplay∞
0exp/parenleftbigg
−β
4x−γx/parenrightbigg
dx=/radicalBigg
β
γK1/parenleftBig/radicalbig
βγ/parenrightBig
[Reβ≥0,Reγ>0] ET I 146(25)
2.11/integraldisplay∞
−∞exp/bracketleftBigg
−/parenleftbigg
x−b
x/parenrightbigg2n/bracketrightBigg
dx=1
nΓ/parenleftbigg1
2n/parenrightbigg
[b≥0]
3.325/integraldisplay∞
0exp/parenleftbigg
−ax2−b
x2/parenrightbigg
dx=1
2/radicalbiggπ
aexp/parenleftBig
−2√
ab/parenrightBig
[a>0,b > 0] FI II 644
3.326
1.8/integraldisplay∞
0exp(−xμ)dx=1
μΓ/parenleftbigg1
μ/parenrightbigg
[Reμ>0] BI (26)(4)
2.10/integraldisplay∞
0xmexp (−βxn)dx=Γ(γ)
nβγγ=m+1
n[Reβ>0,Rem>0,Ren>0]
3.∗/integraldisplay∞
0(x−a)exp(−β(x−b)n)dx=Γ/parenleftbig2
n,β(−b)n/parenrightbig
nβ2/n−(a−b)Γ/parenleftbig1
n,β(−b)n/parenrightbig
nβ1/n
[Ren>0,Reβ>0,|argb|<π]
4.∗/integraldisplayu
0(x−a)exp(−β(x−b)n)dx=Γ/parenleftbig2
n,β(−b)n/parenrightbig
−Γ/parenleftbig2
n,β(u−b)n/parenrightbig
nβ2/n
−(a−b)Γ/parenleftbig1
n,β(−b)n/parenrightbig
−Γ/parenleftbig1
n,β(u−b)n/parenrightbig
nβ1/n
[Ren>0,Reβ>0,|argb|<π , |arg(u−b)|<π]
5.∗/integraldisplay∞
u(x−a)exp(−β(x−b)n)dx=Γ/parenleftbig2
n,β(−b)n/parenrightbig
nβ2/n−(a−b)Γ/parenleftbig1
n,β(u−b)n/parenrightbig
nβ1/n
[Ren>0,Reβ>0,|arg(u−b)|<π]
338 Exponential Functions 3.327
Exponentials of exponentials
3.327/integraldisplay∞
0exp(−aenx)dx=−1
nEi(−a)[ n≥1,Rea≥0,a/negationslash=0 ] LI (26)(5)
3.328/integraldisplay∞
−∞exp(−ex)eμxdx=Γ (μ)[ R e μ>0] NH 145(14)
3.329/integraldisplay∞
0/bracketleftBigg
aexp (−ceax)
1−e−ax−bexp/parenleftbig
−cebx/parenrightbig
1−e−bx/bracketrightBigg
dx=e−clnb
a[a>0,b > 0,c > 0] BI (27)(12)
3.331
1./integraldisplay∞
0exp/parenleftbig
−βe−x−μx/parenrightbig
dx=β−μγ(μ, β)[ R e μ>0] ET I 147(36)
2./integraldisplay∞
0exp(−βex−μx)dx=βμΓ(−μ, β)[ R e β>0] ET I 147(37)
3.11/integraldisplay∞
0/parenleftbig
1−e−x/parenrightbigν−1exp/parenleftbig
βe−x−μx/parenrightbig
dx=B (μ, ν)β−μ+ν
2eβ
2Mν−μ
2,ν+μ−1
2(β)
[Reμ>0,Reν>0] ET I 147(38)
4./integraldisplay∞
0/parenleftbig
1−e−x/parenrightbigν−1exp (−βex−μx)dx=Γ (ν)βμ−1
2e−β
2W1−μ−2ν
2,−μ
2(β)
[Reβ>0,Reν>0] ET I 147(39)
3.332/integraldisplay∞
0/parenleftbig
1−e−x/parenrightbigν−1/parenleftbig
1−λe−x/parenrightbig−/rho1exp/parenleftbig
βe−x−μx/parenrightbig
dx=B (μ, ν)Φ1(μ, /rho1, ν, λ, β )
[Reμ>0,Reν>0,|arg(1−λ)|<π]ET I 147(40)
3.333
1.3/integraldisplay∞
−∞e−μxdx
exp (e−x)−1=Γ (μ)ζ(μ)[ R e μ>1] ET I 121(24)
2.3/integraldisplay∞
−∞e−μxdx
exp (e−x)+1=/parenleftbig
1−21−μ/parenrightbig
Γ(μ)ζ(μ)[ R e μ>0,μ/negationslash=1 ]
=l n2 [ μ=1 ]
ET I 121(25)
3.∗/integraldisplay∞
0/parenleftbiggtanh(x)
x3−1
x2cosh2(x)/parenrightbigg
dx=7ζ(3)
π2
3.33411/integraldisplay∞
0(ex−1)ν−1exp/bracketleftbigg
−β
ex−1−μx/bracketrightbigg
dx=Γ (μ−ν+1 )eβ
2βν−1
2W ν−2μ−1
2,−ν
2(β)
[Reβ>0,Reμ>Reν−1]
ET I 137(41)
Exponentials of hyperbolic functions
3.335/integraldisplay∞
0/parenleftbig
eνx+e−νxcosνπ/parenrightbig
exp (−βsinhx)dx=−π[Eν(β)+Yν(β)]
[Reβ>0] EH II 35(34)
3.338 Exponentials of more complicated arguments 339
3.336
1./integraldisplay∞
0exp(−νx−βsinhx)dx=πcosecνπ[Jν(β)−Jν(β)]
/bracketleftBig
|argβ|<π
2and|argβ|=π
2for Re ν>0;νis not an integer/bracketrightBig
WA 341(2)
2./integraldisplay∞
0exp(nx−βsinhx)dx=1
2[Sn(β)−πEn(β)−πYn(β)]
[Reβ>0;n=0,1,2,...]WA 342(6)
3./integraldisplay∞
0exp(−nx−βsinhx)dx=1
2(−1)n+1[Sn(β)+πEn(β)+πYn(β)]
[Reβ>0;n=0,1,2,...]
EH II 84(47)
3.337
1./integraldisplay∞
−∞exp(−αx−βcoshx)dx=2Kα(β)/bracketleftBig
|argβ|<π
2/bracketrightBig
WA 201(7)
2./integraldisplay∞
−∞exp(−νx+iβcoshx)dx=iπeiνπ
2H(1)
ν(β)[ 0 <argz<π] EH II 21(27)
3./integraldisplay∞
−∞exp(−νx−iβcoshx)dx=−iπe−iνπ
2H(2)
ν(β)[ −π<argz<0] EH II 21(30)
Exponentials of trigonometric functions and logarithms
3.338
1./integraldisplayπ
0{expi[(ν−1)x−βsinx]−expi[(ν+1 )x−βsinx]}dx=2π/bracketleftbig
J/prime
ν(β)+iE/prime
ν(β)/bracketrightbig
[Reβ>0] EH II 36
2./integraldisplayπ
0exp [±i(νx−βsinx)]dx=π[Jν(β)±iEν(β)] [Re β>0] EH II 35(32)
3.10/integraldisplay∞
0exp[−γ(x−βsinx)]dx=1
γ+2∞/summationdisplay
k=1γJk(kβ)
γ2+k2[Reγ>0] WA 619(4)
4.6/integraldisplayπ
−πexp/bracketleftbigga+bsinx+ccosx
1+psinx+qcosx/bracketrightbigg
1+psinx+qcosxdx=2π/radicalbig
1−p2−q2e−αI0(β),
withα=bp+cq−a
1−p2−q2;β=/radicalBigg
α2−a2−b2−c2
1−p2−q2;/bracketleftbig
p2+q2<1/bracketrightbig
5.∗/integraldisplayπ/4
0exp/bracketleftBigg
−∞/summationdisplay
n=1tan2nx
n+1
2/bracketrightBigg
dx=l n√
2
340 Exponential Functions 3.339
3.3396/integraldisplayπ
0exp (zcosx)dx=πI0(z) BI (277)(2)a
3.341/integraldisplayπ
2
0exp (−ptanx)dx=c i (p)sinp−si(p)cos(p)[ p>0] BI (271)(2)a
3.34211/integraldisplay1
0exp (−pxlnx)dx=/integraldisplay1
0x−pxdx=∞/summationdisplay
k=1pk−1
kkBI (29)(1)
3.35 Combinations of exponentials and rational functions
3.351
1.8/integraldisplayu
0xne−μxdx=n!
μn+1−e−uμn/summationdisplay
k=0n!
k!uk
μn−k+1=μ−n−1γ(n+1,μ u)
[u>0,Reμ>0,n=0,1,2,...]
ET I 134(5)
2.11/integraldisplay∞
uxne−μxdx=e−uμn/summationdisplay
k=0n!
k!uk
μn−k+1=μ−n−1Γ(n+1,μ u)
[u>0,Reμ>0,n=0,1,2,...]
ET I 33(4)
3./integraldisplay∞
0xne−μxdx=n!μ−n−1[Reμ>0] ET I 133(3)
4./integraldisplay∞
ue−pxdx
xn+1=(−1)n+1pnEi(−pu)
n!+e−pu
unn−1/summationdisplay
k=0(−1)kpkuk
n(n−1)...(n−k)
[p>0] NT 21(3)
5./integraldisplay∞
1e−μxdx
x=−Ei(−μ)[ R e μ>0] BI (104)(10)
6./integraldisplayu
−∞ex
xdx=l i(eu)=E i ( u)[ u<0]
7.9/integraldisplayu
0xe−μxdx=1
μ2−1
μ2e−μu(1 +μu)[ u>0]
8.11/integraldisplayu
0x2e−μxdx=2
μ3−1
μ3e−μu/parenleftbig
2+2μu+μ2u2/parenrightbig
[u>0]
9.7/integraldisplayu
0x3e−μxdx=6
μ4−1
μ4e−μu/parenleftbig
6+6μu+3μ2u2+μ3u3/parenrightbig
[u>0]
3.352
1./integraldisplayu
0e−μxdx
x+β=eμβ[Ei(−μu−μβ)−Ei(−μβ)] [ u≥0,|argβ|<π] ET II 217(12)
2./integraldisplay∞
ue−μxdx
x+β=−eβμEi(−μu−μβ)[ u≥0,|arg(u+β)|<π , Reμ>0]
ET I 134(6), JA
3.354 Exponentials and rational functions 341
3./integraldisplayv
ue−μxdx
x+α=eαμ{Ei[−(α+v)μ]−Ei[−(α+u)μ]}[−α<n , and−α>v , Reμ>0]
ET I 134 (7)
4./integraldisplay∞
0e−μxdx
x+β=−eβμEi(−μβ)[ |argβ|<π , Reμ>0] ET II 217(11)
5.7/integraldisplay∞
ue−pxdx
a−x=e−paEi(pa−pu)
/bracketleftbig
p>0,a < u ;f o ra>u , one should replace Ei( pa−pu) in this formula with Ei(pa−pu)/bracketrightbig
ET II 251(37)
6.8/integraldisplay∞
0e−μxdx
a−x=e−μaEi(aμ)
[a<0,Reμ>0] BI (91)(4)
7./integraldisplay∞
−∞eipxdx
x−a=iπeiap[p>0] ET II 251(38)
3.353
1./integraldisplay∞
ue−μxdx
(x+β)n=e−uμn−1/summationdisplay
k=1(k−1)!(−μ)n−k−1
(n−1)!(u+β)k−(−μ)n−1
(n−1)!eβμEi[−(u+β)μ]
[n≥2,|arg(u+β)|<π , Reμ>0]
ET I 134(10)
2.7/integraldisplay∞
0e−μxdx
(x+β)n=1
(n−1)!n−1/summationdisplay
k=1(k−1)!(−μ)n−k−1β−k−(−μ)n−1
(n−1)!eβμEi(−βμ)
[n≥2,|argβ|<π , Reμ>0]
ET I 134(9), BI (92)(2)
3./integraldisplay∞
0e−pxdx
(a+x)2=peαpEi(−ap)+1
a[p>0,a > 0]
LI (281)(28), LI (281)(29)
4./integraldisplay1
0xex
(1 +x)2dx=e
2−1. BI (80)(6)
5.7/integraldisplay∞
0xne−μx
x+βdx=(−1)n−1βneβμEi(−βμ)+n/summationdisplay
k=1(k−1)!(−β)n−kμ−k
[|argβ|<π , Reμ>0]
BI (91)(3)a, LET I 135(11)
3.354
1./integraldisplay∞
0e−μxdx
β2+x2=1
β[ci(βμ)sinβμ−si(βμ)cosβμ][ R e β>0,Reμ>0] BI (91)(7)
2./integraldisplay∞
0xe−μxdx
β2+x2=−ci(βμ)cosβμ−si(βμ)sinβμ [Reβ>0,Reμ>0] BI (91)(8)
3.7/integraldisplay∞
0e−μxdx
β2−x2=1
2β/bracketleftbig
e−βμEi(βμ)−eβμEi(−βμ)/bracketrightbig
[|arg (±β)|<π , Reμ>0]BI (91)(14)
342 Exponential Functions 3.355
4./integraldisplay∞
0xe−μxdx
β2−x2=1
2/bracketleftbig
e−βμEi(βμ)+eβμEi(−βμ)/bracketrightbig
/bracketleftbig
|arg (±β)|<π , Reμ>0; for β>0 one should replace Ei( βμ) in this formula with Ei(βμ)/bracketrightbig
BI (91)(15)
5.8/integraldisplay∞
−∞e−ipxdx
a2+x2=π
ae−|ap|[a/negationslash=0,preal] ET I 118(1)a
3.355
1./integraldisplay∞
0e−μxdx
(β2+x2)2=1
2β3{ci(βμ)sinβμ−si(βμ)cosβμ}−βμ[ci(βμ)cosβμ+s i (βμ)sinβμ]
LI (92)(6)
2./integraldisplay∞
0xe−μxdx
(β2+x2)2=1
2β2{−βμ[ci(βμ)sinβμ−si(βμ)cosβμ]}
[Reβ>0,Reμ>0] BI (92)(7)
3.3/integraldisplay∞
0e−pxdx
(a2−x2)2=1
4a3/bracketleftbig
(ap−1)eapEi(−ap)+( 1+ ap)e−apEi(ap)/bracketrightbig
/bracketleftbig
Im/parenleftbig
a2/parenrightbig
>0,p > 0/bracketrightbig
BI (92)(8)
4.3/integraldisplay∞
0xe−pxdx
(a2−x2)2=1
4a2/braceleftbig
−2+ap/bracketleftbig
e−apEi(ap)−eapEi(−ap)/bracketrightbig/bracerightbig
/bracketleftbig
Im/parenleftbig
a2/parenrightbig
>0,p > 0/bracketrightbig
LI (92)(9)
3.356
1./integraldisplay∞
0x2n+1e−px
a2+x2dx=(−1)n−1a2n[ci(ap)cosap+s i (ap)sinap]
+1
p2nn/summationdisplay
k=1(2n−2k+1 ) !/parenleftbig
−a2p2/parenrightbigk−1
[p>0] BI (91)(12)
2./integraldisplay∞
0x2ne−px
a2+x2dx=(−1)na2n−1[ci(ap)sinap−si(ap)cosap]+1
p2n−1n/summationdisplay
k=1(2n−2k)!/parenleftbig
−a2p2/parenrightbigk−1
[p>0] BI (91)(11)
3./integraldisplay∞
0x2n+1e−px
a2−x2dx=1
2a2n/bracketleftbig
eapEi(−ap)+e−apEi(ap)/bracketrightbig
−1
p2nn/summationdisplay
k=1(2n−2k+1 ) !/parenleftbig
a2p2/parenrightbigk−1
[p>0] BI (91)(17)
4./integraldisplay∞
0x2ne−px
a2−x2dx=1
2a2n−1/bracketleftbig
e−apEi(ap)−eapEi(−ap)/bracketrightbig
−1
p2n−1n/summationdisplay
k=1(2n−2k)!/parenleftbig
a2p2/parenrightbigk−1
[p>0] BI (91)(16)
3.358 Exponentials and rational functions 343
3.357
1./integraldisplay∞
0e−μxdx
a3+a2x+ax2+x3=1
2a2{ci(aμ)(sinaμ+c o s aμ)
+si(aμ)(sinaμ−cosaμ)−eaμEi(−aμ)}
[Reμ>0,a > 0] BI (92)(18)
2./integraldisplay∞
0xe−μxdx
a3+a2x+ax2+x3=1
2a{ci(aμ)(sinaμ−cosaμ)
−si(aμ)(sinaμ+c o s aμ)−eaμEi(−aμ)}
[Reμ>0,a > 0] BI (92)(19)
3./integraldisplay∞
0x2e−μxdx
a3+a2x+ax2+x3=1
2{−ci(aμ)(sinaμ+c o s aμ)
−si(aμ)(sinaμ−cosaμ)−eaμEi(−aμ)}
[Reμ>0,a > 0] BI (92)(20)
4./integraldisplay∞
0e−μxdx
a3−a2x+ax2−x3=1
2a2{ci(aμ)(sinaμ−cosaμ)
−si(aμ)(sinaμ+c o s aμ)+e−aμEi(aμ)/bracerightbig
[Reμ>0,a > 0] BI (92)(21)
5./integraldisplay∞
0xe−μxdx
a3−a2x+ax2−x3=1
2a{−ci(aμ)(sinaμ+c o s aμ)
−si(aμ)(sinaμ−cosaμ)+e−aμEi(aμ)/bracerightbig
[Reμ>0,a > 0] BI (92)(22)
6./integraldisplay∞
0x2e−μxdx
a3−a2x+ax2−x3=1
2{ci(aμ)(cosaμ−sinaμ)
+s i (aμ)(cosaμ+s i naμ)+e−aμEi(aμ)/bracerightbig
[Reμ>0,a > 0] BI (92)(23)
3.358
1./integraldisplay∞
0e−px
a4−x4dx=1
4a3/braceleftbig
e−apEi(ap)−eapEi(−ap)+2c i ( ap)sinap−2s i(ap)cosap/bracerightbig
[p>0,a > 0] BI (91)(18)
2./integraldisplay∞
0xe−pxdx
a4−x4=1
4a2/braceleftbig
eapEi(−ap)+e−apEi(ap)−2c i(ap)cosap−2s i(ap)sinap/bracerightbig
[p>0,a > 0] BI (91)(19)
3./integraldisplay∞
0x2e−pxdx
a4−x4=1
4a/braceleftbig
e−apEi(ap)−eapEi(−ap)−2c i(ap)sinap+2s i ( ap)cosap/bracerightbig
[p>0,a > 0] BI (91)(20)
4./integraldisplay∞
0x3e−pxdx
a4−x4=1
4/braceleftbig
eapEi(−ap)+e−apEi(ap)+2c i ( ap)cosap+2s i ( ap)sinap/bracerightbig
[p>0,a > 0] BI (91)(21)
344 Exponential Functions 3.359
5./integraldisplay∞
0x4ne−px
a4−x4dx=1
4a4n−3/bracketleftbig
e−apEi(ap)−eapEi(−ap)+2c i ( ap)sinap−2s i(ap)cosap/bracketrightbig
−1
p4n−3n/summationdisplay
k=1(4n−4k)!/parenleftbig
a4p4/parenrightbigk−1
[p>0,a > 0] BI (91)(22)
6./integraldisplay∞
0x4n+1e−px
a4−x4dx=1
4a4n−2/bracketleftbig
eapEi(−ap)+e−apEi(ap)−2c i(ap)cosap−2s i(ap)sinap/bracketrightbig
−1
p4n−2n/summationdisplay
k=1(4n−4k+1 ) !/parenleftbig
a4p4/parenrightbigk−1
[p>0,a > 0] BI (91)(23)
7./integraldisplay∞
0x4n+2e−px
a4−x4dx=1
4a4n−1/bracketleftbig
e−apEi(ap)−eapEi(−ap)−2c i(ap)sinap+2s i ( ap)cosap/bracketrightbig
−1
p4n−1n/summationdisplay
k=1(4n−4k+2 ) !/parenleftbig
a4p4/parenrightbigk−1
[p>0,a > 0] BI (91)(24)
8./integraldisplay∞
0x4n+3e−px
a4−x4dx=1
4a4n/bracketleftbig
eapEi(−ap)+e−apEi(ap)+2c i ( ap)cosap+2s i ( ap)sinap/bracketrightbig
−1
p4nn/summationdisplay
k=1(4n−4k+3 ) !/parenleftbig
a4p4/parenrightbigk−1
[p>0,a > 0] BI (91)(25)
3.359/integraldisplay∞
−∞(i−x)n
(i+x)ne−ipx
i+x2dx=(−1)n−12πpe−pLn−1(2p)f o r p>0;
=0 f o r p<0.
ET I 118(2)
3.36–3.37 Combinations of exponentials and algebraic functions
3.361
1.8/integraldisplayu
0e−qx
√xdx=/radicalbiggπ
qΦ(√qu)[ q>0]
2.8/integraldisplay∞
0e−qx
√xdx=/radicalbiggπ
q[q>0] BI(98)(10)
3.8/integraldisplay∞
−1e−qx
√1+xdx=eq/radicalbiggπ
q[q>0] BI (104)(16)
3.362
1./integraldisplay∞
1e−μxdx√x−1=/radicalbiggπ
μe−μ[Reμ>0] BI (104)(11)a
2./integraldisplay∞
0e−μxdx√x+β=/radicalbiggπ
μeβμ/bracketleftBig
1−Φ/parenleftBig/radicalbig
βμ/parenrightBig/bracketrightBig
[Reμ>0,|argβ|<π] ET I 135(18)
3.371 Exponentials and algebraic functions 345
3.363
1./integraldisplay∞
u√x−u
xe−μxdx=/radicalbiggπ
μe−uμ−π√u[1−Φ(√uμ)]
[Reμ>0] ET I 136(23)
2./integraldisplay∞
ue−μxdx
x√x−u=π√u[1−Φ(√uμ)] [ u>0,Reμ≥0] ET I 136(26)
3.364
1./integraldisplay2
0e−pxdx/radicalbig
x(2−x)=πe−pI0(p)[ p>0] GW (312)(7a)
2./integraldisplay1
−1e2xdx√
1−x2=πI0(2) BI (277)(2)a
3./integraldisplay∞
0e−pxdx/radicalbig
x(x+a)=eap
2K0/parenleftBigap
2/parenrightBig
[a>0,p > 0] GW (312)(8a)
3.365
1./integraldisplayu
0xe−μxdx√
u2−x2=πu
2[L1(μu)−I1(μu)] +u [u>0,Reμ>0] ET I 136(28)
2./integraldisplay∞
uxe−μxdx√
x2−u2=uK1(uμ)[ u>0,Reμ>0] ET I 136(29)
3.366
1./integraldisplay2u
0(u−x)e−μxdx√
2ux−x2=πue−uμI1(uμ)[ R e μ>0] ET I 136(31)
2./integraldisplay∞
0(x+β)e−μxdx/radicalbig
x2+2βx=βeβμK1(βμ)[ R e μ>0,|argβ|<π] ET I 136(30)
3./integraldisplay∞
0xe−μxdx/radicalbig
x2+β2=βπ
2[H1(βμ)−Y1(βμ)]−β/bracketleftBig
|argβ|<π
2,Reμ>0/bracketrightBig
ET I 136(27)
3.367/integraldisplay∞
0e−μxdx
(1 + cos t+x)√
x2+2x=exp/parenleftbig
2μcos2t
2/parenrightbig
sint/parenleftbigg
t−sint/integraldisplayu
0K0(v)e−vcostdv/parenrightbigg
[Reμ>0] ET I 136(33)
3.368/integraldisplay∞
0e−μxdx
x+/radicalbig
x2+β2=π
2βμ[H1(βμ)−Y1(βμ)]−1
β2μ2
/bracketleftBig
|argβ|<π
2,Reμ>0/bracketrightBig
ET I 136(32)
3.36911/integraldisplay∞
0e−μxdx/radicalbig
(x+a)3=2√a−2√πμeaμ(1−Φ(√aμ)) [ |arga|<π , Reμ>0] ET I 135(20)
3.37111/integraldisplay∞
0xn−1
2e−μxdx=√π·1
2·3
2...2n−1
2μ−n−1
2
=√π2−nμ−n−1/2(2n−1)!! [ n≥0]
[Reμ>0] ET I 135(17)
346 Exponential Functions 3.372
3.372/integraldisplay∞
0xn−1
2(2 +x)n−1
2e−pxdx=(2n−1)!!
pnepKn(p)[p>0,n=0,1,2,...] GW (312)(8)
3.373/integraldisplay∞
0/bracketleftBig/parenleftBig
x+/radicalbig
x2+β2/parenrightBign
+/parenleftBig
x−/radicalbig
x2+β2/parenrightBign/bracketrightBig
e−μxdx=2βn+1On(βμ)
[Reμ>0] WA 05(1)
3.374
1./integraldisplay∞
0/parenleftbig
x+√
1+x2/parenrightbign
√
1+x2e−μxdx=1
2[Sn(μ)−πEn(μ)−πYn(μ)]
[Reμ>0] ET I 37(35)
2./integraldisplay∞
0/parenleftbig
x−√
1+x2/parenrightbign
√
1+x2e−μxdx=−1
2[Sn(μ)+πEn(μ)+πYn(μ)]
[Reμ>0] ET I 137(36)
3.38–3.39 Combinations of exponentials and arbitrary powers
3.381
1./integraldisplayu
0xν−1e−μxdx=μ−νγ(ν,μu)[ R e ν>0] EH I 266(22), EH II 133(1)
2./integraldisplayu
0xp−1e−xdx=∞/summationdisplay
k=0(−1)kup+k
k!(p+k)
=e−u∞/summationdisplay
k=0up+k
p(p+1 )...(p+k)
AD 6.705
3.8/integraldisplay∞
uxν−1e−μxdx=μ−νΓ(ν,μu)[ u>0,Reμ>0]
EH I 256(21), EH II 133(2)
4./integraldisplay∞
0xν−1e−μxdx=1
μνΓ(ν)[ R e μ>0,Reν>0] FI II 779
5./integraldisplay∞
0xν−1e−(p+iq)xdx=Γ (ν)/parenleftbig
p2+q2/parenrightbig−ν
2exp/parenleftbigg
−iνarctanq
p/parenrightbigg
[p>0,Reν>0a n d p=0,0<Reν<1]EH I 12(32)
6./integraldisplay∞
ue−x
xνdx=u−ν
2e−u
2W−ν
2,(1−ν)
2(u)[ u>0] WH
7./integraldisplay∞
0xk−1eiμxdx=Γ(k)
(−iμ)k[0<Re(k)<1,μ/negationslash=0 ]
GH2 62 (313.14)
8.∗/integraldisplayu
0xme−βxndx=γ(v,βun)
nβvv=m+1
n[u>0,Rev>0,Ren>0,Reβ>0]
9.∗/integraldisplay∞
uxme−βxndx=Γ(v,βun)
nβvv=m+1
n[u>0,Rev>0,Ren>0,Reβ>0]
3.383 Exponentials and arbitrary powers 347
10.∗/integraldisplay∞
0xme−βxndx=γ(v,βun)+Γ( v,βun)
nβv
v=m+1
n[u>0,Rev>0,Ren>0,Reβ>0] See also 3.326 1
11.∗/integraldisplay∞
−∞x2me−βx2ndx=2/integraldisplay∞
0x2me−βx2ndx=2(γ(v,βun)+Γ( v,βun))
nβv=Γ(v)
nβv
v=2m+1
2n[u>0,Rev>0,Ren>0,Reβ>0]
3.382
1.6/integraldisplayu
0(u−x)νe−μxdx=(−μ)−ν−1e−uμγ(ν+1,−uμ)[ R e ν>−1,u > 0] ET I 137(6)
2./integraldisplay∞
u(x−u)νe−μxdx=μ−ν−1e−uμΓ(ν+1 ) [ u>0,Reν>−1,Reμ>0]
ET I 137(5), ET II 202(11)
3./integraldisplay∞
0(1 +x)−νe−μxdx=μν
2−1eμ
2W−ν
2,(1−ν)
2(μ)[ R e μ>0] WH
4./integraldisplay∞
0(x+β)νe−μxdx=μ−ν−1eβμΓ(ν+1,βμ)[ |argβ|<π , Reμ>0]
ET I 137(4), ET II 233(10)
5./integraldisplayu
0(a+x)μ−1e−xdx=ea[γ(μ, a+u)−γ(μ, a)] [Re μ>0] EH II 139
6./integraldisplay∞
−∞(β+ix)−νe−ipxdx= 0 [for p>0]
=2π(−p)ν−1eβp
Γ(ν)[forp<0]
[Reν>0,Reβ>0] ET I 118(4)
7./integraldisplay∞
−∞(β−ix)−νe−ipxdx=2πpν−1e−βp
Γ(ν)[forp>0]
= 0 [for p<0]
[Reν>0,Reβ>0] ET I 118(3)
3.383
1.11/integraldisplayu
0xν−1(u−x)μ−1eβxdx=B (μ, ν)uμ+ν−1
1F1(ν;μ+ν;βu)
[Reμ>0,Reν>0] ET II 187(14)
2.11/integraldisplayu
0xμ−1(u−x)μ−1eβxdx=√π/parenleftbiggu
β/parenrightbiggμ−1
2
exp/parenleftbiggβu
2/parenrightbigg
Γ(μ)Iμ−1
2/parenleftbiggβu
2/parenrightbigg
[Reμ>0] ET II 187(13)
3./integraldisplay∞
uxμ−1(x−u)μ−1e−βxdx=1√π/parenleftbiggu
β/parenrightbiggμ−1
2
Γ(μ)exp/parenleftbigg
−βu
2/parenrightbigg
Kμ−1
2/parenleftbiggβu
2/parenrightbigg
[Reμ>0,Reβu > 0] ET II 202(12)
348 Exponential Functions 3.384
4.11/integraldisplay∞
uxν−1(x−u)μ−1e−βxdx=β−μ+ν
2uμ+ν−2
2Γ(μ)exp/parenleftbigg
−βu
2/parenrightbigg
W ν−μ
2,1−μ−ν
2(βu)
[Reμ>0,Reβu > 0] ET II 202(13)
5.11/integraldisplay∞
0e−pxxq−1(1 +ax)−νdx
=π2
pqΓ(ν)sin[π(q−ν)]/bracketleftBigg/parenleftBigp
a/parenrightBigνLν−q
−ν/parenleftbigp
a/parenrightbig
sin(πν)Γ(1−q)−/parenleftBigp
a/parenrightBigqLq−ν
−q/parenleftbigp
a/parenrightbig
sin(πq)Γ(1−ν)/bracketrightBigg
[ν/negationslash=±1,±2,...]
=Γ(q)
pq[ν=0 ]
[Req>0,Rep>0,Rea>0]
6./integraldisplay∞
0xν−1(x+β)−ν+1
2e−μxdx=2ν−1
2Γ(ν)μ−1
2eβμ
2D1−2ν/parenleftBig/radicalbig
2βμ/parenrightBig
[|argβ|<π , Reν>0,Reμ≥0,μ/negationslash=0 ] ET I 39(20), EH II 119(2)a
7./integraldisplay∞
0xν−1(x+β)−ν−1
2e−μxdx=2νΓ(ν)β−1
2eβμ
2D−2ν/parenleftBig/radicalbig
2βμ/parenrightBig
[|argβ|<π , Reν>0,Reμ≥0]
ET I 139(21), EH II 119(1)a
8./integraldisplay∞
0xν−1(x+β)ν−1e−μxdx=1√π/parenleftbiggβ
μ/parenrightbiggν−1
2
eβμ
2Γ(ν)K1
2−ν/parenleftbiggβμ
2/parenrightbigg
[|argβ|<π , Reμ>0,Reν>0]
ET II 233(11), EH II 19(16)a, EH II 82(22)a
9./integraldisplay∞
u(x−u)νe−μx
xdx=uνΓ(ν+1 )Γ ( −ν,uμ)[ u>0,Reν>−1,Reμ>0]
ET I 138(8)
10./integraldisplay∞
0xν−1e−μx
x+βdx=βν−1eβμΓ(ν)Γ(1−ν,βμ)[ |argβ|<π , Reμ>0,Reν>0]
EH II 137(3)
3.384
1./integraldisplay1
−1(1−x)ν−1(1 +x)μ−1e−ipxdx=2μ+ν−1B(μ, ν)eip
1F1(μ;ν+μ;−2ip)
[Reν>0,Reμ>0] ET I 119(13)
2./integraldisplayv
u(x−u)2μ−1(v−x)2ν−1e−pxdx
=B ( 2 μ,2ν)(v−u)μ+ν−1p−μ−νexp/parenleftbigg
−pu+v
2/parenrightbigg
Mμ−ν,μ+ν−1
2(vp−up)
[v>u> 0,Reμ>0,Reν>0]ET I 139(23)
3.385 Exponentials and arbitrary powers 349
3./integraldisplay∞
u(x+β)2ν−1(x−u)2/rho1−1e−μxdx
=(u+β)ν+/rho1−1
μν+/rho1exp/bracketleftbigg(β−u)μ
2/bracketrightbigg
Γ(2/rho1)Wν−/rho1,ν+/rho1−1
2(uμ+βμ)
[u>0,|arg(β+u)|<π , Reμ>0,Re/rho1>0]ET I 139(22)
4./integraldisplay∞
u(x+β)ν(x−u)−νe−μxdx=1
μνπcosec( νπ)e−(β+u)μ
2k2ν/bracketleftbigg(β+u)μ
2/bracketrightbigg
[ν/negationslash=0,u > 0,|arg(u+β)|<π , Reμ>0,Reν<1]ET I 139(17)
5./integraldisplay∞
u(x−u)ν−1(x+u)−ν+1
2e−μxdx=1√μ2ν−1
2Γ(ν)D1−2ν(2√uμ)
[u>0,Reμ>0,Reν>0]
ET I 139(18)
6./integraldisplay∞
u(x−u)ν−1(x+u)−ν−1
2e−μxdx=1√u2ν−1
2Γ(ν)D−2ν(2√uμ)
[u>0,Reμ≥0,Reν>0]
ET I 139(19)
7.6/integraldisplay∞
−∞(β−ix)−μ(γ−ix)−νe−ipxdx=2πe−βppμ+ν−1
Γ(μ+ν)1F1(ν;μ+ν;(β−γ)p)[ f o r p>0]
= 0 [for p<0]
[Reβ>0,Reγ>0,Re(μ+ν)>1]ET I 119(10)
8.6/integraldisplay∞
−∞(β+ix)−μ(γ+ix)−νe−ipxdx= 0 [for p>0]
=2πeγp(−p)μ+ν−1
Γ(μ+ν)1F1[μ;μ+ν;(β−γ)p][ f o r p<0]
[Reβ>0,Reγ>0,Re(μ+ν)>1]ET I 19(11)
9.6/integraldisplay∞
−∞(β+ix)−2μ(γ−ix)−2νe−ipxdx
=2π(β+γ)−μ−νpμ+ν−1
Γ(2ν)exp/parenleftbiggβ−γ
2p/parenrightbigg
Wν−μ,1
2−ν−μ(βp+γp)[ f o r p>0]
=2π(β+γ)−μ−ν(−p)μ+ν−1
Γ(2μ)exp/parenleftbiggβ−γ
2p/parenrightbigg
Wμ−ν,1
2−ν−μ(−βp−γp)[ f o r p<0]
/bracketleftbig
Reβ>0,Reγ>0,Re(μ+ν)>1
2/bracketrightbig
ET I 19(12)
3.38511/integraldisplay1
0xν−1(1−x)λ−1(1−βx)−/rho1e−μxdx=B (ν,λ)Φ1(ν,/rho1,λ +ν,−μ, β)
[Reλ>0,Reν>0,|arg(1−β)|<π]ET I 39(24)
350 Exponential Functions 3.386
3.386
1./integraldisplay∞
−∞(ix)ν0n/productdisplay
k=1(βk+ix)νke−ipxdx
β0−ix=2πe−β0pβν0
0n/productdisplay
k=1(β0+βk)νk
/bracketleftBigg
Reν0>−1,Reβk>0,n/summationdisplay
k=0Reνk<1,argix=π
2signx, p > 0/bracketrightBigg
ET I 118(8)
2./integraldisplay∞
−∞(ix)ν0n/productdisplay
k=1(βk+ix)νke−ipxdx
β0+ix=0
/bracketleftBigg
Reν0>−1,Reβk>0,n/summationdisplay
k=0Reνk<1,argix=π
2signx, p > 0/bracketrightBigg
ET I 119(9)
3.387
1.6/integraldisplay1
−1/parenleftbig
1−x2/parenrightbigν−1e−μxdx=√π/parenleftbigg2
μ/parenrightbiggν−1
2
Γ(ν)Iν−1
2(μ)
/bracketleftBig
Reν>0,|argμ|<π
2/bracketrightBig
WA 172(2)a
2.6/integraldisplay1
−1/parenleftbig
1−x2/parenrightbigν−1eiμxdx=√π/parenleftbigg2
μ/parenrightbiggν−1
2
Γ(ν)Jν−1
2(μ)
[Reν>0] WA 25(3), WA 48(4)a
3./integraldisplay∞
1/parenleftbig
x2−1/parenrightbigν−1e−μxdx=1√π/parenleftbigg2
μ/parenrightbiggν−1
2
Γ(ν)Kν−1
2(μ)
/bracketleftBig
|argμ|<π
2,Reν>0/bracketrightBig
WA 190(4)a
4./integraldisplay∞
1/parenleftbig
x2−1/parenrightbigν−1eiμxdx
=i√π
2/parenleftbigg2
μ/parenrightbiggν−1
2
Γ(ν)H(1)
1
2−ν(μ) [Imμ>0,Reν>0]EH II 83(28)a
=−i√π
2/parenleftbigg
−2
μ/parenrightbiggν−1
2
Γ(ν)H(2)
1
2−ν(−μ)[Imμ<0,Reν>0]EH II 83(29)a
5./integraldisplayu
0/parenleftbig
u2−x2/parenrightbigν−1eμxdx=√π
2/parenleftbigg2u
μ/parenrightbiggν−1
2
Γ(ν)/bracketleftBig
Iν−1
2(uμ)+Lν−1
2(uμ)/bracketrightBig
[u>0,Reν>0] ET II 188(20)a
6./integraldisplay∞
u/parenleftbig
x2−u2/parenrightbigν−1e−μxdx=1√π/parenleftbigg2u
μ/parenrightbiggν−1
2
Γ(ν)Kν−1
2(uμ)
[u>0,Reμ>0,Reν>0]
ET II 203(17)a
3.389 Exponentials and arbitrary powers 351
7.11/integraldisplay∞
0/parenleftbig
x2+u2/parenrightbigν−1e−μxdx=√π
2/parenleftbigg2u
μ/parenrightbiggν−1
2
Γ(ν)/bracketleftBig
Hν−1
2(uμ)−Yν−1
2(uμ)/bracketrightBig
[|argu|<π , Reμ>0] ET I 138(10)
3.388
1./integraldisplay2u
0/parenleftbig
2ux−x2/parenrightbigν−1e−μxdx=√π/parenleftbigg2u
μ/parenrightbiggν−1
2
e−uμΓ(ν)Iν−1
2(uμ)
[u>0,Reν>0] ET I 138(14)
2./integraldisplay∞
0/parenleftbig
2βx+x2/parenrightbigν−1e−μxdx=1√π/parenleftbigg2β
μ/parenrightbiggν−1
2
eβμΓ(ν)Kν−1
2(βμ)
[|argβ|<π , Reν>0,Reμ>0]
ET I 138(13)
3./integraldisplay∞
0/parenleftbig
x2+ix/parenrightbigν−1e−μxdx=−i√πeiμ
2
2μν−1
2Γ(ν)H(2)
ν−1
2/parenleftBigμ
2/parenrightBig
[Reμ>0,Reν>0] ET I 138(15)
4./integraldisplay∞
0/parenleftbig
x2−ix/parenrightbigν−1e−μxdx=i√πe−iμ
2
2μν−1
2Γ(ν)H(1)
ν−1
2/parenleftBigμ
2/parenrightBig
[Reμ>0,Reν>0] ET I 138(16)
3.389
1./integraldisplayu
0x2ν−1/parenleftbig
u2−x2/parenrightbig/rho1−1eμxdx=1
2B(ν,/rho1)u2ν+2/rho1−2
1F2/parenleftbigg
ν;1
2,ν+/rho1;μ2u2
4/parenrightbigg
+μ
2B/parenleftbigg
ν+1
2,/rho1/parenrightbigg
u2ν+2/rho1−1
1F2/parenleftbigg
ν+1
2;3
2,ν+/rho1+1
2;μ2u2
4/parenrightbigg
[Re/rho1>0,Reν>0] ET II 188(21)
2.7/integraldisplay∞
0x2ν−1/parenleftbig
u2+x2/parenrightbig/rho1−1e−μxdx=u2ν+2/rho1−2
2√πΓ(1−/rho1)G31
13/parenleftbiggμ2u2
4/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−ν
1−/rho1−ν,0,1
2/parenrightbigg
/bracketleftBig
|argu|<π
2,Reμ>0,Reν>0/bracketrightBig
ET II 234(15)a
3.7/integraldisplayu
0x/parenleftbig
u2−x2/parenrightbigν−1eμxdx=u2ν
2ν+√π
2/parenleftBigμ
2/parenrightBig1
2−ν
uν+1
2Γ(ν)/bracketleftBig
Iν+1
2(μu)+Lν+1
2(μu)/bracketrightBig
[Reν>0] ET II 188(19)a
4./integraldisplay∞
ux/parenleftbig
x2−u2/parenrightbigν−1e−μxdx=2ν−1
2/parenleftbig√π/parenrightbig−1μ1
2−νuν+1
2Γ(ν)Kν+1
2(uμ)
[Re(uμ)>0] ET II 203(16)a
5./integraldisplay∞
−∞(ix)−νe−ipxdx
β2+x2=πβ−ν−1e−|p|β
/bracketleftBig
|ν|<1,Reβ>0,argix=π
2signx/bracketrightBig
ET I 118(5)
352 Exponential Functions 3.391
6./integraldisplay∞
0xνe−μx
β2+x2dx=1
2Γ(ν)βν−1/bracketleftbigg
exp/parenleftbigg
iμβ+i(ν−1)π
2/parenrightbigg
×Γ(1−ν,iβμ ) + exp/parenleftbigg
−iβμ−i(ν−1)π
2/parenrightbigg
Γ( 1−ν,−iβμ)/bracketrightbigg
[Reβ>0,Reμ>0,Reν>−1]ET II 218(22)
7./integraldisplay∞
0xν−1e−μxdx
1+x2=πcosec( νπ)Vν(2μ,0) [Re μ>0,Reν>0] ET I 138(9)
8./integraldisplay∞
−∞(β+ix)−νe−ipx
γ2+x2dx=π
γ(β+γ)−νe−pγ
[Reν>−1,p > 0,Reβ>0,Reγ>0]ET I 118(6)
9.6/integraldisplay∞
−∞(β−ix)−νe−ipx
γ2+x2dx=π
γ(β+γ)−νeγp
[p<0,Reβ>0,Reγ>0,Reν>−1]ET I 118(7)
3.391/integraldisplay∞
0/bracketleftbigg/parenleftBig/radicalbig
x+2β+√x/parenrightBig2ν
−/parenleftBig/radicalbig
x+2β−√x/parenrightBig2ν/bracketrightbigg
e−μxdx=2ν+1ν
μβνeβμKν(βμ)
[|argβ|<π , Reμ>0] ET I 140(30)
3.392
1./integraldisplay∞
0/parenleftBig
x+/radicalbig
1+x2/parenrightBigν
e−μxdx=1
μS1,ν(μ)+ν
μS0,ν(μ)
[Reμ>0] ET I 140(25)
2./integraldisplay∞
0/parenleftBig/radicalbig
1+x2−x/parenrightBigν
e−μxdx=1
μS1,ν(μ)−ν
μS0,ν(μ)
[Reμ>0] ET I 140(26)
3./integraldisplay∞
0/parenleftbig
x+√
1+x2/parenrightbigν
√
1+x2e−μxdx=πcosecνπ[J−ν(μ)−J−ν(μ)]
[Reμ>0] ET I 140(27), EH II 35(33)
4./integraldisplay∞
0/parenleftbig√
1+x2−x/parenrightbigν
√
1+x2e−μxdx=S0,ν(μ)−νS−1,ν(μ)[ R e μ>0] ET I 140(28)
3.393/integraldisplay∞
0/parenleftBig
x+/radicalbig
x2+4β2/parenrightBig2ν
/radicalbig
x3+4β2xe−μxdx
=/radicalbig
μπ3
22ν+3/2β2ν/bracketleftbig
Jν+1/4(βμ)Yν−1/4(βμ)−Jν−1/4(βμ)Yν+1/4(βμ)/bracketrightbig
[Reβ>0,Reμ>0] ET I 140(33)
3.394/integraldisplay∞
0/parenleftbig
1+√
1+x2/parenrightbigν+1/2
xν+1√
1+x2e−μxdx=√
2Γ (−ν)Dν/parenleftBig/radicalbig
2iμ/parenrightBig
Dν/parenleftBig/radicalbig
−2iμ/parenrightBig
[Reμ≥0,Reν<0] ET I 140(32)
3.411 Rational functions of powers and exponentials 353
3.395
1./integraldisplay∞
1/parenleftbig√
x2−1+x/parenrightbigν+/parenleftbig√
x2−1+x/parenrightbig−ν
√
x2−1e−μxdx=2Kν(μ)
[Reμ>0] ET I 140(29)
2./integraldisplay∞
1/parenleftbig
x+√
x2−1/parenrightbig2ν+/parenleftbig
x−√
x2−1/parenrightbig2ν
/radicalbig
x(x2−1)e−μxdx=/radicalbigg
2μ
πKν+1/4/parenleftBigμ
2/parenrightBig
Kν−1/4/parenleftBigμ
2/parenrightBig
[Reμ>0] ET I 140(34)
3./integraldisplay∞
0/parenleftbig
x+√
x2+1/parenrightbigν+c o s νπ/parenleftbig
x+√
x2+1/parenrightbig−ν
√
x2+1e−μxdx=−π[Eν(μ)+Yν(μ)]
[Reμ>0] EH II 35(34)
3.41–3.44 Combinations of rational functions of powers and exponentials
3.411
1./integraldisplay∞
0xν−1dx
eμx−1=1
μνΓ(ν)ζ(ν)[ R e μ>0,Reν>1] FI II 792a
2./integraldisplay∞
0x2n−1dx
epx−1=(−1)n−1/parenleftbigg2π
p/parenrightbigg2nB2n
4n[n=1,2,...] FI II 721a
3./integraldisplay∞
0xν−1dx
eμx+1=1
μν/parenleftbig
1−21−ν/parenrightbig
Γ(ν)ζ(ν)[ R e μ>0,Reν>0] FI II 792a, WH
4./integraldisplay∞
0x2n−1dx
epx+1=/parenleftbig
1−21−2n/parenrightbig/parenleftbigg2π
p/parenrightbigg2n|B2n|
4n[n=1,2,...] BI(83)(2), EH I 39(25)
5./integraldisplayln 2
0xdx
1−e−x=π2
12BI (104)(5)
6.8/integraldisplay∞
0xν−1e−μx
1−βe−xdx=Γ (ν)∞/summationdisplay
n=0(μ+n)−νβn=Γ (ν)Φ(β,ν,μ )
[Reμ>0 and either |β|≤1,β/negationslash=1,Reν>0; or β=1,Reν>1]EH I 27(3)
7.11/integraldisplay∞
0xν−1e−μx
1−e−βxdx=1
βνΓ(ν)ζ/parenleftbigg
ν,μ
β/parenrightbigg
[Reβ>0,Reμ>0,Reν>1]
ET I 144(10)
8./integraldisplay∞
0xn−1e−px
1+exdx=(n−1)!∞/summationdisplay
k=1(−1)k−1
(p+k)n[p>−1;n=1,2,...] BI (83)(9)
9./integraldisplay∞
0xe−xdx
ex−1=π2
6−1 (cf. 4.231 3) BI (82)(1)
10./integraldisplay∞
0xe−2xdx
e−x+1=1−π2
12(cf.4.251 6) BI (82)(2)
354 Exponential Functions 3.411
11./integraldisplay∞
0xe−3x
e−x+1dx=π2
12−3
4(cf.4.251 5) BI (82)(3)
12.11/integraldisplay∞
0xe−(2n−1)x
1+exdx=−π2
12+2n−1/summationdisplay
k=1(−1)k−1
k2(cf.4.251 6) BI (82)(5)
13.11/integraldisplay∞
0xe−2nx
1+exdx=π2
12+2n/summationdisplay
k=1(−1)k
k2(cf.4.251 5) BI (82)(4)
14.7/integraldisplay∞
0x2e−nx
1−e−xdx=2∞/summationdisplay
k=n1
k3=2/parenleftBigg
ζ(3)−n−1/summationdisplay
k=11
k3/parenrightBigg
[n=1,2,...] (cf. 4.261 12)
BI (82)(9)
15.7/integraldisplay∞
0x2e−nx
1+e−xdx=2∞/summationdisplay
k=n(−1)n+k
k3=(−1)n+1/parenleftBigg
3
2ζ(3) + 2n−1/summationdisplay
k=1(−1)k
k3/parenrightBigg
[n=1,2,...] (cf. 4.261 11)
LI (82)(10)
16./integraldisplay∞
−∞x2e−μx
1+e−xdx=π3csc3μπ/parenleftbig
2−sin2μπ/parenrightbig
[0<Reμ<1] ET I 120(17)a
17./integraldisplay∞
0x3e−nx
1−e−xdx=π4
15−6n−1/summationdisplay
k=11
k4(cf.4.262 5) BI (82)(12)
18.11/integraldisplay∞
0x3e−nx
1+e−xdx=6∞/summationdisplay
k=n(−1)n+k
k4=(−1)n+1/parenleftBigg
7
120π4+6n−1/summationdisplay
k=1(−1)k
k4/parenrightBigg
(cf.4.262 4) LI (82)(13)
19.9/integraldisplay∞
0e−px/parenleftbig
e−x−1/parenrightbigndx
x=−n/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig
ln(p+n−k) LI (89)(10)
20.9/integraldisplay∞
0e−px/parenleftbig
e−x−1/parenrightbigndx
x2=n/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig
(p+n−k)ln(p+n−k) LI (89)(15)
21./integraldisplay∞
0xn−11−e−mx
1−exdx=(n−1)!m/summationdisplay
k=11
kn(cf.4.272 11) LI (83)(8)
22.7/integraldisplay∞
0xp−1
erx−qdx=1
qrpΓ(p)∞/summationdisplay
k=1qk
kp=Γ (p)r−pΦ(q,p,1)
[p>0,r > 0,−1<q< 1]
BI (83)(5)
23./integraldisplay∞
−∞xeμxdx
β+ex=πβμ−1cosec( μπ)[lnβ−πcot(μπ)] [ |argβ|<π , 0<Reμ<1]
BI (101)(5), ET I 120(16)a
24./integraldisplay∞
−∞xeμx
eνx−1dx=/parenleftBigπ
νcosecμπ
ν/parenrightBig2
[Reν>Reμ>0] (cf. 4.254 2)
LI (101)(3)
3.414 Rational functions of powers and exponentials 355
25./integraldisplay∞
0x1+e−x
ex−1dx=π2
3−1 (cf. 4.231 4) BI (82)(6)
26./integraldisplay∞
0x1−e−x
1+e−3xe−xdx=2π2
27LI (82)(7)a
27./integraldisplay∞
01−e−μx
1+exdx
x=l n/bracketleftBigg
Γ/parenleftbigμ
2+1/parenrightbig
Γ/parenleftbigμ+1
2/parenrightbig√π/bracketrightBigg
[Reμ>−1] BI (93)(4)
28./integraldisplay∞
0e−νx−e−μx
e−x+1dx
x=l nΓ/parenleftbigν
2/parenrightbig
Γ/parenleftbigμ+1
2/parenrightbig
Γ/parenleftbigμ
2/parenrightbig
Γ/parenleftbigν+1
2/parenrightbig [Reμ>0,Reν>0] BI (93)(6)
29./integraldisplay∞
−∞epx−eqx
1+erxdx
x=l n/bracketleftBig
tanpπ
2rcotqπ
2r/bracketrightBig
[|r|>|p|,|r|>|q|,r p > 0,r q > 0]
BI (103)(3)
30./integraldisplay∞
−∞epx−eqx
1−erxdx
x=l n/bracketleftBig
sinpπ
rcosecqπ
r/bracketrightBig
[|r|>|p|,|r|>|q|,r p > 0,r q > 0]
BI (103)(4)
31./integraldisplay∞
0e−qx+e(q−p)x
1−e−pxxdx=/parenleftbiggπ
pcosecqπ
p/parenrightbigg2
[0<q<p ] BI (82)(8)
32./integraldisplay∞
0e−px−e(p−q)x
e−qx+1dx
x=l nc o tpπ
2q[0<p<q ] BI (93)(7)
3.412/integraldisplay∞
0/braceleftbigga+be−px
cepx+g+he−px−a+be−qx
ceqx+g+he−qx/bracerightbiggdx
x=a+b
c+g+hlnp
q
[p>0,q > 0] BI (96)(7)
3.413
1./integraldisplay∞
0/parenleftbig
1−e−βx/parenrightbig
(1−e−γx)e−μx
1−e−xdx
x=l nΓ(μ)Γ(β+γ+μ)
Γ(μ+β)Γ(μ+γ)
[Reμ>0,Reμ>−Reβ,Reμ>−Reγ,Reμ>−Re(β+γ)] (cf. 4.267 25)
BI (93)(13)
2./integraldisplay∞
0/braceleftbig
1−e(q−p)x/bracerightbig2
eqx−e(q−2p)xdx
x=l nc o s e cqπ
2p[0<q<p ] BI (95)(6)
3./integraldisplay∞
0e−px−e−qx
1+e−x1+e−(2n+1)x
xdx
=l n/braceleftbiggq(q+2 ) (q+4 )···(q+2n)(p+1 ) (p+3 )···(p+2n−1)
p(p+2 ) (p+4 )···(p+2n)(q+1 ) (q+3 )···(q+2n−1)/bracerightbigg
[Rep>−2n,Req>−2n] (cf. 4.267 14) BI (93)(11)
3.414/integraldisplay∞
0/parenleftbig
1−e−βx/parenrightbig
(1−e−γx)/parenleftbig
1−e−δx/parenrightbig
e−μx
1−e−xdx
x=l nΓ(μ)Γ(μ+β+γ)Γ(μ+β+δ)Γ(μ+γ+δ)
Γ(μ+β)Γ(μ+γ)Γ(μ+δ)Γ(μ+β+γ+δ)
[2 Reμ>|Reβ|+|Reγ|+|Reδ|] (cf. 4.267 31) BI (93)(14), ET I 145(17)
356 Exponential Functions 3.415
3.415
1./integraldisplay∞
0xdx
(x2+β2)(eμx−1)=1
2/bracketleftbigg
ln/parenleftbiggβμ
2π/parenrightbigg
−π
βμ−ψ/parenleftbiggβμ
2π/parenrightbigg/bracketrightbigg
[Reβ>0,Reμ>0]
BI (97)(20), EH I 18(27)
2.11/integraldisplay∞
0xdx
(x2+β2)2(e2πx−1)=−1
8β3−1
4β2+1
4βψ/prime(β)
∼1
4β4∞/summationdisplay
k=0|B2k+2|
β2k
[asymptotic expansion for Re β>0]BI(97)(22), EH I 22(12)
3.11/integraldisplay∞
0xdx
(x2+β2)(eμx+1 )=1
2/bracketleftbigg
ψ/parenleftbiggβμ
2π+1
2/parenrightbigg
−ln/parenleftbiggβμ
2π/parenrightbigg/bracketrightbigg
[Reβ>0,Reμ>0]
4.8/integraldisplay∞
0xdx
(x2+β2)2(e2πx+1 )=1
4β2−1
4βψ/prime/parenleftbigg
β+1
2/parenrightbigg
[Reβ>0,Reμ>0]
3.416
1./integraldisplay∞
0(1 +ix)2n−(1−ix)2n
idx
e2πx−1=1
22n−1
2n+1[n=1,2,...] BI (88)(4)
2./integraldisplay∞
0(1 +ix)2n−(1−ix)2n
idx
eπx+1=1
2n+1[n=1,2,...] BI (87)(1)
3.8/integraldisplay∞
0(1 +ix)2n−1−(1−ix)2n−1
idx
eπx+1=1
2n/bracketleftBig
1−22nB2n/bracketrightBig
[n=1,2,...] BI (87)(2)
3.417
1./integraldisplay∞
−∞xdx
a2ex+b2e−x=π
2ablnb
a[ab >0] (cf. 4.231 8) BI (101)(1)
2./integraldisplay∞
−∞xdx
a2ex−b2e−x=π2
4ab(cf.4.231 10) LI (101)(2)
3.418
1.6/integraldisplay∞
0xdx
ex+e−x−1=1
3/bracketleftbigg
ψ/prime/parenleftbigg1
3/parenrightbigg
−2
3π2/bracketrightbigg
=1.1719536193 ... LI (88)(1)
2.6/integraldisplay∞
0xe−xdx
ex+e−x−1=1
6/bracketleftbigg
ψ/prime/parenleftbigg1
3/parenrightbigg
−5
6π2/bracketrightbigg
=0.3118211319 ... LI (88)(2)
3./integraldisplayln 2
0xdx
ex+2e−x−2=π
8ln 2 BI (104)(7)
3.421 Rational functions of powers and exponentials 357
3.419
1./integraldisplay∞
−∞xdx
(β+ex)( 1+ e−x)=(lnβ)2
2(β−1)[|argβ|<π] (cf. 4.232 2)
BI (101)(16)
2./integraldisplay∞
−∞xdx
(β+ex)( 1−e−x)=π2+( l nβ)2
2(β+1 )[|argβ|<π] (cf. 4.232 3)
BI (101)(17)
3./integraldisplay∞
−∞x2dx
(β+ex)( 1−e−x)=/bracketleftBig
π2+( l nβ)2/bracketrightBig
lnβ
3(β+1 )[|argβ|<π] (cf. 4.261 4)
BI (102)(6)
4./integraldisplay∞
−∞x3dx
(β+ex)( 1−e−x)=/bracketleftBig
π2+( l nβ)2/bracketrightBig2
4(β+1 )[|argβ|<π] (cf. 4.262 3)
BI (102)(9)
5./integraldisplay∞
−∞x4dx
(β+ex)( 1−e−x)=/bracketleftBig
π2+( l nβ)2/bracketrightBig2
15(β+1 )/bracketleftBig
7π2+3( l n β)2/bracketrightBig
lnβ
(cf.4.263 1) BI (102)(10)
6.11/integraldisplay∞
−∞x5dx
(β+ex)( 1−e−x)=/bracketleftBig
π2+( l nβ)2/bracketrightBig2
6(β+1 )/bracketleftBig
3π2+( l nβ)2/bracketrightBig
(cf.4.264 3) BI (102)(11)
7./integraldisplay∞
−∞(x−lnβ)xdx
(β−ex)( 1−e−x)=−/bracketleftBig
4π2+( l nβ)2/bracketrightBig
lnβ
6(β−1)[|argβ|<π] (cf. 4.257 4)
BI (102)(7)
3.421
1./integraldisplay∞
0/parenleftbig
e−νx−1/parenrightbign/parenleftbig
e−ρx−1/parenrightbigme−μxdx
x2
=n/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBigm/summationdisplay
l=0(−1)l/parenleftBigm
l/parenrightBig
×{(m−l)ρ+(n−k)ν+μ}ln [(m−l)ρ+(n−k)ν+μ]
[Reν>0,Reμ>0,Reρ>0]BI (89)(17)
2./integraldisplay∞
0/parenleftbig
1−e−νx/parenrightbign/parenleftbig
1−e−ρx/parenrightbig
e−xdx
x3=1
2n/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig
(ρ+kν+1 )2
×ln(ρ+kν+1 )+1
2n/summationdisplay
k=1(−1)k−1/parenleftBign
k/parenrightBig
(kν+1 )2ln(kν+1 )
[n≥2,Reν>0,Reρ>0]BI (89)(31)
358 Exponential Functions 3.422
3./integraldisplay∞
−∞xe−μxdx
(β+e−x)(γ+e−x)=π/parenleftbig
βμ−1lnβ−γμ−1lnγ/parenrightbig
(β−γ)sinμπ+π2/parenleftbig
βμ−1−γμ−1/parenrightbig
cosμπ
(γ−β)sin2μπ
[|argβ|<π , |argγ|<π , β /negationslash=γ.0<Reμ<2]ET I 120(19)
4./integraldisplay∞
0/parenleftbig
e−px−e−qx/parenrightbig/parenleftbig
e−rx−e−sx/parenrightbig
e−xdx
x=l n(p+s+1 ) (q+r+1 )
(p+r+1 ) (q+s+1 )
[p+s>−1,p+r>−1,q > p ] (cf. 4.267 24) BI (89)(11)
5./integraldisplay∞
0/parenleftbig
1−e−px/parenrightbig/parenleftbig
1−e−qx/parenrightbig/parenleftbig
1−e−rx/parenrightbig
e−xdx
x
=(p+q+1 )l n ( p+q+1 )
+(p+r+1 )l n ( p+r+1 )+( q+r+1 )l n ( q+r+1 )
−(p+1 )l n ( p+1 )−(q+1 )l n ( q+1 )−(r+1 )l n ( r+1 )
−(p+q+r)ln(p+q+r)
[p>0,q > 0,r > 0] (cf. 4.268 3)BI (89)(14)
3.422/integraldisplay∞
−∞x(x−a)eμxdx
(β−ex)( 1−e−x)=−π2
ea−1cosec2μπ[(eαμ+1 )l n μ−2πcotμπ(eαμ−1)]
[a>0,|argβ|<π , |Reμ|<1] (cf. 4.257 5)BI (102)(8)a
3.423
1./integraldisplay∞
0xν−1
(ex−1)2dx=Γ (ν)[ζ(ν−1)−ζ(ν)] [Re ν>2] ET I 313(10)
2.6/integraldisplay∞
0xν−1e−μx
(ex−1)2dx=Γ (ν)[ζ(ν−1,μ+2 )−(μ+1 )ζ(ν,μ+2 ) ]
[Reμ>−2,Reν>2] ET I 313(11)
3.8/integraldisplay∞
0xqe−pxdx
(1−ae−px)2=Γ(q+1 )
apq+1∞/summationdisplay
k=1ak
kq[a<1,q > −1,p > 0] BI (85)(13)
4.7/integraldisplay∞
0xν−1e−μx
(1−βe−x)2dx=Γ (ν)[Φ(β;ν−1;μ)−(μ−1)Φ(β;ν;μ)]
[Reν>0,Reμ>0,|arg(1−β)|<π] (cf. 9.550 )ET I 313(12)
5./integraldisplay∞
−∞xexdx
(β+ex)2=1
βlnβ [|argβ|<π] (cf. 4.231 5)
BI (101)(10)
6.∗/integraldisplayt
0x5e−x
(1−e−x)2dx= 120 ζ(5)−∞/summationdisplay
k=1e−kt
k5/parenleftbig
y5+5y4+2 0y3+6 0y2+ 120 y+ 120/parenrightbig
= 120 ζ(5)−t5e−t/2
2s in h( t/2)−5∞/summationdisplay
k=1e−kt
k5/parenleftbig
y4+4y3+1 2y2+2 4y+2 4/parenrightbig
y=kt
3.427 Rational functions of powers and exponentials 359
3.424
1.7/integraldisplay∞
0(1 +a)ex−a
(1−ex)2e−axxndx=n!ζ(n, a)[ a>−1,n=1,2,...] BI (85)(15)
2./integraldisplay∞
0(1 +a)ex+a
(1 +ex)2e−axxndx=n!∞/summationdisplay
k=1(−1)k
(a+k)n[a>−1,n=1,2,...] BI (85)(14)
3./integraldisplay∞
−∞a2ex+b2e−x
(a2ex−b2e−x)2x2dx=π2
2ab[ab >0] BI (102)(3)a
4./integraldisplay∞
−∞a2ex−b2e−x
(a2ex+b2e−x)2x2dx=π
ablnb
a[ab >0] BI (102)(1)
5./integraldisplay∞
0ex−e−x+2
(ex−1)2x2dx=2
3π2−2 BI (85)(7)
3.425
1.7/integraldisplay∞
−∞xexdx
(a2+b2e2x)n=√πΓ/parenleftbig
n−1
2/parenrightbig
4a2n−1bΓ(n)/bracketleftbigg
2lna
2b−C−ψ/parenleftbigg
n−1
2/parenrightbigg/bracketrightbigg
[ab >0,n > 0]
BI(101)(13), LI(101)(13)
2.7/integraldisplay∞
−∞/parenleftbig
a2ex−e−x/parenrightbig
x2dx
(a2ex+e−x)p+1=−1
ap+1B/parenleftBigp
2,p
2/parenrightBig
lna [a>0,p > 0] BI (102)(5)
3.426
1./integraldisplay∞
−∞(ex−ae−x)x2dx
(a+ex)2(1 +e−x)2=(lna)2
a−1BI (102)(12)
2./integraldisplay∞
−∞(ex−ae−x)x2dx
(a+ex)2(1−e−x)2=π2+( l na)2
a+1BI (102)(13)
3.427
1./integraldisplay∞
0/parenleftbigge−x
x+e−μx
e−x−1/parenrightbigg
dx=ψ(μ)[ R e μ>0] (cf. 4.281 4) WH
2.7/integraldisplay∞
0/parenleftbigg1
1−e−x−1
x/parenrightbigg
e−xdx=C (cf.4.281 1) BI (94)(1)
3./integraldisplay∞
0/parenleftbigg1
2−1
1+e−x/parenrightbigge−2x
xdx=1
2lnπ
4BI (94)(5)
4./integraldisplay∞
0/parenleftbigg1
2−1
x+1
ex−1/parenrightbigge−μx
xdx=l nΓ ( μ)−/parenleftbigg
μ−1
2/parenrightbigg
lnμ+μ−1
2ln(2π)
[Reμ>0] WH
5./integraldisplay∞
0/parenleftbigg1
2e−2x−1
ex+1/parenrightbiggdx
x=−1
2lnπ BI (94)(6)
6./integraldisplay∞
0/parenleftbiggeμx−1
1−e−x−μ/parenrightbigge−x
xdx=−ln Γ(μ)−ln sin( πμ)+l n π
[Reμ<1] EH I 21(6)
360 Exponential Functions 3.428
7./integraldisplay∞
0/parenleftbigge−νx
1−e−x−e−μx
x/parenrightbigg
dx=l nμ−ψ(ν) (cf. 4.281 5) BI (94)(3)
8./integraldisplay∞
0/parenleftbiggn
x−e−μx
1−e−x/n/parenrightbigg
e−xdx=nψ(nμ+n)−nlnn
[Reμ>0,n=1,2,...] BI (94)(4)
9./integraldisplay∞
0/parenleftbigg
μ−1−e−μx
1−e−x/parenrightbigge−x
xdx=l nΓ ( μ+1 ) [ R e μ>−1] WH
10./integraldisplay∞
0/parenleftbigg
νe−x−e−μx−e−(μ+ν)x
ex−1/parenrightbiggdx
x=l nΓ(μ+ν+1 )
Γ(μ+1 )
[Reμ>−1,Reν>0] BI (94)(8)
11./integraldisplay∞
0/bracketleftBig
(1−ex)−1+x−1−1/bracketrightBig
e−xzdx=ψ(z)−lnz [Rez>0] EH I 18(24)
3.428
1./integraldisplay∞
0/parenleftbigg
νe−μx−1
μe−x−1
μe−1−e−μνx
1−e−x/parenrightbiggdx
x=1
μln Γ(μν)−νlnμ
[Reμ>0,Reν>0] BI (94)(18)
2./integraldisplay∞
0/parenleftbiggn−1
2+n−1
1−e−x+e(1−μ)x
1−ex/n+e−nμx
1−e−x/parenrightbigg
e−xdx
x=n−1
2ln 2π−/parenleftbigg
nμ+1
2/parenrightbigg
lnn
[Reμ>0,n=1,2,...] BI (94)(14)
3./integraldisplay∞
0/parenleftbigg
nμ−n−1
2−n
1−e−x−e(1−μ)x
1−ex/n/parenrightbigge−x
xdx=n−1/summationdisplay
k=0ln Γ/parenleftbigg
μ−k
n+1/parenrightbigg
[Reμ>0,n=1,2,...] BI (94)(13)
4./integraldisplay∞
0/parenleftbigge−νx
1−ex−e−μνx
1−eμx−ex
1−ex+eμx
1−eμx/parenrightbiggdx
x=νlnμ
[Reμ>0,Reν>0] LI (94)(15)
5./integraldisplay∞
0/bracketleftbigg1
ex−1−μe−μx
1−e−μx+/parenleftbigg
aμ−μ+1
2/parenrightbigg
e−μx+( 1−aμ)e−x/bracketrightbiggdx
x
=μ−1
2ln(2π)+/parenleftbigg1
2−aμ/parenrightbigg
lnμ
[Reμ>0] BI (94)(16)
6./integraldisplay∞
0/bracketleftbigge−νx
1−e−x−e−μνx
1−e−μx−(μ−1)e−μx
1−e−μx−μ−1
2e−μx/bracketrightbiggdx
x=μ−1
2ln(2π)+/parenleftbigg1
2−μν/parenrightbigg
lnμ
[Reμ>0,Reν>0] (cf. 4.267 37) BI (94)(17)
7./integraldisplay∞
0/bracketleftbigg
1−e−x−(1−e−νx)(1−e−μx)
1−e−x/bracketrightbiggdx
x=l nB ( μ, ν)
[Reμ>0,Reν>0] BI (94)(12)
3.429/integraldisplay∞
0/bracketleftbig
e−x−(1 +x)−μ/bracketrightbigdx
x=ψ(μ)[ R e μ>0] NH 184(7)
3.437 Rational functions of powers and exponentials 361
3.431
1./integraldisplay∞
0/parenleftbigg
e−μx−1+μx−1
2μ2x2/parenrightbigg
xν−1dx=−1
ν(ν+1 ) (ν+2 )μνΓ(ν+3 )
[Reμ>0,−2>Reν>−3]
LI (90)(5)
2./integraldisplay∞
0/bracketleftbigg
x−1−1
2x−2(x+2 )/parenleftbig
1−e−x/parenrightbig/bracketrightbigg
e−pxdx=−1+/parenleftbigg
p+1
2/parenrightbigg
ln/parenleftbigg
1+1
p/parenrightbigg
[Rep>0] ET I 144(6)
3.432
1./integraldisplay∞
0xν−1e−mx/parenleftbig
e−x−1/parenrightbigndx=Γ (ν)n/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig1
(n+m−k)ν
[n=0,1,...,Reν>0] LI (90)(10)
2./integraldisplay∞
0/bracketleftBig
xν−1e−x−e−μx/parenleftbig
1−e−x/parenrightbigν−1/bracketrightBig
dx=Γ (ν)−Γ(μ)
Γ(μ+ν)
[Reμ>0,Reν>0] LI (81)(14)
3.433/integraldisplay∞
0xp−1/bracketleftBigg
e−x+n/summationdisplay
k=1(−1)kxk−1
(k−1)!/bracketrightBigg
dx=Γ (p)[ −n<p< −n+1,n=0,1,...]
FI II 805
3.434
1./integraldisplay∞
0e−νx−e−μx
xρ+1dx=μρ−νρ
ρΓ(1−ρ)[ R e μ>0,Reν>0,Reρ<1]
BI (90)(6)
2./integraldisplay∞
0e−μx−e−νx
xdx=l nν
μ[Reμ>0,Reν>0] FI II 634
3.435
1./integraldisplay∞
0/braceleftbig
(x+1 )e−x−e−x
2/bracerightbigdx
x=1−ln 2 LI (89)(19)
2.11/integraldisplay∞
01−e−μx
x(x+β)dx=1
β/bracketleftbig
ln (βμ)+C−eβμEi(−βμ)/bracketrightbig
[|argβ|<π , Reμ>0] ET II 217 (18)
3./integraldisplay∞
0/parenleftbigg1
1+x−e−x/parenrightbiggdx
x=C FI II 7 95, 802
4./integraldisplay∞
0/parenleftbigg
e−μx−1
1+ax/parenrightbiggdx
x=l na
μ−C [a>0,Reμ>0] BI (92)(10)
3.436/integraldisplay∞
0/braceleftbigge−npx−e−nqx
n−e−mpx−e−mqx
m/bracerightbiggdx
x2=(q−p)lnm
n[p>0,q > 0] BI (89)(28)
3.437/integraldisplay∞
0/braceleftbigg
pe−x−1−e−px
x/bracerightbiggdx
x=plnp−p [p>0] BI (89)(24)
362 Exponential Functions 3.438
3.438
1./integraldisplay∞
0/braceleftbigg/parenleftbigg1
2+1
x/parenrightbigg
e−x−1
xe−x
2/bracerightbiggdx
x=ln 2−1
2BI (89)(19)
2.7/integraldisplay∞
0/braceleftbiggp2
6e−x−p2
2x−p
x2−1−e−px
x3/bracerightbiggdx
x=p2
6lnp−11
36p3
[p>0] BI (89)(33)
3./integraldisplay∞
0/parenleftbigg
e−x−e−2x−1
xe−2x/parenrightbiggdx
x=1−ln2 BI (89)(25)
4./integraldisplay∞
0/braceleftbigg/parenleftbigg
p−1
2/parenrightbigg
e−x+x+2
2x/parenleftbig
e−px−e−x
2/parenrightbig/bracerightbiggdx
x=/parenleftbigg
p−1
2/parenrightbigg
(lnp−1)
[p>0] BI (89)(22)
3.439/integraldisplay∞
0/braceleftbigg
(p−q)e−rx+1
mx/parenleftbig
e−mpx−e−mqx/parenrightbig/bracerightbiggdx
x=plnp−qlnq−(p−q)/parenleftBig
1+l nr
m/parenrightBig
[p>0,q > 0,r > 0]LI(89)(26), LI(89)(27)
3.441/integraldisplay∞
0/braceleftbig
(p−r)e−qx+(r−q)e−px+(q−p)e−rx/bracerightbigdx
x2=(r−q)plnp+(p−r)qlnq+(q−p)rlnr
[p>0,q > 0,r > 0] (cf. 4.268 6)BI (89)(18)
3.442
1./integraldisplay∞
0/braceleftbigg
1−x+2
2x/parenleftbig
1−e−x/parenrightbig/bracerightbigg
e−qxdx
x=−1+/parenleftbigg
q+1
2/parenrightbigg
lnq+1
q
[q>0] BI (89)(23)
2./integraldisplay∞
0/parenleftbigge−x−1
x+1
1+x/parenrightbiggdx
x=C−1 BI (92)(16)
3./integraldisplay∞
0/parenleftbigg
e−px−1
1+a2x2/parenrightbiggdx
x=−C+l na
p[p>0] BI (92)(11)
3.443
1./integraldisplay∞
0/braceleftbigge−xp2
2−p
x+1−e−px
x2/bracerightbiggdx
x=p2
2lnp−3
4p2[p>0] BI (89)(32)
2./integraldisplay∞
0(1−e−px)ne−qx
x3dx=1
2n/summationdisplay
k=2(−1)k−1/parenleftBign
k/parenrightBig
(q+kp)2ln(q+kp)
[n>2,q > 0,p n +q>0] (cf. 4.268 4)BI (89)(30)
3./integraldisplay∞
0/parenleftbig
1−e−px/parenrightbig2e−qxdx
x2=( 2p+q)ln (2p+q)−2(p+q)ln (p+q)+qlnq
[q>0,2p>−q] (cf. 4.268 2)
BI (89)(13)
3.456 Powers and algebraic functions of exponentials 363
3.45 Combinations of powers and algebraic functions of exponentials
3.451
1./integraldisplay∞
0xe−x√
1−e−xdx=4
3/parenleftbigg4
3−ln2/parenrightbigg
BI (99)(1)
2./integraldisplay∞
0xe−x/radicalbig
1−e−2xdx=π
4/parenleftbigg1
2+l n2/parenrightbigg
(cf.4.241 9) BI (99)(2)
3.452
1./integraldisplay∞
0xdx√ex−1=2πln 2 FI II 643a,BI(99)(4)
2./integraldisplay∞
0x2dx√ex−1=4π/braceleftbigg
(ln2)2+π2
12/bracerightbigg
BI (99)(5)
3./integraldisplay∞
0xe−xdx√ex−1=π
2[2ln 2 −1] BI (99)(6)
4./integraldisplay∞
0xe−xdx√
e2x−1=1−ln2 BI (99)(8)
5./integraldisplay∞
0xe−2xdx√ex−1=3
4π/parenleftbigg
ln2−7
12/parenrightbigg
BI (99)(7)
3.453
1./integraldisplay∞
0xex
a2ex−(a2−b2)dx√ex−1=2π
abln/parenleftbigg
1+b
a/parenrightbigg
[ab >0] (cf. 4.298 17) BI (99)(16)
2./integraldisplay∞
0xexdx
[a2ex−(a2+b2)]√ex−1=2π
abarctanb
a[ab >0] (cf. 4.298 18) BI (99)(17)
3.454
1.11/integraldisplay∞
0xe−2nxdx√
e2x−1=(2n−1)!!
(2n)!!π
2/braceleftBigg
ln2 +2n/summationdisplay
k=1(−1)k
k/bracerightBigg
LI (99)(10)
2./integraldisplay∞
0xe−(2n−1)xdx√
e2x−1=−(2n−2)!!
(2n−1)!!/braceleftBigg
ln2 +2n−1/summationdisplay
k=1(−1)k
k/bracerightBigg
LI (99)(9)
3.455
1./integraldisplay∞
0x2exdx/radicalBig
(ex−1)3=8πln 2 BI (99)(11)
2./integraldisplay∞
0x3exdx/radicalBig
(ex−1)3=2 4π/bracketleftbigg
(ln 2)2+π2
12/bracketrightbigg
BI (99)(12)
3.456
1./integraldisplay∞
0xdx
3√
e3x−1=π
3√
3/bracketleftbigg
ln 3 +π
3√
3/bracketrightbigg
BI (99)(13)
364 Exponential Functions 3.457
2./integraldisplay∞
0xdx
3/radicalBig
(e3x−1)2=π
3√
3/bracketleftbigg
ln 3−π
3√
3/bracketrightbigg
(cf.4.244 3) BI (99)(14)
3.457
1./integraldisplay∞
0xe−x/parenleftbig
1−e−2x/parenrightbign−1/2dx=(2n−1)!!
4·(2n)!!π[C+ψ(n+1 )+2l n2 ]
(cf.4.241 5) BI (99)(3)
2./integraldisplay∞
−∞xexdx
(a+ex)n+3/2=2
(2n+1 )an+1/2[ln(4a)−3C−2ψ(2n)−ψ(n)] BI (101)(12)
3./integraldisplay∞
−∞xdx
(a2ex+e−x)μ=−1
2aμB/parenleftBigμ
2,μ
2/parenrightBig
lna [a>0,Reμ>0] BI (101)(14)
3.458
1.7/integraldisplayln 2
0xex(ex−1)p−1dx=1
p/bracketleftBigg
ln2 +∞/summationdisplay
k=0(−1)k−1
p+k+1/bracketrightBigg
BI (104)(4)
2./integraldisplay∞
−∞xexdx
(a+ex)ν+1=1
νaν[lna−C−ψ(ν)] [ a>0]
=1
νaν/bracketleftBigg
lna−ν−1/summationdisplay
k=11
k/bracketrightBigg
[a>0,ν=1,2,...]
BI (101)(11)
3.46–3.48 Combinations of exponentials of more complicated arguments and powers
3.461
1./integraldisplay∞
ue−p2x2
x2ndx=(−1)n2n−1p2n−1√π
(2n−1)!![1−Φ(pu)]
+e−p2u2
2u2n−1n−1/summationdisplay
k=0(−1)k2k+1(pu)2k
(2n−1)(2n−3)···(2n−2k−1)
[p>0] NT 21(4)
2./integraldisplay∞
0x2ne−px2dx=(2n−1)!!
2(2p)n/radicalbiggπ
p[p>0,n=0,1,...] FI II 743
3./integraldisplay∞
0x2n+1e−px2dx=n!
2pn+1[p>0] BI (81)(7)
4./integraldisplay∞
−∞(x+ai)2ne−x2dx=(2n−1)!!
2n√πn/summationdisplay
k=0(−1)k(2a)2kn!
(2k)!(n−k)!BI (100)(12)
5.11/integraldisplay∞
ue−μx2dx
x2=1
ue−μu2−√μπ[1−Φ(u√μ)]/bracketleftBig
|argμ|<π
2,u > 0/bracketrightBig
ET I 135(19)a
6.∗/integraldisplay∞
0exp/parenleftBig
−a/radicalbig
x2+b2/parenrightBig
dx=bK1(ab)[ R e a>0,Reb>0]
3.462 Exponentials of complicated arguments and powers 365
7.∗/integraldisplay∞
0x2exp/parenleftBig
−a/radicalbig
x2+b2/parenrightBig
dx=2b
a2K1(ab)+b2
aK0(ab)
[Rea>0,Reb>0]
8.∗/integraldisplay∞
0x4exp/parenleftBig
−a/radicalbig
x2+b2/parenrightBig
dx=12b2
a3K2(ab)+3b3
a2K1(ab)
[Rea>0,Reb>0]
9.∗/integraldisplay∞
0x6exp/parenleftBig
−a/radicalbig
x2+b2/parenrightBig
dx=90b3
a4K3(ab)+15b4
a3K2(ab)
[Rea>0,Reb>0]
3.462
1./integraldisplay∞
0xν−1e−βx2−γxdx=( 2β)−ν/2Γ(ν)exp/parenleftbiggγ2
8β/parenrightbigg
D−ν/parenleftbiggγ√2β/parenrightbigg
[Reβ>0,Reν>0]
EH II 119(3)a, ET I 313(13)
2.8/integraldisplay∞
−∞xne−px2+2qxdx=1
2n−1p/radicalbiggπ
pdn−1
dqn−1/parenleftBig
qeq2/p/parenrightBig
[p>0] BI (100)(8)
=n!eq2/p/radicalbiggπ
p/parenleftbiggq
p/parenrightbiggn⌊n/2⌋/summationdisplay
k=01
(n−2k)!(k)!/parenleftbiggp
4q2/parenrightbiggk
[p>0] LI (100)(8)
3.11/integraldisplay∞
−∞(ix)νe−β2x2−iqxdx=2−ν
2√πβ−ν−1exp/parenleftbigg
−q2
8β2/parenrightbigg
Dν/parenleftbiggq
β√
2/parenrightbigg
/bracketleftBig
Reβ2>0,Reν>−1,argix=π
2signx/bracketrightBig
ET I 121(23)
4./integraldisplay∞
−∞xnexp/bracketleftbig
−(x−β)2/bracketrightbig
dx=( 2i)−n√πHn(iβ) EH II 195(31)
5.11/integraldisplay∞
0xe−μx2−2νxdx=1
2μ−ν
2μ/radicalbiggπ
μeν2
μ/bracketleftbigg
1−erf/parenleftbiggν√μ/parenrightbigg/bracketrightbigg
/bracketleftBig
|argν|<π
2,Reμ>0/bracketrightBig
ET I 146(31)a
6./integraldisplay∞
−∞xe−px2+2qxdx=q
p/radicalbiggπ
pexp/parenleftbiggq2
p/parenrightbigg
[Rep>0] BI (100)(7)
7.11/integraldisplay∞
0x2e−μx2−2νxdx=−ν
2μ2+/radicalbiggπ
μ52ν2+μ
4eν2
μ/bracketleftbigg
1−erf/parenleftbiggν√μ/parenrightbigg/bracketrightbigg
/bracketleftBig
|argν|<π
2,Reμ>0/bracketrightBig
ET I 146(32)
8./integraldisplay∞
−∞x2e−μx2+2νxdx=1
2μ/radicalbiggπ
μ/parenleftbigg
1+2ν2
μ/parenrightbigg
eν2
μ [|argν|<π , Reμ>0] BI (100)(8)a
9.∗/integraldisplay∞
0e−βxn±adx=e±a
nβ1/nΓ/parenleftbigg1
n/parenrightbigg
[Reβ>0,Ren>0]
366 Exponential Functions 3.462
10.∗/integraldisplay∞
0(x−a)e−β(x−a)dx=eaβ(1−aβ)
β2[Reβ>0]
11.∗/integraldisplay∞
0(x−a)e−β(x+a)dx=e−aβ(1−aβ)
β2[Reβ>0]
12.∗/integraldisplay∞
0(ax±b)me−pxdx=ame±pb/a
pm+1Γ/parenleftbigg
m+1,±pb
a/parenrightbigg/bracketleftbigg
p>0,/vextendsingle/vextendsingle/vextendsingle/vextendsinglearg/parenleftbiggb
a/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle<π/bracketrightbigg
13.∗/integraldisplay∞
u(ax±b)me−pxdx=ame±pb/a
pm+1Γ/parenleftbigg
m+1,p u±pb
a/parenrightbigg
/bracketleftbigg
p>0,/vextendsingle/vextendsingle/vextendsingle/vextendsinglearg/parenleftbiggb
a±u/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle<π/bracketrightbigg
14.∗/integraldisplayu
0(ax±b)me−pxdx=ame±pb/a
pm+1/bracketleftbigg
Γ/parenleftbigg
m+1,±pb
a/parenrightbigg
−Γ/parenleftbigg
m+1,p u±pb
a/parenrightbigg/bracketrightbigg
/bracketleftbigg
u>0,p > 0,/vextendsingle/vextendsingle/vextendsingle/vextendsinglearg/parenleftbiggb
a±u/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle<π/bracketrightbigg
15.
∗/integraldisplay∞
0e−px
(ax±b)ndx=pn−1e±pb/a
anΓ/parenleftbigg
−n+1,±pb
a/parenrightbigg/bracketleftbigg
p>0,/vextendsingle/vextendsingle/vextendsingle/vextendsinglearg/parenleftbiggb
a/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle<π/bracketrightbigg
16.∗/integraldisplay∞
ue−px
(ax±b)ndx=pn−1e±pb/a
anΓ/parenleftbigg
−n+1,p u±pb
a/parenrightbigg
/bracketleftbigg
u>0,p > 0,/vextendsingle/vextendsingle/vextendsingle/vextendsinglearg/parenleftbiggb
a±u/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle<π/bracketrightbigg
17.
∗/integraldisplayu
0e−px
(ax±b)ndx=pn−1e±pb/a
an/bracketleftbigg
Γ/parenleftbigg
−n+1,±pb
a/parenrightbigg
−Γ/parenleftbigg
−n+1,p u±pb
a/parenrightbigg/bracketrightbigg
/bracketleftbigg
u>0,p > 0,/vextendsingle/vextendsingle/vextendsingle/vextendsinglearg/parenleftbiggb
a±u/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle<π/bracketrightbigg
18.
∗/integraldisplay∞
0/parenleftbiggx−a
b/parenrightbiggj
exp/parenleftBigg
−β/parenleftbiggx−a
b/parenrightbiggk/parenrightBigg
dx=bΓ/parenleftBig
j+1
k,β/parenleftbig
−a
b/parenrightbigk/parenrightBig
kβ(j+1)/k
/bracketleftBig
arg/parenleftBig
−a
b/parenrightBig
>0,Reb>0,Reβ>0,Rek>0/bracketrightBig
19.∗/integraldisplay∞
ue−βxn
xmdx=Γ(z,βun)
nβzz=1−m
n[u>0,Reβ>0,Ren>0,Rez>0]
20.∗/integraldisplay∞
0exp/parenleftbig
−a√
x+b2/parenrightbig
√
x2+b2dx=K0(ab)[ R e a>0,Reb>0]
21.∗/integraldisplay∞
0x2exp/parenleftbig
−a√
x+b2/parenrightbig
√
x2+b2dx=b
aK1(ab)[ R e a>0,Reb>0]
22.∗/integraldisplay∞
0x4exp/parenleftbig
−a√
x+b2/parenrightbig
√
x2+b2dx=3b2
a2K1(ab)[ R e a>0,Reb>0]
23.∗/integraldisplay∞
0x6exp/parenleftbig
−a√
x+b2/parenrightbig
√
x2+b2dx=15b3
a3K3(ab)[ R e a>0,Reb>0]
3.471 Exponentials of complicated arguments and powers 367
24.∗/integraldisplay∞
0x2nexp/parenleftbig
−a√
x+b2/parenrightbig
√
x2+b2dx=( 2n−1)!!/parenleftbiggb
a/parenrightbiggn
Kn(ab)
[Rea>0,Reb>0]
25.∗/integraldisplay∞
0exp/parenleftbig
−px2/parenrightbig
√
a2+x2dx=1
2exp/parenleftbigga2p
2/parenrightbigg
K0/parenleftbigga2p
2/parenrightbigg
[Rea>0,Reb>0]
3.463/integraldisplay∞
0/parenleftBig
e−x2−e−x/parenrightBigdx
x=1
2C BI (89)(5)
3.464/integraldisplay∞
0/parenleftBig
e−μx2−e−νx2/parenrightBigdx
x2=√π/parenleftbig√ν−√μ/parenrightbig
[Reμ>0,Reν>0] FI II 645
3.465/integraldisplay∞
0/parenleftbig
1+2βx2/parenrightbig
e−μx2dx=μ+β
2/radicalbiggπ
μ3[Reμ>0] ET I 136(24)a
3.466
1./integraldisplay∞
0e−μ2x2
x2+β2dx=[ 1−Φ(βμ)]π
2βeβ2μ2/bracketleftBig
Reβ>0,|argμ|<π
4/bracketrightBig
NT 19(13)
2./integraldisplay∞
0x2e−μ2x2
x2+β2dx=√π
2μ−πβ
2eμ2β2[1−Φ(βμ)]/bracketleftBig
Reβ>0,|argμ|<π
4/bracketrightBig
ET II 217(16)
3./integraldisplay1
0ex2−1
x2dx=∞/summationdisplay
k=11
k!(2k−1)FI II 683
3.467/integraldisplay∞
0/parenleftbigg
e−x2−1
1+x2/parenrightbiggdx
x=−1
2C BI (92)(12)
3.468
1./integraldisplay∞
u√
2e−x2
√
x2−u2dx
x=π
4u[1−Φ(u)]2[u>0] NT 33(17)
2./integraldisplay∞
0xe−μx2dx√
a2+x2=1
2/radicalbiggπ
μea2μ[1−Φ(a√μ)] [Re μ>0,a > 0] NT 19(11)
3.469
1./integraldisplay∞
0e−μx4−2νx2dx=1
4/radicalbigg2ν
μexp/parenleftbiggν2
2μ/parenrightbigg
K1
4/parenleftbiggν2
2μ/parenrightbigg
[Reμ≥0] ET I 146(23)
2./integraldisplay∞
0/parenleftBig
e−x4−e−x/parenrightBigdx
x=3
4C BI (89)(7)
3./integraldisplay∞
0/parenleftBig
e−x4−e−x2/parenrightBigdx
x=1
4C BI (89)(6)
3.471
1./integraldisplayu
0exp/parenleftbigg
−β
x/parenrightbiggdx
x2=1
βexp/parenleftbigg
−β
u/parenrightbigg
ET II 188(22)
2./integraldisplayu
0xν−1(u−x)μ−1e−β
xdx=βν−1
2u2μ+ν−1
2exp/parenleftbigg
−β
2u/parenrightbigg
Γ(μ)W1−2μ−ν
2,ν
2/parenleftbiggβ
u/parenrightbigg
[Reμ>0,Reβ>0,u > 0]
ET II 187(18)
368 Exponential Functions 3.471
3./integraldisplayu
0x−μ−1(u−x)μ−1e−β
xdx=β−μuμ−1Γ(μ)exp/parenleftbigg
−β
u/parenrightbigg
[Reμ>0,u > 0] ET II 187(16)
4./integraldisplayu
0x−2μ(u−x)μ−1e−β
xdx=1√πuβ1
2−μe−β
2uΓ(μ)Kμ−1
2/parenleftbiggβ
2u/parenrightbigg
[u>0,Reβ>0,Reμ>0]
ET II 187(17)
5./integraldisplay∞
uxν−1(x−u)μ−1eβ
xdx=B ( 1 −μ−ν,μ)uμ+ν−1
1F1/parenleftbigg
1−μ−ν;1−ν;β
u/parenrightbigg
[0<Reμ<Re(1−ν),u > 0]
ET II 203(15)
6./integraldisplay∞
ux−2μ(x−u)μ−1eβ
xdx=/radicalbiggπ
uβ1
2−μΓ(μ)exp/parenleftbiggβ
2u/parenrightbigg
Iμ−1
2/parenleftbiggβ
2u/parenrightbigg
[Reμ>0,u > 0] ET II 202(14)
7./integraldisplay∞
0xν−1(x+γ)μ−1e−β
xdx=βν−1
2γν−1
2+μΓ(1−μ−ν)eβ
2γW ν−1
2+μ,−ν
2/parenleftbiggβ
γ/parenrightbigg
[|argγ|<π , Re(1−μ)>Reν>0]
ET II 234(13)a
8./integraldisplayu
0x−2μ/parenleftbig
u2−x2/parenrightbigμ−1e−β
xdx=1√π/parenleftbigg2
β/parenrightbiggμ−1
2
uμ−3
2Γ(μ)Kμ−1
2/parenleftbiggβ
u/parenrightbigg
[Reβ>0,u > 0,Reμ>0]
ET II 188(23)a
9./integraldisplay∞
0xν−1e−β
x−γxdx=2/parenleftbiggβ
γ/parenrightbiggν
2
Kν/parenleftBig
2/radicalbig
βγ/parenrightBig
[Reβ>0,Reγ>0]
ET II 82(23)a, LET I 146(29)
10./integraldisplay∞
0xν−1exp/bracketleftbiggiμ
2/parenleftbigg
x−β2
x/parenrightbigg/bracketrightbigg
dx=2βνeiνπ
2K−ν(βμ)
/bracketleftbig
Imμ>0,Im/parenleftbig
β2μ/parenrightbig
<0; note that K−ν≡Kν/bracketrightbig
EH II 82(24)
11./integraldisplay∞
0xν−1exp/bracketleftbiggiμ
2/parenleftbigg
x+β2
x/parenrightbigg/bracketrightbigg
dx=iπβνe−iνπ
2H(1)
−ν(βμ)
/bracketleftbig
Imμ>0,Im/parenleftbig
β2μ/parenrightbig
>0/bracketrightbig
EH II 21(33)
12./integraldisplay∞
0xν−1exp/parenleftbigg
−x−μ2
4x/parenrightbigg
dx=2/parenleftBigμ
2/parenrightBigν
K−ν(μ)
/bracketleftBig
|argμ|<π
2,Reμ2>0; note that K−ν≡Kν/bracketrightBig
WA 203(15)
13./integraldisplay∞
0xν−1e−β
x
x+γdx=γν−1eβ
γΓ(1−ν)Γ/parenleftbigg
ν,β
γ/parenrightbigg
[|argγ|<π , Reβ>0,Reν<1]
ET II 218(19)
3.475 Exponentials of complicated arguments and powers 369
14./integraldisplay1
0exp/parenleftbig
1−1
x/parenrightbig
−xν
x(1−x)dx=ψ(ν)[ R e ν>0] BI (80)(7)
15./integraldisplay∞
0x−1
2e−γx−β/xdx=/radicalbiggπ
γe−2√βγ[Reβ≥0,Reγ>0] ET 245 (5.6.1)
16./integraldisplay∞
0xn−1
2e−px−q/xdx=(−1)n√π∂n
∂pn/parenleftBig
p−1/2e−2√pq/parenrightBig
[Rep>0,Req>0]
PBM 344 (2.3.16(2))
3.472
1./integraldisplay∞
0/parenleftBig
exp/parenleftBig
−a
x2/parenrightBig
−1/parenrightBig
e−μx2dx=1
2/radicalbiggπ
μ[exp (−2√aμ)−1]
[Reμ>0,Rea>0] ET I 146(30)
2./integraldisplay∞
0x2exp/parenleftBig
−a
x2−μx2/parenrightBig
dx=1
4/radicalbiggπ
μ3(1 + 2√aμ)e x p(−2√aμ)
[Reμ>0,Rea>0] ET I 146(26)
3./integraldisplay∞
0exp/parenleftBig
−a
x2−μx2/parenrightBigdx
x2=1
2/radicalbiggπ
aexp (−2√aμ)[ R e μ>0,a > 0] ET I 146(28)a
4./integraldisplay∞
0exp/bracketleftbigg
−1
2a/parenleftbigg
x2+1
x2/parenrightbigg/bracketrightbiggdx
x4=/radicalbiggaπ
2(1 +a)e−1/a[a>0] BI (98)(14)
5./integraldisplay∞
0x−n−1/2e−px−q/xdx=(−1)n/radicalbiggπ
p∂n
∂qne−2√pq[Rep>0,Req>0]
PBM 344 (2.3.16(3))
3.473/integraldisplay∞
0exp(−xn)x(m+1/2)n−1dx=(2m−1)!!
2mn√π BI (98)(6)
3.474
1./integraldisplay1
0/braceleftbiggnexp (1 −x−n)
1−xn−xnp
1−x/bracerightbiggdx
x=1
nn/summationdisplay
k=1ψ/parenleftbigg
p+k−1
n/parenrightbigg
[p>0] BI (80)(8)
2./integraldisplay1
0/braceleftBigg
nexp (1 −x−n)
1−xn−exp/parenleftbig
1−1
x/parenrightbig
1−x/bracerightBigg
dx
x=−lnn BI (80)(9)
3.475
1.7/integraldisplay∞
0/braceleftbigg
exp/parenleftBig
−x2n/parenrightBig
−1
1+x2n+1/bracerightbiggdx
x=−1
2nC [n∈Z] BI (92)(14)
2./integraldisplay∞
0/braceleftbigg
exp/parenleftBig
−x2n/parenrightBig
−1
1+x2/bracerightbiggdx
x=−2−nC BI (92)(13)
3./integraldisplay∞
0/braceleftBig
exp/parenleftBig
−x2n/parenrightBig
−e−x/bracerightBigdx
x=/parenleftbig
1−2−n/parenrightbig
C BI (89)(8)
370 Exponential Functions 3.476
3.476
1./integraldisplay∞
0[exp (−νxp)−exp (−μxp)]dx
x=1
plnμ
ν[Reμ>0,Reν>0] BI (89)(3)
2./integraldisplay∞
0[exp (−xp)−exp (−xq)]dx
x=p−q
pqC [p>0,q > 0] BI (89)(9)
3.477
1.10/integraldisplay∞
−∞e−a|x|
x−udx=e−auγ(0,−au)−eauγ(0,au)[ R e a>0,Imu/negationslash=0,argu/negationslash=0 ] MC
2.8/integraldisplay∞
−∞signxexp (−a|x|)
x−udx=−[exp(a|u|)Ei(−a|u|)−exp (−a|u|)Ei(a|u|)]
[a>0] ET II 251(36)
3.478
1./integraldisplay∞
0xν−1exp (−μxp)dx=1
pμ−ν
pΓ/parenleftbiggν
p/parenrightbigg
[Reμ>0,Reν>0,p > 0]
BI(81)(8)a, ET I 313(15, 16)
2./integraldisplay∞
0xν−1[1−exp (−μxp)]dx=−1
|p|μ−ν
pΓ/parenleftbiggν
p/parenrightbigg
[Reμ>0a n d −p<Reν<0f o rp>0,0<Reν<−pforp<0]ET I 313(18, 19)
3.11/integraldisplayu
0xν−1(u−x)μ−1exp (βxn)dx=B (μ, ν)uμ+ν−1
nFn/parenleftbiggν
n,ν+1
n,...,ν+n−1
n;
μ+ν
n,μ+ν+1
n,...,μ+ν+n−1
n;βun/parenrightbigg
[Reμ>0,Reν>0,n=2,3,...]ET II 187(15)
4./integraldisplay∞
0xν−1exp/parenleftbig
−βxp−γx−p/parenrightbig
dx=2
p/parenleftbiggγ
β/parenrightbiggν
2p
Kν
p/parenleftBig
2/radicalbig
βγ/parenrightBig
[Reβ>0,Reγ>0] ET I 313(17)
3.479
1./integraldisplay∞
0xν−1exp/parenleftbig
−β√1+x/parenrightbig
√1+xdx=2√π/parenleftbiggβ
2/parenrightbigg1
2−ν
Γ(ν)K1
2−ν(β)
[Reβ>0,Reν>0] ET I 313(14)
2.11/integraldisplay∞
0xν−1exp/parenleftbig
iμ√
1+x2/parenrightbig
√
1+x2dx=i√π
2/parenleftBigμ
2/parenrightBig1−ν
2Γ/parenleftBigν
2/parenrightBig
H(1)
1−ν
2(μ)
[Imμ>0,Reν>0] EH II 83(30)
3.481
1./integraldisplay∞
−∞xexexp (−μex)dx=−1
μ(C+l nμ)[ R e μ>0] BI (100)(13)
3.511 Hyperbolic functions 371
2./integraldisplay∞
−∞xexexp/parenleftbig
−μe2x/parenrightbig
dx=−1
4[C+l n ( 4 μ)]/radicalbiggπ
μ[Reμ>0] BI (100)(14)
3.482
1.3/integraldisplay∞
0exp(nx−βsinhx)dx=1
2[Sn(β)−πEn(β)−πYn(β)]
[Reβ>0] ET I 168(11)
2./integraldisplay∞
0exp(−nx−βsinhx)dx=(−1)n+11
2[Sn(β)+πEn(β)+πYn(β)]
[Reβ>0] ET I 168(12)
3./integraldisplay∞
0exp(−νx−βsinhx)dx=π
sinνπ[Jν(β)−Jν(β)]
[Reβ>0] ET I 168(13)
3.483/integraldisplay∞
−∞exp (νarcsinh x−iax)√
1+x2dx=⎧
⎪⎪⎨
⎪⎪⎩2e x p/parenleftbigg
−iνπ
2/parenrightbigg
Kν(a)f o r a>0,
2e x p/parenleftbiggiνπ
2/parenrightbigg
Kν(−a)f o r a<0[|Reν|<1]
ET I 122(32)
3.484/integraldisplay∞
0/bracketleftbigg/parenleftbigg
1+a
qx/parenrightbiggqx
−/parenleftbigg
1+a
px/parenrightbiggpx/bracketrightbiggdx
x=(ea−1) lnq
p[p>0,q > 0] BI (89)(34)
3.485/integraldisplayπ/2
0exp/parenleftbig
−tan2x/parenrightbig
dx=πe
2[1−Φ(1)]
3.4866/integraldisplay1
0x−xdx=/integraldisplay1
0e−xlnxdx=∞/summationdisplay
k=1k−k=1.2912859970627 ... FI II 483
3.487
1.∗/integraldisplayπ/4
0exp/bracketleftBigg
−∞/summationdisplay
k=0/parenleftbiggtan2k+1x
k+1
2/parenrightbigg/bracketrightBigg
dx=l n2
3.5 Hyperbolic Functions
3.51 Hyperbolic functions
3.511
1./integraldisplay∞
0dx
coshax=π
2a[a>0]
2./integraldisplay∞
0sinhax
sinhbxdx=π
2btanaπ
2b[b>|a|] BI (27)(10)a
3./integraldisplay∞
0sinhax
coshbxdx=π
2bsecaπ
2b−1
bβ/parenleftbigga+b
2b/parenrightbigg
[b>|a|] GW (351)(3b)
4./integraldisplay∞
0coshax
coshbxdx=π
2bsecaπ
2b[b>|a|] BI (4)(14)a
372 Hyperbolic Functions 3.512
5./integraldisplay∞
0sinhaxcoshbx
sinhcxdx=π
2csinaπ
c
cosaπ
c+c o sbπ
c[c>|a|+|b|] BI (27)(11)
6./integraldisplay∞
0coshaxcoshbx
coshcxdx=π
ccosaπ
2ccosbπ
2c
cosaπ
c+c o sbπ
c[c>|a|+|b|] BI (27)(5)a
7./integraldisplay∞
0sinhaxsinhbx
coshcxdx=π
csinaπ
2csinbπ
2c
cosaπ
c+c o sbπ
c[c>|a|+|b|] BI (27)(6)a
8.11/integraldisplay∞
0dx
cosh2x=1 BI (98)(25)
9./integraldisplay∞
−∞sinh2ax
sinh2xdx=1−aπcotaπ/bracketleftbig
a2<1/bracketrightbig
BI (16)(3)a
10./integraldisplay∞
0sinhaxsinhbx
cosh2bxdx=aπ
2b2secaπ
2b[b>|a|] BI (27)(16)a
3.512
1./integraldisplay∞
0cosh2 βx
cosh2νaxdx=4ν−1
aB/parenleftbigg
ν+β
a,ν−β
a/parenrightbigg
[Re(ν±β)>0,a > 0,β > 0]
LI(27)(17)a, EH I 11(26)
2./integraldisplay∞
0sinhμx
coshνxdx=1
2B/parenleftbiggμ+1
2,ν−μ
2/parenrightbigg
[Reμ>−1,Re(μ−ν)<0]
EH I 11(23)
3.513
1./integraldisplay∞
0dx
a+bsinhx=1√
a2+b2lna+b+√
a2+b2
a+b−√
a2+b2[ab/negationslash=0 ] GW (351)(8)
2./integraldisplay∞
0dx
a+bcoshx=2√
b2−a2arctan√
b2−a2
a+b/bracketleftbig
b2>a2/bracketrightbig
=1√
a2−b2lna+b+√
a2−b2
a+b−√
a2−b2/bracketleftbig
b2<a2/bracketrightbig
GW (351)(7)
3./integraldisplay∞
0dx
asinhx+bcoshx=2√
b2−a2arctan√
b2−a2
a+b/bracketleftbig
b2>a2/bracketrightbig
=1√
a2−b2lna+b+√
a2−b2
a+b−√
a2−b2/bracketleftbig
a2>b2/bracketrightbig
GW (351)(9)
3.516 Hyperbolic functions 373
4./integraldisplay∞
0dx
a+bcoshx+csinhx=2√
b2−a2−c2/bracketleftBigg
arctan√
b2−a2−c2
a+b+c+/epsilon1π/bracketrightBigg
⎡
⎢⎢⎢⎣when b2>a2+c2;a n d⎧
⎪⎪⎪⎨
⎪⎪⎪⎩/epsilon1=0 f o r( b−a)(a+b+c)>0
|/epsilon1|=1 f o r( b−a)(a+b+c)<0
/epsilon1=1 f o r a<b +c
/epsilon1=−1f o r a>b +c⎤
⎥⎥⎥⎦
=1
√
a2−b2+c2lna+b+c+√
a2−b2+c2
a+b+c−√
a2−b2+c2/bracketleftbig
b2<a2+c2,a2/negationslash=b2/bracketrightbig
=1
clna+c
a
[a=b/negationslash=0,c/negationslash=0 ]
=2(a−b)
c(a−b−c)/bracketleftbig
b2=a2+c2,c(a−b−c)<0/bracketrightbig
GW (351)(6)
3.514
1./integraldisplay∞
0dx
coshax+c o s t=t
acosect [0<t<π , a> 0] BI (27)(22)a
2./integraldisplay∞
0coshax−cost1
coshbx−cost2dx=π
bsina(πt2)
b
sint2sina
bπ−πt2
bsint2cost1
[0<|a|<b , 0<t2<π]BI (6)(20)a
3./integraldisplay∞
0coshaxdx
(coshx+c o s t)2=π(−costsinat+asintcosat)
sin3tsinaπ
/bracketleftbig
0<a2<1,0<t<π/bracketrightbig
BI (6)(18)a
4./integraldisplay∞
0sinhaxsinhbx
(coshax+c o s t)2dx=bπ
a2cosectcosecbπ
asinbt
a[0<|b|<a , 0<t<π ]BI (27)(27)a
3.515/integraldisplay∞
−∞/parenleftBigg
1−√
2c os h x√
cosh 2 x/parenrightBigg
dx=−ln 2 BI (21)(12)a
3.516
1./integraldisplay∞
0dx/parenleftbig
z+√
z2−1c os h x/parenrightbigμ=1
2/integraldisplay∞
−∞dx/parenleftbig
z+√
z2−1c os h x/parenrightbigμ=Qμ−1(z)
[Reμ>−1]
For a suitable choice of a single-valued branch of the integrand, this formula is valid for arbitrary
values of zin the z-plane cut from −1t o+ 1p r o v i d e d μ<0. Ifμ>0, this formula ceases to be valid for
points at which the denominator vanishes. CO, WH
1./integraldisplay∞
0dx
/parenleftBig
β+/radicalbig
β2−1c os h x/parenrightBign+1=Qn(β) EH II 181(32)
374 Hyperbolic Functions 3.517
2./integraldisplay∞
0coshγxdx
/parenleftBig
β+/radicalbig
β2−1c os h x/parenrightBigν+1=e−iγπΓ(ν−γ+1 )Qγ
ν(β)
Γ(ν+1 )
[Re(ν±γ)>−1,ν/negationslash=−1,−2,−3,...]
EH I 157(12)
3./integraldisplay∞
0sinh2μxdx
/parenleftBig
β+/radicalbig
β2−1c os h x/parenrightBigν+1=2μe−iμπΓ(ν−2μ+1 )Γ/parenleftbig
μ+1
2/parenrightbig
√π(β2−1)μ
2Γ(ν+1 )Qμ
ν−μ(β)
[Re(ν−2μ+1 )>0,Re(ν+1 )>0]
EH I 155(2)
3.517
1./integraldisplay∞
0cosh/parenleftbig
γ+1
2/parenrightbig
xdx
(β+c o s h x)ν+1
2=/radicalbiggπ
2/parenleftbig
β2−1/parenrightbig−ν
2Γ(ν+γ+1 )Γ ( ν−γ)P−ν
γ(β)
Γ/parenleftbig
ν+1
2/parenrightbig
[Re(ν−γ)>0,Re(ν+γ+1 )>0]
EH I 156(11)
2./integraldisplaya
0cosh/parenleftbig
γ+1
2/parenrightbig
xdx
(cosha−coshx)ν+1
2=/radicalbiggπ
2Γ/parenleftbig1
2−ν/parenrightbig
sinhνaPν
γ(cosha)
/bracketleftbig
Reν<1
2,a > 0/bracketrightbig
EH I 156(8)
3.518
1./integraldisplay∞
0sinh2μxdx
(cosha+s i n h acoshx)ν+1=2μe−iμπ
√πsinhμaΓ(ν−2μ+1 )Γ/parenleftbig
μ+1
2/parenrightbig
Γ(ν+1 )Qμ
ν−μ(cosha)
[Re(ν+1 )>0,Re(ν−2μ+1 )>0,a > 0]EH I 155(3)a
2.10/integraldisplay∞
0sinh2μ+1xdx
(β+c o s h x)ν+1=2μ/parenleftbig
β2−1/parenrightbigμ−ν
2Γ(ν−2μ)Γ(μ+1 )Pμ−ν
μ(β)
[Re(ν−μ)>Reμ>−1,βdoes not lie on the ray ( −∞,+1) of the real axis] EH I 155(1)
3./integraldisplay∞
0sinh2μ−1xcoshxdx/parenleftbig
1+asinh2x/parenrightbigν=1
2a−μB(μ, ν−μ)[ R e ν>Reμ>0,a > 0] EH I 11(22)
4.7/integraldisplay∞
0sinhμ−1x(coshx+1 )ν−1dx
(β+c o s h x)/rho1 =2μ+ν−ρB/parenleftbigg1
2μ, /rho1+2−μ−ν/parenrightbigg
×2F1/parenleftbigg
/rho1, /rho1+2−μ−ν;2−1
2μ−ν;1
2−1
2β/parenrightbigg
[Reμ>0,Re(/rho1−μ−ν)>−2,|arg(1 + β)|<π]EH I 115(11)
5.6/integraldisplay∞
0sinhμ−1x(coshx−1)ν−1dx
(β+c o s h x)/rho1 =2−(2−μ−ν+/rho1)
2F1/parenleftbigg
/rho1,2−μ−ν+/rho1;1+/rho1−μ
2;1−β
2/parenrightbigg
×B/parenleftBig
2−μ−ν+/rho1,−1+ν+μ
2/parenrightBig
[β/negationslash∈(−∞,−1),Re(2 + /rho1)R e(μ+ν),Re(2ν+μ)>2]EH I 115(10)
3.522 Hyperbolic and algebraic functions 375
6.7/integraldisplay∞
0sinhμ−1xcoshν−1x/parenleftbig
cosh2x−β/parenrightbig/rho1dx=2F1/parenleftbigg
/rho1,1+/rho1−μ+ν
2;1+/rho1−ν
2;β/parenrightbigg
2B/parenleftbiggμ
2,1+/rho1−μ+ν
2/parenrightbigg
[β/negationslash∈(1,∞),Reμ>0,2R e ( 1+ /rho1)>Re(μ+ν)]EH I 115(9)
3.519/integraldisplayπ/2
0sinh [( r−p)] tanx
sinh (rtanx)dx=π∞/summationdisplay
k=11
kπ+rsinpkπ
r/bracketleftbig
p2<r2/bracketrightbig
BI (274)(13)
3.52–3.53 Combinations of hyperbolic functions and algebraic functions
3.521
1./integraldisplay∞
0xdx
sinhax=π2
4a2[a>0] GW (352)(2b)
2./integraldisplay∞
0xdx
coshx=2G=πln 2−4L/parenleftBigπ
4/parenrightBig
=1.831931188 ... LI III 225(103a), BI(84)(1)a
3./integraldisplay∞
1dx
xsinhax=−2∞/summationdisplay
k=0Ei[−(2k+1 )a][ a>0] LI (104)(14)
4./integraldisplay∞
1dx
xcoshax=2∞/summationdisplay
k=0(−1)k+1Ei[−(2k+1 )a][ a>0] LI (104)(13)
3.522
1./integraldisplay∞
0xdx
(b2+x2)sin h ax=π
2ab+π∞/summationdisplay
k=1(−1)k
ab+kπ[a>0,b > 0]
2./integraldisplay∞
0xdx
(b2+x2)sin h πx=1
2b−β(b+1 ) [ b>0] BI(97)(16), GW(352)(8)
3./integraldisplay∞
0dx
(b2+x2)cosh ax=2π
b∞/summationdisplay
k=1(−1)k−1
2ab+( 2k−1)π[a>0,b > 0] BI (97)(5)
4./integraldisplay∞
0dx
(b2+x2)cosh πx=1
bβ/parenleftbigg
b+1
2/parenrightbigg
[b>0] BI (97)(4)
5./integraldisplay∞
0xdx
(1 +x2)sin h πx=l n2 −1
2BI (97)(7)
6./integraldisplay∞
0dx
(1 +x2)cosh πx=2−π
2BI (97)(1)
7./integraldisplay∞
0xdx
(1 +x2)sin hπx
2=π
2−1 BI (97)(8)
8./integraldisplay∞
0dx
(1 +x2)coshπx
2=l n2 BI (97)(2)
9./integraldisplay∞
0xdx
(1 +x2)sin hπx
4=1√
2/bracketleftBig
π+2l n/parenleftBig√
2+1/parenrightBig/bracketrightBig
−2 BI (97)(9)
376 Hyperbolic Functions 3.523
10./integraldisplay∞
0dx
(1 +x2)coshπx
4=1√
2/bracketleftBig
π−2ln/parenleftBig√
2+1/parenrightBig/bracketrightBig
BI (97)(3)
3.523
1./integraldisplay∞
0xβ−1
sinhaxdx=2β−1
2β−1aβΓ(β)ζ(β)[ R e β>1,a > 0] WH
2./integraldisplay∞
0x2n−1
sinhaxdx=22n−1
2n/parenleftBigπ
a/parenrightBig2n
|B2n| [a>0,n=1,2,...]
WH, GW(352)(2a)
3./integraldisplay∞
0xβ−1
coshaxdx=2
(2a)βΓ(β)Φ/parenleftbigg
−1,β,1
2/parenrightbigg
=2
(2a)βΓ(β)∞/summationdisplay
k=0(−1)k/parenleftbigg2
2k+1/parenrightbiggβ
[Reβ>0,a > 0]EH I 35, ET I 322(1)
4./integraldisplay∞
0x2n
coshaxdx=/parenleftBigπ
2a/parenrightBig2n+1
|E2n| [a>0] BI(84)(12)a, GW(352)(1a)
5./integraldisplay∞
0x2dx
coshx=π3
8(cf.4.261 6) BI (84)(3)
6./integraldisplay∞
0x3dx
sinhx=π4
8(cf.4.262 1a n d2 ) BI (84)(5)
7./integraldisplay∞
0x4dx
coshx=5
32π5BI (84)(7)
8./integraldisplay∞
0x5
sinhxdx=π6
4BI (84)(8)
9./integraldisplay∞
0x6
coshxdx=61
128π7BI (84)(9)
10./integraldisplay∞
0x7
sinhxdx=17
16π8BI (84)(10)
11./integraldisplay∞
0x1/2dx
coshx=√π∞/summationdisplay
k=0(−1)k 1
(2k+1 )3/2BI (98)(7)a
12./integraldisplay∞
0dx
x1/2coshx=2√π∞/summationdisplay
k=0(−1)k
(2k+1 )1/2BI (98)(25)a
3.524
1./integraldisplay∞
0xμ−1sinhβx
sinhγxdx=Γ(μ)
(2γ)μ/braceleftbigg
ζ/bracketleftbigg
μ,1
2/parenleftbigg
1−β
γ/parenrightbigg/bracketrightbigg
−ζ/bracketleftbigg
μ,1
2/parenleftbigg
1+β
γ/parenrightbigg/bracketrightbigg/bracerightbigg
[Reγ>|Reβ|,Reμ>−1]
ET I 323(10)
2.11/integraldisplay∞
0x2msinhax
sinhbxdx=π
2bd2m
da2m/parenleftBig
tanaπ
2b/parenrightBig
[b>|a|] BI (112)(20)a
3.524 Hyperbolic and algebraic functions 377
3./integraldisplay∞
0sinhax
sinhbxdx
xp=Γ ( 1 −p)∞/summationdisplay
k=0/braceleftbigg1
[b(2k+1 )−a]1−p−1
[b(2k+1 )+ a]1−p/bracerightbigg
[b>|a|,p < 1] BI (131)(2)a
4.11/integraldisplay∞
0x2m+1sinhax
coshbxdx=π
2bd2m+1
da2m+1/parenleftBig
secaπ
2b/parenrightBig
[b>|a|] BI (112)(18)a
5./integraldisplay∞
0xμ−1coshβx
sinhγxdx=Γ(μ)
(2γ)μ/braceleftbigg
ζ/bracketleftbigg
μ,1
2/parenleftbigg
1−β
γ/parenrightbigg/bracketrightbigg
+ζ/bracketleftbigg
μ,1
2/parenleftbigg
1+β
γ/parenrightbigg/bracketrightbigg/bracerightbigg
[Reγ>|Reβ|,Reμ>1]
ET I 323(12)
6./integraldisplay∞
0x2mcoshax
coshbxdx=π
2bd2m
da2m/parenleftBig
secaπ
2b/parenrightBig
[b>|a|] BI(112)(17)
7./integraldisplay∞
0coshax
coshbx·dx
xp=Γ ( 1 −p)∞/summationdisplay
k=0(−1)k/braceleftbigg1
[b(2k+1 )−a]1−p+1
[b(2k+1 )+ a]1−p/bracerightbigg
[b>|a|,p < 1] BI(131)(1)a
8./integraldisplay∞
0x2m+1coshax
sinhbxdx=π
2bd2m+1
da2m+1/parenleftBig
tanaπ
2b/parenrightBig
[b>|a|] BI (112)(19)a
9.8/integraldisplay∞
0x2sinhax
sinhbxdx=π3
4b3sinaπ
2bsec3aπ
2b[b>|a|] BI (84)(18)
10./integraldisplay∞
0x4sinhax
sinhbxdx=8/parenleftBigπ
2bsecaπ
2b/parenrightBig5
·sinaπ
2b·/parenleftBig
2+s i n2aπ
2b/parenrightBig
[b>|a|] BI (82)(17)a
11./integraldisplay∞
0x6sinhax
sinhbxdx=1 6/parenleftBigπ
2bsecaπ
2b/parenrightBig7
sinaπ
2b/parenleftBig
45−30cos2aπ
2b+ 2cos4aπ
2b/parenrightBig
[b>|a|] BI (82)(21)a
12./integraldisplay∞
0xsinhax
coshbxdx=π2
4b2sinaπ
2bsec2aπ
2b[b>|a|] BI (84)(15)a
13./integraldisplay∞
0x3sinhax
coshbxdx=/parenleftBigπ
2bsecaπ
2b/parenrightBig4
sinaπ
2b·/parenleftBig
6−cos2aπ
2b/parenrightBig
[b>|a|] BI (82)(14)a
14./integraldisplay∞
0x5sinhax
coshbxdx=/parenleftBigπ
2bsecaπ
2b/parenrightBig6
sinaπ
2b/parenleftBig
120−60cos2aπ
2b+c o s4aπ
2b/parenrightBig
[b>|a|] BI (82)(18)a
15./integraldisplay∞
0x7sinhax
coshbxdx=/parenleftBigπ
2bsecaπ
2b/parenrightBig8
sinaπ
2b/parenleftBig
5040−4200cos2aπ
2b+ 546cos4aπ
2b−cos6aπ
2b/parenrightBig
[b>|a|] BI (82)(22)a
16./integraldisplay∞
0xcoshax
sinhbxdx=/parenleftBigπ
2bsecaπ
2b/parenrightBig2
[b>|a|] BI (84)(16)a
378 Hyperbolic Functions 3.525
17./integraldisplay∞
0x3coshax
sinhbxdx=2/parenleftBigπ
2bsecaπ
2b/parenrightBig4/parenleftBig
1+2s i n2aπ
2b/parenrightBig
[b>|a|] BI (82)(15)a
18./integraldisplay∞
0x5coshax
sinhbxdx=8/parenleftBigπ
2bsecaπ
2b/parenrightBig6/parenleftBig
15−15cos2aπ
2b+ 2cos4aπ
2b/parenrightBig
[b>|a|] BI (82)(19)a
19./integraldisplay∞
0x7coshax
sinhbxdx=1 6/parenleftBigπ
2bsecaπ
2b/parenrightBig8/parenleftBig
315−420cos2aπ
2b+ 126cos4aπ
2b−4c os6aπ
2b/parenrightBig
[b>|a|] BI(82)(23)a
20./integraldisplay∞
0x2coshax
coshbxdx=π3
8b3/parenleftBig
2s e c3aπ
2b−secaπ
2b/parenrightBig
[b>|a|] BI (84)(17)a
21./integraldisplay∞
0x4coshax
coshbxdx=/parenleftBigπ
2bsecaπ
2b/parenrightBig5/parenleftBig
24−20cos2aπ
2b+c o s4aπ
2b/parenrightBig
[b>|a|] BI (82)(16)a
22./integraldisplay∞
0x6coshax
coshbxdx=/parenleftBigπ
2bsecaπ
2b/parenrightBig7/parenleftBig
720−840cos2aπ
2b+ 182cos4aπ
2b−cos6aπ
2b/parenrightBig
[b>|a|] BI (82)(20)a
23./integraldisplay∞
0sinhax
coshbx·dx
x=l nt a n/parenleftBigaπ
4b+π
4/parenrightBig
[b>|a|] BI (95)(3)a
3.525
1./integraldisplay∞
0sinhax
sinhπx·dx
1+x2=−a
2cosa+1
2sinaln[2(1 + cos a)]
[π≥|a|] BI (97)(10)a
2./integraldisplay∞
0sinhax
sinhπ
2x·dx
1+x2=π
2sina+1
2cosaln1−sina
1+s i n a[π≥2|a|] BI (97)(11)a
3./integraldisplay∞
0coshax
sinhπx·xdx
1+x2=1
2(asina−1) +1
2cosaln[2(1 + cos a)]
[π>|a|] BI (97)(12)a
4./integraldisplay∞
0coshax
sinhπ
2x·xdx
1+x2=π
2cosa−1+1
2sinaln1+s i n a
1−sina
/bracketleftBigπ
2>|a|/bracketrightBig
BI (97)(13)a
5./integraldisplay∞
0sinhax
coshπx·xdx
1+x2=−2s ina
2+π
2sina−cosalntana+π
4
[π>|a|] GW (352)(12)
6./integraldisplay∞
0coshax
coshπx·dx
1+x2= 2cosa
2−π
2cosa−sinalntana+π
4
[π>|a|] GW (352)(11)
3.527 Hyperbolic and algebraic functions 379
7./integraldisplay∞
0sinhax
sinhbx·dx
c2+x2=π
c∞/summationdisplay
k=1sink(b−a)
bπ
bc+kπ[b≥|a|] BI (97)(18)
8./integraldisplay∞
0coshax
sinhbx·xdx
c2+x2=π
2bc+π∞/summationdisplay
k=1cosk(b−a)
bπ
bc+kπ[b>|a|] BI (97)(19)
3.526
1./integraldisplay∞
0sinhaxcoshbx
coshcx·dx
x=1
2ln/braceleftbigg
tan(a+b+c)π
4ccot(b+c−a)π
4c/bracerightbigg
[c>|a|+|b|] BI (93)(10)a
2./integraldisplay∞
0sinh2ax
sinhbx·dx
x=1
2lnseca
bπ [b>|2a|] BI (95)(5)a
3./integraldisplay∞
0xμ−1
sinhβxcoshγxdx=Γ(μ)
(2γ)μ/braceleftbigg
Φ/bracketleftbigg
−1,μ ,1
2/parenleftbigg
1+β
γ/parenrightbigg/bracketrightbigg
+Φ/bracketleftbigg
−1,μ ,1
2/parenleftbigg
1−β
γ/parenrightbigg/bracketrightbigg/bracerightbigg
[Reγ>|Reβ|,Reμ>0]
ET I 323(11)
3.527
1./integraldisplay∞
0xμ−1
sinh2axdx=4
(2a)μΓ(μ)ζ(μ−1) [Re a>0,Reμ>2] BI (86)(7)a
2./integraldisplay∞
0x2m
sinh2axdx=π2m
a2m+1|B2m| [a>0,m =1,2,...] BI(86)(5)a
3.6/integraldisplay∞
0xμ−1
cosh2axdx=4
(2a)μ/parenleftbig
1−22−μ/parenrightbig
Γ(μ)ζ(μ−1) [Re a>0,Reμ>0,μ/negationslash=2 ]
=1
a2ln 2 [Re a>0,μ=2 ]
BI (86)(6)a
4./integraldisplay∞
0xdx
cosh2ax=ln2
a2[a/negationslash=0 ] LO III 396
5./integraldisplay∞
0x2m
cosh2axdx=/parenleftbig
22m−2/parenrightbig
π2m
(2a)2ma|B2m| [a>0,m =1,2,...] BI(86)(2)a
6./integraldisplay∞
0xμ−1sinhax
cosh2axdx=2Γ (μ)
aμ∞/summationdisplay
k=0(−1)k
(2k+1 )μ−1[Reμ>1,a > 0] BI (86)(15)a
7./integraldisplay∞
0xsinhax
cosh2axdx=π
2a2[a>0] BI (86)(8)a
8./integraldisplay∞
0x2m+1sinhax
cosh2axdx=2m+1
a/parenleftBigπ
2a/parenrightBig2m+1
|E2m| [a>0,m =0,1,...] BI (86)(12)a
9./integraldisplay∞
0x2m+1coshax
sinh2axdx=22m+1−1
a2(2a)2m(2m+1 ) !ζ(2m+1 )
[a/negationslash=0,m =1,2,...] BI (86)(13)a
380 Hyperbolic Functions 3.528
10.11/integraldisplay∞
0x2mcoshax
sinh2axdx=22m−1
a/parenleftBigπ
a/parenrightBig2m
|B2m| [a>0,m =1,2,...] BI (86)(14)a
11.8/integraldisplay∞
0xsinhax
cosh2μ+1axdx=√π
4μa2Γ(μ)
Γ/parenleftbig
μ+1
2/parenrightbig [μ>0,a > 0] LI (86)(9)
12./integraldisplay∞
−∞x2dx
sinh2x=π2
3BI (102)(2)a
13./integraldisplay∞
0x2coshax
sinh2axdx=π2
2a3[a>0] BI (86)(11)a
14.11/integraldisplay∞
0x2sinhx
cosh2xdx=4G [a/negationslash=0 ] BI (86)(10)a
15.10/integraldisplay∞
0tanhx
2dx
coshx=l n2 BI (93)(17)a
16.∗/integraldisplay∞
0xμ−1coshax
sinh2ax=2Γ(μ)ζ(μ−1)
aμ/parenleftbig
1−21−μ/parenrightbig
3.528
1./integraldisplay∞
0(1 +xi)2n−1−(1−xi)2n−1
isinhπx
2dx=2 BI (87)(8)
2./integraldisplay∞
0(1 +xi)2n−(1−xi)2n
isinhπx
2dx=(−1)n+12|E2n|+2 [ n=0,1,...] BI (87)(7)
3.529
1./integraldisplay∞
0/parenleftbigg1
sinhx−1
x/parenrightbiggdx
x=−ln 2 BI (94)(10)a
2./integraldisplay∞
0coshax−1
sinhbx·dx
x=−ln cosaπ
2b[b>|a|] GW (352)(66)
3./integraldisplay∞
0/parenleftbigga
sinhax−b
sinhbx/parenrightbiggdx
x=(b−a)ln2 BI (94)(11)a
3.531
1.7/integraldisplay∞
0xdx
2c os h x−1=4√
3/bracketleftBigπ
3ln 2−L/parenleftBigπ
3/parenrightBig/bracketrightBig
=1.1719536193 ...
[see8.26forL(x)] LI (88)(1)
2.10/integraldisplay∞
0xdx
cosh 2 x+c o s2 t=tln2−L(t)
sin 2tLO III 402
3./integraldisplay∞
0x2dx
coshx+c o s t=t
3·π2−t2
sint[0<t<π ] BI (88)(3)a
4./integraldisplay∞
0x4dx
coshx+c o s t=t
15/parenleftbig
π2−t2/parenrightbig/parenleftbig
7π2−3t2/parenrightbig
sint[0<t<π ] BI (88)(4)a
3.533 Hyperbolic and algebraic functions 381
5.3/integraldisplay∞
0x2mdx
coshx−cos 2aπ=2 ( 2m)! cosec2 aπ∞/summationdisplay
k=1sin 2kaπ
k2m+1/bracketleftbig
0<a< 1,a/negationslash=1
2/bracketrightbig
=2/parenleftbig
22m−1−1/parenrightbig
π2m|B2m|/bracketleftbig
a=1
2/bracketrightbig
BI (88)(5)a
6.3/integraldisplay∞
0xμ−1dx
coshx−cost
=iΓ(μ)
sint/bracketleftbig
e−itΦ/parenleftbig
e−it,μ ,1/parenrightbig
−eitΦ/parenleftbig
eit,μ ,1/parenrightbig/bracketrightbig[Reμ>0,0<t< 2π, t /negationslash=π]ET I 323(5)
=/parenleftbig
2−23−μ/parenrightbig
Γ(μ)ζ(μ−1) [μ/negationslash=2,t=π]
=2l n2 [μ=2,t=π]
7./integraldisplay∞
0xμdx
coshx+c o s t=2Γ (μ+1 )
sint∞/summationdisplay
k=1(−1)k−1sinkt
kμ+1[μ>−1,0<t<π ] BII (96)(14)a
8./integraldisplayu
0xdx
cosh 2 x−cos2t=1
2cosec2 t[L(θ+t)−L(θ−t)−2L(t)]
[θ= arctan(tanh ucott),t/negationslash=nπ]
LO III 402
3.532
1.11/integraldisplay∞
0xndx
acoshx+bsinhx=2n!
a+b∞/summationdisplay
k=01
(2k+1 )n+1/parenleftbiggb−a
b+a/parenrightbiggk
[a>0,b > 0,n > −1]GW (352)(5)
2./integraldisplayu
0xcoshxdx
cosh 2 x−cos2t=1
2cosect/braceleftbigg
L/parenleftbiggθ+t
2/parenrightbigg
−L/parenleftbiggθ−t
2/parenrightbigg
+L/parenleftbigg
π−ψ+t
2/parenrightbigg
+L/parenleftbiggψ−t
2/parenrightbigg
−2L/parenleftbiggt
2/parenrightbigg
−2L/parenleftbiggπ−t
2/parenrightbigg/bracerightbigg
/bracketleftbigg
tanθ
2=t a n hu
2cott
2,tanψ
2=c o t hu
2cott
2;t/negationslash=nπ/bracketrightbigg
LO III 288a
3.533
1./integraldisplay∞
0xcoshxdx
cosh 2 x−cos2t=c o s e c t/bracketleftbiggπ
2ln 2−L/parenleftbiggt
2/parenrightbigg
−L/parenleftbigg(π−t)
2/parenrightbigg/bracketrightbigg
[t/negationslash=mπ] LO III 403
2.6/integraldisplay∞
0xsinhaxdx
(coshax−cost)2=π−t
a2cosect [a>0,0<t<π ] (cf. 3.514 1)
BI (88)(11)a
3./integraldisplay∞
0x3sinhxdx
(coshx+c o s t)2=t/parenleftbig
π2−t2/parenrightbig
sint[0<t<π ] (cf. 3.531 3)
BI (88)(13)
382 Hyperbolic Functions 3.534
4.11/integraldisplay∞
0x2m+1 sinhxdx
(coshx−cos 2aπ)2=2 ( 2m+1 ) !c o s e c2 aπ∞/summationdisplay
k=1sin 2kaπ
k2m+1/bracketleftbig
0<a< 1,a/negationslash=1
2/bracketrightbig
=2 ( 2m+1 )/parenleftbig
22m−1−1/parenrightbig
π2m|B2m|/bracketleftbig
a=1
2/bracketrightbig
BI (88)(14)
3.534
1./integraldisplay1
0/radicalbig
1−x2coshaxdx =π
2aI1(a) WA 94(9)
2./integraldisplay1
0coshax√
1−x2dx=π
2I0(a) WA 94(9)
3.535/integraldisplay1
0x√
cosh 2 a−cosh 2 ax·dx
sinhax=π
2√
2a2·arcsin(tanh a)
sinha[a>0] BI (80)(11)
3.536
1.11/integraldisplay∞
0x2
cosh2xdx=π2
12BI (98)(7)
2./integraldisplay∞
0x2tanhx2dx
cosh2x=√π
2∞/summationdisplay
k=0(−1)k
√
2k+1BI (98)(8)
3./integraldisplay∞
0sinh (νarcsinh x)xμ−1
√
1+x2dx=sinμπ
2sinνπ
2
2μπΓ(μ)Γ/parenleftbigg1−μ−ν
2/parenrightbigg
×Γ/parenleftbigg1−μ+ν
2/parenrightbigg
[−1<Reμ<1−|Reν|]ET I 324(14)
4./integraldisplay∞
0cosh(νarccosh x)xμ−1
√
1+x2dx=cosμπ
2cosνπ
2
2μπΓ(μ)Γ/parenleftbigg1−μ−ν
2/parenrightbigg
×Γ/parenleftbigg1−μ+ν
2/parenrightbigg
[0<Reμ<1−|Reν|] ET I 324(15)
3.54 Combinations of hyperbolic functions and exponentials
3.541
1./integraldisplay∞
0e−μxsinhνβxdx =1
2ν+1βB/parenleftbiggμ
2β−ν
2,ν+1/parenrightbigg
[Reβ>0,Reν>−1,Reμ>Reβν]
EH I 11(25), ET I 163(5)
2./integraldisplay∞
0e−μxsinhβx
sinhbxdx=1
2b/bracketleftbigg
ψ/parenleftbigg1
2+μ+β
2b/parenrightbigg
−ψ/parenleftbigg1
2+μ−β
2b/parenrightbigg/bracketrightbigg
[Re(μ+b±β)>0] EH I 16(14)a
3./integraldisplay∞
−∞e−μxsinhμx
sinhβxdx=π
2βtanμπ
β[Reβ>2|Reμ|] BI (18)(6)
4./integraldisplay∞
0e−xsinhax
sinhxdx=1
a−π
2cotaπ
2[0<a< 2] BI (4)(3)
5./integraldisplay∞
0e−pxdx
(coshpx)2q+1=22q−2
pB(q,q)−1
2qp[p>0,q > 0] LI (27)(19)
3.545 Hyperbolic functions and exponentials 383
6./integraldisplay∞
0e−μxdx
coshx=β/parenleftbiggμ+1
2/parenrightbigg
[Reμ>−1] ET I 163(7)
7./integraldisplay∞
0e−μxtanhxdx=β/parenleftBigμ
2/parenrightBig
−1
μ[Reμ>0] ET I 163(9)
8./integraldisplay∞
0e−μx
cosh2xdx=μβ/parenleftBigμ
2/parenrightBig
−1[ R e μ>0] ET I 163(8)
9./integraldisplay∞
0e−μxsinhμx
cosh2μxdx=1
μ(1−ln 2) [Re μ>0] LI (27)(15)
10./integraldisplay∞
0e−qxsinhpx
sinhqxdx=1
p−π
2qcotpπ
2q[0<p< 2q] BI (27)(9)a
3.542
1./integraldisplay∞
0e−μx(coshβx−1)νdx=1
2νβB/parenleftbiggμ
β−ν,2ν+1/parenrightbigg
/bracketleftbigg
Reβ>0,Reν>−1
2,Reμ>Reβν/bracketrightbigg
ET I 163(6)
2./integraldisplay∞
0e−μx(coshx−coshu)ν−1dx=−i/radicalbigg
2
πeiπνΓ(ν)sin hν−1
2uQ1
2−ν
μ−1
2(coshu)
[Reν>0,Reμ>Reν−1]
EH I 155(4), ET I 164(23)
3.543
1./integraldisplay∞
−∞e−ibxdx
sinhx+s i n h t=−iπeitb
sinhπbcosht/parenleftbig
coshπb−e−2itb/parenrightbig
[t>0] ET I 121(30)
2./integraldisplay∞
0e−μx
coshx−costdx= 2 cosec t∞/summationdisplay
k=1sinkt
μ+k[Reμ>−1,t/negationslash=2nπ] BI (6)(10)a
3./integraldisplay∞
01−e−xcost
coshx−coste−(μ−1)xdx=2∞/summationdisplay
k=0coskt
μ+k[Reμ>0,t/negationslash=2nπ] BI (6)(9)a
4./integraldisplay∞
0epx−cost
(coshpx+c o s t)2dx=1
p/parenleftbigg
tcosect+1
1 + cos t/parenrightbigg
[p>0] BI (27)(26)a
3.544/integraldisplay∞
uexp/bracketleftbig
−/parenleftbig
n+1
2/parenrightbig
x/bracketrightbig
/radicalbig
2(c os h x−coshu)dx=Qn(coshu),[u>0] EH II 181(33)
3.545
1./integraldisplay∞
0sinhax
epx+1dx=π
2pcosecaπ
p−1
2a[p>a , p> 0] BI (27)(3)
2./integraldisplay∞
0sinhax
epx−1dx=1
2a−π
2pcotaπ
p[p>a , p> 0] BI (27)(9)
384 Hyperbolic Functions 3.546
3.546
1./integraldisplay∞
0e−βx2sinhaxdx =1
2√π√βexpa2
4βΦ/parenleftbigga
2√β/parenrightbigg
[Reβ>0] ET I166(38)a
2./integraldisplay∞
0e−βx2coshaxdx =1
2/radicalbiggπ
βexpa2
4β[Reβ>0] FI II 720a
3./integraldisplay∞
0e−βx2sinh2axdx =1
4/radicalbiggπ
β/parenleftbigg
expa2
β−1/parenrightbigg
[Reβ>0] ET I 166(40)
4./integraldisplay∞
0e−βx2cosh2axdx =1
4/radicalbiggπ
β/parenleftbigg
expa2
β+1/parenrightbigg
[Reβ>0] ET I 166(41)
3.547
1./integraldisplay∞
0exp(−βsinhx)s i n h γxdx =π
2cotγπ
2[Jγ(β)−Jγ(β)]−π
2[Eγ(β)+Yγ(β)] =γS−1,γ(β)
[Reβ>0] WA 341(5), ET I 168(14)a
2./integraldisplay∞
0exp(−βcoshx)sin h γxsinhxdx=γ
βKγ(β)
3./integraldisplay∞
0exp(−βsinhx)c o s h γxdx =π
2tanπγ
2[Jγ(β)−Jγ(β)]−π
2[Eγ(β)+Yγ(β)] =S0,γ(β)
[Reβ>0,γnot an integer]
ET I 168(16)a, WA 341(4), EH II 84(50)
4./integraldisplay∞
0exp(−βcoshx)cosh γxdx =Kγ(β)[ R e β>0] ET I 168(16)a, WA 201(5)
5./integraldisplay∞
0exp(−βsinhx)s i n h γxcoshxdx=γ
βS0,γ(β)[ R e β>0] ET I 168(7), EH II 85(51)
6./integraldisplay∞
0exp(−βsinhx)s i n h [ ( 2 n+1 )x]c os h xdx=O2n+1(β)
[Reβ>0] ET I 167(5)
7./integraldisplay∞
0exp(−βsinhx)c o s h γxcoshxdx=1
βS1,γ(β)[ R e β>0]
8./integraldisplay∞
0exp(−βsinhx)c o s h2 nxcoshxdx=O2n(β)[ R e β>0] ET I 168(6)
9./integraldisplay∞
0exp(−βcoshx)sin h2νxdx=1√π/parenleftbigg2
β/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Kν(β)
/bracketleftbig
Reβ>0,Reν>−1
2/bracketrightbig
EH II 82(20)
10.11/integraldisplay∞
0exp[−2(βcothx+μx)] sinh2νxdx=1
2βνΓ(μ−ν)W−μ,ν−1
2(4β)
[Reβ>0,Reμ>Reν]
11./integraldisplay∞
0exp/parenleftbigg
−β2
2sinhx/parenrightbigg
sinhν−1xcoshνxdx=−πDν/parenleftBig
βeiπ/4/parenrightBig
Dν/parenleftBig
βe−iπ/4/parenrightBig
/bracketleftBig
Reν>0,|argβ|≤π
4/bracketrightBig
EH II 120(10)
3.548 Hyperbolic functions and exponentials 385
12./integraldisplay∞
0exp (2 νx−2βsinhx)√
sinhxdx=1
2/radicalbig
π3β/bracketleftBig
Jν+1
4(β)Jν−1
4(β)+Yν+1
4(β)Yν−1
4(β)/bracketrightBig
[Reβ>0] EH I 169(20)
13./integraldisplay∞
0exp (−2νx−2βsinhx)√
sinhxdx=1
2/radicalbig
π3β/bracketleftBig
Jν+1
4(β)Yν−1
4(β)−Jν−1
4(β)Yν+1
4(β)/bracketrightBig
[Reβ>0] ET I 169(21)
14./integraldisplay∞
0exp (−2βsinhx)s i n h2 νx√
sinhxdx=1
4i/radicalbigg
π3β
2/braceleftBig
eνπiH(1)
1
2+ν(β)H(2)
1
2−ν(β)
−e−νπiH(1)
1
2−ν(β)H(2)
1
2+ν(β)/bracerightBig
[Reβ>0] ET I 170(24)
15./integraldisplay∞
0exp (−2βsinhx)c o s h2 νx√
sinhxdx=1
4/radicalbigg
π3β
2/braceleftBig
eνπiH(1)
1
2+ν(β)H(2)
1
2−ν(β)
+e−νπiH(1)
1
2−ν(β)H(2)
1
2+ν(β)/bracerightBig
[Reβ>0] ET I 170(25)
16./integraldisplay∞
0exp (−2βcoshx)c o s h2 νx√
coshxdx=/radicalbigg
β
πKν+1
4(β)Kν−1
4(β)
[Reβ>0] ET I 170(26)
17.8/integraldisplay∞
0exp [−2β(coshx−1)] cosh 2 νx√
coshxdx=/radicalbigg
β
π·e2βKν+1
4(β)Kν−1
4(β)
[Reβ>0] ET I 170(27)
18./integraldisplay∞
0cos/bracketleftbig/parenleftbig
ν+1
4/parenrightbig
π/bracketrightbig
exp (−2νx−2βsinhx)+s i n/bracketleftbig/parenleftbig
ν+1
4/parenrightbig
π/bracketrightbig
exp (2 νx−2βsinhx)√
sinhxdx
=1
2/radicalbig
π3β/bracketleftBig
J1
4+ν(β)J1
4−ν(β)+Y1
4+ν(β)Y1
4−ν(β)/bracketrightBig
[Reβ>0] ET I 169(22)
19./integraldisplay∞
0sin/bracketleftbig/parenleftbig
ν+1
4/parenrightbig
π/bracketrightbig
exp(−2νx−2βsinhx)−cos/bracketleftbig/parenleftbig
ν+1
4/parenrightbig
π/bracketrightbig
exp (2 νx−2βsinhx)√
sinhxdx
=1
2/radicalbig
π3β/bracketleftBig
J1
4+ν(β)Y1
4−ν(β)−J1
4−ν(β)Y1
4+ν(β)/bracketrightBig
[Reβ>0] ET I 169(23)
20./integraldisplay∞
0exp [−β(coshx−1)] cosh νxsinhx/radicalbig
coshx(coshx−1)dx=eβKν(β)
[Reβ>0] ET I 169(19)
3.548
1./integraldisplay∞
0e−μx4sinhax2dx=π
4/radicalbigga
2μexp/parenleftbigga2
8μ/parenrightbigg
I1
4/parenleftbigga2
8μ/parenrightbigg
[Reμ>0,a≥0] ET I 166(42)
2./integraldisplay∞
0e−μx4coshax2dx=π
4/radicalbigga
2μexp/parenleftbigga2
8μ/parenrightbigg
I−1
4/parenleftbigga2
8μ/parenrightbigg
[Reμ>0,a > 0] ET I 166(43)
386 Hyperbolic Functions 3.549
3.549
1./integraldisplay∞
0e−βxsinh [(2 n+1 )a r c s i n h x]dx=O2n+1(β)[ R e β>0] (cf. 3.547 6)
ET I 167(5)
2./integraldisplay∞
0e−βxcosh (2 narcsinh x)dx=O2n(β)[ R e β>0] (cf. 3.547 8)
ET I 168(6)
3./integraldisplay∞
0e−βxsinh (νarcsinh x)dx=ν
βS0,ν(β)[ R e β>0] (cf. 3.547 5)ET I 168(7)
4./integraldisplay∞
0e−βxcosh (νarcsinh x)dx=1
βS1,ν(β)[ R e β>0] (cf. 3.547 7)
A number of other integrals containing hyperbolic functions and exponentials, depending on arcsinh x
or arccosh x, can be found by first making the substitution x=s i n h torx=c o s h t.
3.55–3.56 Combinations of hyperbolic functions, exponentials, and powers
3.551
1./integraldisplay∞
0xμ−1e−βxsinhγxdx =1
2Γ(μ)/bracketleftbig
(β−γ)−μ−(β+γ)−μ/bracketrightbig
[Reβ>−1,Reβ>|Reγ|]
ET I 164(18)
2./integraldisplay∞
0xμ−1e−βxcoshγxdx =1
2Γ(μ)/bracketleftbig
(β−γ)−μ+(β+γ)−μ/bracketrightbig
[Reμ>0,Reβ>|Reγ|]
ET I 164(19)
3./integraldisplay∞
0xμ−1e−βxcothxdx=Γ (μ)/bracketleftbigg
21−μζ/parenleftbigg
μ,β
2/parenrightbigg
−β−μ/bracketrightbigg
[Reμ>1,Reβ>0] ET I 164(21)
4./integraldisplay∞
0xne−(p+mq)xsinhmqxdx =2−mn!m/summationdisplay
k=0/parenleftBigm
k/parenrightBig(−1)k
(p+2kq)n+1
[p>0,q > 0,m < p +qm]
LI (81)(4)
5.11/integraldisplay1
0e−βx
xsinhγxdx =1
2/bracketleftbigg
lnβ+γ
β−γ+E i (γ−β)−Ei(−γ−β)/bracketrightbigg
[β>γ ] BI (80)(4)
6./integraldisplay∞
0e−βx
xsinhγxdx =1
2lnβ+γ
β−γ[Reβ>|Reγ|] ET I 163(12)
7./integraldisplay∞
1e−βx
xcoshγxdx =1
2[−Ei(γ−β)−Ei(−γ−β)] [Re β>|Reγ|] ET I 164(15)
8.6/integraldisplay∞
0xe−xcothxdx=π2
4−1 BI (82)(6)
3.554 Hyperbolic functions, exponentials, and powers 387
9./integraldisplay∞
0e−βxtanhxdx
x=l nβ
4+2l nΓ/parenleftBig
β
4/parenrightBig
Γ/parenleftbiggβ
4+1
2/parenrightbigg [Reβ>0] ET I 164(16)
10.6/integraldisplay∞
0xe−xcoth(x/2)dx=π2
3−1
3.552
1./integraldisplay∞
0xμ−1e−βx
sinhxdx=21−μΓ(μ)ζ/bracketleftbigg
μ,1
2(β+1 )/bracketrightbigg
[Reμ>1,Reβ>−1] ET I 164(20)
2./integraldisplay∞
0x2m−1e−ax
sinhaxdx=1
2m|B2m|/parenleftBigπ
a/parenrightBig2m
[a>0,m =1,2,...] EH I 38(24)a
3./integraldisplay∞
0xμ−1e−x
coshxdx=21−μ/parenleftbig
1−21−μ/parenrightbig
Γ(μ)ζ(μ)[ R e μ>0,μ/negationslash=1 ]
= ln 2 [if μ=1 ]
EH I 32(5)
4./integraldisplay∞
0x2m−1e−ax
coshaxdx=1−21−2m
2m|B2m|/parenleftBigπ
a/parenrightBig2m
[a>0,m =1,2,...] EH I 39(25)a
5./integraldisplay∞
0x2e−2nx
sinhxdx=4∞/summationdisplay
k=n1
(2k+1 )3[n=0,1,2,...] (cf. 4.261 13)
BI(84)(4)
6.11/integraldisplay∞
0x3e−2nx
sinhxdx=π4
8−12n/summationdisplay
k=11
(2k−1)4[n=0,1,...] (cf. 4.262 6)
BI (84)(6)
3.553
1./integraldisplay∞
0sinh2ax
sinhxe−xdx
x=1
2ln (aπcosecaπ)[ a<1] BI (95)(7)
2.11/integraldisplay∞
0sinh2x
2
coshx·e−xdx
x=1
2ln4
π(cf.4.267 2) BI (95)(4)
3.554
1.11/integraldisplay∞
0e−βx(1−sechx)dx
x=2l nΓ/parenleftBig
β+3
4/parenrightBig
Γ/parenleftbiggβ+1
4/parenrightbigg−lnβ
4[Reβ>0] ET I 164(17)
2./integraldisplay∞
0e−βx/parenleftbigg1
x−cosech x/parenrightbigg
dx=ψ/parenleftbiggβ+1
2/parenrightbigg
−lnβ
2[Reβ>0] ET I 163(10)
3./integraldisplay∞
0/bracketleftBigg
sinh/parenleftbig1
2−β/parenrightbig
x
sinhx
2−(1−2β)e−x/bracketrightBigg
dx
x=2l nΓ ( β)−lnπ+l n( s i n πβ)
[0<Reβ<1] EH I 21(7)
388 Hyperbolic Functions 3.555
4./integraldisplay∞
0e−βx/parenleftbigg1
x−cothx/parenrightbigg
dx=ψ/parenleftbiggβ
2/parenrightbigg
−lnβ
2+1
β[Reβ>0] ET I 163(11)
5./integraldisplay∞
0/braceleftbigg
−sinhqx
sinhx
2+2qe−x/bracerightbiggdx
x=2l nΓ/parenleftbigg
q+1
2/parenrightbigg
+l nc o s πq−lnπ
/bracketleftbig
q2<1
2/bracketrightbig
WH
6./integraldisplay∞
0xμ−1e−βx(cothx−1)dx=21−μΓ(μ)ζ/parenleftbigg
μ,β
2+1/parenrightbigg
[Reβ>0; Re μ>1] ET I 164(22)
3.555
1./integraldisplay∞
0sinh2ax
1−epx·dx
x=1
4ln/parenleftbiggp
2aπsin2aπ
p/parenrightbigg
[0<2|a|<p] (cf. 3.545 2)
BI (93)(15)
2./integraldisplay∞
0sinh2ax
ex+1·dx
x=−1
4ln (aπcotaπ)/bracketleftbig
a<1
2/bracketrightbig
(cf.3.545 1) BI (93)(9)
3.556
1./integraldisplay∞
−∞x1−epx
sinhxdx=−π2
2tan2pπ
2[p<1] (cf. 4.255 3) BI (101)(4)
2./integraldisplay∞
01−e−px
sinhx·1−e−(p+1)x
xdx=2pln 2 [ p>−1] BI (95)(8)
3.557
1./integraldisplay∞
0e−px−e−qx
coshx−cosm
nπ·dx
x
= 2 cosec/parenleftBigm
nπ/parenrightBign−1/summationdisplay
k=1(−1)k−1sin/parenleftbiggkm
nπ/parenrightbigg
lnΓ/parenleftBig
n+q+k
2n/parenrightBig
Γ/parenleftBig
p+k
2n/parenrightBig
Γ/parenleftBig
n+p+k
2n/parenrightBig
Γ/parenleftBig
q+k
2n/parenrightBig[m+nodd]
= 2 cosec/parenleftBigm
nπ/parenrightBign−1
2/summationdisplay
k=1(−1)k−1sin/parenleftbiggkm
nπ/parenrightbigg
lnΓ/parenleftBig
n+q−k
n/parenrightBig
Γ/parenleftBig
p+k
n/parenrightBig
Γ/parenleftBig
n+p−k
n/parenrightBig
Γ/parenleftBig
q+k
n/parenrightBig[m+neven]
[p>−1,q > −1] BI (96)(1)
2./integraldisplay∞
0(1−e−x)2
coshx+c o sm
nπ·dx
x
= 2 cosec/parenleftBigm
nπ/parenrightBign−1/summationdisplay
k=1(−1)k−1sin/parenleftbiggkm
nπ/parenrightbigg
×ln/bracketleftbig
Γ/parenleftbign+k+1
2n/parenrightbig/bracketrightbig2Γ/parenleftbigk+2
2n/parenrightbig
Γ/parenleftbigk
2n/parenrightbig
/bracketleftbig
Γ/parenleftbigk+1
2n/parenrightbig/bracketrightbig2Γ/parenleftbign+k
2n/parenrightbig
Γ/parenleftbign+k+2
2n/parenrightbig[m+nodd]
= 2 cosec/parenleftBigm
nπ/parenrightBign−1
2/summationdisplay
k=1(−1)k−1sin/parenleftbiggkm
nπ/parenrightbigg
×ln/bracketleftbig
Γ/parenleftbign−k+1
n/parenrightbig/bracketrightbig2Γ/parenleftbigk+2
n/parenrightbig
Γ/parenleftbigk
n/parenrightbig
/bracketleftbig
Γ/parenleftbigk+1
n/parenrightbig/bracketrightbig2Γ/parenleftbign−k
n/parenrightbig
Γ/parenleftbign−k+2
n/parenrightbig[m+neven]
BI (96)(2)
3.558 Hyperbolic functions, exponentials, and powers 389
3./integraldisplay∞
0/bracketleftbigg
e−xtanm
2nπ−e−pxsinm
nπ
coshx+c o sm
nπ/bracketrightbigg
·dx
x
=t a n/parenleftBigm
2nπ/parenrightBig
ln(2n)+2n−1/summationdisplay
k=1(−1)k−1sin/parenleftbiggkm
nπ/parenrightbigg
lnΓ/parenleftBig
p+n+k
2n/parenrightBig
Γ/parenleftBig
p+k
2n/parenrightBig [m+nodd]
=t a n/parenleftBigm
2nπ/parenrightBig
lnn+2n−1
2/summationdisplay
k=1(−1)k−1sin/parenleftbiggkm
nπ/parenrightbigg
lnΓ/parenleftBig
p+n−k
n/parenrightBig
Γ/parenleftBig
p+k
n/parenrightBig [m+neven]
BI (96)(3)
4./integraldisplay∞
01+e−x
coshx+c o s a·dx
x1−p=2s e ca
2Γ(p)∞/summationdisplay
k=1(−1)k−1cos/parenleftbig
k−1
2/parenrightbig
a
kp
[p>0] LI (96)(5)
5./integraldisplay∞
0xqe−x
2coshx
2
coshx+c o s λdx=Γ(q+1 )
cosλ
2∞/summationdisplay
k=1(−1)k−1cos/parenleftbig
k−1
2/parenrightbig
λ
kq+1
[q>−1] LI (96)(5)a
6./integraldisplay∞
0xe−x−cosa
coshx−cosadx=|a|π−a2
2−π2
3BI (88)(8)
7./integraldisplay∞
0x2m+1e−x−cosaπ
coshx−cosaπdx=2·(2m+1 ) !∞/summationdisplay
k=1coskaπ
k2m+2BI (88)(6)
3.558
1./integraldisplay∞
0x1−e−nx
sinh2x
2dx=2nπ2
3−4n−1/summationdisplay
k=1n−k
k2BI (85)(3)
2./integraldisplay∞
0x1−(−1)ne−nx
cosh2x
2dx=nπ2
3+4n−1/summationdisplay
k=1(−1)kn−k
k2LI (85)(1)
3./integraldisplay∞
0x21−e−nx
sinh2x
2dx=8nζ(3)−8n−1/summationdisplay
k=1n−k
k3BI (85)(5)
4./integraldisplay∞
0x2ex1−e−2nx
sinh2xdx=8n∞/summationdisplay
k=11
(2k−1)3−8n−1/summationdisplay
k=1n−k
(2k−1)3LI (85)(6)
5./integraldisplay∞
0x21+(−1)ne−nx
cosh2x
2dx=6nζ(3)−8n−1/summationdisplay
k=1n−k
k3LI (85)(4)
6./integraldisplay∞
0x31−e−nx
sinh2x
2dx=4
15nπ4−24n−1/summationdisplay
k=1n−k
k4BI (85)(9)
7./integraldisplay∞
0x31+(−1)ne−nx
cosh2x
2dx=7
30nπ4+2 4n−1/summationdisplay
k=1(−1)kn−k
k4BI (85)(8)
390 Trigonometric Functions 3.559
3.559/integraldisplay∞
0e−x/bracketleftBigg
a−1
2+(1−e−x)( 1−ax)−xe−x
4s in h2x
2e(2−a)x/bracketrightBigg
dx
x=a−1
2+lnΓ( a)−1
2ln(2π)[ a>0]
BI (96)(6)
3.561/integraldisplay∞
0e−2xtanhx
2
xcoshxdx=2l nπ
2√
2BI (93)(18)
3.562
1./integraldisplay∞
0x2μ−1e−βx2sinhγxdx =1
2Γ(2μ)(2β)−μexp/parenleftbiggγ2
8β/parenrightbigg/bracketleftbigg
D−2μ/parenleftbigg
−γ√2β/parenrightbigg
−D−2μ/parenleftbiggγ√2β/parenrightbigg/bracketrightbigg
/bracketleftbig
Reμ>−1
2,Reβ>0/bracketrightbig
ET I 166(44)
2./integraldisplay∞
0x2μ−1e−βx2coshγxdx =1
2Γ(2μ)(2β)−μexp/parenleftbiggγ2
8β/parenrightbigg/bracketleftbigg
D−2μ/parenleftbigg
−γ√2β/parenrightbigg
+D−2μ/parenleftbiggγ√2β/parenrightbigg/bracketrightbigg
[Reμ>0,Reβ>0] ET I 166(45)
3./integraldisplay∞
0xe−βx2sinhγxdx =γ
4β/radicalbiggπ
βexp/parenleftbiggγ2
4β/parenrightbigg
[Reβ>0] BI(81)(12)a,ET I 165(34)
4./integraldisplay∞
0xe−βx2coshγxdx =γ
4β/radicalbiggπ
βexp/parenleftbiggγ2
4β/parenrightbigg
Φ/parenleftbiggγ
2√β/parenrightbigg
+1
2β
[Reβ>0] ET I 166(35)
5./integraldisplay∞
0x2e−βx2sinhγxdx =√π/parenleftbig
2β+γ2/parenrightbig
8β2√βexp/parenleftbiggγ2
4β/parenrightbigg
Φ/parenleftbiggγ
2√β/parenrightbigg
+γ
4β2
[Reβ>0] ET I 166(36)
6./integraldisplay∞
0x2e−βx2coshγxdx =√π/parenleftbig
2β+γ2/parenrightbig
8β2√βexp/parenleftbiggγ2
4β/parenrightbigg
[Reβ>0] ET I 166(37)
3.6–4.1 Trigonometric Functions
3.61 Rational functions of sines and cosines and trigonometric functions of multiple
angles
3.611
1./integraldisplay2π
0(1−cosx)nsinnxdx =0 BI (68)(10)
2./integraldisplay2π
0(1−cosx)ncosnxdx =(−1)nπ
2n−1BI (68)(11)
3./integraldisplayπ
0(cost+isintcosx)ndx=/integraldisplayπ
0(cost+isintcosx)−n−1dx=πPn(cost) EH I 158(23)a
3.613 Rational functions of sines and cosines 391
3.612
1.6/integraldisplayπ
0sinnxcosmx
sinxdx=0 f o r n≤m;
=πforn>m ,i f m+nis odd and positive
=0 f o r n>m ,i f m+nis even
LI (64)(3)
2./integraldisplayπ
0sinnx
sinxdx=0 f o r neven
=π fornodd
BI (64)(1, 2)
3./integraldisplayπ/2
0sin(2n−1)x
sinxdx=π
2FI II 145
4./integraldisplayπ/2
0sin2nx
sinxdx=2/parenleftbigg
1−1
3+1
5−···+(−1)k−1
2n−1/parenrightbigg
GW (332)(21b)
5./integraldisplayπ
0sin 2nx
cosxdx=2/integraldisplayπ/2
0sin 2nx
cosxdx=(−1)n−14/parenleftbigg
1−1
3+1
5−···+(−1)n−1
2n−1/parenrightbigg
GW (332)(22a)
6./integraldisplayπ
0cos(2n+1 )x
cosxdx=2/integraldisplayπ
2
0cos(2n+1 )x
cosxdx=(−1)nπ GW (332)(22b)
7./integraldisplayπ/2
0sin2nxcosx
sinxdx=π
2LI (45)(17)
3.613
1.6/integraldisplayπ
0cosnxdx
1+acosx=π√
1−a2/parenleftBigg√
1−a2−1
a/parenrightBiggn/bracketleftbig
a2<1,n≥0/bracketrightbig
BI (64)(12)
2.6/integraldisplayπ
0cosnxdx
1−2acosx+a2=πan
1−a2/bracketleftbig
a2<1,n≥0/bracketrightbig
=π
(a2−1)an/bracketleftbig
a2>1,n≥0/bracketrightbig
BI (65)(3)
3./integraldisplayπ
0sinnxsinxdx
1−2acosx+a2=π
2an−1/bracketleftbig
a2<1,n≥1/bracketrightbig
=π
2an+1/bracketleftbig
a2>1,n≥1/bracketrightbig
BI(65)(4), GW(332)(34a)
392 Trigonometric Functions 3.614
4.10/integraldisplayπ
0cosnxcosxdx
1−2acosx+a2=π
2·1+a2
1−a2an−1/bracketleftbig
a2<1,n≥1/bracketrightbig
=π
2an+1·a2+1
a2−1/bracketleftbig
a2>1,n≥1/bracketrightbig
=πa
1−a2/bracketleftbig
n=0,a2<1/bracketrightbig
=π
a(a2−1)/bracketleftbig
n=0,a2>1/bracketrightbig
BI(65)(5), GW(332)(34b)
5./integraldisplayπ
0cos(2n−1)xdx
1−2acos2x+a2=/integraldisplayπ
0cos2nxcosxdx
1−2acos2x+a2=0/bracketleftbig
a2/negationslash=1/bracketrightbig
BI (65)(9, 10)
6./integraldisplayπ
0cos(2n−1)xcos 2xdx
1−2acos2x+a2=0/bracketleftbig
a2/negationslash=1/bracketrightbig
BI (65)(12)
7./integraldisplayπ
0sin 2nxsinxdx
1−2acos2x+a2=/integraldisplayπ
0sin(2n−1)xsin 2xdx
1−2acos2x+a2=0
/bracketleftbig
a2/negationslash=1/bracketrightbig
BI (65)(6, 7)
8./integraldisplayπ
0sin(2n−1)xsinxdx
1−2acos2x+a2=π
2·an−1
1+a/bracketleftbig
a2<1/bracketrightbig
=π
2·1
(1 +a)an/bracketleftbig
a2>1/bracketrightbig
BI (65)(8)
9./integraldisplayπ
0cos(2n−1)xcosxdx
1−2acos2x+a2=π
2·an−1
1−a/bracketleftbig
a2<1/bracketrightbig
=π
2·1
(a−1)an/bracketleftbig
a2>1/bracketrightbig
BI (65)(11)
10./integraldisplayπ
0sinnx−asin(n−1)x
1−2acosx+a2sinmxdx =0 f o r m<n
=π
2am−nform≥n
/bracketleftbig
a2<1/bracketrightbig
LI (65)(13)
11.6/integraldisplayπ
0cosnx−acos(n−1)x
1−2acosx+a2cosmxdx =π
2/parenleftBig
a|m|−n−1/parenrightBig
/bracketleftbig
a2<1/bracketrightbig
BI (65)(14)
12./integraldisplayπ
0sinnx−asin[(n+1 )x]
1−2acosx+a2dx=0/bracketleftbig
a2<1/bracketrightbig
BI (68)(13)
13./integraldisplayπ
0cosnx−acos[(n+1 )x]
1−2acosx+a2dx=πan/bracketleftbig
a2<1/bracketrightbig
BI (68)(14)
3.616 Rational functions of sines and cosines 393
3.6147/integraldisplayπ
0sinx
a2−2abcosx+b2·sinpx·dx
1−2apcospx+a2p
=πbp−1
2ap+1(1−bp)[0<b≤a≤1,p=1,2,3,...]
=πap−1
2b(bp−a2p)/bracketleftbig
0<a≤1,a2<b , p =1,2,3,.../bracketrightbig
BI (66)(9)
3.615
1./integraldisplayπ/2
0cos 2nxdx
1−a2sin2x=(−1)nπ
2√
1−a2/parenleftBigg
1−√
1−a2
a/parenrightBigg2n/bracketleftbig
a2<1/bracketrightbig
BI (47)(27)
2./integraldisplayπ
0cosxsin 2nxdx
1+(a+bsinx)2=−π
bsin/braceleftbigg
2narctan/radicalbiggs
2/bracerightbigg
tan2n/parenleftbigg1
2arccos/radicalbiggs
2a2/parenrightbigg
3./integraldisplayπ
0cosxcos(2n+1 )xdx
1+(a+bsinx)2=π
bcos/braceleftbigg
(2n+1 )a r c t a n/radicalbiggs
2/bracerightbigg
tan2n+1/parenleftbigg1
2arccos/radicalbiggs
2a2/parenrightbigg
where s=−/parenleftbig
1+b2−a2/parenrightbig
+/radicalBig
(1 +b2−a2)2+4a2BI (65)(21, 22)
3.616
1./integraldisplayπ
0/parenleftbig
1−2acosx+a2/parenrightbigndx=πn/summationdisplay
k=0/parenleftBign
k/parenrightBig2
a2kBI (63)(1)
2.10/integraldisplayπ
0dx
(1−2acosx+a2)n=1
2/integraldisplay2π
0dx
(1−2acosx+a2)n
=π
(1−a2)nn−1/summationdisplay
k=0(n+k−1)!
(k!)2(n−k−1)!/parenleftbigga2
1−a2/parenrightbiggk/bracketleftbig
a2<1/bracketrightbig
=π
(a2−1)nn−1/summationdisplay
k=0(n+k−1)!
(k!)2(n−k−1)!1
(a2−1)k/bracketleftbig
a2>1/bracketrightbig
BI (331)(63)
3./integraldisplayπ
0/parenleftbig
1−2acosx+a2/parenrightbigncosnxdx =(−1)nπanBI (63)(2)
4./integraldisplayπ
0/parenleftbig
1−2acosx+a2/parenrightbigncosmxdx
=1
2/integraldisplay2π
0/parenleftbig
1−2acosx+a2/parenrightbigncosmxdx
=0 [ n<m ]
=π(−a)m/parenleftbig
1+a2/parenrightbign−m[(n−m)/2]/summationdisplay
k=0/parenleftBign
k/parenrightBig/parenleftbiggn−k
m+k/parenrightbigg/parenleftbigga
1+a2/parenrightbigg2k
[n≥m]
GW (332)(35a)
5./integraldisplay2π
0sinnxdx
(1−2acos2x+a2)m=0 GW (332)(32a)
394 Trigonometric Functions 3.617
6./integraldisplayπ
0sinxdx
(1−2acos2x+a2)m=1
2(m−1)a/bracketleftbigg1
(1−a)2m−2−1
(1 +a)2m−2/bracketrightbigg
[a/negationslash=0,±1]
GW (332)(32c)
7./integraldisplayπ
0cosnxdx
(1−2acosx+a2)m=1
2/integraldisplay2π
0cosnxdx
(1−2acosx+a2)m
=a2m+n−2π
(1−a2)2m−1m−1/summationdisplay
k=0/parenleftbiggm+n−1
k/parenrightbigg/parenleftbigg2m−k−2
m−1/parenrightbigg/parenleftbigg1−a2
a2/parenrightbiggk/bracketleftbig
a2<1/bracketrightbig
=π
an(a2−1)2m−1m−1/summationdisplay
k=0/parenleftbiggm+n−1
k/parenrightbigg/parenleftbigg2m−k−2
m−1/parenrightbigg/parenleftbig
a2−1/parenrightbigk/bracketleftbig
a2>1/bracketrightbig
GW (332)(31)
8./integraldisplayπ/2
0cos2nxdx
/parenleftbig
a2cos2x+b2sin2x/parenrightbign+1=/parenleftbigg2n
n/parenrightbigg/parenleftbig
b2−a2/parenrightbign
(2ab)2n+1π
[a>0,b > 0] GW (332)(30b)
3.61710/integraldisplayπ
0dx
(1−2acosx+a2)n+1/2=2
|1+a|2n+1Fn/parenleftBigg
2/radicalbig
|a|
|1+a|/parenrightBigg
,|a|/negationslash=1
with
Fn(k)=/integraldisplayπ/2
0dx
/parenleftbig
1−k2sin2x/parenrightbign+1/2
where the Fn(k) satisfies the recurrence relation
Fn+1(k)=Fn(k)+k
2n+1dFn(k)
dk,n =0,1,2,...
and
F0(k)=K(k)≡/integraldisplayπ/2
0dx
/parenleftbig
1−k2sin2x/parenrightbig1/2
is the complete elliptic integral of the first kind.
Introducing the complete elliptic integral of the second kind
E(k)=/integraldisplayπ/2
0/parenleftbig
1−k2sin2x/parenrightbig1/2dx
the derivatives
dK(k)
dk=E(k)
k(1−k2)−K(k)
k,dE(k)
dk=E(k)−K(k)
k
combined with the recurrence relation lead to
F1(k)=F0(k)+kdF0(k)
dk
=K(k)+E(k)
1−k2−K(k)=E(k)
1−k2,
F2(k)=E(k)
1−k2+k
3d
dk/parenleftbiggE(k)
1−k2/parenrightbigg
=1
3( 1−k2)/bracketleftbigg/parenleftbigg4−2k2
1−k2/parenrightbigg
E(k)−K(k)/bracketrightbigg
3.623 Powers of trigonometric functions 395
3.62 Powers of trigonometric functions
3.621
1./integraldisplayπ/2
0sinμ−1xdx=/integraldisplayπ/2
0cosμ−1xdx=2μ−2B/parenleftBigμ
2,μ
2/parenrightBig
FI II 789
2./integraldisplayπ/2
0sin3/2xdx=/integraldisplayπ/2
0cos3/2xdx=1
6√
2π/bracketleftbigg
Γ/parenleftbigg1
4/parenrightbigg/bracketrightbigg2
3./integraldisplayπ/2
0sin2mxdx=/integraldisplayπ/2
0cos2mxdx=(2m−1)!!
(2m)!!π
2FI II 151
4./integraldisplayπ/2
0sin2m+1xdx=/integraldisplayπ/2
0cos2m+1xdx=(2m)!!
(2m+1 ) ! !FI II 151
5./integraldisplayπ/2
0sinμ−1xcosν−1xdx=1
2B/parenleftBigμ
2,ν
2/parenrightBig
[Reμ>0,Reν>0]
LO V 113(50), LO V 122, FI II 788
6.∗/integraldisplayπ/2
0√
sinxd x=/radicalbigg
2
π/parenleftbigg
Γ/parenleftbigg3
4/parenrightbigg/parenrightbigg2
7.∗/integraldisplayπ/2
0dx√
sinx=/parenleftbig
Γ/parenleftbig1
4/parenrightbig/parenrightbig2
2√
2π
3.622
1./integraldisplayπ/2
0tan±μxdx=π
2secμπ
2[|Reμ|<1] BI (42)(1)
2./integraldisplayπ/4
0tanμxdx=1
2β/parenleftbiggμ+1
2/parenrightbigg
[Reμ>−1] BI (34)(1)
3./integraldisplayπ/4
0tan2nxdx=(−1)nπ
4+n−1/summationdisplay
k=0(−1)k
2n−2k−1BI (34)(2)
4.11/integraldisplayπ/4
0tan2n+1xdx=(−1)nln2
2+n−1/summationdisplay
k=0(−1)k
2n−2kBI (34)(3)
3.623
1./integraldisplayπ/2
0tanμ−1xcos2ν−2xdx=/integraldisplayπ/2
0cotμ−1xsin2ν−2xdx=1
2B/parenleftBigμ
2,ν−μ
2/parenrightBig
[0<Reμ<2R eν]BI(42)(6), BI(45)(22)
2.6/integraldisplayπ/4
0tanμxsin2xdx=1+μ
4β/parenleftbiggμ+1
2/parenrightbigg
−1
4[Reμ>−1] BI (34)(4)
3.6/integraldisplayπ/4
0tanμxcos2xdx=1−μ
4β/parenleftbiggμ+1
2/parenrightbigg
+1
4[Reμ>−1] BI (34)(5)
396 Trigonometric Functions 3.624
3.624
1./integraldisplayπ/4
0sinpx
cosp+2xdx=1
p+1[p>−1] GW (331)(34b)
2.3/integraldisplayπ/2
0sinμ−1
2x
cos2μ−1xdx=/integraldisplayπ/2
0cosμ−1
2x
sin2μ−1xdx=1
2/braceleftBigg
Γ/parenleftbigμ
2+1
4/parenrightbig
Γ(1−μ)
Γ/parenleftbig5
4−μ
2/parenrightbig/bracerightBigg
/bracketleftbig
−1
2<Reμ<1/bracketrightbig
LI (55)(12)
3.11/integraldisplayπ/4
0cosn−1
2(2x)
cos2n+1(x)dx=π(2n)!!
22n+1(n!)2BI (38)(3)
4.8/integraldisplayπ/4
0cosμ2x
cos2(μ+1)xdx=22μB(μ+1,μ+1 ) [ R e μ>−1] BI (35)(1)
5./integraldisplayπ/4
0sin2μ−2x
cosμ2xdx=21−2μB(2μ−1,1−μ)=Γ/parenleftbig
μ−1
2/parenrightbig
Γ(1−μ)
2√π/bracketleftbig1
2<Reμ<1/bracketrightbig
BI (35)(4)
6.6/integraldisplayπ/2
0/parenleftbiggsinax
sinx/parenrightbigg2
dx=aπ
2−1
2sinπa[2aβ(a)−1], [a>0]
3.625
1./integraldisplayπ/4
0sin2n−1xcosp2x
cos2p+2n+1xdx=(n−1)!
2·Γ(p+1 )
Γ(p+n+1 )
=(n−1)!
2(p+n)(p+n−1)···(p+1 )=1
2B(n, p+1 )
[p>−1] (cf. 3.251 1) BI (35)(2)
2./integraldisplayπ/4
0sin2nxcosp2x
cos2p+2n+2xdx=1
2B/parenleftbig
n+1
2,p+1/parenrightbig
[p>−1] (cf. 3.251 1) BI (35)(3)
3./integraldisplayπ/4
0sin2n−1xcosm−1
22x
cos2n+2mxdx=(2n−2)!!(2m−1)!!
(2n+2m−1)!!BI (38)(6)
4.8/integraldisplayπ/4
0sin2nxcosm−1
22x
cos2n+2m+1xdx=(2n−1)!!(2m−1)!!
(2n+2m)!!·π
2BI (38)(7)
3.626
1./integraldisplayπ/4
0sin2n−1x
cos2n+2x√
cos 2xdx=(2n−2)!!
(2n+1 ) ! !(cf.3.251 1) BI (38)(4)
2./integraldisplayπ/4
0sin2nx
cos2n+3x√
cos 2xdx=(2n−1)!!
(2n+2 ) ! !·π
2(cf.3.251 1) BI (38)(5)
3.627/integraldisplayπ/2
0tanμx
cosμxdx=/integraldisplayπ/2
0cotμx
sinμxdx=Γ(μ)Γ/parenleftbig1
2−μ/parenrightbig
2μ√πsinμπ
2/bracketleftbig
−1<Reμ<1
2/bracketrightbig
BI (55)(12)a
3.62811/integraldisplayπ
2
0sec2pxsin2p−1xdx=1
2√πΓ(p)Γ/parenleftbig1
2−p/parenrightbig/bracketleftbig
0<p<1
2/bracketrightbig
WA 691
3.631 Powers of trigonometric functions 397
3.63 Powers of trigonometric functions and trigonometric functions of linear
functions
3.631
1./integraldisplayπ
0sinν−1xsinaxdx =πsinaπ
2
2ν−1νB/parenleftbiggν+a+1
2,ν−a+1
2/parenrightbigg
[Reν>0] LO V 121(67a), WA 337a
2.7/integraldisplayπ/2
02s inν−2xsinνxdx =1
1−νcosνπ
2[Reν>1] GW(332)(16d), FI I 152
3.6/integraldisplayπ
0sinνxsinνxdx =2−νπsinνπ
2[Reν>−1] LO V 121(69)
4./integraldisplayπ
0sinnxsin 2mxdx =0 GW (332)(11a)
5./integraldisplayπ
0sin2nxsin(2m+1 )xdx=/integraldisplayπ/2
0sin2nxsin(2m+1 )xdx
=(−1)m2n+1n!(2n−1)!!
(2n−2m−1)!!(2m+2n+1 ) ! ![m≤n]∗
=(−1)n2n+1n!(2m−2n−1)!!(2n−1)!!
(2m+2n+1 ) ! ![m≥n]∗
GW (332)(11b)
6./integraldisplayπ
0sin2n+1xsin(2m+1 )xdx=2/integraldisplayπ/2
0sin2n+1xsin(2m+1 )xdx
=(−1)mπ
22n+1/parenleftbigg2n+1
n−m/parenrightbigg
[n≥m]
=0 [ n<m ]
BI(40)(12), GW(332)(11c)
7./integraldisplayπ
0sinnxcos(2m+1 )xdx=0 GW (332)(12a)
8./integraldisplayπ
0sinν−1xcosaxdx =πcosaπ
2
2ν−1νB/parenleftbiggν+a+1
2,ν−a+1
2/parenrightbigg
[Reν>0] LO V 121(68)a, WA 337a
9./integraldisplayπ/2
0cosν−1xcosaxdx =π
2ννB/parenleftbiggν+a+1
2,ν−a+1
2/parenrightbigg
[Reν>0] GW (332)(9c)
10./integraldisplayπ/2
0sinν−2xcosνxdx =1
ν−1sinνπ
2[Reν>1] GW(332)(16b), FI II 15 2
∗In 3.631.5, for m=nwe should set (2 n−2m−1)!! = 1
398 Trigonometric Functions 3.632
11./integraldisplayπ
0sinνxcosνxdx =π
2νcosνπ
2[Reν>−1] LO V 121(70)a
12./integraldisplayπ
0sin2nxcos 2mxdx =2/integraldisplayπ/2
0sin2nxcos 2mxdx =(−1)m
22n/parenleftbigg2n
n−m/parenrightbigg
π[n≥m]
=0 [ n<m ]
BI(40)(16), GW(332)(12b)
13.7/integraldisplayπ
0sin2n+1xcos 2mxdx
=2/integraldisplayπ/2
0sin2n+1xcos 2mxdx =(−1)m2n+1n!(2n+1 ) ! !
(2m−2n−3)!!(2m+2n+1 ) ! ![n≥m−1]
=(−1)n+12n+1n!(2m−2n+ 3)!!(2 n+1 ) ! !
(2m+2n+1 ) ! ![n<m −1]
GW (332)(12c)
14./integraldisplayπ/2
0cosν−2xsinνxdx =1
ν−1[Reν>1] GW(332)(16c), FI II 152
15./integraldisplayπ
0cosmxsinnxdx =/bracketleftbig
1−(−1)m+n/bracketrightbig/integraldisplayπ/2
0cosmxsinnxdx
=/bracketleftbig
1−(−1)m+n/bracketrightbig/braceleftBiggr−1/summationdisplay
k=0m!
(m−k)!(m+n−2k−2)!!
(m+n)!!+sm!(n−m−2)!!
(m+n)!!/bracerightBigg
⎡
⎢⎣r=/braceleftBigg
mifm≤n
nifm≥ns=⎧
⎪⎨
⎪⎩2i fn−m=4l+2>0
1i fn−m=2l+1>0
0i fn−m=4lorn−m<0⎤
⎥⎦GW (332)(13a)
16./integraldisplayπ/2
0cosnxsinnxdx =1
2n+1n/summationdisplay
k=12k
kFI II 153
17.11/integraldisplayπ
0cosnxsinmxdx =/braceleftBigg/bracketleftbig
1+(−1)m+n/bracketrightbigπ
2n+1/parenleftBign
k/parenrightBig
ifm≤nandn−m=2k
0 otherwise
GW (332)(15a)
18.6/integraldisplayπ
0cosmxcosaxdx =(−1)msinaπ
2m(m+a)2F1/parenleftbigg
−m,−a+m
2;1−a+m
2;−1/parenrightbigg
[a/negationslash=0,±1,±2,...] WA 313
19./integraldisplayπ/2
0cosν−2xcosνxdx =0 [ R e ν>1] GW(332)(16a), FI II 152
20.10/integraldisplayπ/2
0cosnxcosnxdx =π
2n+1[Ren>−1] LO V 122(78), FI II 153
3.632
1./integraldisplayπ
0sinp−1xcos/bracketleftBig
a/parenleftBigπ
2−x/parenrightBig/bracketrightBig
dx=2p−1Γ/parenleftbigp−a
2/parenrightbig
Γ/parenleftbigp+a
2/parenrightbig
Γ(p−a)Γ(p+a)Γ(p)
/bracketleftbig
p2<a2/bracketrightbig
BI (62)(11)
3.634 Powers of trigonometric functions 399
2./integraldisplayπ
2
−π
2cosν−1xsin/bracketleftBig
a/parenleftBig
x+π
2/parenrightBig/bracketrightBig
dx=πsinaπ
2
2ν−1νB/parenleftbiggν+a+1
2,ν−a+1
2/parenrightbigg
[Reν>0] WA 337a
3.10/integraldisplayπ/2
0cospxsin[(p+2n)x]dx=(−1)n−1n−1/summationdisplay
k=0(−1)k2k
p+k+1/parenleftbiggn−1
k/parenrightbigg
[n>0] LI (41)(12)
4./integraldisplayπ
−πcosn−1xcos[m(x−a)]dx=/bracketleftbig
1−(−1)n+m/bracketrightbig
=/integraldisplayπ
2
−π
2cosn−1xcos[m(x−a)]dx
=[1−(−1)n+m]πcosma
2n−1nB/parenleftbiggn+m+1
2,n−m+1
2/parenrightbigg
[n≥m] LO V 123(80), LO V 139(94a)
5./integraldisplayπ/2
0cosp+q−2xcos[(p−q)x]dx=π
2p+q−1(p+q−1)B(p, q)
[p+q>1] WH
3.633
1./integraldisplayπ/2
0cosp−1xsinaxsinxdx=aπ
2p+1p(p+1 )B/parenleftbiggp+a
2+1,p−a
2+1/parenrightbigg LO V 150(110)
2./integraldisplayπ/2
0cosnxsinnxsin 2mxdx =/integraldisplayπ/2
0cosnxcosnxcos 2mxdx =π
2n+2/parenleftBign
m/parenrightBig
BI (42)(19, 20)
3./integraldisplayπ/2
0cosn−1xcos[(n+1 )x]c os2mxdx =π
2n+1/parenleftbiggn−1
m−1/parenrightbigg
[n>m −1] BI (42)(21)
4./integraldisplayπ/2
0cosp+qxcospxcosqxdx =π
2p+q+2/bracketleftbigg
1+1
(p+q+1 )B ( p+1,q+1 )/bracketrightbigg
[p+q>−1] GW (332)(10c)
5.6/integraldisplayπ/2
0cosp+qxsinpxsinqxdx =π
2p+q+2∞/summationdisplay
k=1/parenleftBigp
k/parenrightBig/parenleftBigq
k/parenrightBig
=π
2p+q+2/bracketleftbiggΓ(p+q+1 )
Γ(p+1 )Γ ( q+1 )−1/bracketrightbigg
[p+q>−1] BI (42)(16)
3.634
1./integraldisplayπ/2
0sinμ−1xcosν−1xsin(μ+ν)xdx=s i nμπ
2B(μ, ν)
[Reμ>0,Reν>0]
BI(42)(23), FI II 814a
400 Trigonometric Functions 3.635
2./integraldisplayπ/2
0sinμ−1xcosν−1xcos(μ+ν)xdx=c o sμπ
2B(μ, ν)
[Reμ>0,Reν>0]
BI(42)(24), FI II 814a
3./integraldisplayπ/2
0cosp+n−1xsinpxcos[(n+1 )x]s inxdx=π
2p+n+1Γ(p+n)
n!Γ (p)
[p>−n] BI (42)(15)
3.635
1./integraldisplayπ/4
0cosμ−12xtanxdx=1
4/bracketleftbigg
ψ/parenleftbiggμ+1
2/parenrightbigg
−ψ/parenleftBigμ
2/parenrightBig/bracketrightbigg
[Reμ>0] BI (34)(7)
2.7/integraldisplayπ/2
0cosp+2nxsinpxtanxdx=π
2p+2n+1Γ(p)∞/summationdisplay
k=0/parenleftBign
k/parenrightBigΓ(p+n−k)
(n−k)!
=pπ
2p+2+n+1Γ(p+2n)
Γ(n+1 )Γ ( p+n+1 )
[p>−2n] BI (42)(22)
3./integraldisplayπ/2
0cosn−1xsin[(n+1 )x]c otxdx=π
2BI (45)(18)
3.636
1./integraldisplayπ/2
0tan±μxsin 2xdx=μπ
2cosecμπ
2[0<Reμ<2] BI (45)(20)a
2./integraldisplayπ/2
0tan±μxcos 2xdx=∓μπ
2secμπ
2[|Reμ|<1] BI (45)(21)
3.11/integraldisplayπ/2
0tan2μx
cosxdx=/integraldisplayπ/2
0cot2μx
sinxdx=Γ/parenleftbig
μ+1
2/parenrightbig
Γ(−μ)
2√π/bracketleftbig
−1
2<Reμ<1/bracketrightbig
(cf.3.251 1)
BI (45)(13, 14)
3.637
1./integraldisplayπ/2
0tanpxsinq−2xsinqxdx =−cos(p+q)π
2B(p+q−1,1−p)
[p+q>1>p] GW (332)(15d)
2./integraldisplayπ/2
0tanpxsinq−2xcosqxdx =s i n(p+q)π
2B(p+q−1,1−p)
[p+q>1>p] GW (332)(15b)
3./integraldisplayπ/2
0cotpxcosq−2xsinqxdx =c o spπ
2B(p+q−1,1−p)
[p+q>1>p] GW (332)(15c)
3.642 Trigonometric functions: powers and rational functions 401
4./integraldisplayπ/2
0cotpxcosq−2xcosqxdx =s i npπ
2B(p+q−1,1−p)
[p+q>1>p] GW (332)(15a)
3.638
1./integraldisplayπ/4
0sin2μxdx
cosμ+1
22xcosx=π
2secμπ/bracketleftbig
|Reμ|<1
2/bracketrightbig
(cf.3.192 2)
BI (38)(8)
2./integraldisplayπ/4
0sinμ−1
22xdx
cosμ2xcosx=2
2μ−1·Γ/parenleftbig
μ+1
2/parenrightbig
Γ(1−μ)√πsin/parenleftbigg2μ−1
4π/parenrightbigg
/bracketleftbig
−1
2<Reμ<1/bracketrightbig
BI (38)(17)
3./integraldisplayπ/2
0cosp−1xsinpx
sinxdx=π
2[p>0] GW(332)(17), BI(45)(5)
3.64–3.65 Powers and rational functions of trigonometric functions
3.641
1./integraldisplayπ/2
0sinp−1xcos−px
acosx+bsinxdx=/integraldisplayπ/2
0sin−pxcosp−1x
asinx+bcosxdx=πcosecpπ
a1−pbp
[ab >0,0<p< 1] GW (331)(62)
2./integraldisplayπ/2
0sin1−pxcospx
(sinx+c o s x)3dx=/integraldisplayπ/2
0sinpxcos1−px
(sinx+c o s x)3dx=(1−p)p
2πcosecpπ
[−1<p< 2] BI(48)(5)
3.642
1./integraldisplayπ/2
0sin2μ−1xcos2ν−1xdx
/parenleftbig
a2sin2x+b2cos2x/parenrightbigμ+ν=1
2a2μb2νB(μ, ν)[ R e μ>0,Reν>0] BI (48)(28)
2./integraldisplayπ/2
0sinn−1xcosn−1xdx/parenleftbig
a2cos2x+b2sin2x/parenrightbign=B/parenleftbign
2,n
2/parenrightbig
2(ab)n[ab >0] GW (331)(59a)
3./integraldisplayπ/2
0sin2nxdx
/parenleftbig
a2cos2x+b2sin2x/parenrightbign+1=1
2/integraldisplayπ
0sin2nxdx
/parenleftbig
a2cos2x+b2sin2x/parenrightbign+1
=/integraldisplayπ/2
0cos2nxdx
/parenleftbig
a2sin2x+b2cos2x/parenrightbign+1=1
2/integraldisplayπ
0cos2nxdx
/parenleftbig
a2sin2x+b2cos2x/parenrightbign+1=(2n−1)!!π
2n+1n!ab2n+1
[ab >0] GW (331)(58)
4./integraldisplayπ/2
0cosp+2nxcospxdx
/parenleftbig
a2cos2x+b2sin2x/parenrightbign+1=πn/summationdisplay
k=0/parenleftbigg2n−k
n/parenrightbigg/parenleftbiggp+k−1
k/parenrightbiggbp−1
(2a)2n−k+1(a+b)p+k
[a>0,b > 0,p > −2n−1]
GW (332)(30)
402 Trigonometric Functions 3.643
3.643
1./integraldisplayπ/2
0cospxcospxdx
1−2acos2x+a2=π
2p+1·(1 +a)p−1
1−a/bracketleftbig
a2<1,p > −1/bracketrightbig
GW (332)(33c)
2./integraldisplayπ/2
0sin2nxcosμxcosβx
(1−2acos2x+a2)mdx=(−1)nπ(1−a)2n−2m+1
22m−β−1(1 +a)2m+β+1m−1/summationdisplay
k=0m−k−1/summationdisplay
l=0/parenleftbiggβ
k/parenrightbigg/parenleftbigg2n
l/parenrightbigg
×/parenleftbigg2m−k−l−2
m−1(−2)l/parenrightbigg
(a−1)k
/bracketleftbig
a2<1,β=2m−2n−μ−2,μ > −1/bracketrightbig
GW (332)(33)
3.644
1./integraldisplayπ
0sinmx
p+qcosxdx=2m−2p
q2k/summationdisplay
ν=1/parenleftbiggp2−q2
−4q2/parenrightbiggν−1
B/parenleftbiggm+1−2ν
2,m+1−2ν
2/parenrightbigg
+/parenleftbiggp2−q2
−q2/parenrightbiggk
A
where A=⎧
⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩πp
q2/parenleftBigg
1−/radicalBigg
1−q2
p2/parenrightBigg
ifm=2k+2
1
qlnp+q
p−qifm=2k+1/bracketleftbig
k≥1,q/negationslash=0,p2−q2≥0/bracketrightbig
2./integraldisplayπ
0sinmx
1 + cos xdx=2m−1B/parenleftbiggm−1
2,m+1
2/parenrightbigg
[m≥2]
3./integraldisplayπ
0sinmx
1−cosxdx=2m−1B/parenleftbiggm−1
2,m+1
2/parenrightbigg
[m≥2]
4./integraldisplayπ
0sin2x
p+qcosxdx=pπ
q2/parenleftBigg
1−/radicalBigg
1−q2
p2/parenrightBigg
5./integraldisplayπ
0sin3x
p+qcosxdx=2p
q2+1
q/parenleftbigg
1−p2
q2/parenrightbigg
lnp+q
p−q
3.645/integraldisplayπ
0cosnxdx
(a+bcosx)n+1=π
2n(a+b)n√
a2−b2n/summationdisplay
k=0(−1)k(2n−2k−1)!!(2k−1)!!
(n−k)!k!/parenleftbigga+b
a−b/parenrightbiggk
/bracketleftbig
a2>b2/bracketrightbig
LI (64)(16)
3.646
1./integraldisplayπ/2
0cosnxsinnxsin 2x
1−2acos2x+a2dx=π
4a/bracketleftbigg/parenleftbigg1+a
2/parenrightbiggn
−1
2n/bracketrightbigg/bracketleftbig
a2<1/bracketrightbig
BI (50)(6)
2./integraldisplayπ/2
01−acos2nx
1−2acos2nx+a2cosmxcosmxdx =π
2m+2∞/summationdisplay
k=1/parenleftBigm
kn/parenrightBig
ak+π
2m+1
/bracketleftbig
a2<1/bracketrightbig
LI (50)(7)
3.647/integraldisplayπ/2
0cospxcospxdx
a2sin2x+b2cos2x=π
2b·ap−1
(a+b)p[p>−1,a > 0,b > 0] BI (47)(20)
3.652 Trigonometric functions: powers and rational functions 403
3.648
1./integraldisplayπ/4
0tanlxdx
1 + cosm
nπsin 2x
=1
2ncosecm
nπn−1/summationdisplay
k=0(−1)k−1sinkm
nπ/bracketleftbigg
ψ/parenleftbiggn+l+k
2n/parenrightbigg
−ψ/parenleftbiggl+k
2n/parenrightbigg/bracketrightbigg
[m+nis odd]
=1
ncosecm
nπn−1
2/summationdisplay
k=0(−1)k−1sinkm
nπ/bracketleftbigg
ψ/parenleftbiggn+l−k
n/parenrightbigg
−ψ/parenleftbiggl+k
n/parenrightbigg/bracketrightbigg
[m+nis even]
[lis a natural number] BI (36)(5)
2./integraldisplayπ/2
0tan±μxdx
1 + cos tsin 2x=πcosectsinμtcosec( μπ)/bracketleftbig
|Reμ|<1,t2<π2/bracketrightbig
BI (47)(4)
3.649
1./integraldisplayπ/2
0tan±μxsin 2xdx
1∓2acos2x+a2=π
4acosecμπ
2/bracketleftbigg
1−/parenleftbigg1−a
1+a/parenrightbiggμ/bracketrightbigg/bracketleftbig
a2<1/bracketrightbig
=π
4acosecμπ
2/bracketleftbigg
1+/parenleftbigga−1
a+1/parenrightbiggμ/bracketrightbigg/bracketleftbig
a2>1/bracketrightbig
[−2<Reμ<1] BI (50)(3)
2./integraldisplayπ/2
0tan±μx(1∓acos2x)
1∓2acos2x+a2dx=π
4secμπ
2/bracketleftbigg
1+/parenleftbigg1−a
1+a/parenrightbiggμ/bracketrightbigg/bracketleftbig
a2<1/bracketrightbig
=π
4secμπ
2/bracketleftbigg
1−/parenleftbigga−1
a+1/parenrightbiggμ/bracketrightbigg/bracketleftbig
a2>1/bracketrightbig
[|Reμ|<1] BI (50)(4)
3.651
1./integraldisplayπ/4
0tanμxdx
1+s i n xcosx=1
3/bracketleftbigg
ψ/parenleftbiggμ+2
3/parenrightbigg
−ψ/parenleftbiggμ+1
3/parenrightbigg/bracketrightbigg
[Reμ>−1] BI (36)(3)
2./integraldisplayπ/4
0tanμxdx
1−sinxcosx=1
3/bracketleftbigg
β/parenleftbiggμ+2
3/parenrightbigg
+β/parenleftbiggμ+1
3/parenrightbigg/bracketrightbigg
[Reμ>−1] BI (36)(4)a
3.652
1./integraldisplayπ/2
0tanμxdx
(sinx+c o s x)s i nx=/integraldisplayπ/2
0cotμxdx
(sinx+c o s x)c o sx=πcosecμπ
[0<Reμ<1] BI (49)(1)
2./integraldisplayπ/2
0tanμxdx
(sinx−cosx)s i nx=/integraldisplayπ/2
0cotμxdx
(cosx−sinx)c o sx=−πcotμπ
[0<Reμ<1] BI (49)(2)
3./integraldisplayπ/2
0cotμ+1
2xdx
(sinx+c o s x)c o sx=/integraldisplayπ/2
0tanμ−1
2xdx
(sinx+c o s x)c o sx=πsecμπ
/bracketleftbig
|Reμ|<1
2/bracketrightbig
BI (61)(1, 2)
404 Trigonometric Functions 3.653
3.653
1./integraldisplayπ/2
0tan1−2μxdx
a2cos2x+b2sin2x=/integraldisplayπ/2
0cot1−2μxdx
a2sin2x+b2cos2x=π
2a2μb2−2μsinμπ
[0<Reμ<1] GW (331)(59b)
2.11/integraldisplayπ/2
0tanμxdx
1−asin2x=/integraldisplayπ/2
0cotμxdx
1−acos2x=πsecμπ
2
2/radicalbig
(1−a)μ+1
[|Reμ|<1,a < 1] BI (49)(6)
3./integraldisplayπ/2
0tan±μxdx
1−cos2tsin22x=π
2cosectsecμπ
2cos/bracketleftBig/parenleftBigπ
2−t/parenrightBig
μ/bracketrightBig
/bracketleftbig
|Reμ|<1,t2<π2/bracketrightbig
BI(49)(7), BI(47)(21)
4./integraldisplayπ/2
0tan±μxsin 2x
1−cos2tsin22xdx=πcosec 2 tcosecμπ
2sin/bracketleftBig/parenleftBigπ
2−t/parenrightBig
μ/bracketrightBig
/bracketleftbig
|Reμ|<1,t2<π2/bracketrightbig
BI (47)(22)a
5./integraldisplayπ/2
0tanμxsin2xdx
1−cos2tsin22x=/integraldisplayπ/2
0cotμxcos2xdx
1−cos2tsin22x=π
2cosec 2 tsecμπ
2cos/bracketleftBigμπ
2−(μ+1 )t/bracketrightBig
/bracketleftbig
|Reμ|<1,t2<π2/bracketrightbig
BI(47)(23)a, BI(49)(10)
6./integraldisplayπ/2
0tanμxcos2xdx
1−cos2tsin22x=/integraldisplayπ/2
0cotμxsin2xdx
1−cos2tsin22x=π
2cosec 2 tsecμπ
2cos/bracketleftBigμπ
2−(μ−1)t/bracketrightBig
/bracketleftbig
|Reμ|<1,t2<π2/bracketrightbig
BI(47)(24)a, BI(49)(9)
3.654
1./integraldisplayπ/2
0tanμ+1xcos2xdx
(1 + cos tsin 2x)2=/integraldisplayπ/2
0cotμ+1xsin2xdx
(1 + cos tsin 2x)2=π(μsintcosμt−costsinμt)
2s inμπsin3t/bracketleftbig
|Reμ|<1,t2<π2/bracketrightbig
BI(48)(3), BI(49)(22)
2./integraldisplayπ/2
0tan±μxdx
(sinx+c o s x)2=μπ
sinμπ[0<Reμ<1] BI (56)(9)a
3./integraldisplayπ/2
0tan±(μ−1)xdx
cos2x−sin2x=±π
2cotμπ
2[0<Reμ<2] BI (45)(27, 29)
3.655/integraldisplayπ/2
0tan2μ−1xdx
1−2a/parenleftbig
cost1sin2x+c o s t2cos2x/parenrightbig
+a2=/integraldisplayπ/2
0cot2μ−1xdx
1−2a/parenleftbig
cost1cos2x+c o s t2sin2x/parenrightbig
+a2
=πcosecμπ
(1−2acost2+a2)μ(1−2acost1+a2)1−μ/bracketleftbig
0<Reμ<1,t2
1<π2,t22<π2/bracketrightbig
BI (50)(18)
3.662 Trigonometric functions: powers of linear functions 405
3.656
1./integraldisplayπ/4
0tanμxdx
1−sin2xcos2x=1
12/braceleftbigg
−ψ/parenleftbiggμ+1
6/parenrightbigg
−ψ/parenleftbiggμ+2
6/parenrightbigg
+ψ/parenleftbiggμ+4
6/parenrightbigg
+ψ/parenleftbiggμ+5
6/parenrightbigg
+2ψ/parenleftbiggμ+2
3/parenrightbigg
−2ψ/parenleftbiggμ+1
3/parenrightbigg/bracerightbigg
[Reμ>−1] (cf. 3.651 1a n d2 ) LI (36)(10)
2./integraldisplayπ/2
0tanμ−1xcos2xdx
1−sin2xcos2x=/integraldisplayπ/2
0cotμ−1xsin2xdx
1−sin2xcos2x=π
4√
3cosecμπ
6cosec/parenleftbigg2+μ
6π/parenrightbigg
[0<Reμ<4] LI (47)(26)
3.66 Forms containing powers of linear functions of trigonometric functions
3.661
1./integraldisplay2π
0(asinx+bcosx)2n+1dx=0 BI (68)(9)
2./integraldisplay2π
0(asinx+bcosx)2ndx=(2n−1)!!
(2n)!!·2π/parenleftbig
a2+b2/parenrightbignBI (68)(8)
3./integraldisplayπ
0(a+bcosx)ndx=1
2/integraldisplay2π
0(a+bcosx)ndx=π/parenleftbig
a2−b2/parenrightbign
2Pn/parenleftbigga√
a2−b2/parenrightbigg
=π
2n⌊n/2⌋/summationdisplay
k=0(−1)k(2n−2k)!
k!(n−k)!(n−2k)!an−2k/parenleftbig
a2−b2/parenrightbigk
/bracketleftbig
a2>b2/bracketrightbig
GW (332)(37a)
4./integraldisplayπ
0dx
(a+bcosx)n+1=1
2/integraldisplay2π
0dx
(a+bcosx)n+1=π
(a2−b2)n+1
2Pn/parenleftbigga√
a2−b2/parenrightbigg
=π
2n(a+b)n√
a2−b2n/summationdisplay
k=0(2n−2k−1)!!(2k−1)!!
(n−k)!k!·/parenleftbigga+b
a−b/parenrightbiggk
[a>|b|] GW(332)(38), LI(64)(14)
3.662
1./integraldisplayπ/2
0(secx−1)μsinxdx=/integraldisplayπ/2
0(cosec x−1)μcosxdx=μπcosecμπ
[|Reμ|<1] BI (55)(13)
2./integraldisplayπ/2
0(cosec x−1)μsin 2xdx=( 1−μ)μπcosecμπ [−1<Reμ<2] BI (48)(7)
3./integraldisplayπ/2
0(secx−1)μtanxdx=/integraldisplayπ/2
0(cosec x−1)μcotxdx=−πcosecμπ
[−1<Reμ<0] BI (46)(4,6)
4./integraldisplayπ/4
0(cotx−1)μdx
sin 2x=−π
2cosecμπ [−1<Reμ<0] BI (38)(22)a
406 Trigonometric Functions 3.663
5./integraldisplayπ/4
0(cotx−1)μdx
cos2x=μπcosecμπ [|Reμ|<1] BI (38)(11)a
3.663
1./integraldisplayu
0(cosx−cosu)ν−1
2cosaxdx =/radicalbiggπ
2sinνuΓ/parenleftbigg
ν+1
2/parenrightbigg
P−ν
a−1
2(cosu)
/bracketleftbig
Reν>−1
2;a>0,0<u<π/bracketrightbig
EH I 159(27), ET I 22(28)
2./integraldisplayu
0(cosx−cosu)ν−1cos[(ν+β)x]dx=√πΓ(β+1 )Γ ( ν)Γ(2ν)sin2ν−1u
2νΓ(β+2ν)Γ/parenleftbig
ν+1
2/parenrightbig Cν
β(cosu)
[Reν>0,Reβ>−1,0<u<π ]
EH I 178(23)
3.664
1./integraldisplayπ
0/parenleftBig
z+/radicalbig
z2−1c osx/parenrightBigq
dx=πPq(z)
/bracketleftBig
Rez>0,arg/parenleftBig
z+/radicalbig
z2−1c osx/parenrightBig
=a r g zforx=π
2/bracketrightBig
SM 482
2./integraldisplayπ
0dx/parenleftbig
z+√
z2−1c osx/parenrightbigq=πPq−1(z)
/bracketleftBig
Rez>0,arg/parenleftBig
z+/radicalbig
z2−1c osx/parenrightBig
=a r g zforx=π
2/bracketrightBig
WH
3./integraldisplayπ
0/parenleftBig
z+/radicalbig
z2−1c osx/parenrightBigq
cosnxdx =π
(q+1 ) (q+2 )···(q+n)Pn
q(z)
/bracketleftbigg
Rez>0,arg/parenleftBig
z+/radicalbig
z2−1c osx/parenrightBig
=a r g zforx=π
2,
zlies outside the interval ( −1,1) of the real axis/bracketrightbigg
WH, SM 483(15)
4./integraldisplayπ
0/parenleftBig
z+/radicalbig
z2−1c osx/parenrightBigμ
sin2ν−1xdx
=22ν−1Γ(μ+1 )[ Γ ( ν)]2
Γ(2ν+μ)Cν
μ(z)
=√πΓ(ν)Γ(2ν)Γ(μ+1 )
Γ(2ν+μ)Γ/parenleftbig
ν+1
2/parenrightbigCν
μ(z)=2ν/radicalbiggπ
2/parenleftbig
z2−1/parenrightbig1
4−ν
2Γ(ν)P1
2−ν
μ+ν−1
2(z)
[Reν>0] EH I 155(6)a, EH I 178(22)
5./integraldisplay2π
0/bracketleftBig
β+/radicalbig
β2−1c os (a−x)/bracketrightBigν/parenleftBig
γ+/radicalbig
γ2−1c osx/parenrightBigν−1
dx
=2πPν/parenleftBig
βγ−/radicalbig
β2−1/radicalbig
γ2−1c osa/parenrightBig
[Reβ>0,Reγ>0] EH I 157(18)
3.667 Trigonometric functions: powers of linear functions 407
3.665
1./integraldisplayπ
0sinμ−1xdx
(a+bcosx)μ=2μ−1
/radicalbig
(a2−b2)μB/parenleftBigμ
2,μ
2/parenrightBig
[Reμ>0,0<b<a ] FI II 790a
2./integraldisplayπ
0sin2μ−1xdx
(1 + 2 acosx+a2)ν=B/parenleftbig
μ,1
2/parenrightbig
F/parenleftbig
ν,ν−μ+1
2;μ+1
2;a2/parenrightbig
[Reμ>0,|a|<1] EH I 81(9)
3.666
1./integraldisplayπ
0(β+c o s x)μ−ν−1
2sin2νxdx=2ν+1
2e−iμπ/parenleftbig
β2−1/parenrightbigμ
2Γ/parenleftbig
ν+1
2/parenrightbig
Qμ
ν−1
2(β)
Γ/parenleftbig
ν+μ+1
2/parenrightbig
/bracketleftbig
Re/parenleftbig
ν+μ+1
2/parenrightbig
>0,Reν>−1
2/bracketrightbig
EH I 155(5)a
2.6/integraldisplayπ
0(coshβ+s i n h βcosx)μ+νsin−2νxdx=√π
2νsinhν(β)Γ/parenleftbig1
2−ν/parenrightbig
Pν
μ(coshβ)
/bracketleftbig
Reν<1
2/bracketrightbig
EH I 156(7)
3./integraldisplayπ
0(cost+isintcosx)μsin2ν−1xdx=2ν−1
2√πsin1
2−νtΓ(ν)P1
2−ν
μ+ν−1
2(cost)
/bracketleftbig
Reν>0,t2<π2/bracketrightbig
EH I 158(23)
4./integraldisplay2π
0[cost+isintcos(a−x)]νcosmxdx =i3m2πΓ(ν+1 )
Γ(ν+m+1 )cosmaPm
ν(cost)
/bracketleftBig
0<t<π
2/bracketrightBig
EH I 159(25)
5.10/integraldisplay2π
0[cost+isintcos(a−x)]νsinmxdx =i3m2πΓ(ν+1 )
Γ(ν+m+1 )sinmaPm
ν(cost)
/bracketleftBig
0<t<π
2/bracketrightBig
EH I 159(26)
3.667
1./integraldisplayπ/4
0sinμ−12xdx
(cosx+s i nx)2μ=√π
2μ+1Γ(μ)
Γ/parenleftbig
μ+1
2/parenrightbig [Reμ>0] BI (37)(1)
2./integraldisplayπ/4
0sinμxdx
(cosx−sinx)μ+1cosx=−πcosecμπ [−1<Reμ<0] (cf. 3.192 2)
BI (37)(16)
3./integraldisplayπ/4
0(cosx−sinx)μ
sinμxsin 2xdx=−π
2cosecμπ [−1<Reμ<0] BI (35)(27)
4./integraldisplayπ/4
0sinμxdx
(cosx−sinx)μsin 2x=π
2cosecμπ [0<Reμ<1] LI (37)(20)a
5./integraldisplayπ/4
0sinμxdx
(cosx−sinx)μcos2x=μπcosecμπ [|Reμ|<1] BI (37)(17)
408 Trigonometric Functions 3.668
6./integraldisplayπ/4
0sinμxdx
(cosx−sinx)μ−1cos3x=1−μ
2μπcosecμπ [|Reμ|<1] BI(35)(24), BI(37)(18)
7./integraldisplayπ/2
0sinμ−1xcosν−1x
(sinx+c o s x)μ+νdx=B (μ, ν)[ R e μ>0,Reν>0] BI (48)(8)
3.668
1./integraldisplayπ
4
−π
4/parenleftbiggcosx+s i nx
cosx−sinx/parenrightbiggcos2t
dx=π
2s in(πcos2t)FI II 788
2./integraldisplayv
u(cosu−cosx)μ−1
(cosx−cosv)μ·sinxdx
1−2acosx+a2=/parenleftbig
1−2acosu+a2/parenrightbigμ−1
(1−2acosv+a2)μ·π
sinμπ/bracketleftbig
0<Reμ<1,a2<1/bracketrightbig
BI (73)(2)
3.669/integraldisplayπ/2
0sinp−1xcosq−p−1xdx
(acosx+bsinx)q=/integraldisplayπ/2
0sinq−p−1xcosp−1x
(asinx+bcosx)qdx=B(p, q−p)
aq−pbp
[q>p> 0,a b > 0] BI (331)(9)
3.670
1./integraldisplayπ
0√
a±bcosxd x=/integraldisplayπ/2
−π/2√
a±bcosxdx=2√
a+bK/parenleftBigg/radicalbigg
2b
a+b/parenrightBigg
[a>b> 0]
2.∗/integraldisplayπ
0dx√
a±bcosx=/integraldisplayπ/2
−π/2dx√
a±bsinx=2√
a+bE/parenleftBigg/radicalbigg
2b
a+b/parenrightBigg
[a>b> 0]
3.67 Square roots of expressions containing trigonometric functions
3.671
1./integraldisplayπ/2
0sinαxcosβx/radicalbig
1−k2sin2xd x=1
2B/parenleftbiggα+1
2,β+1
2/parenrightbigg
F/parenleftbiggα+1
2,−1
2;α+β+2
2;k2/parenrightbigg
[α>−1,β > −1,|k|<1]
GW (331)(93)
2./integraldisplayπ/2
0sinαxcosβx/radicalbig
1−k2sin2xdx=1
2B/parenleftbiggα+1
2,β+1
2/parenrightbigg
F/parenleftbiggα+1
2,1
2;α+β+2
2;k2/parenrightbigg
[α>−1,β > −1,|k|<1]
GW (331)(92)
3./integraldisplayπ
0sin2nxdx/radicalbig
1−k2sin2x=π
2n∞/summationdisplay
j=0(2j−1)!! (2 n+2j−1)!!
22jj!(n+j)!k2j/bracketleftbig
k2<1/bracketrightbig
=(2n−1)!!π
2n√
1−k2∞/summationdisplay
j=0[(2j−1)!!]2
22jj!(n+j)!/parenleftbiggk2
k2−1/parenrightbiggj/bracketleftbig
k2<1
2/bracketrightbig
LI (67)(2)
3.676 Trigonometric functions and square roots 409
4.∗/integraldisplayπ
0√
a+bcosxd x=/integraldisplayπ/2
−π/2√
a+bsinxd x=2√
a+bE/parenleftBigg/radicalbigg
2b
a+b/parenrightBigg
[a>b]
5.∗/integraldisplayπ
0dx√
a±bcosx=/integraldisplayπ/2
−π/2dx√
a±bsinx=2
a+bK/parenleftBigg/radicalbigg
2b
a+b/parenrightBigg
[a>b]
3.672
1./integraldisplayπ/4
0sinnx
cosn+1x·dx/radicalbig
cosx(cosx−sinx)=2·(2n)!!
(2n+1 ) ! !BI (39)(5)
2./integraldisplayπ/4
0sinnx
cosn+1x·dx/radicalbig
sinx(cosx−sinx)=(2n−1)!!
(2n)!!π BI (39)(6)
3.673/integraldisplayπ
2
udx√sinx−sinu=√
2K/parenleftbigg
sinπ−2u
4/parenrightbigg
BI (74)(11)
3.674
1.8/integraldisplayπ
2
0dx/radicalbig
1−(p2/2)(1−cos 2x)=K(p), [1>p> 0] BI (67)(5)
2./integraldisplayπ
0sinxdx/radicalbig
1−2pcosx+p2=2/bracketleftbig
p2≤1/bracketrightbig
=2
p/bracketleftbig
p2≥1/bracketrightbig
BI (67)(6)
3.8/integraldisplayπ
0cosxdx/radicalbig
1−2pcosx+p2=1
p/bracketleftbigg1+p2
1+pK/parenleftbigg2√p
1+p/parenrightbigg
−(1 +p)E/parenleftbigg2√p
1+p/parenrightbigg/bracketrightbigg
/bracketleftbig
p2<1/bracketrightbig
BI (67)(7)
3.675
1./integraldisplayπ
usin/parenleftbig
n+1
2/parenrightbig
xdx/radicalbig
2(cos u−cosx)=π
2Pn(cosu) WH
2./integraldisplayu
0cos/parenleftbig
n+1
2/parenrightbig
xdx/radicalbig
2(cos x−cosu)=π
2Pn(cosu) FI II 684, WH
3.676
1./integraldisplayπ/2
0sinxdx/radicalbig
1+p2sin2x=1
parctan p BI (60)(5)
2./integraldisplayπ/2
0tan2x/radicalBig
1−p2sin2xdx=∞ BI (53)(8)
410 Trigonometric Functions 3.677
3./integraldisplayπ/2
0dx/radicalbig
p2cos2x+q2sin2x=1
pK/parenleftBigg/radicalbig
p2−q2
p/parenrightBigg
[0<q<p ] FI II 165
3.677
1./integraldisplayπ/2
0sin2xdx/radicalbig
1+s i n2x=√
2E/parenleftBigg√
2
2/parenrightBigg
−1√
2K/parenleftBigg√
2
2/parenrightBigg
BI (60)(2)
2./integraldisplayπ/2
0cos2xdx/radicalbig
1+s i n2x=√
2/bracketleftBigg
K/parenleftBigg√
2
2/parenrightBigg
−E/parenleftBigg√
2
2/parenrightBigg/bracketrightBigg
BI (60)(3)
3.678
1./integraldisplayπ/4
0/parenleftBig
sec1/22x−1/parenrightBigdx
tanx=l n2 BI (38)(23)
2./integraldisplayπ/4
0tan2xdx/radicalbig
1−k2sin22x=/radicalbig
1−k2−E(k)+1
2K(k) BI (39)(2)
3./integraldisplayu
0/radicalbigg
cos 2x−cos2u
cos2x+1dx=π
2(1−cosu)/bracketleftbigg
u2<π2
4/bracketrightbigg
LI (74)(6)
4./integraldisplayπ/4
0(cosx−sinx)n−1
2
cosn+1x√cosecxdx=(2n−1)!!
(2n)!!π BI (38)(24)
5./integraldisplayπ/4
0(cosx−sinx)n−1
2
cosn+1xtanmx√cosecxd x=(2n−1)!!(2m−1)!!
(2n+2m)!!π BI (38)(25)
3.679
1./integraldisplayπ/2
0cos2x
1−cos2βcos2x·dx/radicalbig
1−k2sin2x
=1
sinβcosβ/radicalbig
1−k/prime2sin2β/braceleftBigπ
2−KE(β,k/prime)−EF(β,k/prime)+KF(β,k/prime)/bracerightBig
∗
MO 138
2./integraldisplayπ/2
0sin2x
1−/parenleftbig
1−k/prime2sin2β/parenrightbig
sin2x·dx/radicalbig
1−k2sin2x
=1
k/prime2sinβcosβ/radicalbig
1−k/prime2sin2β/braceleftBigπ
2−KE(β,k/prime)−EF(β,k/prime)+KF(β,k/prime)/bracerightBig
∗
MO 138
3./integraldisplayπ/2
0sin2x
1−k2sin2βsin2x·dx/radicalbig
1−k2sin2x=KE(β,k)−EF(β,k)
k2sinβcosβ/radicalbig
1−k2sin2βMO 138
∗In 3.679, k/prime=√
1−k2.
3.683 Trigonometric functions: various powers 411
3.68 Various forms of powers of trigonometric functions
3.681
1./integraldisplayπ/2
0sin2μ−1xcos2ν−1xdx/parenleftbig
1−k2sin2x/parenrightbig/rho1=1
2B(μ, ν)F/parenleftbig
/rho1, μ;μ+ν;k2/parenrightbig
[Reμ>0,Reν>0] EH I 115(7)
2./integraldisplayπ/2
0sin2μ−1xcos2ν−1xdx
/parenleftbig
1−k2sin2x/parenrightbigμ+ν=B(μ, ν)
2( 1−k2)μ [Reμ>0,Reν>0] EH I 10(20)
3./integraldisplayπ/2
0sinμxdx
cosμ−3x/parenleftbig
1−k2sin2x/parenrightbigμ
2−1
=Γ/parenleftbigμ+1
2/parenrightbig
Γ/parenleftbig
2−μ
2/parenrightbig
k3/radicalbig
π(μ−1)(μ−3)(μ−5)/braceleftbigg1+(μ−3)k+k2
(1 +k)μ−3−1−(μ−3)k+k2
(1−k)μ−3/bracerightbigg
[−1<Reμ<4] BI (54)(10)
4.8/integraldisplayπ/2
0sinμ+1xdx
cosμx/parenleftbig
1−k2sin2x/parenrightbigμ+1
2=(1−k)−μ−(1 +k)−μ
2kμ√πΓ/parenleftBig
1+μ
2/parenrightBig
Γ/parenleftbigg1−μ
2/parenrightbigg
[−2<Reμ<1] BI (61)(5)
3.682/integraldisplayπ/2
0sinμxcosνx
(a−bcos2x)/rho1dx=1
2a/rho1B/parenleftbiggμ+1
2,ν+1
2/parenrightbigg
F/parenleftbiggν+1
2,/rho1;μ+ν
2+1 ;b
a/parenrightbigg
[Reμ>−1,Reν>−1,a > |b|≥0]
GW (331)(64)
3.683
1./integraldisplayπ/4
0(sinn2x−1) tan/parenleftBigπ
4+x/parenrightBig
dx=/integraldisplayπ/4
0(cosn2x−1)cot xdx=−1
2n/summationdisplay
k=11
k
=−1
2[C+ψ(n+1 ) ]
[n≥0] BI(34)(8), BI(35)(11)
2./integraldisplayπ/4
0(sinμ2x−1)cosecμ2xtan/parenleftBigπ
4+x/parenrightBig
dx=/integraldisplayπ/4
0(cosμ2x−1)secμ2xcotxdx
=1
2[C+ψ(1−μ)]
[Reμ<1] BI (35)(20)
3./integraldisplayπ
4
0/parenleftbig
sin2μ2x−1/parenrightbig
cosecμ2xtan/parenleftBigπ
4+x/parenrightBig
dx=/integraldisplayπ/4
0/parenleftbig
cos2μ2x−1/parenrightbig
secμ2xcotxdx
=−1
2μ+π
2cotμπ
BI (35)(21)
4./integraldisplayπ/4
0(1−secμ2x)c o txdx=/integraldisplayπ/4
0(1−cosecμ2x)t a n/parenleftBigπ
4+x/parenrightBig
dx=1
2[C+ψ(1−μ)]
[Reμ<1] BI (35)(13)
412 Trigonometric Functions 3.684
3.684/integraldisplayπ/4
0(cotμx−1)dx
(cosx−sinx)s i nx=/integraldisplayπ/2
0(tanμx−1)dx
(sinx−cosx)c o sx=−C−ψ(1−μ)[ R e μ<1]
BI (37)(9)
3.685
1./integraldisplayπ/4
0/parenleftbig
sinμ−12x−sinν−12x/parenrightbig
tan/parenleftBigπ
4+x/parenrightBig
dx=/integraldisplayπ/4
0/parenleftbig
cosμ−12x−cosν−12x/parenrightbig
cotxdx
=1
2[ψ(ν)−ψ(μ)]
[Reμ>0,Reν>0]BI(34)(9), BI(35)(12)
2./integraldisplayπ/2
0/parenleftbig
sinμ−1x−sinν−1x/parenrightbigdx
cosx=/integraldisplayπ/2
0/parenleftbig
cosμ−1x−cosν−1x/parenrightbigdx
sinx=1
2/bracketleftBig
ψ/parenleftBigν
2/parenrightBig
−ψ/parenleftBigμ
2/parenrightBig/bracketrightBig
[Reμ>0,Reν>0] BI (46)(2)
3./integraldisplayπ/2
0(sinμx−cosecμx)dx
cosx=/integraldisplayπ/2
0(cosμx−secμx)dx
sinx=−π
2tanμπ
2
[|Reμ|<1] BI (46)(1, 3)
4./integraldisplayπ/4
0(sinμ2x−cosecμ2x)c o t/parenleftBigπ
4+x/parenrightBig
dx=/integraldisplayπ/4
0(cosμ2x−secμ2x)tanxdx
=1
2μ−π
2cosecμπ
[|Reμ|<1] BI (35)(19, 22)
5./integraldisplayπ/4
0(sinμ2x−cosecμ2x)t a n/parenleftBigπ
4+x/parenrightBig
dx=/integraldisplayπ/4
0(cosμ2x−secμ2x)c o txdx
=−1
2μ+π
2cotμπ
[|Reμ|<1] BI (35)(14)
6./integraldisplayπ/4
0/parenleftbig
sinμ−12x+c o s e cμ2x/parenrightbig
cot/parenleftBigπ
4+x/parenrightBig
dx
=/integraldisplayπ/4
0/parenleftbig
cosμ−12x+s e cμ2x/parenrightbig
tanxdx=π
4cosecμπ
[0<Reμ<1] BI (35)(18, 8)
7./integraldisplayπ/4
0/parenleftbig
sinμ−12x−cosecμ2x/parenrightbig
tan/parenleftBigπ
4+x/parenrightBig
dx=/integraldisplayπ/4
0/parenleftbig
cosμ−12x−secμ2x/parenrightbig
cotxdx=π
2cotμπ
[0<Reμ<1] BI(35)(7), LI(34)(10)
3.686/integraldisplayπ/2
0tanxdx
cosμx+s e cμx=/integraldisplayπ/2
0cotxdx
sinμx+c o s e cμx=π
4μBI(47)(28), BI(49)(14)
3.687
1./integraldisplayπ/2
0sinμ−1x+s i nν−1x
cosμ+ν−1xdx=/integraldisplayπ/2
0cosμ−1x+c o sν−1x
sinμ+ν−1xdx=cos/parenleftbigν−μ
4π/parenrightbig
2c os/parenleftbiggν+μ
4π/parenrightbiggB/parenleftBigμ
2,ν
2/parenrightBig
[Reμ>0,Reν>0,Re(μ+ν)<2]
BI (46)(7)
3.688 Trigonometric functions: various powers 413
2./integraldisplayπ/2
0sinμ−1x−sinν−1x
cosμ+ν−1xdx=/integraldisplayπ/2
0cosμ−1x−cosν−1x
sinμ+ν−1xdx=sin/parenleftbigν−μ
4π/parenrightbig
2s in/parenleftbigν+μ
4π/parenrightbigB/parenleftBigμ
2,ν
2/parenrightBig
[Reμ>0,Reν>0,Re(μ+ν)<4]
BI(46)(8)
3./integraldisplayπ/2
0sinμx+s i nνx
sinμ+νx+1cotxdx=/integraldisplayπ
2
0cosμx+c o sνx
cosμ+νx+1tanxdx=π
μ+νsec/parenleftbiggμ−ν
μ+ν·π
2/parenrightbigg
[Reμ>0,Reν>0]
BI (49)(15)a, BI (47)(29)
4./integraldisplayπ/2
0sinμx−sinνx
sinμ+νx−1cotxdx=/integraldisplayπ
2
0cosμx−cosνx
cosμ+νx−1tanxdx=π
μ+νtan/parenleftbiggμ−ν
μ+ν·π
2/parenrightbigg
[Reμ>0,Reν>0]
BI(149)(16)a, BI(47)(30)
5./integraldisplayπ/2
0cosμx+s e cμx
cosνx+s e cνxtanxdx=π
2νsec/parenleftBigμ
ν·π
2/parenrightBig
[|Reν|>|Reμ|] BI (49)(12)
6./integraldisplayπ/2
0cosμx−secμx
cosνx−secνxtanxdx=π
2νtan/parenleftBigμ
ν·π
2/parenrightBig
[|Reν|>|Reμ|] BI (49)(13)
3.688
1./integraldisplayπ/4
0tanνx−tanμx
cosx−sinx·dx
sinx=ψ(μ)−ψ(ν)[ R e μ>0,Reν>0] BI (37)(10)
2./integraldisplayπ/4
0tanμx−tan1−μx
cosx−sinx·dx
sinx=πcotμπ [0<Reμ<1] BI (37)(11)
3./integraldisplayπ/4
0(tanμx+c o tμx)dx=π
2secμπ
2[|Reμ|<1] BI (35)(9)
4./integraldisplayπ/4
0(tanμx−cotμx)t a nxdx=1
μ−π
2cosecμπ
2[0<Reμ<2] BI (35)(15)
5./integraldisplayπ/4
0tanμ−1x−cotμ−1x
cos 2xdx=π
2cotμπ
2[|Reμ|<2] BI (35)(10)
6./integraldisplayπ/4
0tanμx−cotμx
cos2xtanxdx=−1
μ+π
2cotμπ
2[−2<Reμ<0] BI (35)(23)
7./integraldisplayπ/4
0tanμx+c o tμx
1 + cos tsin 2xdx=πcosectcosecμπsinμt [t/negationslash=nπ,|Reμ|<1] BI (36)(6)
8./integraldisplayπ/4
0tanμ−1x+c o tμx
(sinx+c o s x)c o sxdx=πcosecμπ [0<Reμ<1] BI (37)(3)
9./integraldisplayπ/4
0tanμx−cotμx
(sinx+c o s x)c o sxdx=−πcosecμπ+1
μ[0<Reμ<1] BI (37)(4)
10./integraldisplayπ/4
0tanνx−cotμx
(cosx−sinx)c o sxdx=ψ(1−μ)−ψ(1 +ν)[ R e μ<1,Reν>−1] BI (37)(5)
414 Trigonometric Functions 3.689
11./integraldisplayπ/4
0tanμ−1x−cotμx
(cosx−sinx)c o sxdx=πcotμπ [0<Reμ<1] BI (37)(7)
12./integraldisplayπ/4
0tanμx−cotμx
(cosx−sinx)c o sxdx=πcotμπ−1
μ[0<Reμ<1] BI (37)(8)
13./integraldisplayπ/4
01
tanμx+c o tμx·dx
sin 2x=π
8μ[Reμ/negationslash=0 ] BI (37)(12)
14./integraldisplayπ/2
01
(tanμx+c o tμx)ν·dx
tanx=/integraldisplayπ/2
01
(tanμx+c o tμx)ν·dx
sin 2x=√π
22ν+1μΓ(ν)
Γ/parenleftbig
ν+1
2/parenrightbig
[ν>0] BI(49)(25), BI(49)(26)
15./integraldisplayπ/4
0(tanμx−cotμx) (tanνx−cotνx)dx=2πsinμπ
2sinνπ
2
cosμπ+c o s νπ
[|Reμ|<1,|Reν|<1] BI (35)(17)
16./integraldisplayπ/4
0(tanμx+c o tμx) (tanνx+c o tνx)dx=2πcosμπ
2cosνπ
2
cosμπ+c o s νπ
[|Reμ|<1,|Reν|<1] BI (35)(16)
17./integraldisplayπ/4
0(tanμx−cotμx) (tanνx+c o tνx)
cos 2xdx=−πsinμπ
cosμπ+c o s νπ
[|Reμ|<1,|Reν|<1] BI (35)(25)
18./integraldisplayπ/4
0tanνx−cotνx
tanμx−cotμx·dx
sin 2x=π
4μtanνπ
2μ[0<Reν<1] BI (37)(14)
19./integraldisplayπ/4
0tanνx+c o tνx
tanμx+c o tμx·dx
sin 2x=π
4μsecνπ
2μ[0<Reν<1] BI (37)(13)
20./integraldisplayπ/2
0(1 + tan x)ν−1
(1 + tan x)μ+νdx
sinxcosx=ψ(μ+ν)−ψ(μ)[ μ>0,ν > 0] BI (49)(29)
3.689
1./integraldisplayπ/2
0(sinμx+c o s e cμx)c o txdx
sinνx−2c ost+c o s e cνx=π
νcosectcosecμπ
νsinμt
ν
[μ<ν] LI (50)(14)
2./integraldisplayπ/2
0sinμx−2c ost1+c o s e cμx
sinνx+ 2cos t2+c o s e cνx·cotxdx=π
νcosect2cosecμπ
νsinμt2
ν−t2
νcosect2cost1
[(ν>μ> 0) or ( ν<μ< 0) or ( μ>0,ν<0, and μ+ν<0) or ( μ<0,ν>0, and μ+ν>0)]
BI (50)(15)
3.691 Trigonometric functions: complicated arguments 415
3.69–3.71 Trigonometric functions of more complicated arguments
3.691
1./integraldisplay∞
0sin/parenleftbig
ax2/parenrightbig
dx=/integraldisplay∞
0cosax2dx=1
2/radicalbiggπ
2a[a>0] FI II 743a, ET I 64(7)a
2./integraldisplay1
0sin/parenleftbig
ax2/parenrightbig
dx=/radicalbiggπ
2aS/parenleftbig√a/parenrightbig
[a>0]
3./integraldisplay1
0cos/parenleftbig
ax2/parenrightbig
dx=/radicalbiggπ
2aC/parenleftbig√a/parenrightbig
[a>0] ET I 8(5)a
4./integraldisplay∞
0sin/parenleftbig
ax2/parenrightbig
sin 2bxdx =/radicalbiggπ
2a/braceleftbigg
cosb2
aC/parenleftbiggb√a/parenrightbigg
+s i nb2
aS/parenleftbiggb√a/parenrightbigg/bracerightbigg
[a>0,b > 0] ET I 82(1)a
5./integraldisplay∞
0sin/parenleftbig
ax2/parenrightbig
cos 2bxdx =1
2/radicalbiggπ
2a/braceleftbigg
cosb2
a−sinb2
a/bracerightbigg
=1
2/radicalbiggπ
acos/parenleftbiggb2
a+π
4/parenrightbigg
[a>0,b > 0]
ET I 82(18), BI(70)(13) GW(334)(5a)
6./integraldisplay∞
0cosax2sin 2bxdx =/radicalbiggπ
2a/braceleftbigg
sinb2
aC/parenleftbiggb√a/parenrightbigg
−cosb2
aS/parenleftbiggb√a/parenrightbigg/bracerightbigg
[a>0,b > 0] ET I 83(3)a
7./integraldisplay∞
0cosax2cos 2bxdx =1
2/radicalbiggπ
2a/braceleftbigg
cosb2
a+s i nb2
a/bracerightbigg
[a>0,b > 0]
GW(334)(5a), BI(70)(14), ET I 24(7)
8./integraldisplay∞
0(cosax+s i nax)s i n/parenleftbig
b2x2/parenrightbig
dx
=1
2b/radicalbiggπ
2/braceleftbigg/parenleftBig
1+2C/parenleftBiga
2b/parenrightBig/parenrightBig
cos/parenleftbigga2
4b2/parenrightbigg
−/parenleftBig
1−2S/parenleftBiga
2b/parenrightBig/parenrightBig
sin/parenleftbigga2
4b2/parenrightbigg/bracerightbigg
[a>0,b > 0] ET I 85(22)
9./integraldisplay∞
0(cosax+s i nax)c o s/parenleftbig
b2x2/parenrightbig
dx
=1
2b/radicalbiggπ
2/braceleftbigg/parenleftBig
1+2C/parenleftBiga
2b/parenrightBig/parenrightBig
sin/parenleftbigga2
4b2/parenrightbigg
+/parenleftBig
1−2S/parenleftBiga
2b/parenrightBig/parenrightBig
cos/parenleftbigga2
4b2/parenrightbigg/bracerightbigg
[a>0,b > 0] ET I 25(21)
10./integraldisplay∞
0sin/parenleftbig
a2x2/parenrightbig
sin2bxsin 2cxdx =√π
2asin2bc
a2cos/parenleftbiggb2+c2
a2−π
4/parenrightbigg
[a>0,b > 0,c > 0] ET I 84(15)
11./integraldisplay∞
0sin/parenleftbig
a2x2/parenrightbig
cos2bxcos2cxdx =√π
2acos2bc
a2cos/parenleftbiggb2+c2
a2+π
4/parenrightbigg
[a>0,b > 0,c > 0] ET I 84(21)
416 Trigonometric Functions 3.692
12./integraldisplay∞
0cos/parenleftbig
a2x2/parenrightbig
sin 2bxsin2cxdx =√π
2asin2bc
a2sin/parenleftbiggb2+c2
a2−π
4/parenrightbigg
[a>0,b > 0,c > 0] ET I 25(19)
13./integraldisplay∞
0sin/parenleftbig
ax2/parenrightbig
cos/parenleftbig
bx2/parenrightbig
dx=1
4/radicalbiggπ
2/parenleftbigg1√
a+b+1√
a−b/parenrightbigg
[a>b> 0]
=1
4/radicalbiggπ
2/parenleftbigg1√
b+a−1√
b−a/parenrightbigg
[b>a> 0]
BI (177)(21)
14./integraldisplay∞
0/parenleftbig
sin2ax2−sin2bx2/parenrightbig
dx=1
8/parenleftbigg/radicalbiggπ
b−/radicalbiggπ
a/parenrightbigg
[a>0,b > 0] BI (178)(1)
15./integraldisplay∞
0/parenleftbig
cos2ax2−sin2bx2/parenrightbig
dx=1
8/parenleftbigg/radicalbiggπ
b+/radicalbiggπ
a/parenrightbigg
[a>0,b > 0] BI (178)(3)
16./integraldisplay∞
0/parenleftbig
cos2ax2−cos2bx2/parenrightbig
dx=1
8/parenleftbigg/radicalbiggπ
a−/radicalbiggπ
b/parenrightbigg
[a>0,b > 0] BI (178)(5)
17./integraldisplay∞
0/parenleftbig
sin4ax2−sin4bx2/parenrightbig
x=1
64/parenleftBig
8−√
2/parenrightBig/parenleftbigg/radicalbiggπ
b−/radicalbiggπ
a/parenrightbigg
[a>0,b > 0] BI (178)(2)
18./integraldisplay∞
0/parenleftbig
cos4ax2−sin4bx2/parenrightbig
dx=1
8/parenleftbigg/radicalbiggπ
a+/radicalbiggπ
b/parenrightbigg
+1
32/parenleftbigg/radicalbiggπ
2a−/radicalbiggπ
2b/parenrightbigg
[a>0,b > 0] BI (178)(4)
19./integraldisplay∞
0/parenleftbig
cos4ax2−cos4bx2/parenrightbig
dx=1
64/parenleftBig
8+√
2/parenrightBig/parenleftbigg/radicalbiggπ
a−/radicalbiggπ
b/parenrightbigg
[a>0,b > 0] BI (178)(6)
20./integraldisplay∞
0sin2nax2dx=/integraldisplay∞
0cos2nax2dx=∞ BI (177)(5, 6)
21./integraldisplay∞
0sin2n+1/parenleftbig
ax2/parenrightbig
dx=1
22n+1n/summationdisplay
k=0(−1)n+k/parenleftbigg2n+1
k/parenrightbigg/radicalbiggπ
2(2n−2k+1 )a
[a>0] BI (70)(9)
22./integraldisplay∞
0cos2n+1/parenleftbig
ax2/parenrightbig
dx=1
22n+1n/summationdisplay
k=0/parenleftbigg2n+1
k/parenrightbigg/radicalbiggπ
2(2n−2k+1 )a
[a>0] BI(177)(7)a, BI(70)(10)
3.692
1./integraldisplay∞
0/bracketleftbig
sin/parenleftbig
a−x2/parenrightbig
+c o s/parenleftbig
a−x2/parenrightbig/bracketrightbig
dx=/radicalbiggπ
asina GW(333)(30c), BI(178)(7)a
2./integraldisplay∞
0cos/parenleftbiggx2
2−π
8/parenrightbigg
cosaxdx =/radicalbiggπ
2cos/parenleftbigga2
2−π
8/parenrightbigg
[a>0] ET I 24(8)
3.695 Trigonometric functions: complicated arguments 417
3./integraldisplay∞
0sin/bracketleftbig
a/parenleftbig
1−x2/parenrightbig/bracketrightbig
cosbxdx =−1
2/radicalbiggπ
acos/parenleftbigg
a+b2
4a+π
4/parenrightbigg
[a>0] ET I 23(2)
4./integraldisplay∞
0cos/bracketleftbig
a/parenleftbig
1−x2/parenrightbig/bracketrightbig
cosbxdx =1
2/radicalbiggπ
asin/parenleftbigg
a+b2
4a+π
4/parenrightbigg
[a>0] ET I 24(10)
5./integraldisplay∞
0sin/parenleftbigg
ax2+b2
a/parenrightbigg
cos2bxdx =/integraldisplay∞
0cos/parenleftbigg
ax2+b2
a/parenrightbigg
cos2bxdx =1
2/radicalbiggπ
2a
[a>0] BI (70)(19, 20)
6.8/integraldisplay∞
−∞/bracketleftBig
cos/radicalbig
x2−1−cos/radicalbig
x2+1/bracketrightBig
dx=∞/summationdisplay
n=0π/braceleftBig
24n+1[(2n)!]2/parenleftbig
n+1
2/parenrightbig/bracerightBig
3.693
1./integraldisplay∞
0sin/parenleftbig
ax2+2bx/parenrightbig
dx=/radicalbiggπ
2a/braceleftbigg
cosb2
a/parenleftbigg1
2−S2/parenleftbiggb2
a/parenrightbigg/parenrightbigg
−sinb2
a/parenleftbigg1
2−C2/parenleftbiggb2
a/parenrightbigg/parenrightbigg/bracerightbigg
[a>0] BI (70)(3)
2./integraldisplay∞
0cos/parenleftbig
ax2+2bx/parenrightbig
dx=/radicalbiggπ
2a/braceleftbigg
cosb2
a/parenleftbigg1
2−C2/parenleftbiggb2
a/parenrightbigg/parenrightbigg
+s i nb2
a/parenleftbigg1
2−S2/parenleftbiggb2
a/parenrightbigg/parenrightbigg/bracerightbigg
[a>0] BI (70)(4)
3.694
1./integraldisplay∞
0sin/parenleftbig
ax2+2bx+c/parenrightbig
dx=/radicalbiggπ
2acosb2
a/braceleftbigg/parenleftbigg1
2−C2/parenleftbiggb2
a/parenrightbigg/parenrightbigg
sinc+/parenleftbigg1
2−S2/parenleftbiggb2
a/parenrightbigg/parenrightbigg
cosc/bracerightbigg
+/radicalbiggπ
2asinb2
a/braceleftbigg/parenleftbigg1
2−S2/parenleftbiggb2
a/parenrightbigg/parenrightbigg
sinc−/parenleftbigg1
2−C2/parenleftbiggb2
a/parenrightbigg/parenrightbigg
cosc/bracerightbigg
[a>0] GW (334)(4a)
2./integraldisplay∞
0cos/parenleftbig
ax2+2bx+c/parenrightbig
dx=/radicalbiggπ
2acosb2
a/braceleftbigg/parenleftbigg1
2−C2/parenleftbiggb2
a/parenrightbigg/parenrightbigg
cosc−/parenleftbigg1
2−S2/parenleftbiggb2
a/parenrightbigg/parenrightbigg
sinc/bracerightbigg
+/radicalbiggπ
2asinb2
a/braceleftbigg/parenleftbigg1
2−S2/parenleftbiggb2
a/parenrightbigg/parenrightbigg
cosc+/parenleftbigg1
2−C2/parenleftbiggb2
a/parenrightbigg/parenrightbigg
sinc/bracerightbigg
[a>0] GW (334)(4b)
3.695
1./integraldisplay∞
0sin/parenleftbig
a3x3/parenrightbig
sin(bx)dx=π
6a/radicalbigg
b
3a/braceleftBigg
J1
3/parenleftBigg
2b
3a/radicalbigg
b
3a/parenrightBigg
+J−1
3/parenleftBigg
2b
3a/radicalbigg
b
3a/parenrightBigg
−√
3
πK1
3/parenleftBigg
2b
3a/radicalbigg
b
3a/parenrightBigg/bracerightBigg
[a>0,b > 0] ET I 83(5)
2./integraldisplay∞
0cos/parenleftbig
a3x3/parenrightbig
cos(bx)dx=π
6a/radicalbigg
b
3a/braceleftBigg
J1
3/parenleftBigg
2b
3a/radicalbigg
b
3a/parenrightBigg
+J−1
3/parenleftBigg
2b
3a/radicalbigg
b
3a/parenrightBigg
+√
3
πK1
3/parenleftBigg
2b
3a/radicalbigg
b
3a/parenrightBigg/bracerightBigg
[a>0,b > 0] ET I 24(11)
418 Trigonometric Functions 3.696
3.696
1./integraldisplay∞
0sin/parenleftbig
ax4/parenrightbig
sin/parenleftbig
bx2/parenrightbig
dx=−π
4/radicalbigg
b
2asin/parenleftbiggb2
8a−3
8π/parenrightbigg
J1
4/parenleftbiggb2
8a/parenrightbigg
[a>0,b > 0] ET I 83(2)
2./integraldisplay∞
0sin/parenleftbig
ax4/parenrightbig
cos/parenleftbig
bx2/parenrightbig
dx=−π
4/radicalbigg
b
2asin/parenleftbiggb2
8a−π
8/parenrightbigg
J−1
4/parenleftbiggb2
8a/parenrightbigg
[a>0,b > 0] ET I 84(19)
3./integraldisplay∞
0cos/parenleftbig
ax4/parenrightbig
sin/parenleftbig
bx2/parenrightbig
dx=π
4/radicalbigg
b
2acos/parenleftbiggb2
8a−3
8π/parenrightbigg
J1
4/parenleftbiggb2
8a/parenrightbigg
[a>0,b > 0] ET I 83(4), ET I 25(24)
4./integraldisplay∞
0cos/parenleftbig
ax4/parenrightbig
cos/parenleftbig
bx2/parenrightbig
dx=π
4/radicalbigg
b
2acos/parenleftbiggb2
8a−π
8/parenrightbigg
J−1
4/parenleftbiggb2
8a/parenrightbigg
[a>0,b > 0] ET I 25(25)
3.697/integraldisplay∞
0sin/parenleftbigga2
x/parenrightbigg
sin(bx)dx=aπ
2√
bJ1/parenleftBig
2a√
b/parenrightBig
[a>0,b > 0] ET I 83(6)
3.698
1./integraldisplay∞
0sin/parenleftbigga2
x2/parenrightbigg
sin/parenleftbig
b2x2/parenrightbig
dx=1
4b/radicalbiggπ
2/bracketleftbig
sin 2ab−cos 2ab+e−2ab/bracketrightbig
[a>0,b > 0] ET I 83(9)
2.8/integraldisplay∞
0sin/parenleftbigga2
x2/parenrightbigg
cos/parenleftbig
b2x2/parenrightbig
dx=1
4b/radicalbiggπ
2/bracketleftbig
sin2ab+c o s2 ab−e−2ab/bracketrightbig
ET I 24(13)
3./integraldisplay∞
0cos/parenleftbigga2
x2/parenrightbigg
sin/parenleftbig
b2x2/parenrightbig
dx=1
4b/radicalbiggπ
2/bracketleftbig
sin2ab+c o s2 ab+e−2ab/bracketrightbig
[a>0,b > 0] ET I 84(12)
4./integraldisplay∞
0cos/parenleftbigga2
x2/parenrightbigg
cos/parenleftbig
b2x2/parenrightbig
dx=1
4b/radicalbiggπ
2/bracketleftbig
cos2ab−sin 2ab+e−2ab/bracketrightbig
[a>0,b > 0] ET I 24(14)
3.699
1./integraldisplay∞
0sin/parenleftbigg
a2x2+b2
x2/parenrightbigg
dx=√
2π
4a(cos 2ab+s i n2 ab)[ a>0,b > 0] BI (70)(27)
2./integraldisplay∞
0cos/parenleftbigg
a2x2+b2
x2/parenrightbigg
dx=√
2π
4a(cos2ab−sin 2ab)[ a>0,b > 0] BI (70)(28)
3./integraldisplay∞
0sin/parenleftbigg
a2x2−2ab+b2
x2/parenrightbigg
dx=/integraldisplay∞
0cos/parenleftbigg
a2x2−2ab+b2
x2/parenrightbigg
dx=√
2π
4a
[a>0,b > 0]
BI(179)(11, 12)a, ET I 83(6)
3.715 Trigonometric functions: complicated arguments 419
4./integraldisplay∞
0sin/parenleftbigg
a2x2−b2
x2/parenrightbigg
dx=√
2π
4ae−2ab[a>0,b > 0] GW (334)(9b)a
5./integraldisplay∞
0cos/parenleftbigg
a2x2−b2
x2/parenrightbigg
dx=√
2π
4ae−2ab[a>0,b > 0] GW (334)(9b)a
3.711/integraldisplayu
0sin/parenleftBig
a/radicalbig
u2−x2/parenrightBig
cosbxdx =πau
2√
a2+b2J1/parenleftBig
u/radicalbig
a2+b2/parenrightBig
[a>0,b > 0,u > 0]
ET I 27(37)
3.712
1./integraldisplay∞
0sin(axp)dx=Γ/parenleftBig
1
p/parenrightBig
sinπ
2p
pa1
p[a>0,p > 1] EH I 13(40)
2./integraldisplay∞
0cos(axp)dx=Γ/parenleftBig
1
p/parenrightBig
cosπ
2p
pa1
p[a>0,p > 1] EH I 13(39)
3.713
1./integraldisplay∞
0sin(axp+bxq)dx=1
p∞/summationdisplay
k=0(−b)k
k!a−kq+1
pΓ/parenleftbiggkq+1
p/parenrightbigg
sin/bracketleftbiggk(q−p)+1
2pπ/bracketrightbigg
[a>0,b > 0,p > 0,q > 0]
BI (70)(7)
2./integraldisplay∞
0cos(axp+bxq)dx=1
p∞/summationdisplay
k=0(−b)k
k!a−(kq+1)/pΓ/parenleftbiggkq+1
p/parenrightbigg
cos/bracketleftbiggk(q−p)+1
2pπ/bracketrightbigg
[a>0,b > 0,p > 0,q > 0]
BI (70)(8)
3.714
1./integraldisplay∞
0cos(zsinhx)dx=K0(z)[ R e z>0] WA 202(14)
2./integraldisplay∞
0sin(zcoshx)dx=π
2J0(z)[ R e z>0] MO 36
3./integraldisplay∞
0cos(zcoshx)dx=−π
2Y0(z)[ R e z>0] MO 37
4./integraldisplay∞
0cos(zsinhx)c o s h μxdx =c o sμπ
2Kμ(z)[ R e z>0,|Reμ|<1] WA 202(13)
5./integraldisplayπ
0cos(zcoshx)s i n2μxdx=√π/parenleftbigg2
z/parenrightbiggμ
Γ/parenleftbigg
μ+1
2/parenrightbigg
Iμ(z)
/bracketleftbig
Rez>0,Reμ>−1
2/bracketrightbig
WH
3.715
1./integraldisplayπ
0sin (zsinx)s i naxdx =s i naπs0,a(z)=s i n aπ∞/summationdisplay
k=1(−1)k−1z2k−1
(12−a2)( 32−a2)...[(2k−1)2−a2]
[a>0] WA 338(13)
420 Trigonometric Functions 3.715
2./integraldisplayπ
0sin (zsinx)s i nnxdx =1
2/integraldisplayπ
−πsin (zsinx)sinnxdx
=[ 1−(−1)n]/integraldisplayπ/2
0sin(zsinx)sinnxdx =[ 1−(−1)n]π
2Jn(z)
[n=0,±1,±2,...]WA 30(6), GW(334)(153a)
3./integraldisplayπ/2
0sin(zsinx)s i n2xdx=2
z2(sinz−zcosz) LI (43)(14)
4./integraldisplayπ
0sin (zsinx)c o saxdx = (1 + cos aπ)s0,a(z)
= (1 + cos aπ)∞/summationdisplay
k=1(−1)k−1z2k−1
(12−a2)(32−a2)...[(2k−1)2−a2]
[a>0] WA 338(14)
5./integraldisplayπ
0sin (zsinx) cos[(2 n+1 )x]dx=0 GW (334)(53b)
6./integraldisplayπ
0cos(zsinx)s i naxdx =−a(1−cosaπ)s−1,a(z)
=−a(1−cosaπ)/braceleftBigg
−1
a2+∞/summationdisplay
k=1(−1)k−1z2k
a2(22−a2)( 42−a2)...[(2k)2−a2]/bracerightBigg
[a>0] WA 338(12)
7./integraldisplayπ
0cos(zsinx)s i n2nxdx =0 GW (334)(54a)
8./integraldisplayπ
0cos(zsinx)c o saxdx =−asinaπs−1,a(z)
=−asinaπ/braceleftBigg
−1
a2+∞/summationdisplay
k=1(−1)k−1z2k
a2(22−a2)(42−a2)...[(2k)2−a2]/bracerightBigg
[a>0] WA 338(11)
9./integraldisplayπ
0cos(zsinx)c o snxdx =1
2/integraldisplayπ
−πcos(zsinx)c o snxdx
=[ 1+( −1)n]/integraldisplayπ/2
0cos(zsinx)c o snxdx =[ 1+( −1)n]π
2Jn(z)
GW (334)(54b)
10.8/integraldisplayπ/2
0cos(zsinx)c o s2nxdx=π
2(2n−1)!!
znJn(z)[ n=0,1,2,...] FI II 486, WA 35a
11./integraldisplayπ/2
0sin(zcosx)s i n2xdx=2
z2(sinz−zcosz) LI (43)(15)
3.716 Trigonometric functions: complicated arguments 421
12.8/integraldisplayπ/2
0sin(zcosx)c o saxdx =c o saπ
2s0,a(z)=π
4cosecaπ
2[Ja(z)−J−a(z)]
=−π
4secaπ
4[Ea(z)+E−a(z)]
=c o saπ
2∞/summationdisplay
k=1(−1)k−1z2k−1
(12−a2)(32−a2)...[(2k−1)2−a2]
[a>0] WA 339
13./integraldisplayπ
0sin (zcosx)c o snxdx =1
2/integraldisplayπ
−πsin (zcosx)cosnxdx =πsinnπ
2Jn(z) GW (334)(55b)
14./integraldisplayπ/2
0sin(zcosx) cos[(2 n+1 )x]dx=(−1)nπ
2J2n+1(z) WA 30(8)
15.11/integraldisplayπ/2
0sin(acosx)tanxdx=s i (a)+π
2[a>0] BI (43)(17)
16./integraldisplayπ/2
0sin(zcosx)s i n2νxdx=√π
2/parenleftbigg2
z/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Hν(z)
/bracketleftbig
Reν>−1
2/bracketrightbig
WA 358(1)
17.7/integraldisplayπ/2
0cos(zcosx)c o saxdx =−asinaπ
2s−1,a(z)
=π
4secaπ
2[Ja(z)+J−a(z)] =π
4cosecaπ
2[Ea(z)−E−a(z)]
=−asinaπ
2/braceleftBigg
−1
a2+∞/summationdisplay
k=1(−1)k−1z2k
a2(22−a2)( 42−a2)...[(2k)2−a2]/bracerightBigg
[a>0] WA 339
18./integraldisplayπ
0cos(zcosx)c o snxdx =1
2/integraldisplayπ
−πcos(zcosx)cosnxdx =πcosnπ
2Jn(z) GW (334)(56b)
19./integraldisplayπ/2
0cos(zcosx)c o s2 nxdx =(−1)n·π
2J2n(z) WA 30(9)
20./integraldisplayπ/2
0cos(zcosx)s i n2νxdx=√π
2/parenleftbigg2
z/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Jν(z)
/bracketleftbig
Reν>−1
2/bracketrightbig
WA 35, WH
21./integraldisplayπ
0cos(zcosx)s i n2μxdx=√π/parenleftbigg2
z/parenrightbiggμ
Γ/parenleftbigg
μ+1
2/parenrightbigg
Jμ(z)
/bracketleftbig
Reμ>−1
2/bracketrightbig
WH
3.716
1./integraldisplayπ/2
0sin(atanx)dx=1
2/bracketleftbig
e−aEi(a)−eaEi(−a)/bracketrightbig
[a>0] (cf. 3.723 1) BI (43)(1)
2./integraldisplayπ/2
0cos(atanx)dx=π
2e−a[a≥0] BI (43)(2)
422 Trigonometric Functions 3.717
3./integraldisplayπ/2
0sin(atanx)s i n2xdx=aπ
2e−a[a≥0] BI (43)(7)
4./integraldisplayπ/2
0cos(atanx)sin2xdx=1−a
4πe−a[a≥0] BI (43)(8)
5./integraldisplayπ/2
0cos(atanx)cos2xdx=1+a
4πe−a[a≥0] BI (43)(9)
6./integraldisplayπ/2
0sin(atanx)t a nxdx=π
2e−a[a>0] BI (43)(5)
7./integraldisplayπ/2
0cos(atanx)tanxdx=−1
2/bracketleftbig
e−aEi(a)+eaEi(−a)/bracketrightbig
[a>0] (cf. 3.723 5) BI (43)(6)
8./integraldisplayπ/2
0sin(atanx)s i n2xtanxdx=2−a
4πe−a[a>0] BI (43)(11)
9./integraldisplayπ/2
0sin2(atanx)dx=π
4/parenleftbig
1−e−2a/parenrightbig
[a≥0] (cf. 3.742 1) BI (43)(3)
10./integraldisplayπ/2
0cos2(atanx)dx=π
4/parenleftbig
1+e−2a/parenrightbig
[a≥0] (cf. 3.742 3) BI (43)(4)
11./integraldisplayπ/2
0sin2(atanx)c o t2xdx=π
4/parenleftbig
e−2a+2a−1/parenrightbig
[a≥0] BI (43)(19)
12./integraldisplayπ/2
0/bracketleftbig
1−sec2xcos (tan x)/bracketrightbigdx
tanx=C BI (51)(14)
13./integraldisplayπ/2
0sin(acotx)sin2xdx=aπ
2e−a[a≥0] (cf. 3.716 3)
In general, formulas 3.716 remain valid if we replace tan xin the argument of the sine or cosine with
cotxif we also replace sin xwith cos x,c o sxwith sin x, hence tan xwith cot x,c o txwith tan x,s e cx
with cosec x, and cosec xwith sec xin the factors. Analogously,
3.717/integraldisplayπ/2
0sin(acosecx)sin(acotx)dx
cosx=/integraldisplayπ/2
0sin (asecx)sin(atanx)dx
sinx=π
2sina [a≥0]
BI (52)(11, 12)
3.718
1./integraldisplayπ/2
0sin/parenleftBigπ
2p−atanx/parenrightBig
tanp−1xdx=/integraldisplayπ/2
0cos/parenleftBigπ
2p−atanx/parenrightBig
tanpxdx=π
2e−a
/bracketleftbig
p2<1,p/negationslash=0,a≥0/bracketrightbig
BI (44)(5, 6)
2./integraldisplayπ/2
0sin(atanx−νx)s i nν−2xdx=0 [ R e ν>0,a > 0] NH 157(15)
3./integraldisplayπ/2
0sin(ntanx+νx)cosν−1x
sinxdx=π
2[Reν>0] BI (51)(15)
3.722 Trigonometric and rational functions 423
4./integraldisplayπ/2
0cos(atanx−νx)c o sν−2xdx=πe−aaν−1
Γ(ν)[Reν>1,a > 0]
LO V 153(112), NT 157(14)
5./integraldisplayπ/2
0cos(atanx+νx)c o sνxdx=2−ν−1πe−a[Reν>−1,a≥0] BI (44)(4)
6./integraldisplayπ/2
0cos(atanx−γx)c o sνxdx=πaν
2
2ν
2+1·W γ
2,−ν+1
2(2a)
Γ/parenleftbig
1+γ+ν
2/parenrightbig
/bracketleftbigg
a>0,Reν>−1,ν+γ
2/negationslash=−1,−2,.../bracketrightbigg
EH I 274(13)a
7./integraldisplayπ/2
0sinnx−sin(nx−atanx)
sinxcosn−1xdx=/braceleftBigg
π/2[ n=0,a > 0],
π(1−e−a)[n=1,a≥0]
LO V 153(114)
3.719
1.6/integraldisplayπ
0sin (νx−zsinx)dx=πEν(z) WA 336(2)
2./integraldisplayπ
0cos(nx−zsinx)dx=πJn(z) WH
3./integraldisplayπ
0cos(νx−zsinx)dx=πJν(z) WA 336(1)
3.72–3.74 Combinations of trigonometric and rational functions
3.721
1./integraldisplay∞
0sin(ax)
xdx=π
2signa FI II 645
2./integraldisplay∞
1sin(ax)
xdx=−si(a) BI 203(1)
3.8/integraldisplay∞
1cos(ax)
xdx=−ci(a) BI 203(5)
3.722
1./integraldisplay∞
0sin(ax)
x+βdx=c i (aβ)sin(aβ)−cos(aβ)si(aβ)[ |argβ|<π , a> 0]
BI(16)(1), FI II 646a
2.11/integraldisplay∞
−∞sin(ax)
x+βdx=πeiaβ[a>0,Imβ>0]
3./integraldisplay∞
0cos(ax)
x+βdx=−sin(aβ)si(aβ)−cos(aβ)ci(aβ)[ |argβ|<π , a> 0]
ET I 8(7), BI(160)(2)
424 Trigonometric Functions 3.723
4.8/integraldisplay∞
−∞cos(ax)
x+βdx=−iπeiaβ[a>0,Imβ>0]
5.10/integraldisplay∞
0sin(ax)
β−xdx=s i n ( βa)ci(βa)−cos(βa)[si(βa)+π]
[a>0,βnot real and positive]
FI II 646, BI(161)(1)
6.8/integraldisplay∞
−∞sin(ax)
β−xdx=−πeiaβ[a>0,Imβ>0]
7.10/integraldisplay∞
0cos(ax)
β−xdx=−cos(aβ)ci(aβ)+s i n ( aβ)[si(aβ)+π]
[a>0,βnot real and positive]
ET I 8(8), BI(161)(2)a
8.11/integraldisplay∞
−∞cos(ax)
β−xdx=−iπeiaβ[a>0,Imβ>0]
3.723
1.11/integraldisplay∞
0sin(ax)
β2+x2dx=1
2β/bracketleftbig
e−aβEi(aβ)−eaβEi(−aβ)/bracketrightbig
[a>0,β > 0]ET I 65(14), BI(160)(3)
2./integraldisplay∞
0cos(ax)
β2+x2dx=π
2βe−aβ[a≥0,Reβ>0]
FI II 741, 750, ET I 8(11), WH
3./integraldisplay∞
0xsin(ax)
β2+x2dx=π
2e−aβ[a>0,Reβ>0]
FI II 741, 750, ET I 65(15), WH
4./integraldisplay∞
−∞xsin(ax)
β2+x2dx=πe−aβ[a>0,Reβ>0] BI (202)(10)
5.11/integraldisplay∞
0xcos(ax)
β2+x2dx=−1
2/bracketleftbig
e−aβEi(aβ)+eaβEi(−aβ)/bracketrightbig
[a>0,β > 0] BI (160)(6)
6./integraldisplay∞
−∞sin[a(b−x)]
c2+x2dx=π
ce−acsin(ab)[ a>0,b > 0,c > 0] LI (202)(9)
7./integraldisplay∞
−∞cos[a(b−x)]
c2+x2dx=π
ce−accos(ab)[ a>0,b > 0,c > 0] LI (202)(11)a
8./integraldisplay∞
0sin(ax)
β2−x2dx=1
β/bracketleftBig
sin(aβ)ci(aβ)−cos(aβ)/parenleftBig
si(aβ)+π
2/parenrightBig/bracketrightBig
[|argβ|<π , a> 0] BI (161)(3)
9./integraldisplay∞
0cos(ax)
b2−x2dx=π
2bsin(ab)[ a>0,b > 0] BI(161)(5), ET I 9(15)
10./integraldisplay∞
0xsin(ax)
b2−x2dx=−π
2cos(ab)[ a>0] FI II 647, ET II 252(45)
11./integraldisplay∞
0xcos(ax)
β2+x2dx=c o s ( aβ)ci(aβ)+s i n ( aβ)/bracketleftBig
si(aβ)+π
2/bracketrightBig
[|argβ|<π , a> 0] BI (161)(6)
3.726 Trigonometric and rational functions 425
12./integraldisplay∞
−∞sin(ax)
x(x−b)dx=πcos(ab)−1
b[a>0,b > 0] ET II 252(44)
3.724
1./integraldisplay∞
−∞b+cx
p+2qx+x2sin(ax)dx=/parenleftBigg
cq−b/radicalbig
p−q2sin(aq)+ccos(aq)/parenrightBigg
πe−a√
p−q2
/bracketleftbig
a>0,p > q2/bracketrightbig
BI (202)(12)
2./integraldisplay∞
−∞b+cx
p+2qx+x2cos(ax)dx=/parenleftBigg
b−cq/radicalbig
p−q2cos(aq)+csin(aq)/parenrightBigg
πe−a√
p−q2
/bracketleftbig
a>0,p > q2/bracketrightbig
BI (202)(13)
3./integraldisplay∞
−∞cos[(b−1)t]−xcos(bt)
1−2xcost+x2cos(ax)dx=πe−asintsin(bt+acost)
/bracketleftbig
a>0,t2<π2/bracketrightbig
BI (202)(14)
3.725
1./integraldisplay∞
0sin(ax)dx
x(β2+x2)=π
2β2/parenleftbig
1−e−aβ/parenrightbig
[Reβ>0,a > 0] BI (172)(1)
2./integraldisplay∞
0sin(ax)dx
x(b2−x2)=π
2b2(1−cos(ab)) [ a>0] BI (172)(4)
3./integraldisplay∞
0sin(ax)cos(bx)
x(x2+β2)dx=π
2β2e−βbsinh(aβ)[ 0 <a<b ]
=−π
2β2e−aβcosh(bβ)+π
2β2[a>b> 0]
ET I 19(4)
3.726
1.11/integraldisplay∞
0xsin(ax)dx
b3±b2x+bx2±x3
=±1
4b/bracketleftBig
e−abEi(ab)−eabEi(−ab)−2c i(ab)sin (ab)+2c o s ( ab)/parenleftBig
si(ab)+π
2/parenrightBig/bracketrightBig
+πe−ab−πcos(ab)
4b
[a>0,b > 0; if the lower sign is taken, then the integral is a principal value integral]
ET I 65(21)a, BI(176)(10, 13)
2.7/integraldisplay∞
0x2sin(ax)dx
b3±b2x+bx2±x3
=1
4/bracketleftBig
eabEi(−ab)−e−abEi(ab)+2c i ( ab)sin (ab)−2c os (ab)/parenleftBig
si(ab)+π
2/parenrightBig/bracketrightBig
±π/parenleftbig
e−ab+c o s ( ab)/parenrightbig
[a>0,b > 0; if the lower sign is taken, then the integral is a principal value integral]
ET I 66(22), BI(176)(11, 14)
426 Trigonometric Functions 3.727
3.727
1./integraldisplay∞
0cos(ax)
b4+x4dx=π√
2
4b3exp/parenleftbigg
−ab√
2/parenrightbigg/parenleftbigg
cosab√
2+s i nab√
2/parenrightbigg
[a>0,b > 0]BI(160)(25)a, ET I 9(19)
2.8/integraldisplay∞
0sin(ax)
b4−x4dx=1
4b3/bracketleftBig
2s in(ab)ci (ab)−2c os (ab)/parenleftBig
si(ab)+π
2/parenrightBig
+e−abEi(ab)−eabEi(−ab)/bracketrightBig
[a>0,b > 0] BI (161)(12)
3./integraldisplay∞
0cos(ax)
b4−x4dx=π
4b3/bracketleftbig
e−ab+s i n ( ab)/bracketrightbig
[a>0,b > 0] (cf. 3.723 2a n d3.723 9)BI (161)(16)
4./integraldisplay∞
0xsin(ax)
b4+x4dx=π
2b2exp/parenleftbigg
−ab√
2/parenrightbigg
sinab√
2[a>0,b > 0] BI (160)(23)a
5./integraldisplay∞
0xsin(ax)
b4−x4dx=π
4b2/bracketleftbig
e−ab−cos(ab)/bracketrightbig
[a>0,b > 0] BI (161)(13)
6.11/integraldisplay∞
0xcos(ax)
b4−x4dx=1
4b2/bracketleftbigg
2c os (ab)ci(ab)+2s i n ( ab)/parenleftBig
si(ab)+π
2/parenrightBig
−e−abEi(ab)−eabEi(−ab)/bracketrightbigg
[a>0,b > 0] (cf. 3.723 5a n d3.723 11) BI (161)(17)
7./integraldisplay∞
0x2cos(ax)
b4+x4dx=π√
2
4bexp/parenleftbigg
−ab√
2/parenrightbigg/parenleftbigg
cosab√
2−sinab√
2/parenrightbigg
[a>0,b > 0] BI (160)(26)a
8.11/integraldisplay∞
0x2sin(ax)dx
b4−x4=1
4b/bracketleftBig
2s in (ab)ci(ab)
−2c os(ab)/parenleftBig
si(ab)+π
2/parenrightBig
−e−abEi(ab)+eabEi(−ab)/bracketrightBig
[a>0,b > 0] BI (161)(14)
9./integraldisplay∞
0x2cos(ax)
b4−x4dx=π
4b/parenleftbig
sin(ab)−e−ab/parenrightbig
[a>0,b > 0] BI (161)(18)
10./integraldisplay∞
0x3sin(ax)
b4+x4dx=π
2exp/parenleftbigg
−ab√
2/parenrightbigg
cosab√
2[a>0,b > 0] BI (160)(24)
11./integraldisplay∞
0x3sin(ax)
b4−x4dx=−π
4/bracketleftbig
e−ab−cos(ab)/bracketrightbig
[a>0,b > 0] BI (161)(15)
12.7/integraldisplay∞
0x3cos(ax)dx
b4−x4=1
4/bracketleftBig
2c os (ab)ci (ab)+2s i n ( ab)/parenleftBig
si(ab)+π
2/parenrightBig
+e−abEi(ab)+eabEi(−ab)/bracketrightBig
[a>0,b > 0] BI(161)(19)
3.729 Trigonometric and rational functions 427
13./integraldisplay∞
0x3sinax
(x2+b2)3dx=πe−ab
16b/parenleftbig
3a−ba2/parenrightbig
14./integraldisplay∞
0x3sinax
(x2+b2)4dx=πe−aba
96b3/parenleftbig
3+3ab−a2b2/parenrightbig
3.728
1./integraldisplay∞
0cos(ax)dx
(β2+x2)(γ2+x2)=π/parenleftbig
βe−aγ−γe−aβ/parenrightbig
2βγ(β2−γ2)[a>0,Reβ>0,Reγ>0]
BI (175)(1)
2./integraldisplay∞
0xsin(ax)dx
(β2+x2)(γ2+x2)=π/parenleftbig
e−aβ−e−aγ/parenrightbig
2(γ2−β2)[a>0,Reβ>0,Reγ>0]
BI (174)(1)
3./integraldisplay∞
0x2cos(ax)dx
(β2+x2)(γ2+x2)=π/parenleftbig
βe−aβ−γe−aγ/parenrightbig
2(β2−γ2)[a>0,Reβ>0,Reγ>0]
BI (175)(2)
4./integraldisplay∞
0x3sin(ax)dx
(β2+x2)(γ2+x2)=π/parenleftbig
β2e−aβ−γ2e−aγ/parenrightbig
2(β2−γ2)[a>0,Reβ>0,Reγ>0]
BI (174)(2)
5./integraldisplay∞
0cos(ax)dx
(b2−x2)(c2−x2)=π(bsin(ac)−csin(ab))
2bc(b2−c2)[a>0,b > 0,c > 0] BI (175)(3)
6./integraldisplay∞
0xsin(ax)dx
(b2−x2)(c2−x2)=π(cos(ab)−cos(ac))
2(b2−c2)[a>0] BI (174)(3)
7./integraldisplay∞
0x2cos(ax)dx
(b2−x2)(c2−x2)=π(csin(ac)−bsin(ab))
2(b2−c2)[a>0,b > 0,c > 0] BI (175)(4)
8./integraldisplay∞
0x3sin(ax)dx
(b2−x2)(c2−x2)=π/parenleftbig
b2cos(ab)−c2cos(ac)/parenrightbig
2(b2−c2)[a>0,b > 0,c > 0] BI (174)(4)
9./integraldisplay∞
0xsinax
(b2−x2)(c2+x2)dx=π
2e−ac−cosba
a2+c2[a>0,c > 0,breal]
3.729
1./integraldisplay∞
0cos(ax)dx
(b2+x2)2=π
4b3(1 +ab)e−ab[a>0,b > 0] BI (170)(7)
2./integraldisplay∞
0xsin(ax)dx
(b2+x2)2=π
4bae−ab[a>0,b > 0] BI (170)(3)
3./integraldisplay∞
0cos(px)1−x2
(1 +x2)2dx=πp
2e−pBI (43)(10)a
4./integraldisplay∞
0x3sin(ax)dx
(b2+x2)2=π
4(2−ab)e−ab[a>0,b > 0] BI (170)(4)
428 Trigonometric Functions 3.731
3.731 Notation :2A2=√
b4+c2+b2,2B2=√
b4+c2−b2,
1./integraldisplay∞
0cos(ax)dx
(x2+b2)2+c2=π
2ce−aA(Bcos(aB)+Asin(aB))√
b4+c2
[a>0,b > 0,c > 0] BI (176)(3)
2./integraldisplay∞
0xsin(ax)dx
(x2+b2)2+c2=π
2ce−aAsin(aB)[ a>0,b > 0,c > 0] BI (176)(1)
3./integraldisplay∞
0/parenleftbig
x2+b2/parenrightbig
cos(ax)dx
(x2+b2)2+c2=π
2e−aA(Acos(aB)−Bsin(aB))√
b4+c2
[a>0,b > 0,c > 0] BI (176)(4)
4./integraldisplay∞
0x/parenleftbig
x2+b2/parenrightbig
sin(ax)dx
(x2+b2)2+c2=π
2e−aAcos(aB)[ a>0,b > 0,c > 0] BI (176)(2)
3.732
1./integraldisplay∞
0/bracketleftbigg1
β2+(γ−x)2−1
β2+(γ+x)2/bracketrightbigg
sin(ax)dx=π
βe−aβsin(aγ)
[a>0,Reβ>0,γ+iβis not real]
ET I 65(16)
2./integraldisplay∞
0/bracketleftbigg1
β2+(γ−x)2+1
β2+(γ+x)2/bracketrightbigg
cos(ax)dx=π
βe−aβcos(aγ)
[a>0,|Imγ|<Reβ] ET I 8(13)
3./integraldisplay∞
0/bracketleftbiggγ+x
β2+(γ+x)2−γ−x
β2+(γ−x)2/bracketrightbigg
sin(ax)dx=πe−aβcos(aγ)
[a>0,Reβ>0,γ+iβis not real]
LI (175)(17)
4./integraldisplay∞
0/bracketleftbiggγ+x
β2+(γ+x)2+γ−x
β2+(γ−x)2/bracketrightbigg
cos(ax)dx=πe−aβsin(aγ)
[a>0,|Ima|<Reβ] LI (176)(21)
3.733
1./integraldisplay∞
0cos(ax)dx
x4+2b2x2cos 2t+b4=π
2b3exp(−abcost)sin(t+absint)
sin2t/bracketleftBig
a>0,b > 0,|t|<π
2/bracketrightBig
BI (176)(7)
2./integraldisplay∞
0xsin(ax)dx
x4+2b2x2cos 2t+b4=π
2b2exp(−abcost)sin(absint)
sin2t/bracketleftBig
a>0,b > 0,|t|<π
2/bracketrightBig
BI(176)(5), ET I 66(23)
3./integraldisplay∞
0x2cos(ax)dx
x4+2b2x2cos 2t+b4=π
2bexp (−abcost)sin (t−absint)
sin 2t/bracketleftBig
a>0,b > 0,|t|<π
2/bracketrightBig
BI (176)(8)
3.736 Trigonometric and rational functions 429
4./integraldisplay∞
0x3sin(ax)dx
x4+2b2x2cos 2t+b4=π
2exp (−abcost)sin (2t−absint)
sin 2t/bracketleftBig
a>0,b > 0,|t|<π
2/bracketrightBig
BI (176)(6)
5./integraldisplay∞
0sin(ax)dx
x(x4+2b2x2cos2t+b4)=π
2b4/bracketleftbigg
1−exp (−abcost)sin (2t+absint)
sin 2t/bracketrightbigg
/bracketleftBig
a>0,b > 0,|t|<π
2/bracketrightBig
BI (176)(22)
3.734
1./integraldisplay∞
0sin(ax)dx
x(b4+x4)=π
2b4/bracketleftbigg
1−exp/parenleftbigg
−ab√
2/parenrightbigg
cosab√
2/bracketrightbigg
[a>0,b > 0] BI (172)(7)
2./integraldisplay∞
0sin(ax)dx
x(b4−x4)=π
4b4/bracketleftbig
2−e−ab−cos(ab)/bracketrightbig
[a>0,b > 0] BI (172)(10)
3.735/integraldisplay∞
0sin(ax)dx
x(b2+x2)2=π
2b4/bracketleftbigg
1−1
2e−ab(2 +ab)/bracketrightbigg
[a>0,b > 0] WH, BI (172)(22)
3.736
1./integraldisplay∞
0cos(ax)dx
(b2+x2)(b4−x4)=π
8b5/bracketleftbig
sin(ab)+( 2+ ab)e−ab/bracketrightbig
[a>0,b > 0] BI (176)(5)
2./integraldisplay∞
0xsin(ax)dx
(b2+x2)(b4−x4)=π
8b4/bracketleftbig
(1 +ab)e−ab−cos(ab)/bracketrightbig
[a>0,b > 0] BI (174)(5)
3./integraldisplay∞
0x2cos(ax)dx
(b2+x2)(b4−x4)=π
8b3/bracketleftbig
sin(ab)−abe−ab/bracketrightbig
[a>0,b > 0] BI (175)(6)
4./integraldisplay∞
0x3sin(ax)dx
(b2+x2)(b4−x4)=π
8b2/bracketleftbig
(1−ab)e−ab−cos(ab)/bracketrightbig
[a>0,b > 0] BI (174)(6)
5./integraldisplay∞
0x4cos(ax)dx
(b2+x2)(b4−x4)=π
8b/bracketleftbig
sin(ab)+(ab−2)e−ab/bracketrightbig
[a>0,b > 0] BI (175)(7)
6./integraldisplay∞
0x5sin(ax)dx
(b2+x2)(b4−x4)=π
8/bracketleftbig
(ab−3)e−ab−cos(ab)/bracketrightbig
[a>0,b > 0] BI (174)(7)
430 Trigonometric Functions 3.737
3.737
1.8/integraldisplay∞
0cos(ax)dx
(b2+x2)n=πe−ab
(2b)2n−1(n−1)!n−1/summationdisplay
k=0(2n−k−2)!(2ab)k
k!(n−k−1)!
=(−1)n−1π
2b2n−1(n−1)!/bracketleftbiggdn−1
dpn−1/parenleftbigge−ab√p
√p/parenrightbigg/bracketrightbigg
p=1
=(−1)n−1π
2b2n−1(n−1)!/bracketleftbiggdn−1
dpn−1/parenleftbigge−abp
(1 +p)n/parenrightbigg/bracketrightbigg
p=1
[a>0,b > 0]GW(333)(67b), WA 209, WA 192
2./integraldisplay∞
0xsin(ax)dx
(x2+β2)n+1=πae−aβ
22nn!β2n−1n−1/summationdisplay
k=0(2n−k−2)!(2aβ)k
k!(n−k−1)!
=π
2e−aβ[n=0,β≥0]
[a>0,Reβ>0] GW (333)(66c)
3./integraldisplay∞
0sin(ax)dx
x(β2+x2)n+1=π
2β2n+2/bracketleftbigg
1−e−aβ
2nn!Fn(aβ)/bracketrightbigg
/bracketleftbig
a>0,Reβ>0,F 0(z)=1,F 1(z)=z+2,...,F n(z)=(z+2n)Fn−1(z)−zF/prime
n−1(z)/bracketrightbig
GW (333)(66e)
4./integraldisplay∞
0xsin(ax)dx
(b2+x2)3=πa
16b3(1 +ab)e−ab[a>0,b > 0]BI(170)(5), ET I 67(35)a
5./integraldisplay∞
0xsin(ax)dx
(b2+x2)4=πa
96b5/parenleftbig
3+3ab+a2b2/parenrightbig
e−ab[a>0,b > 0]BI(170)(6), ET I 67(35)a
6./integraldisplay∞
0x3sinax
(x2+β2)n+1dx=πe−aβ
22nn!β2n−2/bracketleftbigg
2n−1(2n−3)!!(2−βa)
−n−1/summationdisplay
k=1(2n−k−2)!2k(βa)k−1
k!(n−k−1)!/bracketleftbig
k(k+1 )−2(k+1 )βa+β2a2/bracketrightbig/bracketrightBigg
3.738
1./integraldisplay∞
0xm−1sin(ax)
x2n+β2ndx=−πβm−2n
2nn/summationdisplay
k=1exp/bracketleftbigg
−aβsin(2k−1)π
2n/bracketrightbigg
×cos/braceleftbigg(2k−1)mπ
2n+aβcos(2k−1)π
2n/bracerightbigg
[mis even] ,/bracketleftBig
a>0,|argβ|<π
2n,0<m≤2n/bracketrightBig
ET I 67(38)
2./integraldisplay∞
0xm−1cos(ax)
x2n+β2ndx=πβm−2n
2nn/summationdisplay
k=1exp/bracketleftbigg
−aβsin(2k−1)π
2n/bracketrightbigg
×sin/braceleftbigg(2k−1)mπ
2n+aβcos(2k−1)π
2n/bracerightbigg
[mis odd] ,/bracketleftBig
a>0,|argβ|<π
2n,0<m< 2n+1/bracketrightBig
BI(160)(29)a, ET I 10(29)
3.741 Trigonometric and rational functions 431
3.739
1./integraldisplay∞
0sin(ax)dx
x(x2+22)(x2+42)...(x2+4n2)
=π(−1)n
(2n)!22n+1/bracketleftBigg
2n−1/summationdisplay
k=0(−1)k/parenleftbigg2n
k/parenrightbigg
e2(k−n)a+(−1)n/parenleftbigg2n
n/parenrightbigg/bracketrightBigg
[a>0,n≥0] LI(174)(8)
2./integraldisplay∞
0cos(ax)dx
(x2+12)(x2+32)...[x2+( 2n+1 )2]
=(−1)n
(2n+1 ) !π
22n+1n/summationdisplay
k=0(−1)k/parenleftbigg2n+1
k/parenrightbigg
e(2k−2n−1)a[a≥0,n≥0]
=π2−2n−1
(2n+1 )(n!)2[a=0,n≥0]
BI(175)(8)
3./integraldisplay∞
0xsin(ax)dx
(x2+12)(x2+32)...[x2+( 2n+1 )2]
=π(−1)n
(2n+1 ) ! 22n+1n/summationdisplay
k=0(−1)k/parenleftbigg2n+1
k/parenrightbigg
(2n−2k+1 )e(2k−2n−1)a
[a>0,n≥0] LI (174)(9)
4./integraldisplay∞
0cosaxdx
(x2+22)(x2+42)...(x2+4n2)=π21−2n
(2n)!n/summationdisplay
k=1(−1)kk/parenleftbigg2n
n−k/parenrightbigg
e−2ak
[n≥1,a≥0]
3.741
1./integraldisplay∞
0sin(ax)sin(bx)
xdx=1
4ln/parenleftbigga+b
a−b/parenrightbigg2
[a>0,b > 0,a/negationslash=b] FI II 647
2./integraldisplay∞
0sin(ax)cos(bx)
xdx=π
2[a>b≥0]
=π
4[a=b>0]
=0 [ b>a≥0]
FI II 645
3./integraldisplay∞
0sin(ax)sin(bx)
x2dx=aπ
2[0<a≤b]
=bπ
2[0<b≤a]
BI (157)(1)
432 Trigonometric Functions 3.742
3.742
1./integraldisplay∞
0sin(ax)sin(bx)
β2+x2dx=π
4β/parenleftBig
e−|a−b|β−e−(a+b)β/parenrightBig
[a>0,b > 0,Reβ>0]
=π
2βe−aβsinhbβ [β>0,a≥b≥0]
=π
2βe−bβsinhaβ [β>0,b≥a≥0]
BI(162)(1)a, GW(333)(71a)
2./integraldisplay∞
0sin(ax)cos(bx)
β2+x2dx=1
4βe−aβ/braceleftbig
ebβEi[β(a−b)] +e−bβEi[β(a+b)]/bracerightbig
−1
4βeaβ/braceleftbig
ebβEi[−β(a+b)] +e−bβEi[β(b−a)]/bracerightbig
BI (162)(3)
3./integraldisplay∞
0cos(ax)cos(bx)
β2+x2dx=π
4β/bracketleftBig
e−|a−b|β+e−(a+b)β/bracketrightBig
[a>0,b > 0,Reβ>0]
=π
2βe−aβcoshbβ [β>0,a≥b≥0]
=π
2βe−bβcoshaβ [β>0,b≥a≥0]
BI(163)(1)a, GW(333)(71c)
4./integraldisplay∞
0xcos(ax)cos(bx)
β2+x2dx=−1
4eaβ/braceleftbig
ebβEi[−β(a+b)] +e−bβEi[β(b−a)]/bracerightbig
−1
4e−aβ/braceleftbig
ebβEi[β(a−b)] +e−bβEi[β(a+b)]/bracerightbig
[a/negationslash=b]
=∞ [a=b]
BI (163)(2)
5./integraldisplay∞
0xsin(ax)cos(bx)
x2+β2dx=π
2e−aβcosh(bβ)[ 0 <b<a ]
=π
4e−2aβ[0<b=a]
=−π
2e−bβsinh(aβ)[ 0 <a<b ]
BI (162)(4)
6./integraldisplay∞
0sin(ax)sin(bx)
p2−x2dx=−π
2pcos(ap)sin(bp)[ a>b> 0]
=−π
4psin(2ap)[ a=b>0]
=−π
2psin(ap)cos(bp)[ b>a> 0]
BI (166)(1)
7./integraldisplay∞
0sin(ax)cos(bx)
p2−x2xdx=−π
2cos(ap)cos(bp)[ a>b> 0]
=−π
4cos(2ap)[ a=b>0]
=π
2sin(ap)sin(bp)[ b>a> 0]
BI (166)(2)
3.747 Trigonometric and rational functions 433
8./integraldisplay∞
0cos(ax)cos(bx)
p2−x2dx=π
2psin(ap)cos(bp)[ a>b> 0]
=π
4psin(2ap)[ a=b>0]
=π
2pcos(ap)sin(bp)[ b>a> 0]
BI (166)(3)
3.743
1./integraldisplay∞
0sin(ax)
sin(bx)·dx
x2+β2=π
2β·sinh(aβ)
sinh(bβ)[0<a<b , Reβ>0] ET I 80(21)
2./integraldisplay∞
0sin(ax)
cos(bx)·xdx
x2+β2=−π
2·sinh(aβ)
cosh(bβ)[0<a<b , Reβ>0] ET I 81(30)
3./integraldisplay∞
0cos(ax)
sin(bx)·xdx
x2+β2=π
2·cosh(aβ)
sinh(bβ)[0<a<b , Reβ>0] ET I 23(37)
4./integraldisplay∞
0cos(ax)
cos(bx)·dx
x2+β2=π
2β·cosh(aβ)
cosh(bβ)[0<a<b , Reβ>0] ET I 23(36)
5.6PV/integraldisplay∞
0sin(ax)
sinx·dx
b2−x2=0 i f0 ≤a≤1
=π
bsin(a−1)b if 1≤a≤2
[breal,b/π/negationslash∈Z]
3.7443/integraldisplay∞
0sin(ax)
cos(bx)·dx
x(x2+β2)=π
2β2·sinh(aβ)
cosh(bβ)[0<a<b , Reβ>0] ET I 82(32)
3.7453/integraldisplay∞
0sin(ax)
cos(bx)·dx
x(c2−x2)=0 [ 0 <a<b , c> 0] ET I 82(31)
3.746
1./integraldisplay∞
0dx
xn+1n/productdisplay
k=0sin (akx)=π
2n/productdisplay
k=1ak/bracketleftBigg
a0>n/summationdisplay
k=1ak,a k>0/bracketrightBigg
FI II 646
2./integraldisplay∞
0sin(ax)
xn+1dxn/productdisplay
k=1sin (akx)m/productdisplay
j=1cos (bjx)=π
2n/productdisplay
k=1ak⎡
⎣a>n/summationdisplay
k=1|ak|+m/summationdisplay
j=1|bj|⎤
⎦ WH
3.747
1.7/integraldisplayπ/2
0xm
sinxdx=/parenleftBigπ
2/parenrightBigm/bracketleftBigg
1
m+∞/summationdisplay
k=122k−1−1
42k−1(m+2k)ζ(2k)/bracketrightBigg
=2πG−7
2ζ(3)
[m=2 ] LI (206)(2)
2./integraldisplayπ/2
0xdx
sinx=/integraldisplayπ/2
0/parenleftbigπ
2−x/parenrightbig
dx
cosx=2G BI(204)(18), BI(206)(1), GW(333)(32)
3./integraldisplay∞
0xdx
(x2+b2)sin(ax)=π
2s in h( ab)[b>0] GW (333)(79c)
4./integraldisplayπ
0xtanxdx=−πln 2 BI (218)(4)
434 Trigonometric Functions 3.748
5./integraldisplayπ/2
0xtanxdx=∞ BI (205)(2)
6./integraldisplayπ/4
0xtanxdx=−π
8ln 2 +1
2G=0.1857845358 ... BI (204)(1)
7./integraldisplayπ/2
0xcotxdx=π
2ln 2 FI II 623
8./integraldisplayπ/4
0xcotxdx=π
8ln 2 +1
2G=0.7301810584 ... BI (204)(2)
9./integraldisplayπ/2
0/parenleftBigπ
2−x/parenrightBig
tanxdx=1
2/integraldisplayπ
0/parenleftBigπ
2−x/parenrightBig
tanxdx=π
2ln2 GW(333)(33b), BI(218)(12)
10./integraldisplay∞
0tanaxdx
x=π
2[a>0] LO V 279(5)
11./integraldisplayπ/2
0xcotx
cos2xdx=π
4ln 2 BI (206)(12)
3.748
1./integraldisplayπ/4
0xmtanxdx=1
2/parenleftBigπ
4/parenrightBigm∞/summationdisplay
k=1/parenleftbig
4k−1/parenrightbig
ζ(2k)
42k−1(m+2k)LI (204)(5)
2./integraldisplayπ/2
0xpcotxdx=/parenleftBigπ
2/parenrightBigp/parenleftBigg
1
p−2∞/summationdisplay
k=11
4k(p+2k)ζ(2k)/parenrightBigg
LI (205)(7)
3./integraldisplayπ/4
0xmcotxdx=1
2/parenleftBigπ
4/parenrightBigm/parenleftBigg
2
m−∞/summationdisplay
k=1ζ(2k)
42k−1(m+2k)/parenrightBigg
LI (204)(6)
3.749
1./integraldisplay∞
0xtan(ax)dx
x2+b2=π
e2ab+1[a>0,b > 0] GW (333)(79a)
2./integraldisplay∞
0xcot(ax)dx
x2+b2=π
e2ab−1[a>0,b > 0] GW (333)(79b)
3./integraldisplay∞
0xtan(ax)dx
b2−x2=/integraldisplay∞
0xcot(ax)dx
b2−x2=/integraldisplay∞
0xcosec( ax)dx
b2−x2=∞ BI (161)(7, 8, 9)
3.75 Combinations of trigonometric and algebraic functions
3.751
1./integraldisplay∞
0sin(ax)dx√x+β=/radicalbiggπ
2a/bracketleftBig
cos(aβ)−sin(aβ)+2C/parenleftBig/radicalbig
aβ/parenrightBig
sin(aβ)−2S/parenleftBig/radicalbig
aβ/parenrightBig
cos(aβ)/bracketrightBig
[a>0,|argβ|<π] ET I 65(12)a
2.9/integraldisplay∞
0cos(ax)dx√x+β=/radicalbiggπ
2a/bracketleftBig
cos(aβ)+s i n ( aβ)−2C/parenleftBig/radicalbig
aβ/parenrightBig
cos(aβ)−2S/parenleftBig/radicalbig
aβ/parenrightBig
sin(aβ)/bracketrightBig
[a>0,|argβ|<π] ET I 8(9)a
3.755 Trigonometric and algebraic functions 435
3./integraldisplay∞
usin(ax)√x−udx=/radicalbiggπ
2a[sin(au)+c o s ( au)] [ a>0,u > 0] ET I 65(13)
4./integraldisplay∞
ucos(ax)√x−udx=/radicalbiggπ
2a[cos(au)−sin(au)] [ a>0,u > 0] ET I 8(10)
3.752
1.8/integraldisplay1
0sin(ax)/radicalbig
1−x2dx=∞/summationdisplay
k=0(−1)ka2k+1
(2k−1)!!(2k+3 ) ! !=π
2aH1(a)
[a>0] BI (149)(6)
2./integraldisplay1
0cos(ax)/radicalbig
1−x2dx=π
2aJ1(a) KU 65(6)a
3.753
1.8/integraldisplay1
0sin(ax)dx√
1−x2=∞/summationdisplay
k=0(−1)ka2k+1
[(2k+ 1)!!]2=π
2H0(a)[ a>0] BI (149)(9)
2./integraldisplay1
0cos(ax)dx√
1−x2=π
2J0(a) WA 30(7)a
3./integraldisplay∞
1sin(ax)dx√
x2−1=π
2J0(a)[ a>0] WA 200(14)
4./integraldisplay∞
1cos(ax)√
x2−1dx=−π
2Y0(a) WA 200(15)
5./integraldisplay1
0xsin(ax)√
1−x2dx=π
2J1(a)[ a>0] WA 30(6)
3.754
1./integraldisplay∞
0sin(ax)dx/radicalbig
β2+x2=π
2[I0(aβ)−L0(aβ)] [ a>0,Reβ>0] ET I 66(26)
2./integraldisplay∞
0cos(ax)dx/radicalbig
β2+x2=K0(aβ)[ a>0,Reβ>0]
WA 191(1), GW(333)(78a)
3./integraldisplay∞
0xsin(ax)/radicalBig
(β2+x2)3dx=aK0(aβ)[ a>0,Reβ>0] ET I 66(27)
3.755
1./integraldisplay∞
0/radicalBig/radicalbig
x2+β2−βsin(ax)dx
/radicalbig
x2+β2=/radicalbiggπ
2ae−aβ[a>0] ET I 66(31)
2./integraldisplay∞
0/radicalBig/radicalbig
x2+β2+βcos(ax)dx
/radicalbig
x2+β2=/radicalbiggπ
2ae−aβ[a>0,Reβ>0] ET I 10(25)
436 Trigonometric Functions 3.756
3.756
1./integraldisplay∞
0sin(ax)
xn
2−1n/productdisplay
k=2sin (akx)dx=0/bracketleftBigg
ak>0,a >n/summationdisplay
k=2ak/bracketrightBigg
ET I 80(22)
2./integraldisplay∞
0xn
2−1cos(ax)n/productdisplay
k=1cos(akx)dx=0/bracketleftBigg
ak>0,a >n/summationdisplay
k=1ak/bracketrightBigg
ET I 22(26)
3.757
1.11/integraldisplay∞
0sin(ax)√xdx=/radicalbiggπ
2a[a>0] BI (177)(1)
2.11/integraldisplay∞
0cos(ax)√xdx=/radicalbiggπ
2a[a>0] BI (177)(2)
3.76–3.77 Combinations of trigonometric functions and powers
3.761
1./integraldisplay1
0xμ−1sin(ax)dx=−i
2μ[1F1(μ;μ+1 ;ia)−1F1(μ;μ+1 ;−ia)]
[a>0,Reμ>−1,μ/negationslash=0 ]
ET I 68(2)a
2.8/integraldisplay∞
uxμ−1sinxdx=i
2/bracketleftbig
e−π
2iμΓ(μ, iu)−eπ
2iμΓ(μ,−iu)/bracketrightbig
[Reμ<1] EH II 149(2)
3./integraldisplay∞
1sin(ax)
x2ndx=a2n−1
(2n−1)!/bracketleftBigg2n−1/summationdisplay
k=1(2n−k−1)!
a2n−ksin/parenleftBig
a+(k−1)π
2/parenrightBig
+(−1)nci(a)/bracketrightBigg
[a>0] LI (203)(15)
4./integraldisplay∞
0xμ−1sin(ax)dx=Γ(μ)
aμsinμπ
2=πsecμπ
2
2aμΓ(1−μ)[a>0; 0 <|Reμ|<1]
FI II 809a, BI(150)(1)
5.10/integraldisplayπ
0xmsin(nx)dx=(−1)n+1
nm+1⌊m/2⌋/summationdisplay
k=0(−1)km!
(m−2k)!(nπ)m−2k
−(−1)⌊m/2⌋m!/floorleftbig
m−2/floorleftbigm
2/floorrightbig
−1/floorrightbig
nm+1
GW(333)(6)
6.8/integraldisplay1
0xμ−1cos(ax)dx=1
2μ[1F1(μ;μ+1 ;ia)+1F1(μ;μ+1 ;−ia)]
[a>0,Reμ>0] ET I 11(2)
7./integraldisplay∞
uxμ−1cosxdx=1
2/bracketleftbig
e−π
2iμΓ(μ, iu)s+eπ
2iμΓ(μ,−iu)/bracketrightbig
[Reμ<1] EH II 149(1)
3.763 Trigonometric functions and powers 437
8./integraldisplay∞
1cos(ax)
x2n+1dx=a2n
(2n)!/bracketleftBigg2n/summationdisplay
k=1(2n−k)!
a2n−k+1cos/parenleftBig
a+(k−1)π
2/parenrightBig
+(−1)n+1ci(a)/bracketrightBigg
[a>0] LI (203)(16)
9.8/integraldisplay∞
0xμ−1cos(ax)dx=Γ(μ)
aμcosμπ
2=πcosecμπ
2
2aμΓ(1−μ)[a>0,0<Reμ<1]
FI II 809a, BI(150)(2)
10./integraldisplayπ
0xmcos(nx)dx=(−1)n
nm+1⌊(m−1)/2⌋/summationdisplay
k=0(−1)k m!
(m−2k−1)!(nπ)m−2k−1
+(−1)⌊(m+1)/2⌋2[(m+1 )/2]−m
nm+1·m!
GW (333)(7)
11./integraldisplayπ/2
0xmcosxdx=⌊m/2⌋/summationdisplay
k=0(−1)km!
(m−2k)!/parenleftBigπ
2/parenrightBigm−2k
+(−1)⌊m/2⌋/parenleftBig
2/floorleftBigm
2/floorrightBig
−m/parenrightBig
m!
GW (333)(9c)
12./integraldisplay2nπ
0xmcoskxdx =−m−1/summationdisplay
j=0j!
kj+1/parenleftbiggm
j/parenrightbigg
(2nπ)m−jcosj+1
2π BI (226)(2)
3.762
1./integraldisplay∞
0xμ−1sin(ax)sin(bx)dx=1
2cosμπ
2Γ(μ)/bracketleftBig
|b−a|−μ−(b+a)−μ/bracketrightBig
[a>0,b > 0,a/negationslash=b,−2<Reμ<1]
(forμ=0 ,s e e 3.741 1, for μ=−1, see3.741 3)
BI(149)(7), ET I 321(40)
2./integraldisplay∞
0xμ−1sin(ax)cos(bx)dx=1
2sinμπ
2Γ(μ)/bracketleftBig
(a+b)−μ+|a−b|−μsign(a−b)/bracketrightBig
[a>0,b > 0,|Reμ|<1] (for μ=0s e e 3.741 2)BI(159)(8)a, ET I 321(41)
3./integraldisplay∞
0xμ−1cos(ax)cos(bx)dx=1
2cosμπ
2Γ(μ)/bracketleftBig
(a+b)−μ+|a−b|−μ/bracketrightBig
[a>0,b > 0,0<Reμ<1]
ET I 20(17)
3.763
1./integraldisplay∞
0sin(ax)sin(bx)sin(cx)
xνdx=1
4cosνπ
2Γ(1−ν)/braceleftBig
(c+a−b)ν−1−(c+a+b)ν−1
−|c−a+b|ν−1sign(a−b−c)+|c−a−b|ν−1sign(a+b−c)/bracerightBig
[c>0,0<Reν<4,ν/negationslash=1,2,3,a≥b>0]GW(333)(26a)a, ET I 79(13)
438 Trigonometric Functions 3.764
2./integraldisplay∞
0sin(ax)sin(bx)sin(cx)
xdx=0 [ c<a−bandc>a +b]
=π
8[c=a−bandc=a+b]
=π
4[a−b<c<a +b]
[a≥b>0,c > 0] FI II 645
3./integraldisplay∞
0sin(ax)sin(bx)sin(cx)
x2dx=1
4(c+a+b)ln(c+a+b)
−1
4(c+a−b)ln (c+a−b)−1
4|c−a−b|ln|c−a−b|
×sign(a+b−c)+1
4|c−a+b|ln|c−a+b|sign(a−b−c)
[a≥b>0,c > 0]BI(157)(8)a, ET I 79(11)
4./integraldisplay∞
0sin(ax)sin(bx)sin(cx)
x3dx=πbc
2[0<c<a −bandc>a +b]
=πbc
2−π(a−b−c)2
8[a−b<c<a +b]
[a≥b>0,c > 0]BI(157)(20), ET I 79(12)
3.764
1./integraldisplay∞
0xpsin(ax+b)dx=1
ap+1Γ(1 + p)cos/parenleftBig
b+pπ
2/parenrightBig
[a>0,−1<p< 0] GW (333)(30a)
2./integraldisplay∞
0xpcos(ax+b)dx=−1
ap+1Γ(1 + p)sin/parenleftBig
b+πp
2/parenrightBig
[a>0,−1<p< 0] GW (333)(30b)
3.765
1.10/integraldisplay∞
0sinax
xν(x+b)dx
=a1+νbcosπν
2Γ(−1−ν)1F2/parenleftbigg
1; 1 +ν
2,3
2+ν
2;−1
4a2b2/parenrightbigg
sign(a)
−πcosec( πν)sin(ab)
bν−aνΓ(−ν)1F2/parenleftbigg
1; 1 +ν
2,1+ν
2;−1
4a2b2/parenrightbigg
sign(a)sinπν
2
[Ima=0,−1<Reb<2,argb/negationslash=π]MC
2./integraldisplay∞
0cos(ax)
xν(x+β)dx=Γ(1−ν)
2βν/bracketleftbig
eiaβΓ(ν,iaβ)+e−iaβΓ(ν,−iaβ)/bracketrightbig
[a>0,|Reν|<1,|argβ|<π]
ET II 221(52)
3.766
1.10/integraldisplay∞
0xμ−1sinax
1+x2dx
=−a2−μΓ(μ−2)1F2/parenleftbigg
1;3−μ
2,4−μ
2;a2
4/parenrightbigg
sign(a)sinπμ
2+π
2secπμ
2sinh(a)
[Ima=0,−1<Reμ<3] MC
3.768 Trigonometric functions and powers 439
2./integraldisplay∞
0xμ−1cos(ax)
1+x2dx=π
2cosecμπ
2cosha
+1
2cosμπ
2Γ(μ){exp [−a+iπ(1−μ)]γ(1−μ,−a)−eaγ(1−μ, a)}
[a>0,0<Reμ<3] ET I 319(24)
3.9/integraldisplay∞
0x2μ+1sin(ax)dx
x2+b2=−π
2b2μsec(μπ)sin h( ab)
+sin(μπ)
2a2μΓ(2μ)[1F1(1; 1−2μ;ab)+1F1(1;1−2μ;−ab)]
/bracketleftbig
a>0,−3
2<Reμ<1
2/bracketrightbig
ET II 220(39)
4.9/integraldisplay∞
0x2μ+1cos(ax)dx
x2+b2=−π
2b2(μ+1
2)cosec/bracketleftbigg/parenleftbigg
μ+1
2/parenrightbigg
π/bracketrightbigg
cosh(ab)
+cos/bracketleftbig/parenleftbig
μ+1
2/parenrightbig
π/bracketrightbig
2a2(μ+1
2)Γ/bracketleftbigg
2/parenleftbigg
μ+1
2/parenrightbigg/bracketrightbigg/braceleftbigg
1F1/parenleftbigg
1;1−2/parenleftbigg
μ+1
2/parenrightbigg
;ab/parenrightbigg
+1F1/parenleftbigg
1;1−2/parenleftbigg
μ+1
2/parenrightbigg
;−ab/parenrightbigg/bracerightbigg
/bracketleftbig
a>0,−1<Reμ<1
2/bracketrightbig
ET II 221(56)
3.767
1./integraldisplay∞
0xβ−1sin/parenleftBig
ax−βπ
2/parenrightBig
γ2+x2dx=−π
2γβ−2e−aγ[a>0,Reγ>0,0<Reβ<2]
BI (160)(20)
2./integraldisplay∞
0xβcos/parenleftBig
ax−βπ
2/parenrightBig
γ2+x2dx=π
2γβ−1e−aγ[a>0,Reγ>0,|Reβ|<1]
BI (160)(21)
3./integraldisplay∞
0xβ−1sin/parenleftBig
ax−βπ
2/parenrightBig
x2−b2dx=π
2bβ−2cos/parenleftbigg
ab−πβ
2/parenrightbigg
[a>0,b > 0,0<Reβ<2]
BI (161)(11)
4./integraldisplay∞
0xβcos/parenleftBig
ax−βπ
2/parenrightBig
x2−b2dx=−π
2bβ−1sin/parenleftbigg
ab−πβ
2/parenrightbigg
[a>0,b > 0,|β|<1]
GW (333)(82)
3.768
1./integraldisplay∞
u(x−u)μ−1sin(ax)dx=Γ(μ)
aμsin/parenleftBig
au+μπ
2/parenrightBig
[a>0,0<Reμ<1] ET II 203(19)
2./integraldisplay∞
u(x−u)μ−1cos(ax)dx=Γ(μ)
aμcos/parenleftBig
au+μπ
2/parenrightBig
[a>0,0<Reμ<1] ET II 204(24)
3.11/integraldisplay1
0(1−x)νsin(ax)dx=1
a−Γ(ν+1 )
aν+1Cν(a)=a−ν−1/2sν+1/2,1/2(a)
[a>0,Reν>−1] ET I 11(3)a
440 Trigonometric Functions 3.768
HereCν(a) is the Young’s function given by:
Cν(a)=1
2aν
Γ(ν+1 )[1F1(1;ν+1 ;ia)+1F1(1;ν+1 ;−ia)] =∞/summationdisplay
n=0(−1)naν+2n
Γ(ν+2n+1 ).
4.3/integraldisplay1
0(1−x)νcos(ax)dx=i
2a−ν−1/braceleftbigg
exp/bracketleftbiggi
2(νπ−2a)/bracketrightbigg
γ(ν+1,−ia)
−exp/bracketleftbigg
−i
2(νπ−2a)/bracketrightbigg
γ(ν+1,ia)/bracerightbigg
=Γ (ν+1 )∞/summationdisplay
n=0/parenleftbig
−a2/parenrightbign
Γ(ν+2+2 n)
[a>0,Reν>−1] ET I 11(3)a
5./integraldisplayu
0xν−1(u−x)μ−1sin(ax)dx=uμ+ν−1
2iB(μ, ν)[1F1(ν;μ+ν;iau)−1F1(ν;μ+ν;−iau)]
[a>0,Reμ>0,Reν>−1,ν/negationslash=0 ] ET II 189(26)
6./integraldisplayu
0xν−1(u−x)μ−1cos(ax)dx=uμ+ν−1
2B(μ, ν)[1F1(ν;μ+ν;iau)+1F1(ν;μ+ν;−iau)]
[a>0,Reμ>0,Reν>0]
ET II 189(32)
7./integraldisplayu
0xμ−1(u−x)μ−1sin(ax)dx=√π/parenleftBigu
a/parenrightBigμ−1/2
sinau
2Γ(μ)Jμ−1/2/parenleftBigau
2/parenrightBig
[Reμ>0] ET II 189(25)
8./integraldisplay∞
uxμ−1(x−u)μ−1sin(ax)dx
=√π
2/parenleftBigu
a/parenrightBigμ−1/2
Γ(μ)/bracketleftBig
cosau
2J1/2−μ/parenleftBigau
2/parenrightBig
−sinau
2Y1/2−μ/parenleftBigau
2/parenrightBig/bracketrightBig
/bracketleftbig
a>0,0<Reμ<1
2/bracketrightbig
ET II 203(20)
9./integraldisplayu
0xμ−1(u−x)μ−1cos(ax)dx=√π/parenleftBigu
a/parenrightBigμ−1
2cosau
2Γ(μ)Jμ−1
2/parenleftBigau
2/parenrightBig
[Reμ>0] ET II 189(31)
10./integraldisplay∞
uxμ−1(x−u)μ−1cos(ax)dx=−√π
2/parenleftBigu
a/parenrightBigμ−1
2Γ(μ)/bracketleftBig
sinau
2J1
2−μ/parenleftBigau
2/parenrightBig
−cosau
2Y1
2−μ/parenleftBigau
2/parenrightBig/bracketrightBig
/bracketleftbig
a>0,0<Reμ<1
2/bracketrightbig
ET II 204(25)
11.3/integraldisplay1
0xν−1(1−x)μ−1sin(ax)dx=−i
2B(μ, ν)[1F1(ν;ν+μ;ia)−1F1(ν;ν+μ;−ia)]
[Reμ>0,Reν>−1,ν/negationslash=0 ]
ET I 68 (5)a, ET I 317(5)
12.3/integraldisplay1
0xν−1(1−x)μ−1cos(ax)dx=1
2B(μ, ν)[1F1(ν;ν+μ;ia)+1F1(ν;ν+μ;−ia)]
[Reμ>0,Reν>0] ET I 11(5)
3.771 Trigonometric functions and powers 441
13./integraldisplay1
0xμ(1−x)μsin(2ax)dx=√π
(2a)μ+1
2Γ(μ+1 )Jμ+1
2(a)sina
[a>0,Reμ>−1] ET I 68(4)
14./integraldisplay1
0xμ(1−x)μcos(2ax)dx=√π
(2a)μ+1
2Γ(μ+1 )Jμ+1
2(a)cosa
[a>0,Reμ>−1] ET I 11(4)
3.769
1./integraldisplay∞
0/bracketleftbig
(β+ix)−ν−(β−ix)−ν/bracketrightbig
sin(ax)dx=−πiaν−1e−aβ
Γ(ν)
[a>0,Reβ>0,Reν>0]
ET I 70(15)
2./integraldisplay∞
0/bracketleftbig
(β+ix)−ν+(β−ix)−ν/bracketrightbig
cos(ax)dx=πaν−1e−aβ
Γ(ν)
[a>0,Reβ>0,Reν>0]
ET I 13(19)
3./integraldisplay∞
0x/bracketleftbig
(β+ix)−ν+(β−ix)−ν/bracketrightbig
sin(ax)dx=−πaν−2(ν−1−aβ)
Γ(ν)e−aβ
[a>0,Reβ>0,Reν>0]
ET I 70(16)
4./integraldisplay∞
0x2n/bracketleftbig
(β−ix)−ν−(β+ix)−ν/bracketrightbig
sin(ax)dx=(−1)ni
Γ(ν)(2n)!πaν−2n−1e−aβLν−2n−1
2n (aβ)
[a>0,Reβ>0,0≤2n<Reν]
ET I 70(17)
5./integraldisplay∞
0x2n/bracketleftbig
(β+ix)−ν+(β−ix)−ν/bracketrightbig
cos(ax)dx=(−1)n
Γ(ν)(2n)!πaν−2n−1e−aβLν−2n−1
2n (aβ)
[a>0,Reβ>0,0≤2n<Reν]
ET I 13(20)
6./integraldisplay∞
0x2n+1/bracketleftBig
(β+ix)−ν+(β−ix)−ν/bracketrightBig
sin(ax)dx=(−1)n+1
Γ(ν)(2n+1 ) !πaν−2n−2e−aβLν−2n−2
2n+1(aβ)
[a>0,Reβ>0,−1≤2n+1<Reν]ET I 70(18)
7./integraldisplay∞
0x2n+1/bracketleftBig
(β+ix)−ν−(β−ix)−ν/bracketrightBig
cos(ax)dx=(−1)n+1
Γ(ν)(2n+1 ) !πaν−2n−2e−aβLν−2n−2
2n+1(aβ)
[a>0,Reβ>0,0≤2n<Reν−1]ET I 13(21)
3.771
1./integraldisplay∞
0/parenleftbig
β2+x2/parenrightbigν−1
2sin(ax)dx=√π
2/parenleftbigg2β
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
[I−ν(aβ)−Lν(aβ)]
/bracketleftbig
a>0,Reβ>0,Reν<1
2,ν/negationslash=−1
2,−3
2,−5
2,.../bracketrightbig
EH II 38a, ET I 68(6)
442 Trigonometric Functions 3.771
2./integraldisplay∞
0/parenleftbig
β2+x2/parenrightbigν−1
2cos(ax)dx=1√π/parenleftbigg2β
a/parenrightbiggν
cos(πν)Γ/parenleftbigg
ν+1
2/parenrightbigg
K−ν(aβ)
/bracketleftbigg
a>0,Reβ>0,Reν<1
2/bracketrightbigg
WA 191(1)a, GW(333)(78)a
3./integraldisplayu
0x2ν−1/parenleftbig
u2−x2/parenrightbigμ−1sin(ax)dx
=a
2u2μ+2ν−1B/parenleftbigg
μ, ν+1
2/parenrightbigg
1F2/parenleftbigg
ν+1
2;3
2,μ+ν+1
2;−a2u2
4/parenrightbigg
/bracketleftbig
Reμ>0,Reν>−1
2/bracketrightbig
ET II 189(29)
4./integraldisplayu
0x2ν−1/parenleftbig
u2−x2/parenrightbigμ−1cos(ax)dx=1
2u2μ+2ν−2B(μ, ν)1F2/parenleftbigg
ν;1
2,μ+ν;−a2u2
4/parenrightbigg
[Reμ>0,Reν>0] ET II 190(35)
5.7/integraldisplay∞
0x/parenleftbig
x2+β2/parenrightbigν−1
2sin(ax)dx=1√πβ/parenleftbigg2β
a/parenrightbiggν
cosνπΓ/parenleftbigg
ν+1
2/parenrightbigg
Kν+1(aβ)
=√πβ/parenleftbigg2β
a/parenrightbiggν1
Γ/parenleftbig1
2−ν/parenrightbigKν+1(aβ)
[a>0,Reβ>0,Reν<0]ET I 69(11)
6./integraldisplayu
0/parenleftbig
u2−x2/parenrightbigν−1
2sin(ax)dx=√π
2/parenleftbigg2u
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Hν(au)
/bracketleftbig
a>0,u > 0,Reν>−1
2/bracketrightbig
ET I 69(7), WA 358(1)a
7./integraldisplay∞
u/parenleftbig
x2−u2/parenrightbigν−1
2sin(ax)dx=√π
2/parenleftbigg2u
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
J−ν(au)
/bracketleftbig
a>0,u > 0,|Reν|<1
2/bracketrightbig
EH II 81(12)a, ET I 69(8), WA 187(3)a
8./integraldisplayu
0/parenleftbig
u2−x2/parenrightbigν−1
2cos(ax)dx=√π
2/parenleftbigg2u
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Jν(au)
/bracketleftbig
a>0,u > 0,Reν>−1
2/bracketrightbig
ET I 11(8)
9./integraldisplay∞
u/parenleftbig
x2−u2/parenrightbigν−1
2cos(ax)dx=−√π
2/parenleftbigg2u
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Y−ν(au)
/bracketleftbig
a>0,u > 0,|Reν|<1
2/bracketrightbig
WA 187(4)a, EH II 82(13)a, ET I 11(9)
10./integraldisplayu
0x/parenleftbig
u2−x2/parenrightbigν−1
2sin(ax)dx=√π
2u/parenleftbigg2u
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Jν+1(au)
/bracketleftbig
a>0,u > 0,Reν>−1
2/bracketrightbig
ET I 69(9)
3.772 Trigonometric functions and powers 443
11./integraldisplay∞
ux/parenleftbig
x2−u2/parenrightbigν−1
2sin(ax)dx=√π
2u/parenleftbigg2u
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Y−ν−1(au)
/bracketleftbig
a>0,u > 0,−1
2<Reν<0/bracketrightbig
ET I 69(10)
12.7/integraldisplayu
0x/parenleftbig
u2−x2/parenrightbigν−1
2cos(ax)dx=−uν+1
aνs(ν−1)ν+1(au)
=1
2/parenleftbigg
ν+1
2/parenrightbigg−1
u2ν+1−√π
2u/parenleftbigg2u
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Hν+1(au)
/bracketleftbig
a>0,u > 0,Reν>−1
2/bracketrightbig
ET I 12(10)
13./integraldisplay∞
ux/parenleftbig
x2−u2/parenrightbigν−1/2cos(ax)dx√πu
2/parenleftbigg2u
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
J−ν−1(au)
/bracketleftbig
a>0,u > 0,0<Reν<1
2/bracketrightbig
ET I 12(11)
3.772
1./integraldisplay∞
0/parenleftbig
x2+2βx/parenrightbigν−1/2sin(ax)dx=√π
2/parenleftbigg2β
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
[J−ν(aβ)cos(aβ)+Y−ν(aβ)sin(aβ)]
/bracketleftbig
a>0,|argβ|<π ,1
2>Reν>−3
2/bracketrightbig
ET I 69(12)
2./integraldisplay∞
0/parenleftbig
x2+2βx/parenrightbigν−1/2cos(ax)dx
=−√π
2/parenleftbigg2β
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
[Y−ν(aβ)cos(aβ)−J−ν(aβ)sin(aβ)]
/bracketleftbig
a>0,|Reν|<1
2/bracketrightbig
ET I 12(13)
3./integraldisplay2u
0/parenleftbig
2ux−x2/parenrightbigν−1/2sin(ax)dx=√π/parenleftbigg2u
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
sin(au)Jν(au)
/bracketleftbig
a>0,u > 0,Reν>−1
2/bracketrightbig
ET I 69(13)a
4./integraldisplay∞
2u/parenleftbig
x2−2ux/parenrightbigν−1/2sin(ax)dx=√π
2/parenleftbigg2β
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
[J−ν(au)cos(au)−Y−ν(au)sin(au)]
/bracketleftbig
a>0,u > 0,|Reν|<1
2/bracketrightbig
ET I 70(14)
5./integraldisplay2u
0/parenleftbig
2ux−x2/parenrightbigν−1/2cos(ax)dx=√π/parenleftbigg2u
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Jν(au)cos(au)
/bracketleftbig
a>0,u > 0,Reν>−1
2/bracketrightbig
ET I 12(4)
6./integraldisplay∞
2u/parenleftbig
x2−2ux/parenrightbigν−1/2cos(ax)dx
=−√π
2/parenleftbigg2u
a/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
[J−ν(au)sin(au)+Y−ν(au)cos(au)]
/bracketleftbig
a>0,u > 0,|Reν|<1
2/bracketrightbig
ET I 12(12)
444 Trigonometric Functions 3.773
3.773
1.8/integraldisplay∞
0x2ν
(x2+β2)μ+1sin(ax)dx
=1
2β2ν−2μaB( 1+ ν,μ−ν)1F2/parenleftbigg
ν+1 ;ν+1−μ,3
2;β2a2
4/parenrightbigg
+√πa2μ−2ν+1
4μ−ν+1Γ(ν−μ)
Γ/parenleftbig
μ−ν+3
2/parenrightbig1F2/parenleftbigg
μ+1 ;μ−ν+3
2,μ−ν+1 ;β2a2
4/parenrightbigg
=√π
2Γ (μ+1 )β2ν−2μ−1G21
13/parenleftBigg
a2β2
4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−ν+
1
2
μ−ν+1
2,1
2,0/parenrightBigg
[a>0,Reβ>0,−1<Reν<Reμ+1 ] ET I 71(28)a, ET II 234(17)
2.8/integraldisplay∞
0x2m+1sin(ax)
(z+x2)n+1dx=(−1)n+m
n!·π
2dn
dzn/parenleftBig
zme−a√z/parenrightBig
[a>0,0≤m≤n,|argz|<π]
ET I 68(39)
3./integraldisplay∞
0x2m+1sin(ax)dx
(β2+x2)n+1
2=(−1)m+1√π
2nβnΓ/parenleftbig
n+1
2/parenrightbigd2m+1
da2m+1[anKn(aβ)]
[a>0,Reβ>0,−1≤m≤n]
ET I 67(37)
4./integraldisplay∞
0x2νcos(ax)dx
(x2+β2)μ+1=1
2β2ν−2μ−1B/parenleftbigg
ν+1
2,μ−ν+1
2/parenrightbigg
1F2/parenleftbigg
ν+1
2;ν−μ+1
2,1
2;β2a2
4/parenrightbigg
+√πa2μ−2ν+1
4μ−ν+1Γ/parenleftbig
ν−μ−1
2/parenrightbig
Γ(μ−ν+1 )1F2/parenleftbigg
μ+1 ;μ−ν+1,μ−ν+3
2;β2a2
4/parenrightbigg
=√π
2Γ (μ+1 )β2ν−2μ−1G21
13/parenleftBigg
a2β2
4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−ν+
1
2
μ−ν+1
2,0,1
2/parenrightBigg
/bracketleftbig
a>0,Reβ>0,−1
2<Reν<Reμ+1/bracketrightbig
ET I 14(29)a, ET II 235(19)
5./integraldisplay∞
0x2mcos(ax)dx
(z+x2)n+1=(−1)m+nπ
2·n!·dn
dzn/parenleftBig
zm−1
2e−a√z/parenrightBig
[a>0,n+1>m≥0,|argz|<π]
ET I 10(28)
6.7/integraldisplay∞
0x2mcos(ax)dx
(β2+x2)n+1
2=(−1)m√π
2nβnΓ/parenleftbig
n+1
2/parenrightbig·d2m
da2m{anKn(aβ)}
/bracketleftbig
a>0,Reβ>0,0≤m<n +1
2/bracketrightbig
ET I 14(28)
3.774
1./integraldisplay∞
0sin(ax)dx√
x2+b2/parenleftbig
x+√
x2+b2/parenrightbigν=π
bνsin(νπ)/bracketleftbigg
sinνπ
2Iν(ab)+i
2Jν(iab)−i
2Jν(−iab)/bracketrightbigg
[a>0,b > 0,Reν>−1]
ET I 70(19)
3.775 Trigonometric functions and powers 445
2./integraldisplay∞
0cos(ax)dx√
x2+b2/parenleftbig
x+√
x2+b2/parenrightbigν=π
bνsin(νπ)/bracketleftbigg1
2Jν(iab)+1
2Jν(−iab)−cosνπ
2Iν(ab)/bracketrightbigg
[a>0,b > 0,Reν>−1]
ET I 12(15)
3./integraldisplay∞
0/parenleftBig
x+/radicalbig
x2+β2/parenrightBigν
/radicalbig
x(x2+β2)sin(ax)dx=/radicalbiggaπ
2βνI1
4−ν
2/parenleftbiggaβ
2/parenrightbigg
K1
4+ν
2/parenleftbiggaβ
2/parenrightbigg
/bracketleftbig
a>0,Reβ>0,Reν<3
2/bracketrightbig
ET I 71(23)
4./integraldisplay∞
0/parenleftBig/radicalbig
x2+β2−x/parenrightBigν
/radicalbig
x(x2+β2)cos(ax)dx=/radicalbiggaπ
2βνI−1
4+ν
2/parenleftbiggaβ
2/parenrightbigg
K−1
4−ν
2/parenleftbiggaβ
2/parenrightbigg
/bracketleftbig
a>0,Reβ>0,Reν>−3
2/bracketrightbig
ET I 12(17)
5./integraldisplay∞
0/parenleftBig
β+/radicalbig
x2+β2/parenrightBigν
xν+1
2/radicalbig
x2+β2sin(ax)dx=1
β/radicalbigg
2
aΓ/parenleftbigg3
4−ν
2/parenrightbigg
W ν
2,1
4(aβ)M−ν
2,1
4(aβ)
/bracketleftbig
a>0,Reβ>0,Reν<3
2/bracketrightbig
ET I 71(27)
6./integraldisplay∞
0/parenleftBig
β+/radicalbig
x2+β2/parenrightBigν
xν+1
2/radicalbig
β2+x2cos(ax)dx=1
β√
2aΓ/parenleftbigg1
4−ν
2/parenrightbigg
W ν
2,−1
4(aβ)M−ν
2,−1
4(aβ)
/bracketleftbig
a>0,Reβ>0,Reν<1
2/bracketrightbig
ET I 12(18)
3.775
1./integraldisplay∞
0/parenleftBig/radicalbig
x2+β2+x/parenrightBigν
−/parenleftBig/radicalbig
x2+β2−x/parenrightBigν
/radicalbig
x2+β2sin(ax)dx=2βνsinνπ
2Kν(aβ)
[a>0,Reβ>0,|Reν|<1]
ET I 70(20)
2./integraldisplay∞
0/parenleftBig/radicalbig
x2+β2+x/parenrightBigν
+/parenleftBig/radicalbig
x2+β2−x/parenrightBigν
/radicalbig
x2+β2cos(ax)dx=2βνcosνπ
2Kν(aβ)
[a>0,Reβ>0,|Reν|<1]
ET I 13(22)
3./integraldisplay∞
u/parenleftbig
x+√
x2−u2/parenrightbigν+/parenleftbig
x−√
x2−u2/parenrightbigν
√
x2−u2sin(ax)dx=πuν/bracketleftBig
Jν(au)cosνπ
2−Yν(au)sinνπ
2/bracketrightBig
[a>0,u > 0,|Reν|<1]
ET I 70(22)
4./integraldisplay∞
u/parenleftbig
x+√
x2−u2/parenrightbigν+/parenleftbig
x−√
x2−u2/parenrightbigν
√
x2−u2cos(ax)dx=−πuν/bracketleftBig
Yν(au)cosνπ
2+Jν(au)sinνπ
2/bracketrightBig
[a>0,u > 0,|Reν|<1]
ET I 13(25)
446 Trigonometric Functions 3.776
5./integraldisplayu
0/parenleftbig
x+i√
u2−x2/parenrightbigν+/parenleftbig
x−i√
u2−x2/parenrightbigν
√
u2−x2sin(ax)dx=π
2uνcosecνπ
2[Jν(au)−J−ν(au)]
[a>0,u > 0] ET I 70(21)
6./integraldisplayu
0/parenleftbig
x+i√
u2−x2/parenrightbigν+/parenleftbig
x−i√
u2−x2/parenrightbigν
√
u2−x2cos(ax)dx=π
2uνsecνπ
2[Jν(au)+J−ν(au)]
[a>0,u > 0,|Reν|<1]
ET I 13(24)
7.6/integraldisplay∞
u/parenleftbig
x+√
x2−u2/parenrightbigν+/parenleftbig
x−√
x2−u2/parenrightbigν
/radicalbig
x(x2−u2)sin(ax)dx
=−/radicalbigg/parenleftBigπ
2/parenrightBig3
auν/bracketleftBig
J1/4+ν/2/parenleftBigau
2/parenrightBig
Y1/4−ν/2/parenleftBigau
2/parenrightBig
+J1/4−ν/2/parenleftBigau
2/parenrightBig
Y1/4+ν/2/parenleftBigau
2/parenrightBig/bracketrightBig
/bracketleftbig
a>0,u > 0,|Reν|<3
2/bracketrightbig
ET I 71(25)
8.6/integraldisplay∞
u/parenleftbig
x+√
x2−u2/parenrightbigν+/parenleftbig
x−√
x2−u2/parenrightbigν
/radicalbig
x(x2−u2)cos(ax)dx
=−/radicalbigg/parenleftBigπ
2/parenrightBig3
auν/bracketleftBig
J−1/4+ν/2/parenleftBigau
2/parenrightBig
Y−1/4−ν/2/parenleftBigau
2/parenrightBig
+J−1/4−ν/2/parenleftBigau
2/parenrightBig
Y−1/4+ν/2/parenleftBigau
2/parenrightBig/bracketrightBig
/bracketleftbig
a>0,u > 0,|Reν|<3
2/bracketrightbig
ET I 13(26)
9./integraldisplay∞
0/parenleftBig
x+β+/radicalbig
x2+2βx/parenrightBigν
+/parenleftBig
x+β−/radicalbig
x2+2βx/parenrightBigν
/radicalbig
x2+2βxsin(ax)dx
=πβν/bracketleftBig
Yν(βa)sin/parenleftBig
βa−νπ
2/parenrightBig
+Jν(βa)cos/parenleftBig
βa−νπ
2/parenrightBig/bracketrightBig
[a>0,|argβ|<π , |Reν|<1]ET I 71(26)
10./integraldisplay∞
0/parenleftBig
x+β+/radicalbig
x2+2βx/parenrightBigν
+/parenleftBig
x+β−/radicalbig
x2+2βx/parenrightBigν
/radicalbig
x2+2βxcos(ax)dx
=πβν/bracketleftBig
Jν(βa)sin/parenleftBig
βa−νπ
2/parenrightBig
−Yν(βa)cos/parenleftBig
βa−νπ
2/parenrightBig/bracketrightBig
[a>0,|argβ|<π , |Reν|<1]ET I 13(23)
11./integraldisplay2u
0/parenleftbig√2u+x+i√2u−x/parenrightbig4ν+/parenleftbig√2u+x−i√2u−x/parenrightbig4ν
√
4u2x−x3cos(ax)dx
=( 4u)2νπ3/2/radicalbigga
2Jν−1/4(au)J−ν−1/4(au)
[a>0,u > 0] ET I 14(27)
3.776
1./integraldisplay∞
0a2(b+x)2+p(p+1 )
(b+x)p+2sin(ax)dx=a
bp[a>0,b > 0,p > 0] BI (170)(1)
2./integraldisplay∞
0a2(b+x)2+p(p+1 )
(b+x)p+2cos(ax)dx=p
bp+1[a>0,b > 0,p > 0] BI (170)(2)
3.784 Rational and trigonometric functions 447
3.78–3.81 Rational functions of xand of trigonometric functions
3.781
1./integraldisplay∞
0/parenleftbiggsinx
x−1
1+x/parenrightbiggdx
x=1−C (cf.3.784 4a n d3.781 2) BI (173)(7)
2./integraldisplay∞
0/parenleftbigg
cosx−1
1+x/parenrightbiggdx
x=−C BI (173)(8)
3.782
1./integraldisplayu
01−cosx
xdx−/integraldisplay∞
ucosx
xdx=C+l nu [u>0] GW (333)(31)
2./integraldisplay∞
01−cosax
x2dx=aπ
2[a≥0] BI (158)(1)
3./integraldisplay∞
−∞1−cosax
x(x−b)dx=πsinab
b[a>0,breal,b/negationslash=0 ] ET II 253(48)
3.783
1./integraldisplay∞
0/bracketleftbiggcosx−1
x2+1
2(1 + x)/bracketrightbiggdx
x=1
2C−3
4BI (173)(19)
2./integraldisplay∞
0/parenleftbigg
cosx−1
1+x2/parenrightbiggdx
x=−C EH I 17, BI(273)(21)
3.784
1./integraldisplay∞
0cosax−cosbx
xdx=l nb
a[a>0,b > 0] FI II 635, GW(333)(20)
2./integraldisplay∞
0asinbx−bsinax
x2dx=ablna
b[a>0,b > 0] FI II 647
3./integraldisplay∞
0cosax−cosbx
x2dx=(b−a)π
2[a≥0,b≥0] BI(158)(12), FI II 645
4./integraldisplay∞
0sinx−xcosx
x2dx=1 BI (158)(3)
5./integraldisplay∞
0cosax−cosbx
x(x+β)dx=1
β/bracketleftbigg
ci(aβ)cosaβ+s i (aβ)sinaβ−ci(bβ)cosbβ−si(bβ)sinbβ+l nb
a/bracketrightbigg
[a>0,b > 0,|argβ|<π]ET II 221(49)
6./integraldisplay∞
0cosax+xsinax
1+x2dx=πe−a[a>0] GW (333)(73)
7./integraldisplay∞
0sinax−axcosax
x3dx=π
4a2signa LI (158)(5)
8./integraldisplay∞
0cosax−cosbx
x2(x2+β2)dx=π/bracketleftbig
(b−a)β+e−bβ−e−aβ/bracketrightbig
2β3
[a>0,b > 0,|argβ|<π]
BI(173)(20)a, ET II 222(59)
448 Trigonometric Functions 3.785
9.10/integraldisplay∞
0cosmx
1+a2Tn(x)=π
2n√
1+a2n/summationdisplay
k=1e−msinusinhφ(cosβsinucoshφ+s i nβcosusinhφ)
[u=( 2k−1)π/(2n),φ=a r c s i n h ( 1 /a),β=mcosucoshφ,0<|a|<1]
3.785/integraldisplay∞
01
xn/summationdisplay
k=1akcosbkxdx=−n/summationdisplay
k=1aklnbk/bracketleftBigg
bk>0,n/summationdisplay
k=1ak=0/bracketrightBigg
FI II 649
3.786
1./integraldisplay∞
0(1−cosax)sinbx
x2dx=b
2lnb2−a2
b2+a
2lna+b
a−b
[a>0,b > 0] ET I 81(29)
2.11/integraldisplay∞
0(1−cosax)cosbx
xdx=l n/radicalbig
|a2−b2|
b[a>0,b > 0,a/negationslash=b] FI II 647
3.11/integraldisplay∞
0(1−cosax)cosbx
x2dx=π
2(a−b)[ a<b≤0]
=0 [ 0 <a≤b]
ET I 20(16)
3.787
1./integraldisplay∞
0(cosa−cosnax)sinmx
xdx=π
2(cosa−1) [ m>n a> 0]
=π
2cosa [na > m ]
BI(155)(7)
2./integraldisplay∞
0sin2ax−sin2bx
xdx=1
2lna
b[a>0,b > 0] GW (333)(20b)
3./integraldisplay∞
0x3−sin3x
x5dx=13
32π BI (158)(6)
4./integraldisplay∞
0/parenleftbig
3−4s in2ax/parenrightbig
sin2ax
xdx=1
2ln 2 [ areal,a/negationslash=0 ] HBI (155)(6)
3.788/integraldisplayπ/2
0/parenleftbigg1
x−cotx/parenrightbigg
dx=l nπ
2GW (333)(61)a
3.789/integraldisplayπ/2
04x2cosx+(π−x)x
sinxdx=π2ln 2 LI (206)(10)
3.791
1./integraldisplayπ/2
0xdx
1+s i n x=l n2 GW (333)(55a)
2./integraldisplayπ
0xcosx
1+s i n xdx=πln 2−4G GW (333)(55c)
3./integraldisplayπ/2
0xcosx
1+s i n xdx=πln2−2G GW (333)(55b)
3.792 Rational and trigonometric functions 449
4./integraldisplayπ
0/parenleftbigπ
2−x/parenrightbig
cosx
1−sinxdx=2/integraldisplayπ/2
0/parenleftbigπ
2−x/parenrightbig
cosx
1−sinxdx=πln2 + 4 G=5.8414484669 ...
BI(207)(3), GW(333)(56c)
5./integraldisplayπ/2
0x2dx
1−cosx=−π2
4+πln2 + 4 G=3.3740473667 ... BI (207)(3)
6./integraldisplayπ
0x2dx
1−cosx=4πln2 BI (219)(1)
7./integraldisplayπ/2
0xp+1dx
1−cosx=−/parenleftBigπ
2/parenrightBigp+1
+/parenleftBigπ
2/parenrightBigp
(p+1 )/braceleftBigg
2
p−∞/summationdisplay
k=11
42k−1(p+2k)ζ(2k)/bracerightBigg
[p>0] LI (207)(4)
8./integraldisplayπ/2
0xdx
1 + cos x=π
2−ln2 GW (333)(55a)
9./integraldisplayπ/2
0xsinxdx
1−cosx=π
2ln2 + 2 G GW (333)(56a)
10./integraldisplayπ
0xsinxdx
1−cosx=2πln 2 GW (333)(56b)
11./integraldisplayπ
0x−sinx
1−cosxdx=π
2+/integraldisplayπ/2
0x−sinx
1−cosxdx=2 GW (333)(57a)
12./integraldisplayπ/2
0xsinx
1 + cos xdx=−π
2ln 2 + 2 G GW (333)(55b)
3.792
1./integraldisplayπ
−πdx
1−2acosx+a2=2π
1−a2/bracketleftbig
a2<1/bracketrightbig
FI II 485
2./integraldisplayπ/2
0xcosxdx
1+2asinx+a2=π
2aln(1 + a)−∞/summationdisplay
k=0(−1)ka2k
(2k+1 )2
/bracketleftbig
a2<1/bracketrightbig
LI (241)(2)
3./integraldisplayπ
0xsinxdx
1−2acosx+a2=π
aln(1 + a)/bracketleftbig
a2<1,a/negationslash=0/bracketrightbig
=π
aln/parenleftbigg
1+1
a/parenrightbigg/bracketleftbig
a2<1/bracketrightbig
BI (221)(2)
4./integraldisplay2π
0xsinxdx
1−2acosx+a2=2π
aln(1−a)/bracketleftbig
a2<1,a/negationslash=0/bracketrightbig
=2π
aln/parenleftbigg
1−1
a/parenrightbigg/bracketleftbig
a2>1/bracketrightbig
BI (223)(4)
5./integraldisplay2π
0xsinnxdx
1−2acosx+a2=2π
1−a2/bracketleftBigg
/parenleftbig
a−n−an/parenrightbig
ln(1−a)+n−1/summationdisplay
k=1a−k−ak
n−k/bracketrightBigg
/bracketleftbig
a2<1,a/negationslash=0/bracketrightbig
BI (223)(5)
450 Trigonometric Functions 3.792
6./integraldisplay∞
0sinx
1−2acosx+a2·dx
x=π
4a/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+a
1−a/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1/bracketrightbigg
[areal,a/negationslash=0,a/negationslash=1 ]
GW (333)(62b)
7.8/integraldisplay∞
0sinbx
1−2acosx+a2·dx
x=π
21+a−2a[b]+1
(1−a2)(1−a)[b/negationslash=0,1,2,...]
=π
21+a−ab−ab+1
(1−a2)( 1−a)[b=1,2,...]; [0 <a< 1]
ET I 81(26)
8./integraldisplay∞
0sinxcosbx
1−2acosx+a2·dx
x=π
2(1−a)a[b][b/negationslash=0,1,2,...]
=π
2(1−a)ab+π
4ab−1[b=1,2,3,...];
[0<a< 1,b > 0];(for b=0 ,s e e 3.792 6)ET I 19(5)
9./integraldisplay∞
0(1−acosx)s i nbx
1−2acosx+a2·dx
x=π
2·1−a[b]+1
1−a[b/negationslash=1,2,3,...]
=π
2·1−ab
1−a+πab
4[b=1,2,3,...]
[0<a< 1,b > 0] ET I 82(33)
10.3/integraldisplay∞
01
1−2acosbx+a2dx
β2+x2=π
2β(1−a2)1+ae−bβ
1−ae−bβ
/bracketleftbig
a2<1,b≥0/bracketrightbig
BI (192)(1)
11./integraldisplay∞
01
1−2acosbx+a2dx
β2−x2=aπ
β(1−a2)sinbβ
1−2acosbβ+a2
/bracketleftbig
a2<1,b > 0/bracketrightbig
BI (193)(1)
12./integraldisplay∞
0sinbcx
1−2acosbx+a2xdx
β2+x2=π
2e−βbc−ac
(1−ae−bβ)( 1−aebβ)
/bracketleftbig
a2<1,b > 0,c > 0/bracketrightbig
BI (192)(8)
13./integraldisplay∞
0sinbx
1−2acosbx+a2xdx
β2+x2=π
21
ebβ−a/bracketleftbig
a2<1,b > 0/bracketrightbig
=π
2a1
aebβ−1/bracketleftbig
a2>1,b > 0/bracketrightbig
BI (192)(2)
14./integraldisplay∞
0sinbcx
1−2acosbx+a2xdx
β2−x2=π
2ac−cosβbc
1−2acosβb+a2
/bracketleftbig
a2<1,b > 0,c > 0/bracketrightbig
BI (193)(5)
15./integraldisplay∞
0cosbcx
1−2acosbx+a2dx
β2−x2=π
2β(1−a2)/parenleftbig
1−a2/parenrightbig
sinβbc+2ac+1sinβb
1−2acosβb+a2
/bracketleftbig
a2<1,b > 0,c > 0/bracketrightbig
BI (193)(9)
16./integraldisplay∞
01−acosbx
1−2acosbx+a2dx
1+x2=π
2eb
eb−a/bracketleftbig
a2<1,b > 0/bracketrightbig
FI II 719
3.794 Rational and trigonometric functions 451
17./integraldisplay∞
0cosbx
1−2acosx+a2·dx
x2+β2=π/parenleftbig
eβ−βb+aeβb/parenrightbig
2β(1−a2)(eβ−a)
[0≤b<1,|a|<1,Reβ>0]
ET I 21(21)
18./integraldisplay∞
0sinbxsinx
1−2acosx+a2·dx
x2+β2
=π
2βsinhbβ
eβ−a[0≤b<1]
=π
4β(aeβ−1)/bracketleftBig
ameβ(m+1−b)−e(1−b)β/bracketrightBig
−π
4β(ae−β−1)/bracketleftBig
ame−(m+1−b)β−e−(1−b)β/bracketrightBig
[m≤b≤m+1 ]
[0<a< 1,Reβ>0] ET I 81(27)
19./integraldisplay∞
0(cosx−a)cosbx
1−2acosx+a2·dx
x2+β2=πcoshβb
2β(eβ−a)[0≤b<1,|a|<1,Reβ>0]
ET I 21(23)
20./integraldisplay∞
0sinx
(1−2acos2x+a2)n+1dx
x=/integraldisplay∞
0tanx
(1−2acos2x+a2)n+1dx
x
=/integraldisplay∞
0tanx
(1−2acos4x+a2)n+1dx
x=π
2( 1−a2)2n+1n/summationdisplay
k=0/parenleftBign
k/parenrightBig2
a2k
BI (187)(14)
3.793
1.3/integraldisplay2π
0sinnx−asin[(n+1 )x]
1−2acosx+a2xdx=−2πan/bracketleftBigg
ln(1−a)+n/summationdisplay
k=11
kak/bracketrightBigg
[|a|<1] BI (223)(9)
2./integraldisplay2π
0cosnx−acos[(n+1 )x]
1−2acosx+a2xdx=2πan/bracketleftbig
a2<1/bracketrightbig
BI (223)(13)
3.794
1.3/integraldisplayπ
0xdx
1+a2+2acosx=π2
2( 1−a2)+4
(1−a2)∞/summationdisplay
k=0a2k+1
(2k+1 )2
/bracketleftbig
a2<1/bracketrightbig
2./integraldisplay2π
0xsinnx
1±acosxdx=2π√
1−a2⎡
⎣(∓1)n/parenleftbig
1+√
1−a2/parenrightbign−/parenleftbig
1−√
1−a2/parenrightbign
an
×ln2√1±a√1+a+√1−a+n−1/summationdisplay
k=0(∓1)k
n−k/parenleftbig
1+√
1−a2/parenrightbigk−/parenleftbig
1−√
1−a2/parenrightbigk
ak⎤
⎦
/bracketleftbig
a2<1/bracketrightbig
BI (223)(2)
3.3/integraldisplay2π
0xcosnx
1±acosxdx=2π2
√
1−a2/parenleftBigg
1−√
1−a2
∓a/parenrightBiggn/bracketleftbig
a2<1/bracketrightbig
BI (223)(3)
452 Trigonometric Functions 3.795
4./integraldisplayπ
0xsinxdx
a+bcosx=π
blna+√
a2−b2
2(a−b)[a>|b|>0] GW (333)(53a)
5./integraldisplay2π
0xsinxdx
a+bcosx=2π
blna+√
a2−b2
2(a+b)[a>|b|>0] GW (333)(53b)
6./integraldisplay∞
0sinx
a±bcos2x·dx
x=π
2√
a2−b2/bracketleftbig
a2>b2/bracketrightbig
=0/bracketleftbig
a2<b2/bracketrightbig
BI (181)(1)
3.795/integraldisplay∞
−∞/parenleftbig
b2+c2+x2/parenrightbig
xsinax−/parenleftbig
b2−c2−x2/parenrightbig
csinhac
[x2+(b−c)2][x2+(b+c)2](cosax+c o s h ac)dx=π [c>b> 0]
=2π
eab+1[b>c> 0]
[a>0] BI (202)(18)
3.796
1./integraldisplayπ/2
0cosx±sinx
cosx∓sinxxdx=∓π
4ln 2−G BI (207)(8, 9)
2./integraldisplayπ/4
0cosx−sinx
cosx+s i nxxdx=π
4ln2−1
2G BI (204)(23)
3.797
1./integraldisplayπ/4
0/parenleftBigπ
4−xtanx/parenrightBig
tanxdx=1
2ln2 +π2
32−π
4+π
8ln 2 BI (204)(8)
2./integraldisplayπ/4
0/parenleftbigπ
4−x/parenrightbig
tanxdx
cos2x=−π
8ln2 +1
2G BI (204)(19)
3./integraldisplayπ/4
0π
4−xtanx
cos2xdx=π
8ln2 +1
2G BI (204)(20)
3.798
1.8/integraldisplay∞
0tanx
a+bcos2x·dx
x=π
2√
a2−b2[0<b<a ]
=0 [ 0 <a<b ]
BI (181)(2)
2.8/integraldisplay∞
0tanx
a+bcos4x·dx
x=π
2√
a2−b2[0<b<a ]
=0 [ 0 <a<b ]
BI (181)(3)
3.799
1./integraldisplayπ/2
0xdx
(sinx+acosx)2=a
1+a2π
2−lna
1+a2[a>0] BI (208)(5)
3.812 Rational and trigonometric functions 453
2./integraldisplayπ/4
0xdx
(cosx+asinx)2=1
1+a2ln1+a√
2+π
4·1−a
(1 +a)(1+ a2)
[a>0] BI (204)(24)
3./integraldisplayπ
0acosx+b
(a+bcosx)2x2dx=2π
bln2(a−b)
a+√
a2−b2[a>|b|>0] GW (333)(58a)
3.811
1./integraldisplayπ
0sinx
1−cost1cosx·xdx
1−cost2cosx=πcosect1+t2
2cosect1−t2
2ln1+t a nt1
2
1+t a nt2
2
(cf.3.794 4) BI (222)(5)
2./integraldisplayπ/2
0xdx
(cosx±sinx)s i nx=π
4ln2 +G BI (208))(16, 17)
3./integraldisplayπ/4
0xdx
(cosx+s i nx)s i nx=−π
8ln 2 +G BI (204)(29)
4./integraldisplayπ/4
0xdx
(cosx+s i nx)c o sx=π
8ln 2 BI (204)(28)
5./integraldisplayπ/4
0sinx
sinx+c o s xxdx
cos2x=−π
8ln2 +π
4−1
2ln 2 BI (204)(30)
3.812
1./integraldisplayπ
0xsinxdx
a+bcos2x=π√
abarctan/radicalbigg
b
a[a>0,b > 0]
=π
2√
−abln√a+√
−b√a−√
−b[a>−b>0]
GW (333)(60a)
2./integraldisplayπ/2
0xsin 2xdx
1+acos2x=π
aln1+√1+a
2[a>−1,a/negationslash=0 ] BI (207)(10)
3./integraldisplayπ/2
0xsin 2xdx
1+asin2x=π
aln2/parenleftbig
1+a−√1+a/parenrightbig
2[a>−1,a/negationslash=0 ] BI (207)(2)
4.11/integraldisplayπ
0xdx
a2−cos2x=π2
2a√
a2−1/bracketleftbig
a2>1/bracketrightbig
=0/bracketleftbig
principal value for 0 <a2<1/bracketrightbig
= divergent [ a=0 ]
BI (219)(10)
5.7/integraldisplayπ
0xsinxdx
a2−cos2x=π
2aln/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+a
1−a/vextendsingle/vextendsingle/vextendsingle/vextendsingle[0<a< 1] divergent if a=0
BI (219)(13)
454 Trigonometric Functions 3.813
6.11/integraldisplayπ
0xsin 2xdx
a2−cos2x=πln/braceleftbig
4/parenleftbig
1−a2/parenrightbig/bracerightbig /bracketleftbig
principal value for 0 ≤a2<1/bracketrightbig
=2πln/bracketleftBig
2/parenleftBig
1−a2+a/radicalbig
a2−1/parenrightBig/bracketrightBig/bracketleftbig
a2>1/bracketrightbig
= divergent [ |a|=1 ]
BI (219)(19)
7./integraldisplayπ/2
0xsinxdx
cos2t−sin2x=−2c os e c t∞/summationdisplay
k=0sin(2k+1 )t
(2k+1 )2BI (207)(1)
8./integraldisplayπ
0xsinxdx
1−cos2tsin2x=π(π−2t)cosec 2 t BI (219)(12)
9./integraldisplayπ
0xcosxdx
cos2t−cos2x= 4 cosec t∞/summationdisplay
k=0sin(2k+1 )t
(2k+1 )2BI (219)(17)
10./integraldisplayπ
0xsinxdx
tan2t+c o s2x=π
2(π−2t)cott BI (219)(14)
11./integraldisplay∞
0x(acosx+b)sinxdx
cot2t+c o s2x=2aπlncost
2+πbttant BI (219)(18)
12.∗/integraldisplayπ
0xsinxcosx
a−sin2xdx=−πln2 + ln/bracketleftBigg
1+/radicalbigg
a−1
a/bracketrightBigg
[a>1]
13.∗/integraldisplayπ/2
0ln/parenleftbig
a−sin2x/parenrightbig
dx=−πln2 + iπlnarccos√a [0<a< 1]
14.∗PV/integraldisplayπ/2
0ln/parenleftbig/vextendsingle/vextendsinglea−sin2x/vextendsingle/vextendsingle/parenrightbig
dx=−πln 2 [0 <a< 1]
15.∗PV/integraldisplayπ/2
0ln/parenleftbig/vextendsingle/vextendsinglea−cos2x/vextendsingle/vextendsingle/parenrightbig
dx=−πln2 [0 <a< 1]
3.813
1./integraldisplayπ
0xdx
a2cos2x+b2sin2x=1
4/integraldisplay2π
0xdx
a2cos2x+b2sin2x=π2
2ab
[a>0,b > 0] GW (333)(36)
2./integraldisplay∞
01
β2sin2ax+γ2cos2ax·dx
x2+δ2=πsinh(2 aδ)
4δ/parenleftbig
β2sinh2(aδ)−γ2cosh2(aδ)/parenrightbig/bracketleftbiggβ
γ−γ
β−2
sinh(2 aδ)/bracketrightbigg
/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleargβ
γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle<π , Reδ>0,a > 0/bracketrightbigg
GW(333)(81), ET II 222(63)
3./integraldisplay∞
0sinxdx
x/parenleftbig
a2sin2x+b2cos2x/parenrightbig=π
2ab[ab >0] BI (181)(8)
4./integraldisplay∞
0sin2xdx
x/parenleftbig
a2cos2x+b2sin2x/parenrightbig=π
2b(a+b)[a>0,b > 0] BI (181)(11)
3.814 Rational and trigonometric functions 455
5./integraldisplayπ/2
0xsin 2xdx
a2cos2x+b2sin2x=π
a2−b2lna+b
2b[a>0,b > 0,a/negationslash=b]GW (333)(52a)
6./integraldisplayπ
0xsin 2xdx
a2cos2x+b2sin2x=2π
a2−b2lna+b
2a[a>0,b > 0,a/negationslash=b]GW (333)(52b)
7./integraldisplay∞
0sin 2x
a2cos2x+b2sin2x·dx
x=π
a(a+b)[a>0,b > 0] BI (182)(3)
8./integraldisplay∞
0sin 2ax
β2sin2ax+γ2cos2ax·xdx
x2+δ2=π
2/parenleftbig
β2sinh2(aδ)−γ2cosh2(aδ)/parenrightbig/bracketleftbiggβ−γ
β+γ−e−2aδ/bracketrightbigg
/bracketleftbigg
a>0,/vextendsingle/vextendsingle/vextendsingle/vextendsingleargβ
γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle<π , Reδ>0/bracketrightbigg
ET II 222(64), GW(333)(80)
9./integraldisplay∞
0(1−cosx)s i nx
a2cos2x+b2sin2x·dx
x=π
2b(a+b)[a>0,b > 0] BI (182)(7)a
10./integraldisplay∞
0sinxcos2x
a2cos2x+b2sin2x·dx
x=π
2a(a+b)[a>0,b > 0] BI (182)(4)
11./integraldisplay∞
0sin3x
a2cos2x+b2sin2x·dx
x=π
2b·2
a+b[a>0,b > 0] BI (182)(1)
3.814
1./integraldisplayπ/2
0(1−xcotx)dx
sin2x=π
4BI (206)(9)
2./integraldisplayπ/4
0xtanxdx
(sinx+c o s x)c o sx=−π
8ln2 +π
4−1
2ln2 BI (204)(30)
3./integraldisplay∞
0tanx
a2cos2x+b2sin2xdx
x=π
2ab[a>0,b > 0] BI (181)(9)
4./integraldisplayπ/2
0xcotxdx
a2cos2x+b2sin2x=π
2a2lna+b
b[a>0,b > 0] LI (208)(20)
5./integraldisplayπ/2
0/parenleftbigπ
2−x/parenrightbig
tanxdx
a2cos2x+b2sin2x=1
2/integraldisplayπ
0/parenleftbigπ
2−x/parenrightbig
tanxdx
a2cos2x+b2sin2x
=π
2b2lna+b
a
[a>0,b > 0] GW (333)(59)
6./integraldisplay∞
0sin2xtanx
a2cos2x+b2sin2x·dx
x=π
2b(a+b)[a>0,b > 0] BI (182)(6)
7./integraldisplay∞
0tanx
a2cos22x+b2sin22x·dx
x=π
2ab[a>0,b > 0] BI (181)(10)a
8./integraldisplay∞
0sin22xtanx
a2cos22x+b2sin22x·dx
x=π
2b·1
a+b[a>0,b > 0] BI (182)(2)a
9./integraldisplay∞
0cos22xtanx
a2cos22x+b2sin22x·dx
x=π
2a·1
a+b[a>0,b > 0] BI (182)(5)a
456 Trigonometric Functions 3.815
10./integraldisplay∞
0sin2xcosx
a2cos22x+b2sin22x·dx
xcos 4x=−π
8ba
a2+b2[a>0,b > 0] BI (186)(12)a
11./integraldisplay∞
0sinx
a2cos2x+b2sin2x·dx
xcos 2x=π
2ab·b2−a2
b2+a2[a>0,b > 0] BI (186)(4)a
12./integraldisplay∞
0sinxcosx
a2cos2x+b2sin2x·dx
xcos 2x=π
2a·b
a2+b2[a>0,b > 0] BI (186)(7)a
13./integraldisplay∞
0sinxcos2x
a2cos2x+b2sin2x·dx
xcos 2x=π
2ab·b2
a2+b2[a>0,b > 0] BI (186)(8)a
14./integraldisplay∞
0sin3x
a2cos2x+b2sin2x·dx
xcos 2x=−π
2b·a
a2+b2[a>0,b > 0] BI (186)(10)
15./integraldisplay∞
01−cosx
a2cos2x+b2sin2x·dx
xsinx=π
2ab[a>0,b > 0] BI (186)(3)a
3.815
1./integraldisplayπ/2
0xsin 2xdx/parenleftbig
1+asin2x/parenrightbig/parenleftbig
1+bsin2x/parenrightbig=π
a−bln/braceleftbigg1+√
1+b
1+√1+a·√1+a√
1+b/bracerightbigg
[a>0,b > 0] (cf. 3.812 3)
BI (208)(22)
2./integraldisplayπ/2
0xsin 2xdx/parenleftbig
1+asin2x/parenrightbig
(1 +bcos2x)=π
a+ab+bln/parenleftbig
1+√1+n/parenrightbig√1+a
1+√1+a
[a>0,b > 0] (cf. 3.812 2a n d3 )
BI (208)(24)
3./integraldisplayπ/2
0xsin 2xdx
(1 +acos2x)(1+ bcos2x)=π
a−bln1+√1+a
1+√
1+b
[a>0,b > 0] (cf. 3.812 2)
BI (208)(23)
4./integraldisplayπ/2
0xsin 2xdx/parenleftbig
1−sin2t1cos2x/parenrightbig/parenleftbig
1−sin2t2cos2x/parenrightbig=2π
cos2t1−cos2t2lncost1
2
cost2
2
[−π<t 1<π , −π<t 2<π]
BI (208)(21)
3.816
1./integraldisplayπ
0x2sin 2x
(a2−cos2x)2dx=π2√
a2−1−a
a(a2−1)[a>1] LI (220)(9)
2.7/integraldisplayπ
0/parenleftbig
a2−1−sin2x/parenrightbig
cosx
(a2−cos2x)2x2dx=π
2ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−a
1+a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbig
a
2>1/bracketrightbig
(cf.3.812 5)BI (220)(12)
3.11/integraldisplayπ
0acos2x−sin2x
/parenleftbig
a+s i n2x/parenrightbig2x2dx=−2πln/bracketleftbig
2/parenleftbig
−a+√a√
a+1/parenrightbig/bracketrightbig
/bracketleftBig
a<−1a n d a>0. When a>0, can write√a√
a+1a s/radicalbig
a(a+1 ) ./bracketrightBig
LI (220)(10)
3.818 Rational and trigonometric functions 457
4.11/integraldisplayπ
0acos2x+s i n2x
/parenleftbig
a−sin2x/parenrightbig2x2dx=2πln/bracketleftbig
2/parenleftbig
a−√a√
a+1/parenrightbig/bracketrightbig
/bracketleftBig
a<0a n d a>1. When a>1, can write√a√
a+1a s/radicalbig
a(a+1 ) ./bracketrightBig
(cf.3.812 6)
LI (220)(11)
3.817
1./integraldisplay∞
0sinx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig2·dx
x=π
4·a2+b2
a3b3[ab >0] BI (181)(12)
2./integraldisplay∞
0sinxcosx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig2·dx
x=π
4a3b[ab >0] BI (182)(8)
3./integraldisplay∞
0sin3x
/parenleftbig
a2cos2x+b2sin2x/parenrightbig2·dx
x=π
4ab3[ab >0] BI (181)(15)
4./integraldisplay∞
0sinxcos2x
/parenleftbig
a2cos2x+b2sin2x/parenrightbig2·dx
x=π
4a3b[ab >0] BI (182)(9)
5./integraldisplay∞
0tanx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig2·dx
x=π
4·a2+b2
a3b3[ab >0] BI (181)(13)
6./integraldisplay∞
0tanx
/parenleftbig
a2cos22x+b2sin22x/parenrightbig2·dx
x=π
4a2+b2
a3b3[ab >0] BI (181)(14)
7./integraldisplay∞
0sin2xtanx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig2·dx
x=π
4ab3[ab >0] BI (182)(11)
8./integraldisplay∞
0tanxcos22x
/parenleftbig
a2cos22x+b2sin22x/parenrightbig2·dx
x=π
4a3b[ab >0] BI (182)(10)
3.818
1./integraldisplay∞
0sinx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig3·dx
x=π
16·3a4+2a2b2+3b4
a5b5
[ab >0] BI (181)(16)
2./integraldisplay∞
0sinxcosx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig3·dx
x=π
16·a2+3b2
a5b3[ab >0] BI (182)(13)
3./integraldisplay∞
0sinxcos2x
/parenleftbig
a2cos2x+b2sin2x/parenrightbig3·dx
x=π
16·a2+3b2
a5b3[ab >0] BI (182)(14)
4./integraldisplay∞
0sin3x
/parenleftbig
a2cos2x+b2sin2x/parenrightbig3·dx
x=π
16·3a2+b2
a3b5[ab >0] LI (181)(19)
5./integraldisplay∞
0sin3xcosx
/parenleftbig
a2cos22x+b2sin22x/parenrightbig3·dx
x=π
64·3a2+b2
a3b5[ab >0] BI (182)(17)
458 Trigonometric Functions 3.819
6./integraldisplay∞
0tanx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig3·dx
x=π
163a4+2a2b2+3b4
a5b5
[ab >0] BI (181)(17)
7./integraldisplay∞
0sin2xtanx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig3·dx
x=π
16·3a2+b2
a3b5[ab >0] BI (182)(16)
8./integraldisplay∞
0tanx
/parenleftbig
a2cos22x+b2sin22x/parenrightbig3·dx
x=π
16·3a4+2a2b2+3b4
a5b5
[ab >0] BI (181)(18)
9./integraldisplay∞
0tanxcos22x
/parenleftbig
a2cos22x+b2sin22x/parenrightbig3·dx
x=π
16·a2+3b2
a5b3[ab >0] BI (182)(15)
3.819
1./integraldisplay∞
0sinx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig4·dx
x=π
32·5a6+3a4b2+3a2b4+5b6
a7b7
[ab >0] BI (181)(20)
2./integraldisplay∞
0sinxcosx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig4·dx
x=π
32·a4+2a2b2+5b4
a7b5
[ab >0] BI (182)(18)
3./integraldisplay∞
0sinxcos2x
/parenleftbig
a2cos2x+b2sin2x/parenrightbig4·dx
x=π
32·a4+2a2b2+5b4
a7b5
[ab >0] BI (182)(19)
4./integraldisplay∞
0sin3x
/parenleftbig
a2cos2x+b2sin2x/parenrightbig4·dx
x=π
32·5a4+a2b2+b4
a5b7
[ab >0] BI (181)(23)
5./integraldisplay∞
0sin3xcosx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig4·dx
x=π
32·a2+b2
a5b5[ab >0] BI (182)(26)
6./integraldisplay∞
0sinxcos3x
/parenleftbig
a2cos2x+b2sin2x/parenrightbig4·dx
x=π
32·a2+5b2
a7b3[ab >0] BI (182)(23)
7./integraldisplay∞
0sin3xcos2x
/parenleftbig
a2cos2x+b2sin2x/parenrightbig4·dx
x=π
32·a2+b2
a5b5[ab >0] BI (182)(27)
8./integraldisplay∞
0sinxcos4x
/parenleftbig
a2cos2x+b2sin2x/parenrightbig4·dx
x=π
32·a2+5b2
a7b3[ab >0] BI (182)(24)
9./integraldisplay∞
0sin5x
/parenleftbig
a2cos2x+b2sin2x/parenrightbig4·dx
x=π
32·5a2+b2
a3b7[ab >0] BI (181)(24)
3.821 Powers and trigonometric functions 459
10./integraldisplay∞
0sin3xcosx
/parenleftbig
a2cos22x+b2sin22x/parenrightbig4·dx
x=π
128·5a4+2a2b2+b4
a5b7
[ab >0] BI (182)(22)
11./integraldisplay∞
0sin5xcos3x
/parenleftbig
a2cos22x+b2sin22x/parenrightbig4·dx
x=π
512·5a2+b2
a3b7[ab >0] BI (182)(30)
12./integraldisplay∞
0sin2xtanx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig4·dx
x=π
32·5a4+2a2b2+b4
a5b7
[ab >0] BI (182)(21)
13./integraldisplay∞
0sin4xtanx
/parenleftbig
a2cos2x+b2sin2x/parenrightbig4·dx
x=π
32·5a2+b2
a3b7[ab >0] BI (182)(29)
14./integraldisplay∞
0cos22xtanx
/parenleftbig
a2cos22x+b2sin22x/parenrightbig4·dx
x=π
32·a4+2a2b2+5b4
a7b5
[ab >0] BI (182)(29)
15./integraldisplay∞
0sin34xtanx
/parenleftbig
a2cos22x+b2sin22x/parenrightbig4·dx
x=π
8·a2+b2
a5b5[ab >0] BI (182)(28)
16./integraldisplay∞
0cos42xtanx
/parenleftbig
a2cos22x+b2sin22x/parenrightbig4·dx
x=π
32·a2+5b2
a7b3[ab >0] BI (182)(25)
3.82–3.83 Powers of trigonometric functions combined with other powers
3.821
1./integraldisplayπ
0xsinpxdx=π2
2p+1Γ(p+1 )
/bracketleftBig
Γ/parenleftBigp
2+1/parenrightBig/bracketrightBig2[p>−1] BI(218)(7), LO V 121(71)
2./integraldisplayrπ
0xsinnxdx=π2
2·(2m−1)!!
(2m)!!r2[n=2m]
=(−1)r+1π(2m)!!
(2m+1 ) ! !r [n=2m+1 ]
[ris a natural number] GW (333)(8c)
3.11/integraldisplayπ/2
0xcosnxdx=π2
8(n−1)!!
(n)!!−1
2n−2m−1/summationdisplay
k=0,m−kodd/parenleftBign
k/parenrightBig1
(n−2k)2[n=2m]
=π
2(n−1)!!
(n)!!−1
2n−1m−1/summationdisplay
k=0/parenleftBign
k/parenrightBig1
(n−2k)2[n=2m−1]
GW (333)(9b)
4./integraldisplayπ
0xcos2mxdx=π2
2(2m−1)!!
(2m)!!BI (218)(10)
5./integraldisplaysπ
rπxcos2mxdx=π2
2/parenleftbig
s2−r2/parenrightbig(2m−1)!!
(2m)!!BI (226)(3)
460 Trigonometric Functions 3.822
6./integraldisplay∞
0sinpx
xdx=√π
2·Γ/parenleftbigp
2/parenrightbig
Γ/parenleftbigp+1
2/parenrightbig=2p−2B/parenleftBigp
2,p
2/parenrightBig
[pis a fraction with odd numerator and denominator] L OV2 7 8 ,F II I8 0 8
7./integraldisplay∞
0sin2n+1x
xdx=(2n−1)!!
(2n)!!·π
2BI (151)(4)
8./integraldisplay∞
0sin2nx
xdx=∞ BI (151)(3)
9./integraldisplay∞
0sin2ax
x2dx=aπ
2[a>0] L OV3 0 7 ,3 1 2 ,F II I6 3 2
10./integraldisplay∞
0sin2max
x2dx=(2m−3)!!
(2m−2)!!·aπ
2[a>0] GW (333)(14b)
11./integraldisplay∞
0sin2m+1ax
x3dx=(2m−3)!!
(2m)!!(2m+1 )a2π
4[a>0] GW (333)(14d)
12./integraldisplay∞
0sinpx
xmdx
=p
m−1/integraldisplay∞
0sinp−1x
xm−1cosxdx [p>m −1>0]
=p(p−1)
(m−1)(m−2)/integraldisplay∞
0sinp−2x
xm−2dx−p2
(m−1)(m−2)/integraldisplay∞
0sinpx
xm−2dx [p>m −1>1]
GW (333)(17)
13./integraldisplay∞
0sin2npx√xdx=∞ BI (177)(5)
14./integraldisplay∞
0sin2n+1pxdx√x=1
22n/radicalbiggπ
2pn/summationdisplay
k=0(−1)k/parenleftbigg2n+1
n+k+1/parenrightbigg1√
2k+1BI (177)(7)
3.822
1./integraldisplayπ/2
0xpcosmxdx=−p(p−1)
m2/integraldisplayπ/2
0xp−2cosmxdx+m−1
m/integraldisplayπ/2
0xpcosm−2xdx
[m>1,p > 1] GW (333)(9a)
2./integraldisplay∞
0x−1/2cos2n+1(px)dx=1
22n/radicalbiggπ
2pn/summationdisplay
k=0/parenleftbigg2n+1
n+k+1/parenrightbigg1√
2k+1BI (177)(8)
3.823/integraldisplay∞
0xμ−1sin2axdx =−Γ(μ)cosμπ
2
2μ+1aμ[a>0,−2<Reμ<0]
ET I 319(15), GW(333)(19c)a
3.824
1./integraldisplay∞
0sin2ax
x2+β2dx=π
4β/parenleftbig
1−e−2aβ/parenrightbig
[a>0,Reβ>0] BI (160)(10)
2./integraldisplay∞
0cos2ax
x2+β2dx=π
4β/parenleftbig
1+e−2aβ/parenrightbig
[a>0,Reβ>0] BI (160)(11)
3.824 Powers and trigonometric functions 461
3.7/integraldisplay∞
0sin2mxdx
a2+x2=(−1)m
22m+1·π
2/braceleftBigg
22msinh2ma−2m/summationdisplay
k=0(−1)k/parenleftbigg2m
k/parenrightbigg
sinh[2( m−k)a]/bracerightBigg
[a>0] BI (160)(12)
4.7/integraldisplay∞
0sin2m+1xdx
a2+x2=(−1)m−1
22m+2a/braceleftBigg
e(2m+1)a2m+1/summationdisplay
k=0(−1)k/parenleftbigg2m+1
k/parenrightbigg
e−2kaEi[(2k−2m−1)a]
+e−(2m+1)a2m+1/summationdisplay
k=0(−1)k−1/parenleftbigg2m+1
k/parenrightbigg
e2kaEi[(2m+1−2k)a]/bracerightBigg
[a>0] BI (160)(14)
5.7/integraldisplay∞
0sin2m+1xxdx
a2+x2=π
22m+1e−(2m+1)am/summationdisplay
k=0(−1)m+k/parenleftbigg2m+1
k/parenrightbigg
e2ka
/bracketleftBig
|arga|<π
2/bracketrightBig
,m =0,1,2,...
6.7/integraldisplay∞
0cos2mxdx
a2+x2=π
22m+1a/parenleftbigg2m
m/parenrightbigg
+π
22mm/summationdisplay
k=1/parenleftbigg2m
m+k/parenrightbigg
e−2ka
[a>0] BI (160)(16)
7./integraldisplay∞
0cos2m+1xdx
a2+x2=π
22m+1am/summationdisplay
k=1/parenleftbigg2m+1
m+k+1/parenrightbigg
e−(2k+1)a
[a>0] BI (160)(17)
8./integraldisplay∞
0cos2m+1xxdx
a2+x2=−e−(2m+1)a
22m+22m+1/summationdisplay
k=0/parenleftbigg2m+1
k/parenrightbigg
e2kaEi[(2m−2k+1 )a]
−e(2m+1)a
22m+22m+1/summationdisplay
k=0/parenleftbigg2m+1
k/parenrightbigg
e−2kaEi[(2k−2m−1)a]
BI (160)(18)
9./integraldisplay∞
0cos2ax
b2−x2dx=π
4bsin 2ab [a>0,b > 0] BI (161)(10)
10./integraldisplay∞
0sin2axcos2bx
β2+x2dx=π
8β/bracketleftbigg
1−1
2e−2(a+b)β+e−2bβ−1
2e2(b−a)β−e−2aβ/bracketrightbigg
[a>b]
=π
16β/bracketleftbig
1−e−4aβ/bracketrightbig
[a=b]
=π
8β/bracketleftbigg
1−1
2e−2(a+b)β+e−2bβ−1
2e2(a−b)β−e−2aβ/bracketrightbigg
[a<b]
[a>0,b > 0],(cf.3.824 1a n d3 ) BI (162)(6)
462 Trigonometric Functions 3.825
11./integraldisplay∞
0xsin 2axcos2bx
β2+x2dx=π
8/bracketleftBig
2e−2aβ+e−2(a+b)β+e2(b−a)β/bracketrightBig
[a>0]
=π
8/bracketleftbig
e−4aβ+2e−2aβ/bracketrightbig
[a=b]
=π
8/bracketleftBig
2e−2aβ+e−2(a+b)β−e2(a−b)β/bracketrightBig
[a<b]
LI (162)(5)
3.825
1./integraldisplay∞
0sin2axdx
(b2+x2)(c2+x2)=π/parenleftbig
b−c+ce−2ab−be−2ac/parenrightbig
4bc(b2−c2)
[a>0,b > 0,c > 0] BI (174)(15)
2./integraldisplay∞
0cos2axdx
(b2+x2)(c2+x2)=π/parenleftbig
b−c+be−2ac−ce−2ab/parenrightbig
4bc(b2−c2)
[a>0,b > 0,c > 0] BI (175)(14)
3.3/integraldisplay∞
0sin2axdx
(b2−x2)(c2−x2)=π(csin2ab−bsin 2ac)
4bc(b2−c2)[a>0,b > 0,c > 0,b/negationslash=c]
LI (174)(16)
4.3/integraldisplay∞
0cos2axdx
(b2−x2)(c2−x2)=π(bsin2ac−csin 2ab)
4bc(b2−c2)[a>0,b > 0,c > 0,b/negationslash=c]
LI (175)(15)
3.826
1./integraldisplay∞
0sin2axdx
x2(b2+x2)=π
4b2/bracketleftbigg
2a−1
b/parenleftbig
1−e−2ab/parenrightbig/bracketrightbigg
[a>0,b > 0] BI (172)(13)
2./integraldisplay∞
0sin2axdx
x2(b2−x2)=π
4b2/parenleftbigg
2a−1
bsin 2ab/parenrightbigg
[a>0,b > 0] BII (172)(14)
3.827
1.8/integraldisplay∞
0sin3ax
xνdx=3−3ν−1
4aν−1cosνπ
2Γ(1−ν)[ a<Reν<4,ν/negationslash=1,2,3]
GW (333)(19f)
2.8/integraldisplay∞
0sin3ax
xdx=π
4LO V 277
3./integraldisplay∞
0sin3ax
x2dx=3
4aln 3 BI (156)(2)
4.8/integraldisplay∞
0sin3ax
x3dx=3
8a2π BI(156)(7)a,LO V 312
5./integraldisplay∞
0sin4ax
x2dx=aπ
4[a>0] BI (156)(3)
6./integraldisplay∞
0sin4ax
x3dx=a2ln2 BI (156)(8)
3.828 Powers and trigonometric functions 463
7./integraldisplay∞
0sin4ax
x4dx=a3π
3[a>0] BI(156)(11), LO V 312
8./integraldisplay∞
0sin5ax
x2dx=5
16a(3 ln 3 −ln5) BI (156)(4)
9./integraldisplay∞
0sin5ax
x3dx=5
32a2π [a>0] BI (156)(9)
10./integraldisplay∞
0sin5ax
x4dx=5
96a3(25ln5 −27ln3) BI (156)(12)
11./integraldisplay∞
0sin5ax
x5dx=115
384a4π [a>0] BI(156)(13), LO V 312
12./integraldisplay∞
0sin6ax
x2dx=3
16aπ [a>0] BI (156)(5)
13./integraldisplay∞
0sin6ax
x3dx=3
16a2(8ln2 −3l n3 ) BI (156)(10)
14./integraldisplay∞
0sin6ax
x5dx=1
16a4(27ln3 −32ln2) BI (156)(14)
15./integraldisplay∞
0sin6ax
x6dx=11
40a5π [a>0] LO V 312
3.828 In3.828 1–21 the restrictions a>0,b>0,c>0 apply.
1.8/integraldisplay∞
0sinaxsinbx
xdx=1
2ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglea+b
a−b/vextendsingle/vextendsingle/vextendsingle/vextendsingle[a/negationslash=b] FI II 647
2.8/integraldisplay∞
0sinaxsinbxdx
x2=1
2πmin(a,b) BI (157)(1)
3.8/integraldisplay∞
0sin2axsinbx
xdx=π
4[b<2a]
=π
8[b=2a]
=0 [ b>2a]
BI (151)(10)
4.8/integraldisplay∞
0sin2axcosbx
xdx=1
4ln4a2−b2
b2[2a/negationslash=b] BI (151)(12)
5.8/integraldisplay∞
0sin2axcos2bx
x2dx=1
2πmax(0 ,a−b)
6./integraldisplay∞
0sin 2axcos2bx
xdx=π
2[a>b]
=3
8π [a=b]
=π
4[a<b]
BI (151)(9)
464 Trigonometric Functions 3.828
7.8/integraldisplay∞
0sin2axsinbxsincx
x2dx=π
16(|b−2a−c|−|2a−b−c|+2c)
[a>0,0<c≤b]
BI(157)(9)a, ET I 79(15)
8.8/integraldisplay∞
0sin2axsinbxsincx
xdx=1
8ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle(b+c)
2(2a−b+c)(2a+b−c)
(b−c)2(2a+b+c)(2a−b−c)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
[b/negationslash=c,2a+c/negationslash=b,2a+b/negationslash=c,2a/negationslash=b+c]
LI (152)(2)
9./integraldisplay∞
0sin2axsin2bx
x2dx=π
4a [0≤a≤b]
=π
4b [0≤b≤a]
BI (157)(3)
10.8/integraldisplay∞
0sin2axsin2bx
x4dx=1
6πmin/parenleftbig
a2,b2/parenrightbig
[3 max( a,b)−min(a,b)] BI (157)(27)
11.8/integraldisplay∞
0sin2axcos2bx
x2dx=1
4π[a+m a x ( 0 ,a−b)] BI (157)(6)
12./integraldisplay∞
0sin3axsin 3bx
x4dx=a3π
2[b>a]
=π
16/bracketleftbig
8a3−9(a−b)3/bracketrightbig
[a≤3b≤3a] BI (157)(28)
=9bπ
8/parenleftbig
a2−b2/parenrightbig[3b≤a] LI (157)(28)
13./integraldisplay∞
0sin3axcosbx
xdx=0 [ b>3a]
=−π
16[b=3a]
=−π
8[3a>b>a ]
=π
16[b=a]
=π
4[a>b]
[a>0,b > 0] BI (151)(15)
14.10/integraldisplay∞
0sin3axcos3bx
x2dx=3
16⎛
⎝aln 81−2(a−3b)ln (a−3b)+2 ( a−b)ln (a−b)
+2 (a+b)ln (a+b)−2(a+3b)ln (a+3 )⎞
⎠
[Ima=0,Imb=0 ] MC
3.828 Powers and trigonometric functions 465
15./integraldisplay∞
0sin3axcosbx
x3dx=π
8/parenleftbig
3a2−b2/parenrightbig
[b<a]
=πb2
4[a=b]
=π
16(3a−b)2[a<b< 3a]
=0 [ 3 a<b]
[a>0,b > 0]BI(157)(19), ET I 19(10)
16./integraldisplay∞
0sin3axsinbx
x4dx=bπ
24/parenleftbig
9a2−b2/parenrightbig
[0<b≤a]
=π
48/bracketleftbig
24a3−(3a−b)3/bracketrightbig
[0<a≤b≤3a]
=πa3
2[0<3a≤b]
ET I 79(16)
17./integraldisplay∞
0sin3axsin2bx
xdx=π
8[2b>3a]
=5π
32[2b=3a]
=3π
16[3a>2b>a]
=3π
32[2b=a]
=0 [ a>2b]
[a>0,b > 0] BI (151)(14)
18.8/integraldisplay∞
0sin2axcos3bx
xdx=1
16ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle(2a+b)
3(b−2a)3(2a+3b)(3b−2a)
9b8/vextendsingle/vextendsingle/vextendsingle/vextendsingle
[2a/negationslash=b,2a/negationslash=3b]
BI (151)(13)
19.11/integraldisplay∞
0sin2axsin2bxsin2cx
xdx
=π
32/parenleftbigg
4 sign( c)−2 sign(2 b+c) + 2 sign(2 b−c) + sign(2 a−2b+c)−sign(2 a−2b−c)
+ 2 sign(2 a−c) + sign(2 a+2b+c)−sign(2 a+2b−c)−2 sign(2 a+c)/parenrightbigg
[Ima=0,Imb=0,Imc=0 ] MC
20./integraldisplay∞
0sin2axsin2bxsin 2cxdx
x2
=a−b−c
16ln 4(a−b−c)2−a+b+c
16ln4(a+b+c)2+a+b−c
16ln 4(a+b−c)2
−a−b+c
16ln4(a−b+c)2+a+c
8ln 4(a+c)2−a−c
8ln 4(a−c)2
+b+c
8ln4(b+c)2−b−c
8ln 4(b−c)2−1
2cln2c
[a>0,b > 0,c > 0] BI (157)(10)
466 Trigonometric Functions 3.829
21.8/integraldisplay∞
0sin2axsin3bx
x3dx=3b2π
16[2a>3b]
=a2π
12[2a=3b]
=6b2−(3b−2a)2
32π [3b>2a>b]
=a2π
4[b≥2a]
BI (157)(18)
3.829
1./integraldisplay∞
0xn−sinnx
xn+2dx=π
2n(n+1 ) ![(n−1)/2]/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig
(n−2k)n+1GW (333)(63)
2./integraldisplay∞
0/parenleftbig
1−cos2m−1x/parenrightbigdx
x2=/integraldisplay∞
0/parenleftbig
1−cos2mx/parenrightbigdx
x2=mπ
22m/parenleftbigg2m
m/parenrightbigg
BI (158)(7, 8)
3.831
1./integraldisplay∞
0sin2nax−sin2nbx
xdx=(2n−1)!!
(2n)!!lnb
a[ab >0,n=1,2,...] FI II 651
2./integraldisplay∞
0cos2nax−cos2nbx
xdx=/bracketleftbigg
1−(2n−1)!!
(2n)!!/bracketrightbigg
lnb
a[ab >0,n=0,1,...] FI II 651
3./integraldisplay∞
0cos2m+1ax−cos2m+1bx
xdx=l nb
a[ab >0,m =0,1,...] FI II
4./integraldisplay∞
0cosmaxcosmax−cosmbxcosmbx
xdx=/parenleftbigg
1−1
2m/parenrightbigg
lnb
a
[ab >0,m =0,1,...] LI (155)(8)
3.832
1./integraldisplayπ/2
0xcosp−1xsinaxdx =π
2p+1Γ(p)ψ/parenleftbigp+a+1
2/parenrightbig
−ψ/parenleftbigp−a+1
2/parenrightbig
Γ/parenleftbigp+a+1
2/parenrightbig
Γ/parenleftbigp−a+1
2/parenrightbig
[p>0,−(p+1 )<a<p +1 ]
BI (205)(6)
2.3/integraldisplay∞
0sin2m+1xsin 2mxdx
a2+x2=(−1)mπ
22m+1a/bracketleftBig/parenleftbig
1−e−2a/parenrightbig2m−1/bracketrightBig
sinha
[a>0,m =0,1....] BI (162)(17)
3./integraldisplay∞
0sin2m−1xsin[(2m−1)x]dx
a2+x2=(−1)m+1π
22ma/parenleftbig
1−e−2a/parenrightbig2m−1
[a>0,m =1,2,...] BI (162)(11)
4./integraldisplay∞
0sin2m−1xsin[(2m+1 )x]dx
a2+x2=(−1)m−1π
22mae−2a/parenleftbig
1−e−2a/parenrightbig2m−1
[a>0,m =1,2,...] BI (162)(12)
3.832 Powers and trigonometric functions 467
5./integraldisplay∞
0sin2m+1xsin[3(2 m+1 )x]dx
a2+x2=(−1)mπ
2ae−3(2m+1)asinh2m+1a
[a>0] BI (162)(18)
6.3/integraldisplay∞
0sin2mxsin[(2m−1)x]xdx
a2+x2=(−1)mπ
22m+1ea/bracketleftBig/parenleftbig
1−e−2a/parenrightbig2m−/parenleftbig
1+e−2a/parenrightbig/bracketrightBig
[a≥0,m =0,1,...] BI (162)(13)
7./integraldisplay∞
0sin2mxsin(2mx)xdx
a2+x2=(−1)mπ
22m+1/bracketleftBig/parenleftbig
1−e−2a/parenrightbig2m−1/bracketrightBig
[a>0,m =0,1,...] BI (162)(14)
8./integraldisplay∞
0sin2mxsin[(2m+2 )x]xdx
a2+x2=(−1)mπ
22m+1e−2a/parenleftbig
1−e−2a/parenrightbig2m
[a>0,m =0,1,...] BI (162)(15)
9./integraldisplay∞
0sin2mxsin 4mxxdx
a2+x2=(−1)mπ
2e−4masinh2ma
[a>0,m =1,2,...] BI (162)(16)
10./integraldisplay∞
0sin2mxcosxdx
x2=(2m−3)!!
(2m)!!·π
2[m=1,2,...] GW (333)(15a)
11./integraldisplay∞
0sin2mxcos[(2 m−1)x]dx
a2+x2=(−1)mπ
22ma/bracketleftBig/parenleftbig
1−e−2a/parenrightbig2m−1−1/bracketrightBig
sinha
[a>0,m =1,2,...] BI (162)(25)
12./integraldisplay∞
0sin2mxcos(2mx)dx
a2+x2=(−1)mπ
22m+1a/parenleftbig
1−e−2a/parenrightbig2m
[a>0,m =0,1,...] BI (162)(26)
13./integraldisplay∞
0sin2mxcos[(2 m+2 )x]dx
a2+x2=(−1)mπ
22m+1ae−2a/parenleftbig
1−e−2a/parenrightbig2m
[a>0,m =0,1,...] BI (162)(27)
14./integraldisplay∞
0sin2mxcos 4mxdx
a2+x2=(−1)mπ
2ae−4masinh2ma
[a>0,m =0,1,...] BI (162)(28)
15./integraldisplay∞
0sin2m+1xcosxdx
x=(2m−1)!!
(2m+2 ) ! !·π
2[m=0,1,...] GW (333)(15)
16.3/integraldisplay∞
0sin2m+1xcosxdx
x3=(2m−3)!!
(2m)!!·π
2[m=1,2,...] GW (333)(15b)
17./integraldisplay∞
0sin2m−1xcos[(2 m−1)x]xdx
a2+x2=(−1)mπ
22m/bracketleftBig/parenleftbig
1−e−2a/parenrightbig2m−1−1/bracketrightBig
[m=1,2,..., a> 0] BI (162)(23)
18.3/integraldisplay∞
0sin2m+1xcos 2mxxdx
a2+x2=(−1)m−1π
22m+2/braceleftBig
ea/bracketleftBig/parenleftbig
1−e−2a/parenrightbig2m+1−1/bracketrightBig
−e−a/bracerightBig
[m=0,1,..., a ≥0] BI (162)(29)
468 Trigonometric Functions 3.832
19./integraldisplay∞
0sin2m−1xcos[(2 m+1 )x]xdx
a2+x2=(−1)mπ
22me−2a/parenleftbig
1−e−2a/parenrightbig2m−1
[m=1,2,..., a> 0] BI (162)(24)
20./integraldisplay∞
0sin2m+1xcos[2(2 m+1 )x]xdx
a2+x2=(−1)m−1π
2e−2(2m+1)asinh2m+1a
[m=0,1,..., a> 0] BI (162)(30)
21./integraldisplay∞
0cosmxsinmxxdx
a2+x2=1
2m+1am/summationdisplay
k=1/parenleftBigm
k/parenrightBig/bracketleftbig
e−2kaEi(2ka)−e2kaEi(−2ka)/bracketrightbig
[a>0] BI (162)(8)
22./integraldisplay∞
0cosnsxsinnsxxdx
a2+x2=π
2n+1/bracketleftBig/parenleftbig
1+e−2as/parenrightbign−1/bracketrightBig
[s>0,Rea>0,n≥0]BI (163)(9)
23./integraldisplay∞
0cosnsxsinnsxxdx
a2−x2=π
2/parenleftbig
2−n−cosnascosnas/parenrightbig
[n=0,1,...] BI (166)(10)
24./integraldisplay∞
0cosm−1xsin[(m+1 )x]xdx
a2+x2=π
2me−2a/parenleftbig
1+e−2a/parenrightbigm−1
[a>0,m =1,2,...] BI (163)(6)
25./integraldisplay∞
0cosmxsin[(m+1 )x]xdx
a2+x2=π
2m+1e−a/parenleftbig
1+e−2a/parenrightbigm
[m=0,1,..., a> 0] BI (163)(10)
26.3/integraldisplay∞
0cosmxsin[(m−1)x]xdx
a2+x2=π
2mcosha/bracketleftBig/parenleftbig
1+e−2a/parenrightbigm−1−1/bracketrightBig
[m=0,1,..., a ≥0] BI (163)(7)
27.11/integraldisplay∞
0cosmxsin(3mx)xdx
a2+x2=π
2e−3macoshma [a>0,m =1,2,...] BI (163)(11)
28./integraldisplay∞
0cosnsxcosnxsdx
a2+x2=π
2n+1a/parenleftbig
1+e−2as/parenrightbign[n=0,1,...] BI (163)(16)
29./integraldisplay∞
0cosnsxcosnsxdx
a2−x2=π
2acosnassinnas [n=0,1,...]
30./integraldisplay∞
0cosm−1xcos[(m+1 )x]dx
a2+x2=π
2mae−2a/parenleftbig
1+e−2a/parenrightbigm−1
[m=1,2,..., a> 0] BI (163)(14)
31./integraldisplay∞
0cosmxcos[(m−1)x]dx
a2+x2=π
2m+1aea/bracketleftBig/parenleftbig
1+e−2a/parenrightbigm−/parenleftbig
1−e−2a/parenrightbig/bracketrightBig
[m=0,1,..., a> 0] BI (163)(15)
3.834 Powers and trigonometric functions 469
32./integraldisplay∞
0cosmxcos[(m+1 )x]dx
a2+x2=π
2m+1ae−a/parenleftbig
1+e−2a/parenrightbigm
[m=0,1,..., a> 0] BI (163)(17)
33./integraldisplay∞
0sinpxcosxdx
xq=p
q−1/integraldisplay∞
0sinp−1x
xq−1dx−p+1
q−1/integraldisplay∞
0sinp+1x
xq−1dx [p>q−1>0]
=p(p−1)
(q−1)(q−2)/integraldisplay∞
0sinp−2xcosxdx
xq−2
−(p+1 )2
(q−1)(q−2)/integraldisplay∞
0sinpxcosxdx
xq−2[p>q−1>1]
GW (333)(18)
34./integraldisplay∞
0cos2mxcos 2nxsinxdx
xx=/integraldisplay∞
0cos2m−1xcos 2nxsindx
xx=π
22m+1/parenleftbigg2m
m+n/parenrightbigg
BI (152)(5, 6)
35./integraldisplay∞
0cospaxsinbxcosxdx
x=π
2[b>a p , p> −1] BI (153)(12)
36./integraldisplay∞
0cospaxsinpaxcosxdx
x=π
2p+1(2p−1) [ p>−1] BI (153)(2)
37./integraldisplay∞
0dx
x2/parenleftBiggn/productdisplay
k=1cospkakx/parenrightBigg
sinbxsinx=π
2/bracketleftBigg
b>n/summationdisplay
k=1akpk,a k>0,p k>0/bracketrightBigg
BI (157)(15)
3.833
1.10/integraldisplay∞
0sin2m+1xcos2nxdx
x=/integraldisplay∞
0sin2m+1xcos2n−1xdx
x=(2m−1)!!(2n−1)!!
2m+n+1(m+n)!π
BI (151)(24, 25)
=1
2B/parenleftbigg
m+1
2,n+1
2/parenrightbigg
GW (333)(24)
2./integraldisplay∞
0sin2m+12xcos2n−12xcos2xdx
x=π
2·(2m−1)!!(2n−1)!!
(2m+2n)!!LI (152)(4)
3.834
1./integraldisplay∞
0sin2m+1x
1−2acosx+a2·dx
x=(−1)mπ(1 +a)4m
22m+2a2m+1/braceleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−a
1+a/vextendsingle/vextendsingle/vextendsingle/vextendsingle2m−1
−2m/summationdisplay
k=0(−1)k/parenleftbiggm−1
2
k/parenrightbigg/parenleftbigg4a
(1 +a)2/parenrightbiggk/bracerightBigg
[|a|/negationslash=1 ] GW (333)(62a)
470 Trigonometric Functions 3.835
2./integraldisplay∞
0sin2m+1xcosnx
(1−2acosx+a2)p·dx
x
=n!π
2n+1(2m+n+ 1)!(1 + a)2pn/summationdisplay
k=0(−1)k(2m+2n−2k+ 1)!!(2 m+2k−1)!!
k!(n−k)!
×F/parenleftbigg
m+n−k+3
2,p;2m+n+2 ;4a
(1 +a)2/parenrightbigg
[a/negationslash=±1] GW (333)(62)
3.835
1./integraldisplay∞
0cos2mxcos 2mxsinx
a2cos2x+b2sin2x·dx
x=π
2b2m−1
a(a+b)2m[ab >0] BI (182)(31)a
2./integraldisplay∞
0cos2m−1xcos 2mxsinx
a2cos2x+b2sin2x·dx
x=π
2ab2m−1
(a+b)2m[ab >0] LI (182)(32)a
3.836
1./integraldisplay∞
0/parenleftbiggsinx
x/parenrightbiggnsinmx
xdx=π
2[m≥n] LI (159)(12)
2.11/integraldisplay∞
0/parenleftbiggsinx
x/parenrightbiggn
cosmxdx =nπ
2n⌊1
2(m+n)⌋/summationdisplay
k=0(−1)k(n+m−2k)n−1
k!(n−k)![0≤m<n ]
=0 [ m≥n≥2]
=π
4[m=n=1 ]
GI(159)(14), ET I 20(11)
3./integraldisplay∞
0/parenleftbiggsinx
x/parenrightbiggn−1
sinnxcosxdx
x=π
2[n≥1] BI (159)(20)
4.8/integraldisplay∞
0/parenleftbiggsinx
x/parenrightbiggnsin(anx)
xdx=π
2⎡
⎢⎣1−1
2n−1n!⌊1
2n(1+a)⌋/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig
(n+an−2k)n⎤
⎥⎦
[all real a,n≥1] ET I 20(11)
5.10In(b)=2
π/integraldisplay∞
0/parenleftbiggsinx
x/parenrightbiggn
cosbxdx=n/parenleftbig
2n−1n!/parenrightbig−1⌊r⌋/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig
(n−b−2k)n−1
where 0 ≤b<n,n≥1,r=(n−b)/2, and ⌊r⌋is the largest integer contained in r
LO V 340(14)
6.11/integraldisplay∞
0/parenleftbiggsinx
x/parenrightbiggn
cosanxdx =0 [ a≤−1o ra≥1,n≥2; for n=1s e e 3.741 2]
3.837
1./integraldisplayπ/2
0x2dx
sin2x=πln 2 BI (206)(9)
2./integraldisplayπ/4
0x2dx
sin2x=−π2
16+π
4ln 2 +G=0.8435118417 ... BI (204)(10)
3.839 Powers and trigonometric functions 471
3./integraldisplayπ/4
0x2dx
cos2x=π2
16+π
4ln 2−G GW (333)(35a)
4./integraldisplayπ/4
0xp+1
sin2xdx=−/parenleftBigπ
4/parenrightBigp+1
+(p+1 )/parenleftBigπ
4/parenrightBigp/braceleftBigg
1
p−1
2∞/summationdisplay
k=11
42k−1(p+2k)ζ(2k)/bracerightBigg
[p>0] LI (204)(14)
5./integraldisplayπ/2
0x2cosx
sin2xdx=−π2
4+4G=1.1964612764 ... BI (206)(7)
6./integraldisplayπ/2
0x3cosx
sin3xdx=−π3
16+3
2πln2 BI (206)(8)
7./integraldisplay∞
0cos2nx
cosxsin2nxdx
xm=0/bracketleftbigg
n>m−1
2,m > 0/bracketrightbigg
BI (180)(16)
8./integraldisplay∞
0cos2nx
cosxsin2n+1xdx
xm=0/bracketleftbigg
n>m−2
2,m > 0/bracketrightbigg
BI (180)(17)
9./integraldisplay1
0xdx
cosaxcos[a(1−x)]=1
acoseca·ln seca/bracketleftBig
a<π
2/bracketrightBig
BI (149)(20)
10.3/integraldisplayπ
0xsin(2n+1 )x
sinxdx=1
2π2[n=0,1,2,...]
11.3/integraldisplayπ
0xsin 2nx
sinxdx=−4n/summationdisplay
k=1(2k−1)−2[n=1,2,3,...]
3.838
1./integraldisplayπ/2
0xcosp−1x
sinp+1xdx=π
2psecπp
2[p<1] BI (206)(13)a
2./integraldisplayπ/4
0xsinp−1x
cosp+1xdx=π
4p−1
2pβ/parenleftbiggp+1
2/parenrightbigg
[p>−1] LI (204)(15)
3./integraldisplayπ/4
0xsin2m−1x
cos2m+1xdx=π
8m(1−cosmπ)+1
2mm−1/summationdisplay
k=0(−1)k−1
2m−2k−1BI (204)(17)
4./integraldisplayπ/4
0xsin2mx
cos2m+2xdx=1
2(2m+1 )/bracketleftBigg
π
2+(−1)m−1ln 2 +m−1/summationdisplay
k=0(−1)k−1
m−k/bracketrightBigg
BI (204)(16)
3.839
1.11/integraldisplayπ/4
0xtan2xdx=π
4−π2
32−1
2ln 2 BI (204)(3)
2./integraldisplayπ/4
0xtan3xdx=π
4−1
2+π
8ln2−1
2G BI (204)(7)
3./integraldisplayπ/4
0x2tanx
cos2xdx=1
2ln 2−π
4+π2
16(cf.3.839 1) BI (204)(13)
472 Trigonometric Functions 3.841
4./integraldisplayπ/4
0x2tan2x
cos2xdx=1
3/parenleftbigg
1−π
4ln2−π
2+π2
16+G/parenrightbigg
(cf.3.839 2) BI (204)(12)
5./integraldisplayπ/2
0xcospxtanxdx=π
2p+1p·Γ(p+1 )
/bracketleftBig
Γ/parenleftBigp
2+1/parenrightBig/bracketrightBig2[p>−1] BI (205)(3)
6./integraldisplayπ/2
0xsinpxcotxdx=π
2p−2p−1
pB/parenleftbiggp+1
2,p+1
2/parenrightbigg
[p>−1] BI (206)(11)
7./integraldisplay∞
0sin2nxtanxdx
x=π
2·(2n−1)!!
(2n)!!GW (333)(16)
8./integraldisplay∞
0cossrxtanqxdx
x=π
2[s>−1] BI (151)(26)
9./integraldisplay∞
0cos[(2 n−1)x]
cosx·/parenleftbiggsinx
x/parenrightbigg2n
dx=(−1)n−122n−1
(2n)!·22n−1π|B2n| BI (180)(15)
10./integraldisplay∞
0tanrpxdx
q2+x2=π
2qsecrπ
2tanhrpq/bracketleftbig
r2<1/bracketrightbig
BI (160)(19)
3.84 Integrals containing/radicalbig
1−k2sin2x,√1−k2cos2x, and similar expressions
Notation :k/prime=√
1−k2
3.841
1./integraldisplay∞
0sinx/radicalbig
1−k2sin2xdx
x=E(k) BI (154)(8)
2./integraldisplay∞
0sinx/radicalbig
1−k2cos2xdx
x=E(k) BI (154)(20)
3./integraldisplay∞
0tanx/radicalbig
1−k2sin2xdx
x=E(k) BI (154)(9)
4./integraldisplay∞
0tanx/radicalbig
1−k2cos2xdx
x=E(k) BI (154)(21)
3.842
1.11/integraldisplay∞
0sinx/radicalbig
1+s i n2xdx
x
=/integraldisplay∞
0tanx/radicalbig
1+s i n2x·dx
x
=/integraldisplay∞
0sinx√
1 + cos2xdx
x=/integraldisplay∞
0tanx√
1 + cos2xdx
x=1√
2K/parenleftbigg1√
2/parenrightbigg
≈1.3110287771
BI (183)(4, 5, 9, 10)
2./integraldisplayπ
2
uxcosxdx/radicalbig
sin2x−sin2u=π
2ln (1 + cos u) BI (226)(4)
3.844 Integrals containing/radicalbig
1−k2sin2xfunction ]squareroot and similar expressions 473
3./integraldisplay∞
0sinx/radicalbig
1−k2sin2xdx
x=/integraldisplay∞
0tanx/radicalbig
1−k2sin2xdx
x
=/integraldisplay∞
0sinx√
1−k2cos2xdx
x=/integraldisplay∞
0tanx√
1−k2cos2xdx
x=K(k)
BI (183)(12, 13, 21, 22)
4./integraldisplayπ/2
0xsinxcosx/radicalbig
1−k2sin2xdx=1
2k2[−πk/prime+2E(k)] BI (211)(1)
5./integraldisplayπ/2
0xsinxcosx√
1−k2cos2xdx=1
2k2[π−2E(k)] BI (214)(1)
6./integraldisplayα
0xsinxdx
cos2x/radicalbig
sin2α−sin2x=πsin2α
2
cos2αLO III 284
7./integraldisplayβ
0xsinxdx/parenleftbig
1−sin2αsin2x/parenrightbig/radicalbig
sin2β−sin2x=πlncosα+/radicalbig
1−sin2αsin2β
2c osβcos2α
2
2c osα/radicalbig
1−sin2αsin2βLO III 284
3.843
1./integraldisplay∞
0tanx/radicalbig
1−k2sin22xdx
x=E(k) BI (154)(10)
2./integraldisplay∞
0tanx/radicalbig
1−k2cos22xdx
x=E(k) BI (154)(22)
3.11/integraldisplay∞
0tanx/radicalbig
1+s i n22xdx
x=/integraldisplay∞
0tanx√
1 + cos22xdx
x=1√
2K/parenleftbigg1√
2/parenrightbigg
≈1.3110287771
BI (183)(6, 11)
4./integraldisplay∞
0tanx/radicalbig
1−k2sin22xdx
x=/integraldisplay∞
0tanx√
1−k2cos22xdx
x=K(k) BI (183)(14, 23)
3.844
1./integraldisplay∞
0sinxcosx√
1−k2cos2xdx
x=1
k2[K(k)−E(k)] BI (185)(20)
2./integraldisplay∞
0sinxcos2x√
1−k2cos2x·dx
x=1
k2[K(k)−E(k)] BI (185)(21)
3./integraldisplay∞
0sinxcos3x√
1−k2cos2x·dx
x=1
3k4/bracketleftbig/parenleftbig
2+k2/parenrightbig
K(k)−2/parenleftbig
1+k2/parenrightbig
E(k)/bracketrightbig
BI (185)(22)
4./integraldisplay∞
0sinxcos4x√
1−k2cos2x·dx
x=1
3k4/bracketleftbig/parenleftbig
2+k2/parenrightbig
K(k)−2/parenleftbig
1+k2/parenrightbig
E(k)/bracketrightbig
BI (185)(23)
5./integraldisplay∞
0sin3xcosx√
1−k2cos2x·dx
x=1
3k4/bracketleftbig/parenleftbig
1+k/prime2/parenrightbig
E(k)−2k/prime2K(k)/bracketrightbig
BI (185)(24)
6./integraldisplay∞
0sin3xcos2x√
1−k2cos2x·dx
x=1
3k4/bracketleftbig/parenleftbig
1+k/prime2/parenrightbig
E(k)−2k/prime2K(k)/bracketrightbig
BI (185)(25)
474 Trigonometric Functions 3.845
7./integraldisplay∞
0sin2xtanx√
1−k2cos2x·dx
x=1
k2/bracketleftbig
E(k)−k/prime2K(k)/bracketrightbig
BI (184)(16)
8./integraldisplay∞
0sin4xtanx√
1−k2cos2x·dx
x=1
3k4/bracketleftbig/parenleftbig
2+3k2/parenrightbig
k/prime2K(k)−2/parenleftbig
k/prime2−k2/parenrightbig
E(k)/bracketrightbig
BI (184)(18)
3.845
1.11/integraldisplay∞
0sinxcosx√
1 + cos2x·dx
x=√
2/bracketleftBigg
E/parenleftBigg√
2
2/parenrightBigg
−1
2K/parenleftBigg√
2
2/parenrightBigg/bracketrightBigg
≈0.5990701174 BI (185)(6)
2.11/integraldisplay∞
0sinxcos2x√
1 + cos2x·dx
x=√
2/bracketleftBigg
E/parenleftBigg√
2
2/parenrightBigg
−1
2K/parenleftBigg√
2
2/parenrightBigg/bracketrightBigg
≈0.5990701174 BI (185)(7)
3.11/integraldisplay∞
0sin2xtanx√
1 + cos2x·dx
x=√
2/bracketleftBigg
K/parenleftBigg√
2
2/parenrightBigg
−E/parenleftBigg√
2
2/parenrightBigg/bracketrightBigg
≈0.7119586598 BU (184)(8)
3.846
1./integraldisplay∞
0sinxcosx/radicalbig
1−k2sin2x·dx
x=1
k2/bracketleftbig
E(k)−k/prime2K(k)/bracketrightbig
BI (185)(9)
2./integraldisplay∞
0sinxcos2x/radicalbig
1−k2sin2x·dx
x=1
k2/bracketleftbig
E(k)−k/prime2K(k)/bracketrightbig
BI (185)(10)
3./integraldisplay∞
0sinxcos3x/radicalbig
1−k2sin2x·dx
x=1
3k4/bracketleftbig/parenleftbig
2−3k2/parenrightbig
k/prime2K(k)−2/parenleftbig
k/prime2−k2/parenrightbig
E(k)/bracketrightbig
BI (185)(11)
4./integraldisplay∞
0sinxcos4x/radicalbig
1−k2sin2x·dx
x=1
3k4/bracketleftbig/parenleftbig
2−3k2/parenrightbig
k/prime2K(k)−2/parenleftbig
k/prime2−k2/parenrightbig
E(k)/bracketrightbig
BI (185)(12)
5./integraldisplay∞
0sin3xcosx/radicalbig
1−k2sin2x·dx
x=1
3k4/bracketleftbig/parenleftbig
1+k/prime2/parenrightbig
E(k)−2k/prime2K(k)/bracketrightbig
BI (185)(13)
6./integraldisplay∞
0sin3xcos2x/radicalbig
1−k2sin2x·dx
x=1
3k4/bracketleftbig/parenleftbig
1+k/prime2/parenrightbig
E(k)−2k/prime2K(k)/bracketrightbig
BI (185)(14)
7./integraldisplay∞
0sin2xtanx/radicalbig
1−k2sin2x·dx
x=1
k2[K(k)−E(k)] BI (184)(9)
8./integraldisplay∞
0sin4xtanx/radicalbig
1−k2sin2x·dx
x=1
3k4/bracketleftbig/parenleftbig
2+k2/parenrightbig
K(k)−2/parenleftbig
1+k2/parenrightbig
E(k)/bracketrightbig
BI (184)(11)
3.84711/integraldisplay∞
0sinxcosx/radicalbig
1+s i n2x·dx
x=/integraldisplay∞
0sinxcos2x/radicalbig
1+s i n2x·dx
x=√
2/bracketleftBigg
K/parenleftBigg√
2
2/parenrightBigg
−E/parenleftBigg√
2
2/parenrightBigg/bracketrightBigg
≈0.7119586598
BI (185)(3, 4)
3.848
1./integraldisplay∞
0sin3xcosx/radicalbig
1−k2sin22x·dx
x=1
4k2[K(k)−E(k)] BI (185)(15)
2./integraldisplay∞
0cos22xtanx/radicalbig
1−k2sin22x·dx
x=1
k2/bracketleftbig
E(k)−k/prime2K(k)/bracketrightbig
BI (184)(12)
3.852 Trigonometric functions with powers 475
3./integraldisplay∞
0cos42xtanx/radicalbig
1−k2sin22x·dx
x=1
3k4/bracketleftbig/parenleftbig
2−3k2/parenrightbig
k/prime2K(k)−2/parenleftbig
k/prime2−k2/parenrightbig
E(k)/bracketrightbig
BI (184)(13)
4./integraldisplay∞
0sin24xtanx/radicalbig
1−k2sin22x·dx
x=4
3k4/bracketleftbig/parenleftbig
1+k/prime2/parenrightbig
E(k)−2k/prime2K(k)/bracketrightbig
BI (184)(17)
5./integraldisplay∞
0sin3xcosx√
1−k2cos22x·dx
x=1
4k2/bracketleftbig
E(k)−k/prime2K(k)/bracketrightbig
BI (185)(26)
6./integraldisplay∞
0cos22xtanx√
1−k2cos22x·dx
x=1
k2[K(k)−E(k)] BI (184)(19)
7./integraldisplay∞
0cos42xtanx√
1−k2cos22x·dx
x=1
3k4/bracketleftbig/parenleftbig
2+k2/parenrightbig
K(k)−2/parenleftbig
1+k2/parenrightbig
E(k)/bracketrightbig
BI (184)(20)
3.849
1.11/integraldisplay∞
0sin3xcosx√
1 + cos22x·dx
x=1
2√
2/bracketleftBigg
K/parenleftBigg√
2
2/parenrightBigg
−E/parenleftBigg√
2
2/parenrightBigg/bracketrightBigg
≈0.1779896649 BI (185)(8)
2.11/integraldisplay∞
0sin3xcosx/radicalbig
1+s i n22x·dx
x=√
2
8/bracketleftBigg
2E/parenleftBigg√
2
2/parenrightBigg
−K/parenleftBigg√
2
2/parenrightBigg/bracketrightBigg
≈0.1497675293 BI (185)(5)
3.11/integraldisplay∞
0cos22xtanx/radicalbig
1+s i n22x·dx
x=√
2/bracketleftBigg
K/parenleftBigg√
2
2/parenrightBigg
−E/parenleftBigg√
2
2/parenrightBigg/bracketrightBigg
≈0.7119586598 BI (184)(7)
3.85–3.88 Trigonometric functions of more complicated arguments combined with
powers
3.851
5./integraldisplay∞
0sin/parenleftbig
ax2/parenrightbig
cos(bx)dx
x2=bπ
2/braceleftbigg
S/parenleftbiggb
2√a/parenrightbigg
−C/parenleftbiggb
2√a/parenrightbigg
+√aπsin/parenleftbiggb2
4a+π
4/parenrightbigg/bracerightbigg
[a>0,b > 0],(cf.3.691 7)
ET I 23(3)a
3.852
1./integraldisplay∞
0sin/parenleftbig
ax2/parenrightbig
x2dx=/radicalbiggaπ
2[a≥0] BI (177)(10)a
2./integraldisplay∞
0sin/parenleftbig
ax2/parenrightbig
cos/parenleftbig
bx2/parenrightbigdx
x2=1
2/radicalbiggπ
2/parenleftBig√
a+b+√
a−b/parenrightBig
[a>b> 0]
=1
2√πa [b=a≥0]
=1
2/radicalbiggπ
2/parenleftBig√
a+b−√
b−a/parenrightBig
[b>a> 0],(cf.3.852 1)BI (177)(23)
3./integraldisplay∞
0sin2/parenleftbig
a2x2/parenrightbig
x4dx=2√π
3a3[a≥0] GW (333)(19e)
476 Trigonometric Functions 3.853
4.10/integraldisplay∞
0sin3/parenleftbig
a2x2/parenrightbig
x2dx=a
4/radicalbiggπ
2/parenleftBig
3−√
3/parenrightBig /bracketleftbig
Ima2=0/bracketrightbig
MC
5./integraldisplay∞
0/parenleftbig
sin2x−x2cosx2/parenrightbigdx
x4=1
3/radicalbiggπ
2BI (178)(8)
6./integraldisplay∞
0/parenleftbigg
cos2x−1
1+x2/parenrightbiggdx
x=−1
2C BI (173)(22)
3.853
1./integraldisplay∞
0sin/parenleftbig
ax2/parenrightbig
β2+x2dx=π
2β/bracketleftBig√
2s in/parenleftBig
aβ2+π
4/parenrightBig
C/parenleftbig√aβ/parenrightbig
−√
2c os/parenleftBig
aβ2+π
4/parenrightBig
S/parenleftbig√aβ/parenrightbig
−sin/parenleftbig
aβ2/parenrightbig/bracketrightBig
[a>0,Reβ>0] ET II 219(33)a
2./integraldisplay∞
0cos/parenleftbig
ax2/parenrightbig
β2+x2dx=π
2β/bracketleftBig
cos/parenleftbig
aβ2/parenrightbig
−√
2c os/parenleftBig
aβ2+π
4/parenrightBig
C/parenleftbig√aβ/parenrightbig
−√
2s in/parenleftBig
aβ2+π
4/parenrightBig
S/parenleftbig√aβ/parenrightbig/bracketrightBig
[a>0,Reβ>0] ET II 221(51)a
3./integraldisplay∞
0x2sin/parenleftbig
ax2/parenrightbig
β2+x2dx
=βπ
2/bracketleftBig
sin/parenleftbig
aβ2/parenrightbig
−√
2s in/parenleftBig
aβ2+π
4/parenrightBig
C/parenleftbig√aβ/parenrightbig
+√
2c os/parenleftBig
aβ2+π
4/parenrightBig
S/parenleftbig√aβ/parenrightbig/bracketrightBig
−1
2/radicalbiggπ
2a
[a>0,Reβ>0] ET II 219(32)a
4./integraldisplay∞
0x2cos/parenleftbig
ax2/parenrightbig
β2+x2dx=1
2/radicalbiggπ
2a−βπ
2/braceleftBig
cos/parenleftbig
aβ2/parenrightbig
−√
2c os/parenleftBig
aβ2+π
4/parenrightBig
C/parenleftbig√aβ/parenrightbig
−√
2s in/parenleftBig
aβ2+π
4/parenrightBig
S/parenleftbig√aβ/parenrightbig/bracerightBig
[a>0,Reβ>0] ET II 221(50)a
3.854
1./integraldisplay∞
0/parenleftbig
cos/parenleftbig
ax2/parenrightbig
−sin/parenleftbig
ax2/parenrightbig/parenrightbigdx
x4+b4=πe−ab2
2b3√
2[a>0,b > 0]
LI (178)(11)a, BI (168)(25)
2./integraldisplay∞
0/parenleftbig
cos/parenleftbig
ax2/parenrightbig
+s i n/parenleftbig
ax2/parenrightbig/parenrightbigx2dx
x4+b4=πe−ab2
2b√
2[a>0,b>0] LI (178)(12)
3./integraldisplay∞
0/parenleftbig
cos/parenleftbig
ax2/parenrightbig
+s i n/parenleftbig
ax2/parenrightbig/parenrightbigx2dx
(x4+b4)2=πe−ab2
4√
2b3/parenleftbigg
a+1
2b2/parenrightbigg
[a>0,b > 0] LI (178)(14)
4./integraldisplay∞
0/parenleftbig
cos/parenleftbig
ax2/parenrightbig
−sin/parenleftbig
ax2/parenrightbig/parenrightbigx4dx
(x4+b4)2=πe−ab2
4√
2b/parenleftbigg1
2b2−a/parenrightbigg
[a>0,b > 0] BI (178)(15)
3.856 Trigonometric functions with powers 477
3.855
1./integraldisplay∞
0sin/parenleftbig
ax2/parenrightbig
/radicalbig
β2+x4dx=1
2/radicalbiggaπ
2I1
4/parenleftbiggaβ
2/parenrightbigg
K1
4/parenleftbiggaβ
2/parenrightbigg
[a>0,Reβ>0] ET I 66(28)
2./integraldisplay∞
0cos/parenleftbig
ax2/parenrightbig
/radicalbig
β2+x4dx=1
2/radicalbiggaπ
2I−1
4/parenleftbiggaβ
2/parenrightbigg
K1
4/parenleftbiggaβ
2/parenrightbigg
[a>0,Reβ>0] ET I 9(22)
3./integraldisplayu
0sin/parenleftbig
a2x2/parenrightbig
√
u4−x4dx=a
4/radicalbigg
π3
2/bracketleftbigg
J1
4/parenleftbigga2
u22/parenrightbigg/bracketrightbigg2
[a>0] ET I 66(29)
4./integraldisplay∞
usin/parenleftbig
a2x2/parenrightbig
√
x4−u4dx=−a
4/radicalbigg
π3
2J1
4/parenleftbigga2u2
2/parenrightbigg
Y1
4/parenleftbigga2u2
2/parenrightbigg
[a>0] ET I 66(30)
5./integraldisplayu
0cos/parenleftbig
a2x2/parenrightbig
√
u4−x4dx=a
4/radicalbigg
π3
2/bracketleftbigg
J−1
4/parenleftbigga2u2
2/parenrightbigg/bracketrightbigg2
ET I 9(23)
6./integraldisplay∞
ucos/parenleftbig
a2x2/parenrightbig
√
x4−u4dx=−a
4/radicalbigg
π3
2J−1
4/parenleftbigga2u2
2/parenrightbigg
Y−1
4/parenleftbigga2u2
2/parenrightbigg
ET I 10(24)
3.856
1./integraldisplay∞
0/parenleftBig/radicalbig
β4+x4+x2/parenrightBigν
/radicalbig
β4+x4sin/parenleftbig
a2x2/parenrightbig
dx=a
2/radicalbiggπ
2β2νI1
4−ν
2/parenleftbigga2β2
2/parenrightbigg
K1
4+ν
2/parenleftbigga2β2
2/parenrightbigg
/bracketleftbigg
Reν<3
2,|argβ|<π
4/bracketrightbigg
ET I 71(23)
2./integraldisplay∞
0/parenleftBig/radicalbig
β4+x4+x2/parenrightBigν
/radicalbig
β4+x4cos/parenleftbig
a2x2/parenrightbig
dx=a
2/radicalbiggπ
2β2νI−1
4−ν
2/parenleftbigga2β2
2/parenrightbigg
K−1
4+ν
2/parenleftbigga2β2
2/parenrightbigg
/bracketleftbigg
Reν<3
2,|argβ|<π
4/bracketrightbigg
ET I 12(16)
3./integraldisplay∞
0/parenleftBig/radicalbig
β4+x4−x2/parenrightBigν
/radicalbig
β4+x4cos/parenleftbig
a2x2/parenrightbig
dx=a
2/radicalbiggπ
2β2νI−1
4+ν
2/parenleftbigga2β2
2/parenrightbigg
K−1
4−ν
2/parenleftbigga2β2
2/parenrightbigg
/bracketleftbigg
Reν>−3
2,|argβ|<π
4/bracketrightbigg
ET I 12(17)
4./integraldisplay∞
0sin/parenleftbig
a2x2/parenrightbig
dx
/radicalbig
β4+x4/radicalBig
x2+/radicalbig
β4+x4=sinha2β2
2√
2β2K0/parenleftbigga2β2
2/parenrightbigg
/bracketleftBig
|argβ|<π
4/bracketrightBig
ET I 66(32)
5./integraldisplay∞
0cos/parenleftbig
a2x2/parenrightbig
dx
/radicalbig
β4+x4/radicalbigg/parenleftBig
x2+/radicalbig
β4+x4/parenrightBig3=sinha2β2
2
2√
2β4K1/parenleftbigga2β2
2/parenrightbigg
/bracketleftBig
|argβ|<π
4/bracketrightBig
ET I 10(27)
478 Trigonometric Functions 3.857
6./integraldisplay∞
0/radicalBig/radicalbig
β4+x4+x2
/radicalbig
β4+x4sin/parenleftbig
a2x2/parenrightbig
dx=π
2√
2e−a2β2
2I0/parenleftbigga2β2
2/parenrightbigg
/bracketleftBig
|argβ|<π
4/bracketrightBig
ET I 67(33)
3.857
1./integraldisplay∞
0x2
R1R2/radicalbigg
R2−R1
R2+R1sin/parenleftbig
ax2/parenrightbig
dx=1
2√
bK0(ac)sinab
/bracketleftbigg
R1=/radicalBig
c2+(b−x2)2,R 2=/radicalBig
c2+(b+x2)2,a > 0,c > 0/bracketrightbigg
ET I 67(34)
2./integraldisplay∞
0x2
R1R2/radicalbigg
R2+R1
R2−R1cos/parenleftbig
ax2/parenrightbig
dx=1
2√
bK0(ac)cosab
/bracketleftbigg
R1=/radicalBig
c2+(b−x2)2,R 2=/radicalBig
c2+(b+x2)2,a > 0,c > 0/bracketrightbigg
ET I 10(26)
3.858
1./integraldisplay∞
u/parenleftbig
x2+√
x4−u4/parenrightbigν+/parenleftbig
x2−√
x4−u4/parenrightbigν
√
x4−u4sin/parenleftbig
a2x2/parenrightbig
dx
=−a
4/radicalbigg
π3
au2ν/bracketleftbigg
J1
4+ν
2/parenleftbigga2u2
2/parenrightbigg
Y1
4−ν
2/parenleftbigga2u2
2/parenrightbigg
+J1
4−ν
2/parenleftbigga2u2
2/parenrightbigg
Y1
4+ν
2/parenleftbigga2u2
2/parenrightbigg/bracketrightbigg
/bracketleftbig
Reν<3
2/bracketrightbig
ET I 71(25)
2./integraldisplay∞
u/parenleftbig
x2+√
x4−u4/parenrightbigν+/parenleftbig
x2−√
x4−u4/parenrightbigν
√
x4−u4cos/parenleftbig
a2x2/parenrightbig
dx
=−a
4/radicalbigg
π3
au2ν/bracketleftbigg
J−1
4+ν
2/parenleftbigga2u2
2/parenrightbigg
Y−1
4−ν
2/parenleftbigga2u2
2/parenrightbigg
+J−1
4−ν
2/parenleftbigga2u2
2/parenrightbigg
Y−1
4+ν
2/parenleftbigga2u2
2/parenrightbigg/bracketrightbigg
/bracketleftbig
Reν<3
2/bracketrightbig
ET I 13(26)
3.859/integraldisplay∞
0/bracketleftbigg
cos/parenleftBig
x2n/parenrightBig
−1
1+x2n+1/bracketrightbiggdx
x=−1
2nC BI (173)(24)
3.861
1./integraldisplay∞
0sin2n+1/parenleftbig
ax2/parenrightbigdx
x2m=±√πam−1
2
22n−m+1
2(2m−1)!!n+1/summationdisplay
k=1(−1)k−1/parenleftbigg2n+1
n+k/parenrightbigg
(2k−1)m−1
2
/bracketleftbiggthe + sign is taken when m≡0 (mod 4) or m≡1 (mod 4),
the−sign is taken when m≡2 (mod 4) or m≡3 (mod 4)/bracketrightbigg
BI (177)(19)a
2./integraldisplay∞
0sin2n/parenleftbig
ax2/parenrightbigdx
x2m=±√πam−1
2
22n−2m+1(2m−1)!!n/summationdisplay
k=1(−1)k/parenleftbigg2n
n+k/parenrightbigg
km−1
2
/bracketleftbigg
the + sign is taken when m≡0 (mod 4) or m≡3 (mod 4),
the−sign is taken when m≡2 (mod 4) or m≡1 (mod 4)/bracketrightbigg
BI (177)(18)a, LI (177)(18)
3.862/integraldisplay∞
0/bracketleftbig
cos/parenleftbig
ax2√n/parenrightbig
+s i n/parenleftbig
ax2√n/parenrightbig/bracketrightbig/parenleftbiggsin2x
x2/parenrightbiggn
dx
=√π
(2n−1)!!√
2n/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig/parenleftbig
n−2k+a√n/parenrightbign−1
2
/bracketleftbig
a>√n>0/bracketrightbig
BI (178)(9)
3.866 Trigonometric functions with powers 479
3.863
1./integraldisplay∞
0x2cos/parenleftbig
ax4/parenrightbig
sin/parenleftbig
2bx2/parenrightbig
dx=−π
8/radicalbigg
b3
a3/bracketleftbigg
sin/parenleftbiggb2
2a−π
8/parenrightbigg
J−1
4/parenleftbiggb2
2a/parenrightbigg
+c o s/parenleftbiggb2
2a−π
8/parenrightbigg
J3
4/parenleftbiggb2
2a/parenrightbigg/bracketrightbigg
[a>0,b > 0] ET I 25(22)
2./integraldisplay∞
0x2cos/parenleftbig
ax4/parenrightbig
cos/parenleftbig
2bx2/parenrightbig
dx=−π
8/radicalbigg
b3
a3/bracketleftbigg
sin/parenleftbiggb2
2a+π
8/parenrightbigg
J−3
4/parenleftbiggb2
2a/parenrightbigg
+c o s/parenleftbiggb2
2a+π
8/parenrightbigg
J−1
4/parenleftbiggb2
2a/parenrightbigg/bracketrightbigg
[a>0,b > 0] ET I 25(23)
3.864
1./integraldisplay∞
0sinb
xsinaxdx
x=π
2Y0/parenleftBig
2√
ab/parenrightBig
+K0/parenleftBig
2√
ab/parenrightBig
[a>0,b > 0] WA 204(3)a
2./integraldisplay∞
0cosb
xcosaxdx
x=−π
2Y0/parenleftBig
2√
ab/parenrightBig
+K0/parenleftBig
2√
ab/parenrightBig
[a>0,b > 0]
WA 204(4)a, ET I 24 (12)
3.865
1./integraldisplayu
0/parenleftbig
u2−x2/parenrightbigμ−1
x2μsina
xdx=√π
2/parenleftbigg2
a/parenrightbiggμ−1
2
uμ−3
2Γ(μ)J1
2−μ/parenleftBiga
u/parenrightBig
[a>0,u > 0,0<Reμ<1]
ET II 189(30)
2./integraldisplay∞
u(x−u)μ−1
x2μsina
xdx=/radicalbiggπ
ua1
2−μΓ(μ)sina
2uJμ−1
2/parenleftBiga
2u/parenrightBig
[a>0,u > 0,Reμ>0]
ET II 203(21)
3./integraldisplayu
0/parenleftbig
u2−x2/parenrightbigμ−1
x2μcosa
xdx=−√π
2/parenleftbigg2
a/parenrightbiggμ−1
2
Γ(μ)uμ−3
2Y1
2−μ/parenleftBiga
u/parenrightBig
[a>0,u > 0,0<Reμ<1]
ET II 190(36)
4./integraldisplay∞
u(x−u)μ−1
x2μcosa
xdx=/radicalbiggπ
ua1
2−μΓ(μ)cosa
2uJμ−1
2/parenleftBiga
2u/parenrightBig
[a>0,u > 0,Reμ>0]
ET II 204(26)
3.866
1./integraldisplay∞
0xμ−1sinb2
xsin/parenleftbig
a2x/parenrightbig
dx=π
4/parenleftbiggb
a/parenrightbiggμ
cosecμπ
2[Jμ(2ab)−J−μ(2ab)+I−μ(2ab)−Iμ(2ab)]
[a>0,b > 0,|Reμ|<1]
ET I 322(42)
2./integraldisplay∞
0xμ−1sinb2
xcos/parenleftbig
a2x/parenrightbig
dx=π
4/parenleftbiggb
a/parenrightbiggμ
secμπ
2[Jμ(2ab)+J−μ(2ab)+Iμ(2ab)−I−μ(2ab)]
[a>0,b > 0,|Reμ|<1]
ET I 322(43)
480 Trigonometric Functions 3.867
3./integraldisplay∞
0xμ−1cosb2
xcos/parenleftbig
a2x/parenrightbig
dx=π
4/parenleftbiggb
a/parenrightbiggμ
cosecμπ
2[J−μ(2ab)−Jμ(2ab)+I−μ(2ab)−Iμ(2ab)]
[a>0,b > 0,|Reμ|<1]
ET I 322(44)
3.867
1./integraldisplay1
0cosax−cosa
x
1−x2dx=1
2/integraldisplay∞
0cosax−cosa
x
1−x2dx=π
2sina
[a>0] GW (334)(7a)
2./integraldisplay1
0cosax+c o sa
x
1+x2dx=1
2/integraldisplay∞
0cosax+c o sa
x
1+x2dx=π
2e−a
[a>0] GW (334)(7b)
3.868
1./integraldisplay∞
0sin/parenleftbigg
a2x+b2
x/parenrightbiggdx
x=πJ0(2ab)[ a>0,b > 0]
GW (334)(11a), WA 200(16)
2./integraldisplay∞
0cos/parenleftbigg
a2x+b2
x/parenrightbiggdx
x=−πY0(2ab)[ a>0,b > 0] GW (334)(11a)
3./integraldisplay∞
0sin/parenleftbigg
a2x−b2
x/parenrightbiggdx
x=0 [ a>0,b > 0] GW (334)(11b)
4./integraldisplay∞
0cos/parenleftbigg
a2x−b2
x/parenrightbiggdx
x=2K0(2ab)[ a>0,b > 0] GW (334)(11b)
3.869
1./integraldisplay∞
0sin/parenleftbigg
ax−b
x/parenrightbiggxdx
β2+x2=π
2exp/parenleftbigg
−αβ−b
β/parenrightbigg
[a>0,b > 0,Reβ>0]
ET II 220(42)
2./integraldisplay∞
0cos/parenleftbigg
ax−b
x/parenrightbiggdx
β2+x2=π
2βexp/parenleftbigg
−aβ−b
β/parenrightbigg
[a>0,b > 0,Reβ>0]
ET II 222(58)
3.871
1./integraldisplay∞
0xμ−1sin/bracketleftbigg
a/parenleftbigg
x+b2
x/parenrightbigg/bracketrightbigg
dx=πbμ/bracketleftBig
Jμ(2ab)cosμπ
2−Yμ(2ab)sinμπ
2/bracketrightBig
[a>0,b > 0,Reμ<1]
ET I 319(17)
2./integraldisplay∞
0xμ−1cos/bracketleftbigg
a/parenleftbigg
x+b2
x/parenrightbigg/bracketrightbigg
dx=−πbμ/bracketleftBig
Jμ(2ab)sinμπ
2+Yμ(2ab)cosμπ
2/bracketrightBig
[a>0,b > 0,|Reμ|<1]
ET I 321(35)
3./integraldisplay∞
0xμ−1sin/bracketleftbigg
a/parenleftbigg
x−b2
x/parenrightbigg/bracketrightbigg
dx=2bμKμ(2ab)sinμπ
2[a>0,b > 0,|Reμ|<1]
ET I 319(16)
3.874 Trigonometric functions with powers 481
4./integraldisplay∞
0xμ−1cos/bracketleftbigg
a/parenleftbigg
x−b2
x/parenrightbigg/bracketrightbigg
dx=2bμKμ(2ab)cosμπ
2
[a>0,b > 0,|Reμ|<1]
ET I 321(36)
3.872
1./integraldisplay1
0sin/bracketleftbigg
a/parenleftbigg
x+1
x/parenrightbigg/bracketrightbigg
sin/bracketleftbigg
a/parenleftbigg
x−1
x/parenrightbigg/bracketrightbiggdx
1−x2
=1
2/integraldisplay∞
0sin/bracketleftbigg
a/parenleftbigg
x+1
x/parenrightbigg/bracketrightbigg
sin/bracketleftbigg
a/parenleftbigg
x−1
x/parenrightbigg/bracketrightbiggdx
1−x2=−π
4sin 2a
[a≥0] BI (149)(15), GW (334)(8a)
2./integraldisplay1
0cos/bracketleftbigg
a/parenleftbigg
x+1
x/parenrightbigg/bracketrightbigg
cos/bracketleftbigg
a/parenleftbigg
x−1
x/parenrightbigg/bracketrightbiggdx
1+x2
=1
2/integraldisplay∞
0cos/bracketleftbigg
a/parenleftbigg
x+1
x/parenrightbigg/bracketrightbigg
cos/bracketleftbigg
a/parenleftbigg
x−1
x/parenrightbigg/bracketrightbiggdx
1+x2=π
4e−2a
[a≥0] GW (334)(8b)
3.873
1./integraldisplay∞
0sina2
x2cosb2x2dx
x2=√π
4√
2a/bracketleftbig
sin(2ab) + cos(2 ab)+e−2ab/bracketrightbig
[a>0,b > 0] ET I 24(15)
2./integraldisplay∞
0cosa2
x2cosb2x2dx
x2=√π
4√
2a/bracketleftbig
cos(2ab)−sin(2ab)+e−2ab/bracketrightbig
[a>0,b > 0] ET I 24(16)
3.874
1./integraldisplay∞
0sin/parenleftbigg
a2x2+b2
x2/parenrightbiggdx
x2=√π
2bsin/parenleftBig
2ab+π
4/parenrightBig
[a>0,b > 0]
BI (179)(6)a, GW(334)(10a)
2./integraldisplay∞
0cos/parenleftbigg
a2x2+b2
x2/parenrightbiggdx
x2=√π
2bcos/parenleftBig
2ab+π
4/parenrightBig
[a>0,b > 0]
GI (179)(8)a, GW(334)(10a)
3./integraldisplay∞
0sin/parenleftbigg
a2x2−b2
x2/parenrightbiggdx
x2=−√π
2√
2be−2ab[a≥0,b > 0] GW (335)(10b)
4./integraldisplay∞
0cos/parenleftbigg
a2x2−b2
x2/parenrightbiggdx
x2=√π
2√
2be−2ab[a≥0,b > 0] GW (334)(10b)
5./integraldisplay∞
0sin/parenleftbigg
ax−b
x/parenrightbigg2dx
x2=√
2π
4b[a>0,b > 0] BI (179)(13)a
6./integraldisplay∞
0cos/parenleftbigg
ax−b
x/parenrightbigg2dx
x2=√
2π
4b[a>0,b > 0] BI (179)(14)a
482 Trigonometric Functions 3.875
3.875
1./integraldisplay∞
uxsin/parenleftbig
p√
x2−u2/parenrightbig
x2+a2cosbxdx =π
2exp/parenleftBig
−p/radicalbig
a2+u2/parenrightBig
coshab
[0<b<p ] ET I 27(39)
2./integraldisplay∞
uxsin/parenleftbig
p√
x2−u2/parenrightbig
a2+x2−u2cosbxdx =π
2e−apcos/parenleftBig
b/radicalbig
u2−a2/parenrightBig
[0<b<p , a> 0] ET I 27(38)
3.6/integraldisplay∞
0sin/parenleftbig
p√
a2+x2/parenrightbig
(a2+x2)3/2cosbxdx =πp
2ae−ab[0<p<b , a> 0] ET I 26(29)
3.876
1./integraldisplay∞
0sin/parenleftbig
p√
x2+a2/parenrightbig
√
x2+a2cosbxdx=π
2J0/parenleftBig
a/radicalbig
p2−b2/parenrightBig
[0<b<p ]
=0 [ b>p> 0]
[a>0] ET I 26(30)
2./integraldisplay∞
0cos/parenleftbig
p√
x2+a2/parenrightbig
√
x2+a2cosbxdx=−π
2Y0/parenleftBig
a/radicalbig
p2−b2/parenrightBig
[0<b<p ]
=K0/parenleftBig
a/radicalbig
b2−p2/parenrightBig
[b>p> 0]
[a>0] ET I 26(34)
3./integraldisplay∞
0cos/parenleftbig
p√
x2+a2/parenrightbig
x2+c2cosbxdx =π
2ce−bccos/parenleftBig
p/radicalbig
a2−c2/parenrightBig
[c>0,b > p ] ET I 26(33)
4./integraldisplay∞
0sin/parenleftbig
p√
x2+a2/parenrightbig
(x2+c2)√
x2+a2cosbxdx=π
2ce−bcsin/parenleftbig
p√
a2−c2/parenrightbig
√
a2−c2[c/negationslash=a]
=π
2e−bap
a[c=a]
[b>p , c> 0] ET I 26(31)a
5.6/integraldisplay∞
0cos/parenleftbig
p√
x2+a2/parenrightbig
x2+a2cosbxdx =π
2ae−ab[b>p> 0;a>0] ET I 27(35)a
6.6/integraldisplay∞
0xcos/parenleftbig
p√
x2+a2/parenrightbig
x2+a2sinbxdx =π
2e−ab[a>0, b>p> 0] ET I 85(29)a
7./integraldisplayu
0cos/parenleftbig
p√
u2−x2/parenrightbig
√
u2−x2cosbxdx =π
2J0/parenleftBig
u/radicalbig
b2+p2/parenrightBig
ET I 28(42)
8./integraldisplay∞
ucos/parenleftbig
p√
x2−u2/parenrightbig
√
x2−u2cosbxdx=K0/parenleftBig
u/radicalbig
p2−b2/parenrightBig
[0<b< |p|]
=−π
2Y0/parenleftBig
u/radicalbig
b2−p2/parenrightBig
[b>|p|]
ET I 28(43)
3.881 Trigonometric functions with powers 483
3.877
1./integraldisplayu
0sin/parenleftbig
p√
u2−x2/parenrightbig
4/radicalBig
(u2−x2)3cosbxdx =/radicalbigg
π3p
8J1
4/bracketleftBigu
2/parenleftBig/radicalbig
b2+p2−b/parenrightBig/bracketrightBig
J1
4/bracketleftBigu
2/parenleftBig/radicalbig
b2+p2+b/parenrightBig/bracketrightBig
[b>0,p > 0] ET I 27(40)
2./integraldisplay∞
usin/parenleftbig
p√
x2−u2/parenrightbig
4/radicalBig
(x2−u2)3cosbxdx =−/radicalbigg
π3p
8J1
4/bracketleftBigu
2/parenleftBig
b−/radicalbig
b2−p2/parenrightBig/bracketrightBig
Z1
4/bracketleftBigu
2/parenleftBig
b+/radicalbig
b2−p2/parenrightBig/bracketrightBig
[b>p> 0] ET I 27(41)
3./integraldisplayu
0cos/parenleftbig
p√
u2−x2/parenrightbig
4/radicalBig
(u2−x2)3cosbxdx =/radicalbigg
π3p
8J−1
4/bracketleftBigu
2/parenleftBig/radicalbig
p2+b2−b/parenrightBig/bracketrightBig
J−1
4/bracketleftBigu
2/parenleftBig/radicalbig
p2+b2+b/parenrightBig/bracketrightBig
[u>0,p > 0] ET I 28(44)
4./integraldisplay∞
ucos/parenleftbig
p√
x2−u2/parenrightbig
4/radicalBig
(x2−u2)3cosbxdx =−/radicalbigg
π3p
8J−1
4/bracketleftBigu
2/parenleftBig
b−/radicalbig
b2−p2/parenrightBig/bracketrightBig
Y1
4/bracketleftBigu
2/parenleftBig
b+/radicalbig
b2−p2/parenrightBig/bracketrightBig
[b>p> 0] ET I 28(45)
3.878
1./integraldisplay∞
0sin/parenleftbig
p√
x4+a4/parenrightbig
√
x4+a4cosbx2dx=1
2/radicalbigg/parenleftBigπ
2/parenrightBig3
bJ−1
4/bracketleftbigga2
2/parenleftBig
p−/radicalbig
p2−b2/parenrightBig/bracketrightbigg
J1
4/bracketleftbigga2
2/parenleftBig
p+/radicalbig
p2−b2/parenrightBig/bracketrightbigg
[p>b> 0] ET I 26(32)
2./integraldisplay∞
0cos/parenleftbig
p√
x4+a4/parenrightbig
√
x4+a4cosbx2dx
=−1
2/radicalbigg/parenleftBigπ
2/parenrightBig3
bJ−1
4/bracketleftbigga2
2/parenleftBig
p−/radicalbig
p2−b2/parenrightBig/bracketrightbigg
Y1
4/bracketleftbigga2
2/parenleftBig
p+/radicalbig
p2−b2/parenrightBig/bracketrightbigg
[a>0,p > b > 0] ET I 27(36)
3./integraldisplayu
0cos/parenleftbig
p√
u4−x4/parenrightbig
√
u4−x4cosbx2dx=1
2/radicalbigg/parenleftBigπ
2/parenrightBig3
bJ−1
4/bracketleftbiggu2
2/parenleftBig/radicalbig
p2+b2−p/parenrightBig/bracketrightbigg
J−1
4/bracketleftbiggu2
2/parenleftBig/radicalbig
p2+b2+p/parenrightBig/bracketrightbigg
[p>0,b > 0] ET I 28(46)
3.879/integraldisplay∞
0sinaxpdx
x=π
2p[a>0,p > 0] GW (334)(6)
3.881
1./integraldisplayπ/2
0xsin (atanx)dx=π
4e−a/bracketleftbig
C+l n2 a−e2aEi(−2a)/bracketrightbig
[a>0] BI (205)(9)
2./integraldisplay∞
0sin(atanx)dx
x=π
2/parenleftbig
1−e−a/parenrightbig
[a>0] BI (151)(6)
3./integraldisplay∞
0sin(atanx)c o sxdx
x=π
2/parenleftbig
1−e−a/parenrightbig
[a>0] BI (151)(19)
484 Trigonometric Functions 3.882
4./integraldisplay∞
0cos(atanx)s i nxdx
x=π
2e−a[a>0] BI (151)(20)
5./integraldisplay∞
0sin(atanx)s i n2xdx
x=1+a
2πe−a[a>0] BI (152)(11)
6./integraldisplay∞
0cos(atanx)s i n3xdx
x=1−a
4πe−a[a>0] BI (151)(23)
7./integraldisplay∞
0sin(atanx)t a nx
2cos2xdx
x=1+a
4πe−a[a>0] BI (152)(13)
8./integraldisplayπ/2
0cos(atanx)xdx
sin 2x=−π
4Ei(−a)[ a>0] BI (206)(15)
9./integraldisplayπ/2
0sin(acotx)xdx
sin2x=1−e−a
2aπ [a>0] LI (206)(14)
10./integraldisplayπ/2
0xcos (atanx)t a nxdx=−π
4e−a/bracketleftbig
C+l n2 a+e2aEi(−2a)/bracketrightbig
[a>0] BI (205)(10)
11./integraldisplay∞
0cos(atanx)t a nxdx
x=π
2e−a[a>0] BI (151)(21)
12./integraldisplay∞
0cos(atanx)s i n2xtanxdx
x=1−a
16πe−a[a>0] BI (152)(15)
13./integraldisplay∞
0sin(atanx)t a n2xdx
x=π
2e−a[a>0] BI (152)(9)
14./integraldisplay∞
0cos(atan 2x)tanxdx
x=π
2e−a[a>0] BI (151)(22)
15./integraldisplay∞
0sin(atan2x)c o s22xtanxdx
x=1+a
4πe−a[a>0] BI (152)(13)
16./integraldisplay∞
0sin(atan2x)t a nxtan 2xdx
x=π
2e−a[a>0] BI (152)(10)
17./integraldisplay∞
0sin(atan2x)t a nxcot 2xdx
x=π
2/parenleftbig
1−e−a/parenrightbig
[a>0] BI (180)(6)
3.882
1./integraldisplay∞
0sin/parenleftbig
atan2x/parenrightbigxdx
b2+x2=π
2/bracketleftbig
exp (−atanhb)−e−a/bracketrightbig
[a>0,b > 0] BI (160)(22)
2./integraldisplay∞
0cos/parenleftbig
atan2x/parenrightbig
cosxdx
b2+x2=π
2b/bracketleftbig
coshbexp(−atanhb)−e−asinhb/bracketrightbig
[a>0,b > 0] BI (163)(3)
3./integraldisplay∞
0cos/parenleftbig
atan2x/parenrightbig
cosec2 xxdx
b2+x2=π
2s i n h2 bexp (−atanhb)
[a>0,b > 0] BI (191)(10)
3.892 Trigonometric functions and exponentials 485
4./integraldisplay∞
0cos/parenleftbig
atan2x/parenrightbig
tanxxdx
b2+x2=π
2c os h b/bracketleftbig
e−acoshb−exp (−atanhb)s i n h b/bracketrightbig
[a>0,b > 0] BI (163)(4)
5.11/integraldisplay∞
0cos/parenleftbig
atan2x/parenrightbig
cotxxdx
b2+x2=π
2/bracketleftbig
cothbexp(−atanhb)−e−a/bracketrightbig
[a>0,b > 0] BI (163)(5)
6./integraldisplay∞
0cos/parenleftbig
atan2x/parenrightbig
cot2xxdx
b2+x2=π
2/bracketleftbig
coth2 bexp (−atanhb)−e−a/bracketrightbig
[a>0,b > 0] BI (191)(11)
3.883
1./integraldisplay1
0cos(alnx)dx
(1 +x)2=aπ
2s in h aπBI (404)(4)
2./integraldisplay1
0xμ−1sin(βlnx)dx=−β
β2+μ2[Reμ>|Imβ|] ET I 319(19)
3./integraldisplay1
0xμ−1cos(βlnx)dx=μ
β2+μ2[Reμ>|Imβ|] ET I 321(38)
3.88411/integraldisplay∞
−∞sina/radicalbig
|x|
x−bsignxdx=π/bracketleftBig
exp/parenleftBig
−a/radicalbig
|−b|/parenrightBig
+e x p/parenleftBig
−a/radicalbig
|b|/parenrightBig/bracketrightBig
[a>0,Imb/negationslash=0 ] ET II 253(46)
3.89–3.91 Trigonometric functions and exponentials
3.891
1./integraldisplay2π
0eimxsinnxdx =0 [ m/negationslash=n;o rm=n=0 ]
=πi [m=n/negationslash=0 ]
2./integraldisplay2π
0eimxcosnxdx =0 [ m/negationslash=n]
=π [m=n/negationslash=0 ]
=2π [m=n=0 ]
3.892
1.11/integraldisplayπ
0eiβxsinν−1xdx=πeiβπ
2
2ν−1νB/parenleftbiggν+β+1
2,ν−β+1
2/parenrightbigg
[Reν>−1] NH 158, EH I 12(29)
2./integraldisplayπ
2
−π
2eiβxcosν−1xdx=π
2ν−1νB/parenleftbiggν+β+1
2,ν−β+1
2/parenrightbigg
[Reν>−1] GW (335)(19)
486 Trigonometric Functions 3.893
3.6/integraldisplayπ/2
0ei2βxsin2μxcos2νxdx=1
22μ+2ν+1/braceleftbig
exp/bracketleftbig
iπ/parenleftbig
β−ν−1
2/parenrightbig/bracketrightbig
B(β−μ−ν,2ν+1 )
×F(−2μ, β−μ−ν;1+β−μ+ν;−1) + exp/bracketleftbig
iπ/parenleftbig
μ+1
2/parenrightbig/bracketrightbig
×B(β−μ−ν,2μ+1 )F(−2ν,β−μ−ν;1+β+μ−ν;−1)/bracerightbigg
/bracketleftbig
Reμ>−1
2,Reν>−1
2/bracketrightbig
EH I 80(6)
4./integraldisplayπ
0ei2βxsin2μxcos2νxdx=πexp [iπ(β−ν)]F(−2ν,β−μ−ν;1+β+μ−ν;−1)
4μ+ν(2μ+1 )B ( 1 −β+μ+ν,1+β+μ−ν)
EH I 80(8)
5./integraldisplayπ/2
0ei(μ+ν)xsinμ−1xcosν−1xdx=eiμπ
2B(μ, ν)
=1
2μ+ν−1eiμπ
2/braceleftbigg1
μF(1−ν,1;μ+1 ;−1) +1
νF(1−μ,1;ν+1 ;−1)/bracerightbigg
[Reμ>0,Reν>0] EH I 80(7)
3.893
1.8/integraldisplay∞
0e−pxsin(qx+λ)dx=1
p2+q2(qcosλ+psinλ)[ R e p>0] BI (261)(3)
2.8/integraldisplay∞
0e−pxcos(qx+λ)dx=1
p2+q2(pcosλ−qsinλ)[ R e p>0] BI (261)(4)
3./integraldisplay∞
0e−xcostcos(t−xsint)dx=1 BI (261)(7)
4.8/integraldisplay∞
0e−βxsinax
sinbxdx=R e/braceleftbigg1
2bi/bracketleftbigg
ψ/parenleftbigga+b
2b−iβ
2b/parenrightbigg
−ψ/parenleftbiggb−a
2b−iβ
2b/parenrightbigg/bracketrightbigg/bracerightbigg
[Reβ>0,b/negationslash=0 ] GW (335)(15)
5.8/integraldisplay∞
0e−2pxsin[(2n+1 )x]
sinxdx=1
2p+n/summationdisplay
k=1p
p2+k2[Rep>0] BI (267)(15)
6.8/integraldisplay∞
0e−pxsin2nx
sinxdx=2pn−1/summationdisplay
k=01
p2+( 2k+1 )2[Rep>0] GW (335)(15c)
7./integraldisplay∞
0e−pxcos[(2 n+1 )x]t anxdx=2n+1
p2+( 2n+1 )2+(−1)n2n−1/summationdisplay
k=0(−1)k(2k+1 )
p2+( 2k+1 )2
[p>0] LI (267)(16)
3.894/integraldisplayπ
−π/bracketleftBig
β+/radicalbig
β2−1c osx/bracketrightBigν
einxdx=2πΓ(ν+1 )Pm
ν(β)
Γ(ν+m+1 )
[Reβ>0] ET I 157(15)
3.895
1./integraldisplay∞
0e−βxsin2mxdx=(2m)!
β(β2+22)(β2+42)···[β2+( 2m)2]
[Reβ>0] FI II 615, WA 620a
3.895 Trigonometric functions and exponentials 487
2.10/integraldisplayπ
0e−pxsin2mxdx=(2m)! (1−e−pπ)
p(p2+22)(p2+42)···[p2+( 2m)2]GW (335)(4a)
3.10/integraldisplayπ/2
0e−pxsin2mxdx
=(2m)!
p(p2+22)(p2+42)···[p2+( 2m)2]
×/braceleftBigg
1−e−pπ
2/bracketleftBigg
1+p2
2!+p2/parenleftbig
p2+22/parenrightbig
4!+···+p2/parenleftbig
p2+22/parenrightbig
···/bracketleftbig
p2+( 2m−2)2/bracketrightbig
(2m)!/bracketrightBigg/bracerightBigg
BI (270)(4)
4./integraldisplay∞
0e−βxsin2m+1xdx=(2m+1 ) !
(β2+12)(β2+32)···[β2+( 2m+1 )2]
[Reβ>0] FI II 615, WA 620a
5.10/integraldisplayπ
0e−pxsin2m+1xdx=(2m+ 1)! (1 + e−pπ)
(p2+12)(p2+32)···[p2+( 2m+1 )2]GW (335)(4b)
6.8/integraldisplayπ/2
0e−pxsin2m+1xdx
=(2m+1 ) !
(p2+12)(p2+32)···[p2+( 2m+1 )2]
×/braceleftBigg
1−pe−pπ
2/bracketleftBigg
1+p2+12
3!+···+/parenleftbig
p2+12/parenrightbig/parenleftbig
p2+32/parenrightbig
···/bracketleftbig
p2+( 2m−1)2/bracketrightbig
(2m+1 ) !/bracketrightBigg/bracerightBigg
BI (270)(5)
7./integraldisplay∞
0e−pxcos2mxdx=(2m)!
p(p2+22)···[p2+( 2m)2]
×/braceleftBigg
1+p2
2!+p2/parenleftbig
p2+22/parenrightbig
4!+···+p2/parenleftbig
p2+22/parenrightbig
···/bracketleftbig
p2+( 2m−2)2/bracketrightbig
(2m)!/bracerightBigg
[p>0] BI (262)(3)
8.10/integraldisplayπ/2
0e−pxcos2mxdx
=(2m)!
p(p2+22)···[p2+( 2m)2]
×/braceleftBigg
−e−pπ
2+1+p2
2!+p2/parenleftbig
p2+22/parenrightbig
4!+···+p2/parenleftbig
p2+22/parenrightbig
···/bracketleftbig
p2+( 2m−2)2/bracketrightbig
(2m)!/bracerightBigg
BI (270)(6)
9.7/integraldisplay∞
0e−pxcos2m+1xdx
=(2m+1 ) !p
(p2+12)(p2+32)···[p2+( 2m+1 )2]
×/braceleftBigg
1+p2+12
3!+/parenleftbig
p2+12/parenrightbig/parenleftbig
p2+32/parenrightbig
5!+···+/parenleftbig
p2+12/parenrightbig/parenleftbig
p2+32/parenrightbig
···/bracketleftbig
p2+( 2m−1)2/bracketrightbig
(2m+1 ) !/bracerightBigg
[p>0] BI (262)(4)
488 Trigonometric Functions 3.896
10.11/integraldisplayπ/2
0e−pxcos2m+1xdx
=(2m+1 ) !
(p2+12)(p2+32)···[p2+( 2m+1 )2]
×/braceleftBigg
e−pπ
2+p/bracketleftBigg
1+p2+12
3!+···+/parenleftbig
p2+1/parenrightbig/parenleftbig
p2+32/parenrightbig
···/bracketleftbig
p2+( 2m−1)2/bracketrightbig
(2m+1 ) !/bracketrightBigg/bracerightBigg
BI (270)(7)
11.8/integraldisplay∞
0e−βxsinnax/braceleftbiggsinbx
cosbx/bracerightbigg
dx=2−n−2
a(n+1 )e1
4(1∓1+2n)πi
×⎧
⎨
⎩/parenleftBigg
b+na+iβ
2a
n+1/parenrightBigg−1
±(−1)n/parenleftBigg
b+na−iβ
2a
n+1/parenrightBigg−1⎫
⎬
⎭
[a>0,b > 0,Reβ>0]
12./integraldisplay∞
0e−axcos2mxdx =a2+2m2
a(a2+4m2)DW61 (861.06)
13./integraldisplay∞
0e−axcosmxcosnxdx =a/parenleftbig
a2+m2+n2/parenrightbig
(a2+(m−n)2)(a2+(m+n)2)DW61 (861.15)
14./integraldisplay∞
0e−axsinmxcosnxdx =m/parenleftbig
a2+m2−n2/parenrightbig
(a2+(m−n)2)(a2+(m+n)2)DW61 (861.14)
15./integraldisplay∞
0e−axsin2mxdx =2m
a(a2+4m2)[a>0] DW61 (861.10)
16./integraldisplay∞
0e−axsinmxsinnxdx =2amn
[a2+(m−n)2][a2+(m+n)2]DW61 (861.13)
3.896
1./integraldisplay∞
−∞e−q2x2sin[p(x+λ)]dx=√π
qe−p2
4q2sinpλ BI (269)(2)
2./integraldisplay∞
−∞e−q2x2cos[p(x+λ)]dx=√π
qe−p2
4q2cospλ BI (269)(3)
3./integraldisplay∞
0e−ax2sinbxdx=b
2aexp/parenleftbigg
−b2
4a/parenrightbigg
1F1/parenleftbigg1
2;3
2;b2
4a/parenrightbigg
=b
2a1F1/parenleftbigg
1;3
2;−b2
4a/parenrightbigg
ET I 73(18)
=b
2a∞/summationdisplay
k=11
(2k−1)!!/parenleftbigg
−b2
2a/parenrightbiggk−1
[a>0] FI II 720
4./integraldisplay∞
0e−βx2cosbxdx =1
2/radicalbiggπ
βexp/parenleftbigg
−b2
4β/parenrightbigg
[Reβ>0] BI (263)(2)
3.911 Trigonometric functions and exponentials 489
3.897
1.8/integraldisplay∞
0e−βx2−γxsinbxdx=−i
4/radicalbiggπ
β/braceleftbigg
exp(γ−ib)2
4β/bracketleftbigg
1−Φ/parenleftbiggγ−ib
2√β/parenrightbigg/bracketrightbigg
−exp(γ+ib)2
4β/bracketleftbigg
1−Φ/parenleftbiggγ+ib
2√β/parenrightbigg/bracketrightbigg/bracerightbigg
[Reβ>0] ET I 74(27)
2./integraldisplay∞
0e−βx2−γxcosbxdx=1
4/radicalbiggπ
β/braceleftBigg
exp(γ−ib)2
4β/bracketleftbigg
1−Φ/parenleftbiggγ−ib
2√β/parenrightbigg/bracketrightbigg
+e x p(γ+ib)2
4β/bracketleftbigg
1−Φ/parenleftbiggγ+ib
2√β/parenrightbigg/bracketrightbigg/bracerightbigg
[Reβ>0] ET I 15(16)
3.898
1./integraldisplay∞
0e−βx2sinaxsinbxdx =1
4/radicalbiggπ
β/braceleftbigg
e−(a−b)2
4β−e−(a+b)2
4β/bracerightbigg
[Reβ>0] BI (263)(4)
2./integraldisplay∞
0e−βx2cosaxcosbxdx =1
4/radicalbiggπ
β/braceleftbigg
e−(a−b)2
4β+e−(a+b)2
4β/bracerightbigg
[Reβ>0] BI (263)(5)
3.8/integraldisplay∞
0e−px2sin2axdx =1
4/radicalbiggπ
p/parenleftBig
1−e−a2
p/parenrightBig
[Rep>0] BI (263)(6)
3.899
1.7/integraldisplay∞
0ep2x2sin[(2n+1 )x]
sinxdx=√π
p/bracketleftBigg
1
2+n/summationdisplay
k=1e−(k
p)2/bracketrightBigg
[p>0] BI (267)(17)
2./integraldisplay∞
0e−p2x2cos[(4 n+1 )x]
cosxdx=√π
p/bracketleftBigg
1
2+2n/summationdisplay
k=0(−1)ke−(k
p)2/bracketrightBigg
[p>0] BI (267)(18)
3./integraldisplay∞
0e−px2dx
1−2acosx+a2=/radicalBig
π
p
1−a2/braceleftBigg
1
2+∞/summationdisplay
k=1akexp/parenleftbigg
−k2
4p/parenrightbigg/bracerightBigg/bracketleftbig
a2<1,p > 0/bracketrightbig
EI (266)(1)
=/radicalBig
π
p
a2−1/braceleftBigg
1
2+∞/summationdisplay
k=1a−kexp/parenleftbigg
−k2
4p/parenrightbigg/bracerightBigg/bracketleftbig
a2>1,p > 0/bracketrightbig
LI (266)(1)
3.911
1./integraldisplay∞
0sinax
eβx+1dx=1
2a−π
2βsinhaπ
β[a>0,Reβ>0] BI (264)(1)
2./integraldisplay∞
0sinax
eβx−1dx=π
2βcoth/parenleftbiggπa
β/parenrightbigg
−1
2a[a>0,Reβ>0] BI (264)(2), WH
3.11/integraldisplay∞
0sinax
ex−1ex/2dx=1
2πtanh(aπ)[ a>0] ET I 73(13)
490 Trigonometric Functions 3.912
4./integraldisplay∞
0sinax
1−e−xe−nxdx=π
2−1
2a+π
e2πa−1−n−1/summationdisplay
k=1a
a2+k2
[a>0] BI (264)(8)
5./integraldisplay∞
0sinax
eβx−eγxdx=1
2i(β−γ)/bracketleftbigg
ψ/parenleftbiggβ+ia
β−γ/parenrightbigg
−ψ/parenleftbiggβ−ia
β−γ/parenrightbigg/bracketrightbigg
[Reβ>0,Reγ>0] GW (335)(8)
6./integraldisplay∞
0sinaxdx
eβx(e−x−1)=i
2[ψ(β+ia)−ψ(β−ia)] [Re β>−1] ET 73(15)
3.912
1./integraldisplay∞
0e−βx/parenleftbig
1−e−γx/parenrightbigν−1sinaxdx =−i
2γ/bracketleftbigg
B/parenleftbigg
ν,β−ia
γ/parenrightbigg
−B/parenleftbigg
ν,β+ia
γ/parenrightbigg/bracketrightbigg
[Reβ>0,Reγ>0,Reν>0,a > 0]ET I 73(17)
2./integraldisplay∞
0e−βx/parenleftbig
1−e−γx/parenrightbigν−1cosaxdx =1
2γ/bracketleftbigg
B/parenleftbigg
ν,β−ia
γ/parenrightbigg
+B/parenleftbigg
ν,β+ia
γ/parenrightbigg/bracketrightbigg
[Reβ>0,Reγ>0,Reν>0,a > 0]ET I 15(10)
3.913
1./integraldisplayπ
2
−π
2eiβxcosνx/parenleftbig
β2eix+ν2e−ix/parenrightbigμdx=π2F1/parenleftBig
−μ,β
2−ν
2−μ
2;1+β
2+ν
2−μ
2;β2
ν2/parenrightBig
2ν(ν+1 )B/parenleftbigg
1+β
2+ν
2−μ
2,1−β
2+ν
2+μ
2/parenrightbigg
[Reν>−1,|ν|>|β|] EH I 81(11)a
2.11/integraldisplayπ
2
−π
2e−iuxcosμx/parenleftbig
a2eix+b2e−ix/parenrightbigνdx
=πb2ν2F1/parenleftBig
−ν,−u+μ+ν
2;1+μ−ν−u
2;a2
b2/parenrightBig
2μ(μ+1 )B/parenleftbigg
1−u+ν−μ
2,1+u+μ+ν
2/parenrightbigg/bracketleftbig
fora2<b2/bracketrightbig
=πa2ν2F1/parenleftBig
−ν,u−μ−ν
2;1+μ−ν+u
2;b2
a2/parenrightBig
2μ(μ+1 )B/parenleftbigg
1+u+μ−ν
2,1+μ+ν−u
2/parenrightbigg/bracketleftbig
forb2<a2/bracketrightbig
[Reμ>−1] ET I 122(31)a
3.914
1./integraldisplay∞
0e−β√
γ2+x2cosbxdx =βγ/radicalbig
β2+b2K1/parenleftBig
γ/radicalbig
β2+b2/parenrightBig
[Reβ>0,Reγ>0] ET I 16(26)
2./integraldisplay∞
0/radicalbig
γ2+x2e−β√
γ2+x2cosbxdx =β2γ2
A2K0(γA)+/parenleftbigg2β2γ
A3−γ
A/parenrightbigg
K1(γA)
/bracketleftBig
A=/radicalbig
β2+b2/bracketrightBig
3.915 Trigonometric functions and exponentials 491
3./integraldisplay∞
0/parenleftbig
γ2+x2/parenrightbig
e−β√
γ2+x2cosbxdx
=/parenleftbigg
−3βγ2
A2+4β3γ2
A4/parenrightbigg
K0(γA)+/parenleftbigg
−6βγ
A3+8β3γ
A5+β3γ3
A3/parenrightbigg
K1(γA)
/bracketleftBig
A=/radicalbig
β2+b2/bracketrightBig
4./integraldisplay∞
0e−β√
γ2+x2
/radicalbig
γ2+x2cosbxdx =K0/parenleftBig
γ/radicalbig
β2+b2/parenrightBig
[Reβ>0,Reγ>0,b > 0]
ET I 16(27)
5./integraldisplay∞
0/parenleftBigg
1
β(γ2+x2)3/2+1
γ2+x2/parenrightBigg
e−β√
γ2+x2cosbxdx =1
βγ/radicalbig
β2+b2K1/parenleftBig
γ/radicalbig
β2+b2/parenrightBig
(6.726(4))
6./integraldisplay∞
0xe−β√
γ2+x2sinbxdx =bβγ2
β2+b2K2/parenleftBig
γ/radicalbig
β2+b2/parenrightBig
ET I 175(35)
7./integraldisplay∞
0x/radicalbig
γ2+x2e−β√
γ2+x2sinbxdx
=/parenleftbigg
−bγ2
A2+4bβ2γ2
A4/parenrightbigg
K0(γA)+/parenleftbigg
−2bγ
A3+8bβ2γ
A5+bβ2γ3
A3/parenrightbigg
K1(γA)
/bracketleftBig
A=/radicalbig
β2+b2/bracketrightBig
8./integraldisplay∞
0/parenleftbig
γ2+x2/parenrightbig
e−β√
γ2+x2xsinbxdx=/parenleftbigg
−12bβγ2
A4+24bβ3γ2
A6+bβ3γ4
A4/parenrightbigg
K0(γA)
+/parenleftbigg
−24bβγ
A5+48bβ3γ
A7−3bβγ3
A3+8bβ3γ3
A5/parenrightbigg
K1(γA)
/bracketleftBig
A=/radicalbig
β2+b2/bracketrightBig
9./integraldisplay∞
0xe−β√
γ2+x2
/radicalbig
γ2+x2sinbxdx =γb/radicalbig
β2+b2K1/parenleftBig
γ/radicalbig
β2+b2/parenrightBig
ET I 75(36)
10./integraldisplay∞
0/parenleftBigg
1
β(γ2+x2)3/2+1
γ2+x2/parenrightBigg
e−β√
γ2+x2xsinbxdx =b
βK0/parenleftBig
γ/radicalbig
β2+b2/parenrightBig
(6.726(3))
3.915
1./integraldisplayπ
0eacosxsinxdx=2
asinha GW (337)(15c)
2./integraldisplayπ
0eiβcosxcosnxdx =inπJn(β) EH II 81(2)
3.3/integraldisplayπ
2
−π
2eiβsinxcos2νxdx=√π/parenleftbigg2
β/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Jν(β)/bracketleftbig
Reν>−1
2/bracketrightbig
EH II 81(6)
4./integraldisplayπ
0e±βcosxsin2νxdx=√π/parenleftbigg2
β/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Iν(β)/bracketleftbig
Reν>−1
2/bracketrightbig
GW (337)(15b)
492 Trigonometric Functions 3.916
5./integraldisplayπ
0eiβcosxsin2νxdx=√π/parenleftbigg2
β/parenrightbiggν
Γ/parenleftbigg
ν+1
2/parenrightbigg
Jν(β)/bracketleftbig
Reν>−1
2/bracketrightbig
WA 34(2), WA 60(6)
3.916
1./integraldisplayπ/2
0e−p2tanxsinx
2√cosx
sin 2xdx=/bracketleftbigg
C(p)−1
2/bracketrightbigg2
+/bracketleftbigg
S(p)−1
2/bracketrightbigg2
NT 33(18)a
2./integraldisplayπ/2
0exp (−ptanx)dx
sin2x+acos2x+a=−1
2eapEi(−ap)[ p>0],(cf.3552 4a n d6 )
BI (273)(11)
3./integraldisplayπ/2
0exp (−pcotx)dx
sin2x+acos2x−a=−1
2e−apEi(ap)[ p>0],(cf.3.552 4a n d6 )
BI (273)(12)
4./integraldisplayπ/2
0exp(−ptanx)sin2xdx
(1−a2)−2a2cos2x−(1 +a2)cos22x=−1
4/bracketleftbig
e−apEi(ap)+eapEi(−ap)/bracketrightbig
[p>0] BI (273)(13)
5./integraldisplayπ/2
0exp (−pcotx)s i n2xdx
(1−a2)+2a2cos2x−(1 +a2)cos22x=−1
4/bracketleftbig
e−apEi(ap)+eapEi(−ap)/bracketrightbig
[p>0] BI (273)(14)
3.917
1./integraldisplayπ/2
0e−2βcotxcosν−1/2xsin−(ν+1)xsin/bracketleftbigg
β−/parenleftbigg
ν−1
2/parenrightbigg
x/bracketrightbigg
dx=√π
2(2β)νΓ/parenleftbigg
ν+1
2/parenrightbigg
Jν(β)
/bracketleftbig
Reν>−1
2/bracketrightbig
WA 186(7)
2./integraldisplayπ/2
0e−2βcotxcosν−1/2xsin−(ν+1)xcos/bracketleftbigg
β−/parenleftbigg
ν−1
2/parenrightbigg
x/bracketrightbigg
dx=√π
2(2β)νΓ/parenleftbigg
ν+1
2/parenrightbigg
Yν(β)
/bracketleftbig
Reν>−1
2/bracketrightbig
WA 186(8)
3.918
1./integraldisplayπ/2
0cosμx
sin2μ+2xeiγ(β−μx)−2βcotxdx=iγ
2/radicalbiggπ
2β(2β)−μΓ(μ+1 )H(ε)
μ+1
2(β)
/bracketleftbig
ε=1,2,γ=(−1)ε+1,Reβ>0,Reμ>−1/bracketrightbig
GW (337)(16)
2./integraldisplayπ/2
0cosμxsin(β−μx)
sin2μ+2xe−2βcotxdx=1
2/radicalbiggπ
2β(2β)−μΓ(μ+1 )Jμ+1
2(β)
[Reβ>0,Reμ>−1] WH
3./integraldisplayπ/2
0cosμxcos(β−μx)
sin2μ+2xe−2βcotxdx=−1
2/radicalbiggπ
2β(2β)−μΓ(μ+1 )Yμ+1
2(β)
[Reβ>0,Reμ>−1]GW (337)(17b)
3.922 Trigonometric functions and exponentials 493
3.919
1./integraldisplayπ/2
0sin 2nx
sin2n+2x·dx
exp (2 πcotx)−1=(−1)n−12n−1
4(2n+1 )BI (275)(6), LI (275)(6)
2./integraldisplayπ/2
0sin 2nx
sin2n+2xdx
exp (πcotx)−1=(−1)n−1n
2n+1BI (275)(7), LI (275)(7)
3.92 Trigonometric functions of more complicated arguments combined with
exponentials
3.9216
1./integraldisplay∞
0e−γxcosax2(cosγx−sinγx)dx=/radicalbiggπ
8aexp/parenleftbigg
−γ2
2a/parenrightbigg
[a>0,Reγ≥|Imγ|] ET I 26(28)
2.10/integraldisplayπ/4
0∞/productdisplay
n=1exp/bracketleftbigg
−1
ntan2nx/bracketrightbigg
=π
2−1
3.10/integraldisplayπ/2
0exp/bracketleftBigg
−∞/summationdisplay
n=11
nsin2nx/bracketrightBigg
=/integraldisplayπ/2
0exp/bracketleftBigg
−∞/summationdisplay
n=11
ncos2nx/bracketrightBigg
=π
4
3.922
1./integraldisplay∞
0e−βx2sinax2dx=1
2/integraldisplay∞
−∞e−βx2sinax2dx=/radicalbiggπ
8/radicalBigg/radicalbig
β2+a2−β
β2+a2
=√π
24/radicalbig
β2+a2sin/parenleftbigg1
2arctana
β/parenrightbigg
[Reβ>0,a > 0]FI II 750, BI (263)(8)
2./integraldisplay∞
0e−βx2cosax2dx=1
2/integraldisplay∞
−∞e−βx2cosax2dx=/radicalbiggπ
8/radicalBigg/radicalbig
β2+a2+β
β2+a2
=√π
24/radicalbig
β2+a2cos/parenleftbigg1
2arctana
β/parenrightbigg
[Reβ>0,a > 0]FI II 750, BI (263)(9)
[In formulas 3.922 3a n d4 , a>0,b>0, Reβ>0, and
A=b2
4(a2+β2),B =/radicalbigg
1
2/parenleftBig/radicalbig
β2+a2+β/parenrightBig
,C =/radicalbigg
1
2/parenleftBig/radicalbig
β2+a2−β/parenrightBig
.
Ifais complex, then Re β>|Ima|.]
3./integraldisplay∞
0e−βx2sinax2cosbxdx=−1
2/radicalbiggπ
β2+a2e−Aβ(BsinAa−CcosAa)
=√π
24/radicalbig
β2+a2exp/parenleftbigg
−βb2
4(β2+a2)/parenrightbigg
sin/braceleftbigg1
2arctana
β−ab2
4(β2+a2)/bracerightbigg
LI (263)(10), GW (337)(5)
494 Trigonometric Functions 3.923
4./integraldisplay∞
0e−βx2cosax2cosbxdx=1
2/radicalbiggπ
β2+a2e−Aβ(BcosAa+CsinAa)
=√π
24/radicalbig
β2+a2exp/parenleftbigg
−βb2
4(β2+a2)/parenrightbigg
cos/braceleftbigg1
2arctana
β−ab2
4(β2+a2)/bracerightbigg
LI (263)(11), GW (337)(5)
3.923
1./integraldisplay∞
−∞exp/bracketleftbig
−/parenleftbig
ax2+2bx+c/parenrightbig/bracketrightbig
sin/parenleftbig
px2+2qx+r/parenrightbig
dx
=√π
4/radicalbig
a2+p2expa/parenleftbig
b2−ac/parenrightbig
−/parenleftbig
aq2−2bpq+cp2/parenrightbig
a2+p2
×sin/braceleftBigg
1
2arctanp
a−p/parenleftbig
q2−pr/parenrightbig
−/parenleftbig
b2p−2abq+a2r/parenrightbig
a2+p2/bracerightBigg
[a>0] GW (337)(3), BI (296)(6)
2./integraldisplay∞
−∞exp/bracketleftbig
−/parenleftbig
ax2+2bx+c/parenrightbig/bracketrightbig
cos/parenleftbig
px2+2qx+r/parenrightbig
dx
=√π
4/radicalbig
a2+p2expa/parenleftbig
b2−ac/parenrightbig
−/parenleftbig
aq2−2bpq+cp2/parenrightbig
a2+p2
×cos/braceleftBigg
1
2arctanp
a−p/parenleftbig
q2−pr/parenrightbig
−/parenleftbig
b2p−2abq+a2r/parenrightbig
a2+p2/bracerightBigg
[a>0] GW (337)(3), BI (269)(7)
3.924
1./integraldisplay∞
0e−βx4sinbx2dx=π
4/radicalBigg
b
2βexp/parenleftbigg
−b2
8β/parenrightbigg
I1
4/parenleftbiggb2
8β/parenrightbigg
[Reβ>0,b > 0] ET 73(22)
2./integraldisplay∞
0e−βx4cosbx2dx=π
4/radicalBigg
b
2βexp/parenleftbigg
−b2
8β/parenrightbigg
I−1
4/parenleftbiggb2
8β/parenrightbigg
[Reβ>0,b > 0] ET I 15(12)
3.925
1./integraldisplay∞
0e−p2
x2sin 2a2x2dx=1
2/integraldisplay∞
−∞e−p2
x2sin 2a2x2dx=√π
4ae−2ap(cos2ap+s i n2 ap)
[a>0,b > 0] BI (268)(12)
2./integraldisplay∞
0e−p2
x2cos2a2x2dx=1
2/integraldisplay∞
−∞e−p2
x2cos 2a2x2dx=√π
4ae−2ap(cos2ap−sin 2ap)
[a>0,b > 0] BI (268)(13)
3.932 Trigonometric and exponential functions 495
3.926 Notation :
u=/radicalBigg/radicalbig
a2+β2+β
2,v =/radicalBigg/radicalbig
a2+β2−β
2
1./integraldisplay∞
0e−(βx2+γ
x2)sinax2dx=1
2/radicalbiggπ
a2+β2e−2u√γ[vcos (2v√γ)+usin (2v√γ)]
[Reβ>0,Reγ>0] BI (268)(14)
2./integraldisplay∞
0e−(βx2+γ
x2)cosax2dx=1
2/radicalbiggπ
a2+β2e−2u√γ[ucos (2v√γ)−vsin(2v√γ)]
[Reβ>0,Reγ>0] BI (268)(15)
3.927/integraldisplay∞
0e−p
xsin2a
xdx=aarctan2a
p+p
4lnp2
p2+4a2[a>0,p > 0] LI (268)(4)
3.928
1./integraldisplay∞
0exp/bracketleftbigg
−/parenleftbigg
p2x2+q2
x2/parenrightbigg/bracketrightbigg
sin/parenleftbigg
a2x2+b2
x2/parenrightbigg
dx=√π
2re−2rscos(A+B)sin{A+2rssin(A+B)}
BI (268)(22)
2./integraldisplay∞
0exp/bracketleftbigg
−/parenleftbigg
p2x2+q2
x2/parenrightbigg/bracketrightbigg
cos/parenleftbigg
a2x2+b2
x2/parenrightbigg
dx=√π
2re−2rscos(A+B)cos{A+2rssin(A+B)}
BI (268)(23)
3.929/integraldisplay∞
0/bracketleftBig
e−xcos/parenleftbig
p√x/parenrightbig
+pe−x2sinpx/bracketrightBig
dx=1 LI (268)(3)
Notation : For the formulas in 3.928 :a2+p2>0,r=4/radicalbig
a4+p4,s=4/radicalbig
b4+q4,A=1
2arctana2
p2,a n d
B=1
2arctanb2
q2.
3.93 Trigonometric and exponential functions of trigonometric functions
3.931
1./integraldisplayπ/2
0e−pcosxsin (psinx)dx=E i (−p)−ci(p) NT 13(27)
2./integraldisplayπ
0e−pcosxsin(psinx)dx=−/integraldisplay0
−πe−pcosxsin (psinx)dx=−2s h i(p) GW (337)(11b)
3./integraldisplayπ/2
0e−pcosxcos(psinx)dx=−si(p) NT 13(26)
4./integraldisplayπ/2
0e−pcosxcos(psinx)dx=1
2/integraldisplay2π
0e−pcosxcos(psinx)dx=π GW (337)(11a)
3.932
1./integraldisplayπ
0epcosxsin(psinx)s i nmxdx =1
2/integraldisplay2π
0epcosxsin (psinx)s i nmxdx =π
2·pm
m!
BI (277)(7), GW (337)(13a)
496 Trigonometric Functions 3.933
2./integraldisplayπ
0epcosxcos(psinx)cosmxdx =1
2/integraldisplay2π
0epcosxcos (psinx)cosmxdx =π
2·pm
m!
BI (277)(8), GW (337)(13b)
3.933/integraldisplayπ
0epcosxsin(psinx)c o s e c xdx=πsinhp BI (278)(1)
3.934
1./integraldisplayπ
0epcosxsin(psinx)t a nx
2dx=π(1−ep) BI (271)(8)
2./integraldisplayπ
0epcosxsin(psinx)c o tx
2dx=π(ep−1) BI (272)(5)
3.935/integraldisplayπ
0epcosxcos(psinx)sin 2nx
sinxdx=πn−1/summationdisplay
k=0p2k+1
(2k+1 ) ![p>0] LI (278)(3)
3.936
1./integraldisplay2π
0epcosxcos (psinx−mx)dx=2/integraldisplayπ
0epcosxcos(psinx−mx)dx=2πpm
m!
BI (277)(9), GW (337)(14a)
2./integraldisplay2π
0epsinxsin(pcosx+mx)dx=2πpm
m!sinmπ
2[p>0] GW (337)(14b)
3./integraldisplay2π
0epsinxcos(pcosx+mx)dx=2πpm
m!cosmπ
2[p>0] GW (337)(14b)
4./integraldisplay2π
0ecosxsin (mx−sinx)dx=0 WH
5./integraldisplayπ
0eβcosxcos (ax+βsinx)dx=β−asin(aπ)γ(a,β) EH II 137(2)
3.937 Notation : In formulas 3.937 1a n d2 ,( b−p)2+(a+q)2>0,m=0,1,2,...,A=p2−q2+a2−b2,
B=2 (pq+ab),C=p2+q2−a2−b2,a n d D=2 (ap+bq).
1.11/integraldisplay2π
0exp (pcosx+qsinx)sin(acosx+bsinx−mx)dx
=iπ/bracketleftbig
(b−p)2+(a+q)2/bracketrightbig−m
2/braceleftBig
(A+iB)m/2Im/parenleftBig√
C−iD/parenrightBig
−(A−iB)m/2Im/parenleftBig√
C+iD/parenrightBig/bracerightBig
GW (337)(9b)
2./integraldisplay2π
0exp (pcosx+qsinx)cos(acosx+bsinx−mx)dx
=π/bracketleftbig
(b−p)2+(a+q)2/bracketrightbig−m
2/braceleftBig
(A+iB)m
2Im/parenleftBig√
C−iD/parenrightBig
+(A−iB)m
2Im/parenleftBig√
C+iD/parenrightBig/bracerightBig
GW (337)(9a)
3./integraldisplay2π
0exp (pcosx+qsinx)sin(qcosx−psinx+mx)dx=2π
m!/parenleftbig
p2+q2/parenrightbigm
2sin/parenleftbigg
marctanq
p/parenrightbigg
GW (337)(12)
3.944 Trigonometric and exponentials functions and powers 497
4./integraldisplay2π
0exp (pcosx+qsinx)cos(qcosx−psinx+mx)dx=2π
m!/parenleftbig
p2+q2/parenrightbigm
2cos/parenleftbigg
marctanq
p/parenrightbigg
GW (337)(12)
3.938
1./integraldisplayπ
0er(cospx+cos qx)sin (rsinpx)s i n(rsinqx)dx=π
2∞/summationdisplay
k=11
Γ(pk+1 )Γ ( qk+1 )r(p+q)k
BI (277)(14)
2./integraldisplayπ
0er(cospx+cos qx)cos (rsinpx)c o s(rsinqx)dx=π
2/parenleftBigg
2+∞/summationdisplay
k=1r(p+q)k
Γ(pk+1 )Γ ( qk+1 )/parenrightBigg
BI (277)(15)
3.939
1./integraldisplayπ
0eqcosx sinrx
1−2prcosrx+p2rsin (qsinx)dx=π
2pr∞/summationdisplay
k=1(pq)kr
Γ(kr+1 )
[r>0,0<p< 1] BI (278)(15)
2.3/integraldisplayπ
0eqcosx1−prcosrx
1−2prcosrx+p2rcos(qsinx)dx=π
2/bracketleftBigg
2+∞/summationdisplay
k=1(pq)kr
Γ(kr+1 )/bracketrightBigg
[r>0,0<p< 1] BI (278)(16)
3./integraldisplayπ/2
0epcos 2xcos(psin 2x)dx
cos2x+q2sin2x=π
2qexp/parenleftbigg
pq−1
q+1/parenrightbigg
BI (273)(8)
3.94–3.97 Combinations involving trigonometric functions, exponentials, and powers
3.941
1./integraldisplay∞
0e−pxsinqxdx
x=a r c t a nq
p[p>0] BI (365)(1)
2./integraldisplay∞
0e−pxcosqxdx
x=∞ BI (365)(2)
3.942
1./integraldisplay∞
0e−pxcospxxdx
b4+x4=π
4b2exp/parenleftBig
−bp√
2/parenrightBig
[p>0,b > 0] BI (386)(6)a
2./integraldisplay∞
0e−pxcospxxdx
b4−x4=π
4b2e−bpsinbp [p>0,b > 0] BI (386)(7)a
3.943/integraldisplay∞
0e−βx(1−cosax)dx
x=1
2lna2+β2
β2[Reβ>0] BI (367)(6)
3.944
1./integraldisplayu
0xμ−1e−βxsinδxdx =i
2(β+iδ)−μγ[μ,(β+iδ)u]−i
2(β−iδ)−μγ[μ,(β−iδ)u]
[Reμ>−1] ET I 318(8)
498 Trigonometric Functions 3.944
2./integraldisplay∞
uxμ−1e−βxsinδxdx =i
2(β+iδ)−μΓ[μ,(β+iδ)u]−i
2(β−iδ)−μΓ[μ,(β−iδ)u]
[Reβ>|Imδ|] ET I 318(9)
3./integraldisplayu
0xμ−1e−βxcosδxdx =1
2(β+iδ)−μγ[μ,(β+iδ)u]+1
2(β−iδ)−μγ[μ,(β−iδ)u]
[Reμ>0] ET I 320(28)
4./integraldisplay∞
uxμ−1e−βxcosδxdx =1
2(β+iδ)−μΓ[μ,(β+iδ)u]+1
2(β−iδ)−μΓ[μ,(β−iδ)u]
[Reβ>|Imδ|] ET I 320(29)
5.11/integraldisplay∞
0xμ−1e−βxsinδxdx =Γ(μ)
(β2+δ2)μ/2sin/parenleftbigg
μarctanδ
β/parenrightbigg
[Reμ>−1,Reβ>|Imδ|]
FI II 812, BI (361)(9)
6./integraldisplay∞
0xμ−1e−βxcosδxdx =Γ(μ)
(δ2+β2)μ
2cos/parenleftbigg
μarctanδ
β/parenrightbigg
[Reμ>0,Reβ>|Imδ|]
FI II 812, BI (361)(10)
7./integraldisplay∞
0xμ−1exp (−axcost)s i n(axsint)dx=Γ (μ)a−μsin(μt)
/bracketleftBig
Reμ>−1,a > 0,|t|<π
2/bracketrightBig
EH I 13(36)
8./integraldisplay∞
0xμ−1exp (−axcost)c o s(axsint)dx=Γ (μ)a−μcos(μt)
/bracketleftBig
Reμ>−1,a > 0,|t|<π
2/bracketrightBig
EH I 13(35)
9./integraldisplay∞
0xp−1e−qxsin (qxtant)dx=1
qpΓ(p)cosptsinpt/bracketleftBig
|t|<π
2,q > 0/bracketrightBig
LO V 288(16)
10./integraldisplay∞
0xp−1e−qxcos (qxtant)dx=1
qpΓ(p)cosp(t)cospt
/bracketleftBig
|t|<π
2,q > 0/bracketrightBig
LO V 288(15)
11./integraldisplay∞
0xne−βxsinbxdx=n!/parenleftbiggβ
β2+b2/parenrightbiggn+1/summationdisplay
0≤2k≤n(−1)k/parenleftbiggn+1
2k+1/parenrightbigg/parenleftbiggb
β/parenrightbigg2k+1
=(−1)n∂n
∂βn/parenleftbiggb
b2+β2/parenrightbigg
[Reβ>0,b > 0]GW (336)(3), ET I 72(3)
3.947 Trigonometric and exponentials functions and powers 499
12./integraldisplay∞
0xne−βxcosbxdx=n!/parenleftbiggβ
β2+b2/parenrightbiggn+1/summationdisplay
0≤2k≤n+1(−1)k/parenleftbiggn+1
2k/parenrightbigg/parenleftbiggb
β/parenrightbigg2k
=(−1)n∂n
∂βn/parenleftbiggβ
b2+β2/parenrightbigg
[Reβ>0,b > 0]GW (336)(4), ET I 14(5)
13./integraldisplay∞
0xn−1/2e−βxsinbxdx =(−1)n/radicalbiggπ
2dn
dβn⎛
⎝/radicalBig/radicalbig
β2+b2−β
/radicalbig
β2+b2⎞
⎠
[Reβ>0,b > 0] ET I 72(6)
14./integraldisplay∞
0xn−1/2e−βxcosbxdx =(−1)n/radicalbiggπ
2dn
dβn⎛
⎝/radicalBig/radicalbig
β2+b2+β
/radicalbig
β2+b2⎞
⎠
[Reβ>0,b > 0] ET I 15(6)
3.945
1./integraldisplay∞
0/parenleftbig
e−βxsinax−e−γxsinbx/parenrightbigdx
xr
=Γ ( 1 −r)/braceleftbigg/parenleftbig
b2+γ2/parenrightbigr−1
2sin/bracketleftbigg
(r−1)arctanb
γ/bracketrightbigg
−/parenleftbig
a2+β2/parenrightbigr−1
2sin/bracketleftbigg
(r−1)arctana
β/bracketrightbigg/bracerightbigg
[Reβ>0,Reγ>0,r < 2,r/negationslash=1 ] BI(371)(6)
2./integraldisplay∞
0/parenleftbig
e−βxcosax−e−γxcosbx/parenrightbigdx
xr
=Γ ( 1 −r)/braceleftbigg/parenleftbig
a2+β2/parenrightbigr−1
2cos/bracketleftbigg
(r−1)arctana
β/bracketrightbigg
−/parenleftbig
b2+γ2/parenrightbigr−1
2cos/bracketleftbigg
(r−1)arctanb
γ/bracketrightbigg/bracerightbigg
[Reβ>0,Reγ>0,r < 2,r/negationslash=1 ] BI (371)(7)
3./integraldisplay∞
0/parenleftbig
ae−βxsinbx−be−γxsinax/parenrightbigdx
x2=ab/bracketleftbigg1
2lna2+γ2
b2+β2+γ
aarccotγ
a−β
barccotβ
b/bracketrightbigg
[Reβ>0,Reγ>0] BI (368)(22)
3.946
1./integraldisplay∞
0e−pxsin2m+1axdx
x=(−1)m
22mm/summationdisplay
k=0(−1)k/parenleftbigg2m+1
k/parenrightbigg
arctan(2m−2k+1 )a
p
[m=0,1,..., p> 0] GW (336)(9a)
2./integraldisplay∞
0e−pxsin2maxdx
x=(−1)m+1
22mm−1/summationdisplay
k=0(−1)k/parenleftbigg2m
k/parenrightbigg
ln/bracketleftbig
p2+( 2m−2k)2a2/bracketrightbig
−1
22m/parenleftbigg2m
m/parenrightbigg
lnp
[m=1,2,..., p> 0] GW (336)(9b)
3.947
1./integraldisplay∞
0e−βxsinγxsinaxdx
x=1
4lnβ2+(a+γ)2
β2+(a−γ)2[Reβ>|Imγ|,a > 0] BI (365)(5)
500 Trigonometric Functions 3.948
2.11/integraldisplay∞
0e−pxsinaxsinbxdx
x2=|a+b|
2arctan/parenleftbigg|a+b|
p/parenrightbigg
−|a−b|
2arctan/parenleftbigg|a−b|
p/parenrightbigg
+p
4ln/parenleftbiggp2+(a−b)2
p2+(a+b)2/parenrightbigg
[p>0, for p=0s e e 3.741 3]BI (368)(1), FI II 744
3.11/integraldisplay∞
0e−pxsinaxcosbxdx
x=a r c t a na+b
p+a r c t a na−b
p
[a≥0,p > 0] GW (336)(10b)
3.948
1.11/integraldisplay∞
0e−βx(sinax−sinbx)dx
x=a r c t a na
β−arctanbr
β
[Reβ>0],(cf.3.951 2)
BI (367)(7)
2./integraldisplay∞
0e−βx(cosax−cosbx)dx
x=1
2lnb2+β2
a2+β2[Reβ>0],(cf.3.951 3)
BI (367)(8), FI II 748a
3./integraldisplay∞
0e−βx(cosax−cosbx)dx
x2=β
2lna2+β2
b2+β2+barctanb
β−aarctana
β
[Rep>0] BI (368)(20)
4./integraldisplay∞
0e−βx/parenleftbig
sin2ax−sin2bx/parenrightbigdx
x2=aarctan2a
p−barctan2b
p−p
4lnp2+4a2
p2+4b2
[p>0] BI (368)(25)
5./integraldisplay∞
0e−βx/parenleftbig
cos2ax−cos2bx/parenrightbigdx
x2=−aarctan2a
p+barctan2b
p+p
4lnp2+4a2
p2+4b2
[p>0] BI (368)(26)
3.949
1./integraldisplay∞
0e−pxsinaxsinbxsincxdx
x=−1
4arctana+b+c
p+1
4arctana+b−c
p+1
4arctana−b+c
p
+1
4arctan−a+b+c
p
[p>0] BI (365)(11)
2.8/integraldisplay∞
0e−pxsin2axsinbxdx
x=1
2arctanb
p−1
2/bracketleftbigg1
2arctan2pb
p2+4a2−b2+sπ
2/bracketrightbigg
/bracketleftBigg
s=/braceleftBigg
1f o r p2+4a2−b2<0
0f o r p2+4a2−b2≥0/bracketrightBigg
BI (365)(8)
3./integraldisplay∞
0e−pxsin2axcosbxdx
x=1
8ln/bracketleftbig
p2+( 2a+b)2/bracketrightbig/bracketleftbig
p2+( 2a−b)2/bracketrightbig
(p2+b2)2
[p>0] BI (365)(9)
3.951 Trigonometric and exponentials functions and powers 501
4.8/integraldisplay∞
0e−pxsinaxcos2bxdx
x=1
2arctana
p+1
2/bracketleftbigg1
2arctan2pa
p2+4b2−a2+sπ
2/bracketrightbigg
/bracketleftBigg
s=/braceleftBigg
1f o r p2+4b2−a2<0
0f o r p2+4b2−a2≥0/bracketrightBigg
BI (365)(10)
5./integraldisplay∞
0e−pxsin2axsinbxsincxdx
x=1
8lnp2+(b+c)2
p2+(b−c)2
+1
16ln/bracketleftbig
p2+( 2a−b+c)2/bracketrightbig/bracketleftBig
p2+( 2a+b−c)2/bracketrightBig
[p2+( 2a+b+c)2][p2+( 2a−b−c)2]
[p>0] BI (365)(15)
3.951
1./integraldisplay∞
0/parenleftbig
1−e−x/parenrightbig
cosxdx
x=l n√
2 FI II 745
2./integraldisplay∞
0e−γx−e−βx
xsinbxdx =a r c t a n(β−γ)b
b2+βγ[Reβ>0,Reγ≥0] BI (367)(3)
3./integraldisplay∞
0e−γx−e−βx
xcosbxdx =1
2lnb2+β2
b2+γ2[Reβ>0,Reγ≥0] BI (367)(4)
4.11/integraldisplay∞
0e−γx−e−βx
x2sinbxdx =b
2lnb2+β2
b2+γ2+βarctanb
β−γarctanb
γ
[Reβ>0,Reγ>0] BI (368)(21)a
5./integraldisplay∞
0x
eβx−1cosbxdx =1
2b2−π2
2β2cosech2bπ
β[Reβ>0] ET I 15(18)
6./integraldisplay∞
0/parenleftbigg1
ex−1−1
x/parenrightbigg
cosbxdx =l nb−1
2[ψ(ib)+ψ(−ib)]
[b>0] ET I 15(9)
7./integraldisplay∞
01−cosax
e2πx−1·dx
x=a
4+1
2ln1−e−a
a[a>0] BI (387)(10)
8./integraldisplay∞
0/parenleftbig
e−βx−e−γxcosax/parenrightbigdx
x=1
2lna2+γ2
β2[Reβ>0,Reγ>0] BI (367)(10)
9./integraldisplay∞
0cospx−e−px
b4+x4dx
x=π
2b4exp/parenleftbigg
−1
2bp√
2/parenrightbigg
sin/parenleftbigg1
2bp√
2/parenrightbigg
[p>0] BI (390)(6)
10./integraldisplay∞
0/parenleftbigg1
ex−1−cosx
x/parenrightbigg
dx=C NT 65(8)
11./integraldisplay∞
0/parenleftbigg
ae−px−e−qx
xsinax/parenrightbiggdx
x=a
2lna2+q2
p2+qarctana
q−a
[p>0,q > 0] BI (368)(24)
502 Trigonometric Functions 3.952
12./integraldisplay∞
0x2msinbx
ex−1dx=(−1)m∂2m
∂b2m/bracketleftbiggπ
2cothbπ−1
2b/bracketrightbigg
[b>0] GW (336)(15a)
13./integraldisplay∞
0x2m+1cosbx
ex−1dx=(−1)m∂2m+1
∂b2m+1/bracketleftbiggπ
2cothbπ−1
2b/bracketrightbigg
[b>0] GW (336)(15b)
14./integraldisplay∞
0x2msinbxdx
e(2n+1)cx−e(2n−1)cx=(−1)m∂2m
∂b2m/bracketleftBigg
π
4ctanhbπ
2c−n/summationdisplay
k=1b
b2+( 2k−1)2c2/bracketrightBigg
[b>0] GW (336)(14a)
15./integraldisplay∞
0x2m+1cosbxdx
e(2n+1)cx−e(2n−1)cx=(−1)m∂2m+1
∂b2m+1/bracketleftBigg
π
4ctanhbπ
2c−n/summationdisplay
k=1b
b2+( 2k−1)2c2/bracketrightBigg
[b>0] GW (336)(14b)
16./integraldisplay∞
0x2msinbxdx
e(2n−2)cx=(−1)m∂2m
∂b2m/bracketleftBigg
π
4ccothbπ
2c−1
2b−n−1/summationdisplay
k=1b
b2+( 2k)2c2/bracketrightBigg
[b>0,c > 0] GW (336)(14c)
17./integraldisplay∞
0x2m+1cosbxdx
e2ncx−e(2n−2)cx=(−1)m∂2m+1
∂b2m+1/bracketleftBigg
π
4ccothbπ
2c−1
2b−n−1/summationdisplay
k=1b
b2+( 2k)2c2/bracketrightBigg
[b>0,c > 0] GW (336)(14d)
18./integraldisplay∞
0cosax−cosbx
e(2m+1)px−e(2m−1)pxdx
x=1
2lncoshbπ
2p
coshaπ
2p−1
2m/summationdisplay
k=1lnb2+( 2k−1)2p2
a2+( 2k−1)2p2
[p>0] GW (336)(16a)
19./integraldisplay∞
0cosax−cosbx
e2mpx−e(2m−2)pxdx
x=1
2lnasinhbπ
2p
bsinhaπ
2p−1
2m−1/summationdisplay
k=1lnb2+4k2p2
a2+4k2p2
[p>0] GW (336)(16b)
20./integraldisplay∞
0sinxsinbx
1−ex·dx
x=1
4ln(b+1 )s i n h [ ( b−1)π]
(b−1)sinh[( b+1 )π]/bracketleftbig
b2/negationslash=1/bracketrightbig
LO V 305
21./integraldisplay∞
0sin2ax
1−ex·dx
x=1
4ln2aπ
sinh 2aπLO V 306, BI (387)(5)
3.952
1./integraldisplay∞
0xe−p2x2sinaxdx =a√π
4p3exp/parenleftbigg
−a2
4p2/parenrightbigg
BI (362)(1)
2./integraldisplay∞
0xe−p2x2cosaxdx =1
2p2−a
4p3∞/summationdisplay
k=0(−1)kk!
(2k+1 ) !/parenleftbigga
p/parenrightbigg2k+1
[a>0] BI (362)(2)
3.953 Trigonometric and exponentials functions and powers 503
3./integraldisplay∞
0x2e−p2x2sinaxdx =a
4p4+2p2−a2
8p5∞/summationdisplay
k=0(−1)kk!
(2k+1 ) !/parenleftbigga
p/parenrightbigg2k+1
[a>0] BI (362)(4)
4./integraldisplay∞
0x2e−p2x2cosaxdx =√π2p2−a2
8p5exp/parenleftbigg
−a2
4p2/parenrightbigg
BI (362)(5)
5./integraldisplay∞
0x3e−p2x2sinaxdx =√π6ap2−a3
16p7exp/parenleftbigg
−a2
4p2/parenrightbigg
BI (362)(6)
6.3/integraldisplay∞
0e−p2x2sinaxdx
x=a√π
2p∞/summationdisplay
k=0(−1)k
k!(2k+1 )/parenleftbigga
2p/parenrightbigg2k
=π
2Φ/parenleftbigga
2p/parenrightbigg
BI (365)(21)
7./integraldisplay∞
0xμ−1e−βx2sinγxdx =γe−γ2
4β
2βμ+1
2Γ/parenleftbigg1+μ
2/parenrightbigg
1F1/parenleftbigg
1−μ
2;3
2;γ2
4β/parenrightbigg
[Reβ>0,Reμ>−1] ET I 318(10)
8.10/integraldisplay∞
0xμ−1e−βx2cosaxdx =1
2β−μ/2Γ/parenleftBigμ
2/parenrightBig
e−a2/4β
1F1/parenleftbigg
−μ
2+1
2;1
2;a2
4β/parenrightbigg
[Reβ>0,Reμ>0,a > 0]
ET I 320(30)
9./integraldisplay∞
0x2ne−β2x2cosaxdx =(−1)n√π
2n+1β2n+1exp/parenleftbigg
−a2
8β2/parenrightbigg
D2n/parenleftbigga
β√
2/parenrightbigg
=(−1)n√π
(2β)2n+1exp/parenleftbigg
−a2
4β2/parenrightbigg
H2n/parenleftbigga
2β/parenrightbigg
/bracketleftBig
|argβ|<π
4,a > 0/bracketrightBig
WH, ET I 15(13)
10./integraldisplay∞
0x2n+1e−β2x2sinaxdx =(−1)n√π
2n+3
2β2n+2exp/parenleftbigg
−a2
8β2/parenrightbigg
D2n+1/parenleftbigga
β√
2/parenrightbigg
=(−1)n√π
(2β)2n+2exp/parenleftbigg
−a2
4β2/parenrightbigg
H2n+1/parenleftbigga
2β/parenrightbigg
/bracketleftBig
|argβ|<π
4,a > 0/bracketrightBig
WH, ET I 74(23)
3.953
1./integraldisplay∞
0xμ−1e−γx−βx2sinaxdx
=−i
2(2β)μ
2exp/parenleftbiggγ2−a2
8β/parenrightbigg
Γ(μ)/braceleftbigg
exp/parenleftbigg
−iaγ
4β/parenrightbigg
D−μ/parenleftbiggγ−ia√2β/parenrightbigg
−exp/parenleftbiggiaγ
4β/parenrightbigg
D−μ/parenleftbiggγ+ia√2β/parenrightbigg/bracerightbigg
[Reμ>−1,Reβ>0,a > 0]ET I 318(11)
2./integraldisplay∞
0xμ−1e−γx−βx2cosaxdx
=1
2(2β)μ
2exp/parenleftbiggγ2−a2
8β/parenrightbigg
Γ(μ)/braceleftbigg
exp/parenleftbigg
−iaγ
4β/parenrightbigg
D−μ/parenleftbiggγ−ia√2β/parenrightbigg
+e x p/parenleftbiggiaγ
4β/parenrightbigg
D−μ/parenleftbiggγ+ia√2β/parenrightbigg/bracerightbigg
[Reμ>0,Reβ>0,a > 0]ET I 16(18)
504 Trigonometric Functions 3.954
3./integraldisplay∞
0xe−γx−βx2sinaxdx =i√π
8/radicalbig
β3/braceleftbigg
(γ−ia)exp/bracketleftbigg
−(γ−ia)2
4β/bracketrightbigg/bracketleftbigg
1−Φ/parenleftbiggγ−ia
2√β/parenrightbigg/bracketrightbigg
−(γ+ia)exp/bracketleftbigg
−(γ+ia)2
4β/bracketrightbigg/bracketleftbigg
1−Φ/parenleftbiggγ+ia
2√β/parenrightbigg/bracketrightbigg/bracerightbigg
[Reβ>0,a > 0] ET I 74(28)
4./integraldisplay∞
0xe−γx−βx2cosaxdx =−√π
8/radicalbig
β3/braceleftbigg
(γ−ia)exp(γ−ia)2
4β/bracketleftbigg
1−Φ/parenleftbiggγ−ia
2√β/parenrightbigg/bracketrightbigg
+(γ+ia)exp(γ+ia)2
4β/bracketleftbigg
1−Φ/parenleftbiggγ+ia
2√β/parenrightbigg/bracketrightbigg/bracerightbigg
+1
2β
[Reβ>0,a > 0] ET I 16(17)
3.954
1.11/integraldisplay∞
0e−βx2sinaxxdx
γ2+x2=−π
4eβγ2/bracketleftbigg
2s in h aγ+e−γaΦ/parenleftbigg
γ/radicalbig
β−a
2√β/parenrightbigg
−eγaΦ/parenleftbigg
γ/radicalbig
β+a
2√β/parenrightbigg/bracketrightbigg
[Reβ>0,Reγ>0,a > 0]
ET I 74(26)a
2.11/integraldisplay∞
0e−βx2cosaxdx
γ2+x2=π
4γeβγ2/bracketleftbigg
2c os h aγ−e−γaΦ/parenleftbigg
γ/radicalbig
β−a
2√β/parenrightbigg
−eγaΦ/parenleftbigg
γ/radicalbig
β+a
2√β/parenrightbigg/bracketrightbigg
[Reβ>0,Reγ>0,a > 0]
ET I 15(15)
3.955/integraldisplay∞
0xνe−x2
2cos/parenleftBig
βx−νπ
2/parenrightBig
dx=/radicalbiggπ
2e−β2
4Dν(β)[ R e ν>−1] EH II 120(4)
3.956/integraldisplay∞
0e−x2(2xcosx−sinx)s i nxdx
x2=√πe−1
2eBI (369)(19)
3.957
1./integraldisplay∞
0xμ−1exp/parenleftbigg−β2
4x/parenrightbigg
sinaxdx
=i
2μβμa−μ
2/bracketleftbigg
exp/parenleftbigg
−i
4μπ/parenrightbigg
Kμ/parenleftBig
βeπi
4√a/parenrightBig
−exp/parenleftbiggi
4μπ/parenrightbigg
Kμ/parenleftBig
βe−πi/4√a/parenrightBig/bracketrightbigg
[Reβ>0,Reμ<1,a > 0]ET I 318(12)
2./integraldisplay∞
0xμ−1exp/parenleftbigg−β2
4x/parenrightbigg
cosaxdx
=1
2μβμa−μ
2/bracketleftbigg
exp/parenleftbigg
−i
4μπ/parenrightbigg
Kμ/parenleftBig
βeπi/4√a/parenrightBig
+e x p/parenleftbiggi
4μπ/parenrightbigg
Kμ/parenleftBig
βe−πi/4√a/parenrightBig/bracketrightbigg
[Reβ>0,Reμ<1,a > 0]ET I 320(32)a
3.958
1./integraldisplay∞
−∞xne−(ax2+bx+c)sin(px+q)dx=−/parenleftbigg−1
2a/parenrightbiggn/radicalbiggπ
aexp/parenleftbiggb2−p2
4a−c/parenrightbigg⌊n/2⌋/summationdisplay
k=0n!
(n−2k)!k!ak
×n−2k/summationdisplay
j=0/parenleftbiggn−2k
j/parenrightbigg
bn−2k−jpjsin/parenleftbiggpb
2a−q+π
2j/parenrightbigg
[a>0] GW (37)(1b)
3.963 Trigonometric and exponentials functions and powers 505
2./integraldisplay∞
−∞xne−(ax2+bx+c)cos(px+q)dx=/parenleftbigg−1
2a/parenrightbiggn/radicalbiggπ
aexp/parenleftbiggb2−p2
4a−c/parenrightbigg⌊n/2⌋/summationdisplay
k=0n!
(n−2k)!k!ak
×n−2k/summationdisplay
j=0/parenleftbiggn−2k
j/parenrightbigg
pjcos/parenleftbiggpb
2a−q+π
2j/parenrightbigg
[a>0] GW (337)(1a)
3.959/integraldisplay∞
0xe−p2x2tanaxdx =a√π
p3∞/summationdisplay
k=1(−1)kkexp/parenleftbigg
−a2k2
p2/parenrightbigg
[p>0] BI (362)(15)
3.961
1./integraldisplay∞
0exp/parenleftBig
−β/radicalbig
γ2+x2/parenrightBig
sinaxxdx/radicalbig
γ2+x2=aγ/radicalbig
a2+β2K1/parenleftBig
γ/radicalbig
a2+β2/parenrightBig
[Reβ>0,Reγ>0,a > 0]
ET I 75(36)
2./integraldisplay∞
0exp/bracketleftBig
−β/radicalbig
γ2+x2/bracketrightBig
cosaxdx/radicalbig
γ2+x2=K0/parenleftBig
γ/radicalbig
a2+β2/parenrightBig
[Reβ>0,Reγ>0,a > 0]
ET I 17(27)
3.962
1./integraldisplay∞
0/radicalBig/radicalbig
γ2+x2−γexp/parenleftBig
−β/radicalbig
γ2+x2/parenrightBig
/radicalbig
γ2+x2sinaxdx =/radicalbiggπ
2aexp/parenleftBig
−γ/radicalbig
a2+β2/parenrightBig
/radicalbig
β2+a2/radicalBig
β+/radicalbig
a2+β2
[Reβ>0,Reγ>0,a > 0]
ET I 75(38)
2./integraldisplay∞
0xexp/parenleftBig
−β/radicalbig
γ2+x2/parenrightBig
/radicalbig
γ2+x2/radicalBig/radicalbig
γ2+x2−γcosaxdx =/radicalbiggπ
2/radicalBig
β+/radicalbig
a2+β2
/radicalbig
a2+β2exp/bracketleftBig
−γ/radicalbig
a2+β2/bracketrightBig
[Reβ>0,Reγ>0,a > 0]
ET I 17(29)
3.963
1./integraldisplay∞
0e−tan2xsinx
cos2xdx
x=√π
2BI (391)(1)
2./integraldisplayπ/2
0e−ptanxxdx
cos2x=1
p[ci(p)sinp−cospsi(p)] [ p>0] (cf. 3.339 ) BI (396)(3)
3.8/integraldisplayπ/2
0xe−tan2xsin 4xdx
cos8x=−3
2√π BI (396)(5)
4.8/integraldisplayπ/2
0xe−tan2xsin32xdx
cos8x=2√π BI (396)(6)
506 Trigonometric Functions 3.964
3.964
1./integraldisplayπ/2
0xe−ptanxpsinx−cosx
cos3xdx=−sinpsi(p)−ci(p)cosp
[p>0] LI (396)(4)
2./integraldisplayπ/2
0xe−ptan2xp−cos2x
cos4xcotxdx=1
4/radicalbiggπ
p[p>0] BI (396)(7)
3.8/integraldisplayπ/2
0xe−ptan2xp−2c os2x
cos6xcotxdx=1+2p
8p/radicalbiggπ
p[p>0] BI (396)(8)
3.965
1./integraldisplay∞
0xe−βxsinax2sinβxdx =β
4/radicalbiggπ
2a3e−β2
2a/bracketleftBig
|argβ|<π
4,a > 0/bracketrightBig
ET I 84(17)
2./integraldisplay∞
0xe−βxcosax2cosβxdx =β
4/radicalbiggπ
2a3e−β2
2a [a>0,Reβ>|Imβ|] ET 26(27)
3.966
1./integraldisplay∞
0xe−pxcos/parenleftbig
2x2+px/parenrightbig
dx=0 [ p>0] BI (361)(16)
2./integraldisplay∞
0xe−pxcos/parenleftbig
2x2−px/parenrightbig
dx=p√π
8exp/parenleftbigg
−1
4p2/parenrightbigg
[p>0] BI (361)(17)
3./integraldisplay∞
0x2e−px/bracketleftbig
sin/parenleftbig
2x2+px/parenrightbig
+c o s/parenleftbig
2x2+px/parenrightbig/bracketrightbig
dx=0 [ p>0] BI (361)(18)
4./integraldisplay∞
0x2e−px/bracketleftbig
sin/parenleftbig
2x2−px/parenrightbig
−cos/parenleftbig
2x2−px/parenrightbig/bracketrightbig
dx=√π
16/parenleftbig
2−p2/parenrightbig
exp/parenleftbigg
−1
4p2/parenrightbigg
BI (361)(19)
5.3/integraldisplay∞
0xμ−1e−xcos/parenleftbig
x+ax2/parenrightbig
dx=e1
4aΓ(μ)
(2a)μ
2cosμπ
4D−μ/parenleftbigg1√a/parenrightbigg
[Reμ>0,a > 0] ET I 321(37)
6.6/integraldisplay∞
0xμ−1e−xsin/parenleftbig
x+ax2/parenrightbig
dx=e1
4aΓ(μ)
(2a)μ
2sinμπ
4D−μ/parenleftbigg1√a/parenrightbigg
[Reμ>−1,a > 0] ET I 319(18)
3.967
1./integraldisplay∞
0e−β2
x2sina2x2dx
x2=√π
2βe−√
2aβsin/parenleftBig√
2aβ/parenrightBig
[Reβ>0,a > 0]
ET I 75(30)a, BI(369)(3)a
2./integraldisplay∞
0e−β2
x2cosa2x2dx
x2=√π
2βe−√
2aβcos/parenleftBig√
2aβ/parenrightBig
[Reβ>0,a > 0]
BI (369)(4), ET I 16(20)
3.972 Trigonometric and exponentials functions and powers 507
3./integraldisplay∞
0x2e−βx2cosax2dx=√π
44/radicalBig
(a2+β2)3cos/parenleftbigg3
2arctana
β/parenrightbigg
[Reβ>0] ET I 14(3)a
3.968
1./integraldisplay∞
0e−βx2sinax4dx=−π
8/radicalbigg
β
a/bracketleftbigg
J1
4/parenleftbiggβ2
8a/parenrightbigg
cos/parenleftbiggβ2
8a/parenrightbigg
+π
8+Y1
4/parenleftbiggβ2
8a/parenrightbigg
sin/parenleftbiggβ2
8a/parenrightbigg
+π
8/bracketrightbigg
[Reβ>0,a > 0] ET I 75(34)
2./integraldisplay∞
0e−βx2cosax4dx=π
8/radicalbigg
β
a/bracketleftbigg
J1
4/parenleftbiggβ2
8a/parenrightbigg
sin/parenleftbiggβ2
8a+π
8/parenrightbigg
−Y1
4/parenleftbiggβ2
8a/parenrightbigg
cos/parenleftbiggβ2
8a/parenrightbigg
+π
8/bracketrightbigg
[Reβ>0,a > 0] ET I 16(24)
3.969
1./integraldisplay∞
0e−p2x4+q2x2/bracketleftbig
2pxcos/parenleftbig
2pqx3/parenrightbig
+qsin/parenleftbig
2pqx3/parenrightbig/bracketrightbig
dx=√π
2BI (363)(7)
2./integraldisplay∞
0e−p2x4+q2x2/bracketleftbig
2pxsin/parenleftbig
2pqx3/parenrightbig
−qcos/parenleftbig
2pqx3/parenrightbig/bracketrightbig
dx=0 BI (363)(8)
3.971 Notation : In formulas 3.971 1a n d2 , p≥0,q≥0,r=4/radicalbig
a2+p2,s=4/radicalbig
b2+q2,A=a r c t a na
p,
andB=a r c t a nb
q.
1./integraldisplay∞
0exp/parenleftBig
−px2−q
x2/parenrightBig
sin/parenleftbigg
ax2+b
x2/parenrightbiggdx
x2=1
2/integraldisplay∞
−∞exp/parenleftBig
−px2−q
x2/parenrightBig
sin/parenleftbigg
ax2+b
x2/parenrightbiggdx
x2
=√π
2sexp [−2rscos(A+B)] sin [ A+2rssin(A+B)]
BI (369)(16, 17)
2./integraldisplay∞
0exp/parenleftBig
−px2−q
x2/parenrightBig
cos/parenleftbigg
ax2+b
x2/parenrightbiggdx
x2=1
2/integraldisplay∞
−∞exp/parenleftBig
−px2−q
x2/parenrightBig
cos/parenleftbigg
ax2+b
x2/parenrightbiggdx
x2
=√π
2sexp [−2rscos(A+B)] cos[ A+2rssin(A+B)]
BI (369)(15, 18)
3.972
1./integraldisplay∞
0exp/bracketleftBig
−β/radicalbig
γ4+x4/bracketrightBig
sinax2dx/radicalbig
γ4+x4
=/radicalbiggaπ
8I1/4/bracketleftbiggγ2
2/parenleftBig/radicalbig
β2+a2−β/parenrightBig/bracketrightbigg
K1/4/bracketleftbiggγ2
4/parenleftBig/radicalbig
β2+a2+β/parenrightBig/bracketrightbigg
/bracketleftBig
Reβ>0,|argγ|<π
4,a > 0/bracketrightBig
ET I 75(37)
2./integraldisplay∞
0exp/bracketleftBig
−β/radicalbig
γ4+x4/bracketrightBig
cosax2dx/radicalbig
γ4+x4
=/radicalbiggaπ
8I−1/4/bracketleftbiggγ2
2/parenleftBig/radicalbig
β2+a2−β/parenrightBig/bracketrightbigg
K1/4/bracketleftbiggγ2
4/parenleftBig/radicalbig
β2+a2+β/parenrightBig/bracketrightbigg
/bracketleftBig
Reβ>0,|argγ|<π
4,a > 0/bracketrightBig
ET I 17(28)
508 Trigonometric Functions 3.973
3.973
1./integraldisplay∞
0exp(pcosax)sin(psinax)dx
x=π
2(ep−1) [ p>0,a > 0] W H ,F II I7 2 5
2./integraldisplay∞
0exp(pcosax)sin(psinax+bx)xdx
c2+x2=π
2exp/parenleftbig
−cb+pe−ac/parenrightbig
[a>0,b > 0,c > 0,p > 0]
BI (372)(3)
3./integraldisplay∞
0exp(pcosax)cos(psinax+bx)dx
c2+x2=π
2cexp/parenleftbig
−cb+pe−ac/parenrightbig
[a>0,b > 0,c > 0,p > 0]
BI (372)(4)
4./integraldisplay∞
0exp(pcosx)s i n(psinx+nx)dx
x=π
2ep[p>0] BI (366)(2)
5./integraldisplay∞
0exp(pcosx)s i n(psinx)c o snxdx
x=pn
n!·π
4+π
2∞/summationdisplay
k=n+1pk
k!
[p>0] LI (366)(3)
6./integraldisplay∞
0exp(pcosx)c o s(psinx)sinnxdx
x=π
2n−1/summationdisplay
k=0pk
k!+pn
n!π
4
[p>0] LI (366)(4)
3.974
1./integraldisplay∞
0exp(pcosax)sin(psinax)cosec axdx
b2+x2=π/bracketleftbig
ep−exp/parenleftbig
pe−ab/parenrightbig/bracketrightbig
2bsinhab
[a>0,b > 0,p > 0] BI (391)(4)
2./integraldisplay∞
0[1−exp(pcosax)cos(psinax)] cosec axxdx
b2+x2=π/bracketleftbig
ep−exp/parenleftbig
pe−ab/parenrightbig/bracketrightbig
2s in h ab
[a>0,b > 0,p > 0] BI (391)(5)
3./integraldisplay∞
0exp(pcosax)sin(psinax+ax)cosec axdx
b2+x2=π/bracketleftbig
ep−exp/parenleftbig
pe−ab−ab/parenrightbig/bracketrightbig
2bsinhab
[a>0,b > 0,p > 0] BI (391)(6)
4./integraldisplay∞
0exp(pcosax)cos(psinax+ax)cosec axxdx
b2+x2=π/bracketleftbig
ep−exp/parenleftbig
pe−ab−ab/parenrightbig/bracketrightbig
2s in h ab
[a>0,b > 0,p > 0] BI (391)(7)
5./integraldisplay∞
0exp(pcosax)sin(psinax)xdx
b2−x2=π
2[1−exp (pcosab)cos(psinab)]
[p>0,a > 0] BI (378)(1)
6./integraldisplay∞
0exp(pcosax)cos(psinax)dx
b2−x2=π
2bexp(pcosab)sin(psinab)
[a>0,b > 0,p > 0] BI (378)(2)
3.981 Trigonometric and hyperbolic functions 509
7./integraldisplay∞
0exp(pcosax)sin(psinax)tanaxdx
b2+x2=π
2b·tanhab/bracketleftbig
exp/parenleftbig
pe−ab/parenrightbig
−ep/bracketrightbig
[a>0,b > 0,p > 0] BI (372)(14)
8./integraldisplay∞
0exp(pcosax)sin(psinax)cotaxdx
b2+x2=π
2bcothab/bracketleftbig
ep−exp/parenleftbig
pe−ab/parenrightbig/bracketrightbig
[a>0,b > 0,p > 0] BI (372)(15)
9./integraldisplay∞
0exp(pcosax)sin(psinax)cosec axdx
b2−x2=π
2bcosecab[ep−exp (pcosab)cos(psinab)]
[a>0,b > 0,p > 0] BI (391)(12)
10./integraldisplay∞
0[1−exp(pcosax)cos(psinax)] cosec axxdx
b2−x2=−π
2exp(pcosab)sin(psinab)cosec ab
[a>0,b > 0,p > 0] BI (391)(13)
3.975
1./integraldisplay∞
0sin/parenleftBig
βarctanx
γ/parenrightBig
(γ2+x2)β
2·dx
e2πx−1=1
2ζ(β,γ)−1
4γβ−γ1−β
2(β−1)
[Reβ>1,Reγ>0] WH, ET I 26(7)
2./integraldisplay∞
0sin (βarctan x)
(1 +x2)β
2·dx
e2πx+1=1
2(β−1)−ζ(β)
2β[Reβ>1] EH I 33(13)
3.976/integraldisplay∞
0/parenleftbig
1+x2/parenrightbigβ−1
2e−px2cos [2px+( 2β−1)arctan x]dx=e−p
2pβsinπβΓ(β)
[Reβ>0,p > 0] WH
3.98–3.99 Combinations of trigonometric and hyperbolic functions
3.981
1./integraldisplay∞
0sinax
sinhβxdx=π
2βtanhaπ
2β[Reβ>0,a > 0] BI (264)(16)
2./integraldisplay∞
0sinax
coshβxdx=−π
2βtanhaπ
2β−i
2β/bracketleftbigg
ψ/parenleftbiggβ+ai
4β/parenrightbigg
−ψ/parenleftbiggβ−ai
4β/parenrightbigg/bracketrightbigg
[Reβ>0,a > 0]
GW (335)(12), ET I 88(1)
3./integraldisplay∞
0cosax
coshβxdx=π
2βsechaπ
2β[Reβ>0,all real a] BI (264)(14)
4./integraldisplay∞
0sinaxsinhβx
sinhγxdx=π
2γsinhaπ
γ
coshaπ
γ+c o sβπ
γ+i
2γ/bracketleftbigg
ψ/parenleftbiggβ+γ+ia
2γ/parenrightbigg
−ψ/parenleftbiggβ+γ−ia
2γ/parenrightbigg/bracketrightbigg
[|Reβ|<Reγ, a > 0] ET I 88(5)
5./integraldisplay∞
0cosaxsinhβx
sinhγxdx=π
2γsinπβ
γ
coshaπ
γ+c o sβπ
γ[|Reβ|<Reγ] BI (265)(7)
510 Trigonometric Functions 3.982
6./integraldisplay∞
0sinaxsinhβx
coshγxdx=π
γsinβπ
2γsinhaπ
2γ
coshaπ
γ+c o sβπ
γ[|Reβ|<Reγ, a > 0] BI (265)(2)
7./integraldisplay∞
0cosaxsinhβx
coshγxdx=1
4γ⎡
⎣/braceleftbigg
ψ/parenleftbigg3γ−β+ia
4γ/parenrightbigg
+ψ/parenleftbigg3γ−β−ia
4γ/parenrightbigg
−ψ/parenleftbigg3γ+β−ia
4γ/parenrightbigg/bracerightbigg
−ψ/parenleftbigg3γ+β+ia
4γ/parenrightbigg
+2πsinπβ
γ
cosπβ
γ+c o s hπa
γ⎤
⎦
[|Reβ|<Reγ, a > 0] ET I 31(13)
8./integraldisplay∞
0sinaxcoshβx
sinhγxdx=π
2γ·sinhπa
γ
coshπa
γ+c o sπβ
γ[|Reβ|<Reγ, a > 0] BI (265)(4)
9./integraldisplay∞
0sinaxcoshβx
coshγxdx=i
4γ⎡
⎣ψ/parenleftbigg3γ+β+ia
4γ/parenrightbigg
−ψ/parenleftbigg3γ+β−ai
4γ/parenrightbigg
+ψ/parenleftbigg3γ−β+ia
4γ/parenrightbigg
−ψ/parenleftbigg3γ−β−ai
4γ/parenrightbigg
−2πisinhπa
γ
coshaπ
γ+c o sβπ
γ⎤
⎦
[|Reβ|<Reγ, a > 0] ET I 88(6)
10./integraldisplay∞
0cosaxcoshβx
coshγxdx=π
γcosβπ
2γcoshaπ
2γ
coshaπ
γ+c o sβπ
γ[|Reβ|<Reγ,all real a]BI (265)(6)
11.11/integraldisplayπ/2
0cos2mxcoshβxdx =(2m)! sinhπβ
2
β(β2+22)...[β2+( 2m)2]
[β/negationslash=0 ] WA 620a
12.11/integraldisplayπ/2
0cos2m+1xcoshβxdx =(2m+1 ) !c o s hπβ
2
(β2+12)(β2+32)...[β2+( 2m+1 )2]WA 620a
3.982
1./integraldisplay∞
0cosax
cosh2βxdx=aπ
2β2sinhaπ
2β[Reβ>0,a > 0] BI (264)(16)
2./integraldisplay∞
0sinaxsinhβx
cosh2γxdx=π/parenleftBig
asinβπ
2γcoshaπ
2γ−βcosβπ
2γsinhaπ
2γ/parenrightBig
γ2/parenleftBig
coshaπ
γ−cosβπ
γ/parenrightBig
[|Reβ|<2R eγ, a > 0] ET I 88(9)
3.11/integraldisplay∞
0sin2xcosax
sinh2hxdx=π
4/braceleftbigga+2
1−e−π(a+2)−2a
1−e−πa+a−2
1−e−π(a−2)/bracerightbigg
=I(a)
/bracketleftbigg
I(0) =1
2(πcothπ−1),I(±2) =1
4+π
2(coth 2 π−cothπ)/bracketrightbigg
3.984 Trigonometric and hyperbolic functions 511
3.983
1.6/integraldisplay∞
0cosaxdx
bcoshβx+c=πsin/parenleftBig
a
βarccoshc
b/parenrightBig
β√
c2−b2sinhaπ
β[c>b> 0]
=πsinh/parenleftBig
a
βarccosc
b/parenrightBig
β√
b2−c2sinhaπ
β[b>|c|>0]
[Reβ>0,a > 0] GW (335)(13a)
2./integraldisplay∞
0cosaxdx
coshβx+c o s γ=π
βsinhaγ
β
sinγsinhaπ
β/bracketleftbig
πReβ<Imβγ, a > 0/bracketrightbig
BI (267)(3)
3.3/integraldisplay∞
0cosaxdx
coshx−coshb=−πcothaπsinab
sinhb[a>0,b > 0] ET I 30(8)
4./integraldisplay∞
0cosaxdx
1+2c o s h/parenleftBig/radicalBig
2
3πx/parenrightBig=/radicalbigπ
2
1+2c o s h/parenleftBig/radicalBig
2
3πa/parenrightBig [a>0] ET I 30(9)
5./integraldisplay∞
0sinaxsinhβx
coshγx+c o s δdx=π/braceleftBig
sin/bracketleftBig
β
γ(π−δ)/bracketrightBig
sinh/bracketleftBig
a
γ(π+δ)/bracketrightBig
−sin/bracketleftBig
β
γ(π+δ)/bracketrightBig
sinh/bracketleftBig
a
γ(π−δ)/bracketrightBig/bracerightBig
γsinδ/parenleftBig
cosh2πa
γ−cos2πβ
γ/parenrightBig
[πReγ>|Reγδ|,|Reβ|<Reγ, a > 0]BI (267)(2)
6./integraldisplay∞
0cosaxcoshβx
coshγx+c o s bdx=π/braceleftBig
cos/bracketleftBig
β
γ(π−b)/bracketrightBig
cosh/bracketleftBig
a
γ(π+b)/bracketrightBig
−cos/bracketleftBig
β
γ(π+b)/bracketrightBig
cosh/bracketleftBig
a
γ(π−b)/bracketrightBig/bracerightBig
γsinb/parenleftBig
cosh2πa
γ−cos2πβ
γ/parenrightBig
[|Reβ|<Reγ,0<b<π , a< 0]
BI (267)(6)
7./integraldisplay∞
0cosaxdx
/parenleftBig
β+/radicalbig
β2−1c os h x/parenrightBigν+1=Γ (ν+1−ai)eaπQai
ν(β)
Γ(ν+1 )
[Reν>−1,|arg(β+1 )|<π , a> 0]
ET I 30(10)
3.984
1.6lim
c↑1/integraldisplay∞
0sinaxsinhcx
coshx+c o s bdx=πcoshab
sinhaπ[|b|≤π, a real] BI (267)(1)
2.6lim
c↑1/integraldisplay∞
0cosaxcoshcx
coshx+c o s bdx=−πcotbsinhab
sinhaπ[0<|b|<π , a real] BI (267)(5)
3.8/integraldisplay∞
0sinaxsinhx
2
coshx+c o s βdx=πsinhaβ
2s inβ
2coshaπ[Reβ < π, a > 0] ET I 80(10)
4./integraldisplay∞
0cosaxcoshβ
2x
coshβx+c o s h γdx=πcosaγ
β
2βcoshγ
2coshaπ
β/bracketleftbig
πReβ>/vextendsingle/vextendsingleIm/parenleftbig
βγ/parenrightbig/vextendsingle/vextendsingle/bracketrightbig
ET I 31(16)
5./integraldisplay∞
0sinaxsinhβx
cosh 2 βx+c o s2 axdx=aπ
4(a2+β2)[a>0,Reβ>0] BI (267)(7)
512 Trigonometric Functions 3.985
6./integraldisplay∞
0cosaxcoshβx
cosh 2 βx+c o s2 axdx=βπ
4(a2+β2)[Reβ>0,a > 0] BI (267)(8)
7.8/integraldisplay∞
0sinh2μ−1xcosh2/rho1−2ν+1x/parenleftbig
cosh2x−βsinh2x/parenrightbig/rho1dx=1
2B(μ, ν−μ)2F1(/rho1, μ;ν;β)
[Reν>Reμ>0] EH I 115(12)
3.985
1./integraldisplay∞
0cosaxdx
coshνβx=2ν−2
βΓ(ν)Γ/parenleftbiggν
2+ai
2β/parenrightbigg
Γ/parenleftbiggν
2−ai
2β/parenrightbigg
[Reβ>0,Reν>0,a > 0]
ET I 30(5)
2./integraldisplay∞
0cosaxdx
cosh2nβx=4n−1πa
2(2n−1)!β2sinhaπ
2βn−1/productdisplay
k=1/parenleftbigga2
4β2+k2/parenrightbigg
=πa/parenleftbig
a2+22β2/parenrightbig/parenleftbig
a2+42β2/parenrightbig
···/bracketleftbig
a2+( 2n−2)2β2/bracketrightbig
2(2n−1)!β2nsinhaπ
2β
[n≥2,a > 0] ET I 30(3)
3./integraldisplay∞
0cosaxdx
cosh2n+1βx=π22n−1
(2n)!βcoshaπ
2βn/productdisplay
k=1/bracketleftBigg
a2
4β2+/parenleftbigg2k−1
2/parenrightbigg2/bracketrightBigg
=π/parenleftbig
a2+β2/parenrightbig/parenleftbig
a2+32β2/parenrightbig
···/bracketleftbig
a2+( 2n−1)2β2/bracketrightbig
2(2n)!β2n+1coshaπ
2β
[Reβ>0,n=0,1,..., all real a]ET I 30(4)
3.986
1./integraldisplay∞
0sinβxsinγx
coshδxdx=π
δ·sinhβπ
2δsinhγπ
2δ
coshβ
δπ+c o s hγ
δπ[|Im(β+γ)|<Reδ] BI (264)(19)
2./integraldisplay∞
0sinαxcosβx
sinhγxdx=πsinhπα
γ
2γ/parenleftBig
coshαπ
γ+c o s hβπ
γ/parenrightBig [|Im(α+β)|<Reγ] LI (264)(20)
3./integraldisplay∞
0cosβxcosγx
coshδxdx=π
δ·coshβπ
2δcoshγπ
2δ
coshβπ
δ+c o s hγπ
δ[|Im(β+γ)|<Reδ] BI (264)(21)
4.3/integraldisplay∞
0sin2βx
sinh2πxdx=β
π(e2β−1)+β−1
2π=βcothβ−1
2π
[|Imβ|<π] EH I 44(3)
3.987
1./integraldisplay∞
0sinax(1−tanhβx)dx=1
a−π
2βsinhαπ
2β[Reβ>0] ET I 88(4)a
2./integraldisplay∞
0sinax(cothβx−1)dx=π
2βcothaπ
2β−1
a[Reβ>0] ET I 88(3)
3.991 Trigonometric and hyperbolic functions 513
3.988
1./integraldisplayπ/2
0cosaxsinh (2 bcosx)√cosxdx=π
2√
πbIα
2+1
4(b)I−a
2+1
4(b)
[a>0] ET I 37(66)
2./integraldisplayπ/2
0cosaxcosh (2 bcosx)√cosxdx=π
2√
πbIa
2−1
4(b)I−a
2−1
4(b)
[a>0] ET I 37(67)
3./integraldisplay∞
0cosaxdx√
coshxcosb=πP−1
2+ia(cosb)
√
2c os h aπ[a>0,b > 0] ET I 30(7)
3.989
1./integraldisplay∞
0sina2x2
πsinbx
sinhaxdx=π
2asinπb2
4a2cosechπb
2a[a>0,b > 0] ET I 93(44)
2./integraldisplay∞
0cosa2x2
πsinbx
sinhaxdx=π
2acoshπb
a−cosπb2
4a2
sinhπb
2a[a>0,b > 0] ET I 93(45)
3./integraldisplay∞
0sinx2
πcosax
coshxdx=π
2cosa2π
4−1√
2
coshaπ
2ET I 36(54)
4./integraldisplay∞
0cosx2
πcosax
coshxdx=π
2·sina2π
4+1√
2
coshaπ
2ET I 36(55)
5./integraldisplay∞
0sin/parenleftbig
πax2/parenrightbig
cosbx
coshπxdx=−∞/summationdisplay
k=0exp/bracketleftbig
−/parenleftbig
k+1
2/parenrightbig
b/bracketrightbig
sin/bracketleftBig/parenleftbig
k+1
2/parenrightbig2πa/bracketrightBig
+1√a∞/summationdisplay
k=0exp/bracketleftBigg
−b/parenleftbig
k+1
2/parenrightbig
a/bracketrightBigg
sin/bracketleftBigg
π
4−b2
4πa+/parenleftbig
k+1
2/parenrightbig2π
a/bracketrightBigg
[a>0,b > 0] ET I 36(56)
6./integraldisplay∞
0cos/parenleftbig
πax2/parenrightbig
cosbx
coshπxdx=∞/summationdisplay
k=0(−1)kexp/bracketleftbigg
−/parenleftbigg
k+1
2/parenrightbigg
b/bracketrightbigg
cos/bracketleftBigg/parenleftbigg
k+1
2/parenrightbigg2
πa/bracketrightBigg
+1√a∞/summationdisplay
k=0exp/bracketleftBigg
−b/parenleftbig
k+1
2/parenrightbig
a/bracketrightBigg
cos/bracketleftBigg
π
4−b2
4πa+/parenleftbig
k+1
2/parenrightbig2π
a/bracketrightBigg
[a>0,b > 0] ET I 36(57)
3.991
1./integraldisplay∞
0sinπx2sinaxcothπxdx =1
2tanha
2sin/parenleftbiggπ
4+a2
4π/parenrightbigg
ET I 93(42)
2.11/integraldisplay∞
0cosπx2sinaxcothπxdx =1
2tanha
2/bracketleftbigg
1−cos/parenleftbiggπ
4+a2
4π/parenrightbigg/bracketrightbigg
ET I 93(43)
514 Trigonometric Functions 3.992
3.992
1./integraldisplay∞
0sinπx2cosax
1+2c o s h/parenleftbigg2√
3πx/parenrightbiggdx=−√
3+cos/parenleftBig
π
12−a2
4π/parenrightBig
4c os ha√
3−2ET I 37(60)
2./integraldisplay∞
0cosπx2cosax
1+2c o s h/parenleftbigg2√
3πx/parenrightbiggdx=1−sin/parenleftBig
π
12−a2
4π/parenrightBig
4c os ha√
3−2ET I 37(61)
3.993/integraldisplay∞
0sin2x+c o s x2
cosh (√πx)cosaxdx =√π
2·sin2a+c o s a2
cosh (√πa)ET I 37(58)
3.994
1./integraldisplay∞
0sin (2acoshx)c o sbx√
coshxdx=−π
4√aπ/bracketleftBig
J1
4+ib
2(a)Y1
4−ib
2(a)+J1
4−ib
2(a)Y1
4+ib
2(a)/bracketrightBig
[a>0,b > 0] ET I 37(62)
2./integraldisplay∞
0cos(2acoshx)c o sbx√
coshxdx=−π
4√aπ/bracketleftBig
J−1
4+ib
2(a)Y−1
4−ib
2(a)+J−1
4−ib
2(a)Y−1
4+ib
2(a)/bracketrightBig
[a>0,b > 0] ET I 37(63)
3./integraldisplay∞
0sin (2asinhx)s i nbx√
sinhxdx=−i
2√πa/bracketleftBig
I1
4−ib
2(a)K−1
4+ib
2(a)−I1
4+ib
2(a)K1
4−ib
2(a)/bracketrightBig
[a>0,b > 0] ET I 93(47)
4./integraldisplay∞
0cos(2asinhx)s i nbx√
sinhxdx=−i
2√πa/bracketleftBig
I−1
4−ib
2(a)K−1
4+ib
2(a)−I−1
4+ib
2(a)K−1
4−ib
2(a)/bracketrightBig
[a>0,b > 0] ET I 93(48)
5./integraldisplay∞
0sin (2asinhx)c o sbx√
sinhxdx=√πa
2/bracketleftBig
I1
4−ib
2(a)K1
4+ib
2(a)+I1
4+ib
2(a)K1
4−ib
2(a)/bracketrightBig
[a>0,b > 0] ET I 37(64)
6./integraldisplay∞
0cos(2asinhx)c o sbx√
sinhxdx=√πa
2/bracketleftBig
I−1
4−ib
2(a)K−1
4+ib
2(a)+I−1
4+ib
2(a)K−1
4−ib
2(a)/bracketrightBig
[a>0,b > 0] ET I 37(65)
7./integraldisplay∞
0sin(acoshx)s i n(asinhx)dx
sinhx=π
2sina [a>0] BI (264)(22)
3.995
1./integraldisplayπ/2
0sin/parenleftbig
2acos2x/parenrightbig
cosh(asin2x)
b2cos2x+c2sin2xdx=π
2bcsin2ac
b+c
[b>0,c > 0] BI (273)(9)
2./integraldisplayπ/2
0cos/parenleftbig
2acos2x/parenrightbig
cosh (asin 2x)
b2cos2x+c2sin2xdx=π
2bccos2ac
b+c
[b>0,c > 0] BI (273)(10)
3.997 Trigonometric and hyperbolic functions 515
3.996
1./integraldisplay∞
0sin(asinhx)s i n h βxdx =s i nβπ
2Kβ(a)[ |Reβ|<1,a > 0] EH II 82(26)
2./integraldisplay∞
0cos(asinhx)c o s h βxdx =c o sβπ
2Kβ(a)[ |Reβ|<1,a > 0] WA 202(13)
3./integraldisplayπ/2
0cos(asinx)c os h( βcosx)dx=π
2J0/parenleftBig/radicalbig
a2−β2/parenrightBig
MO 40
4./integraldisplay∞
0sin/parenleftbig
acoshx−1
2βπ/parenrightbig
coshβxdx =π
2Jβ(a)[ |Reβ|<1,a > 0] WA 199(12)
5./integraldisplay∞
0cos/parenleftbig
acoshx−1
2βπ/parenrightbig
coshβxdx =−π
2Yβ(a)[ |Reβ|<1,a > 0] WA 199(13)
3.997
1./integraldisplayπ/2
0sinνxsinh (βcosx)dx=√π
2/parenleftbigg2
β/parenrightbiggν
2
Γ/parenleftbiggν+1
2/parenrightbigg
Lν
2(β)
[Reν>−1] EH II 38(53)
2./integraldisplayπ
0sinνxcosh (βcosx)dx=√π/parenleftbigg2
β/parenrightbiggν
2
Γ/parenleftbiggν+1
2/parenrightbigg
Iν
2(β)
[Reν>−1] WH
3./integraldisplayπ/2
0dx
cosh(tan x)c o sx√
sin 2x=√
2π∞/summationdisplay
k=0(−1)k
√
2k+1BI (276)(13)
4./integraldisplayπ/2
0tanqx
cosh(tan x) + cos λdx
sin 2x=Γ(q)
sinλ∞/summationdisplay
k=1(−1)k−1sinkλ
kq
[q>0] BI (275)(20)
516 Trigonometric Functions 4.111
4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers
4.111
1./integraldisplay∞
0sinax
sinhβx·x2mdx=(−1)mπ
2β·∂2m
∂a2m/parenleftbigg
tanhaπ
2β/parenrightbigg
[Reβ>0] (cf. 3.981 1)
GW (336)(17a)
2./integraldisplay∞
0cosax
sinhβx·x2m+1dx=(−1)mπ
2β∂2m+1
∂a2m+1/parenleftbigg
tanhaπ
2β/parenrightbigg
[Reβ>0] (cf. 3.981 1)
GW (336)(17b)
3./integraldisplay∞
0sinax
coshβx·x2m+1dx=(−1)m+1π
2β·∂2m+1
∂a2m+1/parenleftBigg
1
coshaπ
2β/parenrightBigg
[Reβ>0] (cf. 3.981 3)
GW (336)(18b)
4./integraldisplay∞
0cosax
coshβx·x2mdx=(−1)mπ
2β·∂2m
∂a2m/parenleftBigg
1
coshaπ
2β/parenrightBigg
[Reβ>0] (cf. 3.981 3)
GW (336)(18a)
5./integraldisplay∞
0xsin 2ax
coshβxdx=π2
4β2·sinhaπ
β
cosh2aπ
β[Reβ>0,a > 0] BI (364)(6)a
6./integraldisplay∞
0xcos 2ax
sinhβxdx=π2
4β2·1
cosh2aπ
β[Reβ>0,a > 0] BI (364)(1)a
7./integraldisplay∞
0sinax
coshβxdx
x= 2arctan/parenleftbigg
expπa
2β/parenrightbigg
−π
2[Reβ>0,a > 0]
BI (387)(1), ET I 89(13), LI (298)(17)
4.112
1./integraldisplay∞
0/parenleftbig
x2+β2/parenrightbigcosax
coshπx
2βdx=2β3
cosh3aβ[Reβ>0,a > 0] ET I 32(19)
2./integraldisplay∞
0x/parenleftbig
x2+4β2/parenrightbigcosax
sinhπx
2βdx=6β4
cosh4aβ[Reβ>0,a > 0] ET I 32(20)
4.113 Trigonometric and hyperbolic functions and powers 517
4.113
1./integraldisplay∞
0sinax
sinhπx·dx
x2+β2=−1
2β2−πe−aβ
βsinπβ
+1
2β2/bracketleftbig
2F1/parenleftbig
1,−β;1−β;−e−a/parenrightbig
+2F1/parenleftbig
1,β;1+β:−e−a/parenrightbig/bracketrightbig
=1
2β2−πe−aβ
2βsinπβ−∞/summationdisplay
k=1(−1)ke−ak
k2−β2
[Reβ>0,β/negationslash=0,1,2,..., a> 0]ET I 90(18)
2./integraldisplay∞
0sinax
sinhπx·dx
x2+m2=(−1)mae−ma
2m+1
2mm−1/summationdisplay
k=1(−1)ke−ka
m−k+(−1)me−ma
2mln/parenleftbig
1+e−a/parenrightbig
+1
2m!dm−1
dzm−1/bracketleftbigg(1 +z)m−1
zln(1 + z)/bracketrightbigg
z=e−a
[a>0] ET I 89(17)
3./integraldisplay∞
0sinax
sinhπx·dx
1+x2=1
2/integraldisplay∞
−∞sinax
sinhπxdx
1+x2=−a
2cosha+sinh aln/parenleftBig
2c os ha
2/parenrightBig
GW (336)(21b)
4./integraldisplay∞
0sinax
sinhπ
2x·dx
1+x2=1
2/integraldisplay∞
−∞sinax
sinhπ
2x·dx
1+x2=π
2sinha−coshaarctan(sinh a)
GW (336)(21a)
5./integraldisplay∞
0sinax
sinhπ
4x·dx
1+x2=−π√
2e−a+sinha√
2ln2c os h a+√
2
2c os h a−√
2+√
2c os h aarctan√
2
2s in h a
[a>0] LI (389)(1)
6./integraldisplay∞
0sinax
coshπ
4x·xdx
1+x2=π√
2e−a+sinha√
2ln2c os h a+√
2
2c os h a−√
2−√
2c os h aarctan/parenleftbigg1√
2s in h a/parenrightbigg
[a>0] BI (388)(1)
7./integraldisplay∞
0cosax
sinhπx·xdx
1+x2=−1
2+a
2e−a+c o s h aln/parenleftbig
1+e−a/parenrightbig
[a>0] BI (389)(14), ET I 32(24)
8./integraldisplay∞
0cosax
sinhπ
2x·xdx
1+x2=2s i n h aarctan/parenleftbig
e−a/parenrightbig
+π
2e−a−1
[a>0] BI (389)(11)
9.11/integraldisplay∞
0cosax
coshπx·dx
x2+β2=πe−aβ
2βcos(βπ)−∞/summationdisplay
k=0(−1)ke−(k+1/2)a
/parenleftbig
k+1
2/parenrightbig2−β2
[Reβ>0,a > 0] ET I 32(26)
10.11/integraldisplay∞
0cosax
coshπx·dx
x2+/parenleftbig
m+1
2/parenrightbig2=(−1)me−aβ/parenleftbig
aβ+1
2/parenrightbig
2β2−∞/summationdisplay
k=0(−1)ke−(k+1/2)a
/parenleftbig
k+1
2/parenrightbig2−β2
[Reβ>0,a > 0] ET I 32(25)
518 Trigonometric Functions 4.114
11./integraldisplay∞
0cosax
coshπx·dx
1+x2=2c o s ha
2−/bracketleftbig
eaarctan/parenleftbig
e−a
2/parenrightbig
+e−aarctan/parenleftbig
ea
2/parenrightbig/bracketrightbig
[a>0] ET I 32(21)
12./integraldisplay∞
0cosax
coshπ
2x·dx
1+x2=ae−a+c o s h aln/parenleftbig
1+e−2a/parenrightbig
[a>0] BI (388)(6)
13./integraldisplay∞
0cosax
coshπ
4x·dx
1+x2=π√
2e−a+2s in h a√
2arctan/parenleftbigg1√
2s in h a/parenrightbigg
−cosha√
2ln2c os h a+√
2
2c os h a−√
2
[a>0] BI (388)(5)
4.114
1./integraldisplay∞
0sinax
xsinhβx
sinhγxdx=a r c t a n/parenleftbigg
tanβπ
2γtanhaπ
2γ/parenrightbigg
[|Reβ|<Reγ, a > 0] BI (387)(6)a
2./integraldisplay∞
0cosax
xsinhβx
coshγxdx=1
2lncoshaπ
2γ+s i nβπ
2γ
coshaπ
2γ−sinβπ
2γ[|Reβ|<Reγ] ET I 33(34)
4.115
1./integraldisplay∞
0xsinax
x2+b2·sinhβx
sinhπxdx=π
2e−absinbβ
sinbπ+∞/summationdisplay
k=1(−1)kke−aksinkβ
k2−b2
[0<Reβ<π , a> 0,b > 0]
BI (389)(23)
2./integraldisplay∞
0xsinax
x2+1·sinhβx
sinhπxdx=1
2e−a(asinβ−βcosβ)−1
2sinhasinβln/bracketleftbig
1+2e−acosβ+e−2a/bracketrightbig
+cosh acosβarctansinβ
ea+c o s β
[|Reβ|<π , a> 0] LI (389)(10)
3./integraldisplay∞
0xsinax
x2+1·sinhβx
sinhπ
2xdx
=π
2e−asinβ+1
2cosβsinhalncosha+s i nβ
cosha−sinβ−sinβcoshaarctan/parenleftbiggcosβ
sinha/parenrightbigg
/bracketleftBig
|Reβ|<π
2,a > 0/bracketrightBig
BI (389)(8)
4./integraldisplay∞
0cosax
x2+b2·sinhβx
sinhπxdx=π
2b·e−absinbβ
sinbπ+∞/summationdisplay
k=1(−1)ke−aksinkβ
k2−b2
[0<Reβ<π , a> 0,b > 0]
BI (389)(22)
5./integraldisplay∞
0cosax
x2+1·sinhβx
sinhπxdx=1
2e−a(asinβ−βcosβ)+1
2coshasinβln/parenleftbig
1+2e−acosβ+e−2a/parenrightbig
−sinhacosβarctansinβ
ea+c o s β
[|Reβ|<π , a> 0,b > 0]BI (389)(20)a
4.115 Trigonometric and hyperbolic functions and powers 519
6./integraldisplay∞
0cosax
x2+1·sinhβx
sinhπ
2xdx=π
2e−asinβ−1
2coshacosβlncosha+s i nβ
cosha−sinβ+s i n h asinβarctancosβ
sinha
/bracketleftBig
|Reβ|<π
2,a > 0,b > 0/bracketrightBig
BI (389)(18)
7./integraldisplay∞
0sinax
x2+1
4·sinhβx
coshπxdx=e−a
2/parenleftbigg
asinβ
2−βcosβ
2/parenrightbigg
−sinha
2sinβ
2ln/parenleftbig
1+2e−acosβ+e−2a/parenrightbig
+cosha
2cosβ
2arctansinβ
1+e−acosβ
[|Reβ|<π , a> 0] ET I 91(26)
8./integraldisplay∞
0sinax
x2+β2·coshγx
sinhπxdx=1
2β2−π
2β·e−aβcosβγ
sinβπ+∞/summationdisplay
k=1(−1)k−1e−akcoskγ
k2−β2
[0≤Reβ,|Reγ|<π , a> 0]
BI (389)(21)
9./integraldisplay∞
0sinax
x2+1·coshβx
sinhπxdx=−1
2e−a(acosβ+βsinβ)+1
2sinhacosβln/parenleftbig
1+2e−acosβ+e−2a/parenrightbig
+cosh asinβarctansinβ
ea+c o s β
[|Reβ|<π , a> 0]ET I 91(25), LI (389)(9)
10./integraldisplay∞
0sinax
x2+1·coshβx
sinhπ
2xdx=−π
2e−acosβ+1
2sinhasinβlncosha+s i nβ
cosha−sinβ+cosh acosβarctancosβ
sinha
/bracketleftBig
|Reβ|<π
2,a > 0/bracketrightBig
BI (389)(7)
11./integraldisplay∞
0xcosax
x2+b2·coshβx
sinhπxdx=π
2·e−abcosbβ
sinbπ+∞/summationdisplay
k=1(−1)kke−akcoskβ
k2−b2
[|Reβ|<π , a> 0] BI (389)(24)
12./integraldisplay∞
0xcosax
x2+1·coshβx
sinhπxdx=1
2e−a(acosβ+βsinβ)
−1
2+1
2coshacosβln/bracketleftbig
1+2e−acosβ+e−2a/bracketrightbig
+sinh asinβarctansinβ
ea+c o s β
[|Reβ|<π , a> 0] BI (389)(19)
13./integraldisplay∞
0xcosax
x2+1·coshβx
sinhπ
2xdx=−1+π
2e−acosβ+1
2coshasinβlncosha+s i nβ
cosha−sinβ
+sinh acosβarctancosβ
sinha/bracketleftBig
|Reβ|<π
2,a > 0/bracketrightBig
BI (389)(17)
520 Trigonometric Functions 4.116
14./integraldisplay∞
0cosax
x2+1·coshβx
coshπ
2xdx=ae−acosβ+βe−asinβ+s i n h asinβarctane−2asin 2β
1+e−2acos 2β
+1
2coshacosβln/parenleftbig
1+2e−2acos 2β+e−4a/parenrightbig
/bracketleftBig
|Reβ|<π
2,a>0/bracketrightBig
ET I 34(37)
4.116
1.6/integraldisplay∞
0xcos 2axtanhxdx the integral is divergent BI (364)(2)
2./integraldisplay∞
0cosaxtanhβxdx
x=l nc o t haπ
4β[Reβ>0,a > 0] BI (387)(8)
4.117
1./integraldisplay∞
0sinax
1+x2tanhπx
2dx=acosha−sinhaln(2 sinh a)
[a>0] BI (388)(3)
2./integraldisplay∞
0sinax
1+x2tanhπx
4dx=−π
2ea+s i n h alncotha
2+2c o s h aarctan( ea) BI (388)(4)
3./integraldisplay∞
0sinax
1+x2cothπxdx =a
2e−a−sinhaln/parenleftbig
1−e−a/parenrightbig
[a>0] BI (389)(5)
4./integraldisplay∞
0sinax
1+x2cothπ
2xdx=s i n h aln cotha
2[a>0] BI (389)(6)
5./integraldisplay∞
0xcosax
1+x2tanhπ
2xdx=−ae−a−coshaln/parenleftbig
1−e−2a/parenrightbig
[a>0] BI (388)(7)
6./integraldisplay∞
0xcosax
1+x2tanhπ
4xdx−π
2ea+c o s h alncotha
2+2s i n h aarctan( ea)
[a>0] BI (388)(8)
7./integraldisplay∞
0xcosax
1+x2cothπxdx =−a
2e−a−1
2−coshaln/parenleftbig
1−e−a/parenrightbig
BI (389)(15)a, ET I 33(31)a
8./integraldisplay∞
0xcosax
1+x2cothπ
2xdx=−1+c o s h alncotha
2[a>0] BI (389)(12)
9./integraldisplay∞
0xcosax
1+x2cothπ
4xdx=−2+π
2e−a+c o s h alncotha
2+2s i n h aarctan/parenleftbig
e−a/parenrightbig
[a>0] BI (389)(13)
4.1188/integraldisplay∞
0xsinax
cosh2xdx=π
21
sinh1
2πa/parenleftbigg1
2πacoth1
2πa−1/parenrightbigg
ET I 89(14)
4.119/integraldisplay∞
01−cospx
sinhqx·dx
x=l n/parenleftbigg
coshpπ
2q/parenrightbigg
BI (387)(2)a
4.123 Trigonometric and hyperbolic functions and powers 521
4.121
1./integraldisplay∞
0sinax−sinbx
coshβx·dx
x= 2arctanexpaπ
2β−expbπ
2β
1 + exp(a+b)π
2β
[Reβ>0] GW (336)(19b)
2./integraldisplay∞
0cosax−cosbx
sinhβx·dx
x=l ncoshbπ
2β
coshaπ
2β[Reβ>0] GW (336)(19a)
4.122
1.6/integraldisplay∞
0cosβxsinγx
coshδx·dx
x=a r c t a nsinhγπ
2δ
coshβπ
2δ[Reδ>|Imβ|+|Imγ|] ET I 93(46)a
2./integraldisplay∞
0sin2axcoshβx
sinhx·dx
x=1
4lncosh 2 aπ+c o s βπ
1 + cos βπ[|Reβ|<1] BI (387)(7)
4.123
1./integraldisplay∞
0sinx
coshax+c o s x·xdx
x2−π2=a r c t a n1
a−1
aBI (390)(1)
2./integraldisplay∞
0sinx
coshax−cosx·xdx
x2−π2=a
1+a2−arctan1
aBI (390)(2)
3./integraldisplay∞
0sin 2x
cosh 2 ax−cos2x·xdx
x2−π2=1
2a·1+2a2
1+a2−arctan1
aBI (390)(4)
4./integraldisplay∞
0coshaxsinx
cosh 2 ax−cos2x·xdx
x2−π2=−1
2a(1 +a2)LI (390)(3)
5./integraldisplay∞
0cosax
coshπx+c o s πβ·dx
x2+γ2=πe−aγ
2γ(cosγπ+c o s βπ)
+1
sinhβπ∞/summationdisplay
k=0/braceleftbigge−(2k+1−β)a
γ2−(2k+1−β)2−e−(2k+1+β)a
γ2−(2k+1+ β)2/bracerightbigg
[0<Reβ<1,Reγ>0,a > 0]ET I 33(27)
6./integraldisplay∞
0sinaxsinhbx
cos2ax+c o s h2 bxxp−1dx=Γ(p)
(a2+b2)p
2sin/parenleftBig
parctana
b/parenrightBig∞/summationdisplay
k=0(−1)k
(2k+1 )p
[p>0] BI (364)(8)
7./integraldisplay∞
0sinax2sinπx
2sinhπx
2
cosπx+c o s h πx·xdx=1
4/bracketleftbigg∂ϑ1(z|q)
∂z/bracketrightbigg
z=0,q=e−2a
[a>0] ET I 93(49)
522 Trigonometric Functions 4.124
4.124
1./integraldisplay1
0cospxcosh/parenleftbig
q√
1−x2/parenrightbig
√
1−x2dx=π
2J0/parenleftBig/radicalbig
p2−q2/parenrightBig
MO (40)
2./integraldisplay∞
ucosaxcosh/radicalbig
β(u2−x2)·dx√
u2−x2=π
2J0/parenleftBigg
u/radicalbig
a2−β2/parenrightBigg
ET I 34(38)
4.125
1./integraldisplay∞
0sinh (asinx)c o s(acosx)s i nxsin 2nxdx
x=(−1)n−1a2n−1
(2n−1)!π
8/bracketleftbigg
1+a2
2n(2n+1 )/bracketrightbigg
LI (367)(14)
2./integraldisplay∞
0cosh(asinx)c o s(acosx)s i nxcos(2n−1)xdx
x=(−1)n−1a2(n−1)
[2(n−1)]!π
8/bracketleftbigg
1−a2
2n(2n−1)/bracketrightbigg
LI (367)(15)
3./integraldisplay∞
0sinh (asinx)c o s(acosx)c o sxcos 2nxdx
x=π
2∞/summationdisplay
k=n+1(−1)ka2k+1
(2k+1 ) !+(−1)na2n+1
(2n+1 ) !3π
8
+(−1)n−1a2n−1
(2n−1)!π
8
LI (367)(21)
4.126
1./integraldisplay∞
0sin(acosbx)s in h( asinbx)xdx
c2−x2=π
2[cos(acosbc)cosh( asinbc)−1]
[b>0] BI (381)(2)
2./integraldisplay∞
0sin(acosbx)c os h( asinbx)dx
c2−x2=π
2ccos(acosbc)s in h( asinbc)
[b>0,c > 0] BI (381)(1)
3./integraldisplay∞
0cos(acosbx)s i n h( asinbx)xdx
c2−x2=π
2[acosbc−sin(acosbc)cosh( asinbc)]
[b>0] BI (381)(4)
4./integraldisplay∞
0cos(acosbx)c os h( asinbx)dx
c2−x2=−π
2csin (acosbc)s in h( asinbc)
[b>0] BI (381)(3)
4.13 Combinations of trigonometric and hyperbolic functions and exponentials
4.131
1./integraldisplay∞
0sinaxsinhνγxe−βxdx=−iΓ(ν+1 )
2ν+2γ⎧
⎨
⎩Γ/parenleftBig
β−νγ−ai
2γ/parenrightBig
Γ/parenleftBig
β+νγ−ai
2γ+1/parenrightBig−Γ/parenleftBig
β−νγ+ai
2γ/parenrightBig
Γ/parenleftBig
β+γν+ai
2γ+1/parenrightBig⎫
⎬
⎭
[Reν>−2,Reγ>0,|Re(γν)|<Reβ]ET I 91(30)a
4.133 Trigonometric and hyperbolic functions and exponentials 523
2./integraldisplay∞
0cosaxsinhνγxe−βxdx=Γ(ν+1 )
2ν+2γ⎧
⎨
⎩Γ/parenleftBig
β−νγ−ai
2γ/parenrightBig
Γ/parenleftBig
β+γν−ai
2γ+1/parenrightBig−Γ/parenleftBig
β−νγ+ai
2γ/parenrightBig
Γ/parenleftBig
β+νγ+ai
2γ+1/parenrightBig⎫
⎬
⎭
[Reν>−1,Reγ>0,|Re(γν)|<Reβ]ET I 34(40)a
3./integraldisplay∞
0e−βxsinax
sinhγxdx=∞/summationdisplay
k=12a
a2+[β+( 2k−1)γ]2 BI (264)(9)a
=1
2γi/bracketleftbigg
ψ/parenleftbiggβ+γ+ia
2γ/parenrightbigg
−ψ/parenleftbiggβ+γ−ia
2γ/parenrightbigg/bracketrightbigg
[Reβ>|Reγ|]ET I 91(28)
4./integraldisplay∞
0e−xsinax
sinhxdx=π
2cothaπ
2−1
aET I 91(29)
4.132
1./integraldisplay∞
0sinaxsinhβx
eγx−1dx=−a
2(a2+β2)+π
2γ·sinh2πa
γ
cosh2πa
γ−cos2πβ
γ
+i
2γ/bracketleftbigg
ψ/parenleftbiggβ
γ+ia
γ+1/parenrightbigg
−ψ/parenleftbiggβ
γ−ia
γ+1/parenrightbigg/bracketrightbigg
[Reγ>|Reβ|,a>0] ET I 92(33)
2./integraldisplay∞
0sinaxcoshβx
eγx−1dx=−a
2(a2+β2)+π
2γ·sinh2πa
γ
cosh2πa
γ−cos2πβ
γ
[Reγ>|Reβ|]BI (265)(5)a, ET I 92(34)
3./integraldisplay∞
0sinaxcoshβx
eγx+1dx=a
2(a2+β2)−π
γ·sinhaπ
γcosβπ
γ
cosh2aπ
γ−cos2βπ
γ
[Reγ>|Reβ|] ET I 92(35)
4./integraldisplay∞
0cosaxsinhβx
eγx−1dx=β
2(a2+β2)−π
2γ·sin2πβ
γ
cosh2aπ
γ−cos2βπ
γ
[Reγ>|Reβ|] LI (265)(8)
5./integraldisplay∞
0cosaxsinhβx
eγx+1dx=−β
2(a2+β2)+π
γsinπβ
γcoshπa
γ
cosh2aπ
γ−cos2βπ
γ
[Reγ>|Reβ|] ET I 34(39)
4.133
1.11/integraldisplay∞
0sinaxsinhβxexp/parenleftbigg
−x2
4γ/parenrightbigg
dx=√πγexp/bracketleftbig
γ/parenleftbig
β2−a2/parenrightbig/bracketrightbig
sin(2aβγ)
[Reγ>0] ET I 92(37)
2.11/integraldisplay∞
0cosaxcoshβxexp/parenleftbigg
−x2
4γ/parenrightbigg
dx=√πγexp/bracketleftbig
γ/parenleftbig
β2−a2/parenrightbig/bracketrightbig
cos(2aβγ)
[Reγ>0] ET I 35(41)
524 Trigonometric Functions 4.134
4.134
1./integraldisplay∞
0e−βx2(coshx−cosx)dx=/radicalbiggπ
βcosh1
4β[Reβ>0] ME 24
2./integraldisplay∞
0e−βx2(coshx−cosx)dx=/radicalbiggπ
βsinh1
4β[Reβ>0] ME 24
4.135
1./integraldisplay∞
0sinax2cosh 2 γxe−βx2dx=1
24/radicalBigg
π2
a2+β2exp/parenleftbigg
−βγ2
a2+β2/parenrightbigg
sin/parenleftbiggaγ2
a2+β2+1
2arctana
β/parenrightbigg
[Reβ>0] LI (268)(7)
2./integraldisplay∞
0cosax2cosh 2 γxe−βx2dx=1
24/radicalBigg
π2
a2+β2exp/parenleftbigg
−βγ2
a2+β2/parenrightbigg
cos/parenleftbiggaγ2
a2+β2+1
2arctana
β/parenrightbigg
[Reβ>0] LI (268)(8)
4.136
1./integraldisplay∞
0/parenleftbig
sinh2x+s i nx2/parenrightbig
e−βx4dx=√
2π
4√βI1
4/parenleftbigg1
8β/parenrightbigg
cosh1
8β
[Reβ>0] ME 24
2./integraldisplay∞
0/parenleftbig
sinh2x−sinx2/parenrightbig
e−βx4dx=√
2π
4√βI1
4/parenleftbigg1
8β/parenrightbigg
sinh1
8β
[Reβ>0] ME 24
3./integraldisplay∞
0/parenleftbig
cosh2x+c o s x2/parenrightbig
e−βx4dx=√
2π
4√βI−1
4/parenleftbigg1
8β/parenrightbigg
cosh1
8β
[Reβ>0] ME 24
4./integraldisplay∞
0/parenleftbig
cosh2x−cosx2/parenrightbig
e−βx4dx=√
2π
4√βI−1
4/parenleftbigg1
8β/parenrightbigg
sinh1
8β
[Reβ>0] ME 24
4.137
1./integraldisplay∞
0sin2x2sinh 2x2e−βx4dx=π
4/radicalbig
128β2J−1
4/parenleftbigg1
β/parenrightbigg
cos/parenleftbigg1
β+π
4/parenrightbigg
[Reβ>0] MI 32
2./integraldisplay∞
0sin2x2cosh 2 x2e−βx4dx=π
4/radicalbig
128β2J1
4/parenleftbigg1
β/parenrightbigg
cos/parenleftbigg1
β−π
4/parenrightbigg
[Reβ>0] MI 32
3./integraldisplay∞
0cos2x2sinh 2x2e−βx4dx=−π
4/radicalbig
128β2J1
4/parenleftbigg1
β/parenrightbigg
sin/parenleftbigg1
β−π
4/parenrightbigg
[Reβ>0] MI 32
4.141 Trigonometric and hyperbolic functions, exponentials, and powers 525
4./integraldisplay∞
0cos2x2cosh2 x2e−βx4dx=π
4/radicalbig
128β2J−1
4/parenleftbigg1
β/parenrightbigg
sin/parenleftbigg1
β+π
4/parenrightbigg
[Reβ>0] MI 32
4.138
1./integraldisplay∞
0/parenleftbig
sin22xcosh 2 x2+c o s2 x2sinh 2x2/parenrightbig
e−βx4dx=π
4/radicalbig
32β2J1
4/parenleftbigg1
β/parenrightbigg
cos/parenleftbigg1
β/parenrightbigg
[Reβ>0] MI 32
2./integraldisplay∞
0/parenleftbig
sin22xcosh 2 x2−cos 2x2sinh 2x2/parenrightbig
e−βx4dx=π
4/radicalbig
32β2J1
4/parenleftbigg1
β/parenrightbigg
sin/parenleftbigg1
β/parenrightbigg
[Reβ>0] MI 32
3./integraldisplay∞
0/parenleftbig
cos22xcosh 2 x2+s i n2 x2sinh 2x2/parenrightbig
e−βx4dx=π
4/radicalbig
32β2J−1
4/parenleftbigg1
β/parenrightbigg
cos/parenleftbigg1
β/parenrightbigg
[Reβ>0] MI 32
4./integraldisplay∞
0/parenleftbig
cos22xcosh 2 x2−sin 2x2sinh 2x2/parenrightbig
e−βx4dx=π
4/radicalbig
32β2J−1
4/parenleftbigg1
β/parenrightbigg
sin/parenleftbigg1
β/parenrightbigg
[Reβ>0] MI 32
4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and
powers
4.141
1./integraldisplay∞
0xe−βx2coshxsinxdx=1
4/radicalbiggπ
β3/parenleftbigg
cos1
2β+s i n1
2β/parenrightbigg
[Reβ>0] MI 32
2./integraldisplay∞
0xe−βx2sinhxcosxdx=1
4/radicalbiggπ
β3/parenleftbigg
cos1
2β−sin1
2β/parenrightbigg
[Reβ>0] MI 32
3./integraldisplay∞
0x2e−βx2coshxcosxdx=1
4/radicalbiggπ
β3/parenleftbigg
cos1
2β−1
βsin1
2β/parenrightbigg
[Reβ>0] MI 32
4./integraldisplay∞
0x2e−βx2sinhxsinxdx=1
4/radicalbiggπ
β3/parenleftbigg
sin1
2β+1
βcos1
2β/parenrightbigg
[Reβ>0] MI 32
526 Trigonometric Functions 4.142
4.142
1./integraldisplay∞
0xe−βx2(sinhx+s i nx)dx=1
2/radicalbiggπ
β3cosh1
4β[Reβ>0] ME 24
2./integraldisplay∞
0xe−βx2(sinhx−sinx)dx=1
2/radicalbiggπ
β3sinh1
4β[Reβ>0] ME 24
3./integraldisplay∞
0x2e−βx2(coshx+c o s x)dx=1
2/radicalbiggπ
β3/parenleftbigg
cosh1
4β+1
2βsinh1
4β/parenrightbigg
[Reβ>0] ME 24
4./integraldisplay∞
0x2e−βx2(coshx−cosx)dx=1
2/radicalbiggπ
β3/parenleftbigg
sinh1
4β+1
2βcosh1
4β/parenrightbigg
[Reβ>0] ME 24
4.143
1./integraldisplay∞
0xe−βx2(coshxsinx+s i n h xcosx)dx=1
2β/radicalbiggπ
βcos1
2β
[Reβ>0] MI 32
2./integraldisplay∞
0xe−βx2(coshxsinx−sinhxcosx)dx=1
2β/radicalbiggπ
βsin1
2β
[Reβ>0] MI 32
4.144/integraldisplay∞
0e−x2sinhx2cosaxdx
x2=/radicalbiggπ
2e−a2
8−πa
4/bracketleftbigg
1−Φ/parenleftbigga√
8/parenrightbigg/bracketrightbigg
[a>0] ET I 35(44)
4.145
1./integraldisplay∞
0xe−βx2cosh (2 axsint)sin(2 axcost)dx=a
2/radicalbiggπ
β3exp/parenleftbigg
−a2
βcos2t/parenrightbigg
cos/parenleftbigg
t−a2
βsin 2t/parenrightbigg
[Reβ>0] BI (363)(5)
2./integraldisplay∞
0xe−βx2sinh (2 axsint)c o s( 2 axcost)dx=a
2/radicalbiggπ
β3exp/parenleftbigg
−a2
βcos2t/parenrightbigg
sin/parenleftbigg
t−a2
βsin 2t/parenrightbigg
[Reβ>0] BI (363)(6)
4.14610
1.8/integraldisplay∞
0e−βx2sinhaxsinbxdx =1
2/radicalbiggπ
βexp/parenleftbigga2−b2
4β/parenrightbigg
sinab
2β
[Reβ>0]
2.8/integraldisplay∞
0e−βx2coshaxcosbxdx =1
2/radicalbiggπ
βexp/parenleftbigga2−b2
4β/parenrightbigg
cosab
2β
[Reβ>0]
4.212 Logarithmic functions 527
3./integraldisplay∞
0xe−βx2coshaxsinaxdx =a
4β/radicalbiggπ
β/parenleftbigg
cosa2
2β+s i na2
2β/parenrightbigg
[Reβ>0]
4./integraldisplay∞
0xe−βx2sinhaxcosaxdx =a
4β/radicalbiggπ
β/parenleftbigg
cosa2
2β−sina2
2β/parenrightbigg
[Reβ>0]
5.8/integraldisplay∞
0x2e−βx2coshaxsinaxdx =1
4/radicalbiggπ
β3/parenleftbigg
sina2
2β+a2
βcosa2
2β/parenrightbigg
[Reβ>0]
6.8/integraldisplay∞
0x2e−βx2coshaxcosaxdx =1
4/radicalbiggπ
β3/parenleftbigg
cosa2
2β−a2
βsina2
2β/parenrightbigg
[Reβ>0]
4.2–4.4 Logarithmic Functions
4.21 Logarithmic functions
4.211
1./integraldisplay∞
edx
ln1
x=−∞ BI (33)(9)
2./integraldisplayu
0dx
lnx=l iu F II I I6 5 3 ,F II I6 0 6
4.212
1.7/integraldisplay1
0dx
a+l nx=e−aEi(a)[ a>0] BI (31)(4)
2./integraldisplay1
0dx
a−lnx=−eaEi(−a)[ a>0] BI (31)(5)
3.7/integraldisplay1
0dx
(a+l nx)2=−1
a+e−aEi(a)[ a≥0] BI (31)(14)
4./integraldisplay1
0dx
(a−lnx)2=1
a+eaEi(−a)[ a>0] BI (31)(16)
5.8/integraldisplay1
0lnxdx
(a+l nx)2=1+( 1 −a)e−aEi(a)[ a≥0] BI (31)(15)
6./integraldisplay1
0lnxdx
(a−lnx)2=1+( 1+ a)eaEi(−a)[ a>0] BI (31)(17)
7./integraldisplaye
1lnxdx
(1 + ln x)2=e
2−1 BI (33)(10)
528 Logarithmic Functions 4.213
8.7/integraldisplay1
0dx
(a+l nx)n=1
(n−1)!e−aEi(a)−1
(n−1)!n−1/summationdisplay
k=1(n−k−1)!ak−n
[a≥0] BI (31))(22)
9./integraldisplay1
0dx
(a−lnx)n=(−1)n
(n−1)!eaEi(−a)+(−1)n−1
(n−1)!n−1/summationdisplay
k=1(n−k−1)!(−a)k−n
[a>0,nodd] BI (31)(23)
In integrals of the form/integraldisplay(lnx)m
[an+( l nx)n]ldx, it is convenient to make the substitution x=e−t.
Results 4.212 3,4.212 5, and 4.212 8[ f o r n>1] and 4.213 6,4.213 8 below are divergent but may
be considered to be valid if defined as follows:/integraldisplaya
0f(z)dz
(z−z0)n=1
(n−1)!/parenleftbiggd
dz0/parenrightbiggn−1/bracketleftbigg
PV/integraldisplaya
0f(z)
z−z0dz/bracketrightbigg
where a>z 0>0,n=1,2,3,...and PV indicates the Cauchy principal value.
4.213
1./integraldisplay1
0dx
a2+( l nx)2=1
a[ci(a)sina−si(a)cosa][ a>0] BI (31)(6)
2.7/integraldisplay1
0dx
a2−(lnx)2=1
2a/bracketleftbig
e−aEi(a)−eaEi(−a)/bracketrightbig
[a>0],(cf.4.212 1a n d2 )
BI (31)(8)
3./integraldisplay1
0lnxdx
a2+( l nx)2=c i (a)cosa+s i (a)sina [a>0] BI (31)(7)
4.7/integraldisplay1
0lnxdx
a2−(lnx)2=−1
2/bracketleftbig
e−aEi(a)+eaEi(−a)/bracketrightbig
[a>0],(cf.4.212 1a n d2 )
BI (31)(9)
5./integraldisplay1
0dx
/bracketleftBig
a2+( l nx)2/bracketrightBig2=1
2a3[ci(a)sina−si(a)cosa]−1
2a2[ci(a)cosa+s i (a)sina]
[a>0] LI (31)(18)
6.8/integraldisplay1
0dx
/bracketleftBig
a2−(lnx)2/bracketrightBig2is divergent
7./integraldisplay1
0lnxdx
/bracketleftBig
a2+( l nx)2/bracketrightBig2=1
2a[ci(a)sina−si(a)cosa]−1
2a2
[a>0] BI (31)(19)
8.8/integraldisplay1
0lnxdx
/bracketleftBig
a2−(lnx)2/bracketrightBig2is divergent
4.221 Logarithms of more complicated arguments 529
4.214
1./integraldisplay1
0dx
a4−(lnx)4=−1
4a3/bracketleftbig
eaEi(−a)−e−aEi(a)−2c i (a)sina+2s i ( a)cosa/bracketrightbig
[a>0] BI (31)(10)
2./integraldisplay1
0lnxdx
a4−(lnx)4=−1
4a2/bracketleftbig
eaEi(−a)+e−aEi(a)−2c i (a)cosa−2s i(a)sina/bracketrightbig
[a>0] BI (31)(11)
3./integraldisplay1
0(lnx)2dx
a4−(lnx)4=−1
4a/bracketleftbig
eaEi(−a)−e−aEi(a)+2c i ( a)sina−2s i(a)cosa/bracketrightbig
[a>0] BI (31)(12)
4.7/integraldisplay1
0(lnx)3dx
a4−(lnx)4=−1
4/bracketleftbig
eaEi(−a)+e−aEi(a)+2c i ( a)cosa+2s i ( a)sina/bracketrightbig
[a>0] BI (31)(13)
4.215
1./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggμ−1
dx=Γ (μ)[ R e μ>0] FI II 778
2./integraldisplay1
0dx/parenleftbigg
ln1
x/parenrightbiggμ=π
Γ(μ)cosecμπ [Reμ<1] BI (31)(1)
3./integraldisplay1
0/radicalbigg
ln1
xdx=√π
2BI (32)(1)
4./integraldisplay1
0dx/radicalbigg
ln1
x=√π BI (32)(3)
4.216
1./integraldisplay1/e
0dx/radicalBig
(lnx)2−1=K0(1) GW (32)(2)
2.∗/integraldisplay1/e
0dx√
−lnx−1=√π
e
4.22 Logarithms of more complicated arguments
4.221
1./integraldisplay1
0lnxln(1−x)dx=2−π2
6BI (30)(7)
2./integraldisplay1
0lnxln(1 + x)dx=2−π2
12−2l n2 BI (30)(8)
530 Logarithmic Functions 4.222
3./integraldisplay1
0ln1−ax
1−adx
lnx=−∞/summationdisplay
k=1akln(1 + k)
k[a<1] BI (31)(3)
4.222
1./integraldisplay∞
0lna2+x2
b2+x2dx=(a−b)π [a>0,b > 0] GW (322)(20)
2./integraldisplay∞
0lnxlna2+x2
b2+x2dx=π(b−a)+πlnaa
bb[a>0,b > 0] BI (33)(1)
3./integraldisplay∞
0lnxln/parenleftbigg
1+b2
x2/parenrightbigg
dx=πb(lnb−1) [ b>0] BI (33)(2)
4./integraldisplay∞
0ln/parenleftbig
1+a2x2/parenrightbig
ln/parenleftbigg
1+b2
x2/parenrightbigg
dx=2π/bracketleftbigg1+ab
aln(1 + ab)−b/bracketrightbigg
[a>0,b > 0] BI (33)(3)
5./integraldisplay∞
0ln/parenleftbig
a2+x2/parenrightbig
ln/parenleftbigg
1+b2
x2/parenrightbigg
dx=2π[(a+b)ln (a+b)−alna−b]
[a>0,b > 0] BI (33)(4)
6./integraldisplay∞
0ln/parenleftbigg
1+a2
x2/parenrightbigg
ln/parenleftbigg
1+b2
x2/parenrightbigg
dx=2π[(a+b)ln (a+b)−alna−blnb]
[a>0,b > 0] BI (33)(5)
7./integraldisplay∞
0ln/parenleftbigg
a2+1
x2/parenrightbigg
ln/parenleftbigg
1+b2
x2/parenrightbigg
dx=2π/bracketleftbigg1+ab
aln(1 + ab)−blnb/bracketrightbigg
[a>0,b > 0] BI (33)(7)
8.∗/integraldisplay∞
0ln(1 + ax)xbe−xdx=b/summationdisplay
m=0b!
(b−m)!/bracketleftBigg
(−1)b−m−1
ab−me1/aEi/parenleftbigg
−1
a/parenrightbigg
+b−m/summationdisplay
k=1(k−1)!
(−a)b−m−k/bracketrightBigg
[b>0,an integer]
4.223
1./integraldisplay∞
0ln/parenleftbig
1+e−x/parenrightbig
dx=π2
12BI (256)(10)
2./integraldisplay∞
0ln/parenleftbig
1−e−x/parenrightbig
dx=−π2
6BI (256)(11)
3./integraldisplay∞
0ln/parenleftbig
1+2e−xcost+e−2x/parenrightbig
dx=π2
6−t2
2[|t|<π] BI (256)(18)
4.224
1./integraldisplayu
0ln sinxdx=L/parenleftBigπ
2−u/parenrightBig
−L/parenleftBigπ
2/parenrightBig
LO III 186(15)
2./integraldisplayπ/4
0lnsinxdx=−π
4ln2−1
2G BI (285)(1)
4.225 Logarithms of more complicated arguments 531
3./integraldisplayπ/2
0lnsinxdx=1
2/integraldisplayπ
0ln sinxdx=−π
2ln2 FI II 629,643
4./integraldisplayu
0ln cos xdx=−L(u) LO III 184(10)
5./integraldisplayπ/4
0lncos xdx=−π
4ln 2 +1
2G BI (286)(1)
6./integraldisplayπ/2
0lncos xdx=−π
2ln 2 BI 306(1)
7./integraldisplayπ/2
0(lnsin x)2dx=π
2/bracketleftbigg
(ln2)2+π2
12/bracketrightbigg
BI (305)(19)
8./integraldisplayπ/2
0(lncos x)2dx=π
2/bracketleftbigg
(ln 2)2+π2
12/bracketrightbigg
BI (306)(14)
9.8/integraldisplayπ
0ln (a+bcosx)dx=πlna+√
a2−b2
2[a≥|b|>0] GW (322)(15)
10./integraldisplayπ
0ln (1±sinx)dx=−πln 2±4G GW (322)(16a)
11.7/integraldisplayπ/2
0ln(1 + asinx)dx=π
2lna
2+2G+2∞/summationdisplay
k=1bk
kk/summationdisplay
n=1(−1)n+1
2n−1[a>0] b=1−a
1+a
=−π
2ln2 + 2 G [a=1 ]
12./integraldisplayπ
0ln (1 + acosx)dx=πln/parenleftBigg
1+√
1−a2
2/parenrightBigg
/bracketleftbig
a2≤1/bracketrightbig
BI (330)(1)
12 (1)/integraldisplayπ
0ln (1 + acosx)2dx=⎧
⎪⎨
⎪⎩2πln/parenleftBigg
1+√
1−a2
2/parenrightBigg
fora2≤1
π
2lna2
4fora2≥1
13./integraldisplayπ/2
0ln/parenleftbig
1+2asinx+a2/parenrightbig
dx=∞/summationdisplay
k=022k(k!)2
(2k+1 )·(2k+1 ) ! !/parenleftbigg2a
1+a2/parenrightbigg2k+1
/bracketleftbig
a2≤1/bracketrightbig
BI (308)(24)
14.11/integraldisplaynπ
0ln/parenleftbig
a2−2abcosx+b2/parenrightbig
dx=2nπln [max( |a|,|b|)]
[ab >0] F II I1 4 2 ,1 6 3 ,6 8 8
15.8/integraldisplaynπ
0ln/parenleftbig
1−2acosx+a2/parenrightbig
dx=0/bracketleftbig
a2≤1/bracketrightbig
=nπlna2/bracketleftbig
a2≥1/bracketrightbig
4.225
1./integraldisplayπ/4
0ln(cos x−sinx)dx=−π
8ln 2−1
2G GW (322)(9b)
532 Logarithmic Functions 4.226
2./integraldisplayπ/4
0ln(cos x+s i nx)dx=1
2/integraldisplayπ/2
0ln(cos x+s i nx)dx=−π
8ln2 +1
2G GW (322)(9a)
3./integraldisplay2π
0ln(1 + asinx+bcosx)dx=2πln1+√
1−a2−b2
2/bracketleftbig
a2+b2<1/bracketrightbig
BI (332)(2)
4./integraldisplay2π
0ln/parenleftbig
1+a2+b2+2asinx+2bcosx/parenrightbig
dx=0/bracketleftbig
a2+b2≤1/bracketrightbig
=2πln/parenleftbig
a2+b2/parenrightbig/bracketleftbig
a2+b2≥1/bracketrightbig
BI (322)(3)
4.226
1./integraldisplayπ/2
0ln/parenleftbig
a2−sin2x/parenrightbig2dx=−2πln2/bracketleftbig
a2≤1/bracketrightbig
=2πlna+√
a2−1
2=2π(arccosh a−ln2) [ a>1]
FI II 644, 687
2./integraldisplayπ/2
0ln/parenleftbig
1+asin2x/parenrightbig
dx=1
2/integraldisplayπ
0ln/parenleftbig
1+asin2x/parenrightbig
dx=/integraldisplayπ/2
0ln/parenleftbig
1+acos2x/parenrightbig
dx
=1
2/integraldisplayπ
0ln/parenleftbig
1+acos2x/parenrightbig
dx=πln1+√1+a
2
[a≥−1] BI (308)(15), GW(322)(12)
3./integraldisplayu
0ln/parenleftbig
1−sin2αsin2x/parenrightbig
dx=(π−2θ)lnc otα
2+2uln/parenleftbigg1
2sinα/parenrightbigg
−π
2ln2
+L(θ+u)−L(θ−u)+L/parenleftBigπ
2−2u/parenrightBig
/bracketleftBig
cotθ=c o s αtanu;−π≤α≤π,−π
2≤u≤π
2/bracketrightBig
LO III 287
4./integraldisplayπ/2
0ln/bracketleftbig
1−cos2x/parenleftbig
sin2α−sin2βsin2x/parenrightbig/bracketrightbig
dx=πln/bracketleftBigg
1
2/parenleftBigg
cos2α
2+/radicalbigg
cos4α
2+s i n2β
2cos2β
2/parenrightBigg/bracketrightBigg
[α>β> 0] LO III 283
5./integraldisplayu
0ln/parenleftbigg
1−sin2x
sin2α/parenrightbigg
dx=−uln sin2α−L/parenleftBigπ
2−α+u/parenrightBig
+L/parenleftBigπ
2−α−u/parenrightBig
/bracketleftBig
−π
2≤u≤π
2,|sinu|≤|sinα|/bracketrightBig
LO III 287
6./integraldisplayπ/2
0ln/parenleftbig
a2cos2x+b2sin2x/parenrightbig
dx=1
2/integraldisplayπ
0ln/parenleftbig
a2cos2x+b2sin2x/parenrightbig
dx=πlna+b
2
[a>0,b > 0] GW (322)(13)
4.227 Logarithms of more complicated arguments 533
7./integraldisplayπ/2
0ln1+s i n tcos2x
1−sintcos2xdx=πln1+s i nt
2
cost
2=πlncotπ−t
4
/bracketleftBig
|t|<π
2/bracketrightBig
LO III 283
4.227
1./integraldisplayu
0ln tan xdx=L(u)+L/parenleftBigπ
2−u/parenrightBig
−L/parenleftBigπ
2/parenrightBig
LO III 186(16)
2./integraldisplayπ/4
0lntan xdx=−/integraldisplayπ
2π
4lntan xdx=−G BI (286)(11)
3./integraldisplayπ/2
0ln(atanx)dx=π
2lna [a>0] BI (307)(2)
4.7/integraldisplayπ/4
0(lntan x)ndx=n!(−1)n∞/summationdisplay
k=0(−1)k
(2k+1 )n+1
=1
2/parenleftBigπ
2/parenrightBign+1
|En| [neven]
BI (286)(21)
5.7/integraldisplayπ/2
0(lntan x)2ndx=2 ( 2n)!∞/summationdisplay
k=0(−1)k
(2k+1 )2n+1=/parenleftBigπ
2/parenrightBig2n+1
|E2n| BI (307)(15)
6./integraldisplayπ/2
0(lntan x)2n+1dx=0 BI (307)(14)
7./integraldisplayπ/4
0(lntan x)2dx=π3
16BI (286)(16)
8./integraldisplayπ/4
0(lntan x)4dx=5
64π5BI (286)(19)
9./integraldisplayπ/4
0ln(1 + tan x)dx=π
8ln 2 BI (287)(1)
10./integraldisplayπ/2
0ln(1 + tan x)dx=π
4ln 2 +G BI (308)(9)
11./integraldisplayπ/4
0ln(1−tanx)dx=π
8ln 2−G BI (287)(2)
12.11/integraldisplayπ/2
0(ln(1−tanx))2dx=π
2ln2−2G BI (308)(10)
13./integraldisplayπ/4
0ln(1 + cot x)dx=π
8ln2 +G BI (287)(3)
14./integraldisplayπ/4
0ln(cot x−1)dx=π
8ln2 BI (287)(4)
534 Logarithmic Functions 4.228
15./integraldisplayπ/4
0ln(tan x+c o t x)dx=1
2/integraldisplayπ/2
0ln(tan x+c o t x)dx=π
2ln2 BI (287)(5), BI (308)(11)
16.11/integraldisplayπ/4
0(ln(cot x−tanx))2dx=1
2/integraldisplayπ/2
0(ln(cot x−tanx))2dx=π
2ln 2
BI (287)(6), BI (308)(12)
17./integraldisplayπ/2
0ln/parenleftbig
a2+b2tan2x/parenrightbig
dx=1
2/integraldisplayπ
0ln/parenleftbig
a2+b2tan2x/parenrightbig
dx=πln(a+b)
[a>0,b > 0] GW (322)(17)
4.228
1./integraldisplayπ/2
0ln/parenleftBig
sintsinx+/radicalbig
1−cos2tsin2x/parenrightBig
dx=π
2ln2−2L/parenleftbiggt
2/parenrightbigg
−2L/parenleftbiggπ−t
2/parenrightbigg
LO III 290
2./integraldisplayu
0ln/parenleftBig
cosx+/radicalbig
cos2x−cos2t/parenrightBig
dx=−/parenleftBigπ
2−t−ϕ/parenrightBig
ln cos t+1
2L(u+ϕ)−1
2L(u−ϕ)−L(ϕ)
/bracketleftbigg
cosϕ=sinu
sint0≤u≤t≤π
2/bracketrightbigg
LO III 290
3./integraldisplayt
0ln/parenleftBig
cosx+/radicalbig
cos2x−cos2t/parenrightBig
dx=−/parenleftBigπ
2−t/parenrightBig
ln cos t LO III 285
4./integraldisplayu
0lnsinu+s i ntcosx/radicalbig
sin2u−sin2x
sinu−sintcosx/radicalbig
sin2u−sin2xdx=πln/bracketleftBigg
tant
2sinu+/radicalbigg
tan2t
2sin2u+1/bracketrightBigg
[t>0,u > 0] LO III 283
5./integraldisplayπ/4
0√
lncot xd x=√π
2∞/summationdisplay
k=0(−1)k
/radicalbig
(2k+1 )3BI (297)(9)
6./integraldisplayπ/4
0dx√
lncot x=√π∞/summationdisplay
k=0(−1)k
√
2k+1BI (304)(24)
7./integraldisplayπ/4
0ln/parenleftBig√
tanx+√
cotx/parenrightBig
dx=1
2/integraldisplayπ/2
0ln/parenleftBig√
tanx+√
cotx/parenrightBig
dx=π
8ln2 +1
2G
BI (287)(7), BI (308)(22)
8./integraldisplayπ/4
0ln2/parenleftBig√
cotx−√
tanx/parenrightBig
dx=1
2/integraldisplayπ/2
0ln2/parenleftBig√
cotx−√
tanx/parenrightBig
dx=π
4ln2−G
BI (287)(8), BI (308)(23)
4.229
1./integraldisplay1
0ln/parenleftbigg
ln1
x/parenrightbigg
dx=−C FI II 807
2.11PV/integraldisplay1
0dx
ln/parenleftbigg
ln1
x/parenrightbigg=P V/integraldisplay∞
0e−u
lnudu≈−0.154479 BI (31)(2)
4.231 Logarithms and rational functions 535
3./integraldisplay1
0ln/parenleftbigg
ln1
x/parenrightbiggdx/radicalbigg
ln1
x=−(C+ 2ln 2)√π BI (32)(4)
4.11/integraldisplay1
0ln/parenleftbigg
ln1
x/parenrightbigg/parenleftbigg
ln1
x/parenrightbiggμ−1
dx=ψ(μ)Γ(μ)[ R e μ>0] BI (30)(10)
If the integrand contains (ln ln1
x), it is convenient to make the substitution ln1
x=uso that
x=e−u.
5.7/integraldisplay1
0ln (a+l nx)dx=l na−e−aEi(a)[ a>0] BI (30)(5)
6./integraldisplay1
0ln (a−lnx)dx=l na−eaEi(−a)[ a>0] BI (30)(6)
7./integraldisplayπ/2
π/4lnlntan xdx=π
2ln/parenleftBigg
Γ/parenleftbig3
4/parenrightbig
Γ/parenleftbig1
4/parenrightbig√
2π/parenrightBigg
BI (308)(28)
4.23 Combinations of logarithms and rational functions
4.231
1./integraldisplay1
0lnx
1+xdx=−π2
12FI II 483a
2./integraldisplay1
0lnx
1−xdx=−π2
6FI II 714
3./integraldisplay1
0xlnx
1−xdx=1−π2
6BI (108)(7)
4./integraldisplay1
01+x
1−xlnxdx=1−π2
3BI (108)(9)
5.11/integraldisplay∞
0lnxdx
(x+a)2=lna
a[0<a] BI (139)(1)
6./integraldisplay1
0lnx
(1 +x)2dx=−ln 2 BI (111)(1)
7.7/integraldisplay∞
0lnxdx
(a2+b2x2)n=Γ/parenleftbig
n−1
2/parenrightbig√π
4(n−1)!a2n−1b/bracketleftbigg
2lna
2b−C−ψ/parenleftbigg
n−1
2/parenrightbigg/bracketrightbigg
[a>0,b > 0] LI (139)(3)
8./integraldisplay∞
0lnxdx
a2+b2x2=π
2ablna
b[ab >0] BI (135)(6)
9./integraldisplay∞
0lnpx
q2+x2dx=π
2qlnpq [p>0,q > 0] BI (135)(4)
10./integraldisplay∞
0lnxdx
a2−b2x2=−π2
4ab[ab >0]
536 Logarithmic Functions 4.232
11./integraldisplaya
0lnxdx
x2+a2=πlna
4a−G
a[a>0] GW (324)(7b)
12./integraldisplay1
0lnx
1+x2dx=−/integraldisplay∞
1lnx
1+x2dx=−G FI II 482, 614
13./integraldisplay1
0lnxdx
1−x2=−π2
8BI (108)(11)
14./integraldisplay1
0xlnx
1+x2dx=−π2
48GW (324)(7b)
15./integraldisplay1
0xlnx
1−x2dx=−π2
24
16./integraldisplay1
0lnx1−x2n+2
(1−x2)2dx=−(n+1 )π2
8+n/summationdisplay
k=1n−k+1
(2k−1)2BI (111)(5)
17./integraldisplay1
0lnx1+(−1)nxn+1
(1 +x)2dx=−(n+1 )π2
12−n/summationdisplay
k=1(−1)kn−k+1
k2BI (111)(2)
18./integraldisplay1
0lnx1−xn+1
(1−x)2dx=−(n+1 )π2
6+n/summationdisplay
k=1n−k+1
k2BI (111)(3)
19.∗/integraldisplay1
0xlnx
1+xdx=−1+π2
2
20.∗/integraldisplay1
0(1−x)lnx
1+xdx=1−π2
6
4.232
1./integraldisplayυ
ulnxdx
(x+u)(x+υ)=lnuυ
2(υ−u)ln(u+υ)2
4uυBI (145)(32)
2./integraldisplay∞
0lnxdx
(x+β)(x+γ)=(lnβ)2−(lnγ)2
2(β−γ)[|argβ|<π , |argγ|<π]
ET II 218(24)
3./integraldisplay∞
0lnx
x+adx
x−1=π2+( l na)2
2(a+1 )[a>0] BI (140)(10)
4.233
1.3/integraldisplay1
0lnxdx
1+x+x2=2
9/bracketleftbigg2π2
3−ψ/prime/parenleftbigg1
3/parenrightbigg/bracketrightbigg
=−0.7813024129 ... LI (113)(1)
2.3/integraldisplay1
0lnxdx
1−x+x2=1
3/bracketleftbigg2π2
3−ψ/prime/parenleftbigg1
3/parenrightbigg/bracketrightbigg
=−1.17195361934 ... LI (113)(2)
3.11/integraldisplay1
0xlnxdx
1+x+x2=−1
9/bracketleftbigg7π2
6−ψ/prime/parenleftbigg1
3/parenrightbigg/bracketrightbigg
=−0.15766014917 ... LI (113)(2)
4.3/integraldisplay1
0xlnxdx
1−x+x2=1
6/bracketleftbigg5π2
6−ψ/prime/parenleftbigg1
3/parenrightbigg/bracketrightbigg
=−0.3118211319 ... LI (113)(4)
4.236 Logarithms and rational functions 537
5./integraldisplay∞
0lnxdx
x2+2xacost+a2=tlna
asint[a>0,0<t<π ] GW (324)(13c)
4.234
1.11/integraldisplay∞
1lnxdx
(1 +x2)2=G
2−π
8BI (144)(18)a
2./integraldisplay1
0xlnxdx
(1 +x2)2=−1
4ln2 BI (111)(4)
3./integraldisplay∞
01+x2
(1−x2)2lnxdx=0 BI (142)(2)a
4./integraldisplay∞
01−x2
(1 +x2)2lnxdx=−π
2BI (142)(1)a
5./integraldisplay1
0x2lnxdx
(1−x2)( 1+ x4)=−π2
16/parenleftbig
2+√
2/parenrightbig BI (112)(21)
6./integraldisplay∞
0lnxdx
(a2+b2x2)(1+ x2)=bπ
2a(b2−a2)lna
b[ab >0] BI (317)(16)a
7./integraldisplay∞
0lnx
x2+a2·dx
1+b2x2=π
2( 1−a2b2)/parenleftbigg1
alna+blnb/parenrightbigg
[a>0,b > 0] LI (140)(12)
8./integraldisplay∞
0x2lnxdx
(a2+b2x2)(1+ x2)=aπ
2b(b2−a2)lnb
a[ab >0] LI (140)(12), BI (317)(15)a
4.235
1./integraldisplay∞
0lnx(1−x)xn−2
1−x2ndx=−π2
4n2tan2π
2n[n>1] BI (135)(10)
2./integraldisplay∞
0lnx/parenleftbig
1−x2/parenrightbig
xm−1
1−x2ndx=−π2sin/parenleftbigm+1
n/parenrightbig
πsin/parenleftbigπ
n/parenrightbig
4n2sin2/parenleftbigmπ
2n/parenrightbig
sin2/parenleftbigm+2
2nπ/parenrightbig LI (135)(12)
3.11/integraldisplay∞
0lnx/parenleftbig
1−x2/parenrightbig
xn−3
1−x2ndx=−π2
4n2tan2/parenleftBigπ
n/parenrightBig
[n>2] BI (135)(11)
4./integraldisplay1
0lnxxm−1+xn−m−1
1−xndx=−π2
n2sin2/parenleftbigm
nπ/parenrightbig [n>m ] BI (108)(15)
4.236
1./integraldisplay1
0/braceleftbigg1+(p−1)lnx
1−x+xlnx
(1−x)2/bracerightbigg
xp−1dx=−1+ψ/prime(p)
[p>0] BI (111)(6)a, GW (326)(13)
2./integraldisplay1
0/bracketleftbigg1
1−x+xlnx
(1−x)2/bracketrightbigg
dx=π2
6−1 GW (326)(13a)
538 Logarithmic Functions 4.241
4.24 Combinations of logarithms and algebraic functions
4.241
1./integraldisplay1
0x2nlnx√
1−x2dx=(2n−1)!!
(2n)!!·π
2/parenleftBigg2n/summationdisplay
k=1(−1)k−1
k−ln2/parenrightBigg
BI (118)(5)a
2./integraldisplay1
0x2n+1lnx√
1−x2dx=(2n)!!
(2n+1 ) ! !/parenleftBigg
ln 2 +2n+1/summationdisplay
k=1(−1)k
k/parenrightBigg
BI (118)(5)a
3./integraldisplay1
0x2n/radicalbig
1−x2lnxdx=(2n−1)!!
(2n+2 ) ! !·π
2/parenleftBigg2n/summationdisplay
k=1(−1)k−1
k−1
2n+2−ln2/parenrightBigg
LI (117)(4), GW (324)(53a)
4./integraldisplay1
0x2n+1/radicalbig
1−x2lnxdx=(2n)!!
(2n+3 ) ! !/parenleftBigg
ln2 +2n+1/summationdisplay
k=1(−1)k
k−1
2n+3/parenrightBigg
BI (117)(5), GW (324)(53b)
5./integraldisplay1
0lnx·/radicalBig
(1−x2)2n−1dx=−(2n−1)!!
4·(2n)!!π[ψ(n+1 )+ C+l n4 ] BI (117)(3)
6./integraldisplay/radicalBigg1
2
0lnxdx√
1−x2=−π
4ln2−1
2G BI (145)(1)
7./integraldisplay1
0lnxdx√
1−x2=−π
2ln2 FI II 614, 643
8./integraldisplay∞
1lnxdx
x2√
x2−1=1−ln 2 BI (144)(17)
9./integraldisplay1
0/radicalbig
1−x2lnxdx=−π
8−π
4ln 2 BI (117)(1), GW (324)(53c)
10./integraldisplay1
0x/radicalbig
1−x2lnxdx=1
3ln2−4
9BI (117)(2)
11./integraldisplay1
0lnxdx/radicalbig
x(1−x2)=−√
2π
8/bracketleftbigg
Γ/parenleftbigg1
4/parenrightbigg/bracketrightbigg2
GW (324)(54a)
4.242
1./integraldisplay∞
0lnxdx/radicalbig
(a2+x2)(x2+b2)=1
2aK/parenleftBigg√
a2−b2
a/parenrightBigg
lnab
[a>b> 0] BY (800.04)
2./integraldisplayb
0lnxdx/radicalbig
(a2+x2)(b2−x2)=1
2√
a2+b2/bracketleftbigg
K/parenleftbiggb√
a2+b2/parenrightbigg
lnab−π
2K/parenleftbigga√
a2+b2/parenrightbigg/bracketrightbigg
[a>0,b > 0] BY (800.02)
4.247 Logarithms and algebraic functions 539
3./integraldisplay∞
blnxdx/radicalbig
(x2+a2)(x2−b2)=1
2√
a2+b2/bracketleftbigg
K/parenleftbigga√
a2+b2/parenrightbigg
lnab+π
2K/parenleftbiggb√
a2+b2/parenrightbigg/bracketrightbigg
[a>0,b > 0] BY (800.06)
4./integraldisplayb
0lnxdx/radicalbig
(a2−x2)(b2−x2)=1
2a/bracketleftBigg
K/parenleftbiggb
a/parenrightbigg
lnab−π
2K/parenleftBigg√
a2−b2
a/parenrightBigg/bracketrightBigg
[a>b> 0] BY (800.01)
5./integraldisplaya
blnxdx/radicalbig
(a2−x2)(x2−b2)=1
2aK/parenleftBigg√
a2−b2
a/parenrightBigg
lnab BY (800.03)
6./integraldisplay∞
alnxdx/radicalbig
(x2−a2)(x2−b2)=1
2a/bracketleftBigg
K/parenleftbiggb
a/parenrightbigg
lnab+π
2K/parenleftBigg√
a2−b2
a/parenrightBigg/bracketrightBigg
[a>b> 0] BY (800.05)
4.243/integraldisplay1
0xlnx√
1−x4dx=−π
8ln 2 GW (324)(56b)
4.244
1./integraldisplay1
0lnxdx
3/radicalBig
x(1−x2)2=−1
8/bracketleftbigg
Γ/parenleftbigg1
3/parenrightbigg/bracketrightbigg3
GW (324)(54b)
2./integraldisplay1
0lnxdx
3√
1−x3=−π
3√
3/parenleftbigg
ln3 +π
3√
3/parenrightbigg
BI (118)(7)
3./integraldisplay1
0xlnxdx
3/radicalBig
(1−x3)2=π
3√
3/parenleftbiggπ
3√
3−ln3/parenrightbigg
BI (118)(8)
4.245
1./integraldisplay1
0x4n+1lnx√
1−x4dx=(2n−1)!!
(2n)!!·π
8/parenleftBigg2n/summationdisplay
k=1(−1)k−1
k−ln 2/parenrightBigg
GW (324)(56a)
2./integraldisplay1
0x4n+3lnx√
1−x4dx=(2n)!!
4·(2n+1 ) ! !/parenleftBigg
ln2 +2n+1/summationdisplay
k=1(−1)k
k/parenrightBigg
GW (324)(56c)
4.246/integraldisplay1
0/parenleftbig
1−x2/parenrightbign−1
2lnxdx=−(2n−1)!!
(2n)!!·π
4/bracketleftBigg
2l n2+n/summationdisplay
k=11
k/bracketrightBigg
GW (324)(55)
4.247
1.6/integraldisplay1
0lnx
n√
1−x2ndx=−πB/parenleftbigg1
2n,1
2n/parenrightbigg
8n2sinπ
2n[n>1] GW (324)(54c)a
2.6/integraldisplay1
0lnxdx
n/radicalbig
xn−1(1−x2)=−πB/parenleftbigg1
2n,1
2n/parenrightbigg
8s inπ
2nGW (324)(54)
540 Logarithmic Functions 4.251
4.25 Combinations of logarithms and powers
4.251
1./integraldisplay∞
0xμ−1lnx
β+xdx=πβμ−1
sinμπ(lnβ−πcotμπ)[ |argβ|<π , 0<Reμ<1]
BI (135)(1)
2./integraldisplay∞
0xμ−1lnx
a−xdx=πaμ−1/parenleftbigg
cotμπlna−π
sin2μπ/parenrightbigg
[a>0,0<Reμ<1] ET I 314(5)
3.10/integraldisplay1
0xμ−1lnx
x+1dx=β/prime(μ)[ R e μ>0] GW (324)(6), ET I 314(3)
4./integraldisplay1
0xμ−1lnx
1−xdx=−ψ/prime(μ)=−ζ(2,μ)[ R e μ>0] BI (108)(8)
5.11/integraldisplay1
0lnxx2n
1+xdx=−π2
12+2n/summationdisplay
k=1(−1)k−1
k2BI (108)(4)
6.11/integraldisplay1
0lnxx2n−1
1+xdx=π2
12+2n−1/summationdisplay
k=1(−1)k
k2BI (108)(5)
4.252
1./integraldisplay∞
0xμ−1lnx
(x+β)(x+γ)dx=π
(γ−β)sinμπ/bracketleftbig
βμ−1lnβ−γμ−1lnγ−πcotμπ/parenleftbig
βμ−1−γμ−1/parenrightbig/bracketrightbig
[|argβ|<π , |argγ|<π , 0<Reμ<2,μ/negationslash=1 ] BI (140)(9)a, ET 314(6)
2./integraldisplay∞
0xμ−1lnxdx
(x+β)(x−1)=π
(β+1 )s i n2μπ/bracketleftbig
π−βμ−1(sinμπlnβ−πcosμπ)/bracketrightbig
[|argβ|<π , 0<Reμ<2,μ/negationslash=1 ]
BI (140)(11)
3./integraldisplay∞
0xp−1lnx
1−x2dx=−π2
4cosec2pπ
2[0<p< 2] (see also 4.254 2)
4.6/integraldisplay∞
0xμ−1lnx
(x+a)2dx=(1−μ)aμ−2π
sinμπ/parenleftbigg
lna−πcotμπ+1
μ−1/parenrightbigg
[|arga|<π 0<Reμ<2(μ/negationslash=1 ) ]
GW (324)(13b)
4.253
1.8/integraldisplay1
0xμ−1(1−xr)ν−1lnxdx=1
r2B/parenleftBigμ
r,ν/parenrightBig/bracketleftBig
ψ/parenleftBigμ
r/parenrightBig
−ψ/parenleftBigμ
r+ν/parenrightBig/bracketrightBig
[Reμ>0,Reν>0,r > 0]
GW (324)(3b)a, BI (107)(5)a
2./integraldisplay1
0xp−1
(1−x)p+1lnxdx=−π
pcosecpπ [0<p< 1] bi (319)(10)a
4.255 Logarithms and powers 541
3./integraldisplay∞
u(x−u)μ−1lnxdx
xλ=uμ−λB(λ−μ, μ)[lnu+ψ(λ)−ψ(λ−μ)]
[0<Reμ<Reλ] ET II 203(18)
4.11/integraldisplay∞
0lnx/parenleftbiggx
a2+x2/parenrightbiggpdx
x=lna
2apB/parenleftBigp
2,p
2/parenrightBig
[a>0,p > 0] BI (140)(6)
5./integraldisplay∞
1(x−1)p−1lnxdx=π
pcosecπp [−1<p< 0] BI (289)(12)a
6.7/integraldisplay∞
0lnxdx
(a+x)μ+1=1
μaμ(lna−C−ψ(μ)) [Re μ>0,a/negationslash=0,|arga|<π]
NT 68(7)
7.7/integraldisplay∞
0lnxdx
(a+x)n+1
2=2
(2n−1)an−1
2/parenleftBigg
lna+2l n2 −2n−1/summationdisplay
k=11
2k−1/parenrightBigg
[|arga|<π , n =1,2,...]BI (142)(5)
4.254
1./integraldisplay1
0xp−1lnx
1−xqdx=−1
q2ψ/prime/parenleftbiggp
q/parenrightbigg
[p>0,q > 0] GW (324)(5)
2./integraldisplay∞
0xp−1lnx
1−xqdx=−π2
q2sin2pπ
q[0<p<q ] BI (135)(8)
3./integraldisplay∞
0lnx
xq−1dx
xp=π2
q2sin2p−1
qπ[p<1,p+q>1] BI (140)(2)
4.3/integraldisplay1
0xp−1lnx
1+xqdx=1
q2β/prime/parenleftbiggp
q/parenrightbigg
[p>0,q > 0] GW (324)(7)
5./integraldisplay∞
0xp−1lnx
1+xqdx=−π2
q2cospπ
q
sin2pπ
q[0<p<q ] BI (135)(7)
6./integraldisplay1
0xq−1lnx
1−x2qdx=−π2
8q2[q>0] BI (108)(12)
4.255
1./integraldisplay1
0lnx/parenleftbig
1−x2/parenrightbig
xp−2
1+x2pdx=−/parenleftbiggπ
2p/parenrightbigg2sinπ
2p
cos2/parenleftBig
π
2p/parenrightBig [p>1] BI (108)(13)
2./integraldisplay1
0lnx/parenleftbig
1+x2/parenrightbig
xp−2
1−x2pdx=−/parenleftbiggπ
2p/parenrightbigg2
sec2/parenleftbiggπ
2p/parenrightbigg
[p>1] BI (108)(14)
3./integraldisplay∞
0lnx1−xp
1−x2dx=π2
4tan2/parenleftBigpπ
2/parenrightBig
[p<1] BI (140)(3)
542 Logarithmic Functions 4.256
4.256/integraldisplay1
0ln1
xxμ−1dx
n/radicalBig
(1−xn)n−m=1
n2B/parenleftBigμ
n,m
n/parenrightBig/bracketleftbigg
ψ/parenleftbiggμ+m
n/parenrightbigg
−ψ/parenleftBigμ
n/parenrightBig/bracketrightbigg
[Reμ>0] LI (118)(12)
4.257
1./integraldisplay∞
0xνlnx
βdx
(x+β)(x+γ)=π/bracketleftBig
γνlnγ
β+π(βν−γν)c o tνπ/bracketrightBig
sinνπ(γ−β)
[|argβ|<π , |argγ|<π , |Reν|<1]
ET II 219(30)
2./integraldisplay∞
0lnx
q/parenleftbiggxp
q2p+x2p/parenrightbiggdx
x=0 [ q>0] BI (140)(4)a
3./integraldisplay∞
0lnx
q/parenleftbiggxp
q2p+x2p/parenrightbiggrdx
q2+x2=0 [ q>0] BI (140)(4)a
4./integraldisplay∞
0lnxlnx
adx
(x−1)(x−a)=/bracketleftBig
4π2+( l na)2/bracketrightBig
lna
6(a−1)[a>0] (for a=1s e e 4.261 5)
BI (141)(5)
5./integraldisplay∞
0lnxlnx
axpdx
(x−1)(x−a)=π2[(ap+1 )l n a−2π(ap−1) cot pπ]
(a−1)sin2pπ/bracketleftbig
p2<1,a > 0/bracketrightbig
BI (141)(6)
4.26–4.27 Combinations involving powers of the logarithm and other powers
4.261
1.7/integraldisplay1
0(lnx)2 dx
1+2xcost+x2=t/parenleftbig
π2−t2/parenrightbig
6s int[0≤t≤π] BI (113)(7)
2./integraldisplay1
0(lnx)2dx
x2−x+1=1
2/integraldisplay∞
0(lnx)2dx
x2−x+1=10π3
81√
3GW (324)(16c)
3./integraldisplay1
0(lnx)2dx
x2+x+1=1
2/integraldisplay∞
0(lnx)2dx
x2+x+1=8π3
81√
3GW (324)(16b)
4./integraldisplay∞
0(lnx)2 dx
(x−1)(x+a)=/bracketleftBig
π2+( l na)2/bracketrightBig
lna
3(1 + a)[a>0] BI (141)(1)
5./integraldisplay∞
0(lnx)2dx
(1−x)2=2
3π2BI (139)(4)
6./integraldisplay1
0(lnx)2dx
1+x2=π3
16BI (109)(3)
7./integraldisplay1
0(lnx)21+x2
1+x4dx=1
2/integraldisplay∞
0(lnx)21+x2
1+x4dx=3√
2
64π3BI (109)(5), BI (135)(13)
8.11/integraldisplay1
0(lnx)21−x
1−x6dx=8√
3π3+ 351 ζ(3)
486
4.261 Powers and logarithms 543
9./integraldisplay1
0(lnx)2dx√
1−x2=π
2/bracketleftbigg
(ln2)2+π2
12/bracketrightbigg
BI (118)(13)
10./integraldisplay∞
0(lnx)2xμ−1
1+xdx=π3/parenleftbig
2−sin2μπ/parenrightbig
sin3μπ[0<Reμ<1] ET I 315(10)
11.7/integraldisplay1
0(lnx)2xndx
1+x=2∞/summationdisplay
k=n(−1)n+k
(k+1 )3=(−1)n/parenleftBigg
3
2ζ(3) + 2n/summationdisplay
k=1(−1)k
k3/parenrightBigg
[n=0,1,...] BI (109)(1)
12.7/integraldisplay1
0(lnx)2xndx
1−x=2∞/summationdisplay
k=n1
(k+1 )3=2/parenleftBigg
ζ(3)−n/summationdisplay
k=11
k3/parenrightBigg
[n=0,1,...] BI (109)(2)
13.11/integraldisplay1
0(lnx)2x2ndx
1−x2=2∞/summationdisplay
k=n1
(2k+1 )3=7
4ζ(3)−2n/summationdisplay
k=11
(2k−1)3
[n=0,1,...] BI (109)(4)
14./integraldisplay∞
0(lnx)2 xp−1dx
x2+2xcost+1=πsin(1−p)t
sintsinpπ/braceleftbig
π2−t2+2πcotpπ[πcotpπ+tcot(1−p)t]/bracerightbig
[0<t<π , 0<p< 2,p/negationslash=1 ]
GW (324)(17)
15./integraldisplay1
0(lnx)2x2ndx√
1−x2=(2n−1)!!
2·(2n)!!π⎧
⎨
⎩π2
12+2n/summationdisplay
k=1(−1)k
k2+/bracketleftBigg2n/summationdisplay
k=1(−1)k
k+l n2/bracketrightBigg2⎫
⎬
⎭GW (324)(60a)
16./integraldisplay1
0(lnx)2x2n+1dx√
1−x2=(2n)!!
(2n+1 ) ! !⎧
⎨
⎩−π2
12−2n+1/summationdisplay
k=1(−1)k
k2+/bracketleftBigg2n+1/summationdisplay
k=1(−1)k
k+l n2/bracketrightBigg2⎫
⎬
⎭
GW (324)(60b)
17.7/integraldisplay1
0(lnx)2xμ−1(1−x)ν−1dx=B (μ, ν)/braceleftBig
[ψ(μ)−ψ(ν+μ)]2+ψ/prime(μ)−ψ/prime(μ+ν)/bracerightBig
[Reμ>0,Reν>0] ET I 315(11)
18./integraldisplay1
0(lnx)21−xn+1
(1−x)2dx=2 (n+1 )ζ(3)−2n/summationdisplay
k=1n−k+1
k3LI (111)(8)
19./integraldisplay1
0(lnx)21+(−1)nxn+1
(1 +x)2dx=3
2(n+1 )ζ(3)−2n/summationdisplay
k=1(−1)k−1n−k+1
k3LI (111)(7)
20.7/integraldisplay1
0(lnx)21−x2n+2
(1−x2)2dx=7
4(n+1 )ζ(3)−2n/summationdisplay
k=1n−k+1
(2k−1)3
[n=0,1,...] LI (111)(9)
21./integraldisplay1
0(lnx)2xp−1(1−xr)q−1dx=1
r3B/parenleftBigp
r,q/parenrightBig/braceleftbigg
ψ/prime/parenleftBigp
r/parenrightBig
−ψ/prime/parenleftBigp
r+q/parenrightBig
+/bracketleftBig
ψ/parenleftBigp
r/parenrightBig
−ψ/parenleftBigp
r+q/parenrightBig/bracketrightBig2/bracerightbigg
[p>0,q > 0,r > 0] GW (324)(8a)
544 Logarithmic Functions 4.262
4.262
1./integraldisplay1
0(lnx)3dx
1+x=−7
120π4BI (109)(9)
2./integraldisplay1
0(lnx)3dx
1−x=−π4
15BI (109)(11)
3./integraldisplay∞
0(lnx)3 dx
(x+a)(x−1)=/bracketleftBig
π2+( l na)2/bracketrightBig2
4(a+1 )[a>0] BI (141)(2)
4./integraldisplay1
0(lnx)3xndx
1+x=(−1)n+1/bracketleftBigg
7π4
120−6n−1/summationdisplay
k=0(−1)k
(k+1 )4/bracketrightBigg
[n=1,2,...] BI (109)(10)
5./integraldisplay1
0(lnx)3xndx
1−x=−π4
15+6n−1/summationdisplay
k=01
(k+1 )4[n=1,2,...] BI (109)(12)
6./integraldisplay1
0(lnx)3x2ndx
1−x2=−π4
16+6n−1/summationdisplay
k=01
(2k+1 )4[n=1,2,...] BI (109)(14)
7./integraldisplay1
0(lnx)31−xn+1
(1−x)2dx=−(n+1 )π4
15+6n/summationdisplay
k=1n−k+1
k4BI (111)(11)
8./integraldisplay1
0(lnx)31+(−1)nxn+1
(1 +x)2dx=−7(n+1 )π4
120+6n/summationdisplay
k=1(−1)k−1n−k+1
k4BI (111)(10)
9./integraldisplay1
0(lnx)31−x2n+2
(1−x2)2dx=−(n+1 )π4
16+6n/summationdisplay
k=1n−k+1
(2k−1)4BI (111)(12)
4.263
1.8/integraldisplay∞
0(lnx)4 dx
(x−1)(x+a)=lna/bracketleftBig
π2+( l na)2/bracketrightBig/bracketleftBig
7π2+3( l n a)2/bracketrightBig
15(1 + a)
[a>0] BI (141)(3)
2./integraldisplay1
0(lnx)4dx
1+x2=5π5
64BI (109)(17)
3./integraldisplay1
0(lnx)4 dx
1+2xcost+x2=t/parenleftbig
π2−t2/parenrightbig/parenleftbig
7π2−3t2/parenrightbig
30 sin t
[|t|<π] BI (113)(8)
4.264
1./integraldisplay1
0(lnx)5dx
1+x=−31π6
252BI (109)(20)
2./integraldisplay1
0(lnx)5dx
1−x=−8π6
63BI (109)(21)
4.267 Powers and logarithms 545
3./integraldisplay∞
0(lnx)5 dx
(x−1)(x+a)=/bracketleftBig
π2+( l na)2/bracketrightBig2/bracketleftBig
3π2+( l na)2/bracketrightBig
6(1 + a)
[a>0] BI (141)(4)
4.265/integraldisplay1
0(lnx)6dx
1+x2=61π7
256BI (109)(25)
4.266
1./integraldisplay1
0(lnx)7dx
1+x=−127π8
240BI (109)(28)
2./integraldisplay1
0(lnx)7dx
1−x=−8π8
15BI (109)(29)
4.267
1./integraldisplay1
01−x
1+xdx
lnx=l n2
πBI (127)(3)
2./integraldisplay1
0(1−x)2
1+x2dx
lnx=l nπ
4BI (128)(2)
3.8/integraldisplay1
0(1−x)2
1+2xcosmx
n+x2·dx
lnx
=1
sin/parenleftbigmπ
n/parenrightbign−1/summationdisplay
k=1(−1)ksin/parenleftbiggkmπ
n/parenrightbigg
ln/braceleftbig
Γ/parenleftbign+k+1
2n/parenrightbig/bracerightbig2Γ/parenleftbigk+2
2n/parenrightbig
Γ/parenleftbigk
2n/parenrightbig
/braceleftbig
Γ/parenleftbigk+1
2n/parenrightbig/bracerightbig2Γ/parenleftbign+k
2n/parenrightbig
Γ/parenleftbign+k+2
2n/parenrightbig [m+nis odd]
=1
sin/parenleftbigmπ
n/parenrightbig⌊1
2(n−1)⌋/summationdisplay
k=1(−1)ksin/parenleftbiggkmπ
n/parenrightbigg
ln/braceleftbig
Γ/parenleftbign−k+1
n/parenrightbig/bracerightbig2Γ/parenleftbigk+2
n/parenrightbig
Γ/parenleftbigk
n/parenrightbig
/braceleftbig
Γ/parenleftbigk+1
n/parenrightbig/bracerightbig2Γ/parenleftbign−k
n/parenrightbig
Γ/parenleftbign−k+2
n/parenrightbig[m+nis even]
[m<n ] BI (130)(3)
4./integraldisplay1
01−x
1+x·1
1+x2·dx
lnx=−ln 2
2BI (130)(16)
5./integraldisplay1
01−x
1+x·x2
1+x2·dx
lnx=l n2√
2
πBI (130)(17)
6.11/integraldisplay1
0(1−x)pdx
lnx=∞/summationdisplay
k=1(−1)k/parenleftBigp
k/parenrightBig
ln(1 + k)[ p≥1] BI (123)(2)
7./integraldisplay1
0/parenleftbigg1−xp
1−x−p/parenrightbiggdx
lnx=l nΓ ( p+1 ) GW (326)(10)
8./integraldisplay1
0xp−1−xq−1
lnxdx=l np
q[p>0,q > 0] FI II 647
9./integraldisplay1
0xp−1−xq−1
lnx·dx
1+x=l nΓ/parenleftbigq
2/parenrightbig
Γ/parenleftbigp+1
2/parenrightbig
Γ/parenleftbigp
2/parenrightbig
Γ/parenleftbigq+1
2/parenrightbig [p>0,q > 0] FI II 186
10./integraldisplay1
0xp−1−x−p
(1 +x)lnxdx=1
2/integraldisplay∞
0xp−1−x−p
(1 +x)lnxdx=l n/parenleftBig
tanpπ
2/parenrightBig
[0<p< 1] FI II 816
546 Logarithmic Functions 4.267
11./integraldisplay1
0(xp−xq)xr−1dx
lnx=l np+r
r+q[r>0,p > 0,q > 0] LI (123)(5)
12./integraldisplay1
0xp−xq
(1−ax)ndx
xlnx=∞/summationdisplay
k=0/parenleftbiggn+k−1
k/parenrightbigg
aklnp+k
q+k/bracketleftbig
p>0,q > 0,a2<1/bracketrightbig
BI (130)(15)
13./integraldisplay1
0(xp−1)(xq−1)dx
lnx=l np+q+1
(p+1 ) (q+1 )[p>−1,q > −1,p+q>−1]
GW (324)(19b)
14./integraldisplay1
0xp−xq
1+x·1+x2n+1
xlnxdx=l nΓ/parenleftBigp
2+n+1/parenrightBig
Γ/parenleftbigq+1
2+n/parenrightbig
Γ/parenleftbigp+1
2/parenrightbig
Γ/parenleftbigq
2/parenrightbig
Γ/parenleftBigq
2+n+1/parenrightBig
Γ/parenleftbiggp+1
2+n/parenrightbigg
Γ/parenleftbiggq+1
2/parenrightbigg
Γ/parenleftBigp
2/parenrightBig
[p>0,q > 0] BI (127)(7)
15./integraldisplay1
0xp−xq
1−x·1−xr
lnxdx=l nΓ(q+1 )Γ ( p+r+1 )
Γ(p+1 )Γ ( q+r+1 )
[p>−1,q > −1,p+r>−1,q+r>−1]GW (324)(23)
16./integraldisplay1
0xp−1−xq−1
(1 +xr)lnxdx=l nΓ/parenleftbiggp+r
2r/parenrightbigg
Γ/parenleftbigq
2r/parenrightbig
Γ/parenleftbiggq+r
2r/parenrightbigg
Γ/parenleftBigp
2r/parenrightBig [p>0,q > 0,r > 0] GW (324)(21)
17./integraldisplay1
01−x2p−2q
1+x2pxq−1dx
lnx=l nt a nqπ
4p[0<q<p ] BI (128)(6)
18./integraldisplay∞
0xp−1−xq−1
(1 +xr)l nxdx=l n/parenleftBig
tanpπ
2rcotqπ
2r/parenrightBig
[0<p<r , 0<q<r ]
GW (324)(22), BI (143)(2)
19./integraldisplay∞
0xp−1−xq−1
(1−xr)l nxdx=l n⎛
⎝sinpπ
r
sinqπ
r⎞
⎠ [0<p<r , 0<q<r ] BI (143)(4)
20./integraldisplay1
0xp−1−xq−1
1−x2n·1−x2
lnxdx=l nΓ/parenleftbigp+2
2n/parenrightbig
Γ/parenleftbigq
2n/parenrightbig
Γ/parenleftbigq+2
2n/parenrightbig
Γ/parenleftbigp
2n/parenrightbig [p>0,q > 0] BI (128)(11)
21./integraldisplay1
0xp−1−xq−1
1+x2(2n+1)1+x2
lnxdx=l nΓ/parenleftBig
p+4n+4
4(2n+1)/parenrightBig
Γ/parenleftBig
q+2
4(2n+1)/parenrightBig
Γ/parenleftBig
p+4n+2
4(2n+1)/parenrightBig
Γ/parenleftBig
q
4(2n+1)/parenrightBig
Γ/parenleftBig
q+4n+4
4(2n+1)/parenrightBig
Γ/parenleftBig
p+2
4(2n+1)/parenrightBig
Γ/parenleftBig
q+4n+2
4(2n+1)/parenrightBig
Γ/parenleftBig
p
4(2n+1)/parenrightBig
[p>0,q > 0] BI (128)(7)
22./integraldisplay∞
0xp−1−xq−1
1+x2(2n+1)·1+x2
lnxdx=l n/braceleftbigg
tanpπ
4(2n+1 )·tan(p+2 )π
4(2n+1 )·cotqπ
4(2n+1 )·cot(q+2 )π
4(2n+1 )/bracerightbigg
[0<p< 4n,0<q< 4n]BI (143)(5)
4.267 Powers and logarithms 547
23./integraldisplay∞
0xp−1−xq−1
1−x2n1−x2
lnxdx=l nsinpπ
2n·sin(q+2)π
2n
sinqπ
2n·sin(p+2 )π
2n
[0<p< 2n,0<q< 2n]BI (143)(6)
24./integraldisplay1
0(1−xp)(1−xq)xr−1dx
lnx=l n(p+q+r)r
(p+r)(q+r)[p>0,q > 0,r > 0] BI (123)(8)
25./integraldisplay1
0(1−xp)(1−xq)xr−1dx
(1−x)lnx=l nΓ(p+r)Γ(q+r)
Γ(p+q+r)Γ(r)
[r>0,r+p>0,r+q>0,r+p+q>0]FI II 815a
26./integraldisplay1
0(1−xp)(1−xq)( 1−xr)dx
lnx=l n(p+q+1 ) (q+r+1 ) (r+p+1 )
(p+q+r+1 ) (p+1 ) (q+1 ) (r+1 )
[p>−1,q > −1,r > −1,p+q>−1,p+r>−1,q+r>−1,p+q+r>−1]
GW (324)(19c)
27./integraldisplay1
0(1−xp)(1−xq)( 1−xr)dx
(1−x)lnx=l nΓ(p+1 )Γ ( q+1 )Γ ( r+1 )Γ ( p+q+r+1 )
Γ(p+q+1 )Γ ( p+r+1 )Γ ( q+r+1 )
[p>−1,q > −1,r > −1,p+q>−1,p+r>−1,q+r>−1,p+q+r>−1]
FI II 815
28./integraldisplay1
0(1−xp)(1−xq)( 1−xr)xs−1dx
lnx=l n(p+q+s)(p+r+s)(q+r+s)s
(p+s)(q+s)(r+s)(p+q+r+s)
[p>0,q > 0,r > 0,s > 0]
BI (123)(10)
29./integraldisplay1
0(1−xp)(1−xq)xs−1dx
(1−xr)l nx=l nΓ/parenleftbigp+s
r/parenrightbig
Γ/parenleftbigq+s
r/parenrightbig
Γ/parenleftbigs
r/parenrightbig
Γ/parenleftbigp+q+s
r/parenrightbig
[p>0,q > 0,r > 0,s > 0]
GW (324)(23a)
30./integraldisplay∞
0(1−xp)( 1−xq)xs−1dx
(1−xp+q+2s)l nx=2/integraldisplay1
0(1−xp)(1−xq)xs−1dx
(1−xp+q+2s)lnx
=2l n/parenleftbigg
sinsπ
p+q+2scosec(p+s)π
p+q+2s/parenrightbigg
[s>0,s+p>0,s+p+q>0]GW (324)(23b)a
31./integraldisplay1
0(1−xp)(1−xq)( 1−xr)xs−1dx
(1−x)lnx=l nΓ(p+s)Γ(q+s)Γ(r+s)Γ(p+q+r+s)
Γ(p+q+s)4Γ(p+r+s)Γ(q+r+s)Γ(s)
[p>0,q > 0,r > 0,s > 0]∗BI (127)(11)
32./integraldisplay1
0(1−xp)(1−xq)( 1−xr)xs−1dx
(1−xt)l nx=l nΓ/parenleftbigp+s
t/parenrightbig
Γ/parenleftbigq+s
t/parenrightbig
Γ/parenleftbigr+s
t/parenrightbig
Γ/parenleftbigp+q+r+s
t/parenrightbig
Γ/parenleftbigp+q+s
t/parenrightbig
Γ/parenleftbigq+r+s
t/parenrightbig
Γ/parenleftbigp+r+s
t/parenrightbig
Γ/parenleftbigs
t/parenrightbig
[p>0,q > 0,r > 0,s > 0,t > 0]∗GW (324)(23b)
∗In 4.267.31 the restrictions can be somewhat weakened by writing, for example, s>0,p+s>0,q+s>0,r+s>0,
p+q+s>0,p+r+s>0,q+r+s>0,p+q+r+s>0, in4.267 31 and 32.
548 Logarithmic Functions 4.268
33./integraldisplay1
0/braceleftbiggxp−xp+q
1−x−q/bracerightbiggdx
lnx=l nΓ(p+q+1 )
Γ(p+1 )[p>−1,p+q>−1] BI (127)(19)
34./integraldisplay1
0/braceleftbiggxμ−x
x−1−x(μ−1)/bracerightbiggdx
xlnx=l nΓ ( μ)[ R e μ>0] WH, BI (127)(18)
35./integraldisplay1
0/braceleftbigg
1−x−(1−xp)( 1−xq)
1−x/bracerightbiggdx
xlnx=−ln{B(p, q)}
[p>0,q > 0] BI (130)(18)
36./integraldisplay1
0/braceleftbiggxp−1
1−x−xpq−1
1−xq−1
x(1−x)+1
x(1−xq)/bracerightbiggdx
lnx=qlnp
[p>0] BI (130)(20)
37./integraldisplay1
0/braceleftbiggxq−1
1−x−xpq−1
1−xp−p−1
1−xpxp−1−p−1
2xp−1/bracerightbiggdx
lnx=1−p
2ln(2π)+/parenleftbigg
pq−1
2/parenrightbigg
lnp
[p>0,q > 0] BI (130)(22)
38./integraldisplay1
0(1−xp)(1−xq)−(1−x)2
x(1−x)lnxdx=l nB ( p, q)[ p>0,q > 0] GW (324)(24)
39.6/integraldisplay1
0(xp−1)ndx
lnx=n/summationdisplay
k=0/parenleftbiggn
n−k/parenrightbigg
(−1)n−kln(pk+1 )
[n>0,p n > −1]
GW (324)(19d), BI (123)(12)a
40.6/integraldisplay1
0(1−xp)n
1−xdx
lnx=n/summationdisplay
k=0(−1)k−1ln Γ[(n−k)p+1 ] [ n>1,p n > −1] BI (127)(12)
41./integraldisplay1
0(xp−1)nxq−1dx
lnx=n/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig
ln[q+(n−k)p]
[n>0,q > 0,p n > −q]
BI (123)(12)
42.6/integraldisplay1
0(1−xp)nxq−1 dx
(1−x)lnx=n/summationdisplay
k=0(−1)k−1ln Γ[(n−k)p+q]
[n>1,q > 0,p n > −q]
BI (127)(13)
43.10/integraldisplay1
0(xp−1)n(xq−1)mxr−1dx
lnx=n/summationdisplay
j=0(−1)j/parenleftbiggn
j/parenrightbiggm/summationdisplay
k=0(−1)k/parenleftBigm
k/parenrightBig
ln[r+(m−k)q+(n−j)p]
[n≥0,m≥0,n+m>0,r > 0,p n +qm+r>0]BI (123)(16)
4.268
1./integraldisplay1
0(xp−xq)(1−xr)
(lnx)2dx=(p+1 )l n ( p+1 )−(q+1 )l n ( q+1 )
−(p+r+1 )l n ( p+r+1 )+( q+r+1 )l n ( q+r+1 )
[p>−1,q > −1,p+r>−1,q+r>−1]GW (324)(26)
4.269 Powers and logarithms 549
2./integraldisplay1
0(xp−xq)2dx
(lnx)2=( 2p+1 )l n ( 2 p+1 )+( 2 q+1 )l n ( 2 q+1 )−2(p+q+1 )l n ( p+q+1 )
/bracketleftbig
p>−1
2,q > −1
2/bracketrightbig
GW (324)(26a)
3./integraldisplay1
0(1−xp)(1−xq)( 1−xr)dx
(lnx)2
=(p+q+1 )l n ( p+q+1 )+( q+r+1 )l n ( q+r+1 )+( p+r+1 )l n ( p+r+1 )
−(p+1 )l n ( p+1 )−(q+1 )l n ( q+1 )−(r+1 )l n ( r+1 )−(p+q+r)ln(p+q+r)
[p>−1,q > −1,r > −1,p+q>−1,p+r>−1,q+r>−1,p+q+r>0]
BI (124)(4)
4./integraldisplay1
0(1−xp)nxq−1dx
(lnx)2=1
2n/summationdisplay
k=0(−1)k/parenleftBign
k/parenrightBig
(pk+q)2ln(pk+q)
/bracketleftBig
q>0,p > −q
n/bracketrightBig
BI (124)(14)
5./integraldisplay1
0(1−xp)n(1−xq)mxr−1dx
(lnx)2=⎛
⎝n/summationdisplay
j=0(−1)j/parenleftbiggn
j/parenrightbigg⎞
⎠/parenleftBiggm/summationdisplay
k=0(−1)k/parenleftBigm
k/parenrightBig/parenrightBigg
×[(m−k)q+(n−j)p+r]ln [(m−k)q+(n−j)p+r]
[r>0,m q +r>0,n p +r>0,m q +np+r>0]BI (124)(8)
6./integraldisplay1
0/bracketleftbig
(q−r)xp−1+(r−p)xq−1+(p−q)xr−1/bracketrightbigdx
(lnx)2
=(q−r)plnp+(r−p)qlnq+(p−q)rlnr
[p>0,q > 0,r > 0] BI (124)(9)
7./integraldisplay1
0/bracketleftbiggxp−1
(p−q)(p−r)(p−s)+xq−1
(q−p)(q−r)(q−s)+xr−1
(r−p)(r−q)(r−s)+
+xs−1
(s−p)(s−q)(s−r)/bracketrightbiggdx
(lnx)2=1
2/bracketleftbiggp2lnp
(p−q)(p−r)(p−s)+q2lnq
(q−p)(q−r)(q−s)
+r2lnr
(r−p)(r−q)(r−s)+s2lns
(s−p)(s−q)(s−r)/bracketrightbigg
[p>0,q > 0,r > 0,s > 0]BI (124)(16)
4.269
1./integraldisplay1
0/radicalbigg
ln1
xdx
1+x2=√π
2∞/summationdisplay
k=0(−1)k
/radicalbig
(2k+1 )3BI (115)(33)
2.11/integraldisplay1
0dx/radicalbigg
ln1
x/parenleftbig
1+x2/parenrightbig=√π∞/summationdisplay
k=0(−1)k
√
2k+1BI (133)(2)
550 Logarithmic Functions 4.271
3./integraldisplay1
0/radicalbigg
ln1
xxp−1dx=1
2/radicalbiggπ
p3[p>0] GW (324)(1c)
4./integraldisplay1
0xp−1
/radicalbigg
ln1
xdx=/radicalbiggπ
p[p>0] BI (133)(1)
5./integraldisplay1
0sint−xnsin[(n+1 )t]+xn+1sinnt
1−2xcost+x2·dx/radicalbigg
ln1
x=√πn/summationdisplay
k=1sinkt√
k
[|t|<π] BI (133)(5)
6./integraldisplay1
0cost−x−xn−1cosnt+xncos[(n−1)t]
1−2xcost+x2·dx/radicalbigg
ln1
x=√πn−1/summationdisplay
k=1coskt√
k
[|t|<π] BI (133)(6)
7./integraldisplayv
udx
x·/radicalbigg
lnx
ulnv
x=π [uv >0] BI (145)(37)
4.271
1./integraldisplay1
0(lnx)2ndx
1+x=22n−1
22n·(2n)!ζ(2n+1 ) BI (110)(1)
2./integraldisplay1
0(lnx)2n−1dx
1+x=1−22n−1
2nπ2n|B2n| [n=1,2,...] BI (110)(2)
3./integraldisplay1
0(lnx)2n−1dx
1−x=−1
n22n−2π2n|B2n| [n=1,2,...]BI (110)(5), GW(324)(9a)
4./integraldisplay1
0(lnx)p−1dx
1−x=ei(p−1)πΓ(p)ζ(p)[ p>1] GW (324)(9b)
5./integraldisplay1
0(lnx)ndx
1+x2=(−1)nn!∞/summationdisplay
k=0(−1)k
(2k+1 )n+1BI (110)(11)
6./integraldisplay1
0(lnx)2ndx
1+x2=1
2/integraldisplay∞
0(lnx)2ndx
1+x2=π2n+1
22n+2|E2n| GW (324)(10)a
7./integraldisplay∞
0(lnx)2n+1
1+bx+x2dx=0 [ |b|<2] BI (135)(2)
8./integraldisplay1
0(lnx)2ndx
1−x2=22n+1−1
22n+1·(2n)!ζ(2n+1 ) [ n=1,2,...] BI (110)(12)
9./integraldisplay∞
0(lnx)2ndx
1−x2=0 BI (312)(7)a
10./integraldisplay1
0(lnx)2n−1dx
1−x2=1
2/integraldisplay∞
0(lnx)2n−1dx
1−x2=1−22n
4nπ2n|B2n|
[n=1,2,...]BI (290)(17)a, BI(312)(6)a
4.272 Powers and logarithms 551
11./integraldisplay1
0(lnx)2n−1xdx
1−x2=−1
4nπ2n|B2n| [n=1,2,...] BI (290)(19)a
12./integraldisplay1
0(lnx)2n1+x2
(1−x2)2dx=22n−1
2π2n|B2n| [n=1,2,...] BI (296)(17)a
13./integraldisplay1
0(lnx)2n+1(cos 2aπ−x)dx
1−2xcos 2aπ+x2=−(2n+1 ) !∞/summationdisplay
k=1cos2akπ
k2n+2
[ais not an integer] LI (113)(10)
14.6/integraldisplay∞
0(lnx)n xν−1dx
a2+2axcost+x2=−πcosectdn
dνn/bracketleftbigg
aν−2sin(ν−1)t
sinνπ/bracketrightbigg
[a>0,0<Reν<2,0<|t|<π]
ET I 315(12)
15./integraldisplay1
0(lnx)nxp−1
1−xqdx=−1
qn+1ψ(n)/parenleftbiggp
q/parenrightbigg
[p>0,q > 0] GW (324)(9)
16.3/integraldisplay1
0(lnx)nxp−1
1+xqdx=1
qn+1β(n)/parenleftbiggp
q/parenrightbigg
[p>0,q > 0] GW (324)(10)
4.272
1./integraldisplay1
0/bracketleftbigg
ln/parenleftbigg1
x/parenrightbigg/bracketrightbiggq−1
dx
1+2xcost+x2=c o s e c tΓ(q)∞/summationdisplay
k=1(−1)k−1sinkt
kq[|t|<π ,q< 1] LI (130)(1)
2./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggq−1(1 +x)dx
1+2xcost+x2=s e ct
2·Γ(q)∞/summationdisplay
k=1(−1)k−1cos/bracketleftbig/parenleftbig
k−1
2/parenrightbig
t/bracketrightbig
kq
/bracketleftbig
|t|<π , q<1
2/bracketrightbig
LI (130)(5)
3.9/integraldisplay1
0/bracketleftbigg
ln/parenleftbigg1
x/parenrightbigg/bracketrightbiggμxν−1dx
1−2axcost+x2a2=Γ(μ+1 )
asint∞/summationdisplay
k=1aksinkt
(ν+k−1)μ+1
[a>0,Reμ>0,Reν>0,−π<t<π ]BI (140)(14)a
4./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggr−1cosλ−px
1+p2x2−2pxcosλxq−1dx=Γ (r)∞/summationdisplay
k=1pk−1coskλ
(q+k−1)r
[r>0,q > 0] BI (113)(11)
5./integraldisplay∞
1(lnx)pdx
x2=Γ ( 1+ p)[ p>−1] BI (149)(1)
6./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggμ−1
xν−1dx=1
νμΓ(μ)[ R e μ>0,Reν>0] BI (107)(3)
7./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggn−1
2
xν−1dx=(2n−1)!!
(2ν)n/radicalbiggπ
ν[Reν>0] BI (107)(2)
552 Logarithmic Functions 4.272
8.11/integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggn−1xν−1
1+xdx=Γ/parenleftbigg
3−1
n/parenrightbigg/parenleftBig
p1
n−3−q1
n−3/parenrightBig
[Reν>0] BI (110)(4)
9./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggn−1xν−1
1−xdx=(n−1)!ζ(n, ν)[ R e ν>0] BI (110)(7)
10./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggμ−1
(x−1)n/parenleftbigg
a+nx
x−1/parenrightbigg
xa−1dx=Γ (μ)n/summationdisplay
k=0(−1)kn(n−1)...(n−k+1 )
(a+n−k)μ−1k!
[Reμ>0] LI (110)(10)
11./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggn−11−xm
1−xdx=(n−1)!m/summationdisplay
k=11
knLI (110)(9)
12./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggμ−1xν−1dx
1−x2=Γ (μ)∞/summationdisplay
k=01
(ν+2k)μ=1
2μΓ(μ)ζ/parenleftBig
μ,ν
2/parenrightBig
[Reμ>0,Reν>0] BI (110)(13)
13./integraldisplay1
0xq−x−q
1−x2/parenleftbigg
ln1
x/parenrightbiggp
dx=Γ (p+1 )∞/summationdisplay
k=1/braceleftbigg1
(2k+q−1)p+1−1
(2k−q−1)p+1/bracerightbigg
/bracketleftbig
p>−1,q2<1/bracketrightbig
LI (326)(12)a
14./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggr−1xp−1dx
(1 +xq)s=Γ (r)∞/summationdisplay
k=0/parenleftbigg−s
k/parenrightbigg1
(p+kq)r
[p>0,q > 0,r > 0,0<s<r +2 ]
GW (324)(11)
15./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggn
(1 +xq)mxp−1dx=n!m/summationdisplay
k=0/parenleftBigm
k/parenrightBig1
(p+kq)n+1
[p>0,q > 0] BI (107)(6)
16./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggn
(1−xq)mxp−1dx=n!m/summationdisplay
k=0/parenleftBigm
k/parenrightBig(−1)k
(p+kq)n+1
[p>0,q > 0] BI (107)(7)
17./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbiggp−1xq−1dx
1−axq=1
aqpΓ(p)∞/summationdisplay
k=1ak
kp[p>0,q > 0,a < 1] LI (110)(8)
18./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbigg2−1
n/parenleftbig
xp−1−xq−1/parenrightbig
dx=n
n−1Γ/parenleftbigg1
n/parenrightbigg/parenleftBig
q1−1
n−p1−1
n/parenrightBig
[q>p> 0] BI (133)(4)
19./integraldisplay1
0/parenleftbigg
ln1
x/parenrightbigg2n−1xp−x−p
1−xqxq−1dx=1
p2n∞/summationdisplay
k=n/parenleftbigg2pπ
q/parenrightbiggk|B2k|
2k·(2k−2n)!
/bracketleftBig
p<q
2/bracketrightBig
LI (110)(16)
4.282 Rational functions of lnxand powers 553
4.273/integraldisplayv
u/parenleftBig
lnx
u/parenrightBigp−1/parenleftBig
lnv
x/parenrightBigq−1dx
x=B (p, q)/parenleftBig
lnv
u/parenrightBigp+q−1
[p>0,q > 0,u v > 0] BI (145)(36)
4.274/integraldisplay1
e
0q√xdx
x/radicalbig
−(1 + ln x)=√qπ
q√e[q>0] BI (145)(4)
4.275
1./integraldisplay1
0/bracketleftBigg/parenleftbigg
ln1
x/parenrightbiggq−1
−xp−1(1−x)q−1/bracketrightBigg
dx=Γ(q)
Γ(p+q)[Γ(p+q)−Γ(p)]
[p>0,q > 0] BI (107)(8)
2./integraldisplay1
0/bracketleftbigg
x−/parenleftbigg1
1−lnx/parenrightbiggq/bracketrightbiggdx
xlnx=−ψ(q)[ q>0] BI (126)(5)
4.28 Combinations of rational functions of lnxand powers
4.281
1./integraldisplay1
0/bracketleftbigg1
lnx+1
1−x/bracketrightbigg
dx=C BI (127)(15)
2./integraldisplay∞
1dx
x2(lnp−lnx)=1
pli(p) LA 281(30)
3./integraldisplay1
0xp−1dx
q±lnx=±e∓pqEi(±pq)[ p>0,q > 0] LI (144)(11,12)
4./integraldisplay1
0/bracketleftbigg1
lnx+xμ−1
1−x/bracketrightbigg
dx=−ψ(μ)[ R e μ>0] WH
5./integraldisplay1
0/bracketleftbiggxp−1
lnx+xq−1
1−x/bracketrightbigg
dx=l np−ψ(q)[ p>0,q > 0] BI (127)(17)
6./integraldisplay1
0/bracketleftbigg1
1−x2+1
2xlnx/bracketrightbiggdx
lnx=ln2
2LI (130)(19)
7./integraldisplay1
0/bracketleftbigg
q−1
2+(1−x)(1+ qlnx)+xlnx
(1−x)2xq−1/bracketrightbiggdx
lnx=1
2−q−lnΓ(q)+ln2π
2
[q>0] BI (128)(15)
4.282
1./integraldisplay1
0lnx
4π2+( l nx)2·dx
1−x=1
4−1
2C BI (129)(1)
2./integraldisplay1
01
a2+( l nx)2·dx
1+x2=1
2aβ/parenleftbigg2a+π
4π/parenrightbigg/bracketleftBig
a>−π
2/bracketrightBig
BI (129)(9)
3./integraldisplay1
01
π2+( l nx)2dx
1+x2=4−π
4πBI (129)(6)
4./integraldisplay1
0lnx
π2+( l nx)2·dx
1−x2=1
2/parenleftbigg1
2−ln2/parenrightbigg
BI (129)(10)
554 Logarithmic Functions 4.283
5./integraldisplay1
0lnx
a2+( l nx)2·xdx
1−x2=1
2/bracketleftBigπ
2a+l nπ
a+ψ/parenleftBiga
π/parenrightBig/bracketrightBig
[a>0] BI (129)(14)
6./integraldisplay1
0lnx
π2+( l nx)2·xdx
1−x2=1
2/parenleftbigg1
2−C/parenrightbigg
BI (129)(13)
7./integraldisplay1
01
π2+4( l n x)2·dx
1+x2=ln 2
4πBI (129)(7)
8./integraldisplay1
0lnx
π2+4( l n x)2·dx
1−x2=2−π
16BI (129)(11)
9.10/integraldisplay1
01
π2+1 6( l n x)2·dx
1+x2=1
8π√
2/bracketleftBig
π+2l n/parenleftBig√
2−1/parenrightBig/bracketrightBig
BI (129)(8)
10./integraldisplay1
0lnx
π2+1 6( l n x)2·dx
1−x2=−π
32√
2+1
16+1
16√
2ln/parenleftBig√
2−1/parenrightBig
BI (129)(12)
11./integraldisplay1
0lnx
/bracketleftBig
a2+( l nx)2/bracketrightBig2dx
1−x=−π2
a4∞/summationdisplay
k=1|B2k|/parenleftbigg2π
a/parenrightbigg2k−2
BI (129)(4)
12./integraldisplay1
0lnx
/bracketleftBig
a2+( l nx)2/bracketrightBig2xdx
1−x2=−π2
4a4∞/summationdisplay
k=1|B2k|/parenleftBigπ
a/parenrightBig2k−2
BI (129)(16)
13./integraldisplay1
0xp−x−p
x2−1dx
q2+( l nx)2=2π
q∞/summationdisplay
k=1(−1)k−1sinkpπ
2q+kπ/bracketleftbig
p2<1/bracketrightbig
BI (132)(13)a
4.283
1./integraldisplay1
0/parenleftbiggx−1
lnx−x/parenrightbiggdx
lnx=l n2 −1 BI (132)(17)a
2./integraldisplay1
0/parenleftbigg1
lnx+1
1−x−1
2/parenrightbiggdx
lnx=ln2π
2−1 BI (127)(20)
3./integraldisplay1
0/parenleftbigg1
lnx+x
1−x+x
2/parenrightbiggdx
xlnx=ln2π
2BI (127)(23)
4./integraldisplay1
0/bracketleftBigg
1
(lnx)2−x
(1−x)2/bracketrightBigg
dx=C−1
2GW (326)(8a)
5./integraldisplay1
0/parenleftbigg1
1−x2+1
2lnx−1
2/parenrightbiggdx
lnx=ln2−1
2BI (128)(14)
6./integraldisplay1
0/parenleftbigg1
lnx+1
2·1+x
1−x−lnx/parenrightbiggdx
lnx=ln 2π
2BI (127)(22)
7./integraldisplay1
0/bracketleftbigg1
1−lnx−x/bracketrightbiggdx
xlnx=−C GW (326)(11a)
8./integraldisplay1
0/bracketleftBigg
xq−1
x(lnx)2−q
lnx/bracketrightBigg
dx=qlnq−q [q>0] BI (126)(2)
4.291 Logarithmic functions and powers 555
9./integraldisplay1
0/bracketleftbigg
x+1
alnx−1/bracketrightbiggdx
xlnx=l na
q+C [a>0,q > 0] BI (126)(8)
10./integraldisplay1
0/bracketleftbigg1
lnx+1+x
2(1−x)/bracketrightbiggxp−1
lnxdx=−ln Γ(p)+/parenleftbigg
p−1
2/parenrightbigg
lnp−p+ln 2π
2
[p>0] GW (326)(9)
11./integraldisplay1
0/bracketleftbigg
p−1−1
1−x+/parenleftbigg1
2−1
lnx/parenrightbigg
xp−1/bracketrightbiggdx
lnx=/parenleftbigg1
2−p/parenrightbigg
lnp+p−ln2π
2
[p>0] BI (127)(25)
12./integraldisplay1
0/bracketleftBigg
−1
(lnx)2+(p−2)xp−(p−1)xp−1
(1−x)2/bracketrightBigg
dx=−ψ(p)+p−3
2
[p>0] GW (326)(8)
13./integraldisplay1
0/bracketleftbigg/parenleftbigg
p−1
2/parenrightbigg
x3+1
2/parenleftbigg
1−1
lnx/parenrightbigg/parenleftbig
x2p−1−1/parenrightbig/bracketrightbiggdx
lnx=/parenleftbigg1
2−p/parenrightbigg
(lnp−1)
[p>0] BI (132)(23)a
14./integraldisplay1
0/bracketleftbigg/parenleftbigg
q−1
2/parenrightbiggxp−1−xr−1
lnx+pxpq−1
1−xp−rxrq−1
1−xr/bracketrightbiggdx
lnx=(p−r)/bracketleftbigg1
2−q−lnΓ(q)+ln2π
2/bracketrightbigg
[q>0] BI (132)(13)
4.284
1./integraldisplay1
0/bracketleftBigg
xq−1
x(lnx)3−q
x(lnx)2−q2
2lnx/bracketrightBigg
dx=q2
2lnq−3
4q2
[q>0] BI (126)(3)
2./integraldisplay1
0/bracketleftBigg
xq−1
x(lnx)4−q
x(lnx)3−q2
2x(lnx)2−q3
6lnx/bracketrightBigg
dx=q3
6lnq−11
36q3
[q>0] BI (126)(4)
4.285/integraldisplay1
0xp−1dx
(q+l nx)n=pn−1
(n−1)!e−pqEi(pq)−1
(n−1)!qn−1n−1/summationdisplay
k=1(n−k−1)!(pq)k−1
[p>0,q < 0] BI (125)(21)
In integrals of the form/integraldisplayxa(lnx)ndx
[b±(lnx)m]l, we should make the substitution x=etorx=e−tand
then seek the resulting integrals in 3.351 –3.356 .
4.29–4.32 Combinations of logarithmic functions of more complicated arguments
and powers
4.291
1./integraldisplay1
0ln(1 + x)
xdx=π2
12FI II 483
556 Logarithmic Functions 4.291
2./integraldisplay1
0ln(1−x)
xdx=−π2
6FI II 714
3./integraldisplay1/2
0ln(1−x)
xdx=1
2(ln2)2−π2
12BI (145)(2)
4./integraldisplay1
0ln/parenleftBig
1−x
2/parenrightBigdx
x=1
2(ln2)2−π2
12BI (114)(18)
5./integraldisplay1
0ln1+x
2
1−xdx=1
2(ln 2)2−π2
12BI (115)(1)
6./integraldisplay1
0ln(1 + x)
1+xdx=1
2(ln2)2BI (114)(14)a
7.7/integraldisplay∞
0ln(1 + ax)
1+x2dx=π
4ln/parenleftbig
1+a2/parenrightbig
−/integraldisplaya
0lnud u
1+u2[a>0] GI II (2209)
8./integraldisplay1
0ln(1 + x)
1+x2dx=π
8ln2 FI II 157
9./integraldisplay∞
0ln(1 + x)
1+x2dx=π
4ln2 +G BI (136)(1)
10./integraldisplay1
0ln(1−x)
1+x2dx=π
8ln2−G BI (114)(17)
11./integraldisplay∞
1ln(x−1)
1+x2dx=π
8ln2 BI (144)(4)
12./integraldisplay1
0ln(1 + x)
x(1 +x)dx=π2
12−1
2(ln2)2BI (144)(4)
13./integraldisplay∞
0ln(1 + x)
x(1 +x)dx=π2
6.
BI (141)(9)a
14./integraldisplay1
0ln(1 + x)
(ax+b)2dx=1
a(a−b)lna+b
b+2ln2
b2−a2[a/negationslash=b, ab > 0]
=1
2a2(1−ln2) [ a=b]
LI (114)(5)a
15./integraldisplay∞
0ln(1 + x)
(ax+b)2dx=lna
b
a(a−b)[ab >0] BI (139)(5)
16./integraldisplay1
0ln(a+x)dx
a+x2=1
2√aarccot√aln[(1 + a)a][ a>0] BI (114)(20)
17./integraldisplay∞
0ln(a+x)dx
(b+x)2=alna−blnb
b(a−b)[a>0,b > 0,a/negationslash=b] LI (139)(6)
18./integraldisplaya
0ln(1 + ax)
1+x2dx=1
2arctan aln/parenleftbig
1+a2/parenrightbig
GI II (2195)
4.291 Logarithmic functions and powers 557
19./integraldisplay1
0ln(1 + ax)
1+ax2dx=1
2√aarctan√aln(1 + a)[ a>0] BI (114)(21)
20./integraldisplay1
0ln(ax+b)
(1 +x)2dx=1
a−b/bracketleftbigg1
2(a+b)ln(a+b)−blnb−aln2/bracketrightbigg
[a>0,b > 0,a/negationslash=b] BI (114)(22)
21./integraldisplay∞
0ln(ax+b)
(1 +x)2dx=1
a−b[alna−blnb][ a>0,b > 0] BI (139)(8)
22./integraldisplay∞
0ln(a+x)xdx
(b2+x2)2=1
2(a2+b2)/parenleftbigg
lnb+aπ
2b+a2
b2lna/parenrightbigg
[a>0,b > 0] BI (139)(9)
23./integraldisplay1
0ln(1 + x)1+x2
(1 +x)4dx=−1
3ln 2 +23
72LI (114)(12)
24./integraldisplay1
0ln(1 + x)1+x2
a2+x2·dx
1+a2x2=1
2a(1 +a2)/bracketleftBigπ
2ln/parenleftbig
1+a2/parenrightbig
−2arc t an a·lna/bracketrightBig
[a>0] LI (114)(11)
25./integraldisplay1
0ln(1 + x)1−x2
(ax+b)2dx
(bx+a)2=1
a2−b2/braceleftbigg1
a−b/bracketleftbigga+b
abln(a+b)−1
alnb−1
blna/bracketrightbigg
+4ln2
b2−a2/bracerightbigg
/bracketleftbig
a>0,b > 0,a2/negationslash=b2/bracketrightbig
LI (114)(13)
26./integraldisplay∞
0ln(1 + x)1−x2
(ax+b)2·dx
(bx+a)2=1
ab(a2−b2)lnb
a
[a>0,b > 0] LI (139)(14)
27./integraldisplay1
0ln(1 + ax)1−x2
(1 +x2)2dx=1
2(1 +a)2
1+a2ln(1 + a)−1
2·a
1+a2ln2−π
4·a2
1+a2
[a>−1] BI (114)(23)
28./integraldisplay∞
0ln(a+x)b2−x2
(b2+x2)2dx=1
a2+b2/parenleftbigg
alnb
a−bπ
2/parenrightbigg
[a>0,b > 0] BI (139)(11)
29./integraldisplay∞
0ln2(a−x)b2−x2
(b2+x2)2dx=2
a2+b2/parenleftbigg
alna
b−bπ
2/parenrightbigg
[a>0,b > 0] BI (139)(12)
30./integraldisplay∞
0ln2(a−x)xdx
(b2+x2)2=1
a2+b2/parenleftbigg
lnb−aπ
2b+a2
b2lna/parenrightbigg
[a>0,b > 0] BI (139)(10)
558 Logarithmic Functions 4.292
4.292
1./integraldisplay1
0ln (1±x)√
1−x2dx=−π
2ln2±2G GW (325)(20)
2./integraldisplay1
0xln (1±x)√
1−x2dx=−1±π
2GW (325)(22c)
3./integraldisplaya
−aln(1 + bx)√
a2−x2dx=πln1+√
1−a2b2
2/bracketleftbigg
0≤|b|≤1
a/bracketrightbigg
BI (145)(16, 17)a, GW (325)(21e)
4./integraldisplay1
0xln(1 + ax)√
1−x2dx=−1+π
2·1−√
1−a2
a+√
1−a2
aarcsin a[|a|≤1]
=−1+π
2a+√
a2−1
aln/parenleftBig
a+/radicalbig
a2−1/parenrightBig
[a≥1]
GW (325)(22)
5./integraldisplay1
0ln(1 + ax)
x√
1−x2dx=1
2arcsin a(π−arcsin a)=π2
8−1
2(arccos a)2
[|a|≤1] BI (120)(4), GW (325)(21a)
4.293
1./integraldisplay1
0xμ−1ln(1 + x)dx=1
μ[ln2−β(μ+1 ) ] [ R e μ>−1] BI (106)(4)a
2.6/integraldisplay∞
1xμ−1ln(1 + x)dx=−1
μ[β(−μ) + ln 2] [Re μ<0] ET I 315(17)
3./integraldisplay∞
0xμ−1ln(1 + x)dx=π
μsinμπ[−1<Reμ<0] GW (325)(3)a
4./integraldisplay1
0x2n−1ln(1 + x)dx=1
2n2n/summationdisplay
k=1(−1)k−1
kGW (325)(2b)
5./integraldisplay1
0x2nln(1 + x)dx=1
2n+1/bracketleftBigg
ln 4 +2n+1/summationdisplay
k=1(−1)k
k/bracketrightBigg
GW (325)(2c)
6.11/integraldisplay1
0xn−1
2ln(1 + x)dx=2ln2
2n+1+(−1)n·4
2n+1/bracketleftBigg
π
4−n/summationdisplay
k=0(−1)k
2k+1/bracketrightBigg
GW (325)(2f)
7./integraldisplay∞
0xμ−1ln|1−x|dx=π
μcot(μπ)[ −1<Reμ<0]
BI (134)(4), ET I 315(18)
8./integraldisplay1
0xμ−1ln(1−x)dx=−1
μ[ψ(μ+1 )−ψ(1)] = −1
μ[ψ(μ+1 )+ C]
[Reμ>−1] ET I 316(19)
9.7/integraldisplay∞
1xμ−1ln(x−1)dx=1
μ[πcot(μπ)+ψ(μ+1 )+ C]
[Reμ<0] ET I 316(20)
4.294 Logarithmic functions and powers 559
10./integraldisplay∞
0xμ−1ln(1 + γx)dx=π
μγμsinμπ[−1<Reμ<0,|argγ|<π]
BI (134)(3)
11.11/integraldisplay∞
0xμ−1ln(1 + x)
1+xdx=−π
sinμπ[C+ψ(1−μ)] [ −1<Reμ<1] ET I 316(21)
12./integraldisplay1
0ln(1 + x)
(1 +x)μ+1dx=−ln2
2μμ+2μ−1
2μμ2BI (114)(6)
13./integraldisplay1
0xμ−1ln(1−x)
(1−x)1−νdx=B (μ, ν)[ψ(ν)−ψ(μ+ν)] [Re μ>0,Reν>0] ET I 316(122)
14./integraldisplay∞
0xμ−1ln(γ+x)
(γ+x)νdx=γμ−νB(μ, ν−μ)[ψ(ν)−ψ(ν−μ)+l n γ]
[0<Reμ<Reν] ET I 316(23)
4.294
1./integraldisplay1
0ln(1 + x)(p−1)xp−1−px−p
xdx=2l n2 −π
sinpπ
[0<p< 1] BI (114)(2)
2./integraldisplay1
0ln(1 + x)1+x2n+1
1+xdx=2l n2n/summationdisplay
k=01
2k+1−2n+1/summationdisplay
j=11
jj/summationdisplay
k=1(−1)k−1
kBI (114)(7)
3./integraldisplay1
0ln(1 + x)1−x2n
1+xdx=2l n2 ·n−1/summationdisplay
k=01
2k+1−2n/summationdisplay
j=11
jj/summationdisplay
k=1(−1)k−1
kBI (114)(8)
4./integraldisplay1
0ln(1 + x)1−x2n
1−xdx=2l n2 ·n−1/summationdisplay
k=01
2k+1+2n/summationdisplay
i=1(−1)j
jj/summationdisplay
k=1(−1)k−1
kBI (114)(9)
5./integraldisplay1
0ln(1 + x)1−x2n+1
1−xdx=2l n2n/summationdisplay
k=01
2k+1+2n+1/summationdisplay
j=1(−1)j
jj/summationdisplay
k=1(−1)k−1
kBI (114)(10)
6./integraldisplay1
0ln(1−x)1−(−1)nxn
1−xdx=n/summationdisplay
j=1(−1)j
jj/summationdisplay
k=11
kBI (114)(15)
7./integraldisplay1
0ln(1−x)1−xn
1−xdx=−n/summationdisplay
j=11
jj/summationdisplay
k=11
kBI (114)(16)
8./integraldisplay∞
0ln2(1−x)xpdx=2π
p+1cotpπ [−2<p< −1] BI (134)(13)a
9./integraldisplay1
0[ln(1 + x)]n(1 +x)rdx=(−1)n−1n!
(r+1 )n+1+2r+1n/summationdisplay
k=0(−1)kn!(ln2)n−k
(n−k)!(r+1 )k+1LI (106)(34)a
10./integraldisplay1
0[ln(1−x)]n(1−x)rdx=(−1)nn!
(r+1 )n+1[r>−1] BI (106)(35)a
560 Logarithmic Functions 4.295
11./integraldisplay1
0/parenleftbigg
ln1
1−x2/parenrightbiggn
x2q−1dx=n!
2ζ(n+1,q+1 ) [ −1<q< 0] BI (311)(15)a
12./integraldisplay1
0(lnx)2nln/parenleftbig
1−x2/parenrightbigdx
x=−π2n+2
2(n+ 1)(2 n+1 )|B2n+2| BI (309)(5)a
13.6/integraldisplay1
0/bracketleftbigg
ln1
x/bracketrightbiggm
ln/parenleftbig
1−x2/parenrightbig
dx=−∞/summationdisplay
n=1Γ(m+1 )
n(2n+1 )m+1[m+1>0,n+1>0]
4.295
1./integraldisplay∞
0ln/parenleftbig
μx2+β/parenrightbigdx
γ+x2=π√γln/parenleftBig√μγ+/radicalbig
β/parenrightBig
[Reβ>0,Reμ>0,|argγ|<π]
ET II 218(27)
2./integraldisplay1
0ln/parenleftbig
1+x2/parenrightbigdx
x2=π
2−ln 2 GW (325)(2g)
3./integraldisplay∞
0ln/parenleftbig
1+x2/parenrightbigdx
x2=π GW (325)(4c)
4./integraldisplay∞
0ln/parenleftbig
1+x2/parenrightbigdx
(a+x)2=2a
1+a2/parenleftBigπ
2a+l na/parenrightBig
[a>0] BI (319)(6)a
5./integraldisplay1
0ln/parenleftbig
1+x2/parenrightbigdx
1+x2=π
2ln 2−G BI (114)(24)
6./integraldisplay∞
1ln/parenleftbig
1+x2/parenrightbigdx
1+x2=π
2ln2 +G BI (114)(5)
7./integraldisplay∞
0ln/parenleftbig
a2+b2x2/parenrightbigdx
c2+g2x2=π
cglnag+bc
g[a>0,b > 0,c > 0,g > 0]
BI (136)(11-14)a
8./integraldisplay∞
0ln/parenleftbig
a2+b2x2/parenrightbigdx
c2−g2x2=−π
cgarctanbc
ag[a>0,b > 0,c > 0,g > 0]
BI (136)(15)a
9./integraldisplay∞
0ln/parenleftbig
1+p2x2/parenrightbig
−ln/parenleftbig
1+q2x2/parenrightbig
x2dx=π(p−q)[ p>0,q > 0] FI II 645
10./integraldisplay1
0ln1+a2x2
1+a2dx
1−x2=−(arctan a)2BI (115)(2)
11./integraldisplay1
0ln/parenleftbig
1−x2/parenrightbigdx
x=−π2
12
12./integraldisplay∞
0ln2/parenleftbig
1−x2/parenrightbigdx
x2=0 BI (142)(9)a
13./integraldisplay1
0ln/parenleftbig
1−x2/parenrightbigdx
1+x2=π
4ln 2−G GW (325)(17)
14./integraldisplay∞
1ln/parenleftbig
x2−1/parenrightbigdx
1+x2=π
4ln2 +G BI (144)(6)
4.295 Logarithmic functions and powers 561
15./integraldisplay∞
0ln2/parenleftbig
a2−x2/parenrightbigdx
b2+x2=π
bln/parenleftbig
a2+b2/parenrightbig
[b>0] BI (136)(16)
16./integraldisplay∞
0ln2/parenleftbig
a2−x2/parenrightbigb2−x2
(b2+x2)2dx=−2bπ
a2+b2[b>0] BI (136)(20)
17./integraldisplay1
0ln/parenleftbig
1+x2/parenrightbigdx
x(1 +x2)=1
2/bracketleftbiggπ2
12−1
2(ln2)2/bracketrightbigg
BI (114)(25)
18./integraldisplay∞
0ln/parenleftbig
1+x2/parenrightbigdx
x(1 +x2)=π2
12BI (141)(9)
19./integraldisplay1
0ln/parenleftbig
cos2t+x2sin2t/parenrightbigdx
1−x2=−t2BI (114)(27)a
20./integraldisplay∞
0ln/parenleftbig
a2+b2x2/parenrightbigdx
(c+gx)2=2lnb
cg+b2
a2g2+b2c2/parenleftbigga
bπ+2c
glnc
g+2a2g
b2clna
b/parenrightbigg
[a>0,b > 0,c > 0,g > 0]
BI (139)(16)a
21./integraldisplay1
0ln/parenleftbig
a2+b2x2/parenrightbigdx
(c+gx)2
=2
c(c+g)lna+b2
a2g2+b2c2/bracketleftbigg2a
barccota
b+cb2−ga2
b2(c+g)lna2+b2
a2−2c
glnc+g
c/bracketrightbigg
[a>0,b > 0,c > 0,g > 0]BI (114)(28)a
22.11/integraldisplay∞
0ln/parenleftbig
1+p2x2/parenrightbig
r2+q2x2dx=/integraldisplay∞
0ln/parenleftbig
p2+x2/parenrightbig
q2+r2x2dx=π
qrlnq+pr
r
[qr >0,p > 0]
FI II 745a, BI (318)(1)a, BI (318)(4)a
23./integraldisplay∞
0ln/parenleftbig
1+a2x2/parenrightbig
b2+c2x2dx
d2+g2x2=π
b2g2−c2d2/bracketleftbiggg
dln/parenleftbigg
1+ad
g/parenrightbigg
−c
bln/parenleftbigg
1+ab
c/parenrightbigg/bracketrightbigg
/bracketleftbig
a>0,b > 0,c > 0,d > 0,g > 0,b2g2/negationslash=c2d2/bracketrightbig
BI (141)(10)
24./integraldisplay∞
0ln/parenleftbig
1+a2x2/parenrightbig
b2+c2x2x2dx
d2+g2x2=π
b2g2−c2d2/bracketleftbiggb
cln/parenleftbigg
1+ab
c/parenrightbigg
−d
gln/parenleftbigg
1+ad
g/parenrightbigg/bracketrightbigg
/bracketleftbig
a>0,b > 0,c > 0,d > 0,g > 0,b2g2/negationslash=c2d2/bracketrightbig
BI (141)(11)
25./integraldisplay∞
0ln/parenleftbig
a2+b2x2/parenrightbig dx
(c2+g2x2)2=π
2c3g/parenleftbigg
lnag+bc
g−bc
ag+bc/parenrightbigg
[a>0,b > 0,c > 0,g > 0]
GW (325)(18a)
26./integraldisplay∞
0ln/parenleftbig
a2+b2x2/parenrightbigx2dx
(c2+g2x2)2=π
2cg3/parenleftbigg
lnag+bc
g+bc
ag+bc/parenrightbigg
[a>0,b > 0,c > 0,g > 0]
GW (325)(18b)
562 Logarithmic Functions 4.295
27./integraldisplay1
0ln/parenleftbig
1+ax2/parenrightbig/radicalbig
1−x2dx=π
2/braceleftbigg
ln1+√1+a
2+1
21−√1+a
1+√1+a/bracerightbigg
[a>0] BI (117)(6)
28./integraldisplay1
0ln/parenleftbig
1+a−ax2/parenrightbig/radicalbig
1−x2dx=π
2/braceleftbigg
ln1+√1+a
2−1
21−√1+a
1+√1+a/bracerightbigg
[a>0] BI (117)(7)
29./integraldisplay1
0ln/parenleftbig
1−a2x2/parenrightbigdx√
1−x2=πln1+√
1−a2
2/bracketleftbig
a2<1/bracketrightbig
BI (119)(1)
30.6/integraldisplay1
0ln/parenleftbig
1−a2x2/parenrightbigdx
x√
1−x2=−/parenleftBig
arccos |a|−π
2/parenrightBig2
LI (120)(11)
31./integraldisplay1
0ln/parenleftbig
1−x2/parenrightbig dx/radicalbig
(1−x2)(1−k2x2)=l nk/prime
kK(k)−π
2K(k/prime) BI (120)(12)
32./integraldisplay1
0ln/parenleftbig
1±kx2/parenrightbig dx/radicalbig
(1−x2)( 1−k2x2)=1
2ln2±2k√
kK(k)−π
8K(k/prime) BI (120)(8), BI (120)(14)
33./integraldisplay1
0ln/parenleftbig
1−k2x2/parenrightbig
/radicalbig
(1−x2)(1−k2x2)dx=l nk/primeK(k) BI (119)(27)
34./integraldisplay1
0ln/parenleftbig
1−k2x2/parenrightbig/radicalBigg
1−k2x2
1−x2dx=/parenleftbig
2−k2/parenrightbig
K(k)−(2−lnk/prime)E(k) BI (119)(3)
35./integraldisplay1
0/radicalBigg
1−x2
1−k2x2ln/parenleftbig
1−k2x2/parenrightbig
dx=1
k2/parenleftBig
1+k/prime2−k/prime2lnk/prime/parenrightBig
K(k)−(2−lnk/prime)E(k) BI (119)(7)
36./integraldisplay1
−1ln/parenleftbig
1−x2/parenrightbig dx
(a+bx)√
1−x2=2π√
a2−b2ln√
a2−b2
a+√
a2−b2
[a>0,b > 0,a/negationslash=b] BI (145)(15)
37.8/integraldisplay1
0ln/parenleftbig
1−x2/parenrightbig/parenleftbig
pxp−1−qxq−1/parenrightbig
dx=ψ/parenleftBigq
2+1/parenrightBig
+ψ/parenleftBigp
2+1/parenrightBig
[p>−2,q > −2] BI (106)(15)
38./integraldisplay1
0ln/parenleftbig
1+ax2/parenrightbigdx√
1−x2=πln1+√1+a
2[a≥−1] GW (325)(21b)
39./integraldisplay1
0ln/parenleftbig
1+x2/parenrightbig
xμ−1dx=1
μ/bracketleftBig
ln 2−β/parenleftBigμ
2+1/parenrightBig/bracketrightBig
[Reμ>−2] BI (106)(12)
40./integraldisplay∞
0ln/parenleftbig
1+x2/parenrightbig
xμ−1dx=π
μsinμπ
2[−2<Reμ<0]
BI (311)(4)a, ET I 315(15)
4.298 Logarithmic functions and powers 563
41./integraldisplay∞
0ln/parenleftbig
1+x2/parenrightbigxμ−1dx
1+x
=π
sinμπ/braceleftbigg
ln 2−(1−μ)sinμπ
2β/parenleftbigg1−μ
2/parenrightbigg
−(2−μ)cosμπ
2β/parenleftbigg2−μ
2/parenrightbigg/bracerightbigg
[−2<Reμ<1] ET I 316(25)
4.296
1./integraldisplay1
0ln/parenleftbig
1+2xcost+x2/parenrightbigdx
x=π2
6−t2
2BI (114)(34)
2./integraldisplay∞
−∞ln/parenleftbig
a2−2axcost+x2/parenrightbigdx
1+x2=πln/parenleftbig
1+2a|sint|+a2/parenrightbig
BI (145)(28)
3./integraldisplay∞
0ln/parenleftbig
1+2xcost+x2/parenrightbig
xμ−1dx=2π
μcosμt
sinμπ[|t|<π , −1<Reμ<0]ET I 316(27)
4./integraldisplay∞
0ln/parenleftbiggx2+2axcost+a2
x2−2axcost+a2/parenrightbiggxdx
x2+b2=1
2π2−πt+πarctan/parenleftbig
a2−b2/parenrightbig
cost
(a2+b2)s i nt+2ab
[a>0,b > 0,0<t<π ]
4.297
1./integraldisplay1
0lnax+b
bx+adx
(1 +x)2=1
a−b/bracketleftbigg
(a+b)lna+b
2−alna−blnb/bracketrightbigg
[a>0,b > 0] BI (115)(16)
2./integraldisplay∞
0lnax+b
bx+adx
(1 +x)2=0 [ ab >0] BI (139)(23)
3./integraldisplay1
0ln1−x
xdx
1+x2=π
8ln2 BI (115)(5)
4./integraldisplay1
0ln1+x
1−xdx
1+x2=G BI (115)(17)
5.11/integraldisplay∞
0ln/parenleftbigg1+x
1−x/parenrightbigg2dx
x(1 +x2)=π2
2BI (141)(13)
6./integraldisplayv
ulnv+x
u+xdx
x=1
2/parenleftBig
lnv
u/parenrightBig2
[uv >0] BI (145)(33)
7./integraldisplay∞
0bln(1 + ax)−aln(1 + bx)
x2dx=ablnb
a[a>0,b > 0] FI II 647
8./integraldisplay1
0ln1+ax
1−axdx
x√
1−x2=πarcsin a [|a|≤1] GW (325)(21c), BI (122)(2)
9./integraldisplayv
uln/parenleftbigg1+ax
1−ax/parenrightbiggdx/radicalbig
(x2−u2)(v2−x2)=π
vF/parenleftBig
arcsin av,u
v/parenrightBig
[|av|<1] BI (145)(35)
10.8PV/integraldisplay1
0ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglea+y
a−y/vextendsingle/vextendsingle/vextendsingle/vextendsingledy
y/radicalbig
1−y2=π2
2[0<a≤1]
564 Logarithmic Functions 4.298
4.298
1./integraldisplay∞
0ln1+x2
xx2n−1
1+xdx=ln 2
2n+1
4n2−1
2nβ(2n+1 ) BI (137)(1)
2./integraldisplay∞
0ln1+x2
xx2n
1+xdx=ln 2
2n+1
4n2−1
2nβ(2n+1 ) BI (137)(3)
3./integraldisplay∞
0ln1+x2
xx2n−1
1−xdx=ln 2
2n+1
4n2−1
2nβ(2n+1 ) BI (137)(2)
4./integraldisplay∞
0ln1+x2
xx2n
1−xdx=−ln 2
2n−1
4n2+1
2nβ(2n+1 ) BI (137)(4)
5./integraldisplay∞
0ln1+x2
xx2n−1
1+x2dx=ln2
2n+1
4n2−1
2nβ(2n+1 ) BI (137)(10)
6./integraldisplay1
0ln1+x2
xx2ndx=1
2n+1/braceleftBigg
(−1)nπ
2+l n2 −1
2n+1+2n−1/summationdisplay
k=0(−1)k
2n−2k−1/bracerightBigg
BI (294)(8)
7./integraldisplay1
0ln1+x2
xx2n−1dx=1
2n/braceleftBigg
(−1)n+1ln 2 + ln2 −1
2n+(−1)n+1n−1/summationdisplay
k=1(−1)k
k/bracerightBigg
BI (294)(9)a
8./integraldisplay1
0ln1+x2
xdx
1+x2=π
2ln 2 BI (115)(7)
9./integraldisplay∞
0ln1+x2
xdx
1+x2=πln 2 BI (137)(8)
10./integraldisplay∞
0ln1+x2
xdx
1−x2=0 BI (137)(9)
11./integraldisplay1
0ln1−x2
xdx
1+x2=π
4ln 2 BI (115)(9)
12./integraldisplay∞
1ln1+x2
x+1dx
1+x2=3π
8ln2 BI (144)(8)
13./integraldisplay1
0ln1+x2
x+1dx
1+x2=3π
8ln2−G BI (115)(18)
14./integraldisplay∞
1ln1+x2
x−1dx
1+x2=3π
8ln2 +G BI (144)(9)
15./integraldisplay1
0ln1+x2
1−xdx
1+x2=3π
8ln2 BI (115)(19)
16./integraldisplay∞
0ln1+x2
x2xdx
1+x2=π2
12BI (138)(3)
17./integraldisplay∞
0lna2+b2x2
x2dx
c2+g2x2=π
cglnag+bc
c[a>0,b > 0,c > 0,g > 0]
BI (138)(6, 7, 9, 10)a
18./integraldisplay∞
0lna2+b2x2
x2dx
c2−g2x2=1
cgarctanag
bc[a>0,b>0,c>0,g>0]
BI (138)(8, 11)a
4.311 Logarithmic functions and powers 565
19./integraldisplay∞
0ln1+x2
x2x2dx
(1 +x2)2=π
4(ln4−1) BI (139)(21)
20./integraldisplay1
0ln2/parenleftbigg1−x2
x2/parenrightbigg/radicalbig
1−x2dx=π FI II 643a
21./integraldisplay1
0ln1+2xcost+x2
(1 +x)2dx
x=1
2/integraldisplay∞
0ln1+2xcost+x2
(1 +x)2dx
x=−t2
2
[|t|<π] BI (115)(23), BI (134)(15)
22./integraldisplay∞
0ln1+2xcost+x2
(1 +x)2xp−1dx=−2π(1−cospt)
psinpπ[0<|p|<1,|t|<π] BI (134)(17)
23./integraldisplay1
0ln1+x2sint
1−x2sintdx√
1−x2=πlncot/parenleftbiggπ−t
4/parenrightbigg
[|t|<π] GW (325)(21d)
4.299
1./integraldisplay∞
0ln(x+1 )/parenleftbig
x+a2/parenrightbig
(x+a)2dx
x=( l na)2[a>0] BI (134)(14)
2./integraldisplay1
0ln(1−ax)/parenleftbig
1+ax2/parenrightbig
(1−ax2)2dx
1+ax2=1
2√aarctan√aln(1 + a)
[a>0] BI (115)(25)
3./integraldisplay1
0ln/parenleftbig
1−a2x2/parenrightbig/parenleftbig
1+ax2/parenrightbig
(1−ax2)2dx
1+ax2=1√aarctan√aln(1 + a)
[a>0] BI (115)(26)
4./integraldisplay1
0ln(x+1 )/parenleftbig
x+a2/parenrightbig
(x+a)2xμ−1dx=π(aμ−1)2
μsinμπ[a>0,Reμ>0] BI (134)(16)
4.311
1.11/integraldisplay∞
0ln(1 + xn)
xndx=πcosec/parenleftbigπ
n/parenrightbig
n−1n=2,3,...
2./integraldisplay∞
0ln/parenleftbig
1+x3/parenrightbig dx
1−x+x2=2π√
3ln 3 LI (136)(8)
3./integraldisplay∞
0ln/parenleftbig
1+x3/parenrightbigdx
1+x3=π√
3ln 3−π2
9LI (136)(6)
4./integraldisplay∞
0ln/parenleftbig
1+x3/parenrightbigxdx
1+x3=π√
3ln 3 +π2
9LI (136)(7)
5./integraldisplay∞
0ln/parenleftbig
1+x3/parenrightbig1−x
1+x3dx=−2
9π2BI (136)(9)
6.8/integraldisplay∞
0/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−x
3
a3/vextendsingle/vextendsingle/vextendsingle/vextendsingledx
x3=−π√
3
6a2
566 Logarithmic Functions 4.312
4.312
1./integraldisplay∞
0ln1+x3
x3dx
1+x3=π√
3ln 3 +π2
9BI (138)(12)
2./integraldisplay∞
0ln1+x3
x3xdx
1+x3=π√
3ln 3−π2
9BI (138)(13)
4.313
1./integraldisplay∞
0lnxln/parenleftbig
1+a2x2/parenrightbigdx
x2=πa(1−lna)[ a>0] BI (134)(18)
2./integraldisplay∞
0ln/parenleftbig
1+c2x2/parenrightbig
ln/parenleftbig
a2+b2x2/parenrightbigdx
x2=2π/bracketleftbigg/parenleftbigg
c+b
a/parenrightbigg
ln(b+ac)−b
alnb−clnc/bracketrightbigg
[a>0,b > 0,c > 0]
BI (134)(20, 21)a
3./integraldisplay∞
0ln/parenleftbig
1+c2x2/parenrightbig
ln/parenleftbigg
a2+b2
x2/parenrightbiggdx
x2=2π/bracketleftbigga+bc
bln(a+bc)−a
blna−c/bracketrightbigg
[a>0,a+bc >0] BI (134)(22, 23)a
4./integraldisplay∞
0lnxln1+a2x2
1+b2x2dx
x2=π(a−b)+πlnbb
aa[a>0,b > 0] BI (134)(24)
5./integraldisplay∞
0lnxlna2+2bx+x2
a2−2bx+x2dx
x=2πlnaarcsinb
a[a≥|b|] BI (134)(25)
6./integraldisplay∞
0ln(1 + x)xlnx−x−a
(x+a)2dx
x=(lna)2
2(a−1)[a>0] BI (141)(7)
7./integraldisplay∞
0ln2(1−x)xlnx−x−a
(x+a)2dx
x=π2+( l na)2
1+a[a>0] LI (141)(8)
4.314
1.11/integraldisplay1
0ln(1 + ax)xp−1−xq−1
lnxdx=∞/summationdisplay
k=1(−1)k+1ak
klnp+k
q+k
[|a|<1,p > 0,q > 0] BI (123)(18)
2./integraldisplay∞
0/bracketleftbigg(q−1)x
(1 +x)2−1
x+1+1
(1 +x)q/bracketrightbiggdx
xln(1 + x)=l nΓ ( q)
[q>0] BI (143)(7)
3./integraldisplay1
0xlnx+1−x
x(lnx)2ln(1 + x)dx=l n4
πBI (126)(12)
4./integraldisplay1
0ln/parenleftbig
1−x2/parenrightbig
dx
x/parenleftBig
q2+( l nx)2/parenrightBig=−π
qln Γ/parenleftbiggq+π
π/parenrightbigg
+π
2qln2q+l nq
π−1
[q>0] LI (327)(12)a
4.317 Logarithmic functions and powers 567
4.315
1./integraldisplay1
0ln(1 + x)(lnx)n−1dx
x=(−1)n−1(n−1)!/parenleftbigg
1−1
2n/parenrightbigg
ζ(n+1 ) BI (116)(3)
2./integraldisplay1
0ln(1 + x)(lnx)2ndx
x=22n+1−1
(2n+ 1)(2 n+2 )π2n+2|B2n+2| BI (116)(1)
3./integraldisplay1
0ln(1−x)(lnx)n−1dx
x=(−1)n(n−1)!ζ(n+1 ) BI (116)(4)
4./integraldisplay1
0ln(1−x)(lnx)2ndx
x=−22n
(n+ 1)(2 n+1 )π2n+2|B2n+2| BI (116)(2)
4.316
1./integraldisplay1
0ln (1−axr)/parenleftbigg
ln1
x/parenrightbiggpdx
x=−1
rp+1Γ(p+1 )∞/summationdisplay
k=1ak
kp+2
[p>−1,a < 1,r > 0] BI (116)(7)
2./integraldisplay1
0ln/parenleftbig
1−2axcost+a2x2/parenrightbig/parenleftbigg
ln1
x/parenrightbiggpdx
x=−2Γ(p+1 )∞/summationdisplay
k=1akcoskt
kp+2LI (116)(8)
4.317
1./integraldisplay∞
0ln√
1+x2+a√
1+x2−adx√
1+x2=πarcsin a [|a|<1] BI (142)(11)
2./integraldisplay1
0ln√
1−a2x2−x√
1−a2
1−xdx
x=1
2(arcsin a)2BI (115)(32)
3./integraldisplay1
0ln1 + cos t√
1−x2
1−cost√
1−x2dx
x2+t a n2v=πcottcosv−t
2
sinv+t
2BI (115)(30)
4./integraldisplay1
0ln2/parenleftBigg
x+√
1−x2
x−√
1−x2/parenrightBigg
xdx
1−x2=π2
2BI (115)(31)
5./integraldisplay1
0ln/braceleftBig√
1+kx+√
1−kx/bracerightBigdx/radicalbig
(1−x2)(1−k2x2)=1
4ln(4k)K(k)+π
8K(k/prime) BI (121)(8)
6./integraldisplay1
0ln/braceleftBig√
1+kx−√
1−kx/bracerightBigdx/radicalbig
(1−x2)(1−k2x2)=1
4ln(4k)K(k)+3
8πK(k/prime) BI (121)(9)
7./integraldisplay1
0ln/braceleftBig
1+/radicalbig
1−k2x2/bracerightBigdx/radicalbig
(1−x2)( 1−k2x2)=1
2lnkK(k)+π
4K(k/prime) BI (121)(6)
8./integraldisplay1
0ln/braceleftBig
1−/radicalbig
1−k2x2/bracerightBigdx/radicalbig
(1−x2)( 1−k2x2)=1
2lnkK(k)−3
4πK(k/prime) BI (121)(7)
9./integraldisplay1
0ln1+p√
1−x2
1−p√
1−x2dx
1−x=πarcsin p/bracketleftbig
p2<1/bracketrightbig
BI (115)(29)
568 Logarithmic Functions 4.318
10./integraldisplay1
0ln1+q√
1−k2x2
1−q√
1−k2x2dx/radicalbig
(1−x2)( 1−k2x2)=πF(arcsin q,k/prime)
/bracketleftbig
q2<1/bracketrightbig
BI (122)(15)
11.10/integraldisplay∞
−∞ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+2√
1+x2
1−2√
1+x2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledx
√
1+x2=π2
3
4.318
1./integraldisplay1
0ln (1−xq)
1+( l n x)2dx
x=π/bracketleftbigg
lnΓ/parenleftBigq
2π+1/parenrightBig
−lnq
2+q
2π/parenleftBig
lnq
2π−1/parenrightBig/bracketrightbigg
[q>0] BI (126)(11)
2./integraldisplay∞
0ln(1 + xr)/bracketleftBigg
(p−r)xp−(q−r)xq
lnx+xq−xp
(lnx)2/bracketrightBigg
dx
xr+1=rln/parenleftBig
tanqπ
2rcotpπ
2r/parenrightBig
[p<r , q<r ] BI (143)(9)
In integrals containing ln ( a+bxr), it is useful to make the substitution xr=tand then to seek the
resulting integral in the tables. For example,/integraldisplay∞
0xp−1ln (1 + xr)dx=1
r/integraldisplay∞
0tp
r−1ln(1 + t)dt=π
psinpπ
r(see4.293 3)
4.319
1./integraldisplay∞
0ln/parenleftbig
1−e−2aπx/parenrightbigdx
1+x2=−π/bracketleftbigg1
2ln 2aπ+a(lna−1)−lnΓ(a+1 )/bracketrightbigg
[a>0] BI (354)(6)
2./integraldisplay∞
0ln/parenleftbig
1+e−2aπx/parenrightbigdx
1+x2=π/bracketleftbigg
ln Γ(2 a)−lnΓ(a)+a(1−lna)−/parenleftbigg
2a−1
2/parenrightbigg
ln 2/bracketrightbigg
[a>0] BI (354)(7)
3./integraldisplay∞
0lna+be−px
a+be−qxdx
x=l na
a+blnp
q/bracketleftbiggb
a>−1,p q > 0/bracketrightbigg
FI II 635, BI (354)(1)
4.321
1./integraldisplay∞
−∞xln cosh xdx=0 BI (358)(2)a
2./integraldisplay∞
−∞lncosh xdx
1−x2=0 BI (138)(20)a
4.322
1.11/integraldisplayπ
0xln sinxdx=1
2/integraldisplayπ
0xln cos2xdx=−π2
2ln2 BI (432)(1, 2) FI II 643
2./integraldisplay∞
0lnsin2ax
b2+x2dx=π
bln1−e−2ab
2[a>0,b > 0] GW (338)(28b)
4.324 Logarithmic functions and powers 569
3./integraldisplay∞
0lncos2ax
b2+x2dx=π
bln1+e−2ab
2[a>0,b > 0] GW (338)(28a)
4./integraldisplay∞
0lnsin2ax
b2−x2dx=−π2
2b+aπ [a>0,b > 0] BI (418)(1)
5.11/integraldisplay∞
0lncos2ax
b2−x2dx=∞ BI (418)(2)
6./integraldisplay∞
0lncos2x
x2dx=−π FI II 686
7.7/integraldisplayπ/4
0lnsinxxμ−1dx=−1
2μ/parenleftBigπ
4/parenrightBigμ/bracketleftBigg
ln 2 +2
μ−∞/summationdisplay
k=1ζ(2k)
42k−1(μ+2k)/bracketrightBigg
[Reμ>0] LI (425)(1)
8.7/integraldisplayπ/2
0lnsinxxμ−1dx=−1
μ/parenleftBigπ
2/parenrightBigμ/bracketleftBigg
1
μ−2∞/summationdisplay
k=1ζ(2k)
4k(μ+2k)/bracketrightBigg
[Reμ>0] LI (430)(1)
9./integraldisplayπ/2
0ln(1−cosx)xμ−1dx=−1
μ/parenleftBigπ
2/parenrightBigμ/bracketleftBigg
2
μ−∞/summationdisplay
k=1ζ(2k)
42k−1(μ+2k)/bracketrightBigg
[Reμ>0] LI (430)(2)
10./integraldisplay∞
0ln/parenleftbig
1±2pcosβx+p2/parenrightbigdx
q2+x2=π
qln/parenleftbig
1±pe−βq/parenrightbig/bracketleftbig
p2<1/bracketrightbig
=π
qln/parenleftbig
p±e−βq/parenrightbig/bracketleftbig
p2>1/bracketrightbig
FI II 718a
4.323
1.11/integraldisplayπ
0xln tan2xdx=0 BI (432)(3)
2./integraldisplay∞
0lntan2ax
b2+x2dx=π
blntanh ab [a>0,b > 0] GW (338)(28c)
3./integraldisplay∞
0ln/parenleftbigg1+t a n x
1−tanx/parenrightbigg2dx
x=π2
2GW (338)(26)
4.324
1./integraldisplay∞
0ln/parenleftbigg1+s i n x
1−sinx/parenrightbigg2dx
x=π2GW (338)(25)
2./integraldisplay∞
0ln1+2acospx+a2
1+2acosqx+a2dx
x=l n ( 1+ a)lnq2
p2[−1<a≤1]
=l n/parenleftbigg
1+1
a/parenrightbigg
lnq2
p2[a<−1o ra≥1]
GW (338)(27)
570 Logarithmic Functions 4.325
3./integraldisplay∞
0ln/parenleftbig
a2sin2px+b2cos2px/parenrightbigdx
c2+x2=π
c[ln(asinhcp+bcoshcp)−cp]
[a>0,b > 0,c > 0,p > 0]
GW (338)(29)
4.325
1.3/integraldisplay1
0ln ln/parenleftbigg1
x/parenrightbiggdx
1+x=−Cln2 +∞/summationdisplay
k=2(−1)klnk
k=−Cln2 + 0 .159868905 ···=−1
2(ln 2)2
GW (325)(25a)
2./integraldisplay1
0ln ln/parenleftbigg1
x/parenrightbiggdx
x+eiλ=∞/summationdisplay
k=1(−1)k
ke−ikλ(C+l nk) GW (325)(26)
3./integraldisplay1
0ln ln/parenleftbigg1
x/parenrightbiggdx
(1 +x)2=/integraldisplay∞
1lnlnxdx
(1 +x)2=1
2/bracketleftbigg
ψ/parenleftbigg1
2/parenrightbigg
+l n2 π/bracketrightbigg
=1
2/parenleftBig
lnπ
2−C/parenrightBig
BI (147)(7)
4./integraldisplay1
0ln ln/parenleftbigg1
x/parenrightbiggdx
1+x2=/integraldisplay∞
1lnlnxdx
1+x2=π
2ln√
2πΓ/parenleftbig3
4/parenrightbig
Γ/parenleftbig1
4/parenrightbig BI (148)(1)
5./integraldisplay1
0ln ln/parenleftbigg1
x/parenrightbiggdx
1+x+x2=/integraldisplay∞
1ln lnxdx
1+x+x2=π√
3ln3√
2πΓ/parenleftbig2
3/parenrightbig
Γ/parenleftbig1
3/parenrightbig BI (148)(2)
6./integraldisplay1
0ln ln/parenleftbigg1
x/parenrightbiggdx
1−x+x2=/integraldisplay∞
1ln lnxdx
1−x+x2=2π√
3/bracketleftbigg5
6ln 2π−ln Γ/parenleftbigg1
6/parenrightbigg/bracketrightbigg
BI (148)(5)
7./integraldisplay1
0ln ln/parenleftbigg1
x/parenrightbiggdx
1+2xcost+x2=/integraldisplay∞
1ln lnxdx
1+2xcost+x2=π
2s intln(2π)t/πΓ/parenleftbigg1
2+t
2π/parenrightbigg
Γ/parenleftbigg1
2−t
2π/parenrightbigg
BI (147)(9)
8./integraldisplay1
0ln ln1
xxμ−1dx=−1
μ(C+l nμ)[ R e μ>0] BI (147)(1)
9./integraldisplay∞
1lnlnxxn−2dx
1+x2+x4+···+x2n−2
=π
2ntanπ
2nln 2π+π
nn−1/summationdisplay
k=1(−1)k−1sinkπ
nlnΓ/parenleftbiggn+k
2n/parenrightbigg
Γ/parenleftbiggk
2n/parenrightbigg [nis even]
=π
2ntanπ
2nlnπ+π
nn−1
2/summationdisplay
k=1(−1)k−1sinkπ
nlnΓ/parenleftbiggn−k
n/parenrightbigg
Γ/parenleftbiggk
n/parenrightbigg [nis odd]
BI (148)(4)
4.331 Logarithms and exponentials 571
10./integraldisplay1
0ln ln/parenleftbigg1
x/parenrightbiggdx
(1 +x2)/radicalbigg
ln1
x=/integraldisplay∞
1lnlnxdx
(1 +x2)√
lnx
=√π∞/summationdisplay
k=0(−1)k+1
√
2k+1[ln(2k+1 )+2l n2+ C]
BI (147)(4)
11./integraldisplay1
0ln ln/parenleftbigg1
x/parenrightbiggxμ−1dx/radicalbigg
ln1
x=−(C+l n4 μ)/radicalbiggπ
μ[Reμ>0] BI (147)(3)
12./integraldisplay1
0ln ln/parenleftbigg1
x/parenrightbigg/parenleftbigg
ln1
x/parenrightbiggμ−1
xν−1dx=1
νμΓ(μ)[ψ(μ)−ln(ν)]
[Reμ>0,Reν>0] BI (147)(2)
4.326
1./integraldisplay1
0ln (a−lnx)xμ−1dx=1
μ[lna−eaμEi(−aμ)] [Re μ>0,a > 0] BI (107)(23)
2./integraldisplay1
e
0ln/parenleftbigg
2ln1
x−1/parenrightbiggx2μ−1
lnxdx=−1
2[Ei(−μ)]2[Reμ>0] BI (145)(5)
4.327
1./integraldisplay1
0ln/bracketleftBig
a2+( l nx)2/bracketrightBigdx
1+x2=πln2Γ/parenleftbig2a+3π
4π/parenrightbig
Γ/parenleftbig2a+π
4π/parenrightbig+π
2lnπ
2
/bracketleftBig
a>−π
2/bracketrightBig
BI (147)(10)
2./integraldisplay1
0ln/bracketleftBig
a2+4( l n x)2/bracketrightBigdx
1+x2=πln2Γ/parenleftbiga+3π
4π/parenrightbig
Γ/parenleftbiga+π
4π/parenrightbig+π
2lnπ
[a>−π] BI (147)(16)a
3./integraldisplay∞
0ln/bracketleftBig
a2+( l nx)2/bracketrightBig
xμ−1dx=2
μ[−cosaμci(aμ)−sinaμsi(aμ)+l n a]
[a>0,Reμ>0] GW (325)(28)
If the integrand contains a logarithm whose argument also contains a logarithm, for example, if the
integrand contains lnln1
x, it is useful to make the substitution ln x=tand then seek the transformed
integral in the tables.
4.33–4.34 Combinations of logarithms and exponentials
4.331
1./integraldisplay∞
0e−μxlnxdx=−1
μ(C+l nμ)[ R e μ>0] BI (256)(2)
2./integraldisplay∞
1e−μxlnxdx=−1
μEi(−μ)[ R e μ>0] BI (260)(5)
572 Logarithmic Functions 4.332
3./integraldisplay1
0eμxlnxdx=−1
μ/integraldisplay1
0eμx−1
xdx [μ/negationslash=0 ] GW (324)(81a)
4.332
1./integraldisplay∞
0lnxdx
ex+e−x−1=2π√
3/bracketleftbigg5
6ln2π−lnΓ/parenleftbigg1
6/parenrightbigg/bracketrightbigg
(cf.4.325 6) BI (257)(6)
2./integraldisplay∞
0lnxdx
ex+e−x+1=π√
3ln/bracketleftBigg
Γ/parenleftbig2
3/parenrightbig
Γ/parenleftbig1
3/parenrightbig√
2π/bracketrightBigg
(cf.4.325 5) BI (257)(7)a, LI (260)(3)
4.333/integraldisplay∞
0e−μx2lnxdx=−1
4(C+l n4 μ)/radicalbiggπ
μ[Reμ>0] BI (256)(8), FI II 807a
4.334/integraldisplay∞
0lnxdx
ex2+1+ e−x2=1
2/radicalbiggπ
3∞/summationdisplay
k=1(−1)kC+l n4 k√
ksinkπ
3BI (357)(13)
4.335
1./integraldisplay∞
0e−μx(lnx)2dx=1
μ/bracketleftbiggπ2
6+(C+l nμ)2/bracketrightbigg
[Reμ>0] ET I 149(13)
2./integraldisplay∞
0e−x2(lnx)2dx=√π
8/bracketleftbigg
(C+ 2ln 2)2+π2
2/bracketrightbigg
FI II 808
3.7/integraldisplay∞
0e−μx(lnx)3dx=−1
μ/bracketleftbigg
(C+l nμ)3+π2
2(C+l nμ)−ψ/prime/prime(1)/bracketrightbigg
MI 26
4.336
1.7PV/integraldisplay∞
0e−x
lnxdx=−0.154479567 BI (260)(9)
2./integraldisplay∞
0e−μxdx
π2+( l nx)2=ν/prime(μ)−eμ[Reμ>0] MI 26
4.337
1./integraldisplay∞
0e−μxln(β+x)dx=1
μ/bracketleftbig
lnβ−eμβEi(−βμ)/bracketrightbig
[|argβ|<π , Reμ>0] BI (256)(3)
2./integraldisplay∞
0e−μxln(1 + βx)dx=−1
μeμ
βEi/parenleftbigg
−μ
β/parenrightbigg
[|argβ|<π , Reμ>0] ET I 148(4)
3./integraldisplay∞
0e−μxln|a−x|dx=1
μ/bracketleftbig
lna−e−aμEi(aμ)/bracketrightbig
[a>0,Reμ>0] BI (256)(4)
4.7/integraldisplay∞
0e−μxln/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ
β−x/vextendsingle/vextendsingle/vextendsingle/vextendsingledx=1
μ/bracketleftbig
e−βμEi(βμ)/bracketrightbig
[Reμ>0] MI 26
5.∗/integraldisplay∞
0ln(1 + ax)xζe−xdx=ζ/summationdisplay
μ=0ζ!
(ζ−μ)!/bracketleftBigg
(−1)ζ−μ−1
aζ−μe1/aEi/parenleftbigg
−1
a/parenrightbigg
+ζ−μ/summationdisplay
k=1(k−1)!/parenleftbigg
−1
a/parenrightbiggζ−μ−k/bracketrightBigg
4.338
1./integraldisplay∞
0e−μxln/parenleftbig
β2+x2/parenrightbig
dx=2
μ[lnβ−ci(βμ)cos(βμ)−si(βμ)sin(βμ)]
[Reβ>0,Reμ>0] BI (256)(6)
4.352 Logarithms, exponentials, and powers 573
2./integraldisplay∞
0e−μxln2/parenleftbig
x2−β2/parenrightbig
dx=2
μ/bracketleftbig
ln2β−eβμEi(−βμ)−eβμEi(βμ)/bracketrightbig
[Imβ>0,Reμ>0] BI (256)(5)
4.339/integraldisplay∞
0e−μxln/vextendsingle/vextendsingle/vextendsingle/vextendsinglex+1
x−1/vextendsingle/vextendsingle/vextendsingle/vextendsingledx=1
μ/bracketleftbig
e−μ(ln2μ+γ)−eμEi(−2μ)/bracketrightbig
[Reμ>0] MI 27
4.341/integraldisplay∞
0e−μxln√x+ai+√x−ai√
2adx=π
4μ[H0(aμ)−Y0(aμ)]
[a>0,Reμ>0] ET I 149(20)
4.342
1./integraldisplay∞
0e−2nxln (sinh x)dx=1
2n/bracketleftbigg1
n+l n2 −2β(2n+1 )/bracketrightbigg
BI (256)(17)
2./integraldisplay∞
0e−μxln (cosh x)dx=1
μ/bracketleftbigg
β/parenleftBigμ
2/parenrightBig
−1
μ/bracketrightbigg
[Reμ>0] ET I 165(32)
3.11/integraldisplay∞
0e−μx[ln(sinh x)−lnx]dx=1
μ/bracketleftbigg
lnμ
2−1
μ−ψ/parenleftBigμ
2/parenrightBig/bracketrightbigg
[Reμ>0] ET I 165(33)
4.343/integraldisplayπ
0eμcosx/bracketleftbig
ln/parenleftbig
2μsin2x/parenrightbig
+C/bracketrightbig
dx=−πK0(μ) WA 95(16)
4.35–4.36 Combinations of logarithms, exponentials, and powers
4.351
1./integraldisplay1
0(1−x)e−xlnxdx=1−e
eBI (352)(1)
2./integraldisplay1
0eμx/parenleftbig
μx2+2x/parenrightbig
lnxdx=1
μ2[(1−μ)eμ−1] BI (352)(2)
3./integraldisplay∞
1e−μxlnx
1+xdx=1
2eμ[Ei(−μ)]2[Reμ>0] NT 32(10)
4.352
1./integraldisplay∞
0xν−1e−μxlnxdx=1
μνΓ(ν)[ψ(ν)−lnμ][ R e μ>0,Reν>0]
BI (353)(3), ET I 315(10)a
2./integraldisplay∞
0xne−μxlnxdx=n!
μn+1/bracketleftbigg
1+1
2+1
3+···+1
n−C−lnμ/bracketrightbigg
[Reμ>0] ET I 148(7)
3./integraldisplay∞
0xn−1
2e−μxlnxdx=√π(2n−1)!!
2nμn+1
2/bracketleftbigg
2/parenleftbigg
1+1
3+1
5+···+1
2n−1/parenrightbigg
−C−ln4μ/bracketrightbigg
[Reμ>0] ET I 148(10)
574 Logarithmic Functions 4.353
4./integraldisplay∞
0xμ−1e−xlnxdx=Γ/prime(μ)[ R e μ>0] GW (324)(83a)
4.353
1./integraldisplay∞
0(x−ν)xν−1e−xlnxdx=Γ (ν)[ R e ν>0] GW (324)(84)
2./integraldisplay∞
0/parenleftbigg
μx−n−1
2/parenrightbigg
xn−1
2e−μxlnxdx=(2n−1)!!
(2μ)n/radicalbiggπ
μ
[Reμ>0] BI (357)(2)
3./integraldisplay1
0(μx+n+1 )xneμxlnxdx=eμn/summationdisplay
k=0(−1)k−1 n!
(n−k)!μk+1+(−1)nn!
μn+1
[μ/negationslash=0 ] GW (324)(82)
4.354
1.6/integraldisplay∞
0xν−1lnx
ex+1dx=Γ (ν)∞/summationdisplay
k=1(−1)k−1
kν[ψ(ν)−lnk][ R e ν>0]
=−1
2(ln 2)2[forν=1 ]
GW (324)(86a)
2.7/integraldisplay∞
0xν−1lnx
(ex+1 )2dx=Γ (ν)∞/summationdisplay
k=2(−1)k(k−1)
kν[ψ(ν)−lnk]
[Reν>1] GW (324)(86b)
3./integraldisplay∞
0(x−ν)ex−ν
(ex+1 )2xν−1lnxdx=Γ (ν)∞/summationdisplay
k=1(−1)k−1
kν[Reν>0] GW (324)(87a)
4./integraldisplay∞
0(x−2n)ex−2n
(ex+1 )2x2n−1lnxdx=22n−1−1
2nπ2n|B2n|
[n=1,2,...] GW (324)(87b)
5./integraldisplay∞
0xν−1lnx
(ex+1 )ndx=(−1)nΓ(ν)
(n−1)!∞/summationdisplay
k=n(−1)k(k−1)!
(k−n)!kν[ψ(ν)−lnk]
[Reν>0] GW (324)(86c)
4.355
1./integraldisplay∞
0x2e−μx2lnxdx=1
8μ(2−ln4μ−C)/radicalbiggπ
μ[Reμ>0] BI (357)(1)a
2./integraldisplay∞
0x/parenleftbig
μx2−νx−1/parenrightbig
e−μx2+2νxlnxdx=1
4μ+ν
4μ/radicalbiggπ
μexp/parenleftbiggν2
μ/parenrightbigg/bracketleftbigg
1+Φ/parenleftbiggν√μ/parenrightbigg/bracketrightbigg
[Reμ>0] BI (358)(1)
3./integraldisplay∞
0/parenleftbig
μx2−n/parenrightbig
x2n−1e−μx2lnxdx=(n−1)!
4μn[Reμ>0] BI (353)(4)
4.356 Logarithms, exponentials, and powers 575
4./integraldisplay∞
0/parenleftbig
2μx2−2n−1/parenrightbig
x2ne−μx2lnxdx=(2n−1)!!
2(2μ)n/radicalbiggπ
μ
[Reμ>0] BI (353)(5)
4.356
1./integraldisplay∞
0exp/bracketleftBig
−μ/parenleftBigx
a+a
x/parenrightBig/bracketrightBig
lnxdx
x=2l n aK0(2μ)[ a>0,Reμ>0] GW (324)(91)
2./integraldisplay∞
0exp/parenleftbigg
−ax−b
x/parenrightbigg
lnx/bracketleftbig
2ax2−(2n+1 )x−2b/bracketrightbig
xn−1
2dx
=2/parenleftbiggb
a/parenrightbiggn
2/radicalbiggπ
ae−2√
ab∞/summationdisplay
k=0(n+k)!
(n−k)!(2k)!!/parenleftBig
2√
ab/parenrightBigk
[a>0,b > 0] BI (357)(4)
3./integraldisplay∞
0exp/parenleftbigg
−ax−b
x/parenrightbigg
lnx/bracketleftbig
2ax2+( 2n−1)x−2b/bracketrightbigdx
xn+3
2
=2/parenleftBiga
b/parenrightBign
2/radicalbiggπ
ae−2√
ab∞/summationdisplay
k=0(n+k−1)!
(n−k−1)!(2k)!!/parenleftBig
2√
ab/parenrightBigk
[a>0,b > 0] BI (357)(11)
Forn=1
2:
4./integraldisplay∞
0exp/parenleftBig
−ax−a
x/parenrightBig
lnxax2−b
x2dx=2K0/parenleftBig
2√
ab/parenrightBig
[a>0,b > 0] GW (324)(92c)
Forn=0 :
5./integraldisplay∞
0exp/parenleftbigg
−ax−b
x/parenrightbigg
lnx2ax2−x−2b
x√xdx=2/radicalbiggπ
ae−2√
ab
[a>0,b > 0]
BI (357)(7), GW(324)(92a)
Forn=−1:
6./integraldisplay∞
0exp/parenleftbigg
−ax−b
x/parenrightbigg
lnx2ax2−3x−2b√xdx=1+2√
ab
a/radicalbiggπ
ae−2√
ab
[a>0,b > 0]
LI (357)(6), GW (324)(92b)
7.9/integraldisplay∞
0exp/parenleftbigg
−ax−b
x/parenrightbigg
lnx/parenleftbigg
a−b
x2/parenrightbigg
dx=K0/parenleftBig
2√
ab/parenrightBig
[a>0,b > 0]
576 Logarithmic Functions 4.357
8.9/integraldisplay∞
0exp/parenleftbigg
−ax−b
x/parenrightbigg
lnx/bracketleftbig
2ax2−(2n+1 )x−2b/bracketrightbig
xn−3
2dx
=4/parenleftbiggb
a/parenrightbigg(2n+1)/4
Kn+1
2/parenleftBig
2√
ab/parenrightBig
=2/parenleftbiggb
a/parenrightbiggn
2/radicalbiggπ
ae−2√
abn/summationdisplay
k=0(n+k)!
(n−k)!(2k)!!/parenleftBig
2√
ab/parenrightBigk
[n=0,1,...,a> 0,b > 0]
9.9/integraldisplay∞
0exp/parenleftbigg
−ax−b
x/parenrightbigg
ln/bracketleftbig/parenleftbig
ax2−b/parenrightbig
cos (αlnx)+αxsin (αlnx)/bracketrightbigdx
x2
= 2cos/parenleftBig
αln/radicalbig
b/a/parenrightBig
Kiα/parenleftBig
2√
ab/parenrightBig
[a>0,b > 0,−∞<α< ∞]
10.9/integraldisplay∞
0exp/parenleftbigg
−ax−b
x/parenrightbigg
lnx/bracketleftbig/parenleftbig
ax2−b/parenrightbig
sin (αlnx)−αxcos (αlnx)/bracketrightbigdx
x2
=2s i n/parenleftBig
αln/radicalbig
b/a/parenrightBig
Kiα/parenleftBig
2√
ab/parenrightBig
[a>0,b > 0,−∞<α< ∞]
11.9q/integraldisplay∞
0xαlnx/bracketleftbigg
a−α
x−b
x2/bracketrightbigg
exp/parenleftbigg
−ax−b
x/parenrightbigg
dx=2/parenleftbiggb
a/parenrightbiggα/2
Kα/parenleftBig
2√
ab/parenrightBig
[a>0,b > 0,−∞<α< ∞]
4.357
1./integraldisplay∞
0exp/parenleftbigg
−1+x4
2ax2/parenrightbigg
lnx1+ax2−x4
x2dx=−√
2a3π
2a√e
[a>0] BI (357)(8)
2./integraldisplay∞
0exp/parenleftbigg
−1+x4
2ax2/parenrightbigg
lnxx4+ax2−1
x4dx=√
2a3π
2a√e[a>0] BI (357)(9)
3./integraldisplay∞
0exp/parenleftbigg
−1+x4
2ax2/parenrightbigg
lnxx4+3ax−1
x6dx=(1 +a)√
2a3π
2a√e
[a>0] BI (357)(10)
4.358
1.6/integraldisplay∞
1xν−1e−μx(lnx)mdx=∂m
∂νm/braceleftbig
μ−νΓ(ν,μ)/bracerightbig
[m=0,1,..., Reμ>0,Reν>0]
MI 26
2./integraldisplay∞
0xν−1e−μx(lnx)2dx=Γ(ν)
μν/braceleftBig
[ψ(ν)−lnμ]2+ζ(2,ν)/bracerightBig
[Reμ>0,Reν>0] MI 26
3.9/integraldisplay∞
0xν−1e−μx(lnx)3dx=Γ(ν)
μν/braceleftBig
[ψ(ν)−lnμ]3+3ζ(2,ν)[ψ(ν)−lnμ]−2ζ(3,ν)/bracerightBig
[Reμ>0,Reν>0] MI 26
4.364 Logarithms, exponentials, and powers 577
4.7/integraldisplay∞
0xν−1e−μx(lnx)4dx=Γ(ν)
ν/braceleftbigg
[ψ(ν)−lnμ]4+6ζ(2,ν)[ψ(ν)−lnμ]2
−8ζ(3,ν)[ψ(ν)−lnμ]+3[ζ(2,ν)]2+6ζ(4,ν)/bracerightbigg
[Reμ>0,Reν>0]
5.3/integraldisplay∞
0xν−1e−μx(lnx)ndx=∂n
∂νn/braceleftbig
μ−νΓ(ν)/bracerightbig
[n=0,1,2,...]
4.359
1./integraldisplay∞
0e−μxxp−1−xq−1
lnxdx=1
μ[λ(μ, p−1)−λ(μ, q−1)]
[Reμ>0,p > 0,q > 0] MI 27
2.11/integraldisplay1
0eμxxp−1−xq−1
lnxdx=∞/summationdisplay
k=0μk
k!lnp+k
q+k[p>0,q > 0] BI (352)(9)
4.361
1./integraldisplay∞
0(x+1 )e−μx
π2+( l nx)2dx=ν/prime(μ)−ν/prime/prime(μ)[ R e μ>0] MI 27
2./integraldisplay∞
0e−μxdx
x/bracketleftBig
π2+( l nx)2/bracketrightBig=eμ−ν(μ)[ R e μ>0] MI 27
4.362
1./integraldisplay1
0xexln(1−x)dx=1−e BI (352)(5)a
2./integraldisplay∞
1e−μxln(2x−1)dx
x=1
2/bracketleftBig
Ei/parenleftBig
−μ
2/parenrightBig/bracketrightBig2
[Reμ>0] ET I 148(8)
4.363
1./integraldisplay∞
0e−μxln(a+x)μ(x+a)ln (x+a)−2
x+adx
=1
4/integraldisplay∞
0e−μxln2(a−x)μ(x−a)ln2(x−a)−4
x−adx=( l na)2
[Reμ>0,a > 0] BI (354)(4, 5)
2./integraldisplay1
0x(1−x)(2−x)e−(1−x)2ln(1−x)dx=1−e
4eBI (352)(4)
4.364
1./integraldisplay∞
0e−μxln[(x+a)(x+b)]dx
x+a+b=e(a+b)μ{Ei(−aμ)Ei(−bμ)−ln(ab)Ei [−(a+b)μ]}
[a>0,b > 0,Reμ>0]BI (354)(11)
578 Logarithmic Functions 4.365
2./integraldisplay∞
0e−μxln(x+a+b)/parenleftbigg1
x+a+1
x+b/parenrightbigg
dx
=( 1+l n alnb)ln (a+b)+e−(a+b)μ{Ei(−αμ)Ei(−bμ)}
+(1−ln(ab))Ei[−(a+b)μ]
[a>0,b > 0,Reμ>0]BI (354)(12)
4.365/integraldisplay∞
0/bracketleftbigg
e−x−x
(1 +x)p+1ln(1 + x)/bracketrightbiggdx
x=l np [p>0] BI (354)(15)
4.366
1./integraldisplay∞
0e−μxln/parenleftbigg
1+x2
a2/parenrightbiggdx
x= [ci(aμ)]2+[ s i (aμ)]2[Reμ>0] NT 32(11)a
2./integraldisplay∞
0e−μxln/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−x
2
a2/vextendsingle/vextendsingle/vextendsingle/vextendsingledx
x=E i (aμ)Ei(−aμ)[ R e μ>0] ME 18
3./integraldisplay∞
0xe−μx2ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+x
2
1−x2/vextendsingle/vextendsingle/vextendsingle/vextendsingledx=1
μ[coshμsinh(iμ)−sinhμcosh(iμ)]
[Reμ>0] ; (cf. 4.339 ) MI 27
4.367/integraldisplay∞
0xe−μx2lnx+/radicalbig
x2+2β√2βdx=eβμ
4μK0(βμ)[ |argβ|<π , Reμ>0] ET I 149(19)
4.368/integraldisplay2u
0e−μx2lnx2/parenleftbig
4u2−x2/parenrightbig
u4dx√
4u2−x2=π
2e−2u2μ/bracketleftBigπ
2Y0/parenleftbig
2iu2μ/parenrightbig
−(C−ln 2)J0/parenleftbig
2iu2μ/parenrightbig/bracketrightBig
[Reμ>0] ET I 149(21)a
4.369
1./integraldisplay∞
0xν−1e−μx[ψ(ν)−lnx]dx=Γ(ν)lnμ
μν[Reν>0] ET I 149(12)
2./integraldisplay∞
0xne−μx/braceleftBig/bracketleftbig
lnx−1
2ψ(n+1 )/bracketrightbig2−1
2ψ/prime(n+1 )/bracerightBig
dx
=n!
μn+1/braceleftBigg/bracketleftbigg
lnμ−1
2ψ(n+1 )/bracketrightbigg2
+1
2ψ/prime(n+1 )/bracerightBigg
[Reμ>0] MI 26
4.37 Combinations of logarithms and hyperbolic functions
4.371
1./integraldisplay∞
0lnx
coshxdx=πln/bracketleftBigg√
2πΓ/parenleftbig3
4/parenrightbig
Γ/parenleftbig1
4/parenrightbig/bracketrightBigg
LI (260)(1)a
2./integraldisplay∞
0lnxdx
coshx+c o s t=π
sintln(2π)t/πΓ/parenleftbiggπ+t
2π/parenrightbigg
Γ/parenleftbiggπ−t
2π/parenrightbigg/bracketleftbig
t2<π2/bracketrightbig
BI (257)(7)a
4.374 Logarithms and hyperbolic functions 579
3./integraldisplay∞
0lnxdx
cosh2x=ψ/parenleftbigg1
2/parenrightbigg
+l nπ=l nπ−2l n2−C BI (257)(4)a
4.372
1./integraldisplay∞
1lnxsinhmx
sinhnxdx=π
2ntanmπ
2nln 2π+π
nn−1/summationdisplay
k=1(−1)k−1sinkmπ
nlnΓ/parenleftbign+k
2n/parenrightbig
Γ/parenleftbigk
2n/parenrightbig[m+nis odd]
=π
2ntanmπ
2nlnπ+π
nn−1
2/summationdisplay
k=1(−1)k−1sinkmπ
nlnΓ/parenleftbign−k
n/parenrightbig
Γ/parenleftbigk
n/parenrightbig [m+nis even]
BI (148)(3)a
2./integraldisplay∞
1lnxcoshmx
coshnxdx
=π
2nln2π
cosmπ
2n+π
nn/summationdisplay
k=1(−1)k−1cos(2k−1)mπ
2nlnΓ/parenleftbig2n+2k−1
4n/parenrightbig
Γ/parenleftbig2k−1
4n/parenrightbig [m+nis odd]
=π
2nlnπ
cosmπ
2n+π
nn−1
2/summationdisplay
k=1(−1)k−1cos(2k−1)mπ
2nlnΓ/parenleftbig2n−2k+1
2n/parenrightbig
Γ/parenleftbig2k−1
2n/parenrightbig [m+nis even]
BI (148)(6)a
4.373
1./integraldisplay∞
0ln/parenleftbig
a2+x2/parenrightbig
coshbxdx=π
b/bracketleftBigg
2ln2Γ/parenleftbig2ab+3π
4π/parenrightbig
Γ/parenleftbig2ab+π
4π/parenrightbig−ln2b
π/bracketrightBigg/bracketleftBig
b>0,a > −π
2b/bracketrightBig
. BI (258)(11)a
2./integraldisplay∞
0ln/parenleftbig
1+x2/parenrightbigdx
coshπx
2=2l n4
πBI (258)(1)a
3./integraldisplay∞
0ln/parenleftbig
a2+x2/parenrightbigsinh/parenleftbig2
3πx/parenrightbig
sinhπxdx=2s i nπ
3ln6Γ/parenleftbiga+4
6/parenrightbig
Γ/parenleftbiga+5
6/parenrightbig
Γ/parenleftbiga+1
6/parenrightbig
Γ/parenleftbiga+2
6/parenrightbig
[a>−1]. BI (258)(12)
4./integraldisplay∞
0ln/parenleftbig
1+x2/parenrightbigdx
sinh2ax=2
a/bracketleftbigg
lna
π+π
2a−ψ/parenleftbiggπ+a
π/parenrightbigg/bracketrightbigg
[a>0] BI (258)(5)
5./integraldisplay∞
0ln/parenleftbig
1+x2/parenrightbigcosh/parenleftBigπ
2x/parenrightBig
sinh2/parenleftBigπ
2x/parenrightBigdx=2π−4
πBI (258)(3)
6./integraldisplay∞
0ln/parenleftbig
1+x2/parenrightbigcosh/parenleftBigπ
4x/parenrightBig
sinh2/parenleftBigπ
4x/parenrightBigdx=4√
2−16
π+8√
2
πln/parenleftBig√
2+1/parenrightBig
BI (258)(2)
4.374
1./integraldisplay∞
0ln/parenleftbig
cos2t+e−2xsin2t/parenrightbigdx
sinhx=−2t2BI (259)(10)a
580 Logarithmic Functions 4.375
2./integraldisplay∞
0ln/parenleftbig
a+be−2x/parenrightbigdx
cosh2x=2
(b−a)/bracketleftbigga+b
2ln(a+b)−alna−bln 2/bracketrightbigg
[a>0,a+b>0] LI (259)(14)
4.375
1.11/integraldisplay∞
0lncoshx
2dx
coshx=G−π
4ln 2 BI (259)(11)
2./integraldisplay∞
0lncoth xdx
coshx=π
2ln2 BI (259)(16)
4.376
1./integraldisplay∞
0lnx√xcoshxdx=2√π∞/summationdisplay
k=0(−1)k+1
√
2k+1{ln(2k+1 )+2l n2+ C} BI (147)(4)
2./integraldisplay∞
0lnx(μ+1 )c o s h x−xsinhx
cosh2xxμdx=2Γ ( μ+1 )∞/summationdisplay
k=0(−1)k+1
(2k+1 )μ+1
[Reμ>−1] BI (356)(10)
3./integraldisplay∞
0lnx(n+1 )c o s h x−xsinhx
cosh2xxndx=(−1)n
2nβ(n)/parenleftbigg1
2/parenrightbigg
4./integraldisplay∞
0ln2xnsinh 2ax−ax
sinh2axx2n−1dx=−1
n/parenleftBigπ
a/parenrightBig2n
|B2n|
[n=1,2,...] BI (356)(9)a
5./integraldisplay∞
0lnxaxcoshax−(2n+1 )s i n h ax
sinh2axx2ndx=222n+1−1
(2a)2n+1(2n)!ζ(2n+1 ) BI (356)(14)
6./integraldisplay∞
0lnxaxcoshax−2nsinhax
sinh2axx2n−1dx=22n−1−1
2n|B2n|/parenleftBigπ
a/parenrightBig2n
[n=1,2,...,a> 0] BI (356)(15)
7./integraldisplay∞
0ln(2n+1 )c o s h ax−axsinhax
cosh2axx2ndx=−/parenleftBigπ
2a/parenrightBig2n+1
|E2n|
[a>0] BI (356)(11)
8.6/integraldisplay∞
0lnx2axsinhax−(2n+1 )c o s h ax
cosh3axx2ndx=⎧
⎪⎪⎨
⎪⎪⎩2
a/parenleftbig
22n−1−1/parenrightbig/parenleftBigπ
2a/parenrightBig2n
|B2n|n=1,2,...
1
an=0
[a>0] BI (356)(2)
9.6/integraldisplay∞
0lnx2axcoshax−(2n+1 )s i n h ax
sinh3axx2ndx=1
a/parenleftBigπ
a/parenrightBig2n
|B2n|
[a>0,n=1,2,...] BI (356)(6)a
4.382 Logarithms and trigonometric functions 581
10./integraldisplay∞
0lnxxsinhx−6s in h2/parenleftBigx
2/parenrightBig
−6c os2t
2
(coshx+c o s t)2x2dx=/parenleftbig
π−t2/parenrightbig
t
3s int
[0<t<π ] BI (356)(16)a
11./integraldisplay∞
0ln/parenleftbig
1+x2/parenrightbigcoshπx+πxsinhπx
cosh2πxdx
x2=4−π BI (356)(12)
12./integraldisplay∞
0ln/parenleftbig
1+4x2/parenrightbigcoshπx+πxsinhπx
cosh2πxdx
x2=4l n2 BI (356)(13)
4.377/integraldisplay∞
0ln2xax−n/parenleftbig
1−e−2ax/parenrightbig
sinh2axx2n−1dx=1
2n/parenleftBigπ
a/parenrightBig2n
|B2n|
[n=1,2,...] LI (356)(8)a
4.38–4.41 Logarithms and trigonometric functions
4.381
1./integraldisplay1
0lnxsinaxdx =−1
a[C+l na−ci(a)] [ a>0] GW (338)(2a)
2./integraldisplay1
0lnxcosaxdx =−1
a/bracketleftBig
si(a)+π
2/bracketrightBig
[a>0] BI (284)(2)
3./integraldisplay2π
0lnxsinnxdx =−1
n[C+l n ( 2 nπ)−ci(2nπ)] GW (338)(1a)
4./integraldisplay2π
0lnxcosnxdx =−1
n/bracketleftBig
si(2nπ)+π
2/bracketrightBig
GW (338)(1b)
4.382
1./integraldisplay∞
0ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglex+a
x−a/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinbxdx =π
bsinab [a<0,b > 0] ET I 77(11)
2.10/integraldisplay∞
0ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglex+a
x−a/vextendsingle/vextendsingle/vextendsingle/vextendsinglecosbxdx =2
b/bracketleftBig
cos(ab)/braceleftBig
si(ab)+π
2/bracerightBig
−sin(ab)ci (ab)/bracketrightBig
[a>0,b > 0] ET I 18(9)
3./integraldisplay∞
0lna2+x2
b2+x2coscxdx =π
c/parenleftbig
e−bc−e−ac/parenrightbig
[a>0,b > 0,c > 0]
FI III 648a, BI (337)(5)
4./integraldisplay∞
0lnx2+x+a2
x2−x+a2sinbxdx =2π
bexp/parenleftBigg
−b/radicalbigg
a2−1
4/parenrightBigg
sinb
2
[b>0] ET I 77(12)
5./integraldisplay∞
0ln(x+β)2+γ2
(x−β)2+γ2sinbxdx =2π
be−γbsinβb [Reγ>0,|Imβ|≤Reγ, b > 0]
ET I 77(13)
582 Logarithmic Functions 4.383
4.383
1./integraldisplay∞
0ln/parenleftbig
1+e−βx/parenrightbig
cosbxdx =β
2b2−π
2bsinh/parenleftbiggπb
β/parenrightbigg [Reβ>0,b > 0] ET I 18(13)
2./integraldisplay∞
0ln/parenleftbig
1−e−βx/parenrightbig
cosbxdx =β
2b2−π
2bcoth/parenleftbiggπb
β/parenrightbigg
[Reβ>0,b > 0] ET I 18(14)
4.384
1./integraldisplay1
0ln (sin πx)s i n2nπxdx =0 GW (338)(3a)
2.7/integraldisplay1
0ln (sin πx)s i n ( 2 n+1 )πxdx =2/integraldisplay1/2
0ln(sin πx)s i n ( 2 n+1 )πxdx
=2
(2n+1 )π/bracketleftBigg
ln2−1
2n+1−2n/summationdisplay
k=11
2k−1/bracketrightBigg
GW (338)(3b)
3.6/integraldisplay1
0ln (sin πx)c o s2 nπxdx =2/integraldisplay1/2
0ln(sin πx)c o s2 nπxdx
=−ln 2 [ n=0 ]
=−1
2n[n>0]
GW (338)(3c)
4./integraldisplay1
0ln (sin πx)c o s ( 2 n+1 )πxdx =0 GW (338)(3d)
5./integraldisplayπ/2
0lnsinxsinxdx=l n2 −1 BI (305)(4)
6./integraldisplayπ/2
0lnsinxcosxdx=−1 BI (305)(5)
7./integraldisplayπ/2
0lnsinxcos 2nxdx =⎧
⎪⎨
⎪⎩−π
4n, forn>0
−π
2ln 2,forn=0LI (305)(6)
8./integraldisplayπ
0ln sinxcos[2m(x−n)]dx=−πcos2mn
2mLI (330)(8)
9./integraldisplayπ/2
0lnsinxsin2xdx=π
8(1−ln 4) BI (305)(7)
10./integraldisplayπ/2
0lnsinxcos2xdx=−π
8(1 + ln 4) BI (305)(8)
11./integraldisplayπ/2
0lnsinxsinxcos2xdx=1
9(ln 8−4) BI (305)(9)
12./integraldisplayπ/2
0lnsinxtanxdx=−π2
24BI (305)(11)
4.384 Logarithms and trigonometric functions 583
13./integraldisplayπ/2
0lnsin 2 xsinxdx=/integraldisplayπ/2
0ln sin 2 xcosxdx=2( l n2 −1) BI (305)(16, 17)
14./integraldisplayπ
0ln (1 + pcosx)
cosxdx=πarcsin p/bracketleftbig
p2<1/bracketrightbig
FI II 484
15./integraldisplayπ
0ln sinxdx
1−2acosx+a2=π
1−a2ln1−a2
2/bracketleftbig
a2<1/bracketrightbig
=π
a2−1lna2−1
2a2/bracketleftbig
a2>1/bracketrightbig
BI (331)(8)
16./integraldisplayπ
0ln sinbxdx
1−2acosx+a2=π
1−a2ln1−a2b
2/bracketleftbig
a2<1/bracketrightbig
BI (331)(10)
17./integraldisplayπ
0ln cos bxdx
1−2acosx+a2=π
1−a2ln1+a2b
2/bracketleftbig
a2<1/bracketrightbig
BI (331)(11)
18./integraldisplayπ/2
0lnsinxdx
1−2acos2x+a2=1
2/integraldisplayπ
0lnsinxdx
1−2acos2x+a2
=π
2(1−a2)ln1−a
2/bracketleftbig
a2<1/bracketrightbig
=π
2(a2−1)lna−1
2a/bracketleftbig
a2>1/bracketrightbig
BI (321)(1), BI (331)(13)
19./integraldisplayπ
0ln sinbxdx
1−2acos2x+a2=π
1−a2ln1−ab
2/bracketleftbig
a2<1/bracketrightbig
BI (331)(18)
20./integraldisplayπ
0ln cos bxdx
1−2acos2x+a2=π
1−a2ln1+ab
2/bracketleftbig
a2<1/bracketrightbig
BI (331)(21)
21./integraldisplayπ/2
0lncos xdx
1−2pcos 2x+p2=π
2( 1−p2)ln1+p
2/bracketleftbig
p2<1/bracketrightbig
=π
2(p2−1)lnp+1
2p/bracketleftbig
p2>1/bracketrightbig
BI (321)(8)
22./integraldisplayπ
0ln sinxcosxdx
1−2acosx+a2=π
2a1+a2
1−a2ln/parenleftbig
1−a2/parenrightbig
−aπln2
1−a2/bracketleftbig
a2<1/bracketrightbig
=π
2aa2+1
a2−1lna2−1
a2−πln2
a(a2−1)/bracketleftbig
a2>1/bracketrightbig
LI (331)(9)
23./integraldisplayπ
0ln sinbxcosxdx
1−2acos2x+a2=/integraldisplayπ
0lncos bxcosxdx
1−2acos2x+a2=0
[0<a< 1] BI (331)(19, 22)
24./integraldisplayπ
0ln sinxcos2xdx
1−2acos2x+a2=π
4a1+a
1−aln(1−a)−πln 2
2(1−a)[0<a< 1]
=π
4aa+1
a−1lna−1
a−πln2
2a(a−1)[a>1]
BI (331)(16)
584 Logarithmic Functions 4.385
25./integraldisplayπ/2
0lnsinxcos 2xdx
1−2acos2x+a2=1
2/integraldisplayπ
0lnsinxcos 2xdx
1−2acos2x+a2
=π
2a(1−a2)/braceleftbigg1+a2
2ln(1−a)−a2ln 2/bracerightbigg/bracketleftbig
a2<1/bracketrightbig
=π
2a(a2−1)/braceleftbigg1+a2
2lna−1
a−ln2/bracerightbigg/bracketleftbig
a2>1/bracketrightbig
BI (321)(2), BI (331)(15), LI (321))(2)
26./integraldisplayπ/2
0lncos xcos2xdx
1−2acos2x+a2=π
2a(1−a2)/braceleftbigg1+a2
2ln(1 + a)−a2ln2/bracerightbigg/bracketleftbig
a2<1/bracketrightbig
=π
2a(a2−1)/braceleftbigg1+a2
2ln1+a
a−ln 2/bracerightbigg/bracketleftbig
a2>1/bracketrightbig
BI (321)(9)
4.385
1./integraldisplayπ
0ln sinxdx
a+bcosx=π√
a2−b2ln√
a2−b2
a+√
a2−b2[a>0,a > b ] BI (331)(6)
2./integraldisplayπ/2
0lnsinxdx
(asinx±bcosx)2=/integraldisplayπ/2
0lncos xdx
(acosx±bsinx)2
=1
b(a2+b2)/parenleftbigg
∓alna
b−bπ
2/parenrightbigg
[a>0,b > 0] BI (319)(1,6)a
3./integraldisplayπ/2
0lnsinxdx
a2sin2x+b2cos2x=/integraldisplayπ/2
0lncos xdx
b2sin2x+a2cos2x=π
2ablnb
a+b
[a>0,b > 0] BI (317)(4, 10)
4./integraldisplayπ/2
0lnsinxsin 2xdx
/parenleftbig
asin2x+bcos2x/parenrightbig2=/integraldisplayπ/2
0lncos xsin 2xdx
/parenleftbig
bsin2x+acos2x/parenrightbig2
=1
2b(b−a)lna
b
[a>0,b > 0]BI (319)(3, 7), LI (319)(3)
5./integraldisplayπ/2
0lnsinxa2sin2x−b2cos2x
/parenleftbig
a2sin2x+b2cos2x/parenrightbig2dx=/integraldisplayπ/2
0ln cos xa2cos2x−b2sin2x
/parenleftbig
a2cos2x+b2sin2x/parenrightbig2dx
=π
2b(a+b)
[a>0,b > 0] LI (319)(2, 8)
4.386
1./integraldisplayπ/2
0lnsinxsinx/radicalbig
1+s i n2xdx=/integraldisplayπ/2
0cosxln cos x√
1 + cos2xdx=−π
8ln 2 BI (322)(1, 6)
2./integraldisplayπ/2
0sin3xln sinx/radicalbig
1+s i n2xdx=/integraldisplayπ/2
0cos3xln cos x√
1 + cos2xdx=ln 2−1
4BI (322)(2, 7)
4.387 Logarithms and trigonometric functions 585
3./integraldisplayπ/2
0lnsinxdx/radicalbig
1−k2sin2x=−1
2K(k)lnk−π
4K(k/prime) BI (322)(3)
4./integraldisplayπ/2
0ln cos xdx/radicalbig
1−k2sin2x=1
2K(k)lnk/prime
k−π
4K(k/prime) BI (322)(9)
4.387
1./integraldisplayπ/2
0lnsinxsinμxcosνxdx=/integraldisplayπ/2
0ln cos xcosμxsinνxdx
=1
4B/parenleftbiggμ+1
2,ν+1
2/parenrightbigg/bracketleftbigg
ψ/parenleftbiggμ+1
2/parenrightbigg
−ψ/parenleftbiggμ+ν+2
2/parenrightbigg/bracketrightbigg
[Reμ>−1,Reν>−1]GW (338)(6c)
2./integraldisplayπ/2
0lnsinxsinμ−1xdx=√πΓ/parenleftBigμ
2/parenrightBig
4Γ/parenleftbiggμ+1
2/parenrightbigg/bracketleftbigg
ψ/parenleftBigμ
2/parenrightBig
−ψ/parenleftbiggμ+1
2/parenrightbigg/bracketrightbigg
[Reμ>0] GW (338)(6a)
3./integraldisplayπ/2
0lnsinxcosν−1xdx=√πΓ/parenleftBigν
2/parenrightBig
4Γ/parenleftbiggν+1
2/parenrightbigg/bracketleftbigg
ψ/parenleftBigν
2/parenrightBig
−ψ/parenleftbiggν+1
2/parenrightbigg/bracketrightbigg
[Reν>0] GW (338)(6b)
4./integraldisplayπ/2
0lnsinxsin2nxdx=(2n−1)!!
(2n)!!π
2/braceleftBigg2n/summationdisplay
k=1(−1)k+1
k−ln 2/bracerightBigg
FI II 811
5./integraldisplayπ/2
0lnsinxsin2n+1xdx=(2n)!!
(2n+1 ) ! !/braceleftBigg2n+1/summationdisplay
k=1(−1)k
k+l n2/bracerightBigg
BI (305)(13)
6./integraldisplayπ/2
0lnsinxcos2nxdx=−(2n−1)!!
(2n)!!π
4/bracketleftBiggn/summationdisplay
k=11
k+l n4/bracketrightBigg
=−(2n−1)!!
(2n)!!π
4[C+ψ(n+ 1) + ln 4]
BI (305)(14)
7./integraldisplayπ/2
0lnsinxcos2n+1xdx=−(2n)!!
(2n+1 ) ! !n/summationdisplay
k=01
2k+1
=−(2n)!!
2(2n+1 ) ! !/bracketleftbigg
ψ/parenleftbigg
n+3
2/parenrightbigg
−ψ/parenleftbigg1
2/parenrightbigg/bracketrightbigg
GW (338)(7b)
8./integraldisplayπ/2
0lncos xsin2nxdx=−(2n−1)!!
2n+1·n!π
2{C+2l n2+ ψ(n+1 )}
BI (306)(8)
586 Logarithmic Functions 4.388
9./integraldisplayπ/2
0lncos xcos2nxdx=−(2n−1)!!
2nn!π
2/parenleftBigg
ln2 +2n/summationdisplay
k=1(−1)k
k/parenrightBigg
BI (306)(10)
10./integraldisplayπ/2
0lncos xcos2nxdx=2n−1(n−1)!
(2n−1)!!/bracketleftBigg
ln2 +2n−1/summationdisplay
k=1(−1)k
k/bracketrightBigg
BI (306)(9)
4.388
1./integraldisplayπ/4
0lnsinxsin2nx
cos2n+2xdx=1
2n+1/bracketleftBigg
1
2ln 2 + ( −1)nπ
4+n−1/summationdisplay
k=0(−1)k
2n−2k−1/bracketrightBigg
BI (288)(1)
2./integraldisplayπ/4
0lnsinxsin2n−1x
cos2n+1xdx=1
4n/bracketleftBigg
−ln 2 + ( −1)nln 2 +n−1/summationdisplay
k=1(−1)k
n−k/bracketrightBigg
LI (288)(2)
3./integraldisplayπ/4
0lncos xsin2nx
cos2n+2xdx=1
2n+1/bracketleftBigg
−1
2ln 2 + ( −1)n+1π
4+n/summationdisplay
k=0(−1)k−1
2n−2k+1/bracketrightBigg
BI (288)(10)
4./integraldisplayπ/4
0lncos xsin2n−1x
cos2n+1xdx=1
4n/bracketleftBigg
−ln 2 + ( −1)nln2 +n−1/summationdisplay
k=0(−1)k
n−k/bracketrightBigg
BI (288)(11)
5./integraldisplayπ/2
0lnsinxsinp−1x
cosp+1xdx=−π
2pcosecpπ
2[0<p< 2] BI (310)(4)
6./integraldisplayπ/2
0lnsinxdx
tanp−1xsin 2x=1
4π
p−1secpπ
2/bracketleftbig
p2<1/bracketrightbig
BI (310)(3)
4.389
1./integraldisplayπ
0ln sinxsin2n2xcos2xdx=−(2n−1)!!
(2n)!!π
4n+2BI (330)(9)
2./integraldisplayπ/4
0lnsinxcosn2xsin 2xdx=−1
4(n+1 ){C+ψ(n+2 )+l n2 } BI (285)(2)
3./integraldisplayπ/4
0lncos xcosμ−12xtan 2xdx=1
4(1−μ)β(μ)
[Reμ>0] BI (286)(2)
4./integraldisplayπ/2
0lnsinxsinμ−1xcosxdx=/integraldisplayπ/2
0ln cos xcosμ−1xsinxdx=−1
μ2
[Reμ>0] BI (306)(11)
5.3/integraldisplayπ
2
−π
2ln cos xcospxcospxdx =π
2p+1[C+ψ(p+1 )−2ln2]
[p>−1]
6./integraldisplayπ/2
0lncos xcosp−1xsinpxsinxdx=π
2p+2/bracketleftbigg
C+ψ(p)−1
p−2ln2/bracketrightbigg
[p>0] BI (306)(12)
4.394 Logarithms and trigonometric functions 587
4.391
1./integraldisplayπ/4
0(lncos 2 x)ncosp−12xtanxdx=/integraldisplayπ/4
0(ln sin 2 x)nsinp−12xtan/parenleftBigπ
4−x/parenrightBig
dx=1
2β(n)(p)
[p>0] BI (286)(10), BI (285)(18)
2./integraldisplayπ/4
0(lnsin 2 x)nsinp−12xtan/parenleftBigπ
4+x/parenrightBig
dx=(−1)nn!
2ζ(n+1,p) BI (285)(17)
3./integraldisplayπ/4
0(lncos 2 x)2n−1tanxdx=1−22n−1
4nπ2n|B2n| [n=1,2,...] BI (286)(7)
4./integraldisplayπ/4
0(lncos 2 x)2ntanxdx=22n1
22n+1(2n)!ζ(2n+1 ) BI (286)(8)
4.392
1./integraldisplayπ/4
0ln(sin xcosx)sin2nx
cos2n+2xdx=1
2n+1/bracketleftBigg
(−1)n+1π
2−ln 2 +1
2n+1+2n−1/summationdisplay
k=0(−1)k−1
2n−2k−1/bracketrightBigg
BI (294)(8)
2./integraldisplayπ/4
0ln(sin xcosx)sin2n−1x
cos2n+1xdx=1
2n/bracketleftBigg
(−1)nln 2−ln2 +1
2n+(−1)nn−1/summationdisplay
k=1(−1)k
k/bracketrightBigg
BI (294)(9)
4.393
1./integraldisplayπ/2
0lntan xsinxdx=l n2 BI (307)(3)
2./integraldisplayπ/2
0lntan xcosxdx=−ln 2 BI (307)(4)
3./integraldisplayπ/2
0lntan xsin2xdx=−/integraldisplayπ/2
0ln tan xcos2xdx=π
4BI (307)(5, 6)
4./integraldisplayπ/4
0lntan x
cos 2xdx=−π2
8GW (338)(10b)a
5./integraldisplayπ/2
0sinxln cotx
2dx=l n2 LO III 290
4.394
1./integraldisplayπ/2
0lntan xdx
1−2acos2x+a2=π
2(1−a2)ln1−a
1+a/bracketleftbig
a2<1/bracketrightbig
=π
2(a2−1)lna−1
a+1/bracketleftbig
a2>1/bracketrightbig
BI (321)(15)
2./integraldisplayπ/2
0lntan xcos 2xdx
1−2acos2x+a2=π
4a1+a2
1−a2ln1−a
1+a/bracketleftbig
a2<1/bracketrightbig
=π
4aa2+1
a2−1lna−1
a+1/bracketleftbig
a2>1/bracketrightbig
BI (321)(16)
588 Logarithmic Functions 4.395
3./integraldisplayπ
0lntan bxdx
1−2acos2x+a2=π
1−a2ln1−ab
1+ab[0<a< 1,b > 0] BI (331)(24)
4./integraldisplayπ
0lntan bxcosxdx
1−2acos2x+a2=0 [ 0 <a< 1] BI (331)(25)
5./integraldisplayπ/4
0lntan xcos 2xdx
1−asin2x=−arcsin a
4a(π+a r c s i n a)/bracketleftbig
a2≤1/bracketrightbig
BI (291)(2,3)
6./integraldisplayπ/4
0lntan xcos2xdx
1−a2sin22x=−π
4aarcsin a/bracketleftbig
a2<1/bracketrightbig
BI (291)(9)
7./integraldisplayπ/4
0lntan xcos2xdx
1+a2sin22x=−π
4aarcsinh a=−π
4aln/parenleftBig
a+/radicalbig
1+a2/parenrightBig
/bracketleftbig
a2<1/bracketrightbig
BI (291)(10)
8./integraldisplayu
0sinxln cotx
2
1−cos2αsin2xdx= cosec2 α/braceleftBigπ
2ln2 + L(ϕ−α)−L(ϕ+α)−L/parenleftBigπ
2−2α/parenrightBig/bracerightBig
[tanϕ=c o t αcosu;0<u<π ]
LO III 290
9./integraldisplayπ/4
0ln tan xsin 2xdx
1−cos2tsin22x= cosec2 t/bracketleftBig
L/parenleftBigπ
2−t/parenrightBig
−/parenleftBigπ
2−t/parenrightBig
ln 2/bracketrightBig
LO III 290a
4.395
1./integraldisplayπ/2
0ln tan xdx/radicalbig
1−k2sin2x=−lnk/primeK(k) BI (322)(11)
2./integraldisplayπ/4
ulntan xsin 4xdx
/parenleftbig
sin2u+t a n2υsin22x/parenrightbig/radicalbig
sin22x−sin2u=−π
2cos2υ
sinusinυlnsinυ+√
1−cos2ucos2υ
sinu(1 + sin υ)
/bracketleftBig
0<u<π
2,0<υ<π
2/bracketrightBig
LO III 285a
4.396
1./integraldisplayπ/2
0ln(atanx)s i nμ−12xdx=2μ−2lna/braceleftBig
Γ/parenleftBiga
2/parenrightBig/bracerightBig2
Γ(a)[a>0,Reμ>0] LI (307)(8)
2./integraldisplayπ/2
0lntan xcos2(μ−1)xdx=−√π
4Γ/parenleftbig
u−1
2/parenrightbig
Γ(μ)/bracketleftbigg
C+ψ/parenleftbigg2μ−1
2/parenrightbigg
+l n4/bracketrightbigg
/bracketleftbig
Reμ>1
2/bracketrightbig
BI (307)(9)
3./integraldisplayπ/2
0lntan xcosq−1xcotxsin[(q+1 )x]dx=−π
2[C+ψ(q+1 ) ]
[q>−1] BI (307)(11)
4./integraldisplayπ/2
0lntan xcosq−1xcos[(q+1 )x]dx=−π
2q[q>0] BI (307)(10)
5./integraldisplayπ/4
0(lntan x)ntanpxdx=1
2n+1B(n)/parenleftbiggp+1
2/parenrightbigg
[p>−1] LI (286)(22)
4.397 Logarithms and trigonometric functions 589
6./integraldisplayπ/2
0(lntan x)2n−1dx
cos2x=1−22n
2nπ2n|B2n| [n=1,2,...] BI (312)(6)
7./integraldisplayπ/4
0lntan xtan2n+1xdx=(−1)n+1
4/bracketleftBigg
π2
12+n/summationdisplay
k=1(−1)k
k2/bracketrightBigg
GW (338)(8a)
4.397
1./integraldisplayπ/2
0ln(1 + psinx)dx
sinx=π2
8−1
2/parenleftbig
arccos p2/parenrightbig/bracketleftbig
p2<1/bracketrightbig
BI (313)(1)
2./integraldisplayπ/2
0ln(1 + pcosx)dx
cosx=π2
8−1
2(arccos p)2/bracketleftbig
p2<1/bracketrightbig
BI (313)(8)
3./integraldisplayπ
0ln (1 + pcosx)dx
cosx=πarcsin p/bracketleftbig
p2<1/bracketrightbig
BI (331)(1)
4./integraldisplayπ/2
0cosxln (1 + cos αcosx)
1−cos2αcos2xdx=L/parenleftBigπ
2−α/parenrightBig
−αlnsinα
sinαcosα/bracketleftBig
0<α<π
2/bracketrightBig
LO III 291
5./integraldisplayπ/2
0cosxln (1−cosαcosx)
1−cos2αcos2xdx=L/parenleftBigπ
2−α/parenrightBig
+(π−α)lns in α
sinαcosα/bracketleftBig
0<α<π
2/bracketrightBig
LO III 291
6./integraldisplayπ
0ln/parenleftbig
1−2acosx+a2/parenrightbig
cosnxdx
=1
2/integraldisplay2π
0ln/parenleftbig
1−2acosx+a2/parenrightbig
cosnxdx
=−π
nan/bracketleftbig
a2<1/bracketrightbig
BI (330)(11), BI (332)(5)
=−π
nan/bracketleftbig
a2>1/bracketrightbig
GW (338)(13a)
7./integraldisplayπ
0ln/parenleftbig
1−2acosx+a2/parenrightbig
sinnxsinxdx=1
2/integraldisplay2π
0ln/parenleftbig
1−2acosx+a2/parenrightbig
sinnxsinxdx
=π
2/parenleftbiggan+1
n+1−an−1
n−1/parenrightbigg
/bracketleftbig
a2>1/bracketrightbig
BI (330)(10), BI (332)(4)
8./integraldisplayπ
0ln/parenleftbig
1−2acosx+a2/parenrightbig
sinnxsinxdx=1
2/integraldisplay2π
0ln/parenleftbig
1−2acosx+a2/parenrightbig
cosnxcosxdx
=−π
2/parenleftbiggan+1
n+1+an−1
n−1/parenrightbigg
BI (330)(12), BI (332)(6)
590 Logarithmic Functions 4.398
9./integraldisplayπ
0ln/parenleftbig
1−2acos2x+a2/parenrightbig
cos(2n−1)xdx=0/bracketleftbig
a2<1/bracketrightbig
BI (330)(15)
10./integraldisplayπ
0ln/parenleftbig
1−2acos2x+a2/parenrightbig
sin2nxsinxdx=0/bracketleftbig
a2<1/bracketrightbig
BI (330)(13)
11./integraldisplayπ
0ln/parenleftbig
1−2acos2x+a2/parenrightbig
sin(2n−1)xsinxdx=π
2/parenleftbiggan
n−an−1
n−1/parenrightbigg
/bracketleftbig
a2<1/bracketrightbig
BI (330)(14)
12./integraldisplayπ
0ln/parenleftbig
1−2acos2x+a2/parenrightbig
cos2nxcosxdx=0/bracketleftbig
a2<1/bracketrightbig
BI (330)(16)
13./integraldisplayπ
0ln/parenleftbig
1−2acos2x+a2/parenrightbig
cos(2n−1)xcosxdx=−π
2/parenleftbiggan
n+an−1
n−1/parenrightbigg
/bracketleftbig
a2<1/bracketrightbig
BI (330)(17)
14./integraldisplayπ/2
0ln/parenleftbig
1+2acos2x+a2/parenrightbig
sin2xdx=−aπ
4/bracketleftbig
a2<1/bracketrightbig
=πlna2
4−π
4a/bracketleftbig
a2>1/bracketrightbig
BI (309)(22), LI (309)(22)
15./integraldisplayπ/2
0ln/parenleftbig
1+2acos2x+a2/parenrightbig
cos2xdx=aπ
4/bracketleftbig
a2<1/bracketrightbig
=πlna2
4+π
4a/bracketleftbig
a2>1/bracketrightbig
BI (309)(23), LI (309)(23)
16./integraldisplayπ
0ln/parenleftbig
1−2acosx+a2/parenrightbig
1−2bcosx+b2dx=2πln(1−ab)
1−b2/bracketleftbig
a2≤1,b2<1/bracketrightbig
BI (331)(26)
4.398
1./integraldisplayπ
0ln1+2acosx+a2
1−2acosx+a2sin(2n+1 )xdx=(−1)n2πa2n+1
2n+1/bracketleftbig
a2<1/bracketrightbig
BI (330)(18)
2./integraldisplay2π
0ln1−2acosx+a2
1−2acosnx+a2cosmxdx =2π/parenleftbiggn
mam/n−am
m/parenrightbigg/bracketleftbig
a2≤1/bracketrightbig
=2π/parenleftbiggn
ma−m/n−a−m
m/parenrightbigg/bracketleftbig
a2≥1/bracketrightbig
BI (332)(9)
3./integraldisplayπ
0ln1+2acos2x+a2
1+2acos2nx+a2cotxdx=0 BI (331)(5), LI(331)(5)
4.399
1./integraldisplayπ/2
0ln/parenleftbig
1+asin2x/parenrightbig
sin2xdx=π
2/parenleftbigg
ln1+√1+a
2−1
21−√1+a
1+√1+a/parenrightbigg
[a>−1] BI (309)(14)
4.413 Logarithms and trigonometric functions 591
2./integraldisplayπ/2
0ln/parenleftbig
1+asin2x/parenrightbig
cos2xdx=π
2/parenleftbigg
ln1+√1+a
2+1
21−√1+a
1+√1+a/parenrightbigg
[a>−1] BI (309)(15)
3./integraldisplayπ/2
0ln/parenleftbig
1−cos2βcos2x/parenrightbig
1−cos2αcos2xdx=−π
sinαln1+s i n α
sinα+s i nβ/bracketleftBig
0<β<π
2,0<α<π
2/bracketrightBig
LO III 285
4.411
1./integraldisplayπ
0ln1+s i n x
1 + cos λsinxdx
sinx=λ2/bracketleftbig
λ2<π2/bracketrightbig
BI (331)(2)
2./integraldisplayπ/2
0lnp+qsinax
p−qsinaxdx
sinax=/integraldisplayπ/2
0lnp+qcosax
p−qcosaxdx
cosax=/integraldisplayπ/2
0lnp+qtanax
p−qtanaxdx
tanax=πarcsinq
p
[p>q> 0]
FI II 695a, BI (315)(5, 13,17)a
3./integraldisplayπ/2
0cosx
1−cos2αcos2xln1 + cos βcosx
1−cosβcosxdx=2π
sin 2αlncosα−β
2
sinα+β
2/bracketleftBig
0<α≤β<π
2/bracketrightBig
LO III 284
4.412
1./integraldisplayπ/4
0lntan/parenleftBigπ
4±x/parenrightBigdx
sin 2x=±π2
8BI (293)(1)
2./integraldisplayπ/4
0lntan/parenleftBigπ
4±x/parenrightBigdx
tan 2x=±π2
16BI (293)(2)
3./integraldisplayπ/4
0lntan/parenleftBigπ
4±x/parenrightBig
(ln tan x)2ndx
sin2x=±22n+2−1
4(n+ 1)(2 n+1 )π2n+2|B2n+2| BI (294)(24)
4./integraldisplayπ/4
0lntan/parenleftBigπ
4±x/parenrightBig
(ln tan x)2n−1dx
sin 2x=±1−22n+1
22n+2n(2n)!ζ(2n+1 ) BI (294)(25)
5./integraldisplayπ/4
0lntan/parenleftBigπ
4±x/parenrightBig
(ln sin 2 x)n−1dx
tan 2x=(−1)n−1
2(n−1)!ζ(n+1 ) LI (294)(20)
4.413
1./integraldisplayπ/2
0ln/parenleftbig
p2+q2tan2x/parenrightbig dx
a2sin2x+b2cos2x=π
ablnap+bq
a
[a>0,b > 0,p > 0,q > 0]
BI (318)(1–4)a
2./integraldisplayπ/2
0ln/parenleftbig
1+q2tan2x/parenrightbig 1
p2sin2x+r2cos2xdx
s2sin2x+t2cos2x
=π
p2t2−s2r2/braceleftbiggp2−r2
prln/parenleftbigg
1+qr
p/parenrightbigg
+t2−s2
stln/parenleftbigg
1+qt
s/parenrightbigg/bracerightbigg
[q>0,p > 0,r > 0,s > 0,t > 0]BI (320)(18)
592 Logarithmic Functions 4.414
3./integraldisplayπ/2
0ln/parenleftbig
1+q2tan2x/parenrightbig sin2x
p2sin2x+r2cos2xdx
s2sin2x+t2cos2x
=π
p2t2−s2r2/braceleftbiggt
sln/parenleftbigg
1+qr
p/parenrightbigg
−r
pln/parenleftbigg
1+qt
s/parenrightbigg/bracerightbigg
[q>0,p > 0,r > 0,s > 0,t > 0]BI (320)(20)
4./integraldisplayπ/2
0ln/parenleftbig
1+q2tan2x/parenrightbig cos2x
p2sin2x+r2cos2xdx
s2sin2x+t2cos2x
=π
p2t2−s2r2/braceleftbiggp
rln/parenleftbigg
1+qr
p/parenrightbigg
−s
tln/parenleftbigg
1+qt
s/parenrightbigg/bracerightbigg
[q>0,p > 0,r > 0,s > 0,t > 0]BI (320)(21)
5./integraldisplayπ
0ln tan rxdx
1−2pcosx+p2=π
1−p2ln1−p2r
1+p2r/bracketleftbig
p2<1/bracketrightbig
BI (331)(12)
4.414
1./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbig dx/radicalbig
1−k2sin2x=l nk/primeK(k) BI (323)(1)
2./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbigsin2xdx/radicalbig
1−k2sin2x=1
k2/braceleftbig/parenleftbig
k2−2+l n k/prime/parenrightbig
K(k)+( 2 −lnk/prime)E(k)/bracerightbig
BI (323)(3)
3./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbigcos2x
dx/radicalbig
1−k2sin2x=1
k2/bracketleftBig/parenleftBig
1+k/prime2−k/prime2lnk/prime/parenrightBig
K(k)−(2−lnk/prime)E(k)/bracketrightBig
BI (323)(6)
4./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbig dx/radicalBig/parenleftbig
1−k2sin2x/parenrightbig3=1
k/prime2/bracketleftbig/parenleftbig
k2−2/parenrightbig
K(k)+( 2+l n k/prime)E(k)/bracketrightbig
BI (323)(9)
5./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbigsin2x
dx/radicalBig/parenleftbig
1−k2sin2x/parenrightbig3
=1
k2k/prime2/bracketleftBig
(2 + ln k/prime)E(k)−/parenleftBig
1+k/prime2+k/prime2lnk/prime/parenrightBig
K(k)/bracketrightBig
BI (323)(10)
6./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbig cos2xdx/radicalBig/parenleftbig
1−k2sin2x/parenrightbig3=1
k2/bracketleftBig/parenleftBig
1+k/prime2+l nk/prime/parenrightBig
K(k)−(2 + ln k/prime)E(k)/bracketrightBig
BI (323)(16)
7./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbig/radicalbig
1−k2sin2xd x=/parenleftBig
1+k/prime2/parenrightBig
K(k)−(2−lnk/prime)E(k) BI (324)(18)
4.416 Logarithms and trigonometric functions 593
8./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbig
sin2x/radicalbig
1−k2sin2xdx=1
9k2/braceleftbigg/parenleftBig
−2+1 1 k2−6k4+3k/prime2lnk/prime/parenrightBig
K(k)
+/bracketleftbig
2−10k2−3/parenleftbig
1−2k2/parenrightbig
lnk/prime/bracketrightbig
E(k)/bracerightbigg
BI (324)(20)
9./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbig
cos2x/radicalbig
1−k2sin2xd x=1
9k2/braceleftbigg/parenleftBig
2+7k2−3k4−3k/prime2lnk/prime/parenrightBig
K(k)
−/bracketleftbig
2+8k2−3/parenleftbig
1+k2/parenrightbig
lnk/prime/bracketrightbig
E(k)/bracerightbigg
BI (324)(21), LI (324)(21)
10./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbig sinxcosxdx/radicalBig/parenleftbig
1−k2sin2x/parenrightbig2n+1=2
(2n−1)2k2/braceleftBig
[1 + (2 n−1)lnk/prime]k/prime1−2n−1/bracerightBig
BI (324)(17)
4.415
1./integraldisplay∞
0lnxsinax2dx=−1
4/radicalbiggπ
2a/parenleftBig
ln4a+C−π
2/parenrightBig
[a>0] GW (338)(19)
2./integraldisplay∞
0lnxcosax2dx=−1
4/radicalbiggπ
2a/parenleftBig
ln 4a+C−π
2/parenrightBig
[a>0] GW (338)(19)
4.416
1./integraldisplayπ/2
0cosxln/parenleftBig
1+/radicalbig
sin2β−cos2βtan2αsin2x/parenrightBig
1−sin2αcos2xdx
= cosec2 α{(2α+2γ−π)lnc os β+2L(α)−2L(γ)+L(α+γ)−L(α−γ)}
/bracketleftbigg
cosγ=sinα
sinβ;0<α<β<π
2/bracketrightbigg
LO III 291
2./integraldisplayπ/2
0cosxln/parenleftBig
1−/radicalbig
sin2β−cos2βtan2αsin2x/parenrightBig
1−sin2αcos2xdx
= cosec2 α{(π+2α−2γ)lncos β+2L(α)+2L(γ)−L(α+γ)+L(α−γ)}
/bracketleftbigg
cosγ=sinα
sinβ;0<α<β<π
2/bracketrightbigg
LO III 291
3./integraldisplayπ/2
βln/parenleftBig
sinx+/radicalbig
sin2x−sin2β/parenrightBig
1−cos2αcos2xdx
=−cosecα/braceleftBigg
arctan/parenleftbiggtanβ
sinα/parenrightbigg
ln sinβ+π
2ln1+s i n α
sinα+/radicalbig
1−cos2αcos2β/bracerightBigg
/bracketleftBig
0<α<π , 0<β<π
2/bracketrightBig
LO III 285
4.7/integraldisplayπ/4
0lntan x(lncos2 x)n−1tan2xdx=1
2(−1)n(n−1)!/parenleftBig
1−2−(n+1)/parenrightBig
ζ(n+1 )
BI (287)(20)
594 Logarithmic Functions 4.421
4.42–4.43 Combinations of logarithms, trigonometric functions, and powers
4.421
1./integraldisplay∞
0lnxsinaxdx
x=−π
2(C+l na)[ a>0] FI II 810a
2./integraldisplay∞
0lnaxsinbxxdx
β2+x2=π
2e−bβ/primeln (aβ/prime)−π
4/bracketleftBig
ebβ/primeEi(−bβ/prime)+e−bβ/primeEi(bβ/prime)/bracketrightBig
[β/prime=βsignβ;a>0,b > 0]
ET I 76(5), NT 27(10)a
3./integraldisplay∞
0lnaxcosbxβ/primedx
β2+x2=π
2e−bβ/primeln(aβ/prime)+π
4/bracketleftBig
ebβ/primeEi(−bβ/prime)−e−bβ/primeEi(bβ/prime)/bracketrightBig
[β/prime=βsignβ;a>0,b > 0]
ET I 17(3), NT 27(11)a
4./integraldisplay∞
0lnaxsinbxxdx
x2−c2=π
2{−si(bc)sinbc+c o s bc[lnac−ci(bc)]}
[a>0,b > 0,c > 0] BI (422)(5)
5./integraldisplay∞
0lnaxcosbxdx
x2−c2=π
2c{sinbc[ci(bc)−lnac]−cosbcsi(bc)}
[a>0,b > 0,c > 0] BI (422)(6)
4.422
1./integraldisplay∞
0lnxsinaxxμ−1dx=Γ(μ)
aμsinμπ
2/bracketleftBig
ψ(μ)−lna+π
2cotμπ
2/bracketrightBig
[a>0,|Reμ|<1] BI (411)(5)
2./integraldisplay∞
0lnxcosaxxμ−1dx=Γ(μ)
aμcosμπ
2/bracketleftBig
ψ(μ)−lna−π
2tanμπ
2/bracketrightBig
[a>0,0<Reμ<1] BI (411)(6)
4.423
1./integraldisplay∞
0lnxcosax−cosbx
xdx=l na
b/parenleftbigg
C+1
2lnab/parenrightbigg
[a>0,b > 0] GW (338)(21a)
2./integraldisplay∞
0lnxcosax−cosbx
x2dx=π
2[(a−b)(C−1) +alna−blnb]
[a>0,b > 0] GW (338)(21b)
3./integraldisplay∞
0lnxsin2ax
x2dx=−aπ
2(C+l n2 a−1) [ a>0] GW (338)(20b)
4.424
1./integraldisplay∞
0(lnx)2sinaxdx
x=π
2C2+π3
24+πClna+π
2(lna)2
[a>0] ET I 77(9), FI II 810a
4.429 Logarithms, trigonometric functions, and powers 595
2.6/integraldisplay∞
0(lnx)2sinaxxμ−1dx=Γ(μ)
aμsinμπ
2/bracketleftBig
ψ/prime(μ)+ψ2(μ)+πψ(μ)cotμπ
2−2ψ(μ)lna
−πlnacotμπ
2+( l na)2−1
4π2/bracketrightbigg
[a>0,0<Reμ<1] ET I 77(10)
4.425
1./integraldisplay∞
0ln(1 + x)cosaxdx
x=1
2/braceleftBig
[si(a)]2+ [ci(a)]2/bracerightBig
[a>0] ET I 18(8)
2./integraldisplay∞
0ln2/parenleftbiggb+x
b−x/parenrightbigg
cosaxdx
x=−2πsi(ab)[ a≥0,b > 0] ET I 18(11)
3./integraldisplay∞
0ln/parenleftbig
1+b2x2/parenrightbig
sinaxdx
x=−πEi/parenleftBig
−a
b/parenrightBig
[a>0,b > 0]
GW (338)(24), ET I 77(14)
4./integraldisplay1
0ln/parenleftbig
1−x2/parenrightbig
cos(plnx)dx
x=1
2p2+π
2pcothpπ
2LI (309)(1)a
4.426
1.11/integraldisplay∞
0xlnb2+x2
c2+x2sinaxdx =π
a2/bracketleftbig
(1 +ac)e−ac−(1 +ab)e−ab/bracketrightbig
[b≥0,c≥0,a > 0] GW (338)(23)
2./integraldisplay∞
0lnb2x2+p2
c2x2+p2sinaxdx
x=π/bracketleftBig
Ei/parenleftBig
−ap
c/parenrightBig
−Ei/parenleftBig
−ap
b/parenrightBig/bracketrightBig
[b>0,c > 0,p > 0,a > 0]
ET I 77(15)
4.427/integraldisplay∞
0ln/parenleftBig
x+/radicalbig
β2+x2/parenrightBigsinax/radicalbig
β2+x2dx=π
2K0(aβ)+π
2ln(β)[I0(aβ)−L(aβ)]
[Reβ>0,a > 0] ET I 77(16)
4.428
1./integraldisplay∞
0lncos2axcosbx
x2dx=πbln 2−aπ [a>0,b > 0] ET I 22(29)
2./integraldisplay∞
0ln/parenleftbig
4c os2ax/parenrightbigcosbx
x2+c2dx=π
ccosh(bc)ln/parenleftbig
1+e−2ac/parenrightbig
/bracketleftBig
a<b< 2a<π
c/bracketrightBig
ET I 22(30)
3./integraldisplay∞
0lncos2axsinbx
x(1 +x2)dx=πln/parenleftbig
1+e−2a/parenrightbig
sinhb−πln2/parenleftbig
1−e−b/parenrightbig
[a>0,b > 0] ET I 82(36)
4./integraldisplay∞
0lncos2axcosbx
x2(1 +x2)dx=−πln/parenleftbig
1+e−2a/parenrightbig
coshb+/parenleftbig
b+e−b/parenrightbig
πln2−aπ
[a>0,b > 0] ET I 22(31)
4.429/integraldisplay1
0(1 +x)x
lnxsin (ln x)dx=π
4BI (326)(2)a
596 Logarithmic Functions 4.431
4.431
1./integraldisplay∞
0ln(2±2c osx)sinbx
x2+c2xdx=−πsinh(bc)ln(1±e−c)
[b>0,c > 0] ET I 22(32)
2./integraldisplay∞
0ln(2±2c osx)cosbx
x2+c2dx=π
ccosh(bc)ln/parenleftbig
1±e−c/parenrightbig
[b>0,c > 0] ET I 22(32)
3./integraldisplay∞
0ln/parenleftbig
1+2acosx+a2/parenrightbigsinbx
xdx=−π
2[b]/summationdisplay
k=1(−a)k
k[1 + sign( b−k)]
[0<a< 1,b > 0] ET I 82(25)
4./integraldisplay∞
0ln/parenleftbig
1−2acosx+a2/parenrightbigcosbx
x2+c2dx=π
cln/parenleftbig
1−ae−c/parenrightbig
cosh(bc)+π
c⌊b⌋/summationdisplay
k=1ak
ksinh[c(b−k)]
[|a|<1,b > 0,c > 0] ET I 22(33)
4.432
1./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbig sinx/radicalbig
1−k2sin2xdx
x=/integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbig sinx√
1−k2cos2xdx
x=l nk/primeK(k)
BI ((412, 414))(4)
2./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbigsinxcosx/radicalbig
1−k2sin2xxdx
=1
k2/braceleftbig
πk/prime(1−lnk/prime)+/parenleftbig
2−k2/parenrightbig
K(k)−(4−lnk/prime)E(k)/bracerightbig
BI (426)(3)
3./integraldisplayπ/2
0ln/parenleftbig
1−k2cos2x/parenrightbigsinxcosx√
1−k2cos2xxdx=1
k2/braceleftbig
−π−/parenleftbig
2−k2/parenrightbig
K(k)+( 4 −lnk/prime)E(k)/bracerightbig
BI (426)(6)
4./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbigsinxcosx/radicalbig
1−k2sin2xdx
x=1
k2/braceleftBig/parenleftBig
2−k2−k/prime2lnk/prime/parenrightBig
K(k)−(2−lnk/prime)E(k)/bracerightBig
BI (412)(5)
5./integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbigsinxcosx√
1−k2cos2xdx
x=1
k2/braceleftbig/parenleftbig
k2−2+l n k/prime/parenrightbig
K(k)+( 2 −lnk/prime)E(k)/bracerightbig
BI (414)(5)
4.432 Logarithms, trigonometric functions, and powers 597
6./integraldisplay∞
0ln/parenleftbig
1±ksin2x/parenrightbig sinx/radicalbig
1−k2sin2xdx
x=/integraldisplay∞
0ln/parenleftbig
1±kcos2x/parenrightbig sinx√
1−k2cos2xdx
x
=/integraldisplay∞
0ln/parenleftbig
1±ksin2x/parenrightbig tanx/radicalbig
1−k2sin2xdx
x
=/integraldisplay∞
0ln/parenleftbig
1±kcos2x/parenrightbig tanx√
1−k2cos2xdx
x
=/integraldisplay∞
0ln/parenleftbig
1±ksin22x/parenrightbig tanx/radicalbig
1−k2sin22xdx
x
=/integraldisplay∞
0ln/parenleftbig
1±k2cos22x/parenrightbig tanx√
1−k2cos22xdx
x
=1
2ln2( 1±k)√
kK(k)−π
8K(k/prime)
BI (413)(1–6), BI (415)(1–6)
7./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbig sin3x/radicalbig
1−k2sin2xdx
x=1
k2/braceleftbig/parenleftbig
k2−2+l n k/prime/parenrightbig
K(k)+( 2 −lnk/prime)E(k)/bracerightbig
BI (412)(6)
8./integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbig sin3x√
1−k2cos2xdx
x=1
k2/braceleftBig/parenleftBig
2−k2−k/prime2lnk/prime/parenrightBig
K(k)−(2−lnk/prime)E(k)/bracerightBig
BI (414)(6)a
9./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbigsinxcos2x/radicalbig
1−k2sin2xdx
x=1
k2/braceleftBig/parenleftBig
2−k2−k/prime2lnk/prime/parenrightBig
K(k)−(2−lnk/prime)E(k)/bracerightBig
BI (412)(7)
10./integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbigsinxcos2x√
1−k2cos2xdx
x=1
k2/braceleftbig/parenleftbig
k2−2+l n k/prime/parenrightbig
K(k)+( 2 −lnk/prime)E(k)/bracerightbig
BI (414)(7)
11./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbig tanx/radicalbig
1−k2sin2xdx
x=/integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbig tanx√
1−k2cos2xdx
x=l nk/primeK(k)
BI ((412, 414))(9)
12./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbigsin2xtanx/radicalbig
1−k2sin2xdx
x=1
k2/braceleftbig/parenleftbig
k2−2+l n k/prime/parenrightbig
K(k)+( 2 −lnk/prime)E(k)/bracerightbig
BI (412)(8)
13./integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbigsin2xtanx√
1−k2cos2xdx
x=1
k2/braceleftBig/parenleftBig
2−k2−k/prime2lnk/prime/parenrightBig
K(k)−(2−lnk/prime)E(k)/bracerightBig
BI (414)(8)
14./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbig sin2x/radicalBig/parenleftbig
1−k2sin2x/parenrightbig3dx
x=/integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbig sinx/radicalBig
(1−k2cos2x)3dx
x
=1
k/prime2/braceleftbig/parenleftbig
k2−2/parenrightbig
K(k)+( 2+l n k/prime)E(k)/bracerightbig
BI ((412, 414))(13)
598 Logarithmic Functions 4.432
15./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbigsinxcosx/radicalBig/parenleftbig
1−k2sin2x/parenrightbig3xdx=1
k2/braceleftBig
(1 + ln k/prime)π
k/prime−(2 + ln k/prime)K(k)/bracerightBig
BI (426)(9)
16./integraldisplayπ/2
0ln/parenleftbig
1−k2cos2x/parenrightbigsinxcosx/radicalBig
(1−k2cos2x)3xdx=1
k2{−π+( 2+l n k/prime)K(k)} BI (426)(15)
17./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbigsinxcosx/radicalBig/parenleftbig
1−k2sin2x/parenrightbig3dx
x=/integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbig sin3x/radicalBig
(1−k2cos2x)3dx
x
=1
k2/braceleftbig/parenleftbig
2−k2+l nk/prime/parenrightbig
K(k)−(2 + ln k/prime)E(k)/bracerightbig
BI (412)(14), BI(414)(15)
18./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbig sin3x/radicalBig/parenleftbig
1−k2sin2x/parenrightbig3dx
x
=/integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbigsinxcosx/radicalBig
(1−k2cos2x)3dx
x
=1
k2k/prime2/braceleftBig
(2 + ln k/prime)E(k)−/parenleftBig
2−k2+k/prime2lnk/prime/parenrightBig
K(k)/bracerightBig
BI (412)(15), BI(414)(14)
19./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbigsinxcos2x/radicalBig/parenleftbig
1−k2sin2x/parenrightbig3dx
x=/integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbigsin2xtanx/radicalBig
(1−k2cos2x)3dx
x
=1
k2/braceleftbig/parenleftbig
2−k2+l nk/prime/parenrightbig
K(k)−(2 + ln k/prime)E(k)/bracerightbig
BI (412)(16), BI(414)(17)
20./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbigsin2xtanx/radicalBig/parenleftbig
1−k2sin2x/parenrightbig3dx
x
=/integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbigsinxcos2x/radicalBig
(1−k2cos2x)3dx
x
=1
k2k/prime2/braceleftBig
(2 + ln k/prime)E(k)−/parenleftBig
2−k2+k/prime2lnk/prime/parenrightBig
K(k)/bracerightBig
BI (412)(17), BI(414)(16)
21./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbig tanx/radicalBig/parenleftbig
1−k2sin2x/parenrightbig3dx
x=/integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbig tanx/radicalBig
(1−k2cos2x)3dx
x
=1
k/prime2/braceleftbig/parenleftbig
k2−2/parenrightbig
K(k)+( 2+l n k/prime)E(k)/bracerightbig
BI ((412, 414))(18)
22./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbig/radicalbig
1−k2sin2xsinxdx
x=/integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbig/radicalbig
1−k2cos2xsinxdx
x
=/parenleftbig
2−k2/parenrightbig
K(k)−(2−lnk/prime)E(k)
BI ((412, 414))(1)
4.511 Inverse trigonometric functions 599
23./integraldisplayπ/2
0ln/parenleftbig
1−k2sin2x/parenrightbig/radicalbig
1−k2sin2xsinxcosx·xdx
=1
27k2/braceleftBig
3πk/prime3(1−3lnk/prime)+/parenleftBig
22k/prime2+6k4−3k/prime2lnk/prime/parenrightBig
K(k)/bracerightBig
−/parenleftbig
2−k2/parenrightbig
(14−6lnk/prime)E(k)
BI (426)(1)
24./integraldisplayπ/2
0ln/parenleftbig
1−k2cos2x/parenrightbig/radicalbig
1−k2cos2xsinxcosx·xdx
=1
27k2/braceleftBig
−3π−/parenleftBig
22k/prime2+6k4−3k/prime2lnk/prime/parenrightBig
K(k)+/parenleftbig
2−k2/parenrightbig
(14−6lnk/prime)E(k)/bracerightBig
BI (426)(2)
25./integraldisplay∞
0ln/parenleftbig
1−k2sin2x/parenrightbig/radicalbig
1−k2sin2xtanxdx
x=/integraldisplay∞
0ln/parenleftbig
1−k2cos2x/parenrightbig/radicalbig
1−k2cos2xtanxdx
x
=/parenleftbig
2−k2/parenrightbig
K(k)−(2−lnk/prime)E(k)
((412,414))(2)
26./integraldisplay∞
0ln/parenleftbig
sin2x+k/primecos2x/parenrightbig sinx√
1−k2cos2xdx
x=/integraldisplay∞
0ln/parenleftbig
sin2x+k/primecos2x/parenrightbig tanx√
1−k2cos2xdx
x
=/integraldisplay∞
0ln/parenleftbig
sin22x+k/primecos22x/parenrightbig tanx√
1−k2cos22xdx
x
=1
2ln⎡
⎢⎣2/parenleftBig√
k/prime/parenrightBig3
1+k/prime⎤
⎥⎦K(k)
BI (415)(19–21)
4.44 Combinations of logarithms, trigonometric functions, and exponentials
4.441
1.7/integraldisplay∞
0e−qxsinpxlnxdx=1
p2+q2/bracketleftbigg
qarctanp
q−pC−p
cln/parenleftbig
p2−q2/parenrightbig/bracketrightbigg
[q>0,p > 0] BI (467)(1)
2./integraldisplay∞
0e−qxcospxlnxdx=−1
p2+q2/bracketleftbiggq
2ln/parenleftbig
p2+q2/parenrightbig
+parctanp
q+qC/bracketrightbigg
[q>0] BI (467)(2)
4.442/integraldisplayπ/2
0e−ptanxlncos xdx
sinxcosx=−1
2[ci(p)]2+1
2[si(p)]2[Rep>0] NT 32(11)
4.5 Inverse Trigonometric Functions
4.51 Inverse trigonometric functions
4.511/integraldisplay∞
0arccot pxarccot qxdx =π
2/braceleftbigg1
pln/parenleftbigg
1+p
q/parenrightbigg
+1
qln/parenleftbigg
1+q
p/parenrightbigg/bracerightbigg
[p>0,q > 0] BI (77)(8)
600 Inverse Trigonometric Functions 4.512
4.512/integraldisplayπ
0arctan(cos x)dx=0 BI (345)(1)
4.52 Combinations of arcsines, arccosines, and powers
4.521
1./integraldisplay1
0arcsin x
xdx=π
2ln 2 FI II 614, 623
2./integraldisplay1
0arccos x
1±xdx=∓π
2ln 2 + 2 G BI (231)(7, 8)
3./integraldisplay1
0arcsin xx
1+qx2dx=π
2qln2√1+q
1+√1+q[q>−1] BI (231)(1)
4./integraldisplay1
0arcsin xx
1−p2x2dx=π
2p2ln1+/radicalbig
1−p2
2/radicalbig
1−p2/bracketleftbig
p2<1/bracketrightbig
LI (231)(3)
5./integraldisplay1
0arccos xdx
sin2λ−x2= 2 cosec λ∞/summationdisplay
k=0sin[(2k+1 )λ]
(2k+1 )2BI (231)(10)
6./integraldisplay1
0arcsin xdx
x(1 +qx2)=π
2ln1+√1+q√1+q[q>−1] BI (235)(10)
7./integraldisplay1
0arcsin xx
(1 +qx2)2dx=π
4q√1+q−1
1+q[q>−1] BI (234)(2)
8./integraldisplay1
0arccos xx
(1 +qx2)2dx=π
4q√1+q−1
1+q[q>−1] BI (234)(4)
4.522
1./integraldisplay1
0x/radicalbig
1−k2x2arccos xdx=1
9k2/bracketleftbigg3
2π+k/prime2K(k)−2/parenleftBig
1+k/prime2/parenrightBig
E(k)/bracketrightbigg
BI (236)(9)
2./integraldisplay1
0x/radicalbig
1−k2x2arcsin xdx=1
9k2/bracketleftbigg
−3
2πk/prime3−k/prime2K(k)+2/parenleftBig
1+k/prime2/parenrightBig
E(k)/bracketrightbigg
BI (236)(1)
3./integraldisplay1
0x/radicalbig
k/prime2+k2x2arcsin xdx=1
9k2/bracketleftbigg3
2π+k/prime2K(k)−2/parenleftBig
1+k/prime2/parenrightBig
E(k)/bracketrightbigg
BI(236)(5)
4./integraldisplay1
0xarcsin x√
1−k2x2dx=1
k2/bracketleftBig
−π
2k/prime+E(k)/bracketrightBig
BI (237)(1)
5./integraldisplay1
0xarccos x√
1−k2x2dx=1
k2/bracketleftBigπ
2−E(k)/bracketrightBig
BI (240)(1)
6./integraldisplay1
0xarcsin x/radicalbig
k/prime2+k2x2dx=1
k2/bracketleftBigπ
2−E(k)/bracketrightBig
BI (238)(1)
7./integraldisplay1
0xarccos x/radicalbig
k/prime2+k2x2dx=1
k2/bracketleftBig
−π
2k/prime+E(k)/bracketrightBig
BI (241)(1)
4.531 Arctangents, arccotangents, and powers 601
8./integraldisplay1
0xarcsin xdx
(x2−cos2λ)√
1−x2=2
sinλ∞/summationdisplay
k=0sin[(2k+1 )λ]
(2k+1 )2BI (243)(11)
9./integraldisplay1
0xarcsin kx/radicalbig
(1−x2)(1−k2x2)dx=−π
2klnk/primeBI (239)(1)
10./integraldisplay1
0xarccos kx/radicalbig
(1−x2)(1−k2x2)dx=π
2kln(1 + k) BI (242)(1)
4.523
1./integraldisplay1
0x2narcsin xdx=1
2n+1/bracketleftbiggπ
2−2nn!
(2n+1 ) ! !/bracketrightbigg
BI (229)(1)
2./integraldisplay1
0x2n−1arcsin xdx=π
4n/bracketleftbigg
1−(2n−1)!!
2nn!/bracketrightbigg
BI (229)(2)
3./integraldisplay1
0x2narccos xdx=2nn!
(2n+ 1)(2 n+1 ) ! !BI (229)(4)
4./integraldisplay1
0x2n−1arccos xdx=π
4n(2n−1)!!
2nn!BI (229)(5)
5./integraldisplay1
−1/parenleftbig
1−x2/parenrightbignarccos xdx=π2nn!
(2n+1 ) ! !BI (254)(2)
6./integraldisplay1
−1/parenleftbig
1−x2/parenrightbign−1
2arccos xdx=π2
2(2n−1)!!
2nn!BI (254)(3)
4.524
1./integraldisplay1
0(arcsin x)2 dx
x2√
1−x2=πln2 BI (243)(13)
2./integraldisplay1
0(arccos x)2 dx
/parenleftbig√
1−x2/parenrightbig3=πln 2 BI (244)(9)
4.53–4.54 Combinations of arctangents, arccotangents, and powers
4.531
1./integraldisplay1
0arctan x
xdx=/integraldisplay∞
1arccot x
xdx=G FI II 482, BI (253)(8)
2./integraldisplay∞
0arccot x
1±xdx=±π
4ln2 +G BI (248)(6, 7)
3./integraldisplay1
0arccot x
x(1 +x)dx=−π
8ln2 +G BI (235)(11)
4./integraldisplay∞
0arctan x
1−x2dx=−G. BI (248)(2)
602 Inverse Trigonometric Functions 4.532
5./integraldisplay1
0arctan qxdx
(1 +px)2=1
2q
p2+q2ln(1 +p)2
1+q2+q2−p
(1 +p)(p2+q2)arctan q
[p>−1] BI (243)(7)
6./integraldisplay1
0arccot qxdx
(1 +px)2=1
2q
p2+q2ln1+q2
(1 +p)2+p
p2+q2arctan q+1
1+parccot q
[p>−1] BI (234)(10)
7./integraldisplay1
0arctan x
x(1 +x2)dx=π
8ln2 +1
2G BI (235)(12)
8./integraldisplay∞
0xarctan x
1+x4dx=π2
16BI (248)(3)
9./integraldisplay∞
0xarctan x
1−x4dx=−π
8ln2 BI (248)(4)
10.11/integraldisplay∞
0xarccot x
1−x4dx=π
8ln2 BI (248)(12)
11./integraldisplay∞
0arccot x
x√
1+x2dx=/integraldisplay∞
0arccot x√
1+x2dx=2G BI (251)(3, 10)
12./integraldisplay1
0arctan x
x√
1−x2dx=π
2ln/parenleftBig
1+√
2/parenrightBig
FI II 694
13./integraldisplay1
0xarctan xdx/radicalBig
(1 +x2)/parenleftbig
1+k/prime2x2/parenrightbig=1
k2⎡
⎣F/parenleftBigπ
4,k/parenrightBig
−π
2/radicalBig
2/parenleftbig
1+k/prime2/parenrightbig⎤
⎦ BI (294)(14)
4.532
1./integraldisplay1
0xparctan xdx=1
2(p+1 )/bracketleftBigπ
2−β/parenleftBigp
2+1/parenrightBig/bracketrightBig
[p>−2] BI (229)(7)
2./integraldisplay∞
0xparctan xdx=π
2(p+1 )cosecpπ
2[−1>p> −2] BI (246)(1)
3./integraldisplay1
0xparccot xdx=1
2(p+1 )/bracketleftBigπ
2+β/parenleftBigp
2+1/parenrightBig/bracketrightBig
[p>−1] BI (229)(8)
4./integraldisplay∞
0xparccot xdx=−π
2(p+1 )cosecpπ
2[−1<p< 0] BI (246)(2)
5./integraldisplay∞
0/parenleftbiggxp
1+x2p/parenrightbigg2q
arctan xdx
x=√
π3
22q+2pΓ(q)
Γ/parenleftbig
q+1
2/parenrightbig [q>0] BI (250)(10)
4.533
1./integraldisplay∞
0(1−xarccot x)dx=π
4BI (246)(3)
2./integraldisplay1
0/parenleftBigπ
4−arctan x/parenrightBigdx
1−x=−π
8ln2 +G BI (232)(2)
4.535 Arctangents, arccotangents, and powers 603
3./integraldisplay1
0/parenleftBigπ
4−arctan x/parenrightBig1+x
1−xdx
1+x2=π
8ln 2 +1
2G BI (235)(25)
4./integraldisplay1
0/parenleftbigg
xarccot x−1
xarctan x/parenrightbiggdx
1−x2=−π
4ln2 BI (232)(1)
4.534/integraldisplay∞
0(arctan x)2 dx
x2√
1+x2=/integraldisplay∞
0(arccot x)2xdx√
1+x2=−π2
4+4G BI (251)(9, 17)
4.535
1./integraldisplay1
0arctan px
1+p2xdx=1
2p2arctan pln/parenleftbig
1+p2/parenrightbig
BI (231)(19)
2./integraldisplay1
0arccot px
1+p2xdx=1
p2/braceleftbiggπ
4+1
2arccot p/bracerightbigg
ln/parenleftbig
1+p2/parenrightbig
[p>0] BI (231)(24)
3./integraldisplay∞
0arctan qx
(p+x)2dx=−q
1+p2q2/parenleftBig
lnpq−π
2pq/parenrightBig
[p>0,q > 0] BI (249)(1)
4./integraldisplay∞
0arccot qx
(p+x)2dx=q
1+p2q2/parenleftbigg
lnpq+π
2pq/parenrightbigg
[p>0,q > 0] BI (249)(8)
5./integraldisplay∞
0xarccot px
q2+x2dx=π
2ln1+pq
pq[p>0,q > 0] BI (248)(9)
6./integraldisplay∞
0xarccot pxdx
x2−q2=π
4ln1+p2q2
p2q2[p>0,q > 0] BI (248)(10)
7./integraldisplay∞
0arctan px
x(1 +x2)dx=π
2ln(1 + p)[ p≥0] FI II 745
8./integraldisplay∞
0arctan px
x(1−x2)dx=π
4ln/parenleftbig
1+p2/parenrightbig
[p≥0] BI (250)(6)
9./integraldisplay∞
0arctan qxdx
x(p2+x2)=π
2p2ln(1 + pq)[ p>0,q≥0] BI (250)(3)
10./integraldisplay∞
0arctan qxdx
x(1−p2x2)=π
4lnp2+q2
p2[p≥0] BI (250)(6)
11./integraldisplay∞
0xarctan qx
(p2+x2)2dx=πq
4p(1 +pq)[p>0,q≥0] BI (252)(12)a
12./integraldisplay∞
0xarccot qx
(p2+x2)2dx=π
4p2(1 +pq)[p>0,q≥0] BI (252)(20)a
13./integraldisplay1
0arctan qx
x√
1−x2dx=π
2ln/parenleftBig
q+/radicalbig
1+q2/parenrightBig
BI (244)(11)
14.9/integraldisplay∞
−∞xarctan( αx)dx
(x2+β2)(x2+γ2)=⎧
⎪⎨
⎪⎩π
β2−γ2ln/parenleftbigg1+|αβ|
1+|αγ|/parenrightbigg
sign(α)f o r β/negationslash=γ
πα
2|β|(1 +|αβ|)forβ=γ
forα,β,γ real
604 Inverse Trigonometric Functions 4.536
15.9/integraldisplay∞
−∞xarctan ( α/x)dx
(x2+β2)(x2+γ2)=⎧
⎪⎨
⎪⎩π
β2−γ2ln/parenleftbigg1+|α/γ|
1+|α/β|/parenrightbigg
sign(α)(α,β,γ real; β/negationslash=γ)
πα
2β2(|β|+|α|)(β=γ)
4.536
1./integraldisplay∞
0arctan qxarcsin xdx
x2=1
2qπln1+/radicalbig
1+q2
/radicalbig
1+q2+π
2ln/parenleftBig
q+/radicalbig
1+q2/parenrightBig
−π
2−arctan q
BI (230)(7)
2./integraldisplay∞
0arctan px−arctan qx
xdx=π
2lnp
q[p>0,q > 0] FI II 635
3./integraldisplay∞
0arctan pxarctan qx
x2dx=π
2ln(p+q)p+q
ppqq[p>0,q > 0] FI II 745
4.537
1.8/integraldisplay1
0arctan/parenleftBig/radicalbig
1−x2/parenrightBigdx
1−x2cos2λ=π
2c osλln/bracketleftbigg
cos/parenleftbiggπ−4λ
8/parenrightbigg
cosec/parenleftbiggπ+4λ
8/parenrightbigg/bracketrightbigg
BI (245)(9)
2./integraldisplay1
0arctan/parenleftBig
p/radicalbig
1−x2/parenrightBigdx
1−x2=1
2πln/parenleftBig
p+/radicalbig
1+p2/parenrightBig
[p>0] BI (245)(10)
3./integraldisplay1
0arctan/parenleftBig
tanλ/radicalbig
1−k2x2/parenrightBig/radicalBigg
1−x2
1−k2x2dx=π
2k2/bracketleftBig
E(λ,k)−k/prime2F(γ,k)/bracketrightBig
−π
2k2cotγ/parenleftbigg
1−/radicalBig
1−k2sin2γ/parenrightbigg
BI (245)(12)
4./integraldisplay1
0arctan/parenleftBig
tanλ/radicalbig
1−k2x2/parenrightBig/radicalBigg
1−k2x2
1−x2dx=π
2E(λ,k)−π
2cotλ/parenleftBig
1−/radicalbig
1−k2sin2λ/parenrightBig
BI (245)(11)
5./integraldisplay1
0arctan/parenleftbig
tanλ√
1−k2x2/parenrightbig
/radicalbig
(1−x2)(1−k2x2)dx=π
2F(λ,k) BI (245)(13)
4.538
1./integraldisplay∞
0arctan x2dx
1+x2=/integraldisplay∞
0arctan x3dx
1+x2BI (252)(10, 11)
=/integraldisplay∞
0arccot x2dx
1+x2=/integraldisplay∞
0arccot x3dx
1+x2=π2
8BI (252)(18, 19)
2./integraldisplay∞
01−x2
x2arctan x2dx=π
2/parenleftBig√
2−1/parenrightBig
BI (244)(10)a
4.539/integraldisplay∞
0xs−1arctan/parenleftbig
ae−x/parenrightbig
dx=2−s−1Γ(s)aΦ/parenleftbig
−a2,s+1,1
2/parenrightbig
ET I 222(47)
4.541/integraldisplay∞
0arctan/parenleftbiggpsinqx
1+pcosqx/parenrightbiggxdx
1+x2=π
2ln/parenleftbig
1+pe−q/parenrightbig
[p>−eq] BI (341)(14)a
4.573 Inverse and direct trigonometric functions 605
4.55 Combinations of inverse trigonometric functions and exponentials
4.551
1.9/integraldisplay1
0(arcsin x)e−bxdx=π
2b[I0(b)−L0(b)]−πe−b
2bET I 160(1)
2./integraldisplay1
0x(arcsin x)e−bxdx=π
2b2[L0(b)−I0(b)+bL1(b)−bI1(b)] +1
bET I 161(2)
3.9/integraldisplay∞
0/parenleftBig
arctanx
a/parenrightBig
e−bxdx=1
b[ci(ab)sin(ab)−si(ab)cos(ab)]
[Reb>0] ET I 161(3)
4.9/integraldisplay∞
0/parenleftBig
arccotx
a/parenrightBig
e−bxdx=1
b/bracketleftBigπ
2−ci(ab)sin (ab)+s i ( ab)cos (ab)/bracketrightBig
[Reb>0] ET I 161(4)
4.552/integraldisplay∞
0arctanx
q
e2πx−1dx=1
2/bracketleftbigg
ln Γ(q)−/parenleftbigg
q−1
2/parenrightbigg
lnq+q−1
2ln2π/bracketrightbigg
[q>0] WH
4.553/integraldisplay∞
0/parenleftbigg2
πarccot x−e−px/parenrightbiggdx
x=C+l np [p>0] NT 66(12)
4.56 A combination of the arctangent and a hyperbolic function
4.561/integraldisplay∞
−∞arctan e−x
cosh2qpxdx=1
2/integraldisplay∞
−∞Π(x)
cosh2qpxdx=√
π3
4pΓ(q)
Γ/parenleftbig
q+1
2/parenrightbig
[q>0] LI (282)(10)
4.57 Combinations of inverse and direct trigonometric functions
4.571/integraldisplayπ/2
0arcsin( ksinx)sinxdx/radicalbig
1−k2sin2x=−π
2klnk/primeBI (344)(2)
4.572/integraldisplay∞
0/parenleftbigg2
πarccot x−cospx/parenrightbigg
dx=C+l np [p>0] NT 66(12)
4.573
1./integraldisplay∞
0arccot qxsinpxdx =π
2p/parenleftBig
1−e−p
q/parenrightBig
[p>0,q > 0] BI (347)(1)a
2./integraldisplay∞
0arccot qxcospxdx =1
2p/bracketleftbigg
e−p
qEi/parenleftbiggp
q/parenrightbigg
−ep
qEi/parenleftbigg
−p
q/parenrightbigg/bracketrightbigg
[p>0,q > 0] BI (347)(2)a
3./integraldisplay∞
0arccot rxsinpxdx
1±2qcospx+q2=±π
2pqln1±q
1±qe−p
r/bracketleftbig
p2<1,r > 0,p > 0/bracketrightbig
=±π
2pqlnq±1
q±e−p
r/bracketleftbig
q2>1,r > 0,p > 0/bracketrightbig
BI (347)(10)
606 Inverse Trigonometric Functions 4.574
4./integraldisplay∞
0arccot pxtanxdx
q2cos2x+r2sin2x=π
2r2ln/parenleftbigg
1+r
qtanh1
p/parenrightbigg
[p>0,q > 0,r > 0] BI (347)(9)
4.574
1./integraldisplay∞
0arctan/parenleftbigg2a
x/parenrightbigg
sin(bx)dx=π
be−absinh(ab)[ R e a>0,b > 0] ET I 87(8)
2.7/integraldisplay∞
0arctana
xcos(bx)dx=1
2b/bracketleftbig
e−abEi(ab)−eabEi(−ab)/bracketrightbig
[a>0,b > 0] ET I 29(7)
3./integraldisplay∞
0arctan/bracketleftbigg2ax
x2+c2/bracketrightbigg
sin(bx)dx=π
be−b√
a2+c2sinh(ab)
[b>0] ET I 87(9)
4./integraldisplay∞
0arctan/parenleftbigg2
x2/parenrightbigg
cos(bx)dx=π
be−bsinb [b>0] ET I 29(8)
4.575
1./integraldisplayπ
0arctanpsinx
1−pcosxsinnxdx =π
2npn/bracketleftbig
p2<1/bracketrightbig
BI (345)(4)
2./integraldisplayπ
0arctanpsinx
1−pcosxsinnxcosxdx=π
4/parenleftbiggpn+1
n+1+pn−1
n−1/parenrightbigg
/bracketleftbig
p2<1/bracketrightbig
BI (345)(5)
3./integraldisplayπ
0arctanpsinx
1−pcosxcosnxsinxdx=π
4/parenleftbiggpn+1
n+1−pn−1
n−1/parenrightbigg
/bracketleftbig
p2<1/bracketrightbig
BI (345)(6)
4.576
1./integraldisplayπ
0arctanpsinx
1−pcosxdx
sinx=π
2ln1+p
1−p/bracketleftbig
p2<1/bracketrightbig
BI(346)(1)
2./integraldisplayπ
0arctanpsinx
1−pcosxdx
tanx=−π
2ln/parenleftbig
1−p2/parenrightbig/bracketleftbig
p2<1/bracketrightbig
BI(346)(3)
4.577
1./integraldisplayπ/2
0arctan/parenleftBig
tanλ/radicalbig
1−k2sin2x/parenrightBigsin2xdx/radicalbig
1−k2sin2x
=π
2k2/bracketleftBig
F(λ,k)−E(λ,k) + cot λ/parenleftBig
1−/radicalbig
1−k2sin2λ/parenrightBig/bracketrightBig
BI (344)(4)
2./integraldisplayπ/2
0arctan/parenleftBig
tanλ/radicalbig
1−k2sin2x/parenrightBigcos2xdx/radicalbig
1−k2sin2x
=π
2k2/bracketleftBig
E(λ,k)−k/prime2F(λ,k) + cot λ/parenleftBig/radicalbig
1−k2sin2λ−1/parenrightBig/bracketrightBig
BI (344)(5)
4.601 Change of variables in multiple integrals 607
4.58 A combination involving an inverse and a direct trigonometric function and a
power
4.58110/integraldisplay∞
0arctan xcospxdx
x=/integraldisplay∞
0arctanx
pcosxdx
x=−π
2Ei(−p)
[Re(p)>0] ET I 29(3), NT 25(13)
4.59 Combinations of inverse trigonometric functions and logarithms
4.591
1./integraldisplay1
0arcsin xlnxdx=2−ln2−1
2π BI (339)(1)
2./integraldisplay1
0arccos xlnxdx=l n2 −2 BI (339)(2)
4.592/integraldisplay1
0arccos xdx
lnx=−∞/summationdisplay
k=0(2k−1)!!
2kk!ln(2k+2 )
2k+1BI (339)(8)
4.593
1./integraldisplay1
0arctan xlnxdx=1
2ln 2−π
4+1
48π2BI (339)(3)
2./integraldisplay1
0arccot xlnxdx=−1
48π2−π
4−1
2ln 2 BI (339)(4)
4.594/integraldisplay1
0arctan x(lnx)n−1(lnx+n)dx=n!
(−2)n+1/parenleftbig
2−n−1/parenrightbig
ζ(n+1 ) BI (339)(7)
4.6 Multiple Integrals
4.60 Change of variables in multiple integrals
4.601
1./integraldisplay/integraldisplay
(σ)f(x, y)dxdy =/integraldisplay/integraldisplay
(σ/prime)f[ϕ(u,υ),ψ(u,υ)]|Δ|du dυ
where x=ϕ(u,υ),y=ψ(u,υ), and Δ =∂ϕ
∂u∂ψ
∂υ−∂ψ
∂u∂ϕ
∂υ≡D(ϕ, ψ)
D(u,υ)is the Jacobian determinant
of the functions ϕandψ.
2./integraldisplay/integraldisplay/integraldisplay
(V)f(x, y, z)dxdy dz =/integraldisplay/integraldisplay/integraldisplay
(V/prime)f[ϕ(u,υ,w ),ψ(u,υ,w ),χ(u,υ,w )|Δ|du dυ dw ]
where x=ϕ(u,υ,w ),y=ψ(u,υ,w ), and z=χ(u,υ,w ) and where
Δ=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ϕ
∂u∂ϕ
∂υ∂ϕ
∂w∂ψ
∂u∂ψ
∂υ∂ψ
∂w∂χ
∂u∂χ
∂υ∂χ
∂w/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≡D(ϕ, ψ, χ )
D(u,υ,w )
is the Jacobian determinant of the functions ϕ,ψ,a n d χ.
Here, we assume, both in ( 4.601 1) and in ( 4.601 2) that
608 Multiple Integrals 4.602
(a) the functions ϕ, ψ,a n d χand also their first partial derivatives are continuous in the region
of integration;
(b) the Jacobian does not change sign in this region;(c) there exists a one-to-one correspondence between the old variables x, y,z and the new ones
u,υ,w in the region of integration;
(d) when we change from the variables x, y,z to the variables u,υ,w, the region V(resp. σ)i s
mapped into the region V
/prime(resp. σ/prime).
4.602 Transformation to polar coordinates:
x=rcosϕ, y =rsinϕ;D(x, y)
D(r, ϕ)=r
4.603 Transformation to spherical coordinates:
x=rsinθcosϕ, y =rsinθsinϕ, z =rcosθ,D(x, y, z)
D(r, θ, ϕ)=r2sinθ
4.61 Change of the order of integration and change of variables
4.611
1./integraldisplayα
0dx/integraldisplayx
0f(x, y)dy=/integraldisplayα
0dy/integraldisplayα
yf(x, y)dx
2./integraldisplayα
0dx/integraldisplayβ
αx
0f(x, y)dy=/integraldisplayβ
0dy/integraldisplayα
α
βyf(x, y)dx
4.612
1./integraldisplayR
0dx/integraldisplay√
R2−x2
0f(x, y)dy=/integraldisplayR
0dy/integraldisplay√
R2−y2
0f(x, y)dx
2./integraldisplay2p
0dx/integraldisplayq/p√
2px−x2
0f(x, y)dy=/integraldisplayq
0dy/integraldisplayp/bracketleftBig
1+√
1−(y/q)2/bracketrightBig
p/bracketleftBig
1−√
1−(y/q)2/bracketrightBigf(x, y)dx
4.613 Change of the order of integration and change of variables 609
4.613
1./integraldisplayα
0dx/integraldisplayβ/(β+x)
0f(x, y)dy=/integraldisplayβ/(β+α)
0dy/integraldisplayα
0f(x, y)dx
+/integraldisplay1
β/(β+α)dy/integraldisplayβ(1−y)/y
0f(x, y)dx
2./integraldisplayα
0dx/integraldisplayδ−νx
βxf(x, y)dy=/integraldisplayαβ
0dy/integraldisplayy/β
0f(x, y)dx
+/integraldisplayδ
αβdy/integraldisplay(δ−y)/γ
0f(x, y)dx
/bracketleftbigg
α=δ
β+γ,a > 0,β > 0,γ > 0/bracketrightbigg
3./integraldisplay2α
0dx/integraldisplay3α−x
x2/4αf(x, y)dy=/integraldisplayα
0dy/integraldisplay2√αy
0f(x, y)dx+
+/integraldisplay3α
αdy/integraldisplay3α−y
0f(x, y)dx
4./integraldisplayR
0dx/integraldisplayx+2R
√
R2−x2f(x, y)dy=/integraldisplayR
0dy/integraldisplayR
√
R2−y2f(x, y)dx
+/integraldisplay2R
Rdy/integraldisplayR
0f(x, y)dx
+/integraldisplay3R
2Rdy/integraldisplayR
y−2Rf(x, y)dx
610 Multiple Integrals 4.614
4.614/integraldisplayπ/2
0dϕ/integraldisplay2Rcosϕ
0f(r, ϕ)dr=/integraldisplay2R
0dr/integraldisplayarccosr
2R
0f(r, ϕ)dϕ
4.615/integraldisplayR
0dx/integraldisplay√
R2−x2
0f(x, y)dy=/integraldisplayπ/2
0dϕ/integraldisplayR
0f(rcosϕ, rsinϕ)rd r
4.616/integraldisplay2R
0dx/integraldisplay√
2R−x2
0f(x, y)dy=/integraldisplayπ/2
0dϕ/integraldisplay2Rcosϕ
0f(rcosϕ, rsinϕ)rd r
4.617/integraldisplayβ
αdx/integraldisplayϕ2(x)
ϕ1(x)f(x, y)dy=/integraldisplayβ
0dx/integraldisplayϕ2(x)
0f(x, y)dy−/integraldisplayβ
0dx/integraldisplayϕ1(x)
0f(x, y)dy−/integraldisplayα
0dx/integraldisplayϕ2(x)
0f(x, y)dy
+/integraldisplayα
0dx/integraldisplayϕ1(x)
0f(x, y)dy
[ϕ1(x)≤ϕ2(x)f o rα≤x≤β]
4.618/integraldisplayγ
0dx/integraldisplayϕ(x)
0f(x, y)dy=/integraldisplayγ
0dx/integraldisplay1
0f[x, zϕ(x)]ϕ(x)dz[y=zϕ(x)]
=γ/integraldisplay1
0dz/integraldisplayϕ(γz)
0f(γz,y)dy [x=γz]
4.619/integraldisplayx1
x0dx/integraldisplayy1
y0f(x, y)dy=/integraldisplayx1
x0dx/integraldisplay1
0(y1−y0)f[x, y0+(y1−y0)t]dt
[y=y0+(y1−y0)t]
4.62 Double and triple integrals with constant limits
4.620 General formulas
1./integraldisplayπ
0dω/integraldisplay∞
0f/prime(pcoshx+qcosωsinhx)s i n h xdx=−πsignp/radicalbig
p2−q2f/parenleftBig
signp/radicalbig
p2−q2/parenrightBig
/bracketleftbigg
p2>q2,lim
x→+∞f(x)=0/bracketrightbigg
LO III 389
4.621 Double and triple integrals with constant limits 611
2./integraldisplay2π
0dω/integraldisplay∞
0f/prime[pcoshx+(qcosω+rsinω)sin h x]s in hxdx
=−2πsignp/radicalbig
p2−q2−r2f/parenleftBig
signp/radicalbig
p2−q2−r2/parenrightBig
/bracketleftbigg
p2>q2+r2,lim
x→+∞f(x)=0/bracketrightbigg
LO III 390
3./integraldisplayπ
0/integraldisplayπ
0dxdy
sinxsin2yf/prime/bracketleftbiggp−qcosx
sinxsiny+rcoty/bracketrightbigg
=−2πsignp/radicalbig
p2−q2−r2f/parenleftBig
signp/radicalbig
p2−q2−r2/parenrightBig
/bracketleftbigg
p2>q2+r2,lim
x→+∞f(x)=0/bracketrightbigg
LO III 280
4./integraldisplay∞
−∞dx/integraldisplay∞
−∞f/prime(pcoshxcoshy+qsinhxcoshy+rsinhy)c o s h yd y
=−2πsignp/radicalbig
p2−q2−r2f/parenleftBig
signp/radicalbig
p2−q2−r2/parenrightBig
/bracketleftbigg
p2>q2+r2,lim
x→+∞f(x)=0/bracketrightbigg
LO III 390
5./integraldisplay∞
0dx/integraldisplayπ
0f(pcoshx+qcosωsinhx)sin h2xsinωd ω=2/integraldisplay∞
0f/parenleftBig
signp/radicalbig
p2−q2coshx/parenrightBig
sinh2xdx
/bracketleftbigg
lim
x→+∞f(x)=0/bracketrightbigg
LO III 391
6./integraldisplay∞
0dx/integraldisplay2π
0dω/integraldisplayπ
0f[pcoshx+(qcosω+rsinω)sinθsinhx]s in h2xsinθd θ
=4/integraldisplay∞
0f/parenleftBig
signp/radicalbig
p2−q2−r2coshx/parenrightBig
sinh2xdx
/bracketleftbigg
p2>q2+r2,lim
x→+∞f(x)=0/bracketrightbigg
LO III 390
7./integraldisplay∞
0dx/integraldisplay2π
0dω/integraldisplayπ
0f{pcoshx+[ (qcosω+rsinω)sinθ+scoshθ]s in hx}sinh2xsinθd θ
=4π/integraldisplay∞
0f/parenleftBig
signp/radicalbig
p2−q2−r2−s2coshx/parenrightBig
sinh2xdx
/bracketleftbigg
p2>q2+r2+s2,lim
x→+∞f(x)=0/bracketrightbigg
LO III 391
4.621
1./integraldisplayπ/2
0/integraldisplayπ/2
0siny/radicalbig
1−k2sin2xsin2y
1−k2sin2ydxdy =π
2√
1−k2LO I 252(90)
2./integraldisplayπ/2
0/integraldisplayπ/2
0cosy/radicalbig
1−k2sin2xsin2y
1−k2sin2ydxdy =K(k) LO I 252(91)
3./integraldisplayπ/2
0/integraldisplayπ/2
0sinαsinyd xd y/radicalbig
1−sin2αsin2xsin2y=πα
2LO I 253
612 Multiple Integrals 4.622
4.622
1./integraldisplayπ
0/integraldisplayπ
0/integraldisplayπ
0dxdy dz
1−cosxcosycosz=4πK2/parenleftBigg√
2
2/parenrightBigg
MO 137
2./integraldisplayπ
0/integraldisplayπ
0/integraldisplayπ
0dxdy dz
3−cosycosz−cosxcosz−cosxcosy=√
3πK2/parenleftBig
sinπ
12/parenrightBig
MO 137
3./integraldisplayπ
0/integraldisplayπ
0/integraldisplayπ
0dxdy dz
3−cosx−cosy−cosz=4π/bracketleftBig
18 + 12√
2−10√
3−7√
6/bracketrightBig
K2/bracketleftBig/parenleftBig
2−√
3/parenrightBig/parenleftBig√
3−√
2/parenrightBig/bracketrightBig
MO 137
4.6233/integraldisplay∞
0/integraldisplay∞
0ϕ/parenleftbig
a2x2+b2y2/parenrightbig
dxdy =π
2ab/integraldisplay∞
0ϕ/parenleftbig
x2/parenrightbig
xdx
4.624/integraldisplayπ
0/integraldisplay2π
0f(αcosθ+βsinθcosψ+γsinθsinψ)sinθd θd ψ
=2π/integraldisplayπ
0f(Rcosp)s i npdp=2π/integraldisplay1
−1f(Rt)dt
/bracketleftBig
R=/radicalbig
α2+β2+γ2/bracketrightBig
4.6258pl(a,b)=/integraldisplaya
0dx/integraldisplayb
0dy/parenleftbig
x2+y2+1/parenrightbig−3/2Pl/parenleftBig
1//radicalbig
x2+y2+1/parenrightBig
Then, for even and odd subscripts:
•p2l(a,b)=1
l(2l+1 ) 22lab√
a2+b2+1l−1/summationdisplay
k=0(−1)l−k−122k/parenleftBig
2l+2k
l+k/parenrightBig/parenleftBig
l+k
l−k−1/parenrightBig
/parenleftBig
2k
k/parenrightBig
(2k+1 )
×(2l+2k+1 )k/summationdisplay
j=0/parenleftBig
2j
j/parenrightBig
22j1
(a2+b2+1 )j/parenleftBigg
1
(a2+1 )k−j+1+1
(b2+1 )k−j+1/parenrightBigg
•p2l+1(a,b)=1
22l+1(2l+1 )l/summationdisplay
k=0(−1)l+k
22k/parenleftbiggl
k/parenrightbigg/parenleftbiggl+k+1
k/parenrightbigg/parenleftbigg2l+2k+1
l+k/parenrightbigg
×/braceleftBigg
1
(b2+1 )kb√
b2+1arctan−1a√
b2+1+1
(a2+1 )ka√
a2+1arctan−1b√
a2+1
+abk/summationdisplay
j=122j−1
j/parenleftBig
2j
j/parenrightBig·1
(a2+b2+1 )j/parenleftBigg
1
(a2+1 )k−j+1+1
(b2+1 )k−j+1/parenrightBigg⎫
⎬
⎭
4.63–4.64 Multiple integrals
4.631/integraldisplayx
pdtn−1/integraldisplaytn−1
pdtn−2.../integraldisplayt1
pf(t)dt=1
(n−1)!/integraldisplayx
p(x−t)n−1f(t)dt,
where f(t) is continuous on the interval [ p, q]a n d p≤x≤q. FI II 692
4.635 Multiple integrals 613
4.632
1./integraldisplay/integraldisplay
···/integraldisplay
x1≥0,x2≥0,...,x n≥0
x1+x2+···+xn≤hdx1dx2···dxn=hn
n!
[the volume of an n-dimensional simplex] FI III 472
2./integraldisplay/integraldisplay
···/integraldisplay
x2
1+x2
2+···+x2n≤R2dx1dx2···dxn=√
πn
Γ/parenleftBign
2+1/parenrightBigRn[the volume of an n-dimensional sphere]
FI III 473
4.633/integraldisplay/integraldisplay
···/integraldisplay
x2
1+x2
2+···+x2n≤1dx1dx2···dxn/radicalbig
1−x2
1−x2
2−···− x2n=π(n+1)/2
Γ/parenleftbiggn+1
2/parenrightbigg[n>1]
/bracketleftbig
half-area of the surface of an ( n+ 1)-dimensional sphere x2
1+x2
2+···+x2
n+1=1/bracketrightbig
FI III 474
4.6348/integraldisplay/integraldisplay
···/integraldisplay
x1≥0,x2≥0,...,x n≥0/parenleftBig
x1
q1/parenrightBigα1+/parenleftBig
x2
q2/parenrightBigα2+···+(xn
qn)αn≤1xp1−1
1xp2−1
2···xpn−1
ndx1dx2...d x n
=qp1
1qp2
2...qpnn
α1α2...α nΓ/parenleftbiggp1
α1/parenrightbigg
Γ/parenleftbiggp2
α2/parenrightbigg
...Γ/parenleftbiggpn
αn/parenrightbigg
Γ/parenleftbiggp1
α1+p2
α2+···+pn
αn+1/parenrightbigg
[αi>0,p i>0,q i>0,i=1,2,...,n ]FI III 477
4.635
1.8/integraldisplay/integraldisplay
···/integraldisplay
x1≥0,x2≥0,...,x n≥0/parenleftBig
x1
q1/parenrightBigα1+/parenleftBig
x2
q2/parenrightBigα2+···+(xn
qn)αn≥1f/bracketleftbigg/parenleftbiggx1
q1/parenrightbiggα1
+/parenleftbiggx2
q2/parenrightbiggα2
+···+/parenleftbiggxn
qn/parenrightbiggαn/bracketrightbigg
×xp1−1
1xp2−1
2···xpn−1
ndx1dx2···dxn
=qp1
1qp2
2...qpnn
α1α2···αnΓ/parenleftbiggp1
α1/parenrightbigg
Γ/parenleftbiggp2
α2/parenrightbigg
...Γ/parenleftbiggpn
αn/parenrightbigg
Γ/parenleftbiggp1
α1+p2
α2+···+pn
αn/parenrightbigg/integraldisplay∞
1f(x)xp1
α1+p2
α2+···+pn
αn−1dx
under the assumption that the integral on the right converges absolutely. FI III 487
614 Multiple Integrals 4.636
2.8/integraldisplay/integraldisplay
···/integraldisplay
x1≥0,x2≥0,···,xn≥0/parenleftBig
x1
q1/parenrightBigα1+/parenleftBig
x2
q2/parenrightBigα2+···+(xn
qn)αn≤1f/bracketleftbigg/parenleftbiggx1
q1/parenrightbiggα1
+/parenleftbiggx2
q2/parenrightbiggα2
+···+/parenleftbiggxn
qn/parenrightbiggαn/bracketrightbigg
×xp1−1
1xp2−1
2···xpn−1
ndx1dx2···dxn
=qp1
1qp2
2...qpnn
α1α2...α nΓ/parenleftbiggp1
α1/parenrightbigg
Γ/parenleftbiggp2
α2/parenrightbigg
···Γ/parenleftbiggpn
αn/parenrightbigg
Γ/parenleftbiggp1
α1+p2
α2+···+pn
αn/parenrightbigg/integraldisplay1
0f(x)xp1
α1+p2
α2+···+pn
αn−1dx
under the assumptions that the one-dimensional integral on the right converges absolutely and
that the numbers qi,αi,a n d piare positive. FI III 479
In particular,
3./integraldisplay/integraldisplay
···/integraldisplay
x1≥0,x2≥0,...,x n≥0
x1+x2+···+xn≤1xp1−1
1xp2−1
2...xpn−1
ne−q(x1+x2+···+xn)dx1dx2...d x n
=Γ(p1)Γ(p2)...Γ(pn)
Γ(p1+p2+···+pn)/integraldisplay1
0xp1+p2+···+pn−1e−qxdx
[n>0,p1>0,p2>0,...,p n>0]
4.8/integraldisplay/integraldisplay
···/integraldisplay
x1≥0,x2≥0,···,xn≥0
xα1
1+xα2
2+···+xαn
n≤1xp1−1
1xp2−1
2...xpn−1
n
(1−xα1
1−xα2
2−···− xαnn)μdx1dx2...d x n
=Γ(1−μ)
α1α2...α nΓ/parenleftbiggp1
α1/parenrightbigg
Γ/parenleftbiggp2
α2/parenrightbigg
...Γ/parenleftbiggpn
αn/parenrightbigg
Γ/parenleftbigg
1−μ+p1
α1+p2
α2+···+pn
αn/parenrightbigg
[p1>0,p2>0,...,p n>0,μ < 1]FI III 480
4.636
1.8/integraldisplay/integraldisplay
···/integraldisplay
x1≥0,x2≥0,...,x n≥0
xα1
1+xα2
2+···+xαnn≥1xp1−1
1xp2−1
2...xpn−1
n
(xα1
1+xα2
2+···+xαnn)μdx1dx2...d x n
=1
α1α2...α n/parenleftbigg
μ−p1
α1−p2
α2−···−pn
αn/parenrightbiggΓ/parenleftbiggp1
α1/parenrightbigg
Γ/parenleftbiggp2
α2/parenrightbigg
...Γ/parenleftbiggpn
αn/parenrightbigg
Γ/parenleftbiggp1
α1+p2
α2+···+pn
αn/parenrightbigg
/bracketleftbigg
p1>0,p2>0,...,p n>0;μ>p1
α1+p2
α2+···+pn
αn/bracketrightbigg
FI III 488
4.638 Multiple integrals 615
2.8/integraldisplay/integraldisplay
···/integraldisplay
x1≥0,x2≥0,···,xn≥0
xα1
1+xα2
2+···+xαn
n≤1xp1−1
1xp2−1
2···xpn−1
n
(xα1
1+xα2
2+···+xαnn)μdx1dx2...d x n
=1
α1α2...α n/parenleftbiggp1
α1+p2
α2+···+pn
αn−μ/parenrightbiggΓ/parenleftbiggp1
α1/parenrightbigg
Γ/parenleftbiggp2
α2/parenrightbigg
...Γ/parenleftbiggpn
αn/parenrightbigg
Γ/parenleftbiggp1
α1+p2
α2+···+pn
αn/parenrightbigg
/bracketleftbigg
μ<p1
α1+p2
α2+···+pn
αn/bracketrightbigg
FI III 480
3.8/integraldisplay/integraldisplay
···/integraldisplay
x1≥0,x2≥0,...,x n≥0
xα1
1+xα2
2+···+xαnn≤1xp1−1
1xp2−1
2...xpn−1
n/radicalBigg
1−xα1
1−xα2
2−···− xαnn
1+xα1
1+xα2
2+···+xαnndx1dx2... d x n
=√π
2Γ/parenleftbiggp1
α1/parenrightbigg
Γ/parenleftbiggp1
α2/parenrightbigg
...Γ/parenleftbiggpn
αn/parenrightbigg
α1α2...α n1
Γ(m)⎧
⎪⎪⎨
⎪⎪⎩Γ/parenleftBigm
2/parenrightBig
Γ/parenleftbiggm+1
2/parenrightbigg−Γ/parenleftbiggm+1
2/parenrightbigg
Γ/parenleftbiggm+2
2/parenrightbigg⎫
⎪⎪⎬
⎪⎪⎭,
where m=p
1
α1+p2
α2+···+pn
αn. FI III 480
4.6378/integraldisplay/integraldisplay
···/integraldisplay
x1≥0,x2≥0,...,x n≥0
x1+x2+···+xn≤1f(x1+x2+···+xn)xp1−1
1xp2−1
2...xpn−1
ndx1dx2...d x n
(q1x1+q2x2+···+qnxn+r)p1+p2+···+pn
=Γ(p1)Γ(p2)...Γ(pn)
Γ(p1p2+...p n)/integraldisplay1
0f(x)xp1p2+···pn−1
(q1x+r)p1(q2x+r)p2...(qnx+r)pndx,
[q1≥0,q2≥0,...,q n≥0;r>0]
where f(x) is continuous on the interval (0 ,1).
4.638
1./integraldisplay∞
0/integraldisplay∞
0.../integraldisplay∞
0xp1−1
1xp2−1
2···xpn−1
ne−(q1x1+q2x2+···+qnxn)
(r0+r1x1+r2x2+···+rnxn)s dx1dx2...d x n
=Γ(p1)Γ(p2)...Γ(pn)
Γ(s)/integraldisplay∞
0er0xxs−1dx
(q1r1x)p1(q1r2x)p2...(qnrnx)pn
where pi,qi,ri,a n dsare positive. This result is also valid for r0=0 ,p r o v i d e d p1+p2+···+pn>
s.
2./integraldisplay∞
0/integraldisplay∞
0.../integraldisplay∞
0xp1−1
1xp2−1
2...xpn−1
n
(r0+r1x1+r2x2+···+rnxn)sdx1dx2...d x n
=Γ(p1)Γ(p2)...Γ(pn)Γ(sp1p2−···− pn)
rp1
1rp2
2···rpnnrs−p1−p2−···− pn
0 Γ(s)
[pi>0,r i>0,s > 0]
3.8/integraldisplay∞
0/integraldisplay∞
0.../integraldisplay∞
0xp1−1
1xp2−1
2...xpn−1
n
[1 + (r1x1)q1+(r2x2)q2+···+(rnxn)qn]sdx1dx2...d x n
=Γ/parenleftbiggp1
q1/parenrightbigg
Γ/parenleftbiggp2
q2/parenrightbigg
...Γ/parenleftbiggpn
qn/parenrightbigg
q1q2...q nrp1q1
1rp2q2
2...rpnqnnΓ/parenleftbigg
s−p1
q1−p2
q2−···−pn
qn/parenrightbigg
Γ(s)
[pi>0,q i>0,r i>0,s > 0]
616 Multiple Integrals 4.639
4.639
1./integraldisplay/integraldisplay
···/integraldisplay
x2
1+x2
2+···+x2
n≤1(p1x1+p2x2+···+pnxn)2mdx1dx2...d x n
=(2m−1)!!
2m√
πn
Γ/parenleftBign
2+m+1/parenrightBig/parenleftbig
p2
1+p2
2+···+p2
n/parenrightbigm
FI III 482
2./integraldisplay/integraldisplay
···/integraldisplay
x2
1+x2
2+···+x2
n≤1(p1x1+p2x2+···+pnxn)2m+1dx1dx2...d x n=0 FI III 483
4.641
1.11/integraldisplay/integraldisplay
···/integraldisplay
x2
1+x2
2+···+x2n≤1ep1x1+p2x2+···+pnxndx1dx2...d x n
=√
πn∞/summationdisplay
k=01
k!Γ/parenleftBign
2+k+1/parenrightBig/parenleftbiggp2
1+p2
2+···+p2
n
4/parenrightbiggk
FI III 483
2./integraldisplay/integraldisplay
···/integraldisplay
x2
1+x2
2+···+x2
2n≤1ep1x1p2x2+···p2nx2ndx1dx2...d x 2n=(2π)nIn/parenleftBig/radicalbig
p2
1+p2
2+···+p2
2n/parenrightBig
(p2
1+p2
2+···+p2
2n)n/2
FI III 483a
4.642/integraldisplay/integraldisplay
···/integraldisplay
x2
1+x2
2+···+x2
n≤R2f/parenleftbigg/radicalBig
x2
1+x2
2+···+x2n/parenrightbigg
dx1dx2...d x n=2√
πn
Γ/parenleftBign
2/parenrightBig/integraldisplayR
0xn−1f(x)dx,
where f(x) is a function that is continuous on the interval (0 ,R). FI III 485
4.643/integraldisplay1
0/integraldisplay1
0.../integraldisplay1
0f(x1x2···xn)(1−x1)p1−1(1−x2)p2−1...(1−xn)pn−1
×xp1
2xp1+p2
3···xp1+p2+···+pn−1
n dx1dx2...d x n
=Γ(p1)Γ(p2)...Γ(pn)
Γ(p1+p2+···+pn)/integraldisplay1
0f(x)(1−x)p1+p2+···+pn−1dx
under the assumption that the integral on the right converges absolutely. FI III 488
4.644n−1/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright/integraldisplay/integraldisplay
···/integraldisplay
x2
1+x2
2+···+x2n=1f(p1x1+p2x2+···+pnxn)dx1dx2...d x n−1
|xn|
=2/integraldisplay/integraldisplay
···/integraldisplay
x2
1+x2
2+···+x2
n−1≤1f(p1x1+p2x2+···+pnxn)dx1dx2···dxn−1/radicalBig
1−x2
1−x2
2−···− x2
n−1
=2√
πn−1
Γ/parenleftbiggn−1
2/parenrightbigg/integraldisplayπ
0f/parenleftbigg/radicalBig
p2
1+p2
2+···+p2ncosx/parenrightbigg
sinn−2xdx [n≥3]
where f(x) is continuous on the interval/braceleftBig
−/radicalbig
p2
1+p2
2+···+p2n,/radicalbig
p2
1+p2
2+···+p2n/bracerightBig
. FI III 489
4.648 Multiple integrals 617
4.645 Suppose that two functions f(x1,x2,...,x n)a n d g(x1,x2,...,x n) are continuous in a closed,
bounded region Dand that the smallest and greatest values of the function ginDaremandM,
respectively. Let ϕ(u) denote a function that is continuous for m≤u≤M. We denote by ψ(u)t h e
integral
1. ψ(u)=/integraldisplay/integraldisplay
···/integraldisplay
m≤g(x1,x2,...,x n)≤uf(x1,x2,...,x n)dx1dx2... d x n,
over that portion of the region Don which the inequality m≤g(x1,x2,...,x n)≤uis satisfied.
Then
2./integraldisplay/integraldisplay
···/integraldisplay
m≤g(x1,x2,...,x n)≤Mf(x1,x2,...,x n)ϕ[g(x1,x2,...,x n)]dx1dx2...d x n
=(S)/integraldisplayM
mϕ(u)dψ(u)=(R)/integraldisplayM
mϕ(u)dψ(u)
dudu
where the middle integral must be understood in the sense of Stieltjes. If the derivativedψ
duexists
and is continuous, the Riemann integral on the right exists.
Mmay be + ∞in formulas 4.645 2, in which case/integraltext+∞
mshould be understood to mean lim
M→+∞/integraldisplayM
m.
4.6468/integraldisplay/integraldisplay
···/integraldisplay
x1≥0,x2≥0,...,x n≥0
x1+x2+···+xn≤1xp1−1
1xp2−1
2...xpn−1
n
(q1x1+q2x2+···+qnxn)rdx1dx2...d x n
=Γ(p1)Γ(p2)...Γ(pn)
Γ(p1+p2+···+pn−r+1 )Γ ( r)/integraldisplay∞
0xr−1dx
(1 +q1x)p1(1 +q2x)p2···(1 +qnx)pn
=[p1>0,p2>0,...,p n>0,q1>0,q2>0,...,q n>0,p1+p2+···+pn>r> 0]
FI III 493
4.647/integraldisplay/integraldisplay
···/integraldisplay
0≤x2
1+x2
2+···+x2
n≤1exp/braceleftBigg
p1x1+p2x2+···+pnxn/radicalbig
x2
1+x2
2+···+x2n/bracerightBigg
dx1dx2...d x n
=2√
πn
n(p2
1+p2
2+···+p2n)n
4−1
2In
2−1/parenleftbigg/radicalBig
p2
1+p2
2+···+p2n/parenrightbigg
FI III 495
4.6488/integraldisplay∞
0/integraldisplay∞
0···/integraldisplay∞
0exp/bracketleftbigg
−/parenleftbigg
x1+x2+···+xn+λn+1
x1x2...x n/parenrightbigg/bracketrightbigg
×xc1
n+1−1
1 x2
n+1−1
2 ...xn
n+1−1
n dx1dx2···dxn
=1√n+1(2π)n
2e−(n+1)λ
FI III 496
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5 Indefinite Integrals of Special
Functions
5.1 Elliptic Integrals and Functions
Notation :k/prime=√
1−k2(cf. 8.1).
5.11 Complete elliptic integrals
5.111
1./integraldisplay
K(k)k2p+3dk=1
(2p+3 )2/braceleftbigg
4(p+1 )2/integraldisplay
K(k)k2p+1dk+k2p+2/bracketleftBig
E(k)−(2p+3 )K(k)k/prime2/bracketrightBig/bracerightbigg
BY (610.04)
2./integraldisplay
E(k)k2p+3dk=1
4p2+1 6p+1 5⎧
⎨
⎩4(p+1 )2/integraldisplay
E(k)k2p+1dk
−E(k)k2p+2/bracketleftBig
(2p+3 )k/prime2−2/bracketrightBig
−k2p+2k/prime2K(k)⎫
⎬
⎭
BY (611.04)
5.112
1./integraldisplay
K(k)dk=πk
2⎡
⎣1+∞/summationdisplay
j=1[(2j)!]2k2j
(2j+1 ) 24j(j!)4⎤
⎦ BY (610.00)
2.6/integraldisplay
E(k)dk=πk
2⎡
⎣1−∞/summationdisplay
j=1[(2j)!]2k2j
(4j2−1)24j(j!)4⎤
⎦ BY (611.00)
3./integraldisplay
K(k)kd k=E(k)−k/prime2K(k) BY (610.01)
4./integraldisplay
E(k)kd k=1
3/bracketleftBig/parenleftbig
1+k2/parenrightbig
E(k)−k/prime2K(k)/bracketrightBig
BY (611.01)
5./integraldisplay
K(k)k3dk=1
9/bracketleftBig/parenleftbig
4+k2/parenrightbig
E(k)−k/prime2/parenleftbig
4+3k2/parenrightbig
K(k)/bracketrightBig
BY (610.02)
619
620 Elliptic Integrals and Functions 5.113
6./integraldisplay
E(k)k3dk=1
45/bracketleftBig/parenleftbig
4+k2+9k4/parenrightbig
E(k)−k/prime2/parenleftbig
4+3k2/parenrightbig
K(k)/bracketrightBig
BY 611.02)
7./integraldisplay
K(k)k5dk=1
225/bracketleftBig/parenleftbig
64 + 16 k2+9k4/parenrightbig
E(k)−k/prime2/parenleftbig
64 + 48 k2+4 5k4/parenrightbig
K(k)/bracketrightBig
BY (610.03)
8./integraldisplay
E(k)k5dk=1
1575/bracketleftBig/parenleftbig
64 + 16 k2+9k4+ 225 k6/parenrightbig
E(k)−k/prime2/parenleftbig
64 + 48 k2+4 5k4/parenrightbig
K(k)/bracketrightBig
BY (611.03)
9./integraldisplayK(k)
k2dk=−E(k)
kBY (612.05)
10./integraldisplayE(k)
k2dk=1
k/bracketleftBig
k/prime2K(k)−2E(k)/bracketrightBig
BY (612.02)
11./integraldisplayE(k)
k/prime2dk=kK(k) BY (612.01)
12./integraldisplayE(k)
k4dk=1
9k3/bracketleftBig
2/parenleftbig
k2−2/parenrightbig
E(k)+k/prime2K(k)/bracketrightBig
BY (612.03)
13./integraldisplaykE(k)
k/prime2dk=K(k)−E(k) BY (612.04)
5.113
1./integraldisplay
[K(k)−E(k)]dk
k=−E(k) BY (612.06)
2./integraldisplay/bracketleftBig
E(k)−k/prime2K(k)/bracketrightBigdk
k=2E(k)−k/prime2K(k) BY (612.09)
3./integraldisplay/bracketleftbig/parenleftbig
1+k2/parenrightbig
K(k)−E(k)/bracketrightbigdk
k=−k/prime2K(k) BY (612.12)
4./integraldisplay
[K(k)−E(k)]dk
k2=1
k/bracketleftBig
E(k)−k/prime2K(k)/bracketrightBig
BY (612.07)
5./integraldisplay/bracketleftBig
E(k)−k/prime2K(k)/bracketrightBigdk
k2k/prime2=1
k[K(k)−E(k)]
6./integraldisplay/bracketleftBig/parenleftbig
1+k2/parenrightbig
E(k)−k/prime2K(k)/bracketrightBigdk
kk/prime4=E(k)
k/prime2BY (612.13)
5.114/integraldisplaykK(k)dk
/bracketleftbig
E(k)−k/prime2K(k)/bracketrightbig2=1
k/prime2K(k)−E(k)BY (612.11)
5.115
1./integraldisplay
Π/parenleftBigπ
2,r2,k/parenrightBig
kd k=/parenleftbig
k2−r2/parenrightbig
Π/parenleftBigπ
2,r2,k/parenrightBig
−K(k)+E(k) BY (612.14)
2./integraldisplay/bracketleftBig
K(k)−Π/parenleftBigπ
2,r2,k/parenrightBig/bracketrightBig
kd k=k2K(k)−/parenleftbig
k2−r2/parenrightbig
Π/parenleftBigπ
2,r2,k/parenrightBig
BY (612.15)
3./integraldisplay/bracketleftbiggE(k)
k/prime2+Π/parenleftBigπ
2,r2,k/parenrightBig/bracketrightbigg
kd k=/parenleftbig
k2−r2/parenrightbig
Π/parenleftBigπ
2,r2,k/parenrightBig
BY (612.16)
5.124 Elliptic integrals 621
5.12 Elliptic integrals
5.121/integraldisplayx
0F(x, k)dx/radicalbig
1−k2sin2x=[F(x, k)]2
2/bracketleftBig
0<x≤π
2/bracketrightBig
BY (630.01)
5.12211/integraldisplayx
0E(x, k)/radicalbig
1−k2sin2xdx=[E(x, k)]2
2BY (630.32)
5.123
1./integraldisplayx
0F(x, k)sinxdx=−cosxF(x, k)+1
karcsin( ksinx) BY (630.11)
2./integraldisplayx
0F(x, k)cosxdx=s i nxF(x, k)+1
karccosh/radicalBigg
1−k2sin2x
k/prime2−1
karccosh/parenleftbigg1
k/prime/parenrightbigg
BY (630.21)
5.124
1./integraldisplayx
0E(x, k)sinxdx=−cosxE(x, k)+1
2k/bracketleftBig
ksinx/radicalbig
1−k2sin2x+arcsin( ksinx)/bracketrightBig
BY (630.12)
2./integraldisplayx
0E(x, k)cosxdx=s i nxE(x, k)+1
2k⎡
⎣kcosx/radicalbig
1−k2sin2x
−k/prime2arccosh/radicalBigg
1−k2sin2x
k/prime2−k+k/prime2arccosh/parenleftbigg1
k/prime/parenrightbigg⎤
⎦
BY (630.22)
3.∗/integraldisplaya
0xE(x)dx
(k/prime2+k2x2)2√
a2−x2=π
4/parenleftBigg
a√
1−a2
(k/prime2+k2a2)2+a2E(λ,k)
k/prime2(k/prime2+k2a2)3/2+/parenleftbig
1−a2/parenrightbig
F(λ,k)
(k/prime2+k2a2)3/2/parenrightBigg
λ=a r c s i n/parenleftbigga√
k/prime2+k2a2/parenrightbigg
k/prime=/radicalbig
1−k2 [0<a< 1,0<k< 1]
4.∗/integraldisplaya
0xE(x)dx
(k2−x2)2√
a2−x2=π
4/parenleftBigg
a√
1−a2
k2(k2−a2)+F(φ,k)
k2√
k2−a2+a2E(φ,k)
k2(k2−a2)3/2/parenrightBigg
φ=a r c s i n/parenleftBiga
k/parenrightBig
[0<a<k< 1]
5.∗/integraldisplayπ/2
0E(x, k/prime)sinxcosxdx
/parenleftbig
1−k/prime2cosh2vsin2x/parenrightbig/radicalbig
1−k/prime2sin2x
=1
k/prime2sinhvcoshv/braceleftbigg
E(k/prime)arctan h/parenleftbiggtanhv
k/parenrightbigg
−πtanhv
2−π
2[F(φ,k)−E(φ,k)]/bracerightbigg
φ=a r c s i n/parenleftbiggtanhv
k/parenrightbigg
k/prime=/radicalbig
1−k2 [0<tanhv<k< 1]
622 Elliptic Integrals and Functions 5.124
6.∗/integraldisplayπ/2
0E(x, k)sinxcosxdx
/parenleftbig
1−k2cos2ψsin2x/parenrightbig/radicalbig
1−k2sin2x
=1
k2sinψcosψ/braceleftBigg
E(k)arctan/parenleftbiggtanψ
k/prime/parenrightbigg
−π
2E(β,k)+π
2tanψ/radicalbig
1−k2cos2ψ/parenleftBig
1−/radicalbig
1−k2cos2ψ/parenrightBig/bracerightBigg
β=a r c t a n/parenleftbiggtanψ
k/parenrightbigg
k/prime=/radicalbig
1−k2/bracketleftBig
0<k< 1,0<ψ<π
2/bracketrightBig
7.∗/integraldisplayπ/2
0E(x, k/prime)sinxcosxdx
/parenleftbig
1+k/prime2sinh2μsin2x/parenrightbig/radicalbig
1−k/prime2sin2x
=−1
k/prime2sinhμcoshμ/braceleftbigg
E(k/prime)arctan h( ktanhμ)−π
2/bracketleftbigg
F(φ,k)−E(φ,k)+t a n h μ/radicalBig
1+k/prime2sinh2μ/bracketrightbigg
−π
2cothμ/parenleftbigg
1−/radicalBig
1+k/prime2sinh2μ/parenrightbigg/bracerightbigg
φ= arcsin(tanh μ)k/prime=/radicalbig
1−k2 [0<k< 1,0<tanhμ<1]
8.∗/integraldisplayπ/2
0F(x, k/prime)sinxcosxdx
/parenleftbig
1+k/prime2sinh2μsin2x/parenrightbig/radicalbig
1−k/prime2sin2x
=−1
k/prime2sinhμcoshμ/bracketleftBig
K(k/prime)arctan h( ktanhμ)−π
2F(φ,k)/bracketrightBig
φ= arcsin(tanh μ)k/prime=/radicalbig
1−k2 [0<k< 1,0<tanhμ<1]
9.∗/integraldisplayπ/2
0F(x, k/prime)sinxcosxdx
/parenleftbig
1−k/prime2cosh2νsin2x/parenrightbig/radicalbig
1−k/prime2sin2x
=1
k/prime2sinhνcoshν/bracketleftbigg
K(k/prime)arctan h/parenleftbiggtanhν
k/parenrightbigg
−π
2F(φ,k)/bracketrightbigg
φ=a r c s i n/parenleftbiggtanhν
k/parenrightbigg
k/prime=/radicalbig
1−k2 [0<k< 1,0<tanhν<1]
10.∗/integraldisplayπ/2
0F(x, k)sinxcosxdx
/parenleftbig
1−k2cos2ψsin2x/parenrightbig/radicalbig
1−k2sin2x
=1
k2sinψcosψ/bracketleftbigg
K(k/prime)arctan h/parenleftbiggtanψ
k/prime/parenrightbigg
−π
2F(β,k)/bracketrightbigg
β=a r c t a n/parenleftbiggtanψ
k/prime/parenrightbigg
k/prime=/radicalbig
1−k2 [0<k< 1,0<ψ< 1]
11.∗/integraldisplayb
aln/parenleftbigg/epsilon1+x
/epsilon1−x/parenrightbiggx2dx/radicalbig
(x2−a2)(b2−x2)=π
/epsilon1/parenleftBig
/epsilon12−/radicalbig
(/epsilon12−a2)(/epsilon12−b2)/parenrightBig
+πβ[F(φ,k)−E(φ,k)]
φ=a r c s i n/parenleftbiggβ
/epsilon1/parenrightbigg
k=a
b[0<a<b</epsilon1 ]
5.131 Jacobian elliptic functions 623
5.125
1./integraldisplayx
0Π/parenleftbig
x, α2,k/parenrightbig
sinxdx
=−cosxΠ/parenleftbig
x, α2,k/parenrightbig
+1√
k2−α2arctan⎡
⎣/radicalBigg
k2−α2
1−k2sin2xsinx⎤
⎦/bracketleftbig
α2<k2/bracketrightbig
=−cosxΠ/parenleftbig
x, α2,k/parenrightbig
+1√
α2−k2arctanh⎡
⎣/radicalBigg
α2−k2
1−k2sin2xsinx⎤
⎦/bracketleftbig
α2>k2/bracketrightbig
BY (630.13)
2./integraldisplayx
0Π/parenleftbig
x, α2,k/parenrightbig
cosxdx=s i nxΠ/parenleftbig
x, α2,k/parenrightbig
−f−f0
where
f=1
2/radicalbig
(1−α2)(α2−k2)arctan/bracketleftBigg
2/parenleftbig
1−α2/parenrightbig/parenleftbig
α2−k2/parenrightbig
+/parenleftbig
1−α2sin2x/parenrightbig/parenleftbig
2k2−α2−α2k2/parenrightbig
2α2/radicalbig
(1−α2)(α2−k2)c o sx/radicalbig
1−k2sin2x/bracketrightBigg
for/parenleftbig
1−α2/parenrightbig/parenleftbig
α2−k2/parenrightbig
>0;
=1
2/radicalbig
(α2−1)(α2−k2)ln/bracketleftBigg
2/parenleftbig
α2−1/parenrightbig/parenleftbig
α2−k2/parenrightbig
+/parenleftbig
1−α2sin2x/parenrightbig/parenleftbig
α2+α2k2−2k2/parenrightbig
1−α2sin2x
+2α2/radicalbig
(α2−1)(α2−k2)c o sx/radicalbig
1−k2sin2x
1−α2sin2x/bracketrightBigg
for/parenleftbig
1−α2/parenrightbig/parenleftbig
α2−k2/parenrightbig
<0,
f0is the value of fatx=0 BY (630.23)
Integration with respect to the modulus
5.126/integraldisplay
F(x, k)kd k=E(x, k)−k/prime2F(x, k)+/parenleftBig/radicalbig
1−k2sin2x−1/parenrightBig
cotx BY (613.01)
5.127/integraldisplay
E(x, k)kd k=1
3/bracketleftBig/parenleftbig
1+k2/parenrightbig
E(x, k)−k/prime2F(x, k)+/parenleftBig/radicalbig
1−k2sin2x−1/parenrightBig
cotx/bracketrightBig
BY (613.02)
5.128/integraldisplay
Π/parenleftbig
x, r2,k/parenrightbig
kd k=/parenleftbig
k2−r2/parenrightbig
Π/parenleftbig
x, r2,k/parenrightbig
−F(x, k)+E(x, k)+/parenleftBig/radicalbig
1−k2sin2x−1/parenrightBig
cotx
BY (613.03)
5.13 Jacobian elliptic functions
5.131
1./integraldisplay
snmud u=1
m+1⎡
⎣snm+1ucnudnu+(m+2 )/parenleftbig
1+k2/parenrightbig/integraldisplay
snm+2ud u
−(m+3 )k2/integraldisplay
snm+4ud u⎤
⎦
SI 259, PE(567)
624 Elliptic Integrals and Functions 5.132
2./integraldisplay
cnmud u=1
(m+1 )k/prime2⎡
⎣−cnm+1usnudnu
+(m+2 )/parenleftbig
1−2k2/parenrightbig/integraldisplay
cnm+2ud u+(m+3 )k2/integraldisplay
cnm+4ud u⎤
⎦
PE (568)
3./integraldisplay
dnmud u=1
(m+1 )k/prime2⎡
⎣k2dnm+1usnucnu
+(m+2 )/parenleftbig
2−k2/parenrightbig/integraldisplay
dnm+2ud u−(m+3 )/integraldisplay
dnm+4ud u⎤
⎦
PE (569)
By using formulas 5.131 , we can reduce the integrals (for m/negationslash=1 )/integraltext
snmud u,/integraltext
cnmud u,a n d/integraltext
dnmud uto the integrals 5.132 ,5.133 and5.134 .
5.132
1./integraldisplaydu
snu=l nsnu
cnu+d nuH 87(164)
=l ndnu−cnu
snuSI 266(4)
2./integraldisplaydu
cnu=1
k/primelnk/primesnu+d nu
cnuSI 266(5)
3./integraldisplaydu
dnu=1
k/primearctank/primesnu−cnu
k/primesnu+c nuH 88(166)
=1
k/primearccoscnu
dnuJA
=1
ik/primelncnu+ik/primesnu
dnuSI 266(6)
=1
k/primearcsink/primesnu
dnuJA
5.133
1./integraldisplay
snud u=1
kln(dn u−kcnu) H 87(161)
=1
karccoshdnu−k2cnu
1−k2JA
=1
karcsinh/parenleftbigg
kdnu−cnu
1−k2/parenrightbigg
; JA
=−1
kln (dn u+kcnu) SI 365(1)
2./integraldisplay
cnud u=1
karccos(dn u); H 87(162)
=i
kln(dn u−iksnu); SI 265(2)a, ZH 87(162)
=1
karcsin( ksnu) JA
5.137 Jacobian elliptic functions 625
3./integraldisplay
dnud u= arcsin (sn u); H 87(163)
=a m u=iln (cn u−isnu) SI 266(3), ZH 87(163)
5.134
1./integraldisplay
sn2ud u=1
k2[u−E(amu,k)] PE (564)
2./integraldisplay
cn2ud u=1
k2/bracketleftBig
E(amu,k)−k/prime2u/bracketrightBig
PE (565)
3./integraldisplay
dn2ud u=E(amu,k) PE (566)
5.135
1./integraldisplaysnu
cnudu=1
k/primelndnu+k/prime
cnuSI 266(7)
=1
2k/primelndnu+k/prime
dnu−k/primeH 88(167)
2./integraldisplaysnu
dnudu=i
kk/primelnik/prime−kcnu
dnuSI 266(8)
=1
kk/primearccotkcnu
k/prime
3./integraldisplaycnu
snudu=l n1−dnu
snuSI 266(10)
=1
2ln1−dnu
1+d n uH 88(168)
4./integraldisplaycnu
dnudu=−1
kln1−ksnu
dnuSI 266(9)
=1
2kln1+ksnu
1−ksnu
5./integraldisplaydnu
cnudu=1
2ln1+s n u
1−snuH 88(172)
=l n1+s n u
cnuJA
6./integraldisplaydnu
snudu=1
2ln1−cnu
1+c n uH 87(170)
5.136
1./integraldisplay
snucnud u=−1
k2dnu
2./integraldisplay
snudnud u=−cnu
3./integraldisplay
cnudnud u=s nu
5.137
1./integraldisplaysnu
cn2udu=1
k/prime2dnu
cnuH 88(173)
626 Elliptic Integrals and Functions 5.138
2./integraldisplaysnu
dn2udu=−1
k/prime2cnu
dnuH 88(175)
3./integraldisplaycnu
sn2udu=−dnu
snuH 88(174)
4./integraldisplaycnu
dn2udu=snu
dnuH 88(177)
5./integraldisplaydnu
sn2udu=−cnu
snuH 88(176)
6./integraldisplaydnu
cn2udu=snu
cnuH 88(178)
5.138
1./integraldisplaycnu
snudnudu=l nsnu
dnuH 88(183)
2./integraldisplaysnu
cnudnudu=1
k/prime2lndnu
cnuH 88(182)
3./integraldisplaydnu
snucnudu=l nsnu
cnuH 88(184)
5.139
1.11/integraldisplaycnudnu
snudu=l ns n u H 88(179)
2./integraldisplaysnudnu
cnudu=l n1
cnuH 88(180)
3./integraldisplaysnucnu
dnudu=−1
k2lndnu H 88(181)
5.14 Weierstrass elliptic functions
The invariants g1andg2used below are defined in 8.161.
5.141
1./integraldisplay
℘(u)du=−ζ(u)
2./integraldisplay
℘2(u)du=1
6℘/prime(u)+1
12g2u H 120(192)
3./integraldisplay
℘3(u)du=1
120℘/prime/prime/prime(u)−3
20g2ζ(u)+1
10g3u H 120(193)
4.8/integraldisplaydu
℘(u)−℘(v)=1
℘/prime(v)/bracketleftbigg
2uζ(v)+l nσ(u−v)
σ(u+v)/bracketrightbigg
[℘(v)/negationslash=e1,e2,e3]( s e e 8.162 )
H 120(194)
5./integraldisplayα℘(u)+β
γ℘(u)+δdu=au
γ+αδ−βγ
γ2℘/prime(v)/bracketleftbigg
lnσ(u+v)
σ(u−v)−2uζ(v)/bracketrightbigg
where v=℘−1/parenleftbigg−δ
γ/parenrightbigg
H 120(195)
5.221 The exponential integral function and powers 627
5.2 The Exponential Integral Function
5.21 The exponential integral function
5.211/integraldisplay∞
xEi(−βx)Ei(−γx)dx=/parenleftbigg1
β+1
γ/parenrightbigg
Ei[−(β+γ)x]
−xEi(−βx)Ei(−γx)−e−βx
βEi(−γx)−e−γx
γEi(−βx)
[Re(β+γ)>0] NT 53(2)
5.22 Combinations of the exponential integral function and powers
5.221
1./integraldisplay∞
xEi[−a(x+b)]
xn+1dx=/bracketleftbigg1
xn−(−1)n
bn/bracketrightbiggEi[−a(x+b)]
n+e−ab
nn−1/summationdisplay
k=0(−1)n−k−1
bn−k/integraldisplay∞
xe−ax
xk+1dx
[a>0,b > 0] NT 52(3)
2./integraldisplay∞
xEi[−a(x+b)]
x2dx=/parenleftbigg1
x+1
b/parenrightbigg
Ei[−a(x+b)]−e−abEi(−ax)
b
[a>0,b>0] NT 52(4)
3.∗/integraldisplay
xEi(−ax)dx=x2
2Ei(−ax)+1
2a2e−ax+xe−ax
2a[a>0]
4.∗/integraldisplay
xnEi(−ax)dx=xn+1
n+1Ei(−ax)+n!e−ax
(n+1 )an+1∞/summationdisplay
k=0(ax)k
k!
[a>0]
5.∗/integraldisplay
xEi(−ax)e−bxdx=1
b2Ei[−(a+b)x]−1
b2Ei(−ax)e−bx−x
bEi(−ax)e−bx−1
b(a+b)e−(a+b)x
[a>0,b > 0]
6.∗/integraldisplay
Ei2(−ax)dx=xEi2(−ax)+2
a/bracketleftbig
Ei(−ax)e−ax−Ei(−2ax)/bracketrightbig
[a>0]
7.∗/integraldisplay
xEi2(−ax)dx=x2
2Ei2(−ax)+/parenleftbigg1
a2+x
a/parenrightbigg
Ei(−ax)e−ax−1
a2Ei(−2ax)+1
a2e−2ax
[a>0]
8.∗/integraldisplayu
0Ei(−ax)dx=uEi(−au)+e−au−1
a[a>0]
9.∗/integraldisplay∞
0xEi/parenleftBig
−x
a/parenrightBig
Ei/parenleftBig
−x
b/parenrightBig
dx=/parenleftbigga2+b2
2/parenrightbigg
ln(a+b)−a2
2lna−b2
2lnb−ab
2
[a>0,b > 0]
628 The Sine Integral and the Cosine Integral 5.231
10.∗/integraldisplay∞
0x2Ei/parenleftBig
−x
a/parenrightBig
Ei/parenleftBig
−x
b/parenrightBig
dx=2
3/bracketleftbigg/parenleftbig
a3+b3/parenrightbig
ln(a+b)−a3lna−b3lnb−ab
a+b/parenleftbig
a2−ab+b2/parenrightbig/bracketrightbigg
[a>0,b > 0]
5.23 Combinations of the exponential integral and the exponential
5.231
1./integraldisplayx
0exEi(−x)dx=−lnx−C+exEi(−x) ET II 308(11)
1./integraldisplayx
0e−βxEi(−αx)dx=−1
β/braceleftbigg
e−βxEi(−αx)+l n/parenleftbigg
1+β
α/parenrightbigg
−Ei[−(α+β)x]/bracerightbigg
ET II 308(12)
5.3 The Sine Integral and the Cosine Integral
5.31
1./integraldisplay
cosαxci(βx)dx=sinαxci(βx)
α−si(αx+βx)+s i ( αx−βx)
2αNT 49(1)
2./integraldisplay
sinαxci(βx)dx=−cosαxci(βx)
α+ci(αx+βx) + ci( αx−βx)
2αNT 49(2)
5.32
1./integraldisplay
cosαxsi(βx)dx=sinαxsi(βx)
α+ci(αx+βx)−ci(αx−βx)
2αNT 49(3)
2./integraldisplay
sinαxsi(βx)dx=−cosαxsi(βx)
α+si(αx+βx)−si(αx−βx)
2αNT 49(4)
5.33
1./integraldisplay
ci(αx)ci(βx)dx=xci(αx)ci(βx)+1
2α(si(αx+βx)+s i ( αx−βx))
+1
2β(si(αx+βx)+s i ( βx−αx))−1
αsinαxci(βx)−1
βsinβxci(αx)
NT 53(5)
2./integraldisplay
si(αx)si(βx)dx=xsi(αx)si(βx)−1
2β(si(αx+βx)+s i ( αx−βx))
−1
2α(si(αx+βx)+s i ( βx+αx)) +1
αcosαxsi(βx)+1
βcosβxsi(αx)
NT 54(6)
3./integraldisplay
si(αx)ci(βx)dx=xsi(αx)ci(βx)+1
αcosαxci(βx)
−1
βsinβxsi(αx)−/parenleftbigg1
2α+1
2β/parenrightbigg
ci(αx+βx)−/parenleftbigg1
2α−1
2β/parenrightbigg
ci(αx−βx)
NT 54(10)
5.54 Combinations of the exponential integral and the exponential 629
5.34
1./integraldisplay∞
xsi[a(x+b)]dx
x2=/parenleftbigg1
x+1
b/parenrightbigg
si[a(x+b)]−cosabsi(ax)+s i n abci(ax)
b
[a>0,b > 0] NT 52(6)
2./integraldisplay∞
xci[a(x+b)]dx
x2=/parenleftbigg1
x+1
b/parenrightbigg
ci[a(x+b)] +sinabsi(ax)−cosabci(ax)
b
[a>0,b > 0] NT 52(5)
5.4 The Probability Integral and Fresnel Integrals
5.4111/integraldisplay
Φ(αx)dx=xΦ(αx)+e−α2x2
α√πNT 12(20)a
5.42/integraldisplay
S(αx)dx=xS(αx)+cos2αx2
α√
2πNT 12(22)a
5.43/integraldisplay
C(αx)dx=xC(αx)−sin2αx2
α√
2πNT 12(21)a
5.5 Bessel Functions
Notation :ZandZdenote any of J,N,H(1),H(2). In formulae 5.52–5.56, Zp(x)a n d Zp(x) are arbitrary
Bessel functions of the first, second, or third kinds.
5.51/integraldisplay
Jp(x)dx=2∞/summationdisplay
k=0Jp+2k+1(x) J A ,M O3 0
5.52
1./integraldisplay
xp+1Zp(x)dx=xp+1Zp+1(x) WA 132(1)
2.11/integraldisplay
x−pZp+1(x)dx=−x−pZp(x) WA 132(2)
5.5310/integraldisplay/bracketleftbigg/parenleftbig
α2−β2/parenrightbig
x−p2−q2
x/bracketrightbigg
Zp(αx)Zq(βx)dx
=αxZp+1(αx)Zq(βx)−βxZp(αx)Zq+1(βx)−(p−q)Zp(αx)Zq(βx)
=βxZp(αx)Zq−1(βx)−αxZp−1(αx)Zq(βx)+(p−q)Zp(αx)Zq(βx)
JA, MO 30, WA 134(7)
5.54
1.10/integraldisplay
xZp(αx)Zp(βx)dx=αxZp+1(αx)Zp(βx)−βxZp(αx)Zp+1(βx)
α2−β2
=βxZp(αx)Zp−1(βx)−αxZp−1(αx)Zp(βx)
α2−β2
WA 134(8)
2./integraldisplay
x[Zp(αx)]2dx=x2
2/braceleftBig
[Zp(αx)]2−Zp−1(αx)Zp+1(αx)/bracerightBig
WA 135(11)
630 Bessel Functions 5.55
3.∗/integraldisplay
xZp(ax)Zp(ax)dx=x4
4/bracketleftbig
2Zp(ax)Zp(ax)−Zp−1(ax)Zp+1(ax)−Zp+1(ax)Zp−1(ax)/bracketrightbig
5.5510/integraldisplay1
xZp(αx)Zq(αx)dx=αxZp(αx)Zq+1(αx)−Zp+1(αx)Zq(αx)
p2−q2+Zp(αx)Zq(αx)
p+q
=αxZp−1(αx)Zq(αx)−Zp(αx)Zq−1(αx)
p2−q2−Zp(αx)Zq(αx)
p+q
WA 135(13)
5.56
1./integraldisplay
Z1(x)dx=−Z0(x) JA
2./integraldisplay
xZ0(x)dx=xZ1(x) JA
6–7 Definite Integrals of Special
Functions
6.1 Elliptic Integrals and Functions
Notation :k/prime=√
1−k2(cf. 8.1).
6.11 Forms containing F(x, k)
6.111/integraldisplayπ/2
0F(x, k)cotxdx=π
4K(k/prime)+1
2lnkK(k) BI (350)(1)
6.112
1./integraldisplayπ/2
0F(x, k)sinxcosx
1+ksin2xdx=1
4kK(k)ln(1 +k)√
k
2+π
16kK(k/prime) BI (350)(6)
2./integraldisplayπ/2
0F(x, k)sinxcosx
1−ksin2xdx=1
4kK(k)ln2
(1−k)√
k−π
16kK(k/prime) BI (350)(7)
3./integraldisplayπ/2
0F(x, k)sinxcosx
1−k2sin2xdx=−1
2k2lnk/primeK(k) BI (350)(2)a, BY(802.12)a
6.113
1./integraldisplayπ/2
0F(x, k/prime)sinxcosxdx
cos2x+ksin2x=1
4(1−k)ln2
(1 +k)√
kK(k/prime) BI (350)(5)
2./integraldisplayπ/2
0F(x, k)sinxcosx
1−k2sin2tsin2x·dx/radicalbig
1−k2sin2x
=−1
k2sintcost/bracketleftBig
K(k)a r c t a n( k/primetant)−π
2F(t, k)/bracketrightBig
BI (350)(12)
6.114/integraldisplayv
uF(x, k)dx/radicalBig/parenleftbig
sin2x−sin2u/parenrightbig/parenleftbig
sin2v−sin2x/parenrightbig=1
2c osusinvK(k)K/parenleftBig/radicalbig
1−tan2ucot2v/parenrightBig
/bracketleftbig
k2=1−cot2u·cot2v/bracketrightbig
BI (351)(9)
6.115/integraldisplay1
0F(arcsin x, k)xdx
1+kx2=1
4kK(k)ln(1 +k)√
k
2+π
16kK(k/prime)
(cf.6.112 2) BI (466)(1)
631
632 Elliptic Integrals and Functions 6.121
This and similar formulas can be obtained from formulas 6.111–6.113 by means of the substitution
x=a r c s i n t.
6.12 Forms containing E(x, k)
6.121/integraldisplayπ/2
0E(x, k)sinxcosx
1−k2sin2xdx=1
2k2/braceleftBig/parenleftBig
1+k/prime2/parenrightBig
K(k)−(2 + ln k/prime)E(k)/bracerightBig
BI (350)(4)
6.122/integraldisplayπ/2
0E(x, k)dx/radicalbig
1−k2sin2x=1
2{E(k)K(k)−lnk/prime} BI (350)(10), BY (630.02)
6.123/integraldisplayπ/2
0E(x, k)sinxcosx
1−k2sin2tsin2x·dx/radicalbig
1−k2sin2x
=−1
k2sintcost/bracketleftBig
E(k)arc t an( k/primetant)−π
2E(t, k)+π
2cott/parenleftBig
1−/radicalbig
1−k2sin2t/parenrightBig/bracketrightBig
BI (350)(13)
6.124/integraldisplayv
uE(x, k)dx/radicalBig/parenleftbig
sin2x−sin2u/parenrightbig/parenleftbig
sin2v−sin2x/parenrightbig=1
2c osusinvE(k)K⎛
⎝/radicalBigg
1−tg2u
tg2v⎞
⎠
+k2sinv
2c osuK⎛
⎝/radicalBigg
1−sin22u
sin22v⎞
⎠
/bracketleftbig
k2=1−cot2ucot2v/bracketrightbig
BI (351)(10)
6.13 Integration of elliptic integrals with respect to the modulus
6.131/integraldisplay1
0F(x, k)kd k=1−cosx
sinx=t a nx
2BY (616.03)
6.132/integraldisplay1
0E(x, k)kd k=sin2x+1−cosx
3s inxBY (616.04)
6.133/integraldisplay1
0Π/parenleftbig
x, r2,k/parenrightbig
kd k=t a nx
2−rln/radicalbigg
1+rsinx
1−rsinx−r2Π/parenleftbig
x, r2,0/parenrightbig
BY (616.05)
6.14–6.15 Complete elliptic integrals
6.141
1./integraldisplay1
0K(k)dk=2G FI II 755
2./integraldisplay1
0K(k/prime)dk=π2
4BY (615.03)
6.142/integraldisplay1
0/parenleftBig
K(k)−π
2/parenrightBigdk
k=πln2−2G BY (615.05)
6.1437/integraldisplay1
0K(k)dk
k/prime=K2/parenleftBigg√
2
2/parenrightBigg
=1
16πΓ4/parenleftbigg1
4/parenrightbigg
BY (615.08)
6.144/integraldisplay1
0K(k)dk
1+k=π2
8BY (615.09)
6.161 The theta function 633
6.145/integraldisplay1
0/parenleftbigg
K(k/prime)−ln4
k/parenrightbiggdk
k=1
12/bracketleftBig
24(ln2)2−π2/bracketrightBig
BY (615.13)
6.146 n2/integraldisplay1
0knK(k)dk=(n−1)2/integraldisplay1
0kn−2K(k)dk+1 BY (615.12)
6.147 n/integraldisplay1
0knK(k/prime)dk=(n−1)/integraldisplay1
0kn−2E(k)dk [n>1] (see 6.152 ) BY (615.11)
6.148
1./integraldisplay1
0E(k)dk=1
2+G BY (615.02)
2./integraldisplay1
0E(k/prime)dk=π2
8BY (615.04)
3.∗/integraldisplay1
0E(k)
1+kdk=1
6.149
1./integraldisplay1
0/parenleftBig
E(k)−π
2/parenrightBigdk
k=πln2−2G+1−π
2BY (615.06)
2./integraldisplay1
0(E(k/prime)−1)dk
k=2l n2 −1 BY (615.07)
3.∗/integraldisplay1
0E(k)
1+kdk=1
4.∗/integraldisplay1
0dx
x3/parenleftbigg/radicalbig
a−x2K(x)−E(x)√
1−x2+π
4x2/parenrightbigg
=−π
4ln/parenleftbigg4√e/parenrightbigg
6.151/integraldisplay1
0E(k)dk
k/prime=1
8⎡
⎢⎢⎢⎢⎣4K
2/parenleftBigg√
2
2/parenrightBigg
+π2
K2/parenleftBigg√
2
2/parenrightBigg⎤
⎥⎥⎥⎥⎦ BY (615.10)
6.152 (n+2 )/integraldisplay1
0knE(k/prime)dk=(n+1 )/integraldisplay1
0knK(k/prime)dk [n>1] (see 6.147 ) BY (615.14)
6.1536/integraldisplaya
0K(k)kd k
k/prime2√
a2−k2=π
41√
1−a2ln/parenleftbigg1+a
1−a/parenrightbigg
[0<a< 1] LO I 252
6.154/integraldisplayπ/2
0E(psinx)
1−p2sin2xsinxdx=π
2/radicalbig
1−p2/bracketleftbig
p2>1/bracketrightbig
FI II 489
6.16 The theta function
6.161
1./integraldisplay∞
0xs−1ϑ2/parenleftbig
0|ix2/parenrightbig
dx=2s/parenleftbig
1−2−s/parenrightbig
π−s
2Γ/parenleftbig1
2s/parenrightbig
ζ(s)
[Res>2] ET I 339(20)
2./integraldisplay∞
0xs−1/bracketleftbig
ϑ3/parenleftbig
0|ix2/parenrightbig
−1/bracketrightbig
dx=π−s
2Γ/parenleftbig1
2s/parenrightbig
ζ(s)[ R e s>2] ET I 339(21)
634 Elliptic Integrals and Functions 6.162
3./integraldisplay∞
0xs−1/bracketleftbig
1−ϑ4/parenleftbig
0|ix2/parenrightbig/bracketrightbig
dx=/parenleftbig
1−21−s/parenrightbig
π−1
2sΓ/parenleftbig1
2s/parenrightbig
ζ(s)
[Res>2] ET I 339(22)
4./integraldisplay∞
0xs−1/bracketleftbig
ϑ4/parenleftbig
0|ix2/parenrightbig
+ϑ2/parenleftbig
0|ix2/parenrightbig
−ϑ3/parenleftbig
0|ix2/parenrightbig/bracketrightbig
dx=−(2s−1)/parenleftbig
21−s−1/parenrightbig
π−1
2sΓ/parenleftbig1
2s/parenrightbig
ζ(s)
ET I 339(24)
6.162
1.11/integraldisplay∞
0e−axϑ4/parenleftbiggbπ
2l/vextendsingle/vextendsingle/vextendsingle/vextendsingleiπx
l2/parenrightbigg
dx=l√acosh/parenleftbig
b√a/parenrightbig
cosech/parenleftbig
l√a/parenrightbig
[Rea>0,|b|≤l] ET I 224(1)a
2./integraldisplay∞
0e−axϑ1/parenleftbiggbπ
2l/vextendsingle/vextendsingle/vextendsingle/vextendsingleiπx
l2/parenrightbigg
dx=−l√asinh/parenleftbig
b√a/parenrightbig
sech/parenleftbig
l√a/parenrightbig
[Rea>0,|b|≤l] ET I 224(2)a
3.11/integraldisplay∞
0e−axϑ2/parenleftbigg(l+b)π
2l/vextendsingle/vextendsingle/vextendsingle/vextendsingleiπx
l2/parenrightbigg
dx=−l√asinh/parenleftbig
b√a/parenrightbig
sech/parenleftbig
l√a/parenrightbig
[Rea>0,|b|≤l] ET I 224(3)a
4.11/integraldisplay∞
0e−axϑ3/parenleftbigg(l+b)π
2l/vextendsingle/vextendsingle/vextendsingle/vextendsingleiπx
l2/parenrightbigg
dx=l√acosh/parenleftbig
b√a/parenrightbig
cosech/parenleftbig
l√a/parenrightbig
[Rea>0,|b|≤l] ET I 224(4)a
6.16310
1./integraldisplay∞
0e−(a−μ)xϑ3(π√μx|iπx)dx=1
2√a/bracketleftbig
coth/parenleftbig√a+√μ/parenrightbig
+c o t h/parenleftbig√a−√μ/parenrightbig/bracketrightbig
[Rea>0] ET I 224(7)a
2.10/integraldisplay∞
0ϑ3(iπkx|iπx)e−(k2+l2)xdx=sinh 2l
l(cosh 2 l−cos2k)
6.16411/integraldisplay∞
0/bracketleftbig
ϑ4/parenleftbig
0|ie2x/parenrightbig
+ϑ2/parenleftbig
0|ie2x/parenrightbig
−ϑ3/parenleftbig
0|ie2x/parenrightbig/bracketrightbig
e1
2xcos(ax)dx
=1
2/parenleftBig
21
2+ia−1/parenrightBig/parenleftBig
1−21
2−ia/parenrightBig
π−1
4−1
2iaΓ/parenleftbig1
4+1
2ia/parenrightbig
ζ/parenleftbig1
2+ia/parenrightbig
[a>0] ET I 61(11)
6.165/integraldisplay∞
0e1
2x/bracketleftbig
ϑ3/parenleftbig
0|ie2x/parenrightbig
−1/bracketrightbig
cos(ax)dx
=2
1+4a2/braceleftBig
1+/bracketleftBig/parenleftbig
a2+1
4/parenrightbig
π−1
2ia−1
4Γ/parenleftbig1
2ia+1
4/parenrightbig
ζ/parenleftbig
ia+1
2/parenrightbig/bracketrightBig/bracerightBig
[a>0] ET I 61(12)
6.165 Generalized elliptic integrals 635
6.1710Generalized elliptic integrals
1. Set
Ωj(k)≡/integraldisplayπ
0/bracketleftbig
1−k2cosφ/bracketrightbig−(j+1
2)dφ,
αm(j)=π
(64)mj!
(2j)!(4m+2j)!
(2m+j)!/parenleftbigg1
m!/parenrightbigg2
,λ =π
2/radicalbigg
(2j+1 )k2
1−k2,
then
Ωj(k)=∞/summationdisplay
m=0αm(j)k4m=/radicalbiggπ
(2j+1 )k2/parenleftbig
1−k2/parenrightbig−j⎡
⎣erfλ+1
2(2j+1 )−1/parenleftbigg
1+1
2k2/parenrightbigg
×/braceleftbigg
erfλ−/parenleftbigg2√π/parenrightbigg/parenleftBig
λe−λ2/parenrightBig/parenleftbigg
1+2
3λ2/parenrightbigg/bracerightbigg
−1
12(2j+1 )−2/parenleftbigg
16 +13
k2+1
k4/parenrightbigg
×/braceleftbigg
erfλ−/parenleftbigg2√π/parenrightbigg/parenleftBig
λe−λ2/parenrightBig/parenleftbigg
1+2
3λ2+4
15λ4/parenrightbigg/bracerightbigg
+...⎤
⎦
while for large λ
lim
j→∞Ωj(k)=/radicalbiggπ
(2j+1 )k2/parenleftbig
1−k2/parenrightbig−j
×/bracketleftbigg
1+1
2(2j+1 )−1/braceleftbigg
1+1
2k2/bracerightbigg
−4
3(2j+1 )−2/braceleftbigg
1+13
16k2+1
16k4/bracerightbigg
+.../bracketrightbigg
2. Set
Rμ(k,α,δ )=/integraldisplayπ
0cos2α−1(θ/2)sin2δ−2α−1(θ/2)dθ
[1−k2cosθ]μ+1
2,
0<k< 1,Reδ>Reα>0,Reμ>−1/2,
Mν(μ,α,δ )=(−1)ν2ν/parenleftbig
μ+1
2/parenrightbig
ν
ν!Γ(α)Γ(δ−α+ν)
Γ(δ+ν),
with ( λ)ν=Γ (λ+ν)/Γ(λ),and
Wν(μ,α,δ )=2ν/parenleftbig
μ+1
2/parenrightbig
ν
ν!Γ(α+ν)Γ(δ−α)
Γ(δ+ν),
then:
•for small k:
Rμ(k,α,δ )=/parenleftbig
1−k2/parenrightbig−(μ+1
2)∞/summationdisplay
ν=0/bracketleftbig
k2//parenleftbig
1−k2/parenrightbig/bracketrightbigνMν(μ,α,δ )
=/parenleftbig
1+k2/parenrightbig−(μ+1
2)∞/summationdisplay
ν=0/bracketleftbig
k2//parenleftbig
1+k2/parenrightbig/bracketrightbigνWν(μ,α,δ ),
636 The Exponential Integral Function and Functions Generated by It 6.211
•fork2close to 1:
Rμ(k,α,δ )
=/bracketleftbig
Γ(δ−α)Γ/parenleftbig
μ+α−δ+1
2/parenrightbig
Γ/parenleftbig
μ+1
2/parenrightbig/bracketrightbig/parenleftbig
2k2/parenrightbigα−δ/parenleftbig
1−k2/parenrightbigδ−α−μ−1
2
×/braceleftBig
Γ/parenleftbig
δ−α−μ−1
2/parenrightbig
Γ(α)/bracketleftBig
Γ/parenleftbig
δ−μ−1
2/parenrightbig/parenleftbig
2k2/parenrightbigμ+1
2/bracketrightBig/bracerightBig
/bracketleftbig
Re/parenleftbig
μ+α−δ+1
2/parenrightbig
not an integer/bracketrightbig
=/bracketleftBig
2μ+1
2k2μ+1Γ/parenleftbig
μ+1
2/parenrightbig
Γ(1−α)/bracketrightBig
×∞/summationdisplay
n=0/bracketleftbig
Γ(δ−α+n)Γ ( 1−α+n)Γ/parenleftbig
α−δ+μ−n+1
2/parenrightbig
n!/bracketrightbig/bracketleftbig
2k2//parenleftbig
1−k2/parenrightbig/bracketrightbigα−δ+μ−n+1
2
/bracketleftbig
α−δ+μ+1
2=m,withma non-negative integer/bracketrightbig
6.2–6.3 The Exponential Integral Function and Functions Generated
by It
6.21 The logarithm integral
6.211/integraldisplay1
0li(x)dx=−ln 2 BI (79)(5)
6.212
1./integraldisplay1
0li/parenleftbigg1
x/parenrightbigg
xdx=0 BI (255)(1)
2./integraldisplay1
0li(x)xp−1dx=−1
pln(p+1 ) [ p>−1] BI (255)(2)
3./integraldisplay1
0li(x)dx
xq+1=1
qln(1−q)[ q<1] BI (255)(3)
4./integraldisplay∞
1li(x)dx
xq+1=−1
qln(q−1) [ q>1] BI (255)(4)
6.213
1./integraldisplay1
0li/parenleftbigg1
x/parenrightbigg
sin (alnx)dx=1
1+a2/parenleftBig
alna−π
2/parenrightBig
[a>0] BI (475)(1)
2./integraldisplay∞
1li/parenleftbigg1
x/parenrightbigg
sin(alnx)dx=−1
1+a2/parenleftBigπ
2+alna/parenrightBig
[a>0] BI (475)(9)
3./integraldisplay1
0li/parenleftbigg1
x/parenrightbigg
cos(alnx)dx=−1
1+a2/parenleftBig
lna+π
2a/parenrightBig
[a>0] BI (475)(2)
4./integraldisplay∞
1li/parenleftbigg1
x/parenrightbigg
cos(alnx)dx=1
1+a2/parenleftBig
lna−π
2a/parenrightBig
[a>0] BI (475)(10)
5./integraldisplay1
0li(x)sin(alnx)dx
x=ln/parenleftbig
1+a2/parenrightbig
2a[a>0] BI(479)(1), ET I 98(20)a
6.216 The logarithm integral 637
6./integraldisplay1
0li(x)cos(alnx)dx
x=−arctan a
aBI (479)(2)
7./integraldisplay1
0li(x)sin(alnx)dx
x2=1
1+a2/parenleftBig
alna+π
2/parenrightBig
[a>0] BI (479)(3)
8./integraldisplay∞
1li(x)sin(alnx)dx
x2=1
1+a2/parenleftBigπ
2−alna/parenrightBig
[a>0] BI (479)(13)
9./integraldisplay1
0li(x)cos(alnx)dx
x2=1
1+a2/parenleftBig
lna−π
2a/parenrightBig
[a>0] BI (479)(4)
10./integraldisplay∞
1li(x)cos(alnx)dx
x2=−1
1+a2/parenleftBig
lna+π
2a/parenrightBig
[a>0] BI (479)(14)
11./integraldisplay1
0li(x)sin(alnx)xp−1dx=1
a2+p2/braceleftbigga
2ln/bracketleftbig
(1 +p)2+a2/bracketrightbig
−parctana
1+p/bracerightbigg
[p>0] BI (477)(1)
12./integraldisplay1
0li(x)cos(alnx)xp−1dx=−1
a2+p2/braceleftbigg
aarctana
1+p+p
2ln/bracketleftbig
(1 +p)2+a2/bracketrightbig/bracerightbigg
[p>0] BI (477)(2)
6.214
1./integraldisplay1
0li/parenleftbigg1
x/parenrightbigg/parenleftbigg
ln1
x/parenrightbiggp−1
dx=−πcotpπ·Γ(p)[ 0 <p< 1] BI (340)(1)
2./integraldisplay∞
1li/parenleftbigg1
x/parenrightbigg
(lnx)p−1dx=−π
sinpπΓ(p)[ 0 <p< 1] BI (340)(9)
6.215
1./integraldisplay1
0li(x)xp−1
/radicalBigg
ln/parenleftbigg1
x/parenrightbiggdx=−2/radicalbiggπ
parcsinh√p=−2/radicalbiggπ
pln/parenleftBig√p+/radicalbig
p+1/parenrightBig
[p>0] BI (444)(3)
2./integraldisplay1
0li(x)dx
xp+1/radicalBigg
ln/parenleftbigg1
x/parenrightbigg=−2/radicalbiggπ
parcsin√p [1>p> 0] BI (444)(4)
6.216
1./integraldisplay1
0li(x)/bracketleftbigg
ln/parenleftbigg1
x/parenrightbigg/bracketrightbiggp−1ax
x=−1
pΓ(p)[ 0 <p≤1] BI (444)(1)
2./integraldisplay1
0li(x)/bracketleftbigg
ln/parenleftbigg1
x/parenrightbigg/bracketrightbiggp−1dx
x2=−πΓ(p)
sinpπ[0<p< 1] BI (444)(2)
638 The Exponential Integral Function and Functions Generated by It 6.221
6.22–6.23 The exponential integral function
6.221/integraldisplayp
0Ei(αx)dx=pEi(αp)+1−eαp
αNT 11(7)
6.222/integraldisplay∞
0Ei(−px)Ei(−qx)dx=/parenleftbigg1
p+1
q/parenrightbigg
ln(p+q)−lnq
p−lnp
q
[p>0,q > 0] FI II 653, NT 53(3)
6.223/integraldisplay∞
0Ei(−βx)xμ−1dx=−Γ(μ)
μβμ[Reβ≥0,Reμ>0]
NT 55(7), ET I 325(10)
6.224
1./integraldisplay∞
0Ei(−βx)e−μxdx=−1
μln/parenleftbigg
1+μ
β/parenrightbigg
[Re(β+μ)≥0,μ > 0]
=−1/β [μ=0 ]
FI II 652, NT 48(8)
2./integraldisplay∞
0Ei(ax)e−μxdx=−1
μln/parenleftBigμ
a−1/parenrightBig
[a>0,Reμ>0, μ>a ]
ET I 178(23)a, BI (283)(3)
6.225
1./integraldisplay∞
0Ei/parenleftbig
−x2/parenrightbig
e−μx2dx=−/radicalbiggπ
μarcsinh√μ=−/radicalbiggπ
μln/parenleftBig√μ+/radicalbig
1+μ/parenrightBig
[Reμ>0] BI (283)(5), ET I 178(25)a
2./integraldisplay∞
0Ei/parenleftbig
−x2/parenrightbig
epx2dx=−/radicalbiggπ
parcsin√p [1>p> 0] NT 59(9)a
6.226
1./integraldisplay∞
0Ei/parenleftbigg
−1
4x/parenrightbigg
e−μxdx=−2
μK0(√μ)[ R e μ>0] MI 34
2./integraldisplay∞
0Ei/parenleftbigga2
4x/parenrightbigg
e−μxdx=−2
μK0(a√μ)[ a>0,Reμ>0] MI 34
3./integraldisplay∞
0Ei/parenleftbigg
−1
4x2/parenrightbigg
e−μx2dx=/radicalbiggπ
μEi(−√μ)[ R e μ>0] MI 34
4./integraldisplay∞
0Ei/parenleftbigg
−1
4x2/parenrightbigg
e−μx2+1
4x2dx=/radicalbiggπ
μ[cos√μci√μ−sin√μsi√μ]
[Reμ>0] MI 34
6.227
1./integraldisplay∞
0Ei(−x)e−μxxdx=1
μ(μ+1 )−1
μ2ln(1 + μ)[ R e μ>0] MI 34
6.241 The sine integral and cosine integral functions 639
2./integraldisplay∞
0/bracketleftbigge−axEi(ax)
x−b−eaxEi(−ax)
x+b/bracketrightbigg
dx=0 [ a>0,b < 0]
=π2e−ab[a>0,b > 0]
ET II 253(1)a
6.228
1./integraldisplay∞
0Ei(−x)exxν−1dx=−πΓ(ν)
sinνπ[0<Reν<1] ET II 308(13)
2./integraldisplay∞
0Ei(−βx)e−μxxν−1dx=−Γ(ν)
ν(β+μ)ν2F1/parenleftbigg
1,ν;ν+1 ;μ
β+μ/parenrightbigg
[|argβ|<π , Re(β+μ)>0,Reν>0]ET II 308(14)
6.229/integraldisplay∞
0Ei/parenleftbigg
−1
4x2/parenrightbigg
exp/parenleftbigg
−μx2+1
4x2/parenrightbiggdx
x2=2√π(cos√μsi√μ−sin√μci√μ)
[Reμ>0] MI 34
6.231/integraldisplay∞
−lna/bracketleftbig
Ei(−a)−Ei/parenleftbig
−e−x/parenrightbig/bracketrightbig
e−μxdx=1
μγ(μ, a)[ a<1,Reμ>0] MI 34
6.232
1./integraldisplay∞
0Ei(−ax)sinbxdx =−ln/parenleftbigg
1+b2
a2/parenrightbigg
2b[a>0,b > 0] BI (473)(1)a
2./integraldisplay∞
0Ei(−ax)cosbxdx =−1
barctanb
a[a>0,b > 0] BI (473)(2)a
6.233
1./integraldisplay∞
0Ei(−x)e−μxsinβxdx =−1
β2+μ2/braceleftbiggβ
2ln/bracketleftbig
(1 +μ)2+β2/bracketrightbig
−μarctanβ
1+μ/bracerightbigg
[Reμ>|Imβ|] BI (473)(7)a
2./integraldisplay∞
0Ei(−x)e−μxcosβxdx =−1
β2+μ2/braceleftbiggμ
2ln/bracketleftbig
(1 +μ)2+β2/bracketrightbig
+βarctanβ
1+μ/bracerightbigg
[Reμ>|Imβ|] BI (473)(8)a
6.234/integraldisplay∞
0Ei(−x)lnxdx=C+1 NT 56(10)
6.24–6.26 The sine integral and cosine integral functions
6.241
1./integraldisplay∞
0si(px)si(qx)dx=π
2p[p≥q] BI II 653, NT 54(8)
2./integraldisplay∞
0ci(px)ci(qx)dx=π
2p[p≥q] FI II 653, NT 54(7)
640 The Exponential Integral Function and Functions Generated by It 6.242
3./integraldisplay∞
0si(px)ci(qx)dx=1
4qln/parenleftbiggp+q
p−q/parenrightbigg2
+1
4pln/parenleftbig
p2−q2/parenrightbig2
q4[p/negationslash=q]
=1
qln2 [ p=q]
FI II 653, NT 54(10, 12)
6.242/integraldisplay∞
0ci(ax)
β+xdx=−1
2/braceleftBig
[si(aβ)]2+ [ci(aβ)]2/bracerightBig
[a>0,|argβ|<π] ET II 224(1)
6.243
1./integraldisplay∞
−∞si (a|x|)
x−bsignxdx=πci (a|b|)[ a>0,b > 0] ET II 253(3)
2./integraldisplay∞
−∞ci (a|x|)
x−bdx=−πsignb·si (a|b|)[ a>0] ET II 253(2)
6.244
1.8/integraldisplay∞
0si(px)xdx
q2+x2=π
2Ei(−pq)[ p>0,q > 0] BI (255)(6)
2.8/integraldisplay∞
0si(px)xdx
q2−x2=−π
2ci(pq)[ p>0,q > 0] BI (255)(6)
6.245
1./integraldisplay∞
0ci(px)dx
q2+x2=π
2qEi(−pq)[ p>0,q > 0] BI (255)(7)
2./integraldisplay∞
0ci(px)dx
q2−x2=π
2qsi(pq)[ p>0,q > 0] BI (255)(8)
6.246
1./integraldisplay∞
0si(ax)xμ−1dx=−Γ(μ)
μaμsinμπ
2[a>0,0<Reμ<1]
NT 56(9), ET I 325(12)a
2./integraldisplay∞
0ci(ax)xμ−1dx=−Γ(μ)
μaμcosμπ
2[a>0,0<Reμ<1]
NT 56(8), ET I 325(13)a
6.247
1./integraldisplay∞
0si(βx)e−μxdx=−1
μarctanμ
β[Reμ>0] NT 49(12), ET I 177(18)
2./integraldisplay∞
0ci(βx)e−μxdx=−1
μln/radicalBigg
1+μ2
β2[Reμ>0] NT 49(11), ET I 178(19)a
6.248
1.8/integraldisplay∞
0si(x)e−μx2xdx=π
4μ/bracketleftbigg
Φ/parenleftbigg1
2√μ/parenrightbigg
−1/bracketrightbigg
[Reμ>0] MI 34
6.253 The sine integral and cosine integral functions 641
2./integraldisplay∞
0ci(x)e−μx2dx=1
4/radicalbiggπ
μEi/parenleftbigg
−1
4μ/parenrightbigg
[Reμ>0] MI 34
6.249/integraldisplay∞
0/bracketleftBig
si/parenleftbig
x2/parenrightbig
+π
2/bracketrightBig
e−μxdx=π
μ/braceleftBigg/bracketleftbigg
S/parenleftbiggμ2
4/parenrightbigg
−1
2/bracketrightbigg2
+/bracketleftbigg
C/parenleftbiggμ2
4/parenrightbigg
−1
2/bracketrightbigg2/bracerightBigg
[Reμ>0] ME 26
6.251
1./integraldisplay∞
0si/parenleftbigg1
x/parenrightbigg
e−μxdx=2
μkei (2√μ)[ R e μ>0] MI 34
2./integraldisplay∞
0ci/parenleftbigg1
x/parenrightbigg
e−μxdx=−2
μker (2√μ)[ R e μ>0] MI 34
6.252
1./integraldisplay∞
0sinpxsi(qx)dx=−π
2p/bracketleftbig
p2>q2/bracketrightbig
=−π
4p/bracketleftbig
p2=q2/bracketrightbig
=0/bracketleftbig
p2<q2/bracketrightbig
FI II 652, NT 50(8)
2.6/integraldisplay∞
0cospxsi(qx)dx=−1
4pln/parenleftbiggp+q
p−q/parenrightbigg2/bracketleftbig
p/negationslash=0,p2/negationslash=q2/bracketrightbig
=1
q[p=0 ]
FI II 652, NT 50(10)
3./integraldisplay∞
0sinpxci(qx)dx=−1
4pln/parenleftbiggp2
q2−1/parenrightbigg2/bracketleftbig
p/negationslash=0,p2/negationslash=q2/bracketrightbig
=0 [ p=0 ]
FI II 652, NT 50(9)
4./integraldisplay∞
0cospxci(qx)dx=−π
2p/bracketleftbig
p2>q2/bracketrightbig
=−π
4p/bracketleftbig
p2=q2/bracketrightbig
=0/bracketleftbig
p2<q2/bracketrightbig
FI II 654, NT 50(7)
6.253/integraldisplay∞
0si(ax)sinbx
1−2rcosx+r2dx=−π/parenleftbig
rm+rm+1/parenrightbig
4b(1−r)(1−r2)[b=a−m]
=−π/parenleftbig
2+2r−rm−rm+1/parenrightbig
4b(1−r)(1−r2)[b=a+m]
=−πrm+1
2b(1−r)(1−r2)[a−m−1<b<a −m]
=−π/parenleftbig
1+r−rm+1/parenrightbig
2b(1−r)(1−r2)[a+m<b<a +m+1 ]
ET I 97(10)
642 The Exponential Integral Function and Functions Generated by It 6.254
6.254
1.∗/integraldisplay∞
0ci(x)sin2xdx
x=1
2/bracketleftbigg
L2/parenleftbigg1
2/parenrightbigg
−L2/parenleftbigg
−1
2/parenrightbigg/bracketrightbigg
where L2(x) is the Euler dilogarithm defined as L2(z)=−/integraldisplayz
0log(1−t)
tdta n dt h i si nt u r nc a n
be expressed as L2(z)=Φ ( z,2,1) in terms of the Lerch function defined in 9.550, with zreal.
2.11/integraldisplay∞
0/bracketleftBig
si(ax)+π
2/bracketrightBig
cosbx·dx
x=π
2lna
bH(a−b)
[a>0,b > 0,H(x) is the Heaviside step function] ET I 41(11)
6.255
1./integraldisplay∞
−∞[cosaxci (a|x|)+s i n( a|x|)s i(a|x|)]dx
x−b=−π[signbcosabsi(a|b|)−sinabci(a|b|)]
[a>0] ET II 253(4)
2./integraldisplay∞
−∞[sinaxci (a|x|)−signxcosaxsi (a|x|)]dx
x−b=−π[sin (a|b|)s i(a|b|) + cos abci(a|b|)]
[a>0] ET II 253(5)
6.256
1./integraldisplay∞
0/bracketleftbig
si2(x)+c i2(x)/bracketrightbig
cosaxdx =π
aln(1 + a)[ a>0]
2.∗/integraldisplay∞
0[si(x)cosx−ci(x)sinx]2dx=π
2
3.∗/integraldisplay∞
0si2(x)cos(ax)dx=π
2alog(1 + a)[ 0 ≤a≤2]
4.∗/integraldisplay∞
0ci2(x)cos(ax)dx=π
2alog(1 + a)[ 0 ≤a≤2]
6.257/integraldisplay∞
0si/parenleftBiga
x/parenrightBig
sinbxdx =−π
2bJ0/parenleftBig
2√
ab/parenrightBig
[b>0] ET I 42(18)
6.258
1./integraldisplay∞
0/bracketleftBig
si(ax)+π
2/bracketrightBig
sinbxdx
x2+c2
=π
4c/braceleftbig
e−bc[Ei(bc)−Ei(−ac)] +ebc[Ei(−ac)−Ei(−bc)]/bracerightbig
[0<b≤a, c > 0]
=π
4ce−bc[Ei(ac)−Ei(−ac)] [0 <a≤b, c > 0]
BI (460)(1)
2./integraldisplay∞
0/bracketleftBig
si(ax)+π
2/bracketrightBig
cosbxxdx
x2+c2
=−π
4/braceleftbig
e−bc[Ei(bc)−Ei(−ac)] +ebc[Ei(−bc)−Ei(−ac)]/bracerightbig
[0<b≤a, c > 0]
=π
4e−bc[Ei(−ac)−Ei(ac)] [0 <a≤b, c > 0]
BI (460)(2, 5)
6.262 The sine integral and cosine integral functions 643
6.259
1./integraldisplay∞
0si(ax)sinbxdx
x2+c2=π
2cEi(−ac)sin h( bc)[ 0 <b≤a, c > 0]
=π
4ce−cb[Ei(−bc)+E i ( bc)−Ei(−ac)−Ei(ac)]
+π
2cEi(−bc)sin h( bc)[ 0 <a≤b, c > 0]
ET I 96(8)
2./integraldisplay∞
0ci(ax)sinbxxdx
x2+c2=−π
2sinh(bc)Ei (−ac)[ 0 <b≤a, c > 0]
=−π
2sinh(bc)Ei (−bc)+π
4e−bc[Ei(−bc)+E i ( bc)
−Ei(−ac)−Ei(ac)] [0 <a≤b, c > 0]
BI (460)(3)a, ET I 97(15)a
3./integraldisplay∞
0ci(ax)cosbxdx
x2+c2
=π
2ccoshbcEi(−ac)[ 0 <b≤a, c > 0]
=π
4c/braceleftbig
e−bc[Ei(ac)+E i ( −ac)−Ei(bc)] +ebcEi(−bc)/bracerightbig
[0<a≤b, c > 0]
BI (460)(4), ET I 41(15)
4.∗/integraldisplay∞
0[ci(x)sinx−Si(x)cosx]s i nxxdx
a2+x2=1
8/bracketleftbig
Ei(a)e−a−Ei(−a)ea/bracketrightbig2
[areal]
5.∗/integraldisplay∞
0[ci(x)sinx−Si(x)cosx]2xdx
a2+x2=π3e−|a|
8asinh(a)−π
8|a|/bracketleftbig
Ei(a)e−a−Ei(−a)ea/bracketrightbig2
[areal]
6.261
1./integraldisplay∞
0si(bx)cosaxe−pxdx=−1
2(a2+p2)/bracketleftbigga
2lnp2+(a+b)2
p2+(a−b)2+parctan2bp
b2−a2−p2/bracketrightbigg
[a>0,b > 0,p > 0] ET I 40(8)
2./integraldisplay∞
0si(βx)cosaxe−μxdx=−arctanμ+ai
β
2(μ+ai)−arctanμ−ai
β
2(μ−ai)
[a>0,Reμ>|Imβ|] ET I 40(9)
6.262
1./integraldisplay∞
0ci(bx)sinaxe−μxdx=1
2(a2+μ2)/braceleftBigg
μarctan2aμ
μ2+b2−a2−a
2ln/parenleftbig
μ2+b2−a2/parenrightbig2+4a2μ2
b4/bracerightBigg
[a>0,b > 0,Reμ>0]
ET I 98(16)a
644 The Exponential Integral Function and Functions Generated by It 6.263
2./integraldisplay∞
0ci(bx)cosaxe−pxdx=−1
2(a2+p2)⎧
⎨
⎩p
2ln/bracketleftBig/parenleftbig
b2+p2−a2/parenrightbig2+4a2p2/bracketrightBig
b4+aarctan2ap
b2+p2−a2⎫
⎬
⎭
[a>0,b > 0,Rep>0]ET I 41(16)
3./integraldisplay∞
0ci(βx)cosaxe−μxdx=−ln/bracketleftbigg
1+(μ+ai)2
β2/bracketrightbigg
4(μ+ai)−ln/bracketleftbigg
1+(μ−ai)2
β2/bracketrightbigg
4(μ−ai)
[a>0,Reμ>|Imβ|] ET I 41(17)
6.263
1./integraldisplay∞
0[ci(x)cosx+s i (x)sinx]e−μxdx=−π
2−μlnμ
1+μ2[Reμ>0] ME 26a, ET I 178(21)a
2./integraldisplay∞
0[si(x)cosx−ci(x)sinx]e−μxdx=−π
2μ+l nμ
1+μ2[Reμ>0] ME 26a, ET I 178(20)a
3./integraldisplay∞
0[sinx−xci(x)]e−μxdx=ln/parenleftbig
1+μ2/parenrightbig
2μ2[Reμ>0] ME 26
6.264
1./integraldisplay∞
0si(x)lnxdx=C+1 NT 46(10)
2./integraldisplay∞
0ci(x)lnxdx=π
2NT 56(11)
6.27 The hyperbolic sine integral and hyperbolic cosine integral functions
6.271
1./integraldisplay∞
0shi(x)e−μxdx=1
2μlnμ+1
μ−1=1
μarccoth μ [Reμ>1] MI 34
2.11/integraldisplay∞
0chi(x)e−μxdx=−1
2μln/parenleftbig
μ2−1/parenrightbig
[Reμ>1] MI 34
6.27211/integraldisplay∞
0chi(x)e−px2dx=1
4/radicalbiggπ
pEi/parenleftbigg1
4p/parenrightbigg
[p>0] MI 35
6.273
1.11/integraldisplay∞
0[coshxshi(x)−sinhxchi(x)]e−μxdx=lnμ
μ2−1[Reμ>0] MI 35
2.11/integraldisplay∞
0[coshxchi(x)+s i n h xshi(x)]e−μxdx=μlnμ
1−μ2[Reμ>2] MI 35
6.284 The probability integral 645
6.27411/integraldisplay∞
0[coshxshi(x)−sinhxchi(x)]e−μx2dx=1
4/radicalbiggπ
μe1
4μEi/parenleftbigg
−1
4μ/parenrightbigg
[Reμ>0] MI 35
6.275/integraldisplay∞
0[xchi(x)−sinhx]e−μxdx=−ln/parenleftbig
μ2−1/parenrightbig
2μ2[Reμ>1] MI 35
6.276/integraldisplay∞
0[coshxchi(x)+s i n h xshi(x)]e−μx2xdx=1
8/radicalbiggπ
μ3exp/parenleftbigg1
4μ/parenrightbigg
Ei/parenleftbigg
−1
4μ/parenrightbigg
[Reμ>0] MI 35
6.277
1./integraldisplay∞
0[chi(x) + ci( x)]e−μxdx=−ln/parenleftbig
μ4−1/parenrightbig
2μ[Reμ>1] MI 34
2./integraldisplay∞
0[chi(x)−ci(x)]e−μxdx=1
2μlnμ2+1
μ2−1[Reμ>1] MI 35
6.28–6.31 The probability integral
6.281
1.6/integraldisplay∞
0[1−Φ(px)]x2q−1dx=Γ/parenleftbig
q+1
2/parenrightbig
2√πqp2q[Req>0,Rep>0]
NT 56(12), ET II 306(1)a
2.6/integraldisplay∞
0/bracketleftbigg
1−Φ/parenleftbigg
atα±b
tα/parenrightbigg/bracketrightbigg
dt=2b√π/parenleftbiggb
a/parenrightbigg1−α
2α/bracketleftBig
K1+α
2α(2ab)±K1−α
2α(2ab)/bracketrightBig
e±2ab
[a>0,b > 0,α/negationslash=0 ]
6.282
1./integraldisplay∞
0Φ(qt)e−ptdt=1
p/bracketleftbigg
1−Φ/parenleftbiggp
2q/parenrightbigg/bracketrightbigg
exp/parenleftbiggp2
4q2/parenrightbigg/bracketleftBig
Rep>0,|argq|<π
4/bracketrightBig
MO 175, EH II 148(11)
2./integraldisplay∞
0/bracketleftbigg
Φ/parenleftbigg
x+1
2/parenrightbigg
−Φ/parenleftbigg1
2/parenrightbigg/bracketrightbigg
e−μx+1
4dx=1
(μ+1 ) (μ+2 )exp(μ+1 )2
4/bracketleftbigg
1−Φ/parenleftbiggμ+1
2/parenrightbigg/bracketrightbigg
ME 27
6.283
1./integraldisplay∞
0eβx/bracketleftbig
1−Φ/parenleftbig√αx/parenrightbig/bracketrightbig
dx=1
β/bracketleftbigg√α√α−β−1/bracketrightbigg
[Reα>0,Reβ<Reα]ET II 307(5)
2./integraldisplay∞
0Φ/parenleftbig√qt/parenrightbig
e−ptdt=√q
p1√p+q[Rep>0,Re(q+p)>0]
EH II 148(12)
6.284/integraldisplay∞
0/bracketleftbigg
1−Φ/parenleftbiggq
2√x/parenrightbigg/bracketrightbigg
e−pxdx=1
pe−q√p/bracketleftBig
Rep>0,|argq|<π
4/bracketrightBig
EF 147(235), EH II 148(13)
646 The Exponential Integral Function and Functions Generated by It 6.285
6.285
1./integraldisplay∞
0[1−Φ(x)]e−μ2x2dx=arctan μ√πμ[Reμ>0] MI 37
2./integraldisplay∞
0Φ(iat)e−a2t2−stdt=−1
2ai√πexp/parenleftbiggs2
4a2/parenrightbigg
Ei/parenleftbigg
−s2
4a2/parenrightbigg
/bracketleftBig
Res>0,|arga|<π
4/bracketrightBig
EH II 148(14)a
6.286
1./integraldisplay∞
0[1−Φ(βx)]eμ2x2xν−1dx=Γ/parenleftbiggν+1
2/parenrightbigg
√πνβν2F1/parenleftbiggν
2,ν+1
2;ν
2+1 ;μ2
β2/parenrightbigg
/bracketleftbig
Re2β>Reμ2,Reν>0/bracketrightbig
ET II 306(2)
2./integraldisplay∞
0/bracketleftBigg
1−Φ/parenleftBigg√
2x
2/parenrightBigg/bracketrightBigg
ex2
2xν−1dx=2ν
2−1secνπ
2Γ/parenleftBigν
2/parenrightBig
[0<Reν<1] ET I 325(9)
6.287
1./integraldisplay∞
0Φ(βx)e−μx2xdx=β
2μ/radicalbig
μ+β2/bracketleftbig
Reμ>−Reβ2,Reμ>0/bracketrightbig
ME 27a, ET I 176(4)
2./integraldisplay∞
0[1−Φ(βx)]e−μx2xdx=1
2μ/parenleftBigg
1−β/radicalbig
μ+β2/parenrightBigg
/bracketleftbig
Reμ>−Reβ2,Reμ>0/bracketrightbig
NT 49(14), ET I 177(9)
3.∗I=/integraldisplay∞
−∞r
σ2exp/parenleftBigr
σ2/parenrightBig
Q(rA)Q(rB)dr=1
4−1
2π/bracketleftbigg
αarctan/parenleftbiggA
αB/parenrightbigg
+βarctan/parenleftbiggB
βA/parenrightbigg/bracketrightbigg
B/negationslash=A
=1
4−1
παarctan1
αB=A
Q(x)=1√
2π/integraldisplay∞
xe−t2/2dt=1
2/bracketleftbigg
1−erf/parenleftbiggx√
2/parenrightbigg/bracketrightbigg
,α=/radicalbigg
σ2A2
1+σ2A2,β=/radicalbigg
σ2B2
1+σ2B2,
6.288/integraldisplay∞
0Φ(iax)e−μx2xdx=ai
2μ/radicalbig
μ−a2/bracketleftbig
a>0,Reμ>Rea2/bracketrightbig
MI 37a
6.289
1./integraldisplay∞
0Φ(βx)e(β2−μ2)x2xdx=β
2μ(μ2−β2)/bracketleftBig
Re2μ>Reβ2,|argμ|<π
4/bracketrightBig
ET I 176(5)
2./integraldisplay∞
0[1−Φ(βx)]e(β2−μ2)x2xdx=1
2μ(μ+β)/bracketleftBig
Re2μ>Reβ2,argμ<π
4/bracketrightBig
ET I 177(10)
6.297 The probability integral 647
3./integraldisplay∞
0Φ/parenleftBig√
b−ax/parenrightBig
e−(a+μ)x2xdx=√
b−a
2(μ+a)√μ+b[Reμ>−a>0,b > a ] ME 27
6.291/integraldisplay∞
0Φ(ix)e−(μx+x2)xdx=i√π/bracketleftbigg1
μ+μ
4Ei/parenleftbigg
−μ2
4/parenrightbigg/bracketrightbigg
[Reμ>0] MI 37
6.292/integraldisplay∞
0[1−Φ(x)]e−μ2x2x2dx=1
2√π/braceleftbiggarctan μ
μ3−1
μ2(μ2+1 )/bracerightbigg
/bracketleftBig
|argμ|<π
4/bracketrightBig
MI 37
6.293/integraldisplay∞
0Φ(x)e−μx2dx
x=1
2ln√μ+1+1√μ+1−1= arccoth/radicalbig
μ+1
[Reμ>0] MI 37a
6.294
1./integraldisplay∞
0/bracketleftbigg
1−Φ/parenleftbiggβ
x/parenrightbigg/bracketrightbigg
e−μ2x2xdx=1
2μ2exp(−2βμ)/bracketleftBig
|argβ|<π
4,|argμ|<π
4/bracketrightBig
ET I 177(11)
2./integraldisplay∞
0/bracketleftbigg
1−Φ/parenleftbigg1
x/parenrightbigg/bracketrightbigg
e−μ2x2dx
x=−Ei(−2μ)/bracketleftBig
|argμ|<π
4/bracketrightBig
MI 37
6.295
1./integraldisplay∞
0/bracketleftbigg
1−Φ/parenleftbigg1
x/parenrightbigg/bracketrightbigg
exp/parenleftbigg
−μ2x2+1
x2/parenrightbigg
dx=1√πμ[sin2μci(2μ)−cos2μsi(2μ)]
/bracketleftBig
|argμ|<π
4/bracketrightBig
MI 37
2./integraldisplay∞
0/bracketleftbigg
1−Φ/parenleftbigg1
x/parenrightbigg/bracketrightbigg
exp/parenleftbigg
−μ2x2+1
x2/parenrightbigg
xdx=π
2μ[H1(2μ)−Y1(2μ)]−1
μ2
/bracketleftBig
|argμ|<π
4/bracketrightBig
MI 37
3./integraldisplay∞
0/bracketleftbigg
1−Φ/parenleftbigg1
x/parenrightbigg/bracketrightbigg
exp/parenleftbigg
−μ2x2+1
x2/parenrightbiggdx
x=π
2[H0(2μ)−Y0(2μ)]
/bracketleftBig
|argμ|<π
4/bracketrightBig
MI 37
6.296/integraldisplay∞
0/braceleftBigg
/parenleftbig
x2+a2/parenrightbig/bracketleftbigg
1−Φ/parenleftbigga√
2x/parenrightbigg/bracketrightbigg
−/radicalbigg
2
πax·e−a2
2x2/bracerightBigg
e−μ2x2xdx=1
2μ4e−aμ√
2
/bracketleftBig
|argμ|<π
4,a > 0/bracketrightBig
MI 38a
6.297
1./integraldisplay∞
0/bracketleftbigg
1−Φ/parenleftbigg
γx+β
x/parenrightbigg/bracketrightbigg
e(γ2−μ)x2xdx=1
2√μ/parenleftbig√μ+γ/parenrightbigexp [−2(βγ+β√μ)]
[Reβ>0,Reμ>0] ET I 177(12)a
2./integraldisplay∞
0/bracketleftbigg
1−Φ/parenleftbiggb+2ax2
2x/parenrightbigg/bracketrightbigg
exp/bracketleftbig
−/parenleftbig
μ2−a2/parenrightbig
x2+ab/bracketrightbig
xdx=e−bμ
2μ(μ+a)
[a>0,b > 0,Reμ>0] MI 38
648 The Exponential Integral Function and Functions Generated by It 6.298
3./integraldisplay∞
0/braceleftbigg/bracketleftbigg
1−Φ/parenleftbiggb−2ax2
2x/parenrightbigg/bracketrightbigg
e−ab+/bracketleftbigg
1−Φ/parenleftbiggb+2ax2
2x/parenrightbigg/bracketrightbigg
eab/bracerightbigg
e−μx2xdx=1
μexp/parenleftBig
−b/radicalbig
a2+μ/parenrightBig
[a>0,b > 0,Reμ>0] MI 38
6.298/integraldisplay∞
0/braceleftbigg
2c os h ab−e−abΦ/parenleftbiggb−2ax2
2x/parenrightbigg
−eabΦ/parenleftbiggb+2ax2
2x/parenrightbigg/bracerightbigg
e−(μ−a2)x2xdx=1
μ−a2exp(−b√μ)
[a>0,b > 0,Reμ>0] MI 38
6.299/integraldisplay∞
0cosh(2 νt)exp/bracketleftBig
(acosht)2/bracketrightBig
[1−Φ(acosht)]dt=1
2c os (νπ)exp/parenleftbig1
2a2/parenrightbig
Kν/parenleftbig
a2/parenrightbig
/bracketleftbig
Rea>0,−1
2<Reν<1
2/bracketrightbig
ET II 308(10)
6.311/integraldisplay∞
0[1−Φ(ax)] sinbxdx =1
b/parenleftBig
1−e−b2
4a2/parenrightBig
[a>0,b > 0] ET I 96(4)
6.312/integraldisplay∞
0Φ(ax)sinbx2dx=1
4√
2πb/parenleftBigg
lnb+a2+a√
2b
b+a2−a√
2b+ 2arctana√
2b
b−a2/parenrightBigg
[a>0,b > 0] ET I 96(3)
6.313
1./integraldisplay∞
0sin(βx)/bracketleftbig
1−Φ/parenleftbig√αx/parenrightbig/bracketrightbig
dx=1
β−⎛
⎝α
2
α2+β2⎞
⎠1
2/bracketleftBig/parenleftbig
α2+β2/parenrightbig1
2−α/bracketrightBig−1
2
[Reα>|Imβ|] ET II 307(6)
2./integraldisplay∞
0cos(βx)/bracketleftbig
1−Φ/parenleftbig√αx/parenrightbig/bracketrightbig
dx=⎛
⎝α
2
α2+β2⎞
⎠1
2/bracketleftBig/parenleftbig
α2+β2/parenrightbig1
2+α/bracketrightBig−1
2
[Reα>|Imβ|] ET II 307(7)
6.314
1./integraldisplay∞
0sin(bx)/bracketleftbigg
1−Φ/parenleftbigg/radicalbigga
x/parenrightbigg/bracketrightbigg
dx=b−1exp/bracketleftBig
−(2ab)1
2/bracketrightBig
cos/bracketleftBig
(2ab)1
2/bracketrightBig
[Rea>0,b > 0] ET II 307(8)
2./integraldisplay∞
0cos(bx)/bracketleftbigg
1−Φ/parenleftbigg/radicalbigga
x/parenrightbigg/bracketrightbigg
dx=−b−1exp/bracketleftBig
−(2ab)1
2/bracketrightBig
sin/bracketleftBig
(2ab)1
2/bracketrightBig
[Rea>0,b > 0] ET II 307(9)
6.315
1./integraldisplay∞
0xν−1sin(βx)[1−Φ(αx)]dx=Γ/parenleftbig
1+1
2ν/parenrightbig
β√π(ν+1 )αν+12F2/parenleftbiggν+1
2,ν
2+1 ;3
2,ν+3
2;−β2
4α2/parenrightbigg
[Reα>0,Reν>−1] ET II 307(3)
2./integraldisplay∞
0xν−1cos(βx)[1−Φ(αx)]dx=Γ/parenleftbig1
2+1
2ν/parenrightbig
√πναν2F2/parenleftbiggν
2,ν+1
2;1
2,ν
2+1 ;−β2
4α2/parenrightbigg
[Reα>0,Reν>0] ET II 307(4)
6.323 Fresnel integrals 649
3./integraldisplay∞
0[1−Φ(ax)] cosbx·xdx=1
2a2exp/parenleftbigg
−b2
4a2/parenrightbigg
−1
b2/bracketleftbigg
1−exp/parenleftbigg
−b2
4a2/parenrightbigg/bracketrightbigg
[a>0,b > 0] ET I 40(5)
4./integraldisplay∞
0[Φ(ax)−Φ(bx)] cospxdx
x=1
2/bracketleftbigg
Ei/parenleftbigg
−p2
4b2/parenrightbigg
−Ei/parenleftbiggp2
4a2/parenrightbigg/bracketrightbigg
[a>0,b > 0,p > 0] ET I 40(6)
5./integraldisplay∞
0x−1
2Φ/parenleftbig
a√x/parenrightbig
sinbxdx =1
2√
2πb/braceleftBigg
ln/bracketleftBigg
b+a√
2b+a2
b−a√
2b+a2/bracketrightBigg
+ 2arctan/bracketleftBigg
a√
2b
b−a2/bracketrightBigg/bracerightBigg
[a>0,b > 0] ET I 96(3)
6.316/integraldisplay∞
0e1
2x2/bracketleftbigg
1−Φ/parenleftbiggx√
2/parenrightbigg/bracketrightbigg
sinbxdx =/radicalbiggπ
2eb2
2/bracketleftbigg
1−Φ/parenleftbiggb√
2/parenrightbigg/bracketrightbigg
[b>0] ET I 96(5)
6.3176/integraldisplay∞
0e−a2x2Φ(iax)sinbxdx =i
a√π
2e−b2
4a2 [b>0] ET I 96(2)
6.318/integraldisplay∞
0[1−Φ(x)] si(2px)dx=2
πp/parenleftBig
1−e−p2/parenrightBig
−2√π(1−Φ(p))
[p>0] NT 61(13)a
6.32 Fresnel integrals
6.321
1./integraldisplay∞
0/bracketleftbigg1
2−S(px)/bracketrightbigg
x2q−1dx=√
2Γ/parenleftbig
q+1
2/parenrightbig
sin2q+1
4π
4√πqp2q
/bracketleftbig
0<Req<3
2,p > 0/bracketrightbig
NT 56(14)a
2./integraldisplay∞
0/bracketleftbigg1
2−C(px)/bracketrightbigg
x2q−1dx=√
2Γ/parenleftbig
q+1
2/parenrightbig
cos2q+1
4π
4√πqp2q
/bracketleftbig
0<Req<3
2,p > 0/bracketrightbig
NT 56(13)a
6.322
1./integraldisplay∞
0S(t)e−ptdt=1
p/braceleftbigg
cosp2
4/bracketleftbigg1
2−C/parenleftBigp
2/parenrightBig/bracketrightbigg
+s i np2
4/bracketleftbigg1
2−S/parenleftBigp
2/parenrightBig/bracketrightbigg/bracerightbigg
MO 173a
2./integraldisplay∞
0C(t)e−ptdt=1
p/braceleftbigg
cosp2
4/bracketleftbigg1
2−S/parenleftBigp
2/parenrightBig/bracketrightbigg
−sinp2
4/bracketleftbigg1
2−C/parenleftBigp
2/parenrightBig/bracketrightbigg/bracerightbigg
MO 172a
6.323
1./integraldisplay∞
0S/parenleftBig√
t/parenrightBig
e−ptdx=/parenleftBig/radicalbig
p2+1−p/parenrightBig1
2
2p/radicalbig
p2+1EF 122(58)a
650 The Gamma Function and Functions Generated by It 6.324
2./integraldisplay∞
0C/parenleftBig√
t/parenrightBig
e−ptdt=/parenleftBig/radicalbig
p2+1+ p/parenrightBig1
2
2p/radicalbig
p2+1EF 122(58)a
6.324
1./integraldisplay∞
0/bracketleftbigg1
2−S(x)/bracketrightbigg
sin 2pxdx =1+s i n p2−cosp2
4p[p>0] NT 61(12)a
2./integraldisplay∞
0/bracketleftbigg1
2−C(x)/bracketrightbigg
sin 2pxdx =1−sinp2−cosp2
4p[p>0] NT 61(11)a
6.325
1./integraldisplay∞
0S(x)sinb2x2dx=√π
b2−5
2/bracketleftbig
0<b2<1/bracketrightbig
=0/bracketleftbig
b2>1/bracketrightbig
ET I 98(21)a
2./integraldisplay∞
0C(x)cosb2x2dx=√π
b2−5
2/bracketleftbig
0<b2<1/bracketrightbig
=0/bracketleftbig
b2>1/bracketrightbig
ET I 42(22)
6.326
1./integraldisplay∞
0/bracketleftbigg1
2−S(x)/bracketrightbigg
si(2px)dx=/parenleftBigπ
8/parenrightBig1/2
(S(p)+C(p)−1)−1+s i n p2−cosp2
4p
[p>0] NT 61(15)a
2./integraldisplay∞
0/bracketleftbigg1
2−C(x)/bracketrightbigg
si(2px)dx=/parenleftBigπ
8/parenrightBig1/2
(S(p)−C(p))−1−sinp2−cosp2
4p
[p>0] NT 61(14)a
6.4 The Gamma Function and Functions Generated by It
6.41 The gamma function
6.41111/integraldisplay∞
−∞Γ(α+x)Γ(β−x)dx=−iπ21−α−βΓ(α+β)
[Re(α+β)<1 and either Im α<0<Imβor Im β<0<Imα]
ET II 297(1)
=iπ21−α−βΓ(α+β)
[Re(α+β)<1,Imα<0,Imβ<0]
ET II 297(2)
=0
[Re(α+β)<1,Imα>0,Imβ>0]
ET II 297(3)
6.415 The gamma function 651
6.412/integraldisplayi∞
−i∞Γ(α+s)Γ(β+s)Γ(γ−s)Γ(δ−s)ds=2πiΓ(α+γ)Γ(α+δ)Γ(β+γ)Γ(β+δ)
Γ(α+β+γ+δ)
[Reα,Reβ,Reγ,Reδ>0]
ET II 302(32)
6.413
1./integraldisplay∞
0|Γ(a+ix)Γ(b+ix)|2dx=√πΓ(a)Γ/parenleftbig
a+1
2/parenrightbig
Γ(b)Γ/parenleftbig
b+1
2/parenrightbig
Γ(a+b)
2Γ/parenleftbig
a+b+1
2/parenrightbig
[a>0,b > 0] ET II 302(27)
2./integraldisplay∞
0/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓ(a+ix)
Γ(b+ix)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
dx=√πΓ(a)Γ/parenleftbig
a+1
2/parenrightbig
Γ/parenleftbig
b−a−1
2/parenrightbig
2Γ (b)Γ/parenleftbig
b−1
2/parenrightbig
Γ(b−a)
/bracketleftbig
0<a<b −1
2/bracketrightbig
ET II 302(28)
6.414
1./integraldisplay∞
−∞Γ(α+x)
Γ(β+x)dx=0 [ I m α/negationslash=0,Re(α−β)<−1]
ET II 297(4)
2./integraldisplay∞
−∞dx
Γ(α+x)Γ(β−x)=2α+β−2
Γ(α+β−1)[Re(α+β)>1] ET II 297(5)
3./integraldisplay∞
−∞Γ(γ+x)Γ(δ+x)
Γ(α+x)Γ(β+x)dx=0
[Re(α+β−γ−δ)>1,Imγ,Imδ>0]ET II 299(18)
4./integraldisplay∞
−∞Γ(γ+x)Γ(δ+x)
Γ(α+x)Γ(β+x)dx=±2π2iΓ(α+β−γ−δ−1)
sin[π(γ−δ)] Γ(α−γ)Γ(α−δ)Γ(β−γ)Γ (β−δ)
[Re(α+β−γ−δ)>1, Im γ<0, Im δ<0. In the numerator, we take the plus sign if
Imγ>Imδand the minus sign if Im γ<Imδ.] ET II 300(19)
5./integraldisplay∞
−∞Γ(α−β−γ+x+1 )dx
Γ(α+x)Γ(β−x)Γ(γ+x)=πexp/parenleftbig
±1
2π(δ−γ)i/parenrightbig
Γ(β+γ−1)Γ/parenleftbig1
2(α+β)/parenrightbig
Γ/parenleftbig1
2(γ−δ+1 )/parenrightbig
[Re(β+γ)>1,δ=α−β−γ+1 , I m δ/negationslash= 0. The sign is plus in the argument if the
exponential for Im δ>0a n dm i n u sf o rI m δ<0.] ET II 300(20)
6./integraldisplay∞
−∞dx
Γ(α+x)Γ(β−x)Γ(γ+x)Γ(δ−x)=Γ(α+β+γ+δ−3)
Γ(α+β−1)Γ(β+γ−1)Γ(γ+δ−1)Γ(δ+α−1)
[Re(α+β+γ+δ)>3] ET II 300(21)
6.415
1./integraldisplay−∞
−∞R(x)dx
Γ(α+x)Γ(β−x)Γ(γ+x)Γ(δ−x)
=Γ(α+β+γ+δ−3)
Γ(α+β−1)Γ(β+γ−1)Γ(γ+δ−1)Γ(δ+α−1)/integraldisplay1
0R(t)dt
[Re(α+β+γ+δ)>3,R(x+1 )= R(x)]ET II 301(24)
652 The Gamma Function and Functions Generated by It 6.421
2./integraldisplay∞
−∞R(x)dx
Γ(α+x)Γ(β−x)Γ(γ+x)Γ(δ−x)=/integraldisplay1
0R(t)cos/bracketleftbig1
2π(2t+α−β)/bracketrightbig
dt
Γ/parenleftbiggα+β
2/parenrightbigg
Γ/parenleftbiggγ+δ
2/parenrightbigg
Γ(α+δ−1)
[α+δ=β+γ,Re(α+β+γ+δ)>2,R(x+1 )= −R(x)]ET II 301(25)
6.42 Combinations of the gamma function, the exponential, and powers
6.421
1./integraldisplay∞
−∞Γ(α+x)Γ(β−x)exp[ 2( πn+θ)xi]dx=2πiΓ(α+β)(2 cos θ)−α−βexp[(β−α)iθ]
×[ηn(β)e x p ( 2 nπβi)−ηn(−α)e x p (−2nπαi)]
/bracketleftBigg
Re(α+β)<1,−π
2<θ<π
2,nan integer ,η n(ξ)=/braceleftBigg
0i f/parenleftbig1
2−n/parenrightbig
Imξ>0
sign/parenleftbig1
2−n/parenrightbig
if/parenleftbig1
2−n/parenrightbig
Imξ<0/bracketrightBigg
ET II 298(7)
2./integraldisplay∞
−∞eπicxdx
Γ(α+x)Γ(β−x)Γ(γ+kx)Γ(δ−kx)=0
[Re(α+β+γ+δ)>2,candkare real ,|c|>|k|+1 ] ET II 301(26)
3./integraldisplay∞
−∞Γ(α+x)
Γ(β+x)exp[(2 πn+π−2θ)xi]dx
=2πisign/parenleftbig
n+1
2/parenrightbig(2 cos θ)β−α−1
Γ(β−α)exp[−(2πn+π−θ)αi+θi(β−1)]
/bracketleftBig
Re(β−α)>0,−π
2<θ<π
2,nis an integer ,/parenleftbig
n+1
2/parenrightbig
Imα<0/bracketrightBig
ET II 298(8)
4./integraldisplay∞
−∞Γ(α+x)
Γ(β+x)exp[(2 πn+π−2θ)xi]dx=0
/bracketleftBig
Re(β−α)>0,−π
2<θ<π
2,nis an integer ,/parenleftbig
n+1
2/parenrightbig
Imα>0/bracketrightBig
ET II 297(6)
6.422
1./integraldisplayi∞
−i∞Γ(s−k−λ)Γ/parenleftbig
λ+μ−s+1
2/parenrightbig
Γ/parenleftbig
λ−μ−s+1
2/parenrightbig
zsds
=2πiΓ/parenleftbig1
2−k−μ/parenrightbig
Γ/parenleftbig1
2−k+μ/parenrightbig
zλez
2Wk,μ(z)
/bracketleftbig
Re(k+λ)<0,Reλ>|Reμ|−1
2,|argz|<3
2π/bracketrightbig
ET II 302(29)
2./integraldisplayγ+i∞
γ−i∞Γ(α+s)Γ(−s)Γ(1−c−s)xsds=2πiΓ(α)Γ(α−c+1 ) Ψ ( α,c;x)
/bracketleftbig
−Reα<γ< min (0 ,1−Rec),−3
2π<argx<3
2π/bracketrightbig
EH I 256(5)
6.422 The gamma function, the exponential, and powers 653
3./integraldisplayγ+i∞
γ−i∞Γ(−s)Γ(β+s)tsds=2πiΓ(β)(1 + t)−β[0>γ> Re(1−β),|argt|<π]
EH I 256, BU 75
4./integraldisplay∞i
−∞iΓ/parenleftbiggt−p
2/parenrightbigg
Γ(−t)/parenleftBig√
2/parenrightBigt−p−2
ztdt=2πie1
4z2Γ(−p)Dp(z)
/bracketleftbig
|argz|<3
4π, p is not a positive integer/bracketrightbig
WH
5./integraldisplayi∞
−i∞Γ(s)Γ/parenleftbig1
2ν+1
4−s/parenrightbig
Γ/parenleftbig1
2ν−1
4−s/parenrightbig/parenleftbiggz2
2/parenrightbiggs
ds
=2πi·21
4−1
2νz−1
2e3
4z2Γ/parenleftbig1
2ν+1
4/parenrightbig
Γ/parenleftbig1
2ν−1
4/parenrightbig
Dν(z)
/bracketleftbig
|argz|<3
4π, ν /negationslash=1
2,−1
2,−3
2,.../bracketrightbig
EH II 120
6.3/integraldisplayc+i∞
c−i∞/parenleftbig1
2x/parenrightbig−sΓ/parenleftbig1
2ν+1
2s/parenrightbig/bracketleftbig
Γ/parenleftbig
1+1
2ν−1
2s/parenrightbig/bracketrightbig−1ds=4πiJν(x)
[x>0,−Reν<c< 1] EH II 21(34)
7./integraldisplay−c+i∞
−c−i∞Γ(−ν−s)Γ(−s)/parenleftbig
−1
2iz/parenrightbigν+2sds=−2π2e1
2iνπH(1)
ν(z)
/bracketleftBig
|arg(−iz)|<π
2,0<Reν<c/bracketrightBig
EH II 83(34)
8./integraldisplay−c+i∞
−c−i∞Γ(−ν−s)Γ(−s)/parenleftbig1
2iz/parenrightbigν+2sds=2π2e−1
2iνπH(2)
ν(z)
/bracketleftBig
|arg(iz)|<π
2,0<Reν<c/bracketrightBig
EH II 83(35)
9./integraldisplayi∞
−i∞Γ(−s)/parenleftbig1
2x/parenrightbigν+2s
Γ(ν+s+1 )ds=2πiJν(x)[ x>0,Reν>0] EH II 83(36)
10./integraldisplayi∞
−i∞Γ(−s)Γ(−2ν−s)Γ/parenleftbig
ν+s+1
2/parenrightbig
(−2iz)sds=−π5
2e−i(z−νπ)sec(νπ)(2z)−νH(1)
ν(z)
/bracketleftbig
|arg(−iz)|<3
2π,2ν/negationslash=±1,±3.../bracketrightbig
EH II 83(37)
11./integraldisplayi∞
−i∞Γ(−s)Γ(−2ν−s)Γ/parenleftbig
ν+s+1
2/parenrightbig
(2iz)sds=π5
2ei(z−νπ)sec(νπ)(2z)−νH(2)
ν(z)
/bracketleftbig
|arg(iz)|<3
2π,2ν/negationslash=±1,±3.../bracketrightbig
EH II 84(38)
12./integraldisplayi∞
−i∞Γ(s)Γ/parenleftbig1
2−s−ν/parenrightbig
Γ/parenleftbig1
2−s+ν/parenrightbig
(2z)sds=23
2π3
2iz1
2ezsec(νπ)Kν(z)
/bracketleftbig
|argz|<3
2π,2ν/negationslash=±1,±3,.../bracketrightbig
EH II 84(39)
13./integraldisplay−1
2+i∞
−1
2−i∞Γ(−s)
sΓ(1 + s)x2sds=4π/integraldisplay∞
2xJ0(t)
tdt [x>0] MO 41
654 The Gamma Function and Functions Generated by It 6.423
14./integraldisplayi∞
−i∞Γ(α+s)Γ(β+s)Γ(−s)
Γ(γ+s)(−z)sds=2πiΓ(α)Γ(β)
Γ(γ)F(α,β;γ;z)
[For arg( −z)<π, the path of integration must separate the poles of the integrand at the points
s=0,1,2,3,...from the poles s=−α−nands=−β−n(forn=0,1,2,...).]
15./integraldisplayδ+i∞
δ−i∞Γ(α+s)Γ(−s)
Γ(γ+s)(−z)sds=2πiΓ(α)
Γ(γ)1F1(α;γ;z)
/bracketleftBig
−π
2<arg(−z)<π
2,0>δ> −Reα, γ /negationslash=0,1,2,.../bracketrightBig
EH I 62(15), EH I 256(4)
16./integraldisplayi∞
−i∞/bracketleftBigg
Γ/parenleftbig1
2−s/parenrightbig
Γ(s)/bracketrightBigg2
zsds=2πiz1
2/bracketleftBig
2π−1K0/parenleftBig
4z1
4/parenrightBig
−Y0/parenleftBig
4z1
4/parenrightBig/bracketrightBig
[z>0] ET II 303(33)
17./integraldisplayi∞
−i∞Γ/parenleftbig
λ+μ−s+1
2/parenrightbig
Γ/parenleftbig
λ−μ−s+1
2/parenrightbig
Γ(λ−k−s+1 )zsds=2πizλe−z
2Wk,μ(z)
/bracketleftBig
Reλ>|Reμ|−1
2,|argz|<π
2/bracketrightBig
ET II 302(30)
18./integraldisplayi∞
−i∞Γ(k−λ+s)Γ/parenleftbig
λ+μ−s+1
2/parenrightbig
Γ/parenleftbig
μ−λ+s+1
2/parenrightbig zsds=2πiΓ/parenleftbig
k+μ+1
2/parenrightbig
Γ(2μ+1 )zλe−z
2Mk,μ(z)
/bracketleftBig
Re(k−λ)>0,Re(λ+μ)>−1
2,|argz|<π
2/bracketrightBig
ET II 302(31)
19./integraldisplayi∞
−i∞m/productdisplay
j=1Γ(bj−s)n/productdisplay
j=1Γ(1−aj+s)
q/productdisplay
j=m+1Γ( 1−bj+s)p/productdisplay
j=n+1Γ(aj−s)zsds=2πiGpq
mn/parenleftbigg
z/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
/bracketleftbigg
p+q<2(m+n);|argz|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π;
Reak<1,k=1,...,n ;R e bj>0,j=1,...,m/bracketrightbigg
ET II 303(34)
6.423
1./integraldisplay∞
0e−αx dx
Γ(1 + x)=ν/parenleftbig
e−α/parenrightbig
MI 39, EH III 222(16)
2./integraldisplay∞
0e−αx dx
Γ(x+β+1 )=eβαν/parenleftbig
e−α,β/parenrightbig
MI 39, EH III 222(16)
3./integraldisplay∞
0e−αxxm
Γ(x+1 )dx=μ/parenleftbig
e−α,m/parenrightbig
Γ(m+1 ) [ R e m>−1] MI 39, EH III 222(17)
6.433 Gamma functions and trigonometric functions 655
4./integraldisplay∞
0e−αx xm
Γ(x+n+1 )dx=enαμ/parenleftbig
e−α,m ,n/parenrightbig
Γ(m+1 ) MI 39, EH III 222(17)
6.424/integraldisplay∞
−∞R(x)exp[(2 πn+θ)xi]dx
Γ(α+x)Γ(β−x)=/bracketleftbigg
2c os/parenleftbiggθ
2/parenrightbigg/bracketrightbiggα+β−2
Γ(α+β−1)exp/bracketleftbigg1
2θ(β−α)i/bracketrightbigg/integraldisplay1
0R(t)exp(2 πnti)dt
[Re(α+β)>1,−π<θ<π , n is an integer ,R(x+1 )= R(x)]ET II 299(16)
6.43 Combinations of the gamma function and trigonometric functions
6.431
1./integraldisplay−∞
−∞sinrxdx
Γ(p+x)Γ(q−x)=/parenleftBig
2c osr
2/parenrightBigp+q−2
sinr(q−p)
2
Γ(p+q−1)[|r|<π]
=0 [ |r|>π]
[ris real; Re( p+q)>1]MO 10a, ET II 298(9, 10)
2./integraldisplay∞
−∞cosrxdx
Γ(p+x)Γ(q−x)=/parenleftBig
2c osr
2/parenrightBigp+q−2
cosr(q−p)
2
Γ(p+q−1)[|r|<π]
=0 [ |r|>π]
[ris real; Re( p+q)>1]MO 10a, ET II 299(13, 14)
6.432/integraldisplay∞
−∞sin(mπx)
sin(πx)dx
Γ(α+x)Γ(β−x)=0 [ mis an even integer]
=2α+β−2
Γ(α+β−1)[mis an odd integer]
[Re(α+β)>1] ET II 298(11, 12)
6.433
1./integraldisplay∞
−∞sinπxdx
Γ(α+x)Γ(β−x)Γ(γ+x)Γ(δ−x)=sin/bracketleftBigπ
2(β−α)/bracketrightBig
2Γ/parenleftbiggα+β
2/parenrightbigg
Γ/parenleftbiggγ+δ
2/parenrightbigg
Γ(α+δ−1)
[α+δ=β+γ,Re(α+β+γ+δ)>2]ET II 300(22)
2./integraldisplay∞
−∞cosπxdx
Γ(α+x)Γ(β−x)Γ(γ+x)Γ(δ−x)=cos/bracketleftBigπ
2(β−α)/bracketrightBig
2Γ/parenleftbiggα+β
2/parenrightbigg
Γ/parenleftbiggγ+δ
2/parenrightbigg
Γ(α+δ−1)
[α+δ=β+γ,Re(α+β+γ+δ)>2]ET II 301(23)
656 The Gamma Function and Functions Generated by It 6.441
6.44 The logarithm of the gamma function∗
6.441
1./integraldisplayp+1
plnΓ(x)dx=1
2ln2π+plnp−p FI II 784
2./integraldisplay1
0ln Γ(x)dx=/integraldisplay1
0lnΓ(1 −x)dx=1
2ln2π FI II 783
3./integraldisplay1
0ln Γ(x+q)dx=1
2ln 2π+qlnq−q [q≥0] NH 89(17), ET II 304(40)
4./integraldisplayz
0lnΓ(x+1 )dx=z
2ln2π−z(z+1 )
2+zln Γ(z+1 )−lnG(z+1 ),
where G(z+1 )=( 2 π)z
2exp/parenleftbigg
−z(z+1 )
2−Cz2
2/parenrightbigg∞/productdisplay
k=1/braceleftbigg/parenleftBig
1+z
k/parenrightBigk
exp/parenleftbigg
−z+z2
2k/parenrightbigg/bracerightbigg
WH
5./integraldisplayn
0lnΓ(α+x)dx=n−1/summationdisplay
k=0(a+k)ln(a+k)−na+1
2nln(2π)−1
2n(n−1)
[a≥0;n=1,2,...] ET II 304(41)
6.442/integraldisplay1
0exp(2πnxi)lnΓ( a+x)dx=( 2πni)−1[lna−exp(−2πnai)Ei(2πnai)]
[a>0;n=±1,±2,...]ET II 304(38)
6.443
1./integraldisplay1
0ln Γ(x)sin2πnxdx =1
2πn[ln(2πn)+C] NH 203(5), ET II 304(42)
2./integraldisplay1
0ln Γ(x)sin(2 n+1 )πxdx =1
(2n+1 )π/bracketleftbigg
ln/parenleftBigπ
2/parenrightBig
+2/parenleftbigg
1+1
3+···+1
2n−1/parenrightbigg
+1
2n+1/bracketrightbigg
ET II 305(43)
3./integraldisplay1
0ln Γ(x)cos2 πnxdx =1
4nNH 203(6), ET II 305(44)
4.8/integraldisplay1
0ln Γ(x)cos(2 n+1)πxdx =2
π2/bracketleftBigg
1
(2n+1 )2(C+l n2 π)+2∞/summationdisplay
k=2lnk
4k2−(2n+1 )2/bracketrightBigg
NH 203(6)
5./integraldisplay1
0sin(2πnx)lnΓ( a+x)dx=−(2πn)−1[lna+c o s ( 2 πna)ci(2πna)−sin(2πna)si(2πna)]
[a>0;n=1,2,...] ET II 304(36)
6./integraldisplay1
0cos(2πnx)l nΓ ( a+x)dx=−(2πn)−1[sin(2πna)ci(2πna) + cos(2 πna)si(2πna)]
[a>0;n=1,2,...] ET II 304(37)
∗Here, we are violating our usual order of presentation of the formulas in order to make it easier to examine the
integrals involving the gamma function.
6.457 The incomplete gamma function 657
6.45 The incomplete gamma function
6.451
1./integraldisplay∞
0e−αxγ(β,x)dx=1
αΓ(β)(1 + α)−β[β>0] MI 39
2./integraldisplay∞
0e−αxΓ(β,x)dx=1
αΓ(β)/bracketleftbigg
1−1
(α+1 )β/bracketrightbigg
[β>0] MI 39
6.452
1./integraldisplay∞
0e−μxγ/parenleftbigg
ν,x2
8a2/parenrightbigg
dx=1
μ2−ν−1Γ(2ν)e(aμ)2D−2ν(2aμ)
/bracketleftbigg
|arga|<π
4,Reν>−1
2,Reμ>0/bracketrightbigg
ET I 179(36)
2./integraldisplay∞
0e−μxγ/parenleftbigg1
4,x2
8a2/parenrightbigg
dx=23
4√a√μe(aμ)2K1
4/parenleftbig
a2μ2/parenrightbig/bracketleftBig
|arga|<π
4,Reμ>0/bracketrightBig
ET I 179(35)
6.453/integraldisplay∞
0e−μxΓ/parenleftBig
ν,a
x/parenrightBig
dx=2a1
2νμ1
2ν−1Kν(2√μa)/bracketleftBig
|arga|<π
2,Reμ>0/bracketrightBig
ET I 179(32)
6.454/integraldisplay∞
0e−βxγ/parenleftbig
ν,α√x/parenrightbig
dx=2−1
2νανβ−1
2ν−1Γ(ν)exp/parenleftbiggα2
8β/parenrightbigg
D−ν/parenleftbiggα√2β/parenrightbigg
[Reβ>0,Reν>0]
ET II 309(19), MI 39a
6.455
1./integraldisplay∞
0xμ−1e−βxΓ(ν,αx)dx=ανΓ(μ+ν)
μ(α+β)μ+ν2F1/parenleftbigg
1,μ+ν;μ+1 ;β
α+β/parenrightbigg
[Re(α+β)>0,Reμ>0,Re(μ+ν)>0]ET II 309(16)
2./integraldisplay∞
0xμ−1e−βxγ(ν,αx)dx=ανΓ(μ+ν)
ν(α+β)μ+ν2F1/parenleftbigg
1,μ+ν;ν+1 ;α
α+β/parenrightbigg
[Re(α+β)>0,Reβ>0,Re(μ+ν)>0]ET II 308(15)
6.456
1./integraldisplay∞
0e−αx(4x)ν−1
2γ/parenleftbigg
ν,1
4x/parenrightbigg
dx=√πγ(2ν,√α)
αν+1
2MI 39a
2./integraldisplay∞
0e−αx(4x)ν−1
2Γ/parenleftbigg
ν,1
4x/parenrightbigg
dx=√πΓ( 2ν,√α)
αν+1
2MI 39a
6.457
1./integraldisplay∞
0e−αx(4x)ν
√xγ/parenleftbigg
ν+1,1
4x/parenrightbigg
dx=√πγ(2ν+1,√α)
αν+1
2MI 39
2./integraldisplay∞
0e−αx(4x)ν
√xΓ/parenleftbigg
ν+1,1
4x/parenrightbigg
dx=√πΓ(2ν+1,√α)
αν+1
2MI 39
658 The Gamma Function and Functions Generated by It 6.458
6.458/integraldisplay∞
0x1−2νexp/parenleftbig
αx2/parenrightbig
sin(bx)Γ/parenleftbig
ν,αx2/parenrightbig
dx=π1
22−ναν−1Γ/parenleftbig3
2−ν/parenrightbig
exp/parenleftbiggb2
8α/parenrightbigg
D2ν−2/bracketleftbiggb
(2α)1
2/bracketrightbigg
/bracketleftbigg
|argα|<3π
2,0<Reν<1/bracketrightbigg
ET II 309(18)
6.46–6.47 The function ψ(x)
6.461/integraldisplayx
1ψ(t)dt=l nΓ ( x)
6.462/integraldisplay1
0ψ(α+x)dx=l nα [α>0] ET II 305(1)
6.463/integraldisplay∞
0x−α[C+ψ(1 +x)] =−πcosec( πα)ζ(α)[ 1 <Reα<2] ET II 305(6)
6.464/integraldisplay1
0e2πnxiψ(α+x)dx=e−2πnαiEi(2πnαi)[ α>0;n=±i,±2,...]ET II 305(2)
6.465
1.8/integraldisplay1
0ψ(x)sinπxdx =−2
π/bracketleftBigg
C+l n2 π+2∞/summationdisplay
k=2lnk
4k2−1/bracketrightBigg
(see6.443 4) NH 204
2./integraldisplay1
0ψ(x)sin(2 πnx)dx=−1
2π [n=1,2,...] ET II 305(3)
6.466/integraldisplay∞
0[ψ(α+ix)−ψ(α−ix)] sinxy dx =iπe−αy/parenleftbig
1−e−y/parenrightbig−1
[α>0,y > 0] ET I 96(1)
6.467
1./integraldisplay1
0sin(2πnx)ψ(α+x)dx=s i n ( 2 πnα)ci(2πnα) + cos(2 πnα)si(2πnα)
[α≥0;n=1,2,...] ET II 305(4)
2./integraldisplay1
0cos(2πnx)ψ(α+x)dx=s i n ( 2 πnα)si(2πnα)−cos(2πnα)ci(2πnα)
[α>0;n=1,2,...] ET II 305(5)
6.468/integraldisplay1
0ψ(x)sin2πxdx =−1
2[C+l n ( 2 π)] NH 204
6.469
1./integraldisplay1
0ψ(x)sinπxcosπxdx =−π
4NH 204
2.8/integraldisplay1
0ψ(x)sinπxsin(nπx)dx=n
1−n2[nis even]
=1
2lnn−1
n+1[n>1i so d d ]
NH 204(8)a
6.511 Bessel functions 659
6.471
1./integraldisplay∞
0x−α[lnx−ψ(1 +x)]dx=πcosec( πα)ζ(α)[ 0 <Reα<1] ET II 306(7)
2./integraldisplay∞
0x−α[ln(1 + x)−ψ(1 +x)]dx=πcosec( πα)/bracketleftbig
ζ(α)−(α−1)−1/bracketrightbig
[0<Reα<1] ET II 306(8)
3./integraldisplay∞
0[ψ(x+1 )−lnx]c os ( 2 πxy)dx=1
2[ψ(y+1 )−lny] ET II 306(12)
6.472
1./integraldisplay∞
0x−α/bracketleftbig
(1 +x)−1−ψ/prime(1 +x)/bracketrightbig
dx=−παcosec( πα)/bracketleftbig
ζ(1 +α)−α−1/bracketrightbig
[|Reα|<1] ET II 306(9)
2./integraldisplay∞
0x−α/bracketleftbig
x−1−ψ/prime(1 +x)/bracketrightbig
dx=−παcosec( πα)ζ(1 +α)
[−2<Reα<0] ET II 306(10)
6.473/integraldisplay∞
0x−αψ(n)(1 +x)dx=(−1)n−1πΓ(α+n)
Γ(α)sinπαζ(α+n)
[n=1,2,...;0<Reα<1]
ET II 306(11)
6.5–6.7 Bessel Functions
6.51 Bessel functions
6.511
1./integraldisplay∞
0Jν(bx)dx=1
b[Reν>−1,b > 0] ET II 22(3)
2./integraldisplay∞
0Yν(bx)dx=−1
btan/parenleftBigνπ
2/parenrightBig
[|Reν|<1,b > 0]
WA 432(7), ET II 96(1)
3./integraldisplaya
0Jν(x)dx=2∞/summationdisplay
k=0Jν+2k+1(a)[ R e ν>−1] ET II 333(1)
4./integraldisplaya
0J1
2(t)dt=2S/parenleftbig√a/parenrightbig
WA 599(4)
5./integraldisplaya
0J−1
2(t)dt=2C/parenleftbig√a/parenrightbig
WA 599(3)
6./integraldisplaya
0J0(x)dx=aJ0(a)+πa
2[J1(a)H0(a)−J0(a)H1(a)]
[a>0] ET II 7(2)
660 Bessel Functions 6.512
7./integraldisplaya
0J1(x)dx=1−J0(a)[ a>0] ET II 18(1)
8./integraldisplay∞
aJ0(x)dx=1−aJ0(a)+πa
2[J0(a)H1(a)−J1(a)H0(a)]
[a>0] ET II 7(3)
9./integraldisplay∞
aJ1(x)dx=J0(a)[ a>0] ET II 18(2)
10./integraldisplayb
aYν(x)dx=2∞/summationdisplay
n=0[Yν+2n+1(b)−Yν+2n+1(a)] ET II 339(46)
11./integraldisplaya
0Iν(x)dx=2∞/summationdisplay
n=0(−1)nIν+2n+1(a)[ R e ν>−1] ET II 364(1)
12.∗/integraldisplay∞
0K0(ax)=π
2a[a>0]
13.∗/integraldisplay∞
0K2
0(ax)=π2
4a[a>0]
6.512
1.11/integraldisplay∞
0Jμ(ax)Jν(bx)dx=bνa−ν−1Γ/parenleftbiggμ+ν+1
2/parenrightbigg
Γ(ν+1 )Γ/parenleftbiggμ−ν+1
2/parenrightbiggF/parenleftbiggμ+ν+1
2,ν−μ+1
2;ν+1 ;b2
a2/parenrightbigg
[a>0,b > 0,Re(μ+ν)>−1,b < a .
Fora>b, the positions of μandνshould be reversed.]
ET II 48(6)
2.7/integraldisplay∞
0Jν+n(αt)Jν−n−1(βt)dt=βν−n−1Γ(ν)
αν−nn!Γ (ν−n)F/parenleftbigg
ν,−n;ν−n;β2
α2/parenrightbigg
[0<β<α ]
=(−1)n1
2α[0<β=α]
=0 [ 0 <α<β ]
[Re(ν)>0] MO 50
3.8/integraldisplay∞
0Jν(αx)Jν−1(βx)dx=βν−1
αν[β<α ]
=1
2β[β=α]
=0 [ β>α ]
[Reν>0] WA 444(8), KU (40)a
4./integraldisplay∞
0Jν+2n+1(ax)Jν(bx)dx=bνa−ν−1P(ν,0)
n/parenleftbigg
1−2b2
a2/parenrightbigg
[Reν>−1−n,0<b<a ]
=0 [ R e ν>−1−n,0<a<b ]
ET II 47(5)
6.513 Bessel functions 661
5./integraldisplay∞
0Jν+n(ax)Yν−n(ax)dx=(−1)n+11
2a/bracketleftbig
Reν>−1
2,a > 0,n=0,1,2,.../bracketrightbig
ET II 347(57)
6./integraldisplay∞
0J1(bx)Y0(ax)dx=−b−1
πln/parenleftbigg
1−b2
a2/parenrightbigg
[0<b<a ] ET II 21(31)
7./integraldisplaya
0Jν(x)Jν+1(x)dx=∞/summationdisplay
n=0[Jν+n+1(a)]2[Reν>−1] ET II 338(37)
8.9/integraldisplay∞
0kJn(ka)Jn(kb)dk=1
aδ(b−a)[ n=0,1,...] JAC 110
9.∗/integraldisplay∞
0K0(ax)J1(bx)=1
2bln/parenleftbigg
1+b2
a2/parenrightbigg
[a>0,b > 0]
10.∗/integraldisplay∞
0K0(ax)I1(bx)=−1
2bln/parenleftbigg
1−b2
a2/parenrightbigg
[a>0,b > 0]
6.513
1./integraldisplay∞
0[Jμ(ax)]2Jν(bx)dx=a2μb−2μ−1Γ/parenleftbigg1+ν+2μ
2/parenrightbigg
[Γ(μ+1 ) ]2Γ/parenleftbigg1+ν−2μ
2/parenrightbigg
×⎡
⎢⎢⎣F⎛
⎜⎜⎝1−ν+2μ
2,1+ν+2μ
2;μ+1 ;1−/radicalbigg
1−4a2
b2
2⎞
⎟⎟⎠⎤
⎥⎥⎦2
[Reν+R e2 μ>−1,0<2a<b]ET II 52(33)
2./integraldisplay∞
0[Jμ(ax)]2Kν(bx)dx=b−1
2Γ/parenleftbigg2μ+ν+1
2/parenrightbigg
Γ/parenleftbigg2μ−ν+1
2/parenrightbigg/bracketleftBigg
P−μ
1
2ν−1
2/parenleftBigg/radicalbigg
1+4a2
b2/parenrightBigg/bracketrightBigg2
[2 Reμ>|Reν|−1,Reb>2|Ima|]
ET II 138(18)
3./integraldisplay∞
0Iμ(ax)Kμ(ax)Jν(bx)dx=eμπiΓ/parenleftbiggν+2μ+1
2/parenrightbigg
bΓ/parenleftbiggν−2μ+1
2/parenrightbiggP−μ
1
2ν−1
2/parenleftBigg/radicalbigg
1+4a2
b2/parenrightBigg
Q−μ
1
2ν−1
2/parenleftBigg/radicalbigg
1+4a2
b2/parenrightBigg
[Rea>0,b > 0,Reν>−1,Re(ν+2μ)>−1]ET II 65(20)
4./integraldisplay∞
0Jμ(ax)J−μ(ax)Kν(bx)dx=π
2bsec/parenleftBigνπ
2/parenrightBig
Pμ
1
2ν−1
2/parenleftBigg/radicalbigg
1+4a2
b2/parenrightBigg
P−μ
1
2ν−1
2/parenleftBigg/radicalbigg
1+4a2
b2/parenrightBigg
[|Reν|<1,Reb>2|Ima|]
ET II 138(21)
662 Bessel Functions 6.514
5./integraldisplay∞
0[Kμ(ax)]2Jν(bx)dx=e2μπiΓ/parenleftbigg1+ν+2μ
2/parenrightbigg
bΓ/parenleftbigg1+ν−2μ
2/parenrightbigg/bracketleftBigg
Q−μ
1
2ν−1
2/parenleftBigg/radicalbigg
1+4a2
b2/parenrightBigg/bracketrightBigg2
/bracketleftbig
Rea>0,b > 0,Re/parenleftbig1
2ν±μ/parenrightbig
>−1
2/bracketrightbig
ET II 66(28)
6./integraldisplayz
0Jμ(x)Jν(z−x)dx=2∞/summationdisplay
k=0(−1)kJμ+ν+2k+1(z)[ R e μ>−1,Reν>−1] WA 414(2)
7./integraldisplayz
0Jμ(x)J−μ(z−x)dx=s i nz [−1<Reμ<1] WA 415(4)
8./integraldisplayz
0Jμ(x)J1−μ(z−x)dx=J0(z)−cos(z)[ −1<Reμ<2] WA 415(4)
9.∗/integraldisplay∞
0J2
0(ax)J1(bx)=1
b[b>2a>0]
=2
πbarcsin/parenleftbiggb
2a/parenrightbigg
[2a>b> 0]
6.514
1./integraldisplay∞
0Jν/parenleftBiga
x/parenrightBig
Jν(bx)dx=b−1J2ν/parenleftBig
2√
ab/parenrightBig /bracketleftbig
a>0,b > 0,Reν>−1
2/bracketrightbig
ET II 57(9)
2./integraldisplay∞
0Jν/parenleftBiga
x/parenrightBig
Yν(bx)dx=b−1/bracketleftbigg
Y2ν/parenleftBig
2√
ab/parenrightBig
+2
πK2ν/parenleftBig√
2ab/parenrightBig/bracketrightbigg
/bracketleftbig
a>0,b > 0,−1
2<Reν<3
2/bracketrightbig
ET II 110(12)
3./integraldisplay∞
0Jν/parenleftBiga
x/parenrightBig
Kν(bx)dx=b−1e1
2i(ν+1)πK2ν/bracketleftBig
2e1
4iπ√
ab/bracketrightBig
+b−1e−1
2i(ν+1)πK2ν/bracketleftBig
2e−1
4πi√
ab/bracketrightBig
/bracketleftbig
a>0,Reb>0,|Reν|<5
2/bracketrightbig
ET II 141(31)
4./integraldisplay∞
0Yν/parenleftBiga
x/parenrightBig
Jν(bx)dx=−2b−1
π/bracketleftBig
K2ν/parenleftBig
2√
ab/parenrightBig
−π
2Y2ν/parenleftBig
2√
ab/parenrightBig/bracketrightBig
/bracketleftbig
a>0,b > 0,|Reν|<1
2/bracketrightbig
ET II 62(37)a
5./integraldisplay∞
0Yν/parenleftBiga
x/parenrightBig
Yν(bx)dx=−b−1J2ν/parenleftBig
2√
ab/parenrightBig/bracketleftbig
a>0,b > 0,|Reν|<1
2/bracketrightbig
ET II 110(14)
6./integraldisplay∞
0Yν/parenleftBiga
x/parenrightBig
Kν(bx)dx=−b−1e1
2νπiK2ν/parenleftBig
2e1
4πi√
ab/parenrightBig
−b−1e−1
2νπiK2ν/parenleftBig
2e−1
4πi√
ab/parenrightBig
/bracketleftbig
a>0,Reb>0,|Reν|<5
2/bracketrightbig
ET II 143(37)
6.516 Bessel functions 663
7./integraldisplay∞
0Kν/parenleftBiga
x/parenrightBig
Yν(bx)dx=−2b−1/bracketleftbigg
sin/parenleftbigg3νπ
2/parenrightbigg
ker2ν/parenleftBig
2√
ab/parenrightBig
+c o s/parenleftbigg3νπ
2/parenrightbigg
kei2ν/parenleftBig
2√
ab/parenrightBig/bracketrightbigg
/bracketleftbig
Rea>0,b > 0,|Reν|<1
2/bracketrightbig
ET II 113(28)
8./integraldisplay∞
0Kν/parenleftBiga
x/parenrightBig
Kν(bx)dx=πb−1K2ν/parenleftBig
2√
ab/parenrightBig
[Rea>0,Reb>0] ET II 146(54)
6.515
1./integraldisplay∞
0Jμ/parenleftBiga
x/parenrightBig
Yμ/parenleftBiga
x/parenrightBig
K0(bx)dx=−2b−1J2μ/parenleftBig
2√
ab/parenrightBig
K2μ/parenleftBig
2√
ab/parenrightBig
[a>0,Reb>0] ET II 143(42)
2./integraldisplay∞
0/bracketleftBig
Kμ/parenleftBiga
x/parenrightBig/bracketrightBig2
K0(bx)dx=2πb−1K2μ/parenleftBig
2e1
4πi√
ab/parenrightBig
K2μ/parenleftBig
2e−1
4πi√
ab/parenrightBig
[Rea>0,Reb>0] ET II 147(59)
3./integraldisplay∞
0H(1)
μ/parenleftbigga2
x/parenrightbigg
H(2)
μ/parenleftbigga2
x/parenrightbigg
J0(bx)dx=1 6π−2b−1cosμπK2μ/parenleftBig
2eπi/4a√
b/parenrightBig
K2μ/parenleftBig
2e−πi/4a√
b/parenrightBig
/bracketleftBig
|arga|<π
4,b > 0,|Reμ|<1
4/bracketrightBig
ET II 17(36)
6.516
1./integraldisplay∞
0J2ν/parenleftbig
a√x/parenrightbig
Jν(bx)dx=b−1Jν/parenleftbigga2
4b/parenrightbigg/bracketleftbig
a>0,b > 0,Reν>−1
2/bracketrightbig
ET II 58(16)
2./integraldisplay∞
0J2ν/parenleftbig
a√x/parenrightbig
Yν(bx)dx=−b−1Hν/parenleftbigga2
4b/parenrightbigg/bracketleftbig
a>0,b > 0,Reν>−1
2/bracketrightbig
ET II 111(18)
3./integraldisplay∞
0J2ν/parenleftbig
a√x/parenrightbig
Kν(bx)dx=π
2b−1/bracketleftbigg
Iν/parenleftbigga2
4b/parenrightbigg
−Lν/parenleftbigga2
4b/parenrightbigg/bracketrightbigg
/bracketleftbig
Reb>0,Reν>−1
2/bracketrightbig
ET II 144(45)
4.10/integraldisplay∞
0Y2ν/parenleftbig
a√x/parenrightbig
Jν(bx)dx=1
bJν/parenleftbigga2
4b/parenrightbigg
cot(2πν)−1
2bJ−ν/parenleftbigga2
4b/parenrightbigg
cosec(2 πν)
−23ν−3a2−2νbν−2
π3/2Γ/parenleftbig
ν−1
2/parenrightbig
1F2/parenleftbigg
1;3
2,3
2−ν;a4
64b2/parenrightbigg
[a>0,b > 0] MC
5./integraldisplay∞
0Y2ν/parenleftbig
a√x/parenrightbig
Yν(bx)dx
=b−1
2/bracketleftbigg
sec(νπ)J−ν/parenleftbigga2
4b/parenrightbigg
+ cosec( νπ)H−ν/parenleftbigga2
4b/parenrightbigg
−2 cot(2 νπ)Hν/parenleftbigga2
4b/parenrightbigg/bracketrightbigg
/bracketleftbig
a>0,b > 0,|Reν|<1
2/bracketrightbig
ET II 111(19)
664 Bessel Functions 6.517
6./integraldisplay∞
0Y2ν/parenleftbig
a√x/parenrightbig
Kν(bx)dx=πb−1
2⎡
⎣cosec(2 νπ)L−ν/parenleftbigga2
4b/parenrightbigg
−cot(2νπ)Lν/parenleftbigga2
4b/parenrightbigg
−tan(νπ)Iν/parenleftbigga2
4b/parenrightbigg
−sec(νπ)
πKν/parenleftbigga2
4b/parenrightbigg⎤
⎦
/bracketleftbig
Reb>0,|Reν|<1
2/bracketrightbig
ET II 144(46)
7./integraldisplay∞
0K2ν/parenleftbig
a√x/parenrightbig
Jν(bx)dx=1
4πb−1sec(νπ)/bracketleftbigg
H−ν/parenleftbigga2
4b/parenrightbigg
−Y−ν/parenleftbigga2
4b/parenrightbigg/bracketrightbigg
/bracketleftbig
Rea>0,b > 0,Reν>−1
2/bracketrightbig
ET II 70(22)
8./integraldisplay∞
0K2ν/parenleftbig
a√x/parenrightbig
Yν(bx)dx
=−1
4πb−1/bracketleftbigg
sec(νπ)J−ν/parenleftbigga2
4b/parenrightbigg
−cosec( νπ)H−ν/parenleftbigga2
4b/parenrightbigg
+ 2 cosec(2 νπ)Hν/parenleftbigga2
4b/parenrightbigg/bracketrightbigg
/bracketleftbig
Rea>0,b > 0,|Reν|<1
2/bracketrightbig
ET II 114(34)
9./integraldisplay∞
0K2ν/parenleftbig
a√x/parenrightbig
Kν(bx)dx=πb−1
4c os (νπ)/braceleftbigg
Kν/parenleftbigga2
4b/parenrightbigg
+π
2s in (νπ)/bracketleftbigg
L−ν/parenleftbigga2
4b/parenrightbigg
−Lν/parenleftbigga2
4b/parenrightbigg/bracketrightbigg/bracerightbigg
/bracketleftbig
Reb>0,|Reν|<1
2/bracketrightbig
ET II 147(63)
10./integraldisplay∞
0I2ν/parenleftbig
a√x/parenrightbig
Kν(bx)dx=πb−1
2/bracketleftbigg
Iν/parenleftbigga2
4b/parenrightbigg
+Lν/parenleftbigga2
4b/parenrightbigg/bracketrightbigg
/bracketleftbig
Reb>0,Reν>−1
2/bracketrightbig
ET II 147(60)
6.517/integraldisplayz
0J0/parenleftBig/radicalbig
z2−x2/parenrightBig
dx=s i nz MO 48
6.518/integraldisplay∞
0K2ν(2zsinhx)dx=π2
8c osνπ/parenleftbig
J2
ν(z)+N2
ν(z)/parenrightbig/bracketleftbig
Rez>0,−1
2<Reν<1
2/bracketrightbig
MO 45
6.519
1./integraldisplayπ/2
0J2ν(2zcosx)dx=π
2J2
ν(z)/bracketleftbig
Reν>−1
2/bracketrightbig
WH
2./integraldisplayπ/2
0J2ν(2zsinx)dx=π
2J2
ν(z)/bracketleftbig
Reν>−1
2/bracketrightbig
WA 42(1)a
6.52 Bessel functions combined with xandx2
6.521
1./integraldisplay1
0xJν(αx)Jν(βx)dx=βJν−1(β)Jν(α)−αJν−1(α)Jν(β)
α2−β2[α/negationslash=β, ν > −1]
=αJν(β)J/prime
ν(α)−βJν(α)J/prime
ν(β)
β2−α2[α/negationslash=β, ν > −1]
WH
6.522 Bessel functions combined with xandx2665
2.10/integraldisplay∞
0xKν(ax)Jν(bx)dx=bν
aν(b2+a2)[Rea>0,b > 0,Reν>−1]
ET II 63(2)
3./integraldisplay∞
0xKν(ax)Kν(bx)dx=π(ab)−ν/parenleftbig
a2ν−b2ν/parenrightbig
2s in(νπ)(a2−b2)[|Reν|<1,Re(a+b)>0]
ET II 145(48)
4./integraldisplaya
0xJν(λx)Kν(μx)dx=/parenleftbig
μ2+λ2/parenrightbig−1/bracketleftbigg/parenleftbiggλ
μ/parenrightbiggν
+λaJν+1(λa)Kν(μa)−μaJν(λa)Kν+1(μa)/bracketrightbigg
[Reν>−1] ET II 367(26)
5.∗/integraldisplay∞
0xK1(ax)=π
2a2[a>0]
6.∗/integraldisplay∞
0xK2
0(ax)=1
2a2[a>0]
7.∗/integraldisplay∞
0xK1(ax)J1(bx)=b
a(a2+b2)[a>0,b > 0]
8.∗/integraldisplay∞
0xK0(ax)I0(bx)=1
a2−b2[a>b> 0]
9.∗/integraldisplay∞
0xK1(ax)I1(bx)=b
a(a2−b2)[a>b> 0]
10.∗/integraldisplay∞
0x2K0(ax)=π
2a3[a>0]
11.∗/integraldisplay∞
0x2K1(ax)=2
a3[a>0]
12.∗/integraldisplay∞
0x2K0(ax)J1(bx)=2b
(a2+b2)2[a>0,b > 0]
13.∗/integraldisplay∞
0x2K1(ax)J0(bx)=2a
(a2+b2)2[a>b> 0]
14.∗/integraldisplay∞
0x2K0(ax)I1(bx)=2b
(a2−b2)2[a>b> 0]
15.∗/integraldisplay∞
0x2K1(ax)I0(bx)=2a
(a2−b2)2[a>b> 0]
6.522 Notation :/lscript1=1
2/bracketleftBig/radicalbig
(b+c)2+a2−/radicalbig
(b−c)2+a2/bracketrightBig
,/lscript2=1
2/bracketleftBig/radicalbig
(b+c)2+a2+/radicalbig
(b−c)2+a2/bracketrightBig
1.8/integraldisplay∞
0x[Jμ(ax)]2Kν(bx)dx=Γ/parenleftbig
μ+1
2ν+1/parenrightbig
Γ/parenleftbig
μ−1
2ν+1/parenrightbig
b−2
×/parenleftbig
1+4a2b−2/parenrightbig−1
2P−μ
1
2ν/bracketleftBig/parenleftbig
1+4a2b−2/parenrightbig1
2/bracketrightBig
P−μ
−1
2ν/bracketleftBig/parenleftbig
1+4a2b−2/parenrightbig1
2/bracketrightBig
[Reb>2|Ima|,2R eμ>|Reν|−2]ET II 138(19)
666 Bessel Functions 6.522
2./integraldisplay∞
0x[Kμ(ax)]2Jν(bx)dx=2e2μπiΓ/parenleftbig
1+1
2ν+μ/parenrightbig
b/parenleftbig
4a2+b2/parenrightbig1
2Γ/parenleftbig1
2ν−μ/parenrightbig
×Q−μ
1
2ν/parenleftBig/radicalbig
(1 + 4 a2b−2)/parenrightBig
Q−μ
1
2ν−1/parenleftBig/radicalbig
(1 + 4 a2b−2)/parenrightBig
/bracketleftbig
b>0,Rea>0,Re/parenleftbig1
2ν±μ/parenrightbig
>−1/bracketrightbig
ET II 66(27)a
3.11/integraldisplay
0∞
xK0(ax)Jν(bx)Jν(cx)dx=r−1
1r−1
2(r2−r1)ν(r2−r1)−ν=/lscriptν
1
/lscriptν2(/lscript2
2−/lscript2
1),
/bracketleftBig
r1=/radicalbig
a2+(b−c)2,r2=/radicalbig
a2+(b+c)2,c > 0,Reν>−1,Rea>|Imb|/bracketrightBig
ET II 63(6)
4.10/integraldisplay
0∞
xI0(ax)K0(bx)J0(cx)dx=/parenleftbig
a4+b4+c4−2a2b2+2a2c2+2b2c2/parenrightbig−1
2
[Reb>Rea, c > 0] ET II 16(27)
alternatively, with aandcinterchanged/integraldisplay
0∞
xI0(cx)K0(bx)J0(ax)dx=1
/lscript2
2−/lscript2
1[Reb>Rec, a > 0]
5.10/integraldisplay
0∞
xJ0(ax)K0(bx)J0(cx)dx=/parenleftbig
a4+b4+c4−2a2c2+2a2b2+2b2c2/parenrightbig−1
2
[Reb>|Ima|,c > 0] ET II 15(25)
alternatively, with aandbinterchanged/integraldisplay
0∞
xJ0(bx)K0(ax)J0(cx)dx=1
/lscript2
2−/lscript2
1[Rea>|Imb|,c > 0]
6./integraldisplay∞
0xJ0(ax)Y0(ax)J0(bx)dx=0 [ 0 <b< 2a]
=−2π−1b−1/bracketleftbig
b2−4a2/bracketrightbig−1
2[0<2a<b< ∞]
ET II 15(21)
7./integraldisplay∞
0xJμ(ax)Jμ+1(ax)Kν(bx)dx=Γ/parenleftbigg
μ+3+ν
2/parenrightbigg
Γ/parenleftbigg
μ+3−ν
2/parenrightbigg
b−2/parenleftbig
1+4a2b−2/parenrightbig−1
2
×P1
2ν−1
2
−μ/bracketleftBig/radicalbig
1+4a2b−2/bracketrightBig
P1
2ν−1
2
−μ−1/bracketleftBig/radicalbig
1+4a2b−2/bracketrightBig
[Reb>2|Ima|,2R eμ>|Reν|−3]ET II 138(20)
8./integraldisplay∞
0xKμ−1
2(ax)Kμ+1
2(ax)Jν(bx)dx
=−2e2μπiΓ/parenleftbig1
2ν+μ+1/parenrightbig
bΓ/parenleftbig1
2ν−μ/parenrightbig/parenleftbig
b2+4a2/parenrightbig1
2Q−μ+1
2
1
2ν−1
2/bracketleftBig/parenleftbig
1+4a2b−2/parenrightbig1
2/bracketrightBig
Q−μ−1
2
1
2ν−1
2/bracketleftBig/parenleftbig
1+4a2b−2/parenrightbig1
2/bracketrightBig
/bracketleftbig
b>0,Rea>0,Reν>−1,|Reμ|<1+1
2Reν/bracketrightbig
ET II 67(29)a
9.8/integraldisplay∞
0xI1
2ν(ax)K1
2ν(ax)Jν(bx)dx=b−1/parenleftbig
b2+4a2/parenrightbig−1
2
[b>0,Rea>0,Reν>−1]
ET II 65(16)
6.522 Bessel functions combined with xandx2667
10./integraldisplay∞
0xJ1
2ν(ax)Y1
2ν(ax)Jν(bx)dx
=0 [ a>0,Reν>−1,0<b< 2a]
=−2π−1b−1/parenleftbig
b2−4a2/parenrightbig−1
2[a>0,Reν>−1,2a<b< ∞]
ET II 55(48)
11.8/integraldisplay∞
0xJ1
2(ν+n)(ax)J1
2(ν−n)(ax)Jν(bx)dx
=2π−1b−1/parenleftbig
4a2−b2/parenrightbig−1
2Tn/parenleftbiggb
2a/parenrightbigg
[a>0,Reν>−1,0<b< 2a]
=0 [ a>0,Reν>−1,2a<b]
ET II 52(32)
12./integraldisplay∞
0xI1
2(ν−μ)(ax)K1
2(ν+μ)(ax)Jν(bx)dx=2−μa−μb−1/parenleftbig
b2+4a2/parenrightbig−1
2/bracketleftBig
b+/parenleftbig
b2+4a2/parenrightbig1
2/bracketrightBigμ
[b>0,Rea>0,Reν>−1,Re(ν−μ)>−2]ET II 66(23)
13.8/integraldisplay∞
0xJμ(xasinϕ)Kν−μ(axcosϕcosψ)Jν(xasinψ)dx=(sinϕ)μ(sinψ)ν(cosϕ)ν−μ(cosψ)μ−ν
a2/parenleftbig
1−sin2ϕsin2ψ/parenrightbig
/bracketleftBig
a>0,0<ϕ<π
2,0<ψ<π
2,Reμ>−1,Reν>−1/bracketrightBig
ET II 64(10)
14.8/integraldisplay∞
0xJμ(xasinϕcosψ)Jν−μ(ax)Jν(xacosϕsinψ)dx
=−2π−1a−2sin(μπ)(sinϕ)μ(sinψ)ν(cosϕ)−ν(cosψ)−μ[cos(ϕ+ψ)cos(ϕ−ψ)]−1
/bracketleftBig
a>0,0<ϕ<π
2,0<ψ<1
2π,Reν>−1/bracketrightBig
ET II 54(39)
15.10/integraldisplay∞
0xν+1Jν(bx)Kν(ax)Jν(cx)dx=23ν(abc)νΓ/parenleftbig
ν+1
2/parenrightbig
√π(/lscript2
2−/lscript2
1)2ν+1
[Rea>|Imb|,c > 0]
16.10/integraldisplay∞
0xν+1Iν(cx)Kν(bx)Jν(ax)dx=23ν(abc)νΓ/parenleftbig
ν+1
2/parenrightbig
√π(/lscript2
2−/lscript2
1)2ν+1
[Reb>|Ima|+|Imc|]
17.11/integraldisplay∞
0tν−μ−ρ+1Jμ(ct)Jν(bt)Kρ(at)dt
=21+ν−μ−ρ
cμbνaρΓ(μ−ν+ρ)/integraldisplay/lscript1
0x1+2ν−2ρ/bracketleftbig/parenleftbig
/lscript2
1−x2/parenrightbig/parenleftbig
/lscript2
2−x2/parenrightbig/bracketrightbigμ−ν+ρ−1
(b2−x2)μ−νdx
/lscript1=1
2/bracketleftBig/radicalbig
(b+c)2+a2−/radicalbig
(b−c)2+a2/bracketrightBig
,/lscript2=1
2/bracketleftBig/radicalbig
(b+c)2+a2+/radicalbig
(b−c)2+a2/bracketrightBig
[Rea>|Imb|,c > 0]
668 Bessel Functions 6.523
18.11/integraldisplay∞
0tμ−ν+ρ+1Jμ(ct)Jν(bt)Kρ(at)dt
=21+μ−ν+ρaρ
cμbνΓ(ν−μ−ρ)/integraldisplay/lscript1
0x1+2μ+2ρ/bracketleftbig/parenleftbig
/lscript2
1−x2/parenrightbig/parenleftbig
/lscript2
2−x2/parenrightbig/bracketrightbigν−μ−ρ−1
(c2−x2)ν−μdx
/lscript1=1
2/bracketleftBig/radicalbig
(b+c)2+a2−/radicalbig
(b−c)2+a2/bracketrightBig
,/lscript2=1
2/bracketleftBig/radicalbig
(b+c)2+a2+/radicalbig
(b−c)2+a2/bracketrightBig
[Rea>|Imb|,c > 0]
6.523/integraldisplay∞
0x/bracketleftbig
2π−1K0(ax)−Y0(ax)/bracketrightbig
K0(bx)dx=2π−1/bracketleftBig/parenleftbig
a2+b2/parenrightbig−1+/parenleftbig
b2−a2/parenrightbig−1/bracketrightBig
lnb
a
[Reb>|Ima|,Re(a+b)>0]
ET II 145(50)
6.524
1./integraldisplay∞
0xJ2
ν(ax)Jν(bx)Yν(bx)dx=0/bracketleftbig
0<a<b , Reν>−1
2/bracketrightbig
=−(2πab)−1/bracketleftbig
0<b<a , Reν>−1
2/bracketrightbig
ET II 352(14)
2./integraldisplay∞
0x[J0(ax)K0(bx)]2dx=π
8ab−1
4abarcsin/parenleftbiggb2−a2
b2+a2/parenrightbigg
[a>0,b > 0] ET II 373(9)
6.525 Notation :/lscript1=1
2/bracketleftBig/radicalbig
(b+c)2+a2−/radicalbig
(b−c)2+a2/bracketrightBig
,/lscript2=1
2/bracketleftBig/radicalbig
(b+c)2+a2+/radicalbig
(b−c)2+a2/bracketrightBig
1.10/integraldisplay∞
0x2J1(ax)K0(bx)J0(cx)dx=2a/parenleftbig
a2+b2−c2/parenrightbig/bracketleftBig/parenleftbig
a2+b2+c2/parenrightbig2−4a2c2/bracketrightBig−3
2
[c>0,Reb≥|Ima|,Rea>0]
ET II 15(26)
alternatively, with aandbinterchanged
/integraldisplay∞
0x2J1(bx)K0(ax)J0(cx)dx=2b/parenleftbig
a2+b2−c2/parenrightbig
(/lscript2
2−/lscript2
1)3[Rea>|Imb|,Reb>0,c > 0]
2.10/integraldisplay∞
0x2I0(ax)K1(bx)J0(cx)dx=2b/parenleftbig
b2+c2−a2/parenrightbig/bracketleftBig/parenleftbig
a2+b2+c2/parenrightbig2−4a2b2/bracketrightBig−3
2
[Reb>|Rea|,c > 0] ET II 16(28)
3.10/integraldisplay∞
0x2I0(cx)K0(bx)J0(ax)dx=2b/parenleftbig
a2+b2−c2/parenrightbig
(/lscript2
2−/lscript2
1)3[Rea>|Imb|,c > 0]
6.526
1./integraldisplay∞
0xJ1
2ν/parenleftbig
ax2/parenrightbig
Jν(bx)dx=( 2a)−1J1
2ν/parenleftbiggb2
4a/parenrightbigg
[a>0,b > 0,Reν>−1]ET II 56(1)
6.527 Bessel functions combined with xandx2669
2./integraldisplay∞
0xJ1
2ν/parenleftbig
ax2/parenrightbig
Yν(bx)dx
=( 4a)−1/bracketleftbigg
Y1
2ν/parenleftbiggb2
4a/parenrightbigg
−tan/parenleftBigνπ
2/parenrightBig
J1
2ν/parenleftbiggb2
4a/parenrightbigg
+s e c/parenleftBigνπ
2/parenrightBig
H−1
2ν/parenleftbiggb2
4a/parenrightbigg/bracketrightbigg
[a>0,b > 0,Reν>−1]ET II 109(9)
3./integraldisplay∞
0xJ1
2ν/parenleftbig
ax2/parenrightbig
Kν(bx)dx=π
8acos/parenleftBigνπ
2/parenrightBig/bracketleftbigg
H−1
2ν/parenleftbiggb2
4a/parenrightbigg
−Y−1
2ν/parenleftbiggb2
4a/parenrightbigg/bracketrightbigg
[a>0,Reb>0,Reν>−1]
ET II 140(27)
4./integraldisplay∞
0xY1
2ν/parenleftbig
ax2/parenrightbig
Jν(bx)dx=−(2a)−1H1
2ν/parenleftbiggb2
4a/parenrightbigg
[a>0,Reb>0,Reν>−1]
ET II 61(35)
5./integraldisplay∞
0xY1
2ν/parenleftbig
ax2/parenrightbig
Kν(bx)dx
=π
4asin(νπ)/bracketleftbigg
cos/parenleftBigνπ
2/parenrightBig
H−1
2ν/parenleftbiggb2
4a/parenrightbigg
−sin/parenleftBigνπ
2/parenrightBig
J−1
2ν/parenleftbiggb2
4a/parenrightbigg
−H1
2ν/parenleftbiggb2
4a/parenrightbigg/bracketrightbigg
[a>0,Reb>0,|Reν|<1]ET II 141(28)
6./integraldisplay∞
0xK1
2ν/parenleftbig
ax2/parenrightbig
Jν(bx)dx=π
4a/bracketleftbigg
I1
2ν/parenleftbiggb2
4a/parenrightbigg
−L1
2ν/parenleftbiggb2
4a/parenrightbigg/bracketrightbigg
[Rea>0,b > 0,Reν>−1]
ET II 68(9)
7./integraldisplay∞
0xK1
2ν/parenleftbig
ax2/parenrightbig
Yν(bx)dx=π
4a⎡
⎣cosec( νπ)L−1
2ν/parenleftbiggb2
4a/parenrightbigg
−cot(νπ)L1
2ν/parenleftbiggb2
4a/parenrightbigg
−tan/parenleftBigνπ
2/parenrightBig
I1
2ν/parenleftbiggb2
4a/parenrightbigg
−1
πsec/parenleftBigνπ
2/parenrightBig
K1
2ν/parenleftbiggb2
4a/parenrightbigg⎤
⎦
[Rea>0,b > 0,|Reν|<1]ET II 112(25)
8./integraldisplay∞
0xK1
2ν/parenleftbig
ax2/parenrightbig
Kν(bx)dx
=π
8a/braceleftbigg
sec/parenleftBigνπ
2/parenrightBig
K1
2ν/parenleftbiggb2
4a/parenrightbigg
+πcosec( νπ)/bracketleftbigg
L−1
2ν/parenleftbiggb2
4a/parenrightbigg
−L1
2ν/parenleftbiggb2
4a/parenrightbigg/bracketrightbigg/bracerightbigg
[Rea>0,|Reν|<1] ET II 146(52)
6.527
1./integraldisplay∞
0x2J2ν(2ax)Jν−1
2/parenleftbig
x2/parenrightbig
dx=1
2aJν+1
2/parenleftbig
a2/parenrightbig/bracketleftbig
a>0,Reν>−1
2/bracketrightbig
ET II 355(33)
2./integraldisplay∞
0x2J2ν(2ax)Jν+1
2/parenleftbig
x2/parenrightbig
dx=1
2aJν−1
2/parenleftbig
a2/parenrightbig
[a>0,Reν>−2] ET II 355(35)
3./integraldisplay∞
0x2J2ν(2ax)Yν+1
2/parenleftbig
x2/parenrightbig
dx=−1
2aHν−1
2/parenleftbig
a2/parenrightbig
[a>0,Reν>−2] ET II 355(36)
670 Bessel Functions 6.528
6.528/integraldisplay∞
0xK1
4ν/parenleftbiggx2
4/parenrightbigg
I1
4ν/parenleftbiggx2
4/parenrightbigg
Jν(bx)dx=K1
4ν/parenleftbiggx2
4/parenrightbigg
I1
4ν/parenleftbiggb2
4/parenrightbigg
[b>0,ν > −1] MO 183a
6.529
1./integraldisplay∞
0xJν/parenleftbig
2√ax/parenrightbig
Kν/parenleftbig
2√ax/parenrightbig
Jν(bx)dx=1
2b−2e−2a
b [Rea>0,b > 0,Reν>−1]
ET II 70(23)
2./integraldisplaya
0xJλ(2a)Iλ(2x)Jμ/parenleftBig
2/radicalbig
a2−x2/parenrightBig
Iμ/parenleftBig
2/radicalbig
a2−x2/parenrightBig
dx
=a2λ+2μ+2
2Γ (λ+1 )Γ ( μ+1 )Γ ( λ+μ+2 )
×1F4/parenleftbiggλ+μ+1
2;λ+1,μ+1,λ+μ+1,λ+μ+3
2;−a4/parenrightbigg
[Reλ>−1,Reμ>−1]ET II 376(31)
6.53–6.54 Combinations of Bessel functions and rational functions
6.531
1.10/integraldisplay∞
0Yν(bx)
x+adx
=−πJν(ab)cot(πν)cosec( πν)−πJ−ν(ab)cosec2(πν)+1
νcotπν
21F2/parenleftbigg
1;2−ν
2,2+ν
2;−a2b2
4/parenrightbigg
+ab
ν2−11F2/parenleftbigg
1;3−ν
2,3+ν
2;−a2b2
4/parenrightbigg
tanπν
2
[Reν<1,arga/negationslash=π, b > 0] MC
2./integraldisplay∞
0Yν(bx)
x−adx=π/braceleftBig
cot(νπ)[Yν(ab)+Eν(ab)] +Jν(ab)+2[ c o t ( νπ)]2[Jν(ab)−Jν(ab)]/bracerightBig
[b>0,a > 0,|Reν|<1]
ET II 98(9)
3./integraldisplay∞
0Kν(bx)
x+adx=π2
2[cosec( νπ)]2/bracketleftBig
Iν(ab)+I−ν(ab)−e−1
2iνπJν(iab)−e1
2iνπJ−ν(iab)/bracketrightBig
[Reb>0,|arga|<π , |Reν|<1]
ET II 128(5)
6.532
1.11/integraldisplay∞
0Jν(x)
x2+a2dx=i
a/bracketleftBig
S0,ν(ia)−e−iνπ/2Kν(a)/bracketrightBig
=1
a/bracketleftBig
is0,ν(ia)+π
2sec/parenleftBigνπ
2/parenrightBig
Iν(a)/bracketrightBig
[Rea>0,Reν>−1]
6.536 Bessel functions and rational functions 671
2./integraldisplay∞
0Yν(x)
x2+a2dx=1
cosνπ
2⎡
⎣−π
2atan/parenleftBigνπ
2/parenrightBig
Iν(ab)−1
aKν(ab)
+bsin/parenleftBigνπ
2/parenrightBig
1−ν21F2/parenleftbigg
1;3−ν
2,3+ν
2;a2b2
4/parenrightbigg⎤
⎦
[b>0,Rea>0,|Reν|<1]ET II 99(13)
3./integraldisplay∞
0Yν(bx)
x2−a2dx=π
2a/braceleftBig
Jν(ab)+t a n/parenleftBigνπ
2/parenrightBig/braceleftBig
tan/parenleftBigνπ
2/parenrightBig
[Jν(ab)−Jν(ab)]−Eν(ab)−Yν(ab)/bracerightBig/bracerightBig
[b>0,a > 0,|Reν|<1]
ET II 101(21)
4./integraldisplay∞
0xJ0(ax)
x2+k2dx=K0(ak)[ a>0,Rek>0] WA 466(5)
5./integraldisplay∞
0Y0(ax)
x2+k2dx=−K0(ak)
k[a>0,Rek>0] WA 466(6)
6./integraldisplay∞
0J0(ax)
x2+k2dx=π
2k[I0(ak)−L0(ak)] [ a>0,Rek>0] WA 467(7)
6.533
1./integraldisplayz
0Jp(x)Jq(z−x)dx
x=Jp+q(z)
p[Rep>0,Req>−1] WA 415(3)
2./integraldisplayz
0Jp(x)
xJq(z−x)
z−xdx=/parenleftbigg1
p+1
q/parenrightbiggJp+q(z)
z[Rep>0,Req>0] WA 415(5)
3.11/integraldisplay∞
0[J0(ax)−1]J1(bx)dx
x2=−b
4/bracketleftBig
1+2l na
b/bracketrightBig
[0<b<a ]
=−a2
4b[0<a<b ]
ET II 21(28)a
3b/integraldisplay∞
0[J0(ax)−1]J1(bx)dx
x=⎧
⎪⎪⎨
⎪⎪⎩b
2a2F1/parenleftbigg1
2,1
2;2,b2
a2/parenrightbigg
−1[ 0 <b<a ]
2
πE/parenleftbiggb2
a2/parenrightbigg
−1[ 0 <a<b ]
4./integraldisplay∞
0[1−J0(ax)]J0(bx)dx
x=0 [ 0 <a<b ]
=l na
b[0<b<a ]
ET II 14(16)
6.534/integraldisplay∞
0x3J0(x)
x4−a4dx=1
2K0(a)−1
4πY0(a)[ a>0] ET II 340(5)
6.535/integraldisplay∞
0x
x2+a2[Jν(x)]2dx=Iν(a)Kν(a)[ R e a>0,Reν>−1] ET II 342(26)
6.536/integraldisplay∞
0x3J0(bx)
x4+a4dx= ker( ab)/bracketleftbig
b>0,|arga|<1
4π/bracketrightbig
ET II 8(9), MO 46a
672 Bessel Functions 6.537
6.537/integraldisplay∞
0x2J0(bx)
x4+a4dx=−1
a2kei(ab)/bracketleftBig
b>0,|arga|<π
4/bracketrightBig
MO 46a
6.538
1./integraldisplay∞
0J1(ax)J1(bx)dx
x2=a+b
π/bracketleftBigg
E/parenleftBigg
2i√
ab
|b−a|/parenrightBigg
−K/parenleftBigg
2i√
ab
|b−a|/parenrightBigg/bracketrightBigg
[a>0,b > 0] ET II 21(30)
2.8/integraldisplay∞
0x−1Jν+2n+1(x)Jν+2m+1(x)dx=0 [ m/negationslash=nwithm, nintegers, ν>−1]
=( 4n+2ν+2 )−1[m=n, ν > −1]
EH II 64
6.539
1./integraldisplayb
adx
x[Jν(x)]2=π
2/bracketleftbiggYν(b)
Jν(b)−Yν(a)
Jν(a)/bracketrightbigg
[Jν(x)/negationslash=0 f o r x∈[a,b]]ET II 338(41)
2./integraldisplayb
adx
x[Yν(x)]2=π
2/bracketleftbiggJν(a)
Yν(a)−Jν(b)
Yν(b)/bracketrightbigg
[Yν(x)/negationslash=0 f o r x∈[a,b]]
ET II 339(49)
3./integraldisplayb
adx
xJν(x)Yν(x)=π
2ln/bracketleftbiggJν(a)Yν(b)
Jν(b)Yν(a)/bracketrightbigg
ET II 339(50)
6.541
1./integraldisplay∞
0xJν(ax)Jν(bx)dx
x2+c2=Iν(bc)Kν(ac)[ 0 <b<a , Rec>0,Reν>−1]
=Iν(ac)Kν(bc)[ 0 <a<b , Rec>0,Reν>−1]
ET II 49(10)
2.8/integraldisplay∞
0x1−2nJν(ax)Jν(bx)dx
x2+c2
=/parenleftbigg
−1
c2/parenrightbiggn/bracketleftBigg
Iν(bc)Kν(ac)−1
2/parenleftbiggb
a/parenrightbiggνπ
sin(πν)n−1/summationdisplay
p=0/parenleftbig
a2c2/4/parenrightbigp
p!Γ ( 1−ν+p)n−1−p/summationdisplay
k=0/parenleftbig
b2c2/4/parenrightbigk
k!Γ ( 1−ν+k)/bracketrightBigg
[0<b<a ]
=/parenleftbigg
−1
c2/parenrightbiggn/bracketleftBigg
Iν(bc)Kν(ac)−1
2ν/parenleftbiggb
a/parenrightbiggνn−1/summationdisplay
p=0/parenleftbig
a2c2/4/parenrightbigp
p!(1−ν)pn−1−p/summationdisplay
k=0/parenleftbig
b2c2/4/parenrightbigk
k!(1 +ν)k/bracketrightBigg
[n=1,2,..., Reν>n −1,Rec>0,0<b<a ]
6.544 Bessel functions and rational functions 673
3.8/integraldisplay∞
0xα−1
(x2+z2)ρJμ(cx)Jν(cx)dx=1
2/parenleftBigc
2/parenrightBig2ρ−α
×Γ/bracketleftbigg(μ+ν+α)/2−ρ,1+2ρ−α
(μ−ν−α)/2+ρ+1,(μ+ν−α)/2+ρ+1,(ν−μ−α)/2+ρ+1/bracketrightbigg
×3F4⎛
⎝1−α
2+ρ,1−α
2+ρ, ρ;ρ+1−μ+ν+α
2,ρ+1+μ−ν−α
2,
ρ+1+μ+ν−α
2,ρ+1+ν−μ−α
2;c2z2⎞
⎠+zα−2ρ
2/parenleftBigcz
2/parenrightBigμ+ν
,
Γ/bracketleftbiggρ−(α+μ+ν)/2,(α+μ+ν)/2
ρ, μ+1,ν+1/bracketrightbigg
3F4⎛
⎝1+μ+ν
2,1+μ+ν
2
α+μ+ν
2;1−ρ+α+μ+ν
2,μ+1,ν+1,μ+ν+1 ;c2z2⎞
⎠
/bracketleftbigg
Γ/bracketleftbigga1,...,a p
b1,...,b q/bracketrightbigg
=Γ(a1)...Γ(ap)
Γ(b1)...Γ(bq),c > 0,Rez>0,Re(α+μ+ν)>0; Re( α−2ρ)>1/bracketrightbigg
6.542/integraldisplay∞
0Jν(ax)Yν(bx)−Jν(bx)Yν(ax)
x/braceleftBig
[Jν(bx)]2+[Yν(bx)]2/bracerightBigdx=−π
2/parenleftbiggb
a/parenrightbiggν
[0<b<a ] ET II 352(16)
6.543/integraldisplay∞
0Jμ(bx)/braceleftbigg
cos/bracketleftbigg1
2(ν−μ)π/bracketrightbigg
Jν(ax)−sin/bracketleftbigg1
2(ν−μ)π/bracketrightbigg
Yν(ax)/bracerightbiggxdx
x2+r2=Iμ(br)Kν(ar)
[Rer>0,a≥b>0,Reμ>|Reν|−2]
6.544
1./integraldisplay∞
0Jν/parenleftBiga
x/parenrightBig
Yν/parenleftBigx
b/parenrightBigdx
x2=−1
a/bracketleftbigg2
πK2ν/parenleftbigg2√a√
b/parenrightbigg
−Y2ν/parenleftbigg2√a√
b/parenrightbigg/bracketrightbigg
/bracketleftbig
a>0,b > 0,|Reν|<1
2/bracketrightbig
EI II 357(47)
2./integraldisplay∞
0Jν/parenleftBiga
x/parenrightBig
Jν/parenleftBigx
b/parenrightBigdx
x2=1
aJ2ν/parenleftbigg2√a√
b/parenrightbigg/bracketleftbig
a>0,b > 0,Reν>−1
2/bracketrightbig
ET II 57(10)
3./integraldisplay∞
0Jν/parenleftBiga
x/parenrightBig
Kν/parenleftBigx
b/parenrightBigdx
x2=1
ae1
2iνπK2ν/parenleftbigg2√a√
be1
4iπ/parenrightbigg
+1
ae−1
2iνπK2ν/parenleftbigg2√a√
be−1
4iπ/parenrightbigg
/bracketleftbig
Reb>0,a > 0,|Reν|<1
2/bracketrightbig
ET II 142(32)
4./integraldisplay∞
0Yν/parenleftBiga
x/parenrightBig
Jν/parenleftBigx
b/parenrightBigdx
x2=2
aπ/bracketleftbigg
K2ν/parenleftbigg2√a√
b/parenrightbigg
+π
2Y2ν/parenleftbigg2√a√
b/parenrightbigg/bracketrightbigg
/bracketleftbig
a>0,b > 0,|Reν|<1
2/bracketrightbig
ET II 62(38)
5./integraldisplay∞
0Yν/parenleftBiga
x/parenrightBig
Kν/parenleftBigx
b/parenrightBigdx
x2=4
a/bracketleftbigg
e1
2i(ν+1)πK2ν/parenleftbigg2√a√
be1
4iπ/parenrightbigg
+e−1
2i(ν+1)πK2ν/parenleftbigg2√a√
be−1
4iπ/parenrightbigg/bracketrightbigg
/bracketleftbig
Reb>0,a > 0,|Reν|<1
2/bracketrightbig
ET II 143(38)
674 Bessel Functions 6.551
6./integraldisplay∞
0Kν/parenleftBiga
x/parenrightBig
Jν/parenleftBigx
b/parenrightBigdx
x2=i
a/bracketleftbigg
e1
2νπiK2ν/parenleftbigg
e1
4πi2√a√
b/parenrightbigg
−e−1
2νπiK2ν/parenleftbigg
e−1
4πi2√a√
b/parenrightbigg/bracketrightbigg
/bracketleftbig
Rea>0,b > 0,|Reν|<5
2/bracketrightbig
ET II 70(19)
7./integraldisplay∞
0Kν/parenleftBiga
x/parenrightBig
Yν/parenleftBigx
b/parenrightBigdx
x2=2
a/bracketleftbigg
sin/parenleftbigg3
2πν/parenrightbigg
kei2ν/parenleftbigg2√a√
b/parenrightbigg
−cos/parenleftbigg3
2πν/parenrightbigg
ker2ν/parenleftbigg2√a√
b/parenrightbigg/bracketrightbigg
/bracketleftbig
Rea>0,b > 0,|Reν|<5
2/bracketrightbig
ET II 113(29)
8./integraldisplay∞
0Kν/parenleftBiga
x/parenrightBig
Kν/parenleftBigx
b/parenrightBigdx
x2=π
aK2ν/parenleftbigg2√a√
b/parenrightbigg
[Rea>0,Reb>0] ET II 146(55)
6.55 Combinations of Bessel functions and algebraic functions
6.55110
1./integraldisplay1
0x1/2Jν(xy)dx=√
2y−3/2Γ/parenleftbig3
4+1
2ν/parenrightbig
Γ/parenleftbig1
4+1
2ν/parenrightbig
+y−1/2/bracketleftbig/parenleftbig
ν−1
2/parenrightbig
Jν(y)S−1/2,ν−1(y)−Jν−1(y)S1/2,ν(y)/bracketrightbig
/bracketleftbig
y>0,Reν>−3
2/bracketrightbig
ET II 21(1)
2./integraldisplay∞
1x1/2Jν(xy)dx=y−1/2/bracketleftbig
Jν−1(y)S1/2,ν(y)+/parenleftbig1
2−ν/parenrightbig
Jν(y)S−1/2,ν−1(y)/bracketrightbig
[y>0] ET II 22(2)
6.552
1./integraldisplay∞
0Jν(xy)dx
(x2+a2)1/2=Iν/2/parenleftbig1
2ay/parenrightbig
Kν/2/parenleftbig1
2ay/parenrightbig
[Rea>0,y > 0,Reν>−1]
ET II 23(11), WA 477(3), MO 44
2./integraldisplay∞
0Yν(xy)dx
(x2+a2)1/2=−1
πsec/parenleftbig1
2νπ/parenrightbig
Kν/2/parenleftbig1
2ay/parenrightbig/bracketleftbig
Kν/2/parenleftbig1
2ay/parenrightbig
+πsin/parenleftbig1
2νπ/parenrightbig
Iν/2/parenleftbig1
2ay/parenrightbig/bracketrightbig
[y>0,Rea>0,|Reν|<1]
ET II 100(18)
3./integraldisplay∞
0Kν(xy)dx
(x2+a2)1/2=π2
8sec/parenleftbig1
2νπ/parenrightbig/braceleftBig/bracketleftbig
Jν/2/parenleftbig1
2ay/parenrightbig/bracketrightbig2+/bracketleftbig
Yν/2/parenleftbig1
2ay/parenrightbig/bracketrightbig2/bracerightBig
[Rea>0,Rey>0,|Reν|<1]
ET II 128(6)
4./integraldisplay1
0Jν(xy)dx
(1−x2)1/2=π
2/bracketleftbig
Jν/2/parenleftbig1
2y/parenrightbig/bracketrightbig2[y>0,Reν>−1] ET II 24(22)a
5./integraldisplay1
0Y0(xy)dx
(1−x2)1/2=π
2J0/parenleftbig1
2y/parenrightbig
Y0/parenleftbig1
2y/parenrightbig
[y>0] ET II 102(26)a
6./integraldisplay∞
1Jν(xy)dx
(x2−1)1/2=−π
2Jν/2/parenleftbig1
2y/parenrightbig
Yν/2/parenleftbig1
2y/parenrightbig
[y>0] ET II 24(23)a
6.561 Bessel functions and powers 675
7./integraldisplay∞
1Yν(xy)dx
(x2−1)1/2=π
4/braceleftBig/bracketleftbig
Jν/2/parenleftbig1
2y/parenrightbig/bracketrightbig2−/bracketleftbig
Yν/2/parenleftbig1
2y/parenrightbig/bracketrightbig2/bracerightBig
[y>0] ET II 102(27)
6.553/integraldisplay∞
0x−1/2Iν(x)Kν(x)Kμ(2x)dx=Γ/parenleftbig1
4+1
2μ/parenrightbig
Γ/parenleftbig1
4−1
2μ/parenrightbig
Γ/parenleftbig1
4+ν+1
2μ/parenrightbig
Γ/parenleftbig1
4+ν−1
2μ/parenrightbig
4Γ/parenleftbig3
4+ν+1
2μ/parenrightbig
Γ/parenleftbig3
4+ν−1
2μ/parenrightbig
/bracketleftbig
|Reμ|<1
2,2R eν>|Reμ|−1
2/bracketrightbig
ET II 372(2)
6.554
1./integraldisplay∞
0xJ0(xy)dx
(a2+x2)1/2=y−1e−ay[y>0,Rea>0] ET II 7(4)
2./integraldisplay1
0xJ0(xy)dx
(1−x2)1/2=y−1siny [y>0] ET II 7(5)a
3./integraldisplay∞
1xJ0(xy)dx
(x2−1)1/2=y−1cosy [y>0] ET II 7(6)a
4./integraldisplay∞
0xJ0(xy)dx
(x2+a2)3/2=a−1e−ay[y>0,Rea>0] ET II 7(7)a
5.11/integraldisplay∞
0xν+1Jν(ax)
(x4+4k4)ν+1/2dx=/parenleftbig1
2a/parenrightbigν√π
(2k)2νΓ/parenleftbig
ν+1
2/parenrightbigJν(ak)Kν(ak)
/bracketleftbig
a>0,|argk|>π
4,Reν>−1
2/bracketrightbig
WA 473(1)
6.555/integraldisplay∞
0x1/2J2ν−1/parenleftBig
ax1/2/parenrightBig
Yν(xy)dx=−a
2y2Hν−1/parenleftbigga2
4y/parenrightbigg
/bracketleftbig
a>0,y > 0,Reν>−1
2/bracketrightbig
ET II 111(17)
6.556/integraldisplay∞
0Jν/bracketleftBig
a/parenleftbig
x2+1/parenrightbig1/2/bracketrightBigdx√
x2+1=−π
2Jν/2/parenleftBiga
2/parenrightBig
Yν/2/parenleftBiga
2/parenrightBig
[Reν>−1,a > 0] MO 46
6.56–6.58 Combinations of Bessel functions and powers
6.561
1./integraldisplay1
0xνJν(ax)dx=2ν−1a−νπ1
2Γ/parenleftbig
ν+1
2/parenrightbig
[Jν(a)Hν−1(a)−Hν(a)Jν−1(a)]
/bracketleftbig
Reν>−1
2/bracketrightbig
ET II 333(2)a
2./integraldisplay1
0xνYν(ax)dx=2ν−1a−νπ1
2Γ/parenleftbig
ν+1
2/parenrightbig
[Yν(a)Hν−1(a)−Hν(a)Yν−1(a)]
/bracketleftbig
Reν>−1
2/bracketrightbig
ET II 338(43)a
3./integraldisplay1
0xνIν(ax)dx=2ν−1a−νπ1
2Γ/parenleftbig
ν+1
2/parenrightbig
[Iν(a)Lν−1(a)−Lν(a)Iν−1(a)]
/bracketleftbig
Reν>−1
2/bracketrightbig
ET II 364(2)a
676 Bessel Functions 6.561
4./integraldisplay1
0xνKν(ax)dx=2ν−1a−νπ1
2Γ/parenleftbig
ν+1
2/parenrightbig
[Kν(a)Lν−1(a)+Lν(a)Kν−1(a)]
/bracketleftbig
Reν>−1
2/bracketrightbig
ET II 367(21)a
5./integraldisplay1
0xν+1Jν(ax)dx=a−1Jν+1(a)[ R e ν>−1] ET II 333(3)a
6./integraldisplay1
0xν+1Yν(ax)dx=a−1Yν+1(a)+2ν+1a−ν−2π−1Γ(ν+1 )
[Reν>−1] ET II 339(44)a
7./integraldisplay1
0xν+1Iν(ax)dx=a−1Iν+1(a)[ R e ν>−1] ET II 365(3)a
8./integraldisplay1
0xν+1Kν(ax)dx=2νa−ν−2Γ(ν+1 )−a−1Kν+1(a)
[Reν>−1] ET II 367(22)a
9./integraldisplay1
0x1−νJν(ax)dx=aν−2
2ν−1Γ(ν)−a−1Jν−1(a) ET II 333(4)a
10./integraldisplay1
0x1−νYν(ax)dx=aν−2cot(νπ)
2ν−1Γ(ν)−a−1Yν−1(a)[ R e ν<1] ET II 339(45)a
11./integraldisplay1
0x1−νIν(ax)dx=a−1Iν−1(a)−aν−2
2ν−1Γ(ν)ET II 365(4)a
12./integraldisplay1
0x1−νKν(ax)dx=2−νaν−2Γ(1−ν)−a−1Kν−1(a)
[Reν<1] ET II 367(23)a
13.7/integraldisplay1
0xμJν(ax)dx=2μΓ/parenleftbigν+μ+1
2/parenrightbig
aμ+1Γ/parenleftbigν−μ+1
2/parenrightbig+a−μ{(μ+ν−1)Jν(a)Sμ−1,ν−1(a)−Jν−1(a)Sμ,ν(a)}
[a>0,Re(μ+ν)>−1]ET II 22(8)a
14./integraldisplay∞
0xμJν(ax)dx=2μa−μ−1Γ/parenleftbig1
2+1
2ν+1
2μ/parenrightbig
Γ/parenleftbig1
2+1
2ν−1
2μ/parenrightbig/bracketleftbig
−Reν−1<Reμ<1
2,a > 0/bracketrightbig
EH II 49(19)
15./integraldisplay∞
0xμYν(ax)dx=2μcot/bracketleftbig1
2(ν+1−μ)π/bracketrightbig
a−μ−1Γ/parenleftbig1
2+1
2ν+1
2μ/parenrightbig
Γ/parenleftbig1
2+1
2ν−1
2μ/parenrightbig
/bracketleftbig
|Reν|−1<μ<1
2,a > 0/bracketrightbig
ET II 97(3)a
16./integraldisplay∞
0xμKν(ax)dx=2μ−1a−μ−1Γ/parenleftbigg1+μ+ν
2/parenrightbigg
Γ/parenleftbigg1+μ−ν
2/parenrightbigg
[Re(μ+1±ν)>0,Rea>0]
EH II 51(27)
6.564 Bessel functions and powers 677
17./integraldisplay∞
0Jν(ax)
xν−qdx=Γ/parenleftbig1
2q+1
2/parenrightbig
2ν−qaq−ν+1Γ/parenleftbig
ν−1
2q+1
2/parenrightbig/bracketleftbig
−1<Req<Reν−1
2/bracketrightbig
WA 428(1), KU 144(5)
18./integraldisplay∞
0Yν(x)
xν−μdx=Γ/parenleftbig1
2+1
2μ/parenrightbig
Γ/parenleftbig1
2+1
2μ−ν/parenrightbig
sin/parenleftbig1
2μ−ν/parenrightbig
π
2ν−μπ/bracketleftbig
|Reν|<Re(1 + μ−ν)<3
2/bracketrightbig
WA 430(5)
19./integraldisplay1
0x2m+n+1/2Kn+1/2(αx)dx=/radicalbiggπ
2n/summationdisplay
k=0(n+k)!
k!(n−k)!γ(2m+n−k+1,α)
α2m+n+3/22kSTR
6.562
1./integraldisplay∞
0xμYν(bx)dx
x+a=( 2a)μπ−1/braceleftbig
sin/bracketleftbig1
2π(μ−ν)/bracketrightbig
Γ/bracketleftbig1
2(μ+ν+1 )/bracketrightbig
Γ/bracketleftbig1
2(1 +μ−ν)/bracketrightbig
S−μ,ν(ab)
−2c os/bracketleftbig1
2π(μ−ν)/bracketrightbig
Γ/parenleftbig
1+1
2μ+1
2ν/parenrightbig
Γ/parenleftbig
1+1
2μ−1
2ν/parenrightbig
S−μ−1,ν(ab)/bracerightbig
/bracketleftbig
b>0,|arga|<π , Re (μ±ν)>−1,Reμ<3
2/bracketrightbig
ET II 98(8)
2./integraldisplay∞
0xνJν(ax)
x+kdx=πkν
2c osνπ[H−ν(ak)−Y−ν(ak)]/bracketleftbig
−1
2<Reν<3
2,a > 0,|argk|<π/bracketrightbig
WA 479(7)
3./integraldisplay∞
0xμKν(bx)dx
x+a
=2μ−2Γ/bracketleftbig1
2(μ+ν)/bracketrightbig
Γ/bracketleftbig1
2(μ−ν)/bracketrightbig
b−μ
1F2/parenleftbigg
1;1−μ+ν
2,1−μ−ν
2;a2b2
4/parenrightbigg
−2μ−3Γ/bracketleftbig1
2(μ−ν−1)/bracketrightbig
Γ/bracketleftbig1
2(μ+ν−1)/bracketrightbig
ab1−μ
1F2/parenleftbigg
1;3−μ−ν
2,3−μ+ν
2;a2b2
4/parenrightbigg
−πaμcosec[ π(μ−ν)]{Kν(ab)+πcos(μπ)cosec[ π(ν+μ)]Iν(ab)}
[Reb>0,|arga|<π , Reμ>|Reν|−1]ET II 127(4)
6.563/integraldisplay∞
0x/rho1−1Jν(bx)dx
(x+a)1+μ=πa/rho1−μ−1
sin[(/rho1+ν−μ)π]Γ(μ+1 )
×⎧
⎨
⎩∞/summationdisplay
m=0(−1)m/parenleftbig1
2ab/parenrightbigν+2mΓ(/rho1+ν+2m)
m!Γ (ν+m+1 )Γ( /rho1+ν−μ+2m)
−∞/summationdisplay
m=0/parenleftbig1
2ab/parenrightbigμ+1−/rho1+mΓ(μ+m+1 )
m!Γ/bracketleftbig1
2(μ+ν−/rho1+m+3 )/bracketrightbigsin/bracketleftbig1
2(/rho1+ν−μ−m)π/bracketrightbig
Γ/bracketleftbig1
2(μ−ν−/rho1+m+3 )/bracketrightbig⎫
⎬
⎭
/bracketleftbig
b>0,|arga|<π , Re(/rho1+ν)>0,Re(/rho1−μ)<5
2/bracketrightbig
ET II 23(10), WA 479
6.564
1./integraldisplay∞
0xν+1Jν(bx)dx√
x2+a2=/radicalbigg
2
πbaν+1
2Kν+1
2(ab)/bracketleftbig
Rea>0,b > 0,−1<Reν<1
2/bracketrightbig
ET II 23(15)
678 Bessel Functions 6.565
2./integraldisplay∞
0x1−νJν(bx)dx√
x2+a2=/radicalbiggπ
2ba1
2−ν/bracketleftBig
Iν−1
2(ab)−Lν−1
2(ab)/bracketrightBig
/bracketleftbig
Rea>0,b > 0,Reν>−1
2/bracketrightbig
ET II 23(16)
6.565
1./integraldisplay∞
0x−ν/parenleftbig
x2+a2/parenrightbig−ν−1
2Jν(bx)dx=2νa−2νbνΓ(ν+1 )
Γ(2ν+1 )Iν/parenleftbiggab
2/parenrightbigg
Kν/parenleftbiggab
2/parenrightbigg
/bracketleftbig
Rea>0,b > 0,Reν>−1
2/bracketrightbig
WA 477(4), ET II 23(17)
2./integraldisplay∞
0xν+1/parenleftbig
x2+a2/parenrightbig−ν−1
2Jν(bx)dx=√πbν−1
2νeabΓ/parenleftbig
ν+1
2/parenrightbig
/bracketleftbig
Rea>0,b > 0,Reν>−1
2/bracketrightbig
ET II 24(18)
3./integraldisplay∞
0xν+1/parenleftbig
x2+a2/parenrightbig−ν−3
2Jν(bx)dx=bν√π
2ν+1aeabΓ/parenleftbig
ν+3
2/parenrightbig
[Rea>0,b > 0,Reν>−1]
ET II 24(19)
4./integraldisplay∞
0Jν(bx)xν+1
(x2+a2)μ+1dx=aν−μbμ
2μΓ(μ+1 )Kν−μ(ab)
/bracketleftbig
−1<Reν<Re/parenleftbig
2μ+3
2/parenrightbig
,a > 0,b > 0/bracketrightbig
MO 43
5./integraldisplay∞
0xν+1/parenleftbig
x2+a2/parenrightbigμYν(bx)dx=2ν−1π−1a2μ+2(1 +μ)−1Γ(ν)b−ν
×1F2/parenleftbigg
1;1−ν,2+μ;a2b2
4/parenrightbigg
−2μaμ+ν+1[sin(νπ)]−1
×Γ(μ+1 )b−1−μ[Iμ+ν+1(ab)−2c os (μπ)Kμ+ν+1(ab)]
[b>0,Rea>0,−1<Reν<−2R eμ]ET II 100(19)
6.10/integraldisplay∞
0x1−ν/parenleftbig
x2+a2/parenrightbigμYν(bx)dx=2μa1+μ−νb−1−μπ
Γ(−μ)I−1−μ+ν(ab)cot [π(μ−ν)]cosec( πμ)
−2μa1+μ−νb−1−μπ
Γ(−μ)I1+μ−ν(ab)cosec[ π(μ−ν)]cosec( πν)
+2−1−νa2+2μbν
(1 +μ)πcos(πν)Γ(−μ)1F2/parenleftbigg
1; 2 + μ,1+ν;a2b2
4/parenrightbigg
/bracketleftbig
Reν<1,Re(ν−2μ)>−3,arga2/negationslash=π, b > 0/bracketrightbig
MC
7./integraldisplay∞
0x1+ν/parenleftbig
x2+a2/parenrightbigμKν(bx)dx=2νΓ(ν+1 )aν+μ+1b−1−μSμ−ν,μ+ν+1(ab)
[Rea>0,Reb>0,Reν>−1]
ET II 128(8)
6.567 Bessel functions and powers 679
8.11/integraldisplay∞
0x/rho1−1Jν(ax)
(x2+k2)μ+1dx=aνk/rho1+ν−2μ−2Γ/parenleftbig1
2/rho1+1
2ν/parenrightbig
Γ/parenleftbig
μ+1−1
2/rho1−1
2ν/parenrightbig
2ν+1Γ(μ+1 )Γ ( ν+1 )
×1F2/parenleftbigg/rho1+ν
2;/rho1+ν
2−μ, ν+1 ;a2k2
4/parenrightbigg
+a2μ+2−/rho1Γ/parenleftbig1
2ν+1
2/rho1−μ−1/parenrightbig
22μ+3−/rho1Γ/parenleftbigg
μ+2+1
2ν−1
2/rho1/parenrightbigg
×1F2/parenleftbigg
μ+1 ;μ+2+ν−/rho1
2,μ+2−ν+/rho1
2;a2k2
4/parenrightbigg
/bracketleftbig
a>0,−Reν<Re/rho1<2R eμ+7
2,Rek>0/bracketrightbig
WA 477(1)
6.566
1./integraldisplay∞
0xμYν(bx)dx
x2+a2=2μ−2π−1b1−μ
×cos/bracketleftBigπ
2(μ−ν+1 )/bracketrightBig
Γ/parenleftbig1
2μ+1
2ν−1
2/parenrightbig
Γ/parenleftbig1
2μ−1
2ν−1
2/parenrightbig
×1F2/parenleftbigg
1;2−μ+1+ ν
2,2−μ+1−ν
2;a2b2
4/parenrightbigg
−1
2πaμ−1cosec/bracketleftBigπ
2(μ+ν+1 )/bracketrightBig
cot/bracketleftBigπ
2(μ−ν+1 )/bracketrightBig
Iν(ab)
−aμ−1cosec/bracketleftBigπ
2(μ−ν+1 )/bracketrightBig
Kν(ab)
/bracketleftbig
b>0,Rea>0,|Reν|−1<Reμ<5
2/bracketrightbig
ET II 100(17)
2./integraldisplay∞
0xν+1Jν(ax)dx
x2+b2=bνKν(ab)/bracketleftbig
a>0,Reb>0,−1<Reν<3
2/bracketrightbig
EH II 96(58)
3./integraldisplay∞
0xνKν(ax)dx
x2+b2=π2bν−1
4c osνπ[H−ν(ab)−Y−ν(ab)]
/bracketleftbig
a>0,Reb>0,Reν>−1
2/bracketrightbig
WA 468(9)
4./integraldisplay∞
0x−νKν(ax)dx
x2+b2=π2
4bν+1cosνπ[Hν(ab)−Yν(ab)]
/bracketleftbig
a>0,Reb>0,Reν<1
2/bracketrightbig
WA 468(10)
5./integraldisplay∞
0x−νJν(ax)dx
x2+b2=π
2bν+1[Iν(ab)−Lν(ab)]/bracketleftbig
a>0,Reb>0,Reν>−5
2/bracketrightbig
WA 468(11)
6.567
1./integraldisplay1
0xν+1/parenleftbig
1−x2/parenrightbigμJν(bx)dx=2μΓ(μ+1 )b−(μ+1)Jν+μ+1(b)
[b>0,Reν>−1,Reμ>−1]
ET II 26(33)a
680 Bessel Functions 6.567
2./integraldisplay1
0xν+1/parenleftbig
1−x2/parenrightbigμYν(bx)dx
=b−(μ+1)/bracketleftbig
2μΓ(μ+1 )Yμ+ν+1(b)+2ν+1π−1Γ(ν+1 )Sμ−ν,μ+ν+1(b)/bracketrightbig
[b>0,Reμ>−1,Reν>−1]ET II 103(35)a
3./integraldisplay1
0x1−ν/parenleftbig
1−x2/parenrightbigμJν(bx)dx=21−νSν+μ,μ−ν+1(b)
bμ+1Γ(ν)[b>0,Reμ>−1] ET II 25(31)a
4./integraldisplay1
0x1−ν/parenleftbig
1−x2/parenrightbigμYν(bx)dx=b−(μ+1)/bracketleftbigg
21−νπ−1cos(νπ)Γ( 1−ν)
×Sμ+ν,μ−ν+1(b)−2μcosec( νπ)Γ(μ+1 )Jμ−ν+1(b)/bracketrightbigg
[b>0,Reμ>−1,Reν<1]ET II 104(37)a
5./integraldisplay1
0x1−ν/parenleftbig
1−x2/parenrightbigμKν(bx)dx=2−ν−2bν(μ+1 )−1Γ(−ν)1F2/parenleftbigg
1;ν+1,μ+2 ;b2
4/parenrightbigg
+π2μ−1b−(μ+1)cosec ( νπ)Γ (μ+1 )Iμ−ν+1(b)
[Reμ>−1,Reν<1] ET II 129(12)a
6./integraldisplay1
0x1−νJν(bx)dx√
1−x2=/radicalbiggπ
2bHν−1
2(b)[ b>0] ET II 24(24)a
7./integraldisplay1
0x1+νYν(bx)dx√
1−x2=/radicalbiggπ
2bcosec( νπ)/bracketleftBig
cos(νπ)Jν+1
2(b)−H−ν−1
2(b)/bracketrightBig
[b>0,Reν>−1] ET II 102(28)a
8./integraldisplay1
0x1−νYν(bx)dx√
1−x2=/radicalbiggπ
2b/braceleftBig
cot(νπ)/bracketleftBig
Hν−1
2(b)−Yν−1
2(b)/bracketrightBig
−Jν−1
2(b)/bracerightBig
[b>0,Reν<1] ET II 102(30)a
9./integraldisplay1
0xν/parenleftbig
1−x2/parenrightbigν−1
2Jν(bx)dx=2ν−1√πb−νΓ/parenleftbig
ν+1
2/parenrightbig/bracketleftbigg
Jν/parenleftbiggb
2/parenrightbigg/bracketrightbigg2
/bracketleftbig
b>0,Reν>−1
2/bracketrightbig
ET II 24(25)a
10./integraldisplay1
0xν/parenleftbig
1−x2/parenrightbigν−1
2Yν(bx)dx=2ν−1√πb−νΓ/parenleftbigg
ν+1
2/parenrightbigg
Jν/parenleftbiggb
2/parenrightbigg
Yν/parenleftbiggb
2/parenrightbigg
/bracketleftbig
b>0,Reν>−1
2/bracketrightbig
ET II 102(31)a
11./integraldisplay1
0xν/parenleftbig
1−x2/parenrightbigν−1
2Kν(bx)dx=2ν−1√πb−νΓ/parenleftbigg
ν+1
2/parenrightbigg
Iν/parenleftbiggb
2/parenrightbigg
Kν/parenleftbiggb
2/parenrightbigg
/bracketleftbig
Reν>−1
2/bracketrightbig
ET II 129(10)a
12./integraldisplay1
0xν/parenleftbig
1−x2/parenrightbigν−1
2Iν(bx)dx=2−ν−1√πb−νΓ/parenleftbigg
ν+1
2/parenrightbigg/bracketleftbigg
Iν/parenleftbiggb
2/parenrightbigg/bracketrightbigg2
ET II 365(5)a
13./integraldisplay1
0xν+1/parenleftbig
1−x2/parenrightbig−ν−1
2Jν(bx)dx=2−νbν−1
√πΓ/parenleftbigg1
2−ν/parenrightbigg
sinb
/bracketleftbig
b>0,|Reν|<1
2/bracketrightbig
ET II 25(27)a
6.571 Bessel functions and powers 681
14./integraldisplay∞
1xν/parenleftbig
x2−1/parenrightbigν−1
2Yν(bx)dx=2ν−2√πb−νΓ/parenleftbigg
ν+1
2/parenrightbigg/bracketleftbigg
Jν/parenleftbiggb
2/parenrightbigg
J−ν/parenleftbiggb
2/parenrightbigg
−Yν/parenleftbiggb
2/parenrightbigg
Y−ν/parenleftbiggb
2/parenrightbigg/bracketrightbigg
/bracketleftbig
|Reν|<1
2,b > 0/bracketrightbig
ET II 103(32)a
15./integraldisplay∞
1xν/parenleftbig
x2−1/parenrightbigν−1
2Kν(bx)dx=2ν−1
√πb−νΓ/parenleftbigg
ν+1
2/parenrightbigg/bracketleftbigg
Kν/parenleftbiggb
2/parenrightbigg/bracketrightbigg2
/bracketleftbig
Reb>0,Reν>−1
2/bracketrightbig
ET II 129(11)a
16./integraldisplay∞
1x−ν/parenleftbig
x2−1/parenrightbig−ν−1
2Jν(bx)dx=−2−ν−1√πbνΓ/parenleftbigg1
2−ν/parenrightbigg
Jν/parenleftbiggb
2/parenrightbigg
Yν/parenleftbiggb
2/parenrightbigg
/bracketleftbig
b>0,|Reν|<1
2/bracketrightbig
ET II 25(26)a
17.8/integraldisplay∞
1x−ν+1/parenleftbig
x2−1/parenrightbigν−1
2Jν(bx)dx=2ν
√πb−ν−1Γ/parenleftbigg1
2+ν/parenrightbigg
cosb
/bracketleftbig
b>0,|Reν|<1
2/bracketrightbig
ET II 25(28)
6.568
1./integraldisplay∞
0xνYν(bx)dx
x2−a2=π
2aν−1Jν(ab)/bracketleftbig
a>0,b > 0,−1
2<Reν<5
2/bracketrightbig
ET II 101(22)
2./integraldisplay∞
0xμYν(bx)dx
x2−a2=π
2aμ−1Jν(ab)+2μπ−1aμ−1cos/bracketleftBigπ
2(μ−ν+1 )/bracketrightBig
×Γ/parenleftbiggμ−ν+1
2/parenrightbigg
Γ/parenleftbiggμ+ν+1
2/parenrightbigg
S−μ,ν(ab)
/bracketleftbig
a>0,b > 0,|Reν|−1<Reμ<5
2/bracketrightbig
ET II (101)(25)
6.569/integraldisplay1
0xλ(1−x)μ−1Jν(ax)dx
=Γ(μ)Γ(1+ λ+ν)2−νaν
Γ(ν+1 )Γ ( 1+ λ+μ+ν)
×2F3/parenleftbiggλ+1+ ν
2,λ+2+ ν
2;ν+1,λ+1+ μ+ν
2,λ+2+ μ+ν
2;−a2
4/parenrightbigg
[Reμ>0,Re(λ+ν)>−1]ET II 193(56)a
6.571
1./integraldisplay∞
0/bracketleftBig/parenleftbig
x2+a2/parenrightbig1
2±x/bracketrightBigμ
Jν(bx)dx√
x2+a2=aμI1
2(ν∓μ)/parenleftbiggab
2/parenrightbigg
K1
2(ν±μ)/parenleftbiggab
2/parenrightbigg
/bracketleftbig
Rea>0,b > 0,Reν>−1,Reμ<3
2/bracketrightbig
ET II 26(38)
2./integraldisplay∞
0/bracketleftBig/parenleftbig
x2+a2/parenrightbig1
2−x/bracketrightBigμ
Yν(bx)dx√
x2+a2
=aμ/bracketleftbigg
cot(νπ)I1
2(μ+ν)/parenleftbiggab
2/parenrightbigg
K1
2(μ−ν)/parenleftbiggab
2/parenrightbigg
−cosec( νπ)I1
2(μ−ν)/parenleftbiggab
2/parenrightbigg
K1
2(μ+ν)/parenleftbiggab
2/parenrightbigg/bracketrightbigg
/bracketleftbig
Rea>0,b > 0,Reμ>−3
2,|Reν|<1/bracketrightbig
ET II 104(40)
682 Bessel Functions 6.572
3./integraldisplay∞
0/bracketleftBig/parenleftbig
x2+a2/parenrightbig1
2+x/bracketrightBigμ
Kν(bx)dx√
x2+a2
=π2
4aμcosec( νπ)/bracketleftbigg
J1
2(ν−μ)/parenleftbiggab
2/parenrightbigg
Y−1
2(ν+μ)/parenleftbiggab
2/parenrightbigg
−Y1
2(ν−μ)/parenleftbiggab
2/parenrightbigg
J−1
2(ν+μ)/parenleftbiggab
2/parenrightbigg/bracketrightbigg
[Rea>0,Reb>0] ET II 130(15)
6.572
1./integraldisplay∞
0x−μ/bracketleftBig/parenleftbig
x2+a2/parenrightbig1
2+a/bracketrightBigμ
Jν(bx)dx√
x2+a2=Γ/parenleftbig1+ν−μ
2/parenrightbig
abΓ(ν+1 )W1
2μ,1
2ν(ab)M−1
2μ,1
2ν(ab)
[Rea>0,b > 0,Re(ν−μ)>−1]
ET II 26(40)
2./integraldisplay∞
0x−μ/bracketleftBig/parenleftbig
x2+a2/parenrightbig1
2+a/bracketrightBigμ
Kν(bx)dx√
x2+a2
=Γ/parenleftbig1+ν−μ
2/parenrightbig
Γ/parenleftbig1−ν−μ
2/parenrightbig
2abW1
2μ,1
2ν(iab)W1
2μ,1
2ν(−iab)
[Rea>0,Reb>0,Reμ+|Reν|<1]ET II 130(18), BU 87(6a)
3./integraldisplay∞
0x−μ/bracketleftBig/parenleftbig
x2+a2/parenrightbig1
2−a/bracketrightBigμ
Yν(bx)dx√
x2+a2
=−1
abW−1
2μ,1
2ν(ab)/braceleftBigg
Γ/parenleftbig1+ν+μ
2/parenrightbig
Γ(ν+1 )tan/parenleftbiggν−μ
2π/parenrightbigg
M1
2μ,1
2ν(ab)
+s e c/parenleftbiggν−μ
2π/parenrightbigg
W1
2μ,1
2ν(ab)/bracerightbigg
/bracketleftbig
Rea>0,b > 0,|Reν|<1
2+1
2Reμ/bracketrightbig
ET II 105(42)
6.573
1./integraldisplay∞
0xν−M+1Jν(bx)k/productdisplay
i=1Jμi(aix)dx=0 M=k/summationdisplay
i=1μi
/bracketleftBigg
ai>0,k/summationdisplay
i=1ai<b< ∞,−1<Reν<ReM+1
2k−1
2/bracketrightBigg
ET II 54(42)
2./integraldisplay∞
0xν−M−1Jν(bx)k/productdisplay
i=1Jμi(aix)dx=2ν−M−1b−νΓ(ν)k/productdisplay
i=1aμi
i
Γ(1+ μi),M =k/summationdisplay
i=1μi
/bracketleftBigg
ai>0,k/summationdisplay
i=1ai<b< ∞,0<Reν<ReM+1
2k+3
2/bracketrightBigg
WA 460(16)a, ET II 54(43)
6.576 Bessel functions and powers 683
6.574
1.8/integraldisplay∞
0Jν(αt)Jμ(βt)t−λdt=ανΓ/parenleftbiggν+μ−λ+1
2/parenrightbigg
2λβν−λ+1Γ/parenleftbigg−ν+μ+λ+1
2/parenrightbigg
Γ(ν+1 )
×F/parenleftbiggν+μ−λ+1
2,ν−μ−λ+1
2;ν+1 ;α2
β2/parenrightbigg
[Re(ν+μ−λ+1 )>0,Reλ>−1,0<α<β ]WA 439(2)a, MO 49
If we reverse the positions of νandμand at the same time reverse the positions of αandβ,t h e
function on the right-hand side of this equation will change. Thus, the right-hand side represents
a function ofα
βthat is not analytic atα
β=1 .
Forα=β, we have the following equation:
2./integraldisplay∞
0Jν(αt)Jμ(αt)t−λdt=αλ−1Γ(λ)Γ/parenleftbiggν+μ−λ+1
2/parenrightbigg
2λΓ/parenleftbigg−ν+μ+λ+1
2/parenrightbigg
Γ/parenleftbiggν+μ+λ+1
2/parenrightbigg
Γ/parenleftbiggν−μ+λ+1
2/parenrightbigg
[Re(ν+μ+1 )>Reλ>0,α > 0]
MO 49, WA 441(2)a
Ifμ−ν+λ+1(or ν−μ+λ+1) is a negative integer, the right-hand side of equation 6.574 1( o r
6.574 3) vanishes. The cases in which the hypergeometric function Fin6.574 3( o r6.574 1)
can be reduced to an elementary function are then especially important.
3.∗/integraldisplay∞
0Jν(αt)Jμ(βt)t−λdt=βνΓ/parenleftbiggμ+ν−λ+1
2/parenrightbigg
2λαμ−λ+1Γ/parenleftbiggν−μ+λ+1
2/parenrightbigg
Γ(ν+1 )
×F/parenleftbiggν+μ−λ+1
2,−ν+μ−λ+1
2;μ+1 ;β2
α2/parenrightbigg
[Re(ν+μ−λ+1 )>0,Reλ>−1,0<β<α ]MO 50, WA 440(3)a
Ifμ−ν+λ+1(or ν−μ+λ+1) is a negative integer, the right-hand side of equation 6.754 1( o r
6.574 3) vanishes. The cases in which the hypergeometric function Fin6.754 3( o r6.574 1)
can be reduced to an elementary function are then especially important.
6.575
1.11/integraldisplay∞
0Jν+1(αt)Jμ(βt)tμ−νdt=0 [ α<β ]
=/parenleftbig
α2−β2/parenrightbigν−μβμ
2ν−μαν+1Γ(ν−μ+1 )[α≥β]
[Re(ν+1 )>Reμ>−1] MO 51
2./integraldisplay∞
0Jν(x)Jμ(x)
xν+μdx=√πΓ(ν+μ)
2ν+μΓ/parenleftbig
ν+μ+1
2/parenrightbig
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig
μ+1
2/parenrightbig
[Re(ν+μ)>0]KU 147(17), WA 434(1)
684 Bessel Functions 6.576
6.576
1./integraldisplay∞
0xμ−ν+1Jμ(x)Kν(x)dx=1
2Γ(μ−ν+1 ) [ R e μ>−1,Re(μ−ν)>−1]
ET II 370(47)
2.11/integraldisplay∞
0x−λJν(ax)Jν(bx)dx=aνbνΓ/parenleftbigg
ν+1−λ
2/parenrightbigg
2λ(a+b)2ν−λ+1Γ(ν+1 )Γ/parenleftbigg1+λ
2/parenrightbigg
×F/parenleftbigg
ν+1−λ
2,ν+1
2;2ν+1 ;4ab
(a+b)2/parenrightbigg
[a>0,b > 0,2R eν+1>Reλ>−1]ET II 47(4)
3./integraldisplay∞
0x−λKμ(ax)Jν(bx)dx=bνΓ/parenleftbiggν−λ+μ+1
2/parenrightbigg
Γ/parenleftbiggν−λ−μ+1
2/parenrightbigg
2λ+1aν−λ+1Γ(1 + ν)
×F/parenleftbiggν−λ+μ+1
2,ν−λ−μ+1
2;ν+1 ;−b2
a2/parenrightbigg
[Re(a±ib)>0,Re(ν−λ+1 )>|Reμ|]EH II 52(31), ET II 63(4), WA 449(1)
4./integraldisplay∞
0x−λKμ(ax)Kν(bx)dx=2−2−λa−ν+λ−1bν
Γ(1−λ)Γ/parenleftbigg1−λ+μ+ν
2/parenrightbigg
Γ/parenleftbigg1−λ−μ+ν
2/parenrightbigg
×Γ/parenleftbigg1−λ+μ−ν
2/parenrightbigg
Γ/parenleftbigg1−λ−μ−ν
2/parenrightbigg
×F/parenleftbigg1−λ+μ+ν
2,1−λ−μ+ν
2;1−λ;1−b2
a2/parenrightbigg
[Rea+b>0,Reλ<1−|Reμ|−|Reν|]ET II 145(49), EH II 93(36)
5./integraldisplay∞
0x−λKμ(ax)Iν(bx)dx=bνΓ/parenleftbig1
2−1
2λ+1
2μ+1
2ν/parenrightbig
Γ/parenleftbig1
2−1
2λ−1
2μ+1
2ν/parenrightbig
2λ+1Γ(ν+1 )a−λ+ν+1
×F/parenleftbigg1
2−1
2λ+1
2μ+1
2ν,1
2−1
2λ−1
2μ+1
2ν;ν+1 ;b2
a2/parenrightbigg
[Re(ν+1−λ±μ)>0,a > b ]EH II 93(35)
6./integraldisplay∞
0x−λYμ(ax)Jν(bx)dx=2
πsinπ(ν−μ−λ)
2/integraldisplay∞
0x−λKμ(ax)Iν(bx)dx
[a>b , Reλ>−1,Re (ν−λ+1±μ)>0] (see 6.576 5)EH II 93(37)
7.8/integraldisplay∞
0xμ+ν+1Jμ(ax)Kν(bx)dx=2μ+νaμbνΓ(μ+ν+1 )
(a2+b2)μ+ν+1
[Reμ>|Reν|−1,Reb>|Ima|]
ET 137(16), EH II 93(36)
6.578 Bessel functions and powers 685
6.577
1.8/integraldisplay∞
0xν−μ+1+2 nJμ(ax)Jν(bx)dx
x2+c2=(−1)ncν−μ+2nIμ(ac)Kν(bc)
[a>0,b > a , Rec>0,2+R e μ−2n>Reν>−1−n, n ≥0 an integer] ET II 49(13)
2.8/integraldisplay∞
0xμ−ν+1+2 nJμ(ax)Jν(bx)dx
x2+c2=(−1)ncμ−ν+2nIν(bc)Kμ(ac)
[b>0,a > b , Reν−2n+2>Reμ>−n−1,n≥0 an integer] ET II 49(15)
6.578
1./integraldisplay∞
0x/rho1−1Jλ(ax)Jμ(bx)Jν(cx)dx=2/rho1−1aλbμc−λ−μ−/rho1Γ/parenleftBig
λ+μ+ν+/rho1
2/parenrightBig
Γ(λ+1 )Γ ( μ+1 )Γ/parenleftbigg
1−λ+μ−ν+/rho1
2/parenrightbigg
×F4/parenleftbiggλ+μ−ν+/rho1
2,λ+μ+ν+/rho1
2;λ+1,μ+1 ;a2
c2,b2
c2/parenrightbigg
/bracketleftbigg
Re(λ+μ+ν+/rho1)>0,Re/rho1<5
2,a > 0,b > 0,c > 0,c > a +b/bracketrightbigg
ET II 351(9)
2./integraldisplay∞
0x/rho1−1Jλ(ax)Jμ(bx)Kν(cx)dx
=2/rho1−2aλbμc−/rho1−λ−μ
Γ(λ+1 )Γ ( μ+1 )Γ/parenleftbigg/rho1+λ+μ−ν
2/parenrightbigg
Γ/parenleftbigg/rho1+λ+μ+ν
2/parenrightbigg
×F4/parenleftbigg/rho1+λ+μ−ν
2,/rho1+λ+μ+ν
2;λ+1,μ+1 ;−a2
c2,−b2
c2/parenrightbigg
[Re(/rho1+λ+μ)>|Reν|,Rec>|Ima|+|Imb|]ET II 373(8)
3./integraldisplay∞
0xλ−μ−ν+1Jν(ax)Jμ(bx)Jλ(cx)dx=0
/bracketleftbig
Reλ>−1,Re(λ−μ−ν)<1
2,c > b > 0,0<a<c −b/bracketrightbig
ET II 53(36)
4./integraldisplay∞
0xλ−μ−ν−1Jν(ax)Jμ(bx)Jλ(cx)dx=2λ−μ−ν−1aνbμΓ(λ)
cλΓ(μ+1 )Γ ( ν+1 )/bracketleftbig
Reλ>0,Re(λ−μ−ν)<5
2,c > b > 0,0<a<c −b/bracketrightbig
ET II 53(37)
5./integraldisplay∞
0x1+μYμ(ax)Jν(bx)Jν(cx)dx=0 [ 0 <b<c , 0<a<c −b]
ET II 352(13)
6.11/integraldisplay∞
0xμ+1Kμ(ax)Jν(bx)Jν(cx)dx=1√
2πaμb−μ−1c−μ−1e−(μ+1
2)πi/parenleftbig
u2−1/parenrightbig−1
2μ−1
4Qμ+1
2
ν−1
2(u)
/bracketleftbig
2bcu=a2+b2+c2,Rea>|Imb|+|Imc|,Reν>−1,Re(μ+ν)>−1/bracketrightbig
WA 452(2), ET II 64(12)
7.11/integraldisplay∞
0xμ+1Iν(ax)Kμ(bx)Jν(cx)dx=1√
2πa−μ−1bμc−μ−1e−(μ−1
2ν+1
4)πi/parenleftbig
v2+1/parenrightbig−1
2μ−1
4Qμ+1
2
ν−1
2(iv),
2acv=b2−a2+c2[Reb>|Rea|+|Imc|;R e ν>−1,Re(μ+ν)>−1]ET II 66(22)
686 Bessel Functions 6.578
8.11/integraldisplay∞
0x1−μJμ(ax)Jν(bx)Jν(cx)dx
=/radicalbigg
2
π3a−μ(bc)μ−1(sinhu)μ−1
2sin[(μ−ν)π]e(μ−1
2)πiQ1
2−μ
ν−1
2(coshu)[a>b +c]
=1√
2πa−μ(bc)μ−1(sinv)μ−1
2P1
2−μ
ν−1
2(cosv)[ |b−c|<a<b +c]
=0 [ 0 <a< |b−c|]
/bracketleftbig
2bccoshu=a2−b2−c2,2bccosv=b2+c2−a2,b > 0,c > 0; Re ν>−1,Reμ>−1
2/bracketrightbig
9./integraldisplay∞
0Jν(ax)Jν(bx)Jν(cx)x1−νdx=0 [ 0 <c≤|a−b|orc≥a+b]
=2ν−1Δ2ν−1
(abc)νΓ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig[|a−b|<c<a +b]
Δ=1
4/radicalbig
[c2−(a−b)2][(a+b)2−c2],/bracketleftbig
a>0,b > 0,c > 0; Re ν>−1
2/bracketrightbig
(Δ>0 is equal to the area of a triangle whose sides are a,b,a n d c.)
10.11/integraldisplay∞
0xν+1Kμ(ax)Kμ(bx)Jν(cx)dx=√πcνΓ(ν+μ+1 )Γ ( ν−μ+1 )
23/2(ab)ν+1(u2−1)1
2ν+1
4P−ν−1
2
μ−1
2(u)
/bracketleftbig
2abu=a2+b2+c2,Re(a+b)>|Imc|,Re (ν±μ)>−1,Reν>−1/bracketrightbig
ET II 67(30)
11.11/integraldisplay∞
0xν+1Kμ(ax)Iμ(bx)Jν(cx)dx=(ab)−ν−1cνe−(ν+1
2)πiQν+1
2
μ−1
2(u)
√
2π(u2−1)1
2ν+1
42abu=a2+b2+c2
[Rea>|Reb|+|Imc|;R e ν>−1,Re(μ+ν)>−1]ET II 66(24)
12.8/integraldisplay∞
0xν+1[Jν(ax)]2Yν(bx)dx=0/bracketleftbig
0<b< 2a,|Reν|<1
2/bracketrightbig
=23ν+1a2νb−ν−1
√πΓ/parenleftbig1
2−ν/parenrightbig/parenleftbig
b2−4a2/parenrightbig−ν−1
2/bracketleftbig
0<2a<b , |Reν|<1
2/bracketrightbig
ET II 109(3)
13./integraldisplay∞
0xν+1Jν(ax)Yν(ax)Jν(bx)dx
=0/bracketleftbig
a>0,|Reν|<1
2,0<b< 2a/bracketrightbig
=−23ν+1a2νb−ν−1
√πΓ/parenleftbig1
2−ν/parenrightbig/parenleftbig
b2−4a2/parenrightbig−ν−1
2/bracketleftbig
a>0,2a<b< ∞,|Reν|<1
2/bracketrightbig
ET II 55(49)
14./integraldisplay∞
0xν+1Jμ(xasinψ)Jν(xasinϕ)Kμ(xacosϕcosψ)dx
=2νΓ(μ+ν+1 )( s i n ϕ)ν/parenleftbig
cosα
2/parenrightbig2ν+1
aν+2(cosψ)2ν+2P−μ
ν(cosα)
/bracketleftBig
tan1
2α=t a n ψcosϕ, a > 0,π
2>ϕ> 0,0<ψ<π
2,Reν>−1,Re(μ+ν)>−1/bracketrightBig
ET II 64(11)
6.581 Bessel functions and powers 687
15./integraldisplay∞
0xν+1Jν(ax)Kν(bx)Jν(cx)dx=23ν(abc)νΓ/parenleftbig
ν+1
2/parenrightbig
√π/bracketleftBig
(a2+b2+c2)2−4a2c2/bracketrightBigν+1
2
/bracketleftbig
Reb>|Ima|,c > 0,Reν>−1
2/bracketrightbig
ET II 63(8)
16.8/integraldisplay∞
0xν+1Iν(ax)Kν(bx)Jν(cx)dx=23ν(abc)νΓ/parenleftbig
ν+1
2/parenrightbig
√π/bracketleftBig
(b2−a2+c2)2+4a2c2/bracketrightBigν+1
2
/bracketleftbig
Reb>|Rea|+|Imc|;R e ν>−1
2/bracketrightbig
ET II 65(18)
6.579
1./integraldisplay∞
0x2ν+1Jν(ax)Yν(ax)Jν(bx)Yν(bx)dx
=a2νΓ(3ν+1 )
2πb4ν+2Γ/parenleftbig1
2−ν/parenrightbig
Γ/parenleftbig
2ν+3
2/parenrightbigF/parenleftbigg
ν+1
2,3ν+1 ;2ν+3
2;a2
b2/parenrightbigg
/bracketleftbig
0<a<b , −1
3<Reν<1
2/bracketrightbig
EH II 94(45), ET II 352(15)
2./integraldisplay∞
0x2ν+1Jν(ax)Kν(ax)Jν(bx)Kν(bx)dx
=2ν−3a2νΓ/parenleftbigν+1
2/parenrightbig
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig3ν+1
2/parenrightbig
√πb4ν+2Γ(ν+1 )F/parenleftbigg
ν+1
2,3ν+1
2;2ν+1 ;1 −a4
b4/parenrightbigg
/bracketleftbig
0<a<b , Reν>−1
3/bracketrightbig
ET II 373(10)
3./integraldisplay∞
0x1−2ν[Jν(ax)]4dx=Γ(ν)Γ(2ν)
2π/bracketleftbig
Γ/parenleftbig
ν+1
2/parenrightbig/bracketrightbig2Γ(3ν)[Reν>0] ET II 342(25)
4./integraldisplay∞
0x1−2ν[Jν(ax)]2[Jν(bx)]2dx=a2ν−1Γ(ν)
2πbΓ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig
2ν+1
2/parenrightbigF/parenleftbigg
ν,1
2−ν;2ν+1
2;a2
b2/parenrightbigg
ET II 351(10)
6.581
1./integraldisplaya
0xλ−1Jμ(x)Jν(a−x)dx=2λ∞/summationdisplay
m=0(−1)mΓ(λ+μ+m)Γ(λ+m)
m!Γ (λ)Γ(μ+m+1 )Jλ+μ+ν+2m(a)
[Re(λ+μ)>0,Reν>−1]
ET II 354(25)
2.8/integraldisplaya
0xλ−1(a−x)−1Jμ(x)Jν(a−x)dx
=2λ
aν∞/summationdisplay
m=0(−1)mΓ(λ+μ+m)Γ(λ+m)
m!Γ (λ)Γ(μ+m+1 )(λ+μ+ν+2m)Jλ+μ+ν+2m(a)
[Re(λ+μ)>0,Reν>0]ET II 354(27)
3./integraldisplaya
0xμ(a−x)νJμ(x)Jν(a−x)dx=Γ/parenleftbig
μ+1
2/parenrightbig
Γ/parenleftbig
ν+1
2/parenrightbig
√
2πΓ(μ+ν+1 )aμ+ν+1
2Jμ+ν+1
2(a)
/bracketleftbig
Reμ>−1
2,Reν>−1
2/bracketrightbig
ET II 354(28), EH II 46(6)
688 Bessel Functions 6.582
4./integraldisplaya
0xμ(a−x)ν+1Jμ(x)Jν(a−x)dx=Γ/parenleftbig
μ+1
2/parenrightbig
Γ/parenleftbig
ν+3
2/parenrightbig
√
2πΓ(μ+ν+2 )aμ+ν+3
2Jμ+ν+1
2(a)
/bracketleftbig
Reν>−1,Reμ>−1
2/bracketrightbig
ET II 354(29)
5./integraldisplaya
0xμ(a−x)−μ−1Jμ(x)Jν(a−x)dx=2μΓ/parenleftbig
μ+1
2/parenrightbig
Γ(ν−μ)√πΓ(μ+ν+1 )aμJν(a)
/bracketleftbig
Reν>Reμ>−1
2/bracketrightbig
ET II 355(30)
6.582/integraldisplay∞
0xμ−1|x−b|−μKμ(|x−b|)Kν(x)dx=1√π(2b)−μΓ/parenleftbig1
2−μ/parenrightbig
Γ(μ+ν)Γ(μ−ν)Kν(b)
/bracketleftbig
b>0,Reμ<1
2,Reμ>|Reν|/bracketrightbig
ET II 374(14)
6.583/integraldisplay∞
0xμ−1(x+b)−μKμ(x+b)Kν(x)dx=√πΓ(μ+ν)Γ(μ−ν)
2μbμΓ/parenleftbig
μ+1
2/parenrightbigKν(b)
[|argb|<π , Reμ>|Reν|]
ET II 374(15)
6.584
1.8/integraldisplay∞
0x/rho1−1/bracketleftBig
H(1)
ν(ax)−e/rho1πiH(1)
ν/parenleftbig
axeπi/parenrightbig/bracketrightBig
(x2−r2)m+1dx=πi
2mm!/parenleftbiggd
rd r/parenrightbiggm/bracketleftBig
r/rho1−2H(1)
ν(ar)/bracketrightBig
/bracketleftbig
m=0,1,2,..., Imr>0,a > 0,|Reν|<Re/rho1<2m+7
2/bracketrightbig
WA 465
2.8/integraldisplay∞
0/bracketleftbigg
cos1
2(/rho1−ν)πJν(ax)+s i n1
2(/rho1−ν)πYν(ax)/bracketrightbiggx/rho1−1
(x2+k2)m+1dx
=(−1)m+1
2m·m!/parenleftbiggd
kd k/parenrightbiggm/bracketleftbig
k/rho1−2Kν(ak)/bracketrightbig
/bracketleftbig
m=0,1,2,..., Rek>0,a > 0,|Reν|<Re/rho1<2m+7
2/bracketrightbig
WA 466(2)
3./integraldisplay∞
0{cosνπJν(ax)−sinνπYν(ax)}x1−νdx
(x2+k2)m+1=amKν+m(ak)
2m·m!kν+m
/bracketleftbig
m=0,1,2,..., Rek>0,a > 0,−2m−3
2<Reν<1/bracketrightbig
WA 466(3)
4./integraldisplay∞
0/braceleftbig
cos/bracketleftbig/parenleftbig1
2/rho1−1
2ν−μ/parenrightbig
π/bracketrightbig
Jν(ax)+s i n/bracketleftbig/parenleftbig1
2/rho1−1
2ν−μ/parenrightbig
π/bracketrightbig
Yν(ax)/bracerightbig x/rho1−1
(x2+k2)μ+1dx
=πk/rho1−2μ−2
2s inνπ·Γ(μ+1 )⎡
⎣/parenleftbig1
2ak/parenrightbigνΓ/parenleftbig1
2/rho1+1
2ν/parenrightbig
Γ(ν+1 )Γ/parenleftbig1
2/rho1+1
2ν−μ/parenrightbig1F2/parenleftbigg/rho1+ν
2;/rho1+ν
2−μ, ν+1 ;a2k2
4/parenrightbigg
−/parenleftbig1
2ak/parenrightbig−νΓ/parenleftbig1
2/rho1−1
2ν/parenrightbig
Γ(1−ν)Γ/parenleftbig1
2/rho1−1
2ν−μ/parenrightbig1F2/parenleftbigg/rho1−ν
2;/rho1−ν
2−μ,1−ν;a2k2
4/parenrightbigg⎤
⎦
/bracketleftbig
a>0,Rek>0,|Reν|<Re/rho1<2R eμ+7
2/bracketrightbig
WA 407(1)
6.591 Powers and Bessel functions of complicated arguments 689
5.8/integraldisplay∞
0⎡
⎣n/productdisplay
j=1Jμj(bnx)⎤
⎦⎧
⎨
⎩cos⎡
⎣1
2⎛
⎝/rho1+/summationdisplay
jμj−ν⎞
⎠π⎤
⎦Jν(ax)
+s i n⎡
⎣1
2⎛
⎝/rho1+/summationdisplay
jμj−ν⎞
⎠π⎤
⎦Yν(ax)⎫
⎬
⎭x/rho1−1
x2+k2dx
=−⎡
⎣n/productdisplay
j=1Iμj(bnk)⎤
⎦Kν(ak)k/rho1−2
⎡
⎣Rek>0,a >/summationdisplay
j|Rebj|,Re⎛
⎝/rho1+/summationdisplay
jμj⎞
⎠>|Reν|⎤
⎦WA 472(9)
6.59 Combinations of powers and Bessel functions of more complicated arguments
6.591
1./integraldisplay∞
0x2ν+1
2Jν+1
2/parenleftBiga
x/parenrightBig
Kν(bx)dx=√
2πb−ν−1aν+1
2J1+2ν/parenleftBig√
2ab/parenrightBig
K1+2ν/parenleftBig√
2ab/parenrightBig
[a>0,Reb>0,Reν>−1]
ET II 142(35)
2./integraldisplay∞
0x2ν+1
2Yν+1
2/parenleftBiga
x/parenrightBig
Kν(bx)dx=√
2πb−ν−1aν+1
2Y2ν+1/parenleftBig√
2ab/parenrightBig
K2ν+1/parenleftBig√
2ab/parenrightBig
[a>0,Reb>0,Reν>−1]
ET II 143(41)
3./integraldisplay∞
0x2ν+1
2Kν+1
2/parenleftBiga
x/parenrightBig
Kν(bx)dx=√
2πb−ν−1aν+1
2K2ν+1/parenleftBig
e1
4iπ√
2ab/parenrightBig
K2ν+1/parenleftBig
e−1
4iπ√
2ab/parenrightBig
[Rea>0,Reb>0] ET II 146(56)
4./integraldisplay∞
0x−2ν+1
2Jν−1
2/parenleftBiga
x/parenrightBig
Kν(bx)dx=√
2πbν−1a1
2−νK2ν−1/parenleftBig√
2ab/parenrightBig
×/bracketleftBig
sin(νπ)J2ν−1/parenleftBig√
2ab/parenrightBig
+c o s ( νπ)Y2ν−1/parenleftBig√
2ab/parenrightBig/bracketrightBig
[a>0,Reb>0,Reν<1]ET II 142(34)
5./integraldisplay∞
0x−2ν+1
2Yν−1
2/parenleftBiga
x/parenrightBig
Kν(bx)dx=−/radicalbiggπ
2bν−1a1
2−νsec(νπ)K2ν−1/parenleftBig√
2ab/parenrightBig
×/bracketleftBig
J2ν−1/parenleftBig√
2ab/parenrightBig
−J1−2ν/parenleftBig√
2ab/parenrightBig/bracketrightBig
[a>0,Reν<1] ET II 143(40)
6./integraldisplay∞
0x−2ν+1
2J1
2−ν/parenleftBiga
x/parenrightBig
Jν(bx)dx
=−1
2icosec(2 νπ)bν−1a1
2−ν/bracketleftbig
e2νπiJ1−2ν(u)J2ν−1(v)−e−2νπiJ2ν−1(u)J1−2ν(v)/bracketrightbig
/bracketleftBig
u=/parenleftbig1
2ab/parenrightbig1
2e1
4πi,v=/parenleftbig1
2ab/parenrightbig1
2e−1
4πi,a > 0,b > 0,−1
2<Reν<3/bracketrightBig
ET II 58(12)
690 Bessel Functions 6.592
7./integraldisplay∞
0x−2ν+1
2Kν−1
2/parenleftBiga
x/parenrightBig
Yν(bx)dx=√
2πbν−1a1
2−νY2ν−1/parenleftBig√
2ab/parenrightBig
K2ν−1/parenleftBig√
2ab/parenrightBig
/bracketleftbig
b>0,Rea>0,Reν>1
6/bracketrightbig
ET II 113(30)
8./integraldisplay∞
0x/rho1−1Jμ(ax)Jν/parenleftbiggb
x/parenrightbigg
dx=aν−/rho1bνΓ/parenleftbig1
2μ+1
2/rho1−1
2ν/parenrightbig
22ν−/rho1+1Γ(ν+1 )Γ/parenleftbig1
2μ+1
2ν−1
2/rho1+1/parenrightbig
×0F3/parenleftbigg
ν+1,ν−μ−/rho1
2+1,ν+μ−/rho1
2+1 ;a2b2
16/parenrightbigg
+aμbμ+/rho1Γ/parenleftbig1
2ν−1
2μ−1
2/rho1/parenrightbig
22μ+/rho1+1Γ(μ+1 )Γ/parenleftbig1
2μ+1
2ν+1
2/rho1+1/parenrightbig
×0F3/parenleftbigg
μ+1,μ−ν+/rho1
2+1,ν+μ+/rho1
2+1 ;a2b2
16/parenrightbigg
/bracketleftbig
a>0,b > 0,−Re/parenleftbig
μ+3
2/parenrightbig
<Re/rho1<Re/parenleftbig
ν+3
2/parenrightbig/bracketrightbig
WA 480(1)
6.592
1./integraldisplay∞
0xλ(1−x)μ−1Yν/parenleftbig
a√x/parenrightbig
dx=2−νaνcot(νπ)Γ(μ)Γ/parenleftbig
λ+1+1
2ν/parenrightbig
Γ(1 + ν)Γ/parenleftbig
λ+1+ μ+1
2ν/parenrightbig
×1F2/parenleftbigg
λ+1+1
2ν;1+ν,λ+1+ μ+1
2ν;−a2
4/parenrightbigg
−2νa−νcosec( νπ)Γ(μ)Γ/parenleftbig
λ+1−1
2ν/parenrightbig
Γ(1−ν)Γ/parenleftbig
λ+1+ μ−1
2ν/parenrightbig
×1F2/parenleftbigg
λ−1
2ν+1 ;1 −ν,λ+1+ μ−1
2ν;−a2
4/parenrightbigg
/bracketleftbig
Reλ>−1+1
2|Reν|,Reμ>0/bracketrightbig
ET II 197(76)a
2.10/integraldisplay1
0xλ(1−x)μ−1Kν/parenleftbig
a√x/parenrightbig
dx
=2−ν−1a−νΓ(ν)Γ(μ)Γ/parenleftbig
λ+1−1
2ν/parenrightbig
Γ/parenleftbig
λ+1+ μ−1
2ν/parenrightbig1F2/parenleftbigg
λ+1−1
2ν;1−ν,λ+1+ μ−1
2ν;a2
4/parenrightbigg
+2−1−νaνΓ(−ν)Γ/parenleftbig
λ+1+1
2ν/parenrightbig
Γ(μ)
Γ/parenleftbig
λ+1+ μ+1
2ν/parenrightbig1F2/parenleftbigg
λ+1+1
2ν;1+ν,λ+1+ μ+1
2ν;a2
4/parenrightbigg
=2ν−1
aνΓ(μ)G21
13/parenleftbigga2
4/vextendsingle/vextendsingle/vextendsingle/vextendsingleν
2−λ
ν,0,ν
2−λ−μ/parenrightbigg
OB 159 (3.16)
/bracketleftbig
Reλ>−1+1
2|Reν|,Reμ>0/bracketrightbig
ET II 198(87)a
6.592 Powers and Bessel functions of complicated arguments 691
3.11/integraldisplay∞
1xλ(x−1)μ−1Jν/parenleftbig
a√x/parenrightbig
dx=22λa−2λG20
13/parenleftbigga2
4/vextendsingle/vextendsingle/vextendsingle/vextendsingle0
−μ, λ+1
2ν,λ−1
2ν/parenrightbigg
Γ(μ)
/bracketleftbig
a>0,0<Reμ<3
4−Reλ/bracketrightbig
ET II 205(36)a
4./integraldisplay∞
1xλ(x−1)μ−1Kν/parenleftbig
a√x/parenrightbig
dx=Γ (μ)22λ−1a−2λG30
13/parenleftbigga2
4/vextendsingle/vextendsingle/vextendsingle/vextendsingle0
−μ,
1
2ν+λ,−1
2ν+λ/parenrightbigg
[Rea>0,Reμ>0] ET II 209(60)a
5./integraldisplay1
0x−1
2(1−x)−1
2Jν/parenleftbig
a√x/parenrightbig
dx=π/bracketleftbigg
J1
2ν/parenleftbigg1
2a/parenrightbigg/bracketrightbigg2
[Reν>−1] ET II 194(59)a
6./integraldisplay1
0x−1
2(1−x)−1
2Iν/parenleftbig
a√x/parenrightbig
dx=π/bracketleftbigg
I1
2ν/parenleftbigg1
2a/parenrightbigg/bracketrightbigg2
[Reν>−1] ET II 197(79)
7./integraldisplay1
0x−1
2(1−x)−1
2Kν/parenleftbig
a√x/parenrightbig
dx=1
2πsec/parenleftbigg1
2νπ/parenrightbigg/bracketleftBig
Iν
2/parenleftBiga
2/parenrightBig
+I−ν
2/parenleftBiga
2/parenrightBig/bracketrightBig
Kν
2/parenleftBiga
2/parenrightBig
[|Reν|<1] ET II 198(85)a
8./integraldisplay∞
1x−1
2(x−1)−1
2Kν/parenleftbig
a√x/parenrightbig
dx=/bracketleftBig
Kν
2/parenleftBiga
2/parenrightBig/bracketrightBig2
[Rea>0] ET II 208(56)a
9./integraldisplay1
0x−1
2(1−x)−1
2Yν/parenleftbig
a√x/parenrightbig
dx=π/braceleftbigg
cot(νπ)/bracketleftBig
Jν
2/parenleftBiga
2/parenrightBig/bracketrightBig2
−cosec( νπ)/bracketleftBig
J−ν
2/parenleftBiga
2/parenrightBig/bracketrightBig2/bracerightbigg
[|Reν|<1] ET II 195(68)a
10./integraldisplay∞
1x−1
2ν(x−1)μ−1Jν/parenleftbig
a√x/parenrightbig
dx=Γ (μ)2μa−μJν−μ(a)
/bracketleftbig
a>0,0<Reμ<1
2Reν+3
4/bracketrightbig
ET II 205(34)a
11./integraldisplay∞
1x−1
2ν(x−1)μ−1J−ν/parenleftbig
a√x/parenrightbig
dx=Γ (μ)2μa−μ[cos(νπ)Jν−μ(a)−sin(νπ)Yν−μ(a)]
/bracketleftbig
a>0,0<Reμ<1
2Reν+3
4/bracketrightbig
ET II 205(35)a
12./integraldisplay∞
1x−1
2ν(x−1)μ−1Kν/parenleftbig
a√x/parenrightbig
dx=Γ (μ)2μa−μKν−μ(a)
[Rea>0,Reμ>0] ET II 209(59)a
13./integraldisplay∞
1x−1
2ν(x−1)μ−1Yν/parenleftbig
a√x/parenrightbig
dx=2μa−μYν−μ(a)Γ(μ)
/bracketleftbig
a>0,0<Reμ<1
2Reν+3
4/bracketrightbig
ET II 206(40)a
14./integraldisplay∞
1x−1
2ν(x−1)μ−1H(1)
ν/parenleftbig
a√x/parenrightbig
dx=2μa−μH(1)
ν−μ(a)Γ(μ)
[Reμ>0,Ima>0] ET II 206(45)a
15./integraldisplay∞
1x−1
2ν(x−1)μ−1H(2)
ν/parenleftbig
a√x/parenrightbig
dx=2μa−μH(2)
ν−μ(a)Γ(μ)
[Reμ>0,Ima<0] ET II 207(48)a
692 Bessel Functions 6.593
16./integraldisplay1
0x−1
2ν(1−x)μ−1Jν/parenleftbig
a√x/parenrightbig
dx=22−νa−μ
Γ(ν)sμ+ν−1,μ−ν(a)
[Reμ>0] ET II 194(64)a
17./integraldisplay1
0x−1
2ν(1−x)μ−1Yν/parenleftbig
a√x/parenrightbig
dx=22−νa−μcot(νπ)
Γ(ν)sμ+ν−1,μ−ν(a)
−2μa−μcosec( νπ)Jμ−ν(a)Γ(μ)
[Reμ>0,Reν<1] ET II 196(75)a
6.593
1./integraldisplay∞
0√xJ2ν−1/parenleftbig
a√x/parenrightbig
Jν(bx)dx=1
2ab−2Jν−1/parenleftbigga2
4b/parenrightbigg/bracketleftbig
b>0,Reν>−1
2/bracketrightbig
ET II 58(15)
2./integraldisplay∞
0√xJ2ν−1/parenleftbig
a√x/parenrightbig
Kν(bx)dx=πa
4b2/bracketleftbigg
Iν−1/parenleftbigga2
4b/parenrightbigg
−Lν−1/parenleftbigga2
4b/parenrightbigg/bracketrightbigg
/bracketleftbig
Reb>0,Reν>−1
2/bracketrightbig
ET II 144(44)
6.594
1./integraldisplay∞
0xνI2ν−1/parenleftbig
a√x/parenrightbig
J2ν−1/parenleftbig
a√x/parenrightbig
Kν(bx)dx=√π2−νa2ν−1b−2ν−1
2Jν−1
2/parenleftbigga2
2b/parenrightbigg
[Reb>0,Reν>0] ET II 148(65)
2./integraldisplay∞
0xνI2ν−1/parenleftbig
a√x/parenrightbig
Y2ν−1/parenleftbig
a√x/parenrightbig
Kν(bx)dx
=√π2−ν−1a2ν−1b−2ν−1
2cosec( νπ)
×/bracketleftbigg
H1
2−ν/parenleftbigga2
2b/parenrightbigg
+c o s ( νπ)Jν−1
2/parenleftbigga2
2b/parenrightbigg
+s i n ( νπ)Yν−1
2/parenleftbigga2
2b/parenrightbigg/bracketrightbigg
[Reb>0,Reν>0] ET II 148(66)
3./integraldisplay∞
0xνJ2ν−1/parenleftbig
a√x/parenrightbig
K2ν−1/parenleftbig
a√x/parenrightbig
Kν(bx)dx
=π22−ν−2a2ν−1b−2ν−1
2cosec( νπ)/bracketleftbigg
H1
2−ν/parenleftbigga2
2b/parenrightbigg
−Y1
2−ν/parenleftbigga2
2b/parenrightbigg/bracketrightbigg
[Reb>0,Reν>0] ET II 148(67)
6.595
1./integraldisplay∞
0xν+1Jν(cx)n/productdisplay
i=1z−μi
iJμi(aizi)dx=0 zi=/radicalBig
x2+b2
i
/bracketleftBigg
ai>0,Rebi>0,n/summationdisplay
i=1ai<c;R e/parenleftBigg
1
2n+n/summationdisplay
i=1μi−1
2/parenrightBigg
>Reν>−1/bracketrightBigg
EH II 52(33), ET II 60(26)
2./integraldisplay∞
0xν−1Jν(cx)n/productdisplay
i=1z−μi
iJμi(aizi)dx=2ν−1Γ(ν)c−νn/productdisplay
i=1/bracketleftbig
b−μi
iJμi(aibi)/bracketrightbig
zi=/radicalBig
x2+b2
i
/bracketleftBigg
ai>0,Rebi>0,n/summationdisplay
i=1ai<c , Re/parenleftBigg
1
2n+n/summationdisplay
i=1μi+3
2/parenrightBigg
>Reν>0/bracketrightBigg
EH II 52(34), ET II 60(27)
6.596 Powers and Bessel functions of complicated arguments 693
6.596
1./integraldisplay∞
0Jν/parenleftBig
α/radicalbig
x2+z2/parenrightBigx2μ+1
/radicalbig
(x2+z2)νdx=2μΓ(μ+1 )
αμ+1zν−μ−1Jν−μ−1(αz)
/bracketleftbigg
α>0,Re/parenleftbigg1
2ν−1
4/parenrightbigg
>Reμ>−1/bracketrightbigg
WA 457(5)
2./integraldisplay∞
0Jν/parenleftbig
α√
t2+1/parenrightbig
√
t2+1dt=−π
2Jν
2/parenleftBigα
2/parenrightBig
Yν
2/parenleftBigα
2/parenrightBig
[Reν>−1,α > 0] MO 46
3./integraldisplay∞
0Kν/parenleftBig
α/radicalbig
x2+z2/parenrightBigx2μ+1
/radicalbig
(x2+z2)νdx=2μΓ(μ+1 )
αμ+1zν−μ−1Kν−μ−1(αz)
[α>0,Reμ>−1] WA 457(6)
4.8/integraldisplay∞
0Jν(βx)Jμ−1/braceleftbig
α√
x2+z2/bracerightbig
(x2+z2)1
2μ+1
2xν+1dx=αμ−1zν
2μ−1Γ(μ)Kν(βz)
[α<β , Re(μ+2 )>Reν>−1]
ET II 59(19)
5.8/integraldisplay∞
0Jν(βx)Jμ/braceleftbig
α√
x2+z2/bracerightbig
/radicalbig
(x2+z2)μxν−1dx=2ν−1Γ(ν)
βνJμ(αz)
zμ
[Re(μ+2 )>Reν>0,β > α > 0]
WA 459(12)
6.6/integraldisplay∞
0Jν(βx)Jμ/parenleftbig
α√
x2+z2/parenrightbig
/radicalbig
(x2+z2)μxν+1dx
=0 [ 0 <α<β ]
=βν
αμ/parenleftBigg/radicalbig
α2−β2
z/parenrightBiggμ−ν−1
Jμ−ν−1/braceleftBig
z/radicalbig
α2−β2/bracerightBig
[α>β> 0]
[Reμ>Reν>−1] WA 415(1)
7.8/integraldisplay∞
0Jν(βx)Kμ/parenleftbig
α√
x2+z2/parenrightbig
/radicalbig
(x2+z2)μxν+1dx=βν
αμ/parenleftBigg/radicalbig
α2+β2
z/parenrightBiggμ−ν−1
Kμ−ν−1/parenleftBig
z/radicalbig
α2+β2/parenrightBig
/bracketleftBig
α>0,β > 0,Reν>−1,|argz|<π
2/bracketrightBig
KU 151(31), WA 416(2)
8.8/integraldisplay∞
0Jν(ux)Kμ/parenleftBig
v/radicalbig
x2−y2/parenrightBig/parenleftbig
x2−y2/parenrightbig−μ
2xν+1dx=π
2exp/bracketleftbigg
−iπ/parenleftbigg
μ−ν−1
2/parenrightbigg/bracketrightbigg
·uν
vμ
·/bracketleftBigg√
u2+v2
y/bracketrightBiggμ−ν−1
H(2)
μ−ν−1/parenleftBig
y/radicalbig
u2+v2/parenrightBig
/bracketleftBig
Reμ<1,Reν>−1,u>0,v > 0,y>0;/parenleftbig
x2−y2/parenrightbig1
2α=e1
2απi/parenleftbig
y2−n2/parenrightbig1
2αifx<y/bracketrightBig
694 Bessel Functions 6.597
9.8/integraldisplay∞
0Jν(ux)H(2)
μ/parenleftBig
v/radicalbig
x2+y2/parenrightBig/parenleftbig
x2+y2/parenrightbig−μ
2xν+1dx
=uν
vμ/bracketleftBigg√
v2−u2
y/bracketrightBiggμ−ν−1
H(2)
μ−ν−1/parenleftBig
y/radicalbig
v2−u2/parenrightBig
[u<v]⎡
⎣Reμ>Reν>−1,u > 0,v > 0,y > 0;,arg/radicalbig
v2−u2=0,forv>u
arg/parenleftbig
v2−u2/parenrightbigσ=−πσforv<u,w h e r e σ=1
2orσ=μ−ν−1
2⎤
⎦
MO 43
10.8/integraldisplay∞
0Jν(βx)Jμ/parenleftBig
α/radicalbig
x2+z2/parenrightBig
Jμ/parenleftBig
γ/radicalbig
x2+z2/parenrightBigxν−1
(x2+z2)μdx=2ν−1Γ(ν)
βνJμ(αz)
zμJμ(γz)
zμ/bracketleftbig
α>0;β>α +γ;γ>0,Re/parenleftbig
2μ+5
2/parenrightbig
>Reν>0/bracketrightbig
WA 459(14)
11.8/integraldisplay∞
0Jν(βt)tν−1n/productdisplay
k=1Jμ/parenleftBig
αk/radicalbig
t2+x2/parenrightBig/radicalBig
(t2+x2)−nμdt=2ν−1β−νΓ(ν)n/productdisplay
k=1/bracketleftbig
x−μJμ(αkx)/bracketrightbig
/bracketleftBigg
x>0,α 1>0,α 2>0,...,α n>0,β >n/productdisplay
k=1αk;R e/parenleftbigg
nμ+1
2n+1
2/parenrightbigg
>Reν>0/bracketrightBigg
MO 43
12.8/integraldisplay∞
0J2
μ/parenleftbig√
a2+x2/parenrightbig
(a2+x2)νx2ν−2dx=Γ/parenleftbig
ν−1
2/parenrightbig
2aν+1√πHν(2a)/bracketleftbig
Reν>1
2/bracketrightbig
WA 457(8)
6.597/integraldisplay∞
0tν+1Jμ/bracketleftBig
b/parenleftbig
t2+y2/parenrightbig1
2/bracketrightBig/parenleftbig
t2+y2/parenrightbig−1
2μ/parenleftbig
t2+β2/parenrightbig−1Jν(at)dt
=βνJμ/bracketleftBig
b/parenleftbig
y2−β2/parenrightbig1
2/bracketrightBig/parenleftbig
y2−β2/parenrightbig−1
2μKν(aβ)
[a≥b,Reβ>0,−1<Reν<2+R e μ]EH II 95(56)
6.598/integraldisplay1
0xμ
2(1−x)ν
2Jμ/parenleftbig
a√x/parenrightbig
Jν/parenleftbig
b√
1−x/parenrightbig
dx=2aμbν/parenleftbig
a2+b2/parenrightbig−1
2(ν+μ+1)Jν+μ+1/parenleftBig/radicalbig
a2+b2/parenrightBig
[Reν>−1,Reμ>−1] EH II 46a
6.61 Combinations of Bessel functions and exponentials
6.611
1./integraldisplay∞
0e−αxJν(βx)dx=β−ν/bracketleftBig/radicalbig
α2+β2−α/bracketrightBigν
/radicalbig
α2+β2[Reν>−1,Re (α±iβ)>0]
EH II 49(18), WA 422(8)
6.611 Bessel functions and exponentials 695
2./integraldisplay∞
0e−αxYν(βx)dx=/parenleftbig
α2+β2/parenrightbig−1
2cosec( νπ)
×/braceleftbigg
βν/bracketleftBig/parenleftbig
α2+β2/parenrightbig1
2+α/bracketrightBig−ν
cos(νπ)−β−ν/bracketleftBig/parenleftbig
α2+β2/parenrightbig1
2+α/bracketrightBigν/bracerightbigg
[Reα>0,β > 0,|Reν|<1]MO 179, ET II 105(1)
3./integraldisplay∞
0e−αxKν(βx)dx=π
βsin(νπ)sin(νθ)
sinθ/bracketleftbigg
cosθ=α
β;θ→π
2forβ→∞/bracketrightbigg
ET II 131(22)
=πcosec( νπ)
2/radicalbig
α2−β2/bracketleftbigg
β−ν/parenleftBig
α+/radicalbig
α2−β2/parenrightBigν
−βν/parenleftBig/radicalbig
α2−β2+α/parenrightBig−ν/bracketrightbigg
[|Reν|<1,Re(α+β)>0]
ET I 197(24), MO 180
4.8/integraldisplay∞
0e−αxIν(βx)dx=β−ν/bracketleftBig
α−/radicalbig
α2−β2/bracketrightBigν
/radicalbig
α2−β2[Reν>−1,Reα>|Reβ|]
MO 180, ET I 195(1)
5./integraldisplay∞
0e−αxH(1,2)
ν(βx)dx=/parenleftBig/radicalbig
α2+β2−α/parenrightBigν
βν/radicalbig
α2+β2⎧
⎪⎨
⎪⎩1±i
sin(νπ)⎡
⎢⎣cos(νπ)−/parenleftBig
α+/radicalbig
α2+β2/parenrightBig2ν
b2ν⎤
⎥⎦⎫
⎪⎬
⎪⎭
[−1<Reν<1; a plus sign corresponds to the function H(1)
ν, a minus sign to the function H(2)
ν.]
MO 180, ET I188(54, 55)
6./integraldisplay∞
0e−αxH(1)
0(βx)dx=1/radicalbig
α2+β2⎧
⎨
⎩1−2i
πln⎡
⎣α
β+/radicalBigg
1+/parenleftbiggα
β/parenrightbigg2⎤
⎦⎫
⎬
⎭
[Reα>|Imβ|] MO 180, ET I 188(53)
7./integraldisplay∞
0e−αxH(2)
0(βx)dx=1/radicalbig
α2+β2⎧
⎨
⎩1+2i
πln⎡
⎣α
β+/radicalBigg
1+/parenleftbiggα
β/parenrightbigg2⎤
⎦⎫
⎬
⎭
[Reα>|Imβ|] MO 180, ET I 188(53)
8./integraldisplay∞
0e−αxY0(βx)dx=−2
π/radicalbig
α2+β2lnα+/radicalbig
α2+β2
β
[Reα>|Imβ|] MO 47, ET I 187(44)
9.11/integraldisplay∞
0e−αxK0(βx)dx=arccosα
β/radicalbig
β2−α2[Re(α+β)>0]WA 424, ET II 131(22)
=1/radicalbig
α2−β2ln/parenleftBigg
α
β+/radicalBigg
α2
β2−1/parenrightBigg
[Re(α+β)>0]
MO 48
696 Bessel Functions 6.612
10.10/integraldisplayb
aαd α/integraldisplay∞
0dkJ1(kα)e−k|β|=/integraldisplayb
a/parenleftBigg
1−|β|/radicalbig
α2+β2/parenrightBigg
dα
(see3.241 6)
6.612
1./integraldisplay∞
0e−2αxJ0(x)Y0(x)dx=K/bracketleftBig
α/parenleftbig
α2+1/parenrightbig−1
2/bracketrightBig
π(α2+1 )1
2[Reα>0] ET II 347(58)
2./integraldisplay∞
0e−2αxI0(x)K0(x)dx=1
2K/bracketleftBig/parenleftbig
1−α2/parenrightbig1
2/bracketrightBig
[0<α< 1]
=1
2αK/bracketleftBigg/parenleftbigg
1−1
α2/parenrightbigg1
2/bracketrightBigg
[1<α< ∞]
ET II 370(48)
3./integraldisplay∞
0e−αxJν(βx)Jν(γx)dx=1
π√γβQν−1
2/parenleftbiggα2+β2+γ2
2βγ/parenrightbigg
/bracketleftbig
Re (α±iβ±iγ)>0,γ > 0,Reν>−1
2/bracketrightbig
WA 426(2), ET II 50(17)
4./integraldisplay∞
0e−αx[J0(βx)]2dx=2
π/radicalbig
α2+4β2K/parenleftBigg
2β/radicalbig
α2+4β2/parenrightBigg
MO 178
5./integraldisplay∞
0e−2αxJ2
1(βx)dx=/parenleftbig
2α2+β2/parenrightbig
K/parenleftbigg
β√
α2+β2/parenrightbigg
−2/parenleftbig
α2+β2/parenrightbig
E/parenleftbigg
β√
α2+β2/parenrightbigg
πβ2/radicalbig
α2+β2WA 428(3)
6./integraldisplay∞
0e−3xIl(x)Im(x)In(x)dx=r1g+r2
π2g+r3
where
g=√
3−1
96π3Γ2/parenleftbigg1
24/parenrightbigg
Γ2/parenleftbigg11
24/parenrightbigg
and
6.614 Bessel functions and exponentials 697
(lmn) r1 r2 r3
000 10 0
100 10 −1/3
1105/12 −1/2 0
111 −1/83/4 0
20010/3 2 −2
2103/8 −9/41/3
211 −2/3 20
22073/36 −29/6 0
221 −15/1621/8 0
2225/8 −27/20 0
30035/2 21 −13
310 −79/36 −85/6 4
311 −11/421/2 −2/3
320319/48 −119/8−1/3
321 −125/36269/30 0
32235/16 −213/40 0
33050/3 −1046/25 0
331 −35/3148/5 0
33235/9 −1012/105 0
333 −35/161587/280 0
400994/9542/3 −92
410 −515/16 −879/8115/3
411 −9/2357/5 −12
42012907/120−13903/10−6
421 −229/161251/40 1
42235/3 −1024/35 0
4302641/48−28049/2001/3
431 −1505/36118051/1050 0(lmn) r1 r2 r3
432525/32 −4617/112 0
433 −595/728809/420 0
4406025/36 −620161/1470 0
441 −29175/224131379/400 0
4422975/48 −31231/200 0
443 −539/32119271/2800 0
44477/8 −186003/7700 0
5009287/123005/2 −2077/3
510 −189029/180 −138331/50 348
511275/45751/10 −150
5202897/16 −15123/20 −229/3
521 −937/1227059/30 24
522509/8 −4209/28 0
5303589/18 −1993883 /3075 0
531 −1329/8297981/700 −4/3
5322555/36 −187777/1050 0
533 −2233/48164399/1400 0
54018471/32 −28493109 /19600 −1/3
541 −1390/3286274/245 0
5427777/32 −1715589 /2800 0
543 −5621/724550057 /23100 0
5441155/32 −560001/6160 0
550197045/108 −101441689 /22050 0
551 −12023/818569853 /4900 0
5521683/2 −5718309 /2695 0
553 −5159/162504541 /3080 0
55424563/312 −1527851 /77000 0
555 −9251/20812099711 /107800 0
6.61311/integraldisplay∞
0e−xzJν+1
2/parenleftbiggx2
2/parenrightbigg
dx=Γ(ν+1 )√πD−ν−1/parenleftbig
zeπ
4i/parenrightbig
D−ν−1/parenleftBig
ze−πi
4/parenrightBig
[Reν>−1] MO 122
6.614
1./integraldisplay∞
0e−αxJν/parenleftbig
β√x/parenrightbig
dx=β
4/radicalbiggπ
α3exp/parenleftbigg
−β2
8α/parenrightbigg/bracketleftbigg
I1
2(ν−1)/parenleftbiggβ2
8α/parenrightbigg
−I1
2(ν+1)/parenleftbiggβ2
8α/parenrightbigg/bracketrightbigg
=1
αe−β2/4α[ν=0 ]
MO 178
2./integraldisplay∞
0e−αxY2ν/parenleftBig
2/radicalbig
βx/parenrightBig
dx=e−1
2β
α√αβ/braceleftbigg
cot(νπ)Γ(ν+1 )
Γ(2ν+1 )M1
2,ν/parenleftbiggβ
α/parenrightbigg
−cosec( νπ)W1
2,nu/parenleftbiggβ
α/parenrightbigg/bracerightbigg
[Reα>0,|Reν|<1] ET I 188(50)a
3./integraldisplay∞
0e−αxI2ν/parenleftBig
2/radicalbig
βx/parenrightBig
dx=e1
2β
α√αβΓ(ν+1 )
Γ(2ν+1 )M−1
2,ν/parenleftbiggβ
α/parenrightbigg
[Reα>0,Reν>−1] ET I 197(20)a
698 Bessel Functions 6.615
4./integraldisplay∞
0e−αxK2ν/parenleftBig
2/radicalbig
βx/parenrightBig
dx=e1
2β
α
2√αβΓ(ν+1 )Γ ( 1 −ν)W−1
2,ν/parenleftbiggβ
α/parenrightbigg
[Reα>0,|Reν|<1] ET I 199(37)a
5./integraldisplay∞
0e−αxK1/parenleftbig
β√x/parenrightbig
dx=β
8/radicalbiggπ
α3exp/parenleftbiggβ2
8α/parenrightbigg/bracketleftbigg
K1/parenleftbiggβ2
8α/parenrightbigg
−K0/parenleftbiggβ2
8α/parenrightbigg/bracketrightbigg
MO 181
6.615/integraldisplay∞
0e−αxJν/parenleftbig
2β√x/parenrightbig
Jν/parenleftbig
2γ√x/parenrightbig
dx=1
αIν/parenleftbigg2βγ
α/parenrightbigg
exp/parenleftbigg
−β2+γ2
α/parenrightbigg
[Reν>−1]
MO 178
6.616
1./integraldisplay∞
0e−αxJ0/parenleftBig
β/radicalbig
x2+2γx/parenrightBig
dx=1/radicalbig
α2+β2exp/bracketleftBig
γ/parenleftBig
α−/radicalbig
α2+β2/parenrightBig/bracketrightBig
MO 179
2./integraldisplay∞
1e−αxJ0/parenleftBig
β/radicalbig
x2−1/parenrightBig
dx=1/radicalbig
α2+β2exp/parenleftBig
−/radicalbig
α2+β2/parenrightBig
MO 179
3./integraldisplay∞
−∞eitxH(1)
0/parenleftBig
r/radicalbig
α2−t2/parenrightBig
dt=−2ieiα√
r2+x2
√
r2+x2/bracketleftBig
0≤arg/radicalbig
α2−t2<π , 0≤argα<π ;randxare real/bracketrightBig
MO 49
4./integraldisplay∞
−∞e−itxH(2)
0/parenleftBig
r/radicalbig
α2−t2/parenrightBig
dt=2ie−iα√
r2+x2
√
r2+x2/bracketleftBig
−π<arg/radicalbig
α2−t2≤0,−π<argα≤0,r andxare real/bracketrightBig
MO 49
5.3/integraldisplay1
−1e−axI0/parenleftBig
b/radicalbig
1−x2/parenrightBig
dx=2/parenleftbig
a2+b2/parenrightbig−1/2sinh/radicalbig
a2+b2
[a>0,b > 0]
6.8/integraldisplay∞
0e−xyJ0/bracketleftBig
y/radicalbig
1−x2/bracketrightBig
/(α+y)dy=∞/summationdisplay
n=0n!Pn(x)
αn+1
6.617
1./integraldisplay∞
0Kq−p(2zsinhx)e(p+q)xdx=π2
4sin[( p−q)π][Jp(z)Yq(z)−Jq(z)Yp(z)]
[Rez>0,−1<Re(p−q)<1]
MO 44
2./integraldisplay∞
0K0(2zsinhx)e−2pxdx=−π
4/braceleftbigg
Jp(z)∂Yp(z)
∂p−Yp(z)∂Jp(z)
∂p/bracerightbigg
[Rez>0] MO 44
6.618
1./integraldisplay∞
0e−αx2Jν(βx)dx=√π
2√αexp/parenleftbigg
−β2
8α/parenrightbigg
I1
2ν/parenleftbiggβ2
8α/parenrightbigg
[Reα>0,β > 0,Reν>−1]
WA 432(5), ET II 29(8)
6.621 Bessel functions, exponentials, and powers 699
2./integraldisplay∞
0e−αx2Yν(βx)dx=−√π
2√αexp/parenleftbigg
−β2
8α/parenrightbigg/bracketleftbigg
tanνπ
2I1
2ν/parenleftbiggβ2
8α/parenrightbigg
+1
πsec/parenleftBigνπ
2/parenrightBig
K1
2ν/parenleftbiggβ2
8α/parenrightbigg/bracketrightbigg
[Reα>0,β > 0,|Reν|<1]
WA 432(6), ET II 106(3)
3./integraldisplay∞
0e−αx2Kν(βx)dx=1
4sec/parenleftBigνπ
2/parenrightBig√π√αexp/parenleftbiggβ2
8α/parenrightbigg
K1
2ν/parenleftbiggβ2
8α/parenrightbigg
[Reα>0,|Reν|<1]
EH II 51(28), ET II 132(24)
4./integraldisplay∞
0e−αx2Iν(βx)dx=√π
2√αexp/parenleftbiggβ2
8α/parenrightbigg
I1
2ν/parenleftbiggβ2
8α/parenrightbigg
[Reν>−1,Reα>0] EH II 92(27)
5./integraldisplay∞
0e−αx2Jμ(βx)Jν(βx)dx
=2−ν−μ−1α−ν+μ+1
2βν+μΓ/parenleftbigμ+ν+1
2/parenrightbig
Γ(μ+1 )Γ ( ν+1 )
×3F3/parenleftbiggν+μ+1
2,ν+μ+2
2,ν+μ+1
2;μ+1,ν+1,ν+μ+1 ;−β2
α/parenrightbigg
[Re(ν+μ)>−1,Reα>0]EH II 50(21)a
6.62–6.63 Combinations of Bessel functions, exponentials, and powers
6.621 Notation :
/lscript1=1
2/bracketleftBig/radicalbig
(a+ρ)2+z2−/radicalbig
(a−ρ)2+z2/bracketrightBig
,/lscript2=1
2/bracketleftBig/radicalbig
(a+ρ)2+z2+/radicalbig
(a−ρ)2+z2/bracketrightBig
1./integraldisplay∞
0e−αxJν(βx)xμ−1dx
=/parenleftBig
β
2α/parenrightBigν
Γ(ν+μ)
αμΓ(ν+1 )F/parenleftbiggν+μ
2,ν+μ+1
2;ν+1 ;−β2
α2/parenrightbigg
WA 421(2)
=/parenleftBig
β
2α/parenrightBigν
Γ(ν+μ)
αμΓ(ν+1 )/parenleftbigg
1+β2
α2/parenrightbigg1
2−μ
F/parenleftbiggν−μ+1
2,ν−μ
2+1 ;ν+1 ;−β2
α2/parenrightbigg
WA 421(3)
=/parenleftBig
β
2/parenrightBigν
Γ(ν+μ)
/radicalBig
(α2+β2)ν+μΓ(ν+1 )F/parenleftbiggν+μ
2,1−μ+ν
2;ν+1 ;β2
α2+β2/parenrightbigg
[Re(ν+μ)>0,Re (α+iβ)>0,Re (α−iβ)>0]
WA 421(3)
=/parenleftbig
α2+β2/parenrightbig−1
2μΓ(ν+μ)P−ν
μ−1/bracketleftBig
α/parenleftbig
α2+β2/parenrightbig−1
2/bracketrightBig
[α>0,β > 0,Re(ν+μ)>0]
ET II 29(6)
700 Bessel Functions 6.621
2./integraldisplay∞
0e−αxYν(βx)xμ−1dx
=c o t νπ/parenleftBig
β
2/parenrightBigν
Γ(ν+μ)
/radicalBig
(α2+β2)ν+μΓ(ν+1 )F/parenleftbiggν+μ
2,ν−μ+1
2;ν+1 ;β2
α2+β2/parenrightbigg
−cosecνπ/parenleftBig
β
2/parenrightBig−ν
Γ(μ−ν)
/radicalBig
(α2+β2)μ−νΓ(1−ν)F/parenleftbiggμ−ν
2,1−ν−μ
2;1−ν;β2
α2+β2/parenrightbigg
[Reμ≥|Reν|,Re (α±iβ)>0]
WA 421(4)
=−2
πΓ(ν+μ)/parenleftbig
β2+α2/parenrightbig−1
2μQ−ν
μ−1/bracketleftBig
α/parenleftbig
α2+β2/parenrightbig−1
2/bracketrightBig
[α>0,β > 0,Reμ>|Reν|]
ET II 105(2)
3./integraldisplay∞
0xμ−1e−αxKν(βx)dx=√π(2β)ν
(α+β)μ+νΓ(μ+ν)Γ(μ−ν)
Γ/parenleftbig
μ+1
2/parenrightbigF/parenleftbigg
μ+ν,ν+1
2;μ+1
2;α−β
α+β/parenrightbigg
[Reμ>|Reν|,Re(α+β)>0]
ET II 131(23)a, EH II 50(26)
4./integraldisplay∞
0xm+1e−αxJν(βx)dx=(−1)m+1β−νdm+1
dαm+1⎡
⎣/parenleftBig/radicalbig
α2+β2−α/parenrightBigν
/radicalbig
α2+β2⎤
⎦
[β>0,Reν>−m−2] ET II 28(3)
5.10/integraldisplay∞
0e−zxJ1(ax)J1/2(ρx)x−3/2dx
=1
a/radicalbigg2
πρ/braceleftbigg/lscript1
2/radicalBig
a2−/lscript2
1+a2
2arcsin/parenleftbigg/lscript1
2/parenrightbigg
+z/bracketleftbigg/radicalBig
ρ2−/lscript2
1−ρ/bracketrightbigg/bracerightbigg
[arga>0,argρ>0,argz>0]
6.10/integraldisplay∞
0e−zxJ1(ax)J1/2(ρx)x−1/2dx=1
a/radicalbigg2
πρ/bracketleftbigg
ρ−/radicalBig
ρ2−/lscript2
1/bracketrightbigg
[arga>0,argρ>0,argz>0]
7.10/integraldisplay∞
0e−zxJ1(ax)J1/2(ρx)x1/2dx=1
a/radicalbigg2
πρ/lscript1/radicalbig
a2−/lscript2
1
/lscript2
2−/lscript2
1
[arga>0,argρ>0,argz>0]
8.10/integraldisplay∞
0e−zxJ1(ax)J3/2(ρx)x1/2dx=/radicalbigg
2
π/lscript2
1/radicalbig
ρ2−/lscript2
1
ρ3/2a(/lscript2
2−/lscript2
1)
[arga>0,argρ>0,argz>0]
9.10/integraldisplay∞
0e−zxJ1(ax)J3/2(ρx)x−3/2dx=1√
2π1
ρ3/2a/bracketleftbigg
a2arcsin/parenleftbigg/lscript1
a/parenrightbigg
−/lscript1/radicalBig
a2−/lscript2
1/bracketrightbigg
[arga>0,argρ>0,argz>0]
6.621 Bessel functions, exponentials, and powers 701
10.10/integraldisplay∞
0e−zxJ1(ax)J5/2(ρx)x−1/2dx=1√
2πz
ρ5/2a/bracketleftBigg
/lscript1/radicalBig
a2−/lscript2
1+2a2/lscript1/radicalbig
a2−/lscript2
1−3a2arcsin/parenleftbigg/lscript1
a/parenrightbigg/bracketrightBigg
[arga>0,argρ>0,argz>0]
11.10/integraldisplay∞
0e−zxJ1(ax)J5/2(ρx)x−3/2dx
=1√
2π1
ρ5/2a⎡
⎣/lscript1/radicalbig
a2−/lscript2
1/parenleftbigg7a2
8−a2z2−/lscript4
1
4−5a2/lscript2
1
8/parenrightbigg
−1
2/parenleftbig
/lscript2
1+/lscript2
2/parenrightbig
/lscript1/radicalBig
a2−/lscript2
1+a r c s i n/parenleftbigg/lscript1
a/parenrightbigg/parenleftbigg3
2a2z2+1
2a2ρ2−3a4
8/parenrightbigg⎤
⎦
[arga>0,argρ>0,argz>0]
12.10/integraldisplay∞
0e−zxJ1(ax)J5/2(ρx)x−5/2dx
=1√
2π1
ρ5/2a⎧
⎨
⎩2/bracketleftBig
ρ5/2−/parenleftbig
ρ2−/lscript2
1/parenrightbig5/2/bracketrightBig
15+za2arcsin/parenleftbigg/lscript1
a/parenrightbigg/bracketleftbigg3a2
8−ρ2
2−z2
2/bracketrightbigg
+z/lscript1/radicalBig
a2−/lscript2
1/bracketleftbiggρ2
2−3a2
8+z2
6−/lscript2
1
4/bracketrightbigg
+z3a2/lscript1
3/radicalbig
a2−/lscript2
1⎫
⎬
⎭
[arga>0,argρ>0,argz>0]
13.10/integraldisplay∞
0e−zxJ2(ax)J3/2(ρx)x1/2dx=/radicalbigg
2
πa2ρ3/2/radicalbig
/lscript2
2−ρ2
(/lscript2
2−/lscript2
1)/lscript4
2
[arga>0,argρ>0,argz>0]
14.10/integraldisplay∞
0e−zxJ2(ax)J3/2(ρx)x−1/2dx=/radicalbigg
2
πρ3/2
a2/bracketleftBigg
2
3−/radicalbig
ρ2−/lscript2
1
ρ+/parenleftbig
ρ2−/lscript2
1/parenrightbig3/2
3ρ3/bracketrightBigg
[arga>0,argρ>0,argz>0]
15.10/integraldisplay∞
0e−zxJ3(ax)J1/2(ρx)x−1/2dx
=/radicalbigg2
πρ1
3a3/braceleftbigg
ρ/bracketleftbig
3a2−4ρ2+1 2z2/bracketrightbig
−/radicalBig
ρ2−/lscript2
1/braceleftbig
12/lscript2
2−16ρ2+4/lscript2
1−3a2/bracerightbig/bracerightbigg
[arga>0,argρ>0,argz>0]
16.10/integraldisplay∞
0e−zxJ3(ax)J3/2(ρx)x1/2dx
=/radicalbigg
2
πρ3/2/braceleftBigg
4
a3/bracketleftBigg
2
3−/radicalbig
ρ2−/lscript2
1
ρ+/parenleftbig
ρ2−/lscript2
1/parenrightbig3/2
3ρ2/bracketrightBigg
−a/radicalbig
/lscript2
2−a2
(/lscript2
2−/lscript2
1)/lscript3
2/bracerightBigg
[arga>0,argρ>0,argz>0]
17.10/integraldisplay∞
0e−zxJ3(ax)J3/2(ρx)x−1/2dx=/radicalbigg
2
πρ3/2
3a3/bracketleftBigg/radicalBig
/lscript2
2−ρ2/parenleftBigg
4ρ2/parenleftbig
2ρ2−/lscript2
1/parenrightbig
−/lscript4
1
ρ4/parenrightBigg
−8z/bracketrightBigg
[arga>0,argρ>0,argz>0]
702 Bessel Functions 6.622
18.10/integraldisplay∞
0e−zxJ3(ax)J3/2(ρx)x−3/2dx
=/radicalbigg
2
πρ3/2
3a3/braceleftbigg
a2−4
5ρ2+4z2−/radicalBig
ρ2−/lscript2
1/bracketleftbigg4/lscript2
2
ρ−24ρ
5+8/lscript2
1
5ρ−a2
ρ+/lscript4
1
5ρ3/bracketrightbigg/bracerightbigg
[arga>0,argρ>0,argz>0]
19.10/integraldisplay∞
0e−zxJ3(ax)J3/2(ρx)x−5/2dx
=−/radicalbigg
2
πρ3/2
3a3⎧
⎨
⎩/parenleftbigg
a2−4
5ρ2/parenrightbigg
z+4z3
3
+/radicalBig
/lscript2
2−ρ2/bracketleftbigg
a2+32
15ρ2−12
5/lscript2
1−4
3/lscript2
2+2/lscript4
1
5ρ2+a4/lscript2
1
16ρ4+a2/lscript2
1
24ρ4+/lscript6
1
30ρ4/bracketrightbigg
−a6
16ρ3arcsin/parenleftbiggρ
/lscript2/parenrightbigg⎫
⎬
⎭
[arga>0,argρ>0,argz>0]
6.622
1./integraldisplay∞
0/parenleftbig
J0(x)−e−αx/parenrightbigdx
x=l n2 α [α>0] NT 66(13)
2./integraldisplay∞
0ei(u+x)
u+xJ0(x)dx=π
2iH(1)
0(u) MO 44
3.8/integraldisplay∞
0e−xcoshαIν(x)xμ−1dx=/radicalbigg
2
πe−(μ−1
2)πiQμ−1
2
ν−1
2(coshα)
sinhμ−1
2α
[Re(μ+ν)>0,Re (cosh α)>1]
WA 388(6)a
6.623
1./integraldisplay∞
0e−αxJν(βx)xνdx=(2β)νΓ/parenleftbig
ν+1
2/parenrightbig
√π(α2+β2)ν+1
2/bracketleftbig
Reν>−1
2,Reα>|Imβ|/bracketrightbig
WA 422(5)
2./integraldisplay∞
0e−αxJν(βx)xν+1dx=2α(2β)νΓ/parenleftbig
ν+3
2/parenrightbig
√π(α2+β2)ν+3
2[Reν>−1,Reα>|Imβ|]
WA 422(6)
3./integraldisplay∞
0e−αxJν(βx)dx
x=/parenleftBig/radicalbig
α2+β2−α/parenrightBigν
νβν
[Reν>0; Re α>|Imβ|] (cf. 6.611 1)WA 422(7)
6.624
1./integraldisplay∞
0xe−αxK0(βx)dx=1
α2−β2⎧
⎨
⎩α/radicalbig
α2−β2ln⎡
⎣α
β+/radicalBigg/parenleftbiggα
β/parenrightbigg2
−1⎤
⎦−1⎫
⎬
⎭MO 181
6.625 Bessel functions, exponentials, and powers 703
2./integraldisplay∞
0√xe−αxK±1
2(βx)dx=/radicalbiggπ
2β1
α+βMO 181
3./integraldisplay∞
0e−tz(z2−1)−1/2
Kμ(t)tνdt=Γ(ν−μ+1 )
(z2−1)−1
2(ν+1)eiμπQμ
ν(z)
[Re(ν±μ)>−1] EH II 57(7)
4./integraldisplay∞
0e−tz(z2−1)−1/2
I−μ(t)tνdt=Γ(−ν−μ)
(z2−1)1
2νPμ
ν(z)[ R e ( ν+μ)<0] EH II 57(8)
5./integraldisplay∞
0e−tz(z2−1)−1
2Iμ(t)tνdt=Γ(ν+μ+1 )
(z2−1)−1
2(ν+1)P−μ
ν(z)
[Re(ν+μ)>−1] EH II 57(9)
6./integraldisplay∞
0e−tcosθJμ(tsinθ)tνdt=Γ (ν+μ+1 )P−μ
ν(cosθ)
/bracketleftbig
Re(ν+μ)>−1,0≤θ<1
2π/bracketrightbig
EH II 57(10)
7./integraldisplay∞
0Jν(bx)xν
eπx−1dx=(2b)νΓ/parenleftbig
ν+1
2/parenrightbig
√π∞/summationdisplay
n=11
(n2π2+b2)ν+1
2
[Reν>0,|Imb|<π] WA 423(9)
6.625
1./integraldisplay1
0xλ−ν−1(1−x)μ−1e±iαxJν(αx)dx=2−νανΓ(λ)Γ(μ)
Γ(λ+μ)Γ(ν+1 )2F2/parenleftbigg
λ,ν+1
2;λ+μ,2ν+1 ;±2iα/parenrightbigg
[Reλ>0,Reμ>0] ET II 194(58)a
2./integraldisplay1
0xν(1−x)μ−1e±iαxJν(αx)dx=(2α)νΓ(μ)Γ/parenleftbig
ν+1
2/parenrightbig
√πΓ(μ+2ν+1 )1F1/parenleftbigg
ν+1
2;μ+2ν+1 ;±2iα/parenrightbigg
/bracketleftbig
Reμ>0,Reν>−1
2/bracketrightbig
ET II 194(57)a
3./integraldisplay1
0xν(1−x)μ−1e±αxJν(αx)dx=(2α)νΓ/parenleftbig
ν+1
2/parenrightbig
Γ(μ)√πΓ(μ+2ν+1 )1F1/parenleftbigg
ν+1
2;μ+2ν+1 ;±2α/parenrightbigg
/bracketleftbig
Reμ>0,Reν>−1
2/bracketrightbig
BU 9(16a), ET II 197(77)a
4./integraldisplay1
0xλ−1(1−x)μ−1e±αxIν(αx)dx=/parenleftbig1
2α/parenrightbigνΓ(λ+ν)Γ (μ)
Γ(ν+1 )Γ ( λ+μ+ν)
×2F2/parenleftbigg
ν+1
2,λ+ν;2ν+1,μ+λ+ν;±2α/parenrightbigg
[Reμ>0,Re(λ+ν)>0]ET II 197(78)a
5./integraldisplay1
0xμ−κ(1−x)2κ−1Iμ−κ/parenleftbigg1
2xz/parenrightbigg
e−1
2xzdx=Γ(2κ)√πΓ(1 + 2 μ)ex
2z−κ−1
2Mκ,u(z)
/bracketleftbig
Re/parenleftbig
κ−1
2−μ/parenrightbig
<0,Reκ>0/bracketrightbig
BU 129(14a)
704 Bessel Functions 6.626
6./integraldisplay∞
1x−λ(x−1)μ−1e−αxIν(αx)dx=(2α)λΓ(μ)√πG21
23/parenleftbigg
2α/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2−λ,0
−μ, ν−λ,−ν−λ/parenrightbigg
/bracketleftbig
0<Reμ<1
2+R eλ,Reα>0/bracketrightbig
ET II 207(50)a
7./integraldisplay∞
1x−λ(x−1)μ−1e−αxKν(αx)dx=Γ (μ)√π(2α)λG30
23/parenleftbigg
2α/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,
1
2−λ
−μ, ν−λ,−ν−λ/parenrightbigg
[Reμ>0,Reα>0] ET II 208(55)a
8./integraldisplay∞
1x−ν(x−1)μ−1e−αxIν(αx)dx=(2α)ν−μΓ/parenleftbig1
2−μ+ν/parenrightbig
Γ(μ)√πΓ(1−μ+2ν)
×1F1/parenleftbigg1
2−μ+ν;1−μ+2ν;−2α/parenrightbigg
/bracketleftbig
0<Reμ<1
2+R eν,Reα>0/bracketrightbig
ET II 207(49)a
9./integraldisplay∞
1x−ν(x−1)μ−1e−αxKν(αx)dx=√πΓ(μ)(2α)−1
2μ−1
2e−αW−1
2μ,ν−1
2μ(2α)
[Reμ>0,Reα>0] ET II 208(53)a
10./integraldisplay∞
1x−μ−1
2(x−1)μ−1e−αxKν(αx)dx=√πΓ(μ)(2α)−1
2e−αW−μ,ν(2α)
[Reμ>0,Reα>0] ET II 207(51)a
11.3/integraldisplay1
−1/parenleftbig
1−x2/parenrightbig−1/2xe−axI1/parenleftBig
b/radicalbig
1−x2/parenrightBig
dx=2
b/braceleftBig
sinha−a/parenleftbig
a2+b2/parenrightbig−1/2sinh/radicalbig
a2+b2/bracerightBig
[a>0,b > 0]
6.626
1.11/integraldisplay∞
0xλ−1e−αxJμ(βx)Jν(γx)dx=βμγν
Γ(ν+1 )2−ν−μα−λ−μ−ν∞/summationdisplay
m=0Γ(λ+μ+ν+2m)
m!Γ (μ+m+1 )
×F/parenleftbigg
−m,−μ−m;ν+1 ;γ2
β2/parenrightbigg/parenleftbigg
−β2
4α2/parenrightbiggm
[Re(λ+μ+ν)>0,Re (α±iβ±iγ)>1]EH II 48(15)
2./integraldisplay∞
0e−2αxJν(βx)Jμ(βx)xν+μdx=Γ/parenleftbig
ν+μ+1
2/parenrightbig
βν+μ
√
π3
×/integraldisplayπ
2
0cosν+μϕcos(ν−μ)ϕ
(α2+β2cos2ϕ)ν+μ/radicalbig
α2+β2cos2ϕdϕ
/bracketleftbig
Reα>|Imβ|,Re(ν+μ)>−1
2/bracketrightbig
WA 427(1)
3./integraldisplay∞
0e−2αxJ0(βx)J1(βx)xdx=K/parenleftbigg
β√
α2+β2/parenrightbigg
−E/parenleftbigg
β√
α2+β2/parenrightbigg
2πβ/radicalbig
α2+β2WA 427(2)
4./integraldisplay∞
0e−2αxI0(βx)I1(βx)xdx=1
2πβ/braceleftbiggα
α2−β2E/parenleftbiggβ
α/parenrightbigg
−1
αK/parenleftbiggβ
α/parenrightbigg/bracerightbigg
[Reα>Reβ] WA 428(5)
6.628 Bessel functions, exponentials, and powers 705
5.10/integraldisplay∞
0xν−μ+2ne−zxJμ(αx)Jν(ρx)dx=1√π/parenleftBiga
2/parenrightBigμ−ν−2n−1/parenleftBigρ
a/parenrightBigν
×1
Γ/parenleftbig
μ−ν−n+1
2/parenrightbig∞/summationdisplay
q=0Γ/parenleftbig
ν+n+q+1
2/parenrightbig/parenleftbig
ν−μ+n+1
2/parenrightbig
q
q!Γ/parenleftbig
ν+q+1
2/parenrightbig
×a−2q/integraldisplay/lscript1/ρ
0dx√
1−x2x2ν+2q/parenleftbigg
ρ2+z2
1−x2/parenrightbiggq
where /lscript1=1
2/bracketleftBig/radicalbig
(a+ρ)2+z2−/radicalbig
(a−ρ)2+z2/bracketrightBig/bracketleftbig
μ>ν +2n, n =0,1,..., ν > −1
2/bracketrightbig
6.627/integraldisplay∞
0x−1/2
x+ae−xKν(x)dx=πeaKν(a)√acos(νπ)/bracketleftbig
|arga|<π , |Reν|<1
2/bracketrightbig
ET II 368(29)
6.628
1./integraldisplay∞
0e−xcosβJ−ν(xsinβ)xμdx=Γ (μ−ν+1 )Pν
μ(cosβ)
/bracketleftBig
0<β<π
2,Re(μ−ν)>−1/bracketrightBig
WA 424(3), WH
2./integraldisplay∞
0e−xcosβYν(xsinβ)xμdx=−sinμπ
sin(μ+ν)πΓ(μ−ν+1 )
π
×/bracketleftBig
Qν
μ(cosβ+0·i)e1
2νπi+Qν
μ(cosβ−0·i)e−1
2νπi/bracketrightBig
/bracketleftBig
Re(μ+ν)>−1,0<β<π
2/bracketrightBig
WA 424(4)
3./integraldisplay1
0exu
2(1−x)2ν−1xμ−νJμ−ν/parenleftbiggixu
2/parenrightbigg
dx=22(ν−μ)eπ
2(μ−ν)iB(2ν,2μ−2ν+1 )
Γ(μ−ν+1 )eu
2
uν+1
2Mν,μ(u)
MO 118a
4.8/integraldisplay∞
0e−xcoshαIν(xsinhα)xμdx=Γ (ν+μ+1 )P−ν
μ(coshα)
/bracketleftbig
Re(μ+ν)>−1,|Imα|<1
2π/bracketrightbig
WA 423(1)
5./integraldisplay∞
0e−xcoshαKν(xsinhα)xμdx=sinμπ
sin(ν+μ)πΓ(μ−ν+1 )Qν
μ(coshα)
[Re(μ+1 )>|Reν|] WA 423(2)
6./integraldisplay∞
0e−xcoshαIν(x)xμ−1dx=cosνπ
sin(μ+ν)πQν−1
2
μ−1
2(coshα)
/radicalbigπ
2(sinhα)μ−1
2
[Re(μ+ν)>0,Re (cosh α)>1]
WA 424(6)
7./integraldisplay∞
0e−xcoshαKν(x)xμ−1dx=/radicalbiggπ
2Γ(μ−ν)Γ(μ+ν)P1
2−μ
ν−1
2(coshα)
(sinhα)μ−1
2
[Reμ>|Reν|,Re (cosh α)>−1]
WA 424(7)
706 Bessel Functions 6.629
6.6298/integraldisplay∞
0x−1/2e−xαcosϕcosψJμ(αxsinϕ)Jν(αxsinψ)dx
=Γ/parenleftbig
μ+ν+1
2/parenrightbig
α−1
2P−μ
ν−1
2(cosϕ)P−ν
μ−1
2(cosψ)
/bracketleftbigg
α>0,0<ϕ<π
2,0<ψ<π
2,Re(μ+ν)>−1
2/bracketrightbigg
ET II 50(19)
6.631
1./integraldisplay∞
0xμe−αx2Jν(βx)dx=βνΓ/parenleftbig1
2ν+1
2μ+1
2/parenrightbig
2ν+1α1
2(μ+ν+1)Γ(ν+1 )1F1/parenleftbiggν+μ+1
2;ν+1 ;−β2
4α/parenrightbigg
BU 8(15)
=Γ/parenleftbig1
2ν+1
2μ+1
2/parenrightbig
βα1
2μΓ(ν+1 )exp/parenleftbigg
−β2
8α/parenrightbigg
M1
2μ,1
2ν/parenleftbiggβ2
4α/parenrightbigg
[Reα>0,Re(μ+ν)>−1]
EH II 50(22), ET II 30(14), BU 14(13b)
2./integraldisplay∞
0xμe−αx2Yν(βx)dx
=−α−1
2μβ−1sec/parenleftbiggν−μ
2π/parenrightbigg
exp/parenleftbigg
−β2
8α/parenrightbigg
×/braceleftBigg
Γ/parenleftbig1
2+1
2μ+1
2ν/parenrightbig
Γ(1 + ν)sin/parenleftbiggν−μ
2π/parenrightbigg
M1
2μ,1
2ν/parenleftbiggβ2
4α/parenrightbigg
+W1
2μ,1
2ν/parenleftbiggβ2
4α/parenrightbigg/bracerightBigg
[Reα>0,Reμ>|Reν|−1,β > 0]ET II 106(4)
3./integraldisplay∞
0xμe−αx2Kν(βx)dx=1
2α−1
2μβ−1Γ/parenleftbigg1+ν+μ
2/parenrightbigg
Γ/parenleftbigg1−ν+μ
2/parenrightbigg
exp/parenleftbiggβ2
8α/parenrightbigg
W−1
2μ,1
2ν/parenleftbiggβ2
4α/parenrightbigg
[Reμ>|Reν|−1] ET II 132(25)
4.11/integraldisplay∞
0xν+1e−αx2Jν(βx)dx=βν
(2α)ν+1exp/parenleftbigg
−β2
4α/parenrightbigg
[Reα>0,Reν>−1]
WA 431(4), ET II 29(10)
5./integraldisplay∞
0xν−1e−αx2Jν(βx)dx=2ν−1β−ν/bracketleftbigg
1−γ/parenleftbigg
ν,β2
4α/parenrightbigg/bracketrightbigg
[Reα>0,Reν>0] ET II 30(11)
6./integraldisplay∞
0xν+1e±iαx2Jν(βx)dx=βν
(2α)ν+1exp/bracketleftbigg
±i/parenleftbiggν+1
2π−β2
4α/parenrightbigg/bracketrightbigg
/bracketleftbig
α>0,−1<Reν<1
2,β > 0/bracketrightbig
ET II 30(12)
7./integraldisplay∞
0xe−αx2Jν(βx)dx=√πβ
8α3
2exp/parenleftbigg
−β2
8α/parenrightbigg/bracketleftbigg
I1
2ν−1
2/parenleftbiggβ2
8α/parenrightbigg
−I1
2ν+1
2/parenleftbiggβ2
8α/parenrightbigg/bracketrightbigg
[Reα>0,Reν>−2] ET II 29(9)
6.633 Bessel functions, exponentials, and powers 707
8./integraldisplay1
0xn+1e−αx2In(2αx)dx=1
4α/bracketleftBigg
eα−e−αn/summationdisplay
r=−nIr(2α)/bracketrightBigg
[n=0,1,...] ET II 365(8)a
9./integraldisplay∞
1x1−ne−αx2In(2αx)dx=1
4α/bracketleftBigg
eα−e−αn−1/summationdisplay
r=1−nIr(2α)/bracketrightBigg
[n=1,2,...] ET II 367(20)a
10./integraldisplay∞
0e−x2x2n+μ+1Jμ/parenleftbig
2x√z/parenrightbig
dx=n!
2e−zz1
2μLμ
n(z)[ n=0,1,...;n+R eμ>−1]
BU 135(5)
6.632/integraldisplay∞
0x−1
2exp/bracketleftBig
−/parenleftbig
x2+a2−2axcosϕ/parenrightbig1
2/bracketrightBig/bracketleftbig
x2+a2−2axcosϕ/bracketrightbig−1
2Kν(x)dx
=πa−1
2sec(νπ)Pν−1
2(−cosϕ)Kν(a)
/bracketleftbig
|arga|+|Reϕ|<π , |Reν|<1
2/bracketrightbig
ET II 368(32)
6.633
1./integraldisplay∞
0xλ+1e−αx2Jμ(βx)Jν(γx)dx=βμγνα−μ+ν+λ+2
2
2ν+μ+1Γ(ν+1 )∞/summationdisplay
m=0Γ/parenleftbig
m+1
2ν+1
2μ+1
2λ+1/parenrightbig
m!Γ (m+μ+1 )/parenleftbigg
−β2
4α/parenrightbiggm
×F/parenleftbigg
−m,−μ−m;ν+1 ;γ2
β2/parenrightbigg
[Reα>0,Re(μ+ν+λ)>−2,β>0,γ > 0]EH II 49(20)a, ET II 51(24)a
2./integraldisplay∞
0e−/rho12x2Jp(αx)Jp(βx)xdx=1
2/rho12exp/parenleftbigg
−α2+β2
4/rho12/parenrightbigg
Ip/parenleftbiggαβ
2/rho12/parenrightbigg
/bracketleftBig
Rep>−1,|arg/rho1|<π
4,α > 0,β > 0/bracketrightBig
KU 146(16)a, WA 433(1)
3./integraldisplay∞
0x2ν+1e−αx2Jν(x)Yν(x)dx=−1
2√πα−3
2ν−1
2exp/parenleftbigg
−1
2α/parenrightbigg
W1
2ν,1
2ν/parenleftbigg1
α/parenrightbigg
/bracketleftbig
Reα>0,Reν>−1
2/bracketrightbig
ET II 347(59)
4./integraldisplay∞
0xe−αx2Iν(βx)Jν(γx)dx=1
2αexp/parenleftbiggβ2−γ2
4α/parenrightbigg
Jν/parenleftbiggβγ
2α/parenrightbigg
[Reα>0,Reν>−1] ET II 63(1)
5./integraldisplay∞
0xλ−1e−αx2Jμ(βx)Jν(βx)dx
=2−ν−μ−1α−1
2(ν+λ+μ)βν+μΓ/parenleftbig1
2λ+1
2μ+1
2ν/parenrightbig
Γ(μ+1 )Γ ( ν+1 )
×3F3/bracketleftbiggν
2+μ
2+1
2,ν
2+μ
2+1,ν+μ+λ
2;μ+1,ν+1,μ+ν+1 ;−β2
α/bracketrightbigg
[Re(ν+λ+μ)>0,Reα>0]WA 434, EH II 50(21)
708 Bessel Functions 6.634
6.634/integraldisplay∞
0xe−x2
2a[Iν(x)+I−ν(x)]Kν(x)dx=aeaKν(a)[ R e a>0,−1<Reν<1]
ET II 371(49)
6.635
1./integraldisplay∞
0x−1e−α
xJν(βx)dx=2Jν/parenleftBig/radicalbig
2αβ/parenrightBig
Kν/parenleftBig/radicalbig
2αβ/parenrightBig
[Reα>0,β > 0] ET II 30(15)
2./integraldisplay∞
0x−1e−α
xYν(βx)dx=2Yν/parenleftBig/radicalbig
2αβ/parenrightBig
Kν/parenleftBig/radicalbig
2αβ/parenrightBig
[Reα>0,β > 0] ET II 106(5)
3./integraldisplay∞
0x−1e−α
x−βxJν(γx)dx=2Jν/braceleftbigg√
2α/bracketleftBig/radicalbig
β2+γ2−β/bracketrightBig1
2/bracerightbigg
Kν/braceleftbigg√
2α/bracketleftBig/radicalbig
β2+γ2+β/bracketrightBig1
2/bracerightbigg
[Reα>0,Reβ>0,γ > 0]
ET II 30(16)
6.636/integraldisplay∞
0x−1
2e−α√xJν(βx)dx=√
2√πβΓ/parenleftbig
ν+1
2/parenrightbig
D−ν−1
2/parenleftBig
2−1
2αe1
4πiβ−1
2/parenrightBig
D−ν−1
2/parenleftBig
2−1
2αe−1
4πiβ−1
2/parenrightBig
/bracketleftbig
Reα>0,β > 0,Reν>−1
2/bracketrightbig
ET II 30(17)
6.637
1./integraldisplay∞
0/parenleftbig
β2+x2/parenrightbig−1
2exp/bracketleftBig
−α/parenleftbig
β2+x2/parenrightbig1
2/bracketrightBig
Jν(γx)dx
=I1
2ν/braceleftbigg1
2β/bracketleftBig/parenleftbig
α2+γ2/parenrightbig1
2−α/bracketrightBig/bracerightbigg
K1
2ν/braceleftbigg1
2β/bracketleftBig/parenleftbig
α2+γ2/parenrightbig1
2+α/bracketrightBig/bracerightbigg
[Reα>0,Reβ>0,γ > 0,Reν>−1]ET II 31(20)
2./integraldisplay∞
0/parenleftbig
β2+x2/parenrightbig−1
2exp/bracketleftBig
−α/parenleftbig
β2+x2/parenrightbig1
2/bracketrightBig
Yν(γx)dx
=−sec/parenleftBigνπ
2/parenrightBig
K1
2ν/braceleftbigg1
2β/bracketleftBig/parenleftbig
α2+γ2/parenrightbig1
2+α/bracketrightBig/bracerightbigg
×/parenleftbigg1
πK1
2ν/braceleftbigg1
2β/bracketleftBig/parenleftbig
α2+γ2/parenrightbig1
2+α/bracketrightBig/bracerightbigg
+s i n/parenleftBigνπ
2/parenrightBig
I1
2ν/braceleftbigg1
2β/bracketleftBig/parenleftbig
α2+γ2/parenrightbig1
2−α/bracketrightBig/bracerightbigg/parenrightbigg
[Reα>0,Reβ>0,γ > 0,|Reν|<1]ET II 106(6)
3./integraldisplay∞
0/parenleftbig
x2+β2/parenrightbig−1
2exp/bracketleftBig
−α/parenleftbig
x2+β2/parenrightbig1
2/bracketrightBig
Kν(γx)dx
=1
2sec/parenleftBigνπ
2/parenrightBig
K1
2ν/parenleftbigg1
2β/bracketleftBig
α+/parenleftbig
α2−γ2/parenrightbig1
2/bracketrightBig/parenrightbigg
K1
2ν/parenleftbigg1
2β/bracketleftBig
α−/parenleftbig
α2−γ2/parenrightbig1
2/bracketrightBig/parenrightbigg
[Reα>0,Reβ>0,Re(γ+β)>0,|Reν|<1]ET II 132(26)
6.64 Combinations of Bessel functions of more complicated arguments, exponentials,
and powers
6.641/integraldisplay∞
0√xe−αxJ±1
4/parenleftbig
x2/parenrightbig
dx=√πα
4/bracketleftbigg
H∓1
4/parenleftbiggα2
4/parenrightbigg
−Y∓1
4/parenleftbiggα2
4/parenrightbigg/bracketrightbigg
MI 42
6.645 Bessel functions of complicated arguments, exponentials, and powers 709
6.642
1.10/integraldisplay∞
0x−1e−αxYν/parenleftbigg2
x/parenrightbigg
dx=2Kν/parenleftbig
2√a/parenrightbig
Yν/parenleftbig
2√a/parenrightbig
[Rea>0] MC
2./integraldisplay∞
0x−1e−αxH(1,2)
ν/parenleftbigg2
x/parenrightbigg
dx=H(1,2)
ν/parenleftbig√α/parenrightbig
Kν/parenleftbig√α/parenrightbig
MI 44, EH II 91(26)
6.643
1./integraldisplay∞
0xμ−1
2e−αxJ2ν/parenleftbig
2β√x/parenrightbig
dx=Γ/parenleftbig
μ+ν+1
2/parenrightbig
βΓ(2ν+1 )e−β2
2αα−μMμ,ν/parenleftbiggβ2
α/parenrightbigg
/bracketleftbig
Re/parenleftbig
μ+ν+1
2/parenrightbig
>0/bracketrightbig
BU 14(13a), MI 42a
2./integraldisplay∞
0xμ−1
2e−αxI2ν/parenleftbig
2β√x/parenrightbig
dx=Γ/parenleftbig
μ+ν+1
2/parenrightbig
Γ(2ν+1 )β−1eβ2
2αα−μM−μ,ν/parenleftbiggβ2
α/parenrightbigg
/bracketleftbig
Re/parenleftbig
μ+ν+1
2/parenrightbig
>0/bracketrightbig
MI 45
3./integraldisplay∞
0xμ−1
2e−αxK2ν/parenleftbig
2β√x/parenrightbig
dx=Γ/parenleftbig
μ+ν+1
2/parenrightbig
Γ/parenleftbig
μ−ν+1
2/parenrightbig
2βeβ2
2αα−μW−μ,ν/parenleftbiggβ2
α/parenrightbigg
/bracketleftbig
Re/parenleftbig
μ+ν+1
2/parenrightbig
>0/bracketrightbig
,(cf.6.631 3)
MI 47a
4./integraldisplay∞
0xn+1
2νe−αxJν/parenleftbig
2β√x/parenrightbig
dx=n!βνe−β2
αα−n−ν−1Lν
n/parenleftbiggβ2
α/parenrightbigg
[n+ν>−1] MO 178a
5./integraldisplay∞
0x−1
2e−αxY2ν/parenleftbig
β√x/parenrightbig
dx=−/radicalbiggπ
αexp/parenleftBig
−β2
8α/parenrightBig
cos(νπ)/bracketleftbigg
sin(νπ)Iν/parenleftbiggβ2
8α/parenrightbigg
+1
πKν/parenleftbiggβ2
8α/parenrightbigg/bracketrightbigg
/bracketleftbig
|Reν|<1
2/bracketrightbig
MI 44
6./integraldisplay∞
0x1
2me−αxKm/parenleftbig
2√x/parenrightbig
dx=Γ(m+1 )
2α/parenleftbigg1
α/parenrightbigg1
2m−1
2
e1
2αW−1
2(m+1),−1
2m/parenleftbigg1
α/parenrightbigg
MI 48a
6.644/integraldisplay∞
0e−βxJ2ν/parenleftbig
2a√x/parenrightbig
Jν(bx)dx=e x p/parenleftbigg
−a2β
β2+b2/parenrightbigg
Jν/parenleftbigga2b
β2+b2/parenrightbigg1/radicalbig
β2+b2/bracketleftbig
Reβ>0,b > 0,Reν>−1
2/bracketrightbig
ET II 58(17)
6.645
1./integraldisplay∞
1/parenleftbig
x2−1/parenrightbig−1
2e−αxJν/parenleftBig
β/radicalbig
x2−1/parenrightBig
dx=I1
2ν/bracketleftbigg1
2/parenleftBig/radicalbig
α2+β2−α/parenrightBig/bracketrightbigg
K1
2ν/bracketleftbigg1
2/parenleftBig/radicalbig
α2+β2+α/parenrightBig/bracketrightbigg
MO 179a
2./integraldisplay∞
1/parenleftbig
x2−1/parenrightbig1
2νe−αxJν/parenleftBig
β/radicalbig
x2−1/parenrightBig
dx=/radicalbigg
2
πβν/parenleftbig
α2+β2/parenrightbig−1
2ν−1
4Kν+1
2/parenleftBig/radicalbig
α2+β2/parenrightBig
MO 179a
710 Bessel Functions 6.646
3.3/integraldisplay1
−1/parenleftbig
1−x2/parenrightbig−1/2e−axI1/parenleftBig
b/radicalbig
1−x2/parenrightBig
dx=2
b/parenleftBig
cosh/radicalbig
a2+b2−cosha/parenrightBig
[a>0,b > 0]
6.646
1./integraldisplay∞
1/parenleftbiggx−1
x+1/parenrightbigg1
2ν
e−αxJν/parenleftBig
β/radicalbig
x2−1/parenrightBig
dx=exp/parenleftBig
−/radicalbig
α2+β2/parenrightBig
/radicalbig
α2+β2/parenleftBigg
β
α+/radicalbig
α2+β2/parenrightBiggν
[Reν>−1] E F8 9 ( 5 2 ) ,M O1 7 9
2./integraldisplay∞
1/parenleftbiggx−1
x+1/parenrightbigg1
2ν
e−αxIν/parenleftBig
β/radicalbig
x2−1/parenrightBig
dx=exp/parenleftBig
−/radicalbig
α2−β2/parenrightBig
/radicalbig
α2−β2/parenleftBigg
β
α+/radicalbig
α2−β2/parenrightBiggν
[Reν>−1,α > β ] MO 180
3.7/integraldisplay∞
be−pt/parenleftbiggt−b
t+b/parenrightbiggν/2
Kν/bracketleftBig
a/parenleftbig
t2−b2/parenrightbig1/2/bracketrightBig
dt=Γ(ν+1 )
2saν/bracketleftbig
xνe−bxΓ(−ν,bx)−yνebsΓ(−ν,by)/bracketrightbig
where x=p−s, y =p+s, s =/parenleftbig
p2−a2/parenrightbig1/2[Re(p+a)>0,|Re(ν)|<1].
ME 39a
6.647
1./integraldisplay∞
0x−λ−1
2(β+x)λ−1
2e−αxK2μ/bracketleftBig/radicalbig
x(β+x)/bracketrightBig
dx
=1
βe1
2αβΓ/parenleftbig1
2−λ+μ/parenrightbig
Γ/parenleftbig1
2−λ−μ/parenrightbig
Wλ,μ(z1)Wλ,μ(z2)
z1=1
2β/parenleftBig
α+/radicalbig
α2−1/parenrightBig
,z2=1
2β/parenleftBig
α−/radicalbig
α2−1/parenrightBig
/bracketleftbig
|argβ|<π , Reα>−1,Reλ+|Reμ|<1
2/bracketrightbig
ET II 377(37)
2./integraldisplay∞
0(α+x)−1
2x−1
2e−xcoshtKν/bracketleftBig/radicalbig
x(α+x)/bracketrightBig
dx
=1
2sec/parenleftBigνπ
2/parenrightBig
e1
2αcoshtK1
2ν/parenleftbigg1
4αet/parenrightbigg
K1
2ν/parenleftbigg1
4αe−t/parenrightbigg
[−1<Reν<1] ET II 377(36)
3.11/integraldisplayα
0xλ−1
2(α−x)−λ−1
2e−xsinhtI2μ/bracketleftBig/radicalbig
x(α−x)/bracketrightBig
dx
=e−(α/2) sinh t2Γ/parenleftbig1
2+λ+μ/parenrightbig
Γ/parenleftbig1
2−λ+μ/parenrightbig
α[Γ(2μ+1 ) ]2Mλ,μ/parenleftbigg1
2αet/parenrightbigg
M−λ,μ/parenleftbigg1
2αe−t/parenrightbigg
/bracketleftbig
Reμ>|Reλ|−1
2/bracketrightbig
ET II 377(32)
6.648/integraldisplay∞
−∞e/rho1x/parenleftbiggα+βex
αex+β/parenrightbiggν
K2ν/bracketleftBig/parenleftbig
α2+β2+2αβcoshx/parenrightbig1
2/bracketrightBig
dx=2Kν+/rho1(α)Kν−/rho1(β)
[Reα>0,Reβ>0] ET II 379(45)
6.651 Bessel and exponential functions and powers 711
6.649
1./integraldisplay∞
0Kμ−ν(2zsinhx)e(ν+μ)xdx=π2
4sin[( ν−μ)π][Jν(z)Yμ(z)−Jμ(z)Yν(z)]
[Rez>0,−1<Re(ν−μ)<1]
MO 44
2./integraldisplay∞
0Jν+μ(2xsinht)e(ν−μ)tdt=Kν(x)Iμ(x)
/bracketleftbig
Re(ν−μ)<3
2,Re(ν+μ)>−1,x > 0/bracketrightbig
EH II 97(68)
3./integraldisplay∞
0Yν−μ(2xsinht)e−(ν+μ)tdt=1
sin[π(μ−ν)]{Iμ(x)Kν(x)−cos[(ν−μ)π]Iν(x)Kμ(x)}
/bracketleftbig
|Re(ν−μ)|<1,Re(ν+μ)>−1
2,x > 0/bracketrightbig
EH II 97(73)
4./integraldisplay∞
0K0(2zsinhx)e−2νxdx=−π
4/braceleftbigg
Jν(z)∂Yν(z)
∂ν−Yν(z)∂Jν(z)
∂ν/bracerightbigg
6.65 Combinations of Bessel and exponential functions of more complicated argu-
ments and powers
6.651
1./integraldisplay∞
0xλ+1
2e−1
4α2x2Iμ/parenleftbig1
4α2x2/parenrightbig
Jν(βx)dx
=1√
2π2λ+1β−λ−3
2G21
23/parenleftbiggβ2
2α2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−μ,1+μ
h,
1
2,k/parenrightbigg
h=3
4+1
2λ+1
2ν, k =3
4+1
2λ−1
2ν
/bracketleftBig
|argα|<π
4,β > 0,−3
2−Re(2μ+ν)<Reλ<0/bracketrightBig
ET II 68(8)
2./integraldisplay∞
0xλ+1
2e−1
4α2x2Kμ/parenleftbig1
4α2x2/parenrightbig
Jν(βx)dx
=/radicalbiggπ
22λ+1β−λ−3
2G12
23/parenleftbiggβ2
2α2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−μ,1+μ
h,
1
2,k/parenrightbigg
h=3
4+1
2λ+1
2ν, k =3
4+1
2λ−1
2ν
/bracketleftBig
|argα|<π
4,Re (λ+ν±2μ)>−3
2/bracketrightBig
ET II 69(15)
3./integraldisplay∞
0x2μ−ν+1e−1
4αx2Iμ/parenleftbig1
4αx2/parenrightbig
Jν(βx)dx
=2μ−ν+1
2(πα)−1
2Γ/parenleftbigg1
2+μ/parenrightbiggβν−2μ−1
Γ/parenleftbig1
2−μ+ν/parenrightbig1F1/parenleftbigg1
2+μ;1
2−μ+ν;−β2
2α/parenrightbigg
/bracketleftbig
Reα>0,β > 0,Reν>2R eμ+1
2>−1
2/bracketrightbig
ET II 68(6)
712 Bessel Functions 6.652
4./integraldisplay∞
0x2μ+ν+1e−1
4α2x2Kμ/parenleftbig1
4α2x2/parenrightbig
Jν(βx)dx
=√π2μα−2μ−2ν−2βνΓ( 1+2 μ+ν)
Γ/parenleftbig
μ+ν+3
2/parenrightbig1F1/parenleftbigg
1+2μ+ν;μ+ν+3
2;−β2
2α2/parenrightbigg
/bracketleftbig
|argα|<1
4π,Reν>−1,Re(2μ+ν)>−1,β > 0/bracketrightbig
ET II 69(13)
5./integraldisplay∞
0x2μ+ν+1e−1
2αx2Iμ/parenleftbig1
2αx2/parenrightbig
Kν(βx)dx
=2μ−1
2√πβ−μ−3
2α−1
2μ−1
2ν−1
4Γ(2μ+ν+1 )Γ/parenleftbig
μ+1
2/parenrightbig
exp/parenleftbiggβ2
8α/parenrightbigg
Wk,m/parenleftbiggβ2
4α/parenrightbigg
2k=−3μ−ν−1
2,2m=μ+ν+1
2 /bracketleftbig
Reα>0,Reμ>−1
2,Re (2μ+ν)>−1/bracketrightbig
ET II 146(53)
6./integraldisplay∞
0xe−1
4αx2J1
2ν/parenleftbig1
4βx2/parenrightbig
Jν(γx)dx=2/parenleftbig
α2+β2/parenrightbig−1
2exp/parenleftbigg
−αγ2
α2+β2/parenrightbigg
J1
2ν/parenleftbiggβγ2
α2+β2/parenrightbigg
[γ>0,Reα>|Imβ|,Reν>−1]
ET II 56(2)
7./integraldisplay∞
0xe−1
4αx2I1
2ν/parenleftbig1
4αx2/parenrightbig
Jν(βx)dx=/parenleftbigg1
2πα/parenrightbigg−1
2
β−1exp/parenleftbigg
−β2
2α/parenrightbigg
[Reα>0,β > 0,Reν>−1]
ET II 67(3)
8./integraldisplay∞
0x1−νe−1
4α2x2Iν/parenleftbig1
4α2x2/parenrightbig
Jν(βx)dx=/radicalbigg
2
πβν−1
αexp/parenleftbigg
−β2
4α2/parenrightbigg
D−2ν/parenleftbiggβ
α/parenrightbigg
/bracketleftbig
|argα|<1
4π, β > 0,Reν>−1
2/bracketrightbig
ET II 67(1)
9./integraldisplay∞
0x−ν−1e−1
4α2x2Iν+1/parenleftbig1
4α2x2/parenrightbig
Jν(βx)dx=/radicalbigg
2
πβνexp/parenleftbigg
−β2
4α2/parenrightbigg
D−2ν−3/parenleftbiggβ
α/parenrightbigg
/bracketleftbig
|argα|<1
4π,Reν>−1,β > 0/bracketrightbig
ET II 67(2)
6.652/integraldisplay∞
0x2νe−/parenleftBig
x2
8+αx/parenrightBig
Iν/parenleftbiggx2
8/parenrightbigg
dx=Γ(4ν+1 )
24νΓ(ν+1 )eα2
2
αν+1W−3
2ν,1
2ν/parenleftbig
α2/parenrightbig
/bracketleftbig
Re/parenleftbig
ν+1
4/parenrightbig
>0/bracketrightbig
MI 45
6.653
1./integraldisplay∞
0exp/bracketleftbigg
−1
2x−1
2x/parenleftbig
a2+b2/parenrightbig/bracketrightbigg
Iν/parenleftbiggab
x/parenrightbiggdx
x=2Iν(a)Kν(b)[ 0 <a<b ]
=2Kν(a)Iν(b)[ 0 <b<a ]
[Reν>−1]WA 482(2)a, EH II 53(37), WA 482(3)a
2./integraldisplay∞
0exp/bracketleftbigg
−1
2x−1
2x/parenleftbig
z2+w2/parenrightbig/bracketrightbigg
Kν/parenleftBigzw
x/parenrightBigdx
x=2Kν(z)Kν(w)
[|argz|<π , |argw|<π , arg(z+w)]<1
4πWA 483(1), EH II 53(36)
6.662 Bessel, hyperbolic, and exponential functions 713
6.654/integraldisplay∞
0x−1
2e−β2
8x−αxKν/parenleftbiggβ2
8x/parenrightbigg
dx=√
4πα−1
2K2ν/parenleftbig
β√α/parenrightbig
ME 39
6.655/integraldisplay∞
0x/parenleftbig
β2+x2/parenrightbig−1
2exp/parenleftbigg
−α2β
β2+x2/parenrightbigg
Jν/parenleftbiggα2x
β2+x2/parenrightbigg
Jν(γx)dx=γ−1e−βγJ2ν(2α√γ)
/bracketleftbig
Reβ>0,γ > 0,Reν>−1
2/bracketrightbig
ET II 58(14)
6.656
1./integraldisplay∞
0e−(ξ−z)c o s h tJ2ν/bracketleftBig
2(zξ)1
2sinht/bracketrightBig
dt=Iν(z)Kν(ξ)
/bracketleftbig
Reν>−1
2,Re(ξ−z)>0/bracketrightbig
EH II 98(78)
2./integraldisplay∞
0e−(ξ+z)c o s h tK2ν/bracketleftBig
2(zξ)1
2sinht/bracketrightBig
dt=1
2Kν(z)Kν(ξ)sec(νπ)
/bracketleftbigg
|Reν|<1
2,Re/parenleftBig
z1
2+ξ1
2/parenrightBig2
≥0/bracketrightbigg
EH II 98(79)
6.66 Combinations of Bessel, hyperbolic, and exponential functions
Bessel and hyperbolic functions
6.661
1./integraldisplay∞
0sinh(ax)Kν(bx)dx=π
2cosec/parenleftbigνπ
2/parenrightbig
sin/bracketleftbig
νarcsin/parenleftbiga
b/parenrightbig/bracketrightbig
√
b2−a2
[Reb>|Rea|,|Reν|<2]
ET II 133(32)
2./integraldisplay∞
0cosh(ax)Kν(bx)dx=πcos/bracketleftbig
νarcsin/parenleftbiga
b/parenrightbig/bracketrightbig
2/radicalbig
b2−a2cos/parenleftBigνπ
2/parenrightBig [Reb>|Rea|,|Reν|<1]
ET II 134(33)
6.662 Notation :
/lscript1=1
2/bracketleftBig/radicalbig
(b+c)2+a2−/radicalbig
(b−c)2+a2/bracketrightBig
,/lscript 2=1
2/bracketleftBig/radicalbig
(b+c)2+a2+/radicalbig
(b−c)2+a2/bracketrightBig
1.10/integraldisplay∞
0cosh(βx)K0(αx)J0(γx)dx=K(k)√u+v
u=1
2/braceleftbigg/radicalBig
(α2+β2+γ2)2−4α2β2/bracerightbigg
+α2−β2−γ2
v=1
2/braceleftbigg/radicalBig
(α2+β2+γ2)2−4α2β2/bracerightbigg
−α2+β2+γ2
k2=v(u+v)−1[Reα>|Reβ|,γ > 0]
ET II 15(23)
714 Bessel Functions 6.663
alternatively, with a=γ,b=β,c=α,/integraldisplay∞
0cosh(bx)K0(cx)J0(ax)dx=K(k)/radicalbig
/lscript2
2−/lscript2
1
k2=/lscript2
2−c2
/lscript2
2−/lscript2
1,[Rec>|Reb|,a > 0]
2.10/integraldisplay∞
0sinh(βx)K1(αx)J0(γx)dx=a−1/bracketleftbigg
uE(k)−K(k)E(u)+K(k)snudnu
cnu/bracketrightbigg
cn2u=2γ2/braceleftbigg/bracketleftBig/parenleftbig
α2+β2+γ2/parenrightbig2−4α2β2/bracketrightBig1
2−α2+β2+γ2/bracerightbigg−1
k2=1
2/braceleftbigg
1−/parenleftbig
α2−β2−γ2/parenrightbig/bracketleftBig/parenleftbig
α2+β2+γ2/parenrightbig2−4α2β2/bracketrightBig−1
2/bracerightbigg
[Reα>|Reβ|,γ > 0]
ET II 15(24)
alternatively, with a=γ,b=β,c=α,/integraldisplay∞
0sinh(bx)K1(cx)J0(ax)dx=c−1/bracketleftbigg
uE(k)−K(k)E(u)+K(k)snudnu
cnu/bracketrightbigg
cn2u=a2
/lscript2
2−c2,k2=/lscript2
2−c2
/lscript2
2−/lscript2
1[Rec>|Reb|,a > 0]
6.663
1./integraldisplay∞
0Kν±μ(2zcosht)c os h[( μ∓ν)t]dt=1
2Kμ(z)Kν(z)
[Rez>0] WA 484(1), EH II 54(39)
2./integraldisplay∞
0Yμ+ν(2zcosht)cosh[( μ−ν)t]dt=π
4[Jμ(z)Jν(z)−Yμ(z)Yν(z)]
[z>0] EH II 96(64)
3./integraldisplay∞
0Jμ+ν(2zcosht)c o s h [ ( μ−ν)t]dt=−π
4[Jμ(z)Yν(z)+Jν(z)Yμ(z)]
[z>0] EH II 97(65)
4./integraldisplay∞
0Jμ+ν(2zsinht)cosh[( μ−ν)t]dt=1
2[Iν(z)Kμ(z)+Iμ(z)Kν(z)]
/bracketleftbig
Re(ν+μ)>−1,|Re(μ−ν)|<3
2,z > 0/bracketrightbig
EH II 97(71)
5./integraldisplay∞
0Jμ+ν(2zsinht)sin h[( μ−ν)t]dt=1
2[Iν(z)Kμ(z)−Iμ(z)Kν(z)]
/bracketleftbig
Re(ν+μ)>−1,|Re(μ−ν)|<3
2,z > 0/bracketrightbig
EH II 97(72)
6.664
1./integraldisplay∞
0J0(2zsinht)sin h(2 νt)dt=sin(νπ)
π[Kν(z)]2/bracketleftbig
|Reν|<3
4,z > 0/bracketrightbig
EH II 97(69)
6.668 Bessel, hyperbolic, and exponential functions 715
2./integraldisplay∞
0Y0(2zsinht)cosh(2 νt)dt=−cos(νπ)
π[Kν(z)]2/bracketleftbig
|Reν|<3
4,z > 0/bracketrightbig
EH II 97(70)
3./integraldisplay∞
0Y0(2zsinht)sin h(2 νt)dt=1
π/bracketleftbigg
Iν(z)∂Kν(z)
∂ν−Kν(z)∂Iν(z)
∂ν/bracketrightbigg
−1
πcos(νπ)[Kν(z)]2
/bracketleftbig
|Reν|<3
4,z > 0/bracketrightbig
EH II 97(75)
4./integraldisplay∞
0K0(2zsinht)c o s h2 νtdt=π2
8/braceleftbig
J2
ν(z)+N2
ν(z)/bracerightbig
[Rez>0] MO 44
5./integraldisplay∞
0K2μ(zsinh 2t)coth2νtd t=1
4zΓ/parenleftbigg1
2+μ−ν/parenrightbigg
Γ/parenleftbigg1
2−μ−ν/parenrightbigg
Wν,μ(iz)Wν,μ(−iz)
/bracketleftBig
|argz|≤π
2,|Reμ|+R eν<1
2/bracketrightBig
MO 119
6./integraldisplay∞
0cosh(2 μx)K2ν(2acoshx)dx=1
2Kμ+ν(a)Kμ−ν(a)
[Rea>0] ET II 378(42)
6.665/integraldisplay∞
0sechxcosh(2 λx)I2μ(asechx)dx=Γ/parenleftbig1
2+λ+μ/parenrightbig
Γ/parenleftbig1
2−λ+μ/parenrightbig
2a[Γ(2μ+1 ) ]2Mλ,μ(a)M−λ,μ(a)
/bracketleftbig
|Reλ|−Reμ<1
2/bracketrightbig
ET II 378(43)
Bessel, hyperbolic, and algebraic functions
6.666/integraldisplay∞
0xν+1sinh(αx)cosec h( πx)Jν(βx)dx=2
π∞/summationdisplay
n=1(−1)n−1nν+1sin(nα)Kν(nβ)
[|Reα|<π , Reν>−1]
ET II 41(3), WA 469(12)
6.667
1.3/integraldisplaya
0cosh/parenleftbig√
a2−x2/parenrightbig
sinhtI2ν(x)√
a2−x2dx=π
2Iν/parenleftbigg1
2aet/parenrightbigg
Iν/parenleftbigg1
2ae−t/parenrightbigg
/bracketleftbig
Reν>−1
2/bracketrightbig
ET II 365(10)
2./integraldisplaya
0cosh/parenleftbig√
a2−x2sinht/parenrightbig
K2ν(x)√
a2−x2dx=π2
4cosec( νπ)/bracketleftbig
I−ν/parenleftbig
aet/parenrightbig
I−ν/parenleftbig
ae−t/parenrightbig
−Iν/parenleftbig
aet/parenrightbig
Iν/parenleftbig
ae−t/parenrightbig/bracketrightbig
/bracketleftbig
|Reν|<1
2/bracketrightbig
ET II 367(25)
Exponential, hyperbolic, and Bessel functions
6.668 Notation :
/lscript1=1
2/bracketleftBig/radicalbig
(b+c)2+a2−/radicalbig
(b−c)2+a2/bracketrightBig
,/lscript 2=1
2/bracketleftBig/radicalbig
(b+c)2+a2+/radicalbig
(b−c)2+a2/bracketrightBig
716 Bessel Functions 6.669
1.10/integraldisplay∞
0e−αxsinh(βx)J0(γx)dx=(αβ)1
2r−1
1r−1
2(r2−r1)1
2(r2+r1)−1
2
r1=/radicalbig
γ2+(β−α)2,r 2=/radicalbig
γ2+(β+α)2,[Reα>|Reβ|,γ > 0]ET II 12(52)
alternatively, with a=γ,b=β,c=α,/integraldisplay∞
0e−cxsinh(bx)J0(ax)dx=/lscript1
/lscript2
2−/lscript2
1
[Rec>|Reb|,a > 0]
2.10/integraldisplay∞
0e−αxcosh(βx)J0(γx)dx=(αβ)1
2r−1
1r−1
2(r2−r1)1
2(r2+r1)−1
2
r1=/radicalbig
γ2+(β−α)2,r 2=/radicalbig
γ2+(β+α)2,[Reα>|Reβ|,γ > 0]ET II 12(54)
alternatively, with a=γ,b=β,c=α,/integraldisplay∞
0e−cxcosh(bx)J0(ax)dx=/lscript2
/lscript2
2−/lscript2
1
[Rec>|Reb|,a > 0]
6.669
1./integraldisplay∞
0/bracketleftbigg
coth/parenleftbigg1
2x/parenrightbigg/bracketrightbigg2λ
e−βcoshxJ2μ(αsinhx)dx=Γ/parenleftbig1
2−λ+μ/parenrightbig
αΓ(2μ+1 )M−λ,μ/bracketleftBig/parenleftbig
α2+β2/parenrightbig1
2−β/bracketrightBig
×Wλ,μ/bracketleftBig/parenleftbig
α2+β2/parenrightbig1
2+β/bracketrightBig
/bracketleftbig
Reβ>|Reα|,Re(μ−λ)>−1
2/bracketrightbig
BU 86(5b)a, ET II 363(34)
2./integraldisplay∞
0/bracketleftbigg
coth/parenleftbigg1
2x/parenrightbigg/bracketrightbigg2λ
e−βcoshxY2μ(αsinhx)dx
=−sec[(μ+λ)π]
αWλ,μ/parenleftBig/radicalbig
α2+β2+β/parenrightBig
W−λ,μ/parenleftBig/radicalbig
α2+β2−β/parenrightBig
−tan[(μ+λ)π]Γ/parenleftbig1
2−λ+μ/parenrightbig
αΓ(2μ+1 )Wλ,μ/parenleftBig/radicalbig
α2+β2+β/parenrightBig
M−λ,μ/parenleftBig/radicalbig
α2+β2−β/parenrightBig
/bracketleftbig
Reβ>|Reα|,Reλ<1
2−|Reμ|/bracketrightbig
ET II 363(35)
3./integraldisplay∞
0e−1
2(a1a2)tcoshx/bracketleftbigg
coth/parenleftbigg1
2x/parenrightbigg/bracketrightbigg2ν
K2μ(t√a1a2sinhx)dx
=Γ/parenleftbig1
2+μ−ν/parenrightbig
Γ/parenleftbig1
2−μ−ν/parenrightbig
2t√a1a2Wν,μ(a1t)Wν,μ(a2t)
/bracketleftbigg
Reν<Re1±2μ
2,Re/bracketleftBig
t(√a1+√a2)2/bracketrightBig
>0/bracketrightbigg
BU 85(4a)
4./integraldisplay∞
0e−1
2(a1a2)tcoshx/bracketleftBig
coth/parenleftBigx
2/parenrightBig/bracketrightBig2ν
I2μ(t√a1a2sinhx)dx=Γ/parenleftbig1
2+μ−ν/parenrightbig
t√a1a2Γ(1 + 2 μ)Wν,μ(a1t)Mν,μ(a2t)
/bracketleftbig
Re/parenleftbig1
2+μ−ν/parenrightbig
>0,Reμ>0,a1>a2/bracketrightbig
BU 86(5c)
5./integraldisplay∞
−∞e2νs−x−y
2tanhsI2μ/parenleftbigg√xy
coshs/parenrightbiggds
coshs=Γ/parenleftbig1
2+μ+ν/parenrightbig
Γ/parenleftbig1
2+μ−ν/parenrightbig
√xy[Γ(1 + 2 μ)]2Mν,μ(x)M−ν,μ(y)
/bracketleftbig
Re/parenleftbig
±ν+1
2+μ/parenrightbig
>0/bracketrightbig
BU 83(3a)a
6.671 Bessel and trigonometric functions 717
6./integraldisplay∞
−∞e2νs−x+y
2tanhsJ2μ/parenleftbigg√xy
coshs/parenrightbiggds
coshs=Γ/parenleftbig1
2+μ+ν/parenrightbig
Γ/parenleftbig1
2+μ−ν/parenrightbig
√xy[Γ(1 + 2 μ)]2Mν,μ(x)Mν,μ(y)
/bracketleftbig
Re/parenleftbig
∓ν+1
2+μ/parenrightbig
>0/bracketrightbig
BU 84(3b)a
6.67–6.68 Combinations of Bessel and trigonometric functions
6.671
1./integraldisplay∞
0Jν(αx)sinβxdx =sin/parenleftBig
νarcsinβ
α/parenrightBig
/radicalbig
α2−β2[β<α ]
=∞or 0 [ β=α]
=ανcosνπ
2/radicalbig
β2−α2/parenleftBig
β+/radicalbig
β2−α2/parenrightBigν[β>α ]
[Reν>−2] WA 444(4)
2./integraldisplay∞
0Jν(αx)cosβxdx =cos/parenleftBig
νarcsinβ
α/parenrightBig
/radicalbig
α2−β2[β<α ]
=∞or 0 [ β=α]
=−ανsinνπ
2/radicalbig
β2−α2/parenleftBig
β+/radicalbig
β2−α2/parenrightBigν[β>α ]
[Reν>−1] WA 444(5)
3./integraldisplay∞
0Yν(ax)sin(bx)dx
=c o t/parenleftBigνπ
2/parenrightBig/parenleftbig
a2−b2/parenrightbig−1
2sin/bracketleftbigg
νarcsin/parenleftbiggb
a/parenrightbigg/bracketrightbigg
[0<b<a , |Reν|<2]
=1
2cosec/parenleftBigνπ
2/parenrightBig/parenleftbig
b2−a2/parenrightbig−1
2
×/braceleftbigg
a−νcos(νπ)/bracketleftBig
b−/parenleftbig
b2−a2/parenrightbig1
2/bracketrightBigν
−aν/bracketleftBig
b−/parenleftbig
b2−a2/parenrightbig1
2/bracketrightBig−ν/bracerightbigg
[0<a<b , |Reν|<2]
ET I 103(33)
4./integraldisplay∞
0Yν(ax)cos(bx)dx
=tan/parenleftbigνπ
2/parenrightbig
(a2−b2)1
2cos/bracketleftbigg
νarcsin/parenleftbiggb
a/parenrightbigg/bracketrightbigg
[0<b<a , |Reν|<1]
=−sin/parenleftBigνπ
2/parenrightBig/parenleftbig
b2−a2/parenrightbig−1
2/braceleftbigg
a−ν/bracketleftBig
b−/parenleftbig
b2−a2/parenrightbig1
2/bracketrightBigν
+c o t ( νπ)
+aν/bracketleftBig
b−/parenleftbig
b2−a2/parenrightbig1
2/bracketrightBig−ν
cosec( νπ)/bracerightbigg
[0<a<b , |Reν|<1]
ET I 47(29)
718 Bessel Functions 6.671
5./integraldisplay∞
0Kν(ax)sin(bx)dx
=1
4πa−νcosec/parenleftBigνπ
2/parenrightBig/parenleftbig
a2+b2/parenrightbig−1
2/braceleftbigg/bracketleftBig/parenleftbig
b2+a2/parenrightbig1
2+b/bracketrightBigν
−/bracketleftBig/parenleftbig
b2+a2/parenrightbig1
2−b/bracketrightBigν/bracerightbigg
[Rea>0,b > 0,|Reν|<2,ν/negationslash=0 ] ET I 105(48)
6./integraldisplay∞
0Kν(ax)cos(bx)dx
=π
4/parenleftbig
b2+a2/parenrightbig−1
2sec/parenleftBigνπ
2/parenrightBig/braceleftbigg
a−ν/bracketleftBig
b+/parenleftbig
b2+a2/parenrightbig1
2/bracketrightBigν
+aν/bracketleftBig
b+/parenleftbig
b2+a2/parenrightbig1
2/bracketrightBig−ν/bracerightbigg
[Rea>0,b>0,|Reν|<1]ET I 49(40)
7./integraldisplay∞
0J0(ax)sin(bx)dx=0 [ 0 <b<a ]
=1√
b2−a2[0<a<b ]
ET I 99(1)
8./integraldisplay∞
0J0(ax)cos(bx)dx=1√
a2−b2[0<b<a ]
=∞ [a=b]
=0 [ 0 <a<b ]
ET I 43(1)
9./integraldisplay∞
0J2n+1(ax)sin(bx)dx=(−1)n1√
a2−b2T2n+1/parenleftbiggb
a/parenrightbigg
[0<b<a ]
=0 [ 0 <a<b ]
ET I 99(2)
10./integraldisplay∞
0J2n(ax)cos(bx)dx=(−1)n1√
a2−b2T2n/parenleftbiggb
a/parenrightbigg
[0<b<a ]
=0 [ 0 <a<b ]
ET I 43(2)
11./integraldisplay∞
0Y0(ax)sin(bx)dx=2arc s in/parenleftbigb
a/parenrightbig
π√
a2−b2[0<b<a ]
=2
π1√
b2−a2ln/bracketleftBigg
b
a−/radicalbigg
b2
a2−1/bracketrightBigg
[0<a<b ]
ET I 103(31)
12./integraldisplay∞
0Y0(ax)cos(bx)dx=0 [ 0 <b<a ]
=−1√
b2−a2[0<a<b ]
ET I 47(28)
6.672 Bessel and trigonometric functions 719
13./integraldisplay∞
0K0(βx)sinαxdx =1/radicalbig
α2+β2ln/parenleftBigg
α
β+/radicalBigg
α2
β2+1/parenrightBigg
[α>0,β > 0] WA 425(11)a, MO 48
14.8/integraldisplay∞
0K0(βx)cosαxdx =π
2/radicalbig
α2+β2[α>0] WA 425(10)a, MO 48
6.672
1./integraldisplay∞
0Jν(ax)Jν(bx)sin(cx)dx
=0 [ R e ν>−1,0<c<b −a,0<a<b ]
=1
2√
abPν−1
2/parenleftbiggb2+a2−c2
2ab/parenrightbigg
[Reν>−1,b−a<c<b +a,0<a<b ]
=−cos(νπ)
π√
abQν−1
2/parenleftbigg
−b2+a2−c2
2ab/parenrightbigg
[Reν>−1,b+a<c , 0<a<b ]
ET I 102(27)
2./integraldisplay∞
0Jν(x)J−ν(x)cos(bx)dx=1
2Pν−1
2/parenleftbigg1
2b2−1/parenrightbigg
[0<b< 2]
=0 [ 2 <b]
ET I 46(21)
3./integraldisplay∞
0Kν(ax)Kν(bx)cos(cx)dx=π2
4√
absec(νπ)Pν−1
2/bracketleftbig/parenleftbig
a2+b2+c2/parenrightbig
(2ab)−1/bracketrightbig
/bracketleftbig
Re(a+b)>0,c > 0,|Reν|<1
2/bracketrightbig
ET I 50(51)
4./integraldisplay∞
0Kν(ax)Iν(bx)cos(cx)dx=1
2√
abQν−1
2/parenleftbigga2+b2+c2
2ab/parenrightbigg
/bracketleftbig
Rea>|Reb|,c > 0,Reν>−1
2/bracketrightbig
ET I 49(47)
5./integraldisplay∞
0sin(2ax)[Jν(x)]2dx=1
2Pν−1
2/parenleftbig
1−2a2/parenrightbig
[0<a< 1,Reν>−1]
=1
πcos(νπ)Qν−1
2/parenleftbig
2a2−1/parenrightbig
[a>1,Reν>−1]
ET II 343(30)
6./integraldisplay∞
0cos(2ax)[Jν(x)]2dx=1
πQν−1
2/parenleftbig
1−2a2/parenrightbig/bracketleftbig
0<a< 1,Reν>−1
2/bracketrightbig
=−1
πsin(νπ)Qν−1
2/parenleftbig
2a2−1/parenrightbig/bracketleftbig
a>1,Reν>−1
2/bracketrightbig
ET II 344(32)
7./integraldisplay∞
0sin(2ax)J0(x)Y0(x)dx=0 [ 0 <a< 1]
=−K/bracketleftBig/parenleftbig
1−a−2/parenrightbig1
2/bracketrightBig
πa[a>1]
ET II 348(60)
720 Bessel Functions 6.673
8./integraldisplay∞
0K0(ax)I0(bx)cos(cx)dx=1/radicalbig
c2+(a+b)2K/braceleftBigg
2√
ab/radicalbig
c2+(a+b)2/bracerightBigg
[Rea>|Reb|,c > 0] ET I 49(46)
9./integraldisplay∞
0cos(2ax)J0(x)Y0(x)dx=−1
πK(a)[ 0 <a< 1]
=−1
πaK/parenleftbigg1
a/parenrightbigg
[a>1]
ET II 348(61)
10./integraldisplay∞
0cos(2ax)[Y0(x)]2dx=1
πK/parenleftBig/radicalbig
1−a2/parenrightBig
[0<a< 1]
=2
πaK/parenleftBigg/radicalbigg
1−1
a2/parenrightBigg
[a>1]
ET II 348(62)
6.673
1./integraldisplay∞
0/bracketleftBig
Jν(ax)cos/parenleftBigνπ
2/parenrightBig
−Yν(ax)sin/parenleftBigνπ
2/parenrightBig/bracketrightBig
sin(bx)dx
=0 [ 0 <b<a , |Reν|<2]
=1
2aν√
b2−a2/braceleftbigg/bracketleftBig
b+/parenleftbig
b2−a2/parenrightbig1
2/bracketrightBigν
+/bracketleftBig
b−/parenleftbig
b2−a2/parenrightbig1
2/bracketrightBigν/bracerightbigg
[0<a<b , |Reν|<2]
ET I 104(39)
2./integraldisplay∞
0/bracketleftBig
Yν(ax)cos/parenleftBigνπ
2/parenrightBig
+Jν(ax)sin/parenleftBigνπ
2/parenrightBig/bracketrightBig
cos(bx)dx
=0 [ 0 <b<a , |Reν|<1]
=−1
2aν√
b2−a2/braceleftbigg/bracketleftBig
b+/parenleftbig
b2−a2/parenrightbig1
2/bracketrightBigν
+/bracketleftBig
b−/parenleftbig
b2−a2/parenrightbig1
2/bracketrightBigν/bracerightbigg
[0<a<b , |Reν|<1]
ET I 48(32)
3.∗/integraldisplayπ/2
0[cosxI0(acosx)+I1(acosx)]dx=ea−1
a
6.674
1./integraldisplaya
0sin(a−x)Jν(x)dx=aJν+1(a)−2ν∞/summationdisplay
n=0(−1)nJν+2n+2(a)
[Reν>−1] ET II 334(12)
2./integraldisplaya
0cos(a−x)Jν(x)dx=aJν(a)−2ν∞/summationdisplay
n=0(−1)nJν+2n+1(a)
[Reν>−1] ET II 336(23)
3./integraldisplaya
0sin(a−x)J2n(x)dx=aJ2n+1(a)+(−1)n2n/bracketleftBigg
cosa−J0(a)−2n/summationdisplay
m=1(−1)mJ2m(a)/bracketrightBigg
[n=0,1,2,...] ET II 334(10)
6.676 Bessel and trigonometric functions 721
4./integraldisplaya
0cos(a−x)J2n(x)dx=aJ2n(a)−(−1)n2n/bracketleftBigg
sina−2n−1/summationdisplay
m=0(−1)mJ2m+1(a)/bracketrightBigg
[n=0,1,2,...] ET II 335(21)
5./integraldisplaya
0sin(a−x)J2n+1(x)dx=aJ2n+2(a)+(−1)n(2n+1 )/bracketleftBigg
sina−2n/summationdisplay
m=0(−1)mJ2m+1(a)/bracketrightBigg
[n=0,1,2,...] ET II 334(11)
6./integraldisplaya
0cos(a−x)J2n+1(x)dx=aJ2n+1(a)+(−1)n(2n+1 )/bracketleftBigg
cosa−J0(a)−2n/summationdisplay
m=1(−1)mJ2m(a)/bracketrightBigg
[n=0,1,2,...] ET II 336(22)
7./integraldisplayz
0sin(z−x)J0(x)dx=zJ1(z) WA 415(2)
8./integraldisplayz
0cos(z−x)J0(x)dx=zJ0(z) WA 415(1)
6.675
1./integraldisplay∞
0Jν/parenleftbig
a√x/parenrightbig
sin(bx)dx=a√π
4b3
2/bracketleftbigg
cos/parenleftbigga2
8b−νπ
4/parenrightbigg
J1
2ν−1
2/parenleftbigga2
8b/parenrightbigg
−sin/parenleftbigga2
8b−νπ
4/parenrightbigg
J1
2ν+1
2/parenleftbigga2
8b/parenrightbigg/bracketrightbigg
[a>0,b > 0,Reν>−4]
ET I 110(23)
2./integraldisplay∞
0Jν/parenleftbig
a√x/parenrightbig
cos(bx)dx
=−a√π
4b3
2/bracketleftbigg
sin/parenleftbigga2
8b−νπ
4/parenrightbigg
J1
2ν−1
2/parenleftbigga2
8b/parenrightbigg
+c o s/parenleftbigga2
8b−νπ
4/parenrightbigg
J1
2ν+1
2/parenleftbigga2
8b/parenrightbigg/bracketrightbigg
[a>0,b > 0,Reν>−2]ET I 53(22)a
3./integraldisplay∞
0J0/parenleftbig
a√x/parenrightbig
sin(bx)dx=1
bcos/parenleftbigga2
4b/parenrightbigg
[a>0,b > 0] ET I 110(22)
4./integraldisplay∞
0J0/parenleftbig
a√x/parenrightbig
cos(bx)dx=1
bsin/parenleftbigga2
4b/parenrightbigg
[a>0,b > 0] ET I 53(21)
6.676
1./integraldisplay∞
0Jν/parenleftbig
a√x/parenrightbig
Jν/parenleftbig
b√x/parenrightbig
sin(cx)dx=1
cJν/parenleftbiggab
2c/parenrightbigg
cos/parenleftbigga2+b2
4c−νπ
2/parenrightbigg
[a>0,b > 0,c > 0,Reν>−2]
ET I 111(29)a
2./integraldisplay∞
0Jν/parenleftbig
a√x/parenrightbig
Jν/parenleftbig
b√x/parenrightbig
cos(cx)dx=1
cJν/parenleftbiggab
2c/parenrightbigg
sin/parenleftbigga2+b2
4c−νπ
2/parenrightbigg
[a>0,b > 0,c > 0,Reν>−1]
ET I 54(27)
3./integraldisplay∞
0J0/parenleftbig
a√x/parenrightbig
K0/parenleftbig
a√x/parenrightbig
sin(bx)dx=1
2bK0/parenleftbigga2
2b/parenrightbigg
[Rea>0,b > 0] ET I 111(31)
722 Bessel Functions 6.677
4./integraldisplay∞
0J0/parenleftbig√ax/parenrightbig
K0/parenleftbig√ax/parenrightbig
cos(bx)dx=π
4b/bracketleftBig
I0/parenleftBiga
2b/parenrightBig
−L0/parenleftBiga
2b/parenrightBig/bracketrightBig
[Rea>0,b > 0] ET I 54(29)
5./integraldisplay∞
0K0/parenleftbig√ax/parenrightbig
Y0/parenleftbig√ax/parenrightbig
cos(bx)dx=−1
2bK0/parenleftBiga
2b/parenrightBig/bracketleftbig
Re√a>0,b > 0/bracketrightbig
ET I 54(30)
6./integraldisplay∞
0K0/parenleftBig√axe1
4πi/parenrightBig
K0/parenleftBig√axe−1
4πi/parenrightBig
cos(bx)dx=π2
8b/bracketleftBig
H0/parenleftBiga
2b/parenrightBig
−Y0/parenleftBiga
2b/parenrightBig/bracketrightBig
[Rea>0,b>0] ET I 54(31)
6.677
1./integraldisplay∞
aJ0/parenleftBig
b/radicalbig
x2−a2/parenrightBig
sin(cx)dx=0 [ 0 <c<b ]
=cos/parenleftbig
a√
c2−b2/parenrightbig
√
c2−b2[0<b<c ]
ET I 113(47)
2./integraldisplay∞
aJ0/parenleftBig
b/radicalbig
x2−a2/parenrightBig
cos(cx)dx=exp/parenleftbig
−a√
b2−c2/parenrightbig
√
b2−c2[0<c<b ]
=−sin/parenleftbig
a√
c2−b2/parenrightbig
√
c2−b2[0<b<c ]
ET I 57(48)a
3.6/integraldisplay∞
0J0/parenleftBig
α/radicalbig
x2+z2/parenrightBig
cosβxdx =cosz/radicalbig
α2−β2
/radicalbig
α2−β2[0<β<α , z> 0]
=0 [ 0 <α<β , z> 0]
MO 47a
4./integraldisplay∞
0Y0/parenleftBig
α/radicalbig
x2+z2/parenrightBig
cosβxdx =1/radicalbig
α2−β2sin/parenleftBig
z/radicalbig
α2−β2/parenrightBig
[0<β<α , z> 0]
=−1/radicalbig
β2−α2exp/parenleftBig
−z/radicalbig
β2−α2/parenrightBig
[0<α<β , z> 0]
MO 47a
5./integraldisplay∞
0K0/bracketleftBig
α/radicalbig
x2+β2/bracketrightBig
cos(γx)dx=π
2/radicalbig
α2+γ2exp/parenleftBig
−β/radicalbig
α2+γ2/parenrightBig
[Reα>0,Reβ>0,γ > 0]
ET I 56(43)
6./integraldisplaya
0J0/parenleftBig
b/radicalbig
a2−x2/parenrightBig
cos(cx)dx=sin/parenleftbig
a√
b2+c2/parenrightbig
√
b2+c2[b>0] MO 48a, ET I 57(47)
7./integraldisplay∞
0J0/parenleftBig
b/radicalbig
x2−a2/parenrightBig
cos(cx)dx=cosh/parenleftbig
a√
b2−c2/parenrightbig
√
b2−c2[0<c<b , a> 0]
=0 [ 0 <b<c , a> 0]
ET I 57(49)
6.681 Bessel and trigonometric functions 723
8./integraldisplay∞
0H(1)
0/parenleftBig
α/radicalbig
β2−x2/parenrightBig
cos(γx)dx=−iexp/parenleftBig
iβ/radicalbig
α2+γ2/parenrightBig
/radicalbig
α2+γ2/bracketleftBig
π>arg/radicalbig
β2−x2≥0,α > 0,γ > 0/bracketrightBig
ET I 59(59)
9./integraldisplay∞
0H(2)
0/parenleftBig
α/radicalbig
β2−x2/parenrightBig
cos(γx)dx=iexp/parenleftBig
−iβ/radicalbig
α2+γ2/parenrightBig
/radicalbig
α2+γ2/bracketleftBig
−π<arg/radicalbig
β2−x2≤0,α > 0,γ > 0/bracketrightBig
ET I 58(58)
6.678/integraldisplay∞
0/bracketleftBig
K0/parenleftbig
2√x/parenrightbig
+π
2Y0/parenleftbig
2√x/parenrightbig/bracketrightBig
sin(bx)dx=π
2bsin/parenleftbigg1
b/parenrightbigg
[b>0] ET I 111(34)
6.679
1./integraldisplay∞
0J2ν/bracketleftBig
2bsinh/parenleftBigx
2/parenrightBig/bracketrightBig
sin(bx)dx=−i[Iν−ib(a)Kν+ib(a)−Iν+ib(a)Kν−ib(a)]
[a>0,b > 0,Reν>−1]
ET I 115(59)
2./integraldisplay∞
0J2ν/bracketleftBig
2asinh/parenleftBigx
2/parenrightBig/bracketrightBig
cos(bx)dx=Iν−ib(a)Kν+ib(a)+Iν+ib(a)Kν−ib(a)
/bracketleftbig
a>0,b > 0,Reν>−1
2/bracketrightbig
ET I 59(64)
3./integraldisplay∞
0J2ν/bracketleftBig
2acosh/parenleftBigx
2/parenrightBig/bracketrightBig
cos(bx)dx=−π
2[Jν+ib(a)Yν−ib(a)+Jν−ib(a)Yν+ib(a)] ET I 59(63)
4./integraldisplay∞
0J0/bracketleftBig
2asinh/parenleftBigx
2/parenrightBig/bracketrightBig
sin(bx)dx=2
πsinh(πb)[Kib(a)]2
[a>0,b > 0] ET I 115(58)
5./integraldisplay∞
0J0/bracketleftBig
2asinh/parenleftBigx
2/parenrightBig/bracketrightBig
cos(bx)dx=[Iib(a)+I−ib(a)]Kib(a)
[a>0,b > 0] ET I 59(62)
6./integraldisplay∞
0Y0/bracketleftBig
2asinh/parenleftBigx
2/parenrightBig/bracketrightBig
cos(bx)dx=−2
πcosh(πb)[Kib(a)]2
[a>0,b > 0] ET I 59(65)
7./integraldisplay∞
0K0/bracketleftBig
2asinh/parenleftBigx
2/parenrightBig/bracketrightBig
cos(bx)dx=π2
4/braceleftBig
[Jib(a)]2+[Yib(a)]2/bracerightBig
[Rea>0,b > 0] ET I 59(66)
6.681
1./integraldisplayπ
2
0cos(2μx)J2ν(2acosx)dx=π
2Jν+μ(a)Jν−μ(a)/bracketleftbig
Reν>−1
2/bracketrightbig
ET II 361(23)
2./integraldisplayπ
2
0cos(2μx)Y2ν(2acosx)dx=π
2[cot(2 νπ)Jν+μ(a)Jν−μ(a)−cosec(2 νπ)Jμ−ν(a)J−μ−ν(a)]
/bracketleftbig
|Reν|<1
2/bracketrightbig
ET II 361(24)
724 Bessel Functions 6.682
3./integraldisplayπ
2
0cos(2μx)I2ν(2acosx)dx=π
2Iν−μ(a)Iν+μ(a)/bracketleftbig
Reν>−1
2/bracketrightbig
ET I 59(61)
4./integraldisplayπ
2
0cos(νx)Kν(2acosx)dx=π
2I0(a)Kν(a)[ R e ν<1] WA 484(3)
5./integraldisplayπ
0J0(2zcosx)c o s2 nxdx =(−1)nπJ2
n(z). MO 45
6./integraldisplayπ
0J0(2zsinx)cos2 nxdx =πJ2
n(z). WA 43(3), MO 45
7./integraldisplayπ
2
0cos(2nπ)Y0(2asinx)dx=π
2Jn(a)Yn(a)[ n=0,1,2,...] ET II 360(16)
8./integraldisplayπ
0sin(2μx)J2ν(2asinx)dx=πsin(μπ)Jν−μ(a)Jν+μ(a)
[Reν>−1] ET II 360(13)
9./integraldisplayπ
0cos(2μx)J2ν(2asinx)dx=πcos(μπ)Jν−μ(a)Jν+μ(a)
/bracketleftbig
Reν>−1
2/bracketrightbig
ET II 360(14)
10./integraldisplayπ
2
0Jν+μ(2zcosx) cos[( ν−μ)x]dx=π
2Jν(z)Jμ(z)[ R e ( ν+μ)>−1] MO 42
11./integraldisplayπ
2
0cos[(μ−ν)x]Iμ+ν(2acosx)dx=π
2Iμ(a)Iν(a)[ R e ( μ+ν)>−1]
WA 484(2), ET II 378(39)
12./integraldisplayπ
2
0cos[(μ−ν)x]Kμ+ν(2acosx)dx=π2
4cosec[( μ+ν)π][I−μ(a)I−ν(a)−Iμ(a)Iν(a)]
[|Re(μ+ν)|<1] ET II 378(40)
13.8/integraldisplayπ
2
0Kν−m(2acosx)cos[( m+ν)x]dx=(−1)mπ
2Im(a)Kν(a)
[|Re(ν−m)|<1] WA 485(4)
6.682
1.7/integraldisplayπ
2
0Jν−1
2(xsint)sinν+1
2td t=/radicalbiggπ
2xJν(x)
[νmay be zero, a natural number, one half, or a natural number plus one half; x>0]MO 42a
2./integraldisplayπ
2
0Jν(zsinx)s i nνxcos2νxdx=2ν−1√πΓ/parenleftbigg
ν+1
2/parenrightbigg
z−νJ2
ν/parenleftBigz
2/parenrightBig
/bracketleftbig
Reν>−1
2/bracketrightbig
MO 42a
6.683 Bessel and trigonometric functions 725
6.683
1./integraldisplayπ
2
0Jν(zsinx)Iμ(zcosx)t a nν+1xdx=/parenleftBigz
2/parenrightBigν
Γ/parenleftbiggμ−ν
2/parenrightbigg
Γ/parenleftbiggμ+ν
2+1/parenrightbiggJμ(z)
[Reν>Reμ>−1] WA 407(4)
2./integraldisplayπ
2
0Jν(z1sinx)Jμ(z2cosx)sinν+1xcosμ+1xdx=zν
1zμ
2Jν+μ+1/parenleftBig/radicalbig
z2
1+z2
2/parenrightBig
/radicalBig
(z2
1+z2
2)ν+μ+1
[Reν>−1,Reμ>−1] WA 410(1)
3./integraldisplayπ
2
0Jν/parenleftbig
zcos2x/parenrightbig
Jμ/parenleftbig
zsin2x/parenrightbig
sinxcosxdx=1
z∞/summationdisplay
k=0(−1)kJν+μ+2k+1(z)
[Reν>−1,Reμ>−1] (see also 6.513 6)WA 414(1)
4./integraldisplayπ
2
0Jμ(zsinθ)( s i nθ)1−μ(cosθ)2ν+1dθ=sμ+ν,ν−μ+1(z)
2μ−1zν+1Γ(μ)
[Reν>−1] WA 407(2)
5./integraldisplayπ
2
0Jμ(zsinθ)( s i nθ)1−μdθ=Hμ−1
2(z)
/radicalbigg
2z
πWA 407(3)
6./integraldisplayπ
2
0Jμ(asinθ)( s i nθ)μ+1(cosθ)2/rho1+1dθ=2/rho1Γ(/rho1+1 )a−/rho1−1J/rho1+μ+1(a)
[Re/rho1>−1,Reμ>−1]
WA 406(1), EH II 46(5)
7./integraldisplayπ
2
0Jν(2zsinθ)(sinθ)ν(cosθ)2νdθ
=1
2∞/summationdisplay
m=0(−1)mzν+2mΓ/parenleftbig
ν+m+1
2/parenrightbig
Γ/parenleftbig
ν+1
2/parenrightbig
m!Γ (ν+m+1 )Γ ( 2 ν+m+1 )
=1
2z−ν√πΓ/parenleftbig
ν+1
2/parenrightbig
[Jν(z)]2/bracketleftbig
Reν>−1
2/bracketrightbig
EH II 47(10)
8./integraldisplayπ
2
0Jν(zsinθ)( s i nθ)ν+1(cosθ)−2νdθ=2−νzν−1
√πΓ/parenleftbigg1
2−ν/parenrightbigg
sinz
/bracketleftbig
−1<Reν<1
2/bracketrightbig
EH II 68(39)
9./integraldisplayπ
2
0Jν/parenleftbig
zsin2θ/parenrightbig
Jν/parenleftbig
zcos2θ/parenrightbig
(sinθ)2ν+1(cosθ)2ν+1dθ=Γ/parenleftbig1
2+ν/parenrightbig
J2ν+1
2(z)
22ν+3
2Γ(ν+1 )√z/bracketleftbig
Reν>−1
2/bracketrightbig
WA 409(1)
726 Bessel Functions 6.684
10./integraldisplayπ
2
0Jμ/parenleftbig
zsin2θ/parenrightbig
Jν/parenleftbig
zcos2θ/parenrightbig
sin2μ+1θcos2ν+1θd θ=Γ/parenleftbig
μ+1
2/parenrightbig
Γ/parenleftbig
ν+1
2/parenrightbig
Jμ+ν+1
2(z)
2√πΓ(μ+ν+1 )√
2z/bracketleftbig
Reμ>−1
2,Reν>−1
2/bracketrightbig
WA 417(1)
6.684
1.8/integraldisplayπ
0(sinx)2νJν/parenleftBig/radicalbig
α2+β2−2αβcosx/parenrightBig
/parenleftBig/radicalbig
α2+β2−2αβcosx/parenrightBigνdx=2ν√πΓ/parenleftbigg
ν+1
2/parenrightbiggJν(α)
ανJν(β)
βν
/bracketleftbig
Reν>−1
2/bracketrightbig
ET II 362(27)
2./integraldisplayπ
0(sinx)2νYν/parenleftBig/radicalbig
α2+β2−2αβcosx/parenrightBig
/parenleftBig/radicalbig
α2+β2−2αβcosx/parenrightBigνdx=2ν√πΓ/parenleftbigg
ν+1
2/parenrightbiggJν(α)
ανYν(β)
βν
/bracketleftbig
|α|<|β|,Reν>−1
2/bracketrightbig
ET II 362(28)
6.685/integraldisplayπ
2
0secxcos(2λx)K2μ(asecx)dx=π
2aWλ,μ(a)W−λ,μ(a)[ R e a>0] ET II 378(41)
6.686
1./integraldisplay∞
0sin/parenleftbig
ax2/parenrightbig
Jν(bx)dx=−√π
2√asin/parenleftbiggb2
8a−ν+1
4π/parenrightbigg
J1
2ν/parenleftbiggb2
8a/parenrightbigg
[a>0,b>0,Reν>−3] ET II 34(13)
2./integraldisplay∞
0cos/parenleftbig
ax2/parenrightbig
Jν(bx)dx=√π
2√acos/parenleftbiggb2
8a−ν+1
4π/parenrightbigg
J1
2ν/parenleftbiggb2
8a/parenrightbigg
[a>0,b > 0,Reν>−1]
ET II 38(38)
3./integraldisplay∞
0sin/parenleftbig
ax2/parenrightbig
Yν(bx)dx
=−√π
4√asec/parenleftBigνπ
2/parenrightBig
×/bracketleftbigg
cos/parenleftbiggb2
8a−3ν+1
4π/parenrightbigg
J1
2ν/parenleftbiggb2
8a/parenrightbigg
−sin/parenleftbiggb2
8a+ν−1
4π/parenrightbigg
Y1
2ν/parenleftbiggb2
8a/parenrightbigg/bracketrightbigg
[a>0,b > 0,−3<Reν<3]ET II 107(7)
4./integraldisplay∞
0cos/parenleftbig
ax2/parenrightbig
Yν(bx)dx
=√π
4√asec/parenleftBigνπ
2/parenrightBig
×/bracketleftbigg
sin/parenleftbiggb2
8a−3ν+1
4π/parenrightbigg
J1
2ν/parenleftbiggb2
8a/parenrightbigg
+c o s/parenleftbiggb2
8a+ν−1
4π/parenrightbigg
Y1
2ν/parenleftbiggb2
8a/parenrightbigg/bracketrightbigg
[a>0,b > 0,−1<Reν<1]ET II 107(8)
5./integraldisplay∞
0sin/parenleftbig
ax2/parenrightbig
J1(bx)dx=1
bsinb2
4a[a>0,b > 0] ET II 19(16)
6.693 Bessel and trigonometric functions and powers 727
6./integraldisplay∞
0cos/parenleftbig
ax2/parenrightbig
J1(bx)dx=2
bsin2/parenleftbiggb2
8a/parenrightbigg
[a>0,b > 0] ET II 20(20)
7./integraldisplay∞
0sin2/parenleftbig
ax2/parenrightbig
J1(bx)dx=1
2bcos/parenleftbiggb2
8a/parenrightbigg
[a>0,b > 0] ET II 19(17)
6.687/integraldisplay∞
0cos/parenleftbiggx2
2a/parenrightbigg
K2ν/parenleftbig
xeiπ
4/parenrightbig
K2ν/parenleftbig
xe−iπ
4/parenrightbig
dx
=Γ/parenleftbig1
4+ν/parenrightbig
Γ/parenleftbig1
4−ν/parenrightbig√π
8√aW1
4,ν/parenleftbig
aeiπ
2/parenrightbig
W1
4,ν/parenleftbig
ae−iπ
2/parenrightbig
/bracketleftbig
a>0,|Reν|<1
4/bracketrightbig
ET II 372(1)
6.688
1./integraldisplayπ
2
0Jν(μzsint)cos(μxcost)dt=π
2Jν
2/parenleftBigg
μ√
x2+z2+x
2/parenrightBigg
Jν
2/parenleftBigg
μ√
x2+z2−x
2/parenrightBigg
[Reν>−1,Rez>0] MO 46
2./integraldisplayπ
2
0(sinx)ν+1cos(βcosx)Jν(αsinx)dx=2−1
2√παν/parenleftbig
α2+β2/parenrightbig−1
2ν−1
4Jν+1
2/bracketleftBig/parenleftbig
α2+β2/parenrightbig1
2/bracketrightBig
[Reν>−1] ET II 361(19)
3./integraldisplayπ
2
0cos[(z−ζ)cosθ]J2ν/bracketleftBig
2/radicalbig
zζsinθ/bracketrightBig
dθ=π
2Jν(z)Jν(ζ)
/bracketleftbig
Reν>−1
2/bracketrightbig
EH II 47(8)
6.69–6.74 Combinations of Bessel and trigonometric functions and powers
6.691/integraldisplay∞
0xsin(bx)K0(ax)dx=πb
2/parenleftbig
a2+b2/parenrightbig−3
2[Rea>0,b > 0] ET I 105(47)
6.692
1./integraldisplay∞
0xKν(ax)Iν(bx)sin(cx)dx=−1
2(ab)−3
2c/parenleftbig
u2−1/parenrightbig−1
2Q1
ν−1
2(u),u =( 2ab)−1/parenleftbig
a2+b2+c2/parenrightbig
/bracketleftbig
Rea>|Reb|,c > 0,Reν>−3
2/bracketrightbig
ET I 106(54)
2./integraldisplay∞
0xKν(ax)Kν(bx)sin(cx)dx=π
4(ab)−3
2c/parenleftbig
u2−1/parenrightbig−1
2Γ/parenleftbig3
2+ν/parenrightbig
Γ/parenleftbig3
2−ν/parenrightbig
P−1
ν−1
2(u)
u=( 2ab)−1/parenleftbig
a2+b2+c2/parenrightbig/bracketleftbig
Re(a+b)>0,c > 0,|Reν|<3
2/bracketrightbig
ET I 107(61)
6.693
1./integraldisplay∞
0Jν(αx)sinβxdx
x=1
νsin/parenleftbigg
νarcsinβ
α/parenrightbigg
[β≤α]
=ανsinνπ
2
ν/parenleftBig
β+/radicalbig
β2−α2/parenrightBigν [β≥α]
[Reν>−1] WA 443(2)
728 Bessel Functions 6.693
2.8/integraldisplay∞
0Jν(αx)cosβxdx
x=1
νcos/parenleftbigg
νarcsinβ
α/parenrightbigg
[β≤α]
=ανcosνπ
2
ν/parenleftBig
β+/radicalbig
β2−α2/parenrightBigν [β≥α][ R e ν>0]
WA 443(3)
3./integraldisplay∞
0Yν(ax)sin(bx)dx
x
=−1
νtan/parenleftBigνπ
2/parenrightBig
sin/bracketleftbigg
νarcsin/parenleftbiggb
a/parenrightbigg/bracketrightbigg
[0<b<a , |Reν|<1]
=1
2νsec/parenleftBigνπ
2/parenrightBig/braceleftbigg
a−νcos(νπ)/bracketleftBig
b−/parenleftbig
b2−a2/parenrightbig1
2/bracketrightBigν
−aν/bracketleftBig
b−/parenleftbig
b2−a2/parenrightbig1
2/bracketrightBig−ν/bracerightbigg
[0<a<b , |Reν|<1]
ET I 103(35)
4./integraldisplay∞
0Jν(ax)sin(bx)dx
x2
=√
a2−b2sin/bracketleftbig
νarcsin/parenleftbigb
a/parenrightbig/bracketrightbig
ν2−1−bcos/bracketleftbig
νarcsin/parenleftbigb
a/parenrightbig/bracketrightbig
ν(ν2−1)[0<b<a , Reν>0]
=−aνcos/parenleftbigνπ
2/parenrightbig/bracketleftbig
b+ν√
b2−a2/bracketrightbig
ν(ν2−1)/bracketleftbig
b+√
b2−a2/bracketrightbigν [0<a<b , Reν>0]
ET I 99(6)
5./integraldisplay∞
0Jν(ax)cos(bx)dx
x2
=acos/bracketleftbig
(ν−1)arcsin/parenleftbigb
a/parenrightbig/bracketrightbig
2ν(ν−1)+acos/bracketleftbig
(ν+1 )a r c s i n/parenleftbigb
a/parenrightbig/bracketrightbig
2ν(ν+1 )[0<b<a , Reν>1]
=aνsin/parenleftbigνπ
2/parenrightbig
2ν(ν−1)/bracketleftbig
b+√
b2−a2/bracketrightbigν−1−aν+2sin/parenleftbigνπ
2/parenrightbig
2ν(ν+1 )/bracketleftbig
b+√
b2−a2/bracketrightbigν+1[0<a<b , Reν>1]
ET I 44(6)
6./integraldisplay∞
0J0(αx)sinxdx
x=π
2[0<α< 1]
= arccosec α [α>1]
WH
7./integraldisplay∞
0J0(x)sinβxdx
x=π
2[β>1]
=a r c s i n β/bracketleftbig
β2<1/bracketrightbig
=−π
2[β<−1]
8./integraldisplay∞
0[J0(x)−cosαx]dx
x=l n2 α NT 66(13)
9./integraldisplayz
0Jν(x)sin(z−x)dx
x=2
ν∞/summationdisplay
k=0(−1)kJν+2k+1(z)[ R e ν>0] WA 416(4)
6.697 Bessel and trigonometric functions and powers 729
10./integraldisplayz
0Jν(x)cos(z−x)dx
x=1
νJν(z)+2
ν∞/summationdisplay
k=1(−1)kJν+2k(z)
[Reν>0] WA 416(5)
6.69410/integraldisplay∞
0/bracketleftbiggJ1(ax)
x/bracketrightbigg2
sin(bx)dx
=1
2b−/parenleftbigg4a
3π/parenrightbigg/bracketleftbigg/parenleftbigg
1+b2
4a2/parenrightbigg
E/parenleftbiggb
2a/parenrightbigg
+/parenleftbigg
1−b2
4a2/parenrightbigg
K/parenleftbiggb
2a/parenrightbigg/bracketrightbigg
[0≤b≤2a]ET I 102(22)
=1
2b−2b
3π/bracketleftBigg/parenleftbigg
1+b2
4a2/parenrightbigg
E/parenleftbigg2a
b/parenrightbigg
−/parenleftBigg
1−/parenleftbigg4a2
b2/parenrightbigg−1/parenrightBigg
K/parenleftbigg2a
b/parenrightbigg/bracketrightBigg
[0≤2a≤b]
6.695
1./integraldisplay∞
0sinαx
β2+x2J0(ux)dx=sinhαβ
βK0(βu)[ α>0,Reβ>0,u > α ] MO 46
2./integraldisplay∞
0cosαx
β2+x2J0(ux)dx=π
2e−αβ
βI0(βu)[ α>0,Reβ>0,−α<u<α ]
MO 46
3./integraldisplay∞
0x
x2+β2sin(αx)J0(γx)dx=π
2e−αβI0(γβ)[ α>0,Reβ>0,0<γ<α ]
ET II 10(36)
4./integraldisplay∞
0x
x2+β2cos(αx)J0(γx)dx=c o s h ( αβ)K0(βγ)[ α>0,Reβ>0,α < γ ]
ET II 11(45)
6.696/integraldisplay∞
0[1−cos(αx)]J0(βx)dx
x= arccosh/parenleftbiggα
β/parenrightbigg
[0<β<α ]
=0 [ 0 <α<β ]
ET II 11(43)
6.697
1./integraldisplay∞
−∞sin[α(x+β)]
x+βJ0(x)dx=2/integraldisplayα
0cosβu√
1−u2du [0≤α≤1] WA 463(2)
=πJ0(β)[ 1 ≤α<∞] WA 463(1), ET II 345(42)
2./integraldisplay∞
0sin(x+t)
x+tJ0(t)dt=π
2J0(x)[ x>0] WA 475(4)
3./integraldisplay∞
0cos(x+t)
x+tJ0(t)dt=−π
2Y0(x)[ x>0] WA 475(5)
4./integraldisplay∞
−∞|x|
x+βsin[α(x+β)]J0(bx)dx=0 [ 0 ≤α<b] WA 464(5), ET II 345(43)a
5./integraldisplay∞
−∞sin[α(x+β)]
x+β/bracketleftBig
Jn+1
2(x)/bracketrightBig2
dx=π/bracketleftBig
Jn+1
2(β)/bracketrightBig2
[2≤α<∞,n=0,1,...]
ET II 346(45)
730 Bessel Functions 6.698
6./integraldisplay∞
−∞sin[α(x+β)]
x+βJn+1
2(x)J−n−1
2(x)dx=πJn+1
2(β)J−n−1
2(β)
[2≤α<∞,n=0,1,...]
ET II 346(46)
7./integraldisplay∞
−∞Jμ[a(z+x)]
(z+x)μJν[a(ζ+x)]
(ζ+x)νdx=Γ(μ+ν)√π/radicalBig
2
a
Γ/parenleftbig
μ+1
2/parenrightbig
Γ/parenleftbig
ν+1
2/parenrightbig·Jμ+ν−1
2[a(z−ζ)]
(z−ζ)μ+ν−1
2
[Re(μ+ν)>0] WA 463(3)
6.698
1./integraldisplay∞
0√xJν+1
4(ax)J−ν+1
4(ax)sin(bx)dx=/radicalbigg
2
πbcos/bracketleftbig
2νarccos/parenleftbigb
2a/parenrightbig/bracketrightbig
√
4a2−b2[0<b< 2a]
=0 [ 0 <2a<b]
ET I 102(26)
2./integraldisplay∞
0√xJν−1
4(ax)J−ν−1
4(ax)cos(bx)dx=/radicalbigg
2
πbcos/bracketleftbig
2νarccos/parenleftbigb
2a/parenrightbig/bracketrightbig
√
4a2−b2[0<b< 2a]
=0 [ 0 <2a<b]
ET I 46(24)
3./integraldisplay∞
0√xI1
4−ν/parenleftbigg1
2ax/parenrightbigg
K1
4+ν/parenleftbigg1
2ax/parenrightbigg
sin(bx)dx=/radicalbiggπ
2ba−2ν/parenleftbig
b+√
a2+b2/parenrightbig2ν
√
a2+b2/bracketleftbig
Rea>0,b > 0,Reν<5
4/bracketrightbig
ET I 106(56)
4./integraldisplay∞
0√xI−1
4−ν/parenleftbigg1
2ax/parenrightbigg
K−1
4+ν/parenleftbigg1
2ax/parenrightbigg
cos(bx)dx=/radicalbiggπ
2ba−2ν/parenleftbig
b+√
a2+b2/parenrightbig2ν
√
a2+b2/bracketleftbig
Rea>0,b > 0,Reν<3
4/bracketrightbig
ET I 50(49)
6.699
1./integraldisplay∞
0xλJν(ax)sin(bx)dx=21+λa−(2+λ)bΓ/parenleftbig2+λ+ν
2/parenrightbig
Γ/parenleftbigν−λ
2/parenrightbigF/parenleftbigg2+λ+ν
2,2+λ−ν
2;3
2;b2
a2/parenrightbigg
/bracketleftbig
0<b<a , −Reν−1<1+R e λ<3
2/bracketrightbig
=/parenleftbigg1
2a/parenrightbiggν
b−(ν+λ+1)Γ(ν+λ+1 )
Γ(ν+1 )sin/bracketleftbigg
π/parenleftbigg1+λ+ν
2/parenrightbigg/bracketrightbigg
×F/parenleftbigg2+λ+ν
2,1+λ+ν
2;ν+1 ;a2
b2/parenrightbigg
/bracketleftbig
0<a<b , −Reν−1<1+R e λ<3
2/bracketrightbig
ET I 100(11)
6.699 Bessel and trigonometric functions and powers 731
2./integraldisplay∞
0xλJν(ax)cos(bx)dx
=2λa−(1+λ)Γ/parenleftbig1+λ+ν
2/parenrightbig
Γ/parenleftbigν−λ+1
2/parenrightbig F/parenleftbigg1+λ+ν
2,1+λ−ν
2;1
2;b2
a2/parenrightbigg
/bracketleftbig
0<b<a , −Reν<1+R e λ<3
2/bracketrightbig
=/parenleftbiga
2/parenrightbigνb−(ν+1+λ)Γ( 1+ λ+ν)c o s/bracketleftbigπ
2(1 +λ+ν)/bracketrightbig
Γ(ν+1 )F/parenleftbigg1+λ+ν
2,2+λ+ν
2;ν+1 ;a2
b2/parenrightbigg
/bracketleftbig
0<a<b , −Reν<1+R e λ<3
2/bracketrightbig
ET I 45(13)
3./integraldisplay∞
0xλKμ(ax)sin(bx)dx=2λbΓ/parenleftBig
2+μ+λ
2/parenrightBig
Γ/parenleftBig
2+λ−μ
2/parenrightBig
a2+λF/parenleftbigg2+μ+λ
2,2+λ−μ
2;3
2;−b2
a2/parenrightbigg
[Re(−λ±μ)<2,Rea>0,b > 0]
ET I 106(50)
4./integraldisplay∞
0xλKμ(ax)cos(bx)dx=2λ−1a−λ−1Γ/parenleftbiggμ+λ+1
2/parenrightbigg
Γ/parenleftbigg1+λ−μ
2/parenrightbigg
×F/parenleftbiggμ+λ+1
2,1+λ−μ
2;1
2;−b2
a2/parenrightbigg
[Re (−λ±μ)<1,Rea>0,b > 0]ET I 49(42)
5./integraldisplay∞
0xνsin(ax)Jν(bx)dx=√π2νbν/parenleftbig
a2−b2/parenrightbig−ν−1
2
Γ/parenleftbig1
2−ν/parenrightbig/bracketleftbig
0<b<a , −1<Reν<1
2/bracketrightbig
=0/bracketleftbig
0<a<b , −1<Reν<1
2/bracketrightbig
ET II 32(4)
6./integraldisplay∞
0xνcos(ax)Jν(bx)dx=−2νsin(νπ)√πΓ/parenleftbigg1
2+ν/parenrightbigg
bν/parenleftbig
a2−b2/parenrightbig−ν−1
2/bracketleftbig
0<b<a , |Reν|<1
2/bracketrightbig
=2νbν
√πΓ/parenleftbigg1
2+ν/parenrightbigg/parenleftbig
b2−a2/parenrightbig−ν−1
2/bracketleftbig
0<a<b , |Reν|<1
2/bracketrightbig
ET II 36(29)
7./integraldisplay∞
0xν+1sin(ax)Jν(bx)dx
=−21+νasin(νπ)√πbνΓ/parenleftbigg
ν+3
2/parenrightbigg/parenleftbig
a2−b2/parenrightbig−ν−3
2/bracketleftbig
0<b<a , −3
2<Reν<−1
2/bracketrightbig
=−21+ν
√πabνΓ/parenleftbigg
ν+3
2/parenrightbigg/parenleftbig
b2−a2/parenrightbig−ν−3
2/bracketleftbig
0<a<b , −3
2<Reν<−1
2/bracketrightbig
ET II 32(3)
8./integraldisplay∞
0xν+1cos(ax)Jν(bx)dx=21+ν√πabν/parenleftbig
a2−b2/parenrightbig−ν−3
2
Γ/parenleftbig
−1
2−ν/parenrightbig/bracketleftbig
0<b<a , −1<Reν<−1
2/bracketrightbig
=0/bracketleftbig
0<a<b , −1<Reν<−1
2/bracketrightbig
ET II 36(28)
732 Bessel Functions 6.711
9./integraldisplay1
0xνsin(ax)Jν(ax)dx=1
2ν+1[sinaJν(a)−cosaJν+1(a)]
[Reν>−1] ET II 334(9)a
10./integraldisplay1
0xνcos(ax)Jν(ax)dx=1
2ν+1[cosaJν(a)+s i n aJν+1(a)]
/bracketleftbig
Reγ>−1
2/bracketrightbig
ET II 335(20)
11./integraldisplay∞
0x1+νKν(ax)sin(bx)dx=√π(2a)νΓ/parenleftbigg3
2+ν/parenrightbigg
b/parenleftbig
b2+a2/parenrightbig−3
2−ν
/bracketleftbig
Rea>0,b > 0,Reν>−3
2/bracketrightbig
ET I 105(49)
12./integraldisplay∞
0xμKμ(ax)cos(bx)dx=1
2√π(2a)μΓ/parenleftbigg
μ+1
2/parenrightbigg/parenleftbig
b2+a2/parenrightbig−μ−1
2
/bracketleftbig
Rea>0,b > 0,Reμ>−1
2/bracketrightbig
ET I 49(41)
13./integraldisplay∞
0xνYν−1(ax)sin(bx)dx=0/bracketleftbig
0<b<a , |Reν|<1
2/bracketrightbig
=2ν√πaν−1b
Γ/parenleftbig1
2−ν/parenrightbig/parenleftbig
b2−a2/parenrightbig−ν−1
2/bracketleftbig
0<a<b , |Reν|<1
2/bracketrightbig
ET I 104(36)
14./integraldisplay∞
0xνYν(ax)cos(bx)dx=0/bracketleftbig
0<b<a , |Reν|<1
2/bracketrightbig
=−2ν√πaν/parenleftbig
b2−a2/parenrightbig−ν−1
2
Γ/parenleftbig1
2−ν/parenrightbig/bracketleftbig
0<a<b , |Reν|<1
2/bracketrightbig
ET I 47(30)
6.711
1./integraldisplay∞
0xν−μJμ(ax)Jν(bx)sin(cx)dx=0 [ 0 <c<b −a,−1<Reν<1+R e μ]
ET I 103(28)
2./integraldisplay∞
0xν−μ+1Jμ(ax)Jν(bx)cos(cx)dx=0
[0<c<b −a, a > 0,b > 0,−1<Reν<Reμ]ET I 47(25)
3./integraldisplay∞
0xν−μ−2Jμ(ax)Jν(bx)sin(cx)dx=2ν−μ−1aμb−νcΓ(ν)
Γ(μ+1 )
[0<a , 0<b , 0<c<b −a,0<Reν<Reμ+3 ] ET I 103(29)
4./integraldisplay∞
0x/rho1−μ−1Jμ(ax)J/rho1(bx)cos(cx)dx=2/rho1−μ−1b−/rho1aμΓ(/rho1)
Γ(μ+1 )
[b>0,a > 0,0<c<b −a,0<Re/rho1<Reμ+2 ] ET I 47(26)
6.713 Bessel and trigonometric functions and powers 733
5./integraldisplay∞
0x1−2νsin(2ax)Jν(x)Yν(x)dx=−Γ/parenleftbig3
2−ν/parenrightbig
a
2Γ/parenleftbig
2ν−1
2/parenrightbig
Γ(2−ν)F/parenleftbigg3
2−ν,3
2−2ν;2−ν;a2/parenrightbigg
/bracketleftbig
0<Reν<3
2,0<a< 1/bracketrightbig
ET II 348(63)
6.10/integraldisplay∞
0arg sin( zx)xν−μ−4Jμ(ax)Jν(ρx)dx=zΓ(ν)aμρ−ν
2μ−ν+3Γ(μ+1 )/bracketleftbiggρ2
ν−1−a2
μ+1−2z2
3/bracketrightbigg
7.10/integraldisplay∞
0cos (zx)xν−μ−3Jμ(ax)Jν(ρx)dx=Γ(ν)aμρ−ν
2μ−ν+3Γ(μ+1 )/bracketleftbiggρ2
ν−1−a2
μ+1−2z2/bracketrightbigg
6.712
1./integraldisplay∞
0xν[Jν(ax)cos(ax)+Yν(ax)sin(ax)] sin(bx)dx=√π(2a)ν
Γ/parenleftbig1
2−ν/parenrightbig/parenleftbig
b2+2ab/parenrightbig−ν−1
2
/bracketleftbig
b>0,−1<Reν<1
2/bracketrightbig
ET I 104(40)
2./integraldisplay∞
0xν[Yν(ax)cos(ax)−Jν(ax)sin(ax)] cos( bx)dx=−√π(2a)ν
Γ/parenleftbig1
2−ν/parenrightbig/parenleftbig
b2+2ab/parenrightbig−ν−1
2
ET I 48(35)
3./integraldisplay∞
0xν[Jν(ax)cos(ax)−Yν(ax)sin(ax)] sin(bx)dx
=0/bracketleftbig
0<b< 2a,−1<Reν<1
2/bracketrightbig
=2ν√πbν
Γ/parenleftbig1
2−ν/parenrightbig/parenleftbig
b2−2ab/parenrightbig−ν−1
2/bracketleftbig
2a<b , −1<Reν<1
2/bracketrightbig
ET I 104(41)
4./integraldisplay∞
0xν[Jν(ax)sin(ax)+Yν(ax)cos(ax)] cos( bx)dx
=0/bracketleftbig
0<b< 2a,|Reν|<1
2/bracketrightbig
=−√π(2a)ν
Γ/parenleftbig1
2−ν/parenrightbig/parenleftbig
b2−2ab/parenrightbig−ν−1
2/bracketleftbig
0<2a<b , |Reν|<1
2/bracketrightbig
ET I 48(33)
6.713
1./integraldisplay∞
0x1−2νsin(2ax)/braceleftBig
[Jν(x)]2−[Yν(x)]2/bracerightBig
dx
=sin(2νπ)Γ/parenleftbig3
2−ν/parenrightbig
Γ/parenleftbig3
2−2ν/parenrightbig
a
πΓ(2−ν)F/parenleftbigg3
2−ν,3
2−2ν;2−ν;a2/parenrightbigg
/bracketleftbig
0<Reν<3
4,0<a< 1/bracketrightbig
ET II 348(64)
2./integraldisplay∞
0x2−2νsin(2ax)[Jν(x)Jν−1(x)−Yν(x)Yν−1(x)]dx
=−sin(2νπ)Γ/parenleftbig3
2−ν/parenrightbig
Γ/parenleftbig5
2−2ν/parenrightbig
a
πΓ(2−ν)F/parenleftbigg3
2−ν,5
2−2ν;2−ν;a2/parenrightbigg
/bracketleftbig1
2<Reν<5
4,0<a< 1/bracketrightbig
ET II 348(65)
734 Bessel Functions 6.714
3./integraldisplay∞
0x2−2νsin(2ax)[Jν(x)Yν−1(x)+Yν(x)Jν−1(x)]dx
=−Γ/parenleftbig3
2−ν/parenrightbig
a
Γ/parenleftbig
2ν−3
2/parenrightbig
Γ(2−ν)F/parenleftbigg3
2−ν,5
2−2ν;2−ν;a2/parenrightbigg
/bracketleftbig1
2<Reν<5
2,0<a< 1/bracketrightbig
ET II 349(66)
6.714
1./integraldisplay∞
0sin(2ax)[xνJν(x)]2dx
=a−2νΓ/parenleftbig1
2+ν/parenrightbig
2√πΓ(1−ν)F/parenleftbigg1
2+ν,1
2;1−ν;a2/parenrightbigg/bracketleftbig
0<a< 1,|Reν|<1
2/bracketrightbig
=a−4ν−1Γ/parenleftbig1
2+ν/parenrightbig
2Γ( 1+ ν)Γ/parenleftbig1
2−2ν/parenrightbigF/parenleftbigg1
2+ν,1
2+2ν;1+ν;1
a2/parenrightbigg/bracketleftbig
a>1,|Reν|<1
2/bracketrightbig
ET II 343(31)
2./integraldisplay∞
0cos(2ax)[xνJν(x)]2dx
=a−2νΓ(ν)
2√πΓ/parenleftbig1
2−ν/parenrightbigF/parenleftbigg
ν+1
2,1
2;1−ν;a2/parenrightbigg
+Γ(−ν)Γ/parenleftbig1
2+2ν/parenrightbig
2πΓ/parenleftbig1
2−ν/parenrightbigF/parenleftbigg1
2+ν,1
2+2ν;1+ν;a2/parenrightbigg/bracketleftbig
0<a< 1,−1
4<Reν<1
2/bracketrightbig
=−sin(νπ)a−4ν−1Γ/parenleftbig1
2+2ν/parenrightbig
Γ(1 + ν)Γ/parenleftbig1
2−ν/parenrightbigF/parenleftbigg1
2+ν,1
2+2ν;1+ν;1
a2/parenrightbigg/bracketleftbig
a>1,−1
4<Reν<1
2/bracketrightbig
ET II 344(33)
6.715
1./integraldisplay∞
0xν
x+βsin(x+β)Jν(x)dx=π
2sec(νπ)βνJ−ν(β)
/bracketleftbig
|argβ|<π , |Reν|<1
2/bracketrightbig
ET II 340(8)
2./integraldisplay∞
0xν
x+βcos(x+β)Jν(x)dx=−π
2sec(νπ)βνY−ν(β)
/bracketleftbig
|argβ|<π , |Reν|<1
2/bracketrightbig
ET II 340(9)
6.716
1./integraldisplaya
0xλsin(a−x)Jν(x)dx=2aλ+1∞/summationdisplay
n=0(−1)nΓ(ν−λ+2n)Γ(ν+λ+1 )
Γ(ν−λ)Γ(ν+λ+3+2 n)(ν+2n+1 )Jν+2n+1(a)
[Re(λ+ν)>−1] ET II 335(16)
2./integraldisplaya
0xλcos(a−x)Jν(x)dx=aλ+1Jν(a)
λ+ν+1+2aλ+1
×∞/summationdisplay
n=1(−1)nΓ(ν−λ+2n−1)Γ(ν+λ+1 )
Γ(ν−λ)Γ(ν+λ+2n+2 )(ν+2n)Jν+2n(a)
[Re(λ+ν)>−1] ET II 335(26)
6.721 Bessel and trigonometric functions and powers 735
6.717/integraldisplay∞
−∞sin[a(x+β)]
xν(x+β)Jν+2n(x)dx=πβ−νJν+2n(β)
/bracketleftbig
1≤a<∞,n=0,1,2,...;R e ν>−3
2/bracketrightbig
ET II 345(44)
6.718
1./integraldisplay∞
0xν
x2+β2sin(αx)Jν(γx)dx=βν−1sinh(αβ)Kν(βγ)
/bracketleftbig
0<α≤γ,Reβ>0,−1<Reν<3
2/bracketrightbig
ET II 33(8)
2./integraldisplay∞
0xν+1
x2+β2cos(αx)Jν(γx)dx=βνcosh(αβ)Kν(βγ)
/bracketleftbig
0<α≤γ,Reβ>0,−1<Reν<1
2/bracketrightbig
ET II 37(33)
3./integraldisplay∞
0x1−ν
x2+β2sin(αx)Jν(γx)dx=π
2β−νe−αβIν(βγ)/bracketleftbig
0<γ≤α,Reβ>0,Reν>−1
2/bracketrightbig
ET II 33(9)
4./integraldisplay∞
0x−ν
x2+β2cos(αx)Jν(γx)dx=π
2β−ν−1e−αβIν(βγ)
/bracketleftbig
0<γ≤α,Reβ>0,Reν>−3
2/bracketrightbig
ET II 37(34)
6.719
1.6/integraldisplayα
0sin(βx)√
α2−x2Jν(x)dx=π∞/summationdisplay
n=0(−1)nJ2n+1(αβ)J1
2ν+n+1
2/parenleftbig1
2α/parenrightbig
J1
2ν−n−1
2/parenleftbig1
2α/parenrightbig
[Reν>−2] ET II 335(17)
2./integraldisplayα
0cos(βx)√
α2−x2Jν(x)dx=π
2J0(αβ)/bracketleftBig
J1
2ν/parenleftbig1
2α/parenrightbig/bracketrightBig2
+π∞/summationdisplay
n=1(−1)nJ2n(αβ)J1
2ν+n/parenleftbig1
2α/parenrightbig
J1
2ν−n/parenleftbig1
2α/parenrightbig
[Reν>−1] ET II 336(27)
6.721
1./integraldisplay∞
0√xJ1
4/parenleftbig
a2x2/parenrightbig
sin(bx)dx=2−3/2a−2√
πbJ1
4/parenleftbiggb2
4a2/parenrightbigg
[b>0] ET I 108(1)
2./integraldisplay∞
0√xJ−1
4/parenleftbig
a2x2/parenrightbig
cos(bx)dx=2−3/2a−2√
πbJ−1
4/parenleftbiggb2
4a2/parenrightbigg
[b>0] ET I 51(1)
3./integraldisplay∞
0√xY1
4/parenleftbig
a2x2/parenrightbig
sin(bx)dx=−2−3/2√
πba−2H1
4/parenleftbiggb2
4a2/parenrightbigg
ET I 108(7)
4./integraldisplay∞
0√xY−1
4/parenleftbig
a2x2/parenrightbig
cos(bx)dx=−2−3/2√
πba−2H−1
4/parenleftbiggb2
4a2/parenrightbigg
ET I 52(7)
736 Bessel Functions 6.722
5./integraldisplay∞
0√xK1
4/parenleftbig
a2x2/parenrightbig
sin(bx)dx=2−5/2√
π3ba−2/bracketleftbigg
I1
4/parenleftbiggb2
4a2/parenrightbigg
−L1
4/parenleftbiggb2
4a2/parenrightbigg/bracketrightbigg
/bracketleftBig
|arga|<π
4,b > 0/bracketrightBig
ET I 109(11)
6./integraldisplay∞
0√xK−1
4/parenleftbig
a2x2/parenrightbig
cos(bx)dx=2−5/2√
π3ba−2/bracketleftbigg
I−1
4/parenleftbiggb2
4a2/parenrightbigg
−L−1
4/parenleftbiggb2
4a2/parenrightbigg/bracketrightbigg
[b>0] ET I 52(10)
6.722
1./integraldisplay∞
0√xK1
8+ν/parenleftbig
a2x2/parenrightbig
I1
8−ν/parenleftbig
a2x2/parenrightbig
sin(bx)dx=√
2πb−3/2Γ/parenleftbig5
8−ν/parenrightbig
Γ/parenleftbig5
4/parenrightbigWν,1
8/parenleftbiggb2
8a2/parenrightbigg
M−ν,1
8/parenleftbiggb2
8a2/parenrightbigg
/bracketleftbigg
Reν<5
8,|arga|<π
4,b > 0/bracketrightbigg
ET I 109(13)
2.10/integraldisplay∞
0√xJ−1
8−ν/parenleftbig
a2x2/parenrightbig
J−1
8+ν/parenleftbig
a2x2/parenrightbig
cos(bx)dx
=√π
23/4a3/2Γ/parenleftbig1
4/parenrightbig
Γ/parenleftbig3
4/parenrightbig
Γ/parenleftbig5
8−ν/parenrightbig
Γ/parenleftbig5
8+ν/parenrightbig2F3/parenleftBigg
3
8−ν,3
8+ν;3
8,3
4,7
8;−/parenleftbiggb
4a/parenrightbigg4/parenrightBigg
−1
a2/radicalbigg
2b
πcos(πν)2F3/parenleftBigg
1
2−ν,1
2+ν;1
2,7
8,9
8;−/parenleftbiggb
4a/parenrightbigg4/parenrightBigg
−b5/2ν
15a4/radicalbigg
2
πsin(πν)2F3/parenleftBigg
1−ν,1+ν;11
8,3
2,13
8;−/parenleftbiggb
4a/parenrightbigg4/parenrightBigg
/bracketleftbig
a2>0,Imb=0/bracketrightbig
MC
3./integraldisplay∞
0√xJ1
8−ν/parenleftbig
a2x2/parenrightbig
J1
8+ν/parenleftbig
a2x2/parenrightbig
sin(bx)dx
=/radicalbigg
2
πb−3/2/bracketleftbigg
eπi/8Wν,1
8/parenleftbiggb2eπi/2
8a2/parenrightbigg
W−ν,1
8/parenleftbiggb2eπi/2
8a2/parenrightbigg
+e−iπ/8Wν,1
8/parenleftbiggb2e−πi/2
8a2/parenrightbigg
W−ν,1
8/parenleftBigg
b2e−πi
2
8a2/parenrightBigg⎤
⎦
[b>0] ET I 108(6)
4./integraldisplay∞
0√xK1
8−ν/parenleftbig
a2x2/parenrightbig
I−1
8−ν/parenleftbig
a2x2/parenrightbig
cos(bx)dx
=√
2πb−3/2Γ/parenleftbig3
8−ν/parenrightbig
Γ/parenleftbig3
4/parenrightbigWν,−1
8/parenleftbiggb2
8a2/parenrightbigg
M−ν,−1
8/parenleftbiggb2
8a2/parenrightbigg
/bracketleftbig
Reν<3
8,b > 0/bracketrightbig
ET I 52(12)
6.723/integraldisplay∞
0xJν/parenleftbig
x2/parenrightbig/bracketleftbig
sin(νπ)Jν/parenleftbig
x2/parenrightbig
−cos(νπ)Yν/parenleftbig
x2/parenrightbig/bracketrightbig
J4ν(4ax)dx=1
4Jν/parenleftbig
a2/parenrightbig
J−ν/parenleftbig
a2/parenrightbig
[a>0,Reν>−1] ET II 375(20)
6.726 Bessel and trigonometric functions and powers 737
6.724
1./integraldisplay∞
0x2λJ2ν/parenleftBiga
x/parenrightBig
sin(bx)dx
=√πa2νΓ(λ−ν+1 )b2ν−2λ−1
42ν−λΓ(2ν+1 )Γ/parenleftbigg
ν−λ+1
2/parenrightbigg0F3/parenleftbigg
2ν+1,ν−λ,ν−λ+1
2;a2b2
16/parenrightbigg
+a2λ+2Γ(ν−λ−1)b
22λ+3Γ(ν+λ+2 )0F3/parenleftbigg3
2,λ−ν+2,λ+ν+2 ;a2b2
16/parenrightbigg
/bracketleftbig
−5
4<Reλ<Reν, a > 0,b > 0/bracketrightbig
ET I 109(15)
2./integraldisplay∞
0x2λJ2ν/parenleftBiga
x/parenrightBig
cos(bx)dx
=4λ−2ν√πa2νb2ν−2λ−1Γ/parenleftbig
λ−ν+1
2/parenrightbig
Γ(2ν+1 )Γ ( ν−λ)0F3/parenleftbigg
2ν+1,ν−λ+1
2,ν−λ;a2b2
16/parenrightbigg
+4−λ−1a2λ+1Γ/parenleftbig
ν−λ−1
2/parenrightbig
Γ/parenleftbigg
ν+λ+3
2/parenrightbigg0F3/parenleftbigg1
2,λ−ν+3
2,ν+λ+3
2;a2b2
16/parenrightbigg
/bracketleftbig
−3
4<Reλ<Reν−1
2,a > 0,b > 0/bracketrightbig
ET I 53(14)
6.725
1./integraldisplay∞
0sin(bx)√xJν/parenleftbig
a√x/parenrightbig
dx=−/radicalbiggπ
bsin/parenleftbigga2
8b−νπ
4−π
4/parenrightbigg
Jν
2/parenleftbigga2
8b/parenrightbigg
[Reν>−3,a > 0,b > 0]
ET I 110(27)
2./integraldisplay∞
0cos(bx)√xJν/parenleftbig
a√x/parenrightbig
dx=/radicalbiggπ
bcos/parenleftbigga2
8b−νπ
4−π
4/parenrightbigg
J1
2ν/parenleftbigga2
8b/parenrightbigg
[Reν>−1,a > 0,b > 0]
ET I 54(25)
3./integraldisplay∞
0x1
2νJν/parenleftbig
a√x/parenrightbig
sin(bx)dx=2−νaνb−ν−1cos/parenleftbigga2
4b−νπ
2/parenrightbigg
/bracketleftbig
−2<Reν<1
2,a > 0,b > 0/bracketrightbig
ET I 110(28)
4./integraldisplay∞
0x1
2νJν/parenleftbig
a√x/parenrightbig
cos(bx)dx=2−νb−ν−1aνsin/parenleftbigga2
4b−νπ
2/parenrightbigg
/bracketleftbig
−1<Reν<1
2,a > 0,b > 0/bracketrightbig
ET I 54(26)
6.726
1./integraldisplay∞
0x/parenleftbig
x2+b2/parenrightbig−1
2νJν/parenleftBig
a/radicalbig
x2+b2/parenrightBig
sin(cx)dx
=/radicalbiggπ
2a−νb−ν+3
2c/parenleftbig
a2−c2/parenrightbig1
2ν−3
4Jν−3
2/parenleftBig
b/radicalbig
a2−c2/parenrightBig/bracketleftbig
0<c<a , Reν>1
2/bracketrightbig
=0/bracketleftbig
0<a<c , Reν>1
2/bracketrightbig
ET I 111(37)
738 Bessel Functions 6.727
2./integraldisplay∞
0/parenleftbig
x2+b2/parenrightbig−1
2νJν/parenleftBig
a/radicalbig
x2+b2/parenrightBig
cos(cx)dx
=/radicalbiggπ
2a−νb−ν+1
2/parenleftbig
a2−c2/parenrightbig1
2ν−1
4Jν−1
2/parenleftBig
b/radicalbig
a2−c2/parenrightBig/bracketleftbig
0<c<a , b> 0,Reν>−1
2/bracketrightbig
=0/bracketleftbig
0<a<c , b> 0,Reν>−1
2/bracketrightbig
ET I 55(37)
3./integraldisplay∞
0x/parenleftbig
x2+b2/parenrightbig1
2νK±ν/parenleftBig
a/radicalbig
x2+b2/parenrightBig
sin(cx)dx
=/radicalbiggπ
2aνbν+3
2c/parenleftbig
a2+c2/parenrightbig−1
2ν−3
4K−ν−3
2/parenleftBig
b/radicalbig
a2+c2/parenrightBig
[Rea>0,Reb>0,c > 0]ET I 113(45)
4.11/integraldisplay∞
0/parenleftbig
x2+b2/parenrightbig∓1
2νKν/parenleftBig
a/radicalbig
x2+b2/parenrightBig
cos(cx)dx
=/radicalbiggπ
2a∓νb1
2∓ν/parenleftbig
a2+c2/parenrightbig±1
2ν−1
4K±ν−1
2/parenleftBig
b/radicalbig
a2+c2/parenrightBig
[Rea>0,Reb>0,cis real] ET I 56(45)
5./integraldisplay∞
0/parenleftbig
x2+a2/parenrightbig−1
2νYν/parenleftBig
b/radicalbig
x2+a2/parenrightBig
cos(cx)dx
=/radicalbiggaπ
2(ab)−ν/parenleftbig
b2−c2/parenrightbig1
2ν−1
4Yν−1
2/parenleftBig
a/radicalbig
b2−c2/parenrightBig/bracketleftbig
0<c<b , a> 0,Reν>−1
2/bracketrightbig
=−/radicalbigg
2a
π(ab)−ν/parenleftbig
c2−b2/parenrightbig1
2ν−1
4Kν−1
2/parenleftBig
a/radicalbig
c2−b2/parenrightBig/bracketleftbig
0<b<c , a> 0,Reν>−1
2/bracketrightbig
ET I 56(41)
6.727
1.9/integraldisplaya
0cos(cx)√
a2−x2Jν/parenleftBig
b/radicalbig
a2−x2/parenrightBig
dx=π
2J1
2ν/bracketleftBiga
2/parenleftBig/radicalbig
b2+c2−c/parenrightBig/bracketrightBig
J1
2ν/bracketleftBiga
2/parenleftBig/radicalbig
b2+c2+c/parenrightBig/bracketrightBig
[Reν>−1,c > 0,a > 0]
ET I 113(48)
2./integraldisplay∞
asin(cx)√
x2−a2Jν/parenleftBig
b/radicalbig
x2−a2/parenrightBig
dx=π
2J1
2ν/bracketleftBiga
2/parenleftBig
c−/radicalbig
c2+b2/parenrightBig/bracketrightBig
J−1
2ν/bracketleftBiga
2/parenleftBig
c+/radicalbig
c2+b2/parenrightBig/bracketrightBig
[0<b<c , a> 0,Reν>−1]
ET I 113(49)
3./integraldisplay∞
acos(cx)√
x2−a2Jν/parenleftBig
b/radicalbig
x2−a2/parenrightBig
dx=−π
2J1
2ν/bracketleftBiga
2/parenleftBig
c−/radicalbig
c2−b2/parenrightBig/bracketrightBig
Y−1
2ν/bracketleftBiga
2/parenleftBig
c+/radicalbig
c2−b2/parenrightBig/bracketrightBig
[0<b<c , a> 0,Reν>−1]
ET I 58(54)
4.8/integraldisplaya
0/parenleftbig
a2−x2/parenrightbig1
2νcosxIν/parenleftBig/radicalbig
a2−x2/parenrightBig
dx=√πa2ν+1
2ν+1Γ/parenleftbig
ν+3
2/parenrightbig
/bracketleftbig
Reν>−1
2/bracketrightbig
WA 409(2)
6.731 Bessel and trigonometric functions and powers 739
6.728
1./integraldisplay∞
0xsin/parenleftbig
ax2/parenrightbig
Jν(bx)dx
=√πb
8a3/2/bracketleftbigg
cos/parenleftbiggb2
8a−νπ
4/parenrightbigg
J1
2ν−1
2/parenleftbiggb2
8a/parenrightbigg
−sin/parenleftbiggb2
8a−νπ
4/parenrightbigg
J1
2ν+1
2/parenleftbiggb2
8a/parenrightbigg/bracketrightbigg
[a>0,b > 0,Reν>−4]ET II 34(14)
2./integraldisplay∞
0xcos/parenleftbig
ax2/parenrightbig
Jν(bx)dx
=√πb
8a3/2/bracketleftbigg
cos/parenleftbiggb2
8a−νπ
4/parenrightbigg
J1
2ν+1
2/parenleftbiggb2
8a/parenrightbigg
+s i n/parenleftbiggb2
8a−νπ
4/parenrightbigg
J1
2ν−1
2/parenleftbiggb2
8a/parenrightbigg/bracketrightbigg
[a>0,b > 0,Reν>−2]ET II 38(39)
3./integraldisplay∞
0J0(βx)sin/parenleftbig
αx2/parenrightbig
xdx=1
2αcosβ2
4α[α>0,β > 0] MO 47
4./integraldisplay∞
0J0(βx)cos/parenleftbig
αx2/parenrightbig
xdx=1
2αsinβ2
4α[α>0,β > 0] MO 47
5./integraldisplay∞
0xν+1sin/parenleftbig
ax2/parenrightbig
Jν(bx)dx=bν
2ν+1aν+1cos/parenleftbiggb2
4a−νπ
2/parenrightbigg
/bracketleftbig
a>0,b > 0,−2<Reν<1
2/bracketrightbig
ET II 34(15)
6./integraldisplay∞
0xν+1cos/parenleftbig
ax2/parenrightbig
Jν(bx)dx=bν
2ν+1aν+1sin/parenleftbiggb2
4a−νπ
2/parenrightbigg
/bracketleftbig
a>0,b > 0,−1<Reν<1
2/bracketrightbig
ET II 38(40)
6.729
1./integraldisplay∞
0xsin/parenleftbig
ax2/parenrightbig
Jν(bx)Jν(cx)dx=1
2acos/parenleftbiggb2+c2
4a−νπ
2/parenrightbigg
Jν/parenleftbiggbc
2a/parenrightbigg
[a>0,b > 0,c > 0,Reν>−2]
ET II 51(26)
2./integraldisplay∞
0xcos/parenleftbig
ax2/parenrightbig
Jν(bx)Jν(cx)dx=1
2asin/parenleftbiggb2+c2
4a−νπ
2/parenrightbigg
Jν/parenleftbiggbc
2a/parenrightbigg
[a>0,b > 0,c > 0,Reν>−1]
ET II 51(27)
6.731
1.11/integraldisplay∞
0xsin/parenleftbig
ax2/parenrightbig
Jν/parenleftbig
bx2/parenrightbig
J2ν(2cx)dx
=1
2√
b2−a2sin/parenleftbiggac2
b2−a2/parenrightbigg
Jν/parenleftbiggbc2
b2−a2/parenrightbigg
[0<a<b , Reν>−1]
=1
2√
a2−b2cos/parenleftbiggac2
a2−b2/parenrightbigg
Jν/parenleftbiggbc2
a2−b2/parenrightbigg
[0<b<a , Reν>−1]
ET II 356(41)a
740 Bessel Functions 6.732
2.10/integraldisplay∞
0xcos/parenleftbig
ax2/parenrightbig
Jν/parenleftbig
bx2/parenrightbig
J2ν(2cx)dx
=1
2√
b2−a2cos/parenleftbiggac2
b2−a2/parenrightbigg
Jν/parenleftbiggbc2
b2−a2/parenrightbigg/bracketleftbig
0<a<b , Reν>−1
2/bracketrightbig
=1
2√
a2−b2sin/parenleftbiggac2
a2−b2/parenrightbigg
Jν/parenleftbiggbc2
a2−b2/parenrightbigg/bracketleftbig
0<b<a , Reν>−1
2/bracketrightbig
ET II 356(42)a
6.732/integraldisplay∞
0x2cos/parenleftbiggx2
2a/parenrightbigg
Y1(x)K1(x)dx=−a3K0(a)[ a>0] ET II 371(52)
6.733
1./integraldisplay∞
0sin/parenleftBiga
2x/parenrightBig
[sinxJ0(x) + cos xY0(x)]dx
x=πJ0/parenleftbig√a/parenrightbig
Y0/parenleftbig√a/parenrightbig
[a>0] ET II 346(51)
2./integraldisplay∞
0cos/parenleftBiga
2x/parenrightBig
[sinxY0(x)−cosxJ0(x)]dx
x=πJ0/parenleftbig√a/parenrightbig
Y0/parenleftbig√a/parenrightbig
[a>0] ET II 347(52)
3./integraldisplay∞
0xsin/parenleftBiga
2x/parenrightBig
K0(x)dx=πa
2J1/parenleftbig√a/parenrightbig
K1/parenleftbig√a/parenrightbig
[a>0] ET II 368(34)
4./integraldisplay∞
0xcos/parenleftBiga
2x/parenrightBig
K0(x)dx=−πa
2Y1/parenleftbig√a/parenrightbig
K1/parenleftbig√a/parenrightbig
[a>0] ET II 369(35)
6.734/integraldisplay∞
0cos/parenleftbig
a√x/parenrightbig
Kν(bx)dx√x
=π
2√
bsec(νπ)/bracketleftbigg
Dν−1
2/parenleftbigga√
2b/parenrightbigg
D−ν−1
2/parenleftbigg
−a√
2b/parenrightbigg
+Dν−1
2/parenleftbigg
−a√
2b/parenrightbigg
D−ν−1
2/parenleftbigga√
2b/parenrightbigg/bracketrightbigg
/bracketleftbig
Reb>0,|Reν|<1
2/bracketrightbig
ET II 132(27)
6.735
1./integraldisplay∞
0x1/4sin/parenleftbig
2a√x/parenrightbig
J−1
4(x)dx=√πa3/2J3
4/parenleftbig
a2/parenrightbig
[a>0] ET II 341(10)
2./integraldisplay∞
0x1/4cos/parenleftbig
2a√x/parenrightbig
J1
4(x)dx=√πa3/2J−3
4/parenleftbig
a2/parenrightbig
[a>0] ET II 341(12)
3./integraldisplay∞
0x1/4sin/parenleftbig
2a√x/parenrightbig
J3
4(x)dx=√πa3/2J−1
4/parenleftbig
a2/parenrightbig
[a>0] ET II 341(11)
4./integraldisplay∞
0x1/4cos/parenleftbig
2a√x/parenrightbig
J−3
4(x)dx=√πa3/2J1
4/parenleftbig
a2/parenrightbig
[a>0] ET II 341(13)
6.736
1.11/integraldisplay∞
0x−1/2sinxcos/parenleftbig
4a√x/parenrightbig
J0(x)dx=−2−3/2√π/bracketleftBig
cos/parenleftBig
a2−π
4/parenrightBig
J0/parenleftbig
a2/parenrightbig
−sin/parenleftBig
a2−π
4/parenrightBig
Y0/parenleftbig
a2/parenrightbig/bracketrightBig
[a>0] ET II 341(18)
2./integraldisplay∞
0x−1/2cosxcos/parenleftbig
4a√x/parenrightbig
J0(x)dx=−2−3/2√π/bracketleftBig
sin/parenleftBig
a2−π
4/parenrightBig
J0/parenleftbig
a2/parenrightbig
+c o s/parenleftBig
a2−π
4/parenrightBig
Y0/parenleftbig
a2/parenrightbig/bracketrightBig
[a>0] ET II 342(22)
6.737 Bessel and trigonometric functions and powers 741
3./integraldisplay∞
0x−1/2sinxsin/parenleftbig
4a√x/parenrightbig
J0(x)dx=/radicalbiggπ
2cos/parenleftBig
a2+π
4/parenrightBig
J0/parenleftbig
a2/parenrightbig
[a>0] ET II 341(16)
4./integraldisplay∞
0x−1/2cosxsin/parenleftbig
4a√x/parenrightbig
J0(x)dx=/radicalbiggπ
2cos/parenleftBig
a2−π
4/parenrightBig
J0/parenleftbig
a2/parenrightbig
[a>0] ET II 342(20)
5./integraldisplay∞
0x−1/2sinxcos/parenleftbig
4a√x/parenrightbig
Y0(x)dx=2−3/2√π/bracketleftBig
3s in/parenleftBig
a2−π
4/parenrightBig
J0/parenleftbig
a2/parenrightbig
−cos/parenleftBig
a2−π
4/parenrightBig
Y0/parenleftbig
a2/parenrightbig/bracketrightBig
[a>0] ET II 347(55)
6./integraldisplay∞
0x−1/2cosxcos/parenleftbig
4a√x/parenrightbig
Y0(x)dx
=−2−3/2√π/bracketleftBig
3c os/parenleftBig
a2−π
4/parenrightBig
J0/parenleftbig
a2/parenrightbig
+s i n/parenleftBig
a2−π
4/parenrightBig
Y0/parenleftbig
a2/parenrightbig/bracketrightBig
[a>0] ET II 347(56)
6.737
1./integraldisplay∞
0sin/parenleftbig
a√
x2+b2/parenrightbig
√
x2+b2Jν(cx)dx=π
2J1
2ν/bracketleftbiggb
2/parenleftBig
a−/radicalbig
a2−c2/parenrightBig/bracketrightbigg
J−1
2ν/bracketleftbiggb
2/parenleftBig
a+/radicalbig
a2−c2/parenrightBig/bracketrightbigg
[a>0,Reb>0,c > 0,a > c , Reν>−1]ET II 35(19)
2./integraldisplay∞
0cos/parenleftbig
a√
x2+b2/parenrightbig
√
x2+b2Jν(cx)dx=−π
2J1
2ν/bracketleftbiggb
2/parenleftBig
a−/radicalbig
a2−c2/parenrightBig/bracketrightbigg
Y−1
2ν/bracketleftbiggb
2/parenleftBig
a+/radicalbig
a2−c2/parenrightBig/bracketrightbigg
[a>0,Reb>0,c > 0,a > c , Reν>−1]ET II 39(44)
3./integraldisplaya
0cos/parenleftbig
b√
a2−x2/parenrightbig
√
a2−x2Jν(cx)dx=π
2J1
2ν/bracketleftBiga
2/parenleftBig/radicalbig
b2+c2−b/parenrightBig/bracketrightBig
J1
2ν/bracketleftBiga
2/parenleftBig/radicalbig
b2+c2+b/parenrightBig/bracketrightBig
[c>0,Reν>−1] ET II 39(47)
4./integraldisplaya
0xν+1cos/parenleftbig√
a2−x2/parenrightbig
√
a2−x2Iν(x)dx=√πa2ν+1
2ν+1Γ/parenleftbigg
ν+3
2/parenrightbigg [Reν>−1] ET II 365(9)
5./integraldisplay∞
0xν+1sin/parenleftbig
a√
b2+x2/parenrightbig
√
b2+x2Jν(cx)dx
=/radicalbiggπ
2b1
2+νcν/parenleftbig
a2−c2/parenrightbig−1
4−1
2νJ−ν−1
2/parenleftBig
b/radicalbig
a2−c2/parenrightBig/bracketleftbig
0<c<a , Reb>0,−1<Reν<1
2/bracketrightbig
=0/bracketleftbig
0<a<c , Reb>0,−1<Reν<1
2/bracketrightbig
ET II 35(20)
742 Bessel Functions 6.738
6./integraldisplay∞
0xν+1cos/parenleftbig
a√
x2+b2/parenrightbig
√
x2+b2Jν(cx)dx=−/radicalbiggπ
2b1
2+νcν/parenleftbig
a2−c2/parenrightbig−1
4−1
2νY−ν−1
2/parenleftBig
b/radicalbig
a2−c2/parenrightBig
/bracketleftbigg
0<c<a , Reb>0,−1<Reν<1
2/bracketrightbigg
=/radicalbigg
2
πb1
2+νcν/parenleftbig
c2−a2/parenrightbig−1
4−1
2νKν+1
2/parenleftBig
b/radicalbig
c2−a2/parenrightBig
/bracketleftbigg
0<a<c , Reb>0,−1<Reν<1
2/bracketrightbigg
ET II 39(45)
6.738
1./integraldisplaya
0xν+1sin/parenleftBig
b/radicalbig
a2−x2/parenrightBig
Jν(x)dx=/radicalbiggπ
2aν+3
2b/parenleftbig
1+b2/parenrightbig−1
2ν−3
4Jν+3
2/parenleftBig
a/radicalbig
1+b2/parenrightBig
[Reν>−1] ET II 335(19)
2./integraldisplay∞
0xν+1cos/parenleftBig
a/radicalbig
x2+b2/parenrightBig
Jν(cx)dx
=/radicalbiggπ
2abν+3
2cν/parenleftbig
a2−c2/parenrightbig−1
2ν−3
4/bracketleftBig
cos(πν)Jν+3
2/parenleftBig
b/radicalbig
a2−c2/parenrightBig
−sin(πν)Yν+3
2/parenleftBig
b/radicalbig
a2−c2/parenrightBig/bracketrightBig
/bracketleftbig
0<c<a , Reb>0,−1<Reν<−1
2/bracketrightbig
=0
/bracketleftbig
0<a<c , Reb>0,−1<Reν<−1
2/bracketrightbig
ET II 39(43)
6.739/integraldisplayt
0x−1/2cos/parenleftbig
b√t−x/parenrightbig
√t−xJ2ν/parenleftbig
a√x/parenrightbig
dx=πJν/bracketleftbigg√
t
2/parenleftBig/radicalbig
a2+b2+b/parenrightBig/bracketrightbigg
Jν/bracketleftbigg√
t
2/parenleftBig/radicalbig
a2+b2−b/parenrightBig/bracketrightbigg
/bracketleftbig
Reν>−1
2/bracketrightbig
EH II 47(7)
6.741
1./integraldisplay1
0cos(μarccos x)√
1−x2Jν(ax)dx=π
2J1
2(μ+ν)/parenleftBiga
2/parenrightBig
J1
2(ν−μ)/parenleftBiga
2/parenrightBig
[Re(μ+ν)>−1,a > 0]ET II 41(54)
2./integraldisplay1
0cos[(ν+ 1)arccos x]√
1−x2Jν(ax)dx=/radicalbiggπ
acos/parenleftBiga
2/parenrightBig
Jν+1
2/parenleftBiga
2/parenrightBig
[Reν>−1,a > 0] ET II 40(53)
3./integraldisplay1
0cos[(ν−1)arccos x]√
1−x2Jν(ax)dx=/radicalbiggπ
asin/parenleftBiga
2/parenrightBig
Jν−1
2/parenleftBiga
2/parenrightBig
[Reν>0,a > 0] ET II 40(52)a
6.75 Combinations of Bessel, trigonometric, and exponential functions and powers
6.751 Notation :/lscript1=1
2/bracketleftBig/radicalbig
(b+c)2+a2−/radicalbig
(b−c)2+a2/bracketrightBig
,/lscript2=1
2/bracketleftBig/radicalbig
(b+c)2+a2+/radicalbig
(b−c)2+a2/bracketrightBig
6.752 Combinations of Bessel, trigonometric, and exponential functions and powers 743
1./integraldisplay∞
0e−1
2axsin(bx)I0/parenleftbigg1
2ax/parenrightbigg
dx=1√
2b1√
b2+a2/radicalBig
b+/radicalbig
b2+a2
[Rea>0,b > 0] ET I 105(44)
2./integraldisplay∞
0e−1
2axcos(bx)I0/parenleftbigg1
2ax/parenrightbigg
dx=a√
2b1
√
a2+b2/radicalbig
b+√
a2+b2
[Rea>0,b > 0] ET I 48(38)
3.10/integraldisplay∞
0e−bxcos(ax)J0(cx)dx=/bracketleftbigg/radicalBig
(b2+c2−a2)2+4a2b2+b2+c2−a2/bracketrightbigg1/2
√
2/radicalBig
(b2+c2−a2)2+4a2b2
[c>0] ET II 11(46)
alternatively, with aandbinterchanged,
/integraldisplay∞
0e−axcos(bx)J0(cx)dx=/radicalbig
/lscript2
2−b2
/lscript2
2−/lscript2
1[c>0]
6.752
1.10/integraldisplay∞
0e−axJ0(bx)sin(cx)dx
x=a r c s i n/parenleftBigg
2c/radicalbig
a2+(c+b)2+/radicalbig
a2+(c−b)2/parenrightBigg
=a r c s i n/parenleftbiggc
/lscript2/parenrightbigg
[Rea>|Imb|,c > 0] ET I 101(17)
2.10/integraldisplay∞
0e−axJ1(cx)sin(bx)dx
x=b
c(1−r)=b−/radicalbig
b2−/lscript2
1
c,
/bracketleftbigg
b2=c2
1−r2−a2
r2,c > 0/bracketrightbigg
ET II 19(15)
Notation : For integrals 6.752 3–6.752 5 we define the auxiliary functions
/lscript1(a)≡/lscript1(a,ρ,z)=1
2/bracketleftBig/radicalbig
(a+ρ)2+z2−/radicalbig
(a−ρ)2+z2/bracketrightBig
/lscript2(a)≡/lscript1(a,ρ,z)=1
2/bracketleftBig/radicalbig
(a+ρ)2+z2+/radicalbig
(a−ρ)2+z2/bracketrightBig
when a≥0,ρ≥0, and z≥0.
3.10/radicalbiggπ
2/integraldisplay∞
0e−zxJν+1/2(ax)Jν+1(ρx)√xd x
=a−ν−3/2ρ−ν−1/lscript2ν+2
1/radicalbig
ρ2−/lscript2
1a/parenleftbig
ρ2−/lscript2
1/parenrightbig
/lscript1(/lscript2
2−/lscript2
1)
=aν+1/2ρν+1
/lscript2ν+2
2/radicalbig
/lscript2
2−a2
/lscript2
2−/lscript2
1[Rez>|Ima|+|Imρ|]
744 Bessel Functions 6.753
4.10/radicalbiggπ
2/integraldisplay∞
0e−zxJν+1/2(ax)Jν(ρx)dx√x
=aν+1/2ρν/integraldisplay1//lscript2
01
/lscript2ν
21/radicalbig
1−a2//lscript2
2d/parenleftbigg1
/lscript2/parenrightbigg
=a−ν−1/2ρν/integraldisplaya//lscript2
0x2νdx√
1−x2/bracketleftbig
ν>−1
2,Rez>|Ima|+|Imρ|/bracketrightbig
5.10/integraldisplay∞
0e−zxsin(ax)J1(ρx)dx
x2=/radicalbig
/lscript2
2−a2/parenleftBig
a−/radicalbig
a2−/lscript2
1/parenrightBig2
2aρ+ρ
2arcsin/parenleftbigga
/lscript2/parenrightbigg
[Rez>|Ima|+|Imρ|]
6.753
1.8/integraldisplay∞
0sin (xasinψ)
xe−xacosϕcosψJν(xasinϕ)dx=ν−1/parenleftBig
tanϕ
2/parenrightBigν
sin(νψ)
/bracketleftBig
Reν>−1,a > 0,0<ϕ<π
2,0<ψ<π
2/bracketrightBig
ET II 33(10)
2./integraldisplay∞
0cos(xasinψ)
xe−xacosϕcosψJν(xasinϕ)dx=ν−1/parenleftBig
tanϕ
2/parenrightBigν
cos(νψ)
/bracketleftBig
Reν>0,a > 0,0<ϕ , ψ<π
2/bracketrightBig
ET II 38(35)
3.8/integraldisplay∞
0xν+1e−sxsin(bx)Jν(ax)dx=−2(2a)ν
√πΓ(ν+3
2)R−2ν−3/bracketleftbig
bcos(ν+3
2)ϕ+ssin(ν+3
2)ϕ/bracketrightbig
/bracketleftbigg
Reν>−3
2,Res>|Ima|+|Imb|,
R4=/parenleftbig
s2+a2−b2/parenrightbig2+4b2s2,ϕ=a r g/parenleftbig
s2+a2−b2−2ibs/parenrightbig/bracketrightbigg
4.8/integraldisplay∞
0xν+1e−sxcos(bx)Jν(ax)dx=2(2a)ν
√πΓ(ν+3
2)R−2ν−3/bracketleftbig
scos(ν+3
2)ϕ−bsin(ν+3
2)ϕ/bracketrightbig
,
/bracketleftbigg
Reν>−1,Res>|Ima|+|Imb|,
R4=/parenleftbig
s2+a2−b2/parenrightbig2+4b2s2,ϕ=a r g/parenleftbig
s2+a2−b2−2ibs/parenrightbig/bracketrightbigg
5.10/integraldisplay∞
0xνe−axcosϕcosψsin(axsinψ)Jν(axsinϕ)dx
=2νΓ/parenleftbig
ν+1
2/parenrightbig
√πa−ν−1(sinϕ)ν/parenleftbig
cos2ψ+s i n2ψcos2ϕ/parenrightbig−ν−1
2sin/bracketleftbig/parenleftbig
ν+1
2/parenrightbig
β/bracketrightbig
tanβ
2=t a n ψcosϕ/bracketleftBig
a>0,0<ϕ<π
2,0<ψ<π
2,Reν>−1/bracketrightBig
ET II 34(12)
6.755 Combinations of Bessel, trigonometric, and exponential functions and powers 745
6./integraldisplay∞
0xνe−axcosϕcosψcos(axsinψ)Jν(axsinϕ)dx
=2νΓ/parenleftbig
ν+1
2/parenrightbig
√πa−ν−1(sinϕ)ν/parenleftbig
cos2ψ+s i n2ψcos2ϕ/parenrightbig−ν−1
2cos/bracketleftbig/parenleftbig
ν+1
2/parenrightbig
β/bracketrightbig
tanβ
2=t a n ψcosϕ/bracketleftbigg
a>0,0<ϕ , ψ<π
2,Reν>−1
2/bracketrightbigg
ET II 38(37)
6.754
1./integraldisplay∞
0e−x2sin(bx)I0/parenleftbig
x2/parenrightbig
dx=√π
23/2e−b2
8I0/parenleftbiggb2
8/parenrightbigg
[b>0] ET I 108(9)
2./integraldisplay∞
0e−axcos/parenleftbig
x2/parenrightbig
J0/parenleftbig
x2/parenrightbig
dx=1
4/radicalbiggπ
2/bracketleftbigg
J0/parenleftbigga2
16/parenrightbigg
cos/parenleftbigga2
16−π
4/parenrightbigg
−Y0/parenleftbigga2
16/parenrightbigg
cos/parenleftbigga2
16+π
4/parenrightbigg/bracketrightbigg
[a>0] MI 42
3./integraldisplay∞
0e−axsin/parenleftbig
x2/parenrightbig
J0/parenleftbig
x2/parenrightbig
dx=1
4/radicalbiggπ
2/bracketleftbigg
J0/parenleftbigga2
16/parenrightbigg
sin/parenleftbigga2
16−π
4/parenrightbigg
−Y0/parenleftbigga2
16/parenrightbigg
sin/parenleftbigga2
16+π
4/parenrightbigg/bracketrightbigg
[a>0] MI 42
6.755
1./integraldisplay∞
0x−νe−xsin/parenleftbig
4a√x/parenrightbig
Iν(x)dx=/parenleftBig
23/2a/parenrightBigν−1
e−a2W1
2−3
2ν,1
2−1
2ν/parenleftbig
2a2/parenrightbig
[a>0,Reν>0] ET II 366(14)
2./integraldisplay∞
0x−ν−1
2e−xcos/parenleftbig
4a√x/parenrightbig
Iν(x)dx=23
2ν−1aν−1e−a2W−3
2ν,1
2ν/parenleftbig
2a2/parenrightbig
/bracketleftbig
a>0,Reν>−1
2/bracketrightbig
ET II 366(16)
3./integraldisplay∞
0x−νexsin/parenleftbig
4a√x/parenrightbig
Kν(x)dx=/parenleftBig
23/2a/parenrightBigν−1
πΓ/parenleftbig3
2−2ν/parenrightbig
Γ/parenleftbig1
2+ν/parenrightbigea2W3
2ν−1
2,1
2−1
2ν/parenleftbig
2a2/parenrightbig
/bracketleftbig
a>0,0<Reν<3
4/bracketrightbig
ET II 369(38)
4./integraldisplay∞
0x−ν−1
2excos/parenleftbig
4a√x/parenrightbig
Kν(x)dx=23
2ν−1πaν−1Γ/parenleftbig1
2−2ν/parenrightbig
Γ/parenleftbig1
2+ν/parenrightbigea2W3
2ν,−1
2ν/parenleftbig
2a2/parenrightbig
/bracketleftbig
a>0,−1
2<Reν<1
4/bracketrightbig
ET II 369(42)
5./integraldisplay∞
0x/rho1−3
2e−xsin/parenleftbig
4a√x/parenrightbig
Kν(x)dx=√πaΓ(/rho1+ν)Γ(/rho1−ν)
2/rho1−2Γ/parenleftbig
/rho1+1
2/parenrightbig2F2/parenleftbigg
/rho1+ν,/rho1−ν;3
2,/rho1+1
2;−2a2/parenrightbigg
[Re/rho1>|Reν|] ET II 369(39)
6./integraldisplay∞
0x/rho1−1e−xcos/parenleftbig
4a√x/parenrightbig
Kν(x)dx=√πΓ(/rho1+ν)Γ(/rho1−ν)
2/rho1Γ/parenleftbig
/rho1+1
2/parenrightbig2F2/parenleftbigg
/rho1+ν,/rho1−ν;1
2,/rho1+1
2;−2a2/parenrightbigg
[Re/rho1>|Reν|] ET II 370(43)
7./integraldisplay∞
0x−1/2e−xcos/parenleftbig
4a√x/parenrightbig
I0(x)dx=1√
2πe−a2K0/parenleftbig
a2/parenrightbig
[a>0] ET II 366(15)
746 Bessel Functions 6.756
8./integraldisplay∞
0x−1/2excos/parenleftbig
4a√x/parenrightbig
K0(x)dx=/radicalbiggπ
2ea2K0/parenleftbig
a2/parenrightbig
[a>0] ET II 369(40)
9./integraldisplay∞
0x−1/2e−xcos/parenleftbig
4a√x/parenrightbig
K0(x)dx=1√
2π3/2e−a2I0/parenleftbig
a2/parenrightbig
ET II 369(41)
6.756
1./integraldisplay∞
0x−1
2e−a√xsin/parenleftbig
a√x/parenrightbig
Jν(bx)dx
=i√
2πbΓ/parenleftbigg
ν+1
2/parenrightbigg
D−ν−1
2/parenleftbigga√
b/parenrightbigg/bracketleftbigg
D−ν−1
2/parenleftbiggia√
b/parenrightbigg
−D−ν−1
2/parenleftbigg
−ia√
b/parenrightbigg/bracketrightbigg
[a>0,b > 0,Reν>−1]ET II 34(17)
2./integraldisplay∞
0x−1
2e−a√xcos/parenleftbig
a√x/parenrightbig
Jν(bx)dx
=1√
2πbΓ/parenleftbigg
ν+1
2/parenrightbigg
D−ν−1
2/parenleftbigga√
b/parenrightbigg/bracketleftbigg
D−ν−1
2/parenleftbiggia√
b/parenrightbigg
+D−ν−1
2/parenleftbigg
−ia√
b/parenrightbigg/bracketrightbigg
/bracketleftbig
a>0,b > 0,Reν>−1
2/bracketrightbig
ET II 39(42)
3./integraldisplay∞
0x−1/2e−a√xsin/parenleftbig
a√x/parenrightbig
J0(bx)dx=1
2baI1
4/parenleftbigga2
4b/parenrightbigg
K1
4/parenleftbigga2
4b/parenrightbigg
/bracketleftBig
|arga|<π
4,b > 0/bracketrightBig
ET II 11(40)
4./integraldisplay∞
0x−1/2e−a√xcos/parenleftbig
a√x/parenrightbig
J0(bx)dx=a
2bI−1
4/parenleftbigga2
4b/parenrightbigg
K1
4/parenleftbigga2
4b/parenrightbigg
/bracketleftBig
|arga|<π
4,b > 0/bracketrightBig
ET II 12(49)
6.757
1./integraldisplay∞
0e−bxsin/bracketleftbig
a/parenleftbig
1−e−x/parenrightbig/bracketrightbig
Jν/parenleftbig
ae−x/parenrightbig
dx
=2∞/summationdisplay
n=0(−1)nΓ(ν−b+2n+1 )Γ( ν+b)
Γ(ν−b+1 )Γ ( ν+b+2n+2 )(ν+2n−1)Jν+2n+1(a)
[Reb>−Reν] ET I 193(26)
2./integraldisplay∞
0e−bxcos/bracketleftbig
a/parenleftbig
1−e−x/parenrightbig/bracketrightbig
Jν/parenleftbig
ae−x/parenrightbig
dx
=Jν(a)
ν+b+∞/summationdisplay
n=02(−1)nΓ(ν−b+2n)Γ(ν+b)
Γ(ν−b+1 )Γ ( ν+b+2n+1 )(ν+2n)Jν+2n(a)
[Reb>−Reν] ET I 193(27)
6.758/integraldisplayπ
2
−π
2ei(μ−ν)θ(cosθ)ν+μ(λz)−ν−μJν+μ(λz)dθ
=π(2az)−μ(2bz)−νJμ(az)Jν(bz);λ=/radicalBig
2c osθ(a2eiθ+b2e−iθ)
λ=/radicalBig
2c osθ(a2eiθ+b2e−iθ)[ R e ( ν+μ)>−1]EH II 48(12)
6.775 Bessel functions and the logarithm, or arctangent 747
6.76 Combinations of Bessel, trigonometric, and hyperbolic functions
6.761/integraldisplay∞
0coshxcos (2asinhx)Jν(bex)Jν/parenleftbig
be−x/parenrightbig
dx=J2ν/parenleftbig
2√
b2−a2/parenrightbig
2√
b2−a2[0<a<b , Reν>−1]
=0 [ 0 <b<a , Reν>−1]
ET II 359(10)
6.762/integraldisplay∞
0coshxsin (2asinhx)/bracketleftbig
Jν(bex)Yν/parenleftbig
be−x/parenrightbig
−Yν(bex)Jν/parenleftbig
be−x/parenrightbig/bracketrightbig
dx
=0/bracketleftbig
0<a<b , |Reν|<1
2/bracketrightbig
=−2
πcos(νπ)/parenleftbig
a2−b2/parenrightbig−1/2K2ν/bracketleftBig
2/parenleftbig
a2−b2/parenrightbig1/2/bracketrightBig/bracketleftbig
0<b<a , |Reν|<1
2/bracketrightbig
ET II 360(12)
6.763/integraldisplay∞
0coshxcos (2asinhx)Yν(bex)Yν/parenleftbig
be−x/parenrightbig
dx
=−1
2/parenleftbig
b2−a2/parenrightbig−1/2J2ν/bracketleftBig
2/parenleftbig
b2−a2/parenrightbig1/2/bracketrightBig
[0<a<b , |Reν|<1]
=2
πcos(νπ)/parenleftbig
a2−b2/parenrightbig−1/2K2ν/bracketleftBig
2/parenleftbig
a2−b2/parenrightbig1/2/bracketrightBig
[0<b<a , |Reν|<1]
ET II 360(11)
6.77 Combinations of Bessel functions and the logarithm, or arctangent
6.771/integraldisplay∞
0xμ+1
2lnxJν(ax)dx=2μ−1
2Γ/parenleftbigμ+ν
2+3
4/parenrightbig
Γ/parenleftbigν−μ
2+1
4/parenrightbig
aμ+3
2/bracketleftbigg
ψ/parenleftbiggμ+ν
2+3
4/parenrightbigg
+ψ/parenleftbiggν−μ
2+1
4/parenrightbigg
−lna2
4/bracketrightbigg
/bracketleftbig
a>0,−Reν−3
2<Reμ<0/bracketrightbig
ET II 32(25)
6.772
1./integraldisplay∞
0lnxJ0(ax)dx=−1
a[ln(2a)+C] WA 430(4)a, ET II 10(27)
2./integraldisplay∞
0lnxJ1(ax)dx=−1
a/bracketleftBig
ln/parenleftBiga
2/parenrightBig
+C/bracketrightBig
ET II 19(11)
3./integraldisplay∞
0ln/parenleftbig
a2+x2/parenrightbig
J1(bx)dx=2
b[K0(ab)+l n a] ET II 19(12)
4./integraldisplay∞
0J1(tx)ln/radicalbig
1+t4dt=2
xkerx MO 46
6.773/integraldisplay∞
0ln/parenleftbig
x+√
x2+a2/parenrightbig
√
x2+a2J0(bx)dx=/bracketleftbigg1
2K2
0/parenleftbiggab
2/parenrightbigg
+l naI0/parenleftbiggab
2/parenrightbigg
K0/parenleftbiggab
2/parenrightbigg/bracketrightbigg
[a>0,b > 0] ET II 10(28)
6.774/integraldisplay∞
0ln√
x2+a2+x√
x2+a2−xJ0(bx)dx√
x2+a2=K2
0/parenleftbiggab
2/parenrightbigg
[Rea>0,b > 0] ET II 10(29)
6.775/integraldisplay∞
0x/bracketleftBig
ln/parenleftBig
1+/radicalbig
a2+x2/parenrightBig
−lnx/bracketrightBig
J0(bx)dx=1
b2/parenleftbig
1−e−ab/parenrightbig
[Rea>0,b > 0] ET II 12(55)
748 Bessel Functions 6.776
6.776/integraldisplay∞
0xln/parenleftbigg
1+a2
x2/parenrightbigg
J0(bx)dx=2
b/bracketleftbigg1
b−aK1(ab)/bracketrightbigg
[Rea>0,b > 0] ET II 10(30)
6.777/integraldisplay∞
0J1(tx)arctan t2dt=−2
xkeix MO 46
6.78 Combinations of Bessel and other special functions
6.781/integraldisplay∞
0si(ax)J0(bx)dx=−1
barcsin/parenleftbiggb
a/parenrightbigg
[0<b<a ]
=0 [ 0 <a<b ]
ET II 13(6)
6.782
1./integraldisplay∞
0Ei(−x)J0/parenleftbig
2√zx/parenrightbig
dx=e−z−1
zNT 60(4)
2./integraldisplay∞
0si(x)J0/parenleftbig
2√zx/parenrightbig
dx=−sinz
zNT 60(6)
3./integraldisplay∞
0ci(x)J0/parenleftbig
2√zx/parenrightbig
dx=cosz−1
zNT 60(5)
4./integraldisplay∞
0Ei(−x)J1/parenleftbig
2√zx/parenrightbigdx√x=Ei(−z)−C−lnz√zNT 60(7)
5./integraldisplay∞
0si(x)J1/parenleftbig
2√zx/parenrightbigdx√x=−π
2−si(z)√zNT 60(9)
6./integraldisplay∞
0ci(z)J1/parenleftbig
2√zx/parenrightbigdx√x=ci(z)−C−lnz√zNT 60(8)
7./integraldisplay∞
0Ei(−x)Y0/parenleftbig
2√zx/parenrightbig
dx=C+l nz−e2Ei(−z)
πzNT 63(5)
6.783
1./integraldisplay∞
0xsi/parenleftbig
a2x2/parenrightbig
J0(bx)dx=−2
b2sin/parenleftbiggb2
4a2/parenrightbigg
[a>0] ET II 13(7)a
2./integraldisplay∞
0xci/parenleftbig
a2x2/parenrightbig
J0(bx)dx=2
b2/bracketleftbigg
1−cos/parenleftbiggb2
4a2/parenrightbigg/bracketrightbigg
[a>0] ET II 13(8)a
3./integraldisplay∞
0ci/parenleftbig
a2x2/parenrightbig
J0(bx)dx=1
b/bracketleftbigg
ci/parenleftbiggb2
4a2/parenrightbigg
+l n/parenleftbiggb2
4a2/parenrightbigg
+2C/bracketrightbigg
[a>0] ET II 13(8)a
4./integraldisplay∞
0si/parenleftbig
a2x2/parenrightbig
J1(bx)dx=1
b/bracketleftbigg
−si/parenleftbiggb2
4a2/parenrightbigg
−π
2/bracketrightbigg
[a>0] ET II 20(25)a
6.784
1./integraldisplay∞
0xν+1[1−Φ(ax)]Jν(bx)dx=a−νΓ/parenleftbig
ν+3
2/parenrightbig
b2Γ(ν+2 )exp/parenleftbigg
−b2
8a2/parenrightbigg
M1
2ν+1
2,1
2ν+1
2/parenleftbiggb2
4a2/parenrightbigg
/bracketleftBig
|arga|<π
4,b > 0,Reν>−1/bracketrightBig
ET II 92(22)
6.792 Integration of Bessel functions 749
2./integraldisplay∞
0xν[1−Φ(ax)]Jν(bx)dx=/radicalbigg
2
πa1
2−νΓ/parenleftbig
ν+1
2/parenrightbig
b3/2Γ/parenleftbig
ν+3
2/parenrightbigexp/parenleftbigg
−b2
8a2/parenrightbigg
M1
2ν−1
4,1
2ν+1
4/parenleftbiggb2
4a2/parenrightbigg
/bracketleftbigg
|arga|<π
4,Reν>−1
2,b > 0/bracketrightbigg
ET II 92(23)
6.785/integraldisplay∞
0exp/parenleftBig
a2
2x−x/parenrightBig
x/bracketleftbigg
1−Φ/parenleftbigga√
2x/parenrightbigg/bracketrightbigg
Kν(x)dx=π5/2
4sec(νπ)/braceleftBig
[Jν(a)]2+[Yν(a)]2/bracerightBig
/bracketleftbig
Rea>0,|Reν|<1
2/bracketrightbig
ET II 370(46)
6.786/integraldisplay∞
0xν−2μ+2n+2ex2Γ/parenleftbig
μ, x2/parenrightbig
Yν(bx)dx
=(−1)nΓ/parenleftbig3
2−μ+ν+n/parenrightbig
Γ/parenleftbig3
2−μ+n/parenrightbig
bΓ(1−μ)exp/parenleftbiggb2
8/parenrightbigg
Wμ−1
2ν−n−1,1
2ν/parenleftbiggb2
4/parenrightbigg
/bracketleftbig
nis an integer ,b > 0,Re(ν−μ+n)>−3
2,Re(−μ+n)>−3
2,Reν<1
2−2n/bracketrightbig
ET II 108(2)
6.787/integraldisplay∞
0xν+2n−1
2
B(a+x, a−x)Jν(bx)dx=0
/bracketleftbig
π≤b<∞,−1<Reν<2a−2n−7
2/bracketrightbig
ET II 92(21)
6.79 Integration of Bessel functions with respect to the order
6.791
1./integraldisplay∞
−∞Kix+iy(a)Kix+iz(b)dx=πKiy−iz(a+b)[ |arga|+|argb|<π] ET II 382(21)
2./integraldisplay∞
−∞Jν−x(a)Jμ+x(a)dx=Jμ+ν(2a)[ R e ( μ+ν)>1] ET II 379(1)
3./integraldisplay∞
−∞Jκ+x(a)Jλ−x(a)Jμ+x(a)Jν−x(a)dx
=Γ(κ+λ+μ+ν+1 )
Γ(κ+λ+1 )Γ ( λ+μ+1 )Γ ( μ+ν+1 )Γ( ν+κ+1 )
×4F5⎛
⎝κ+λ+μ+ν+1
2,κ+λ+μ+ν+1
2,κ+λ+μ+ν
2+1,κ+λ+μ+ν
2+1 ;
κ+λ+μ+ν+1,κ+λ+1,λ+μ+1,μ+ν+1,ν+κ+1 ;−4a2⎞
⎠
[Re(κ+λ+μ+ν)>−1] ET II 379(3)
6.792
1./integraldisplay∞
−∞eπxKix+iy(a)Kix+iz(b)dx=πe−πzKi(y−z)(a−b)
[a>b> 0] ET II 382(22)
750 Bessel Functions 6.793
2./integraldisplay∞
−∞ei/rho1xKν+ix(α)Kν−ix(β)dx=π/parenleftbiggαeρ+β
α+βeρ/parenrightbiggν
K2ν/parenleftBig/radicalbig
α2+β2+2αβcosh/rho1/parenrightBig
[|argα|+|argβ|+|Im/rho1|<π]
ET II 382(23)
3./integraldisplay∞
−∞e(π−γ)xKix+iy(a)Kix+iz(b)dx=πe−βy−αzKiy−iz(c)
[0<γ<π , a> 0,b > 0,c > 0, α,β,γ —the angles of the triangle with sides a,b,c]
ET II 382(24), EH II 55(44)a
4.11/integraldisplay∞
−∞e−cxiH(2)
ν−ix(a)H(2)
ν+ix(b)dx=2i/parenleftbiggh
k/parenrightbigg2ν
H(2)
2ν(hk)
h=/radicalBig
ae1
2c+be−1
2c,k=/radicalBig
ae−1
2c+be1
2c[a,b > 0,cis real] ET II 380(11)
5./integraldisplay∞
−∞a−μ−xb−ν+xecxiJμ+x(a)Jν−x(b)dx
=/bracketleftBigg
2c os/parenleftbigc
2/parenrightbig
a2e−1
2ci+b2e1
2ci/bracketrightBigg1
2μ+1
2ν
exp/bracketleftBigc
2(ν−μ)i/bracketrightBig
Jμ+ν/braceleftbigg/bracketleftBig
2c os/parenleftBigc
2/parenrightBig/parenleftBig
a2e−1
2ci+b2e1
2ci/parenrightBig/bracketrightBig1/2/bracerightbigg
[a>0,b > 0,|c|<π , Re(μ+ν)>1]
=0
[a>0,b > 0,|c|≥π,Re(μ+ν)>1]
EH II 54(41), ET II 379(2)
6.793
1./integraldisplay∞
−∞e−cxi[Jν−ix(a)Yν+ix(b)+Yν−ix(a)Jν+ix(b)]dx=−2/parenleftbiggh
k/parenrightbigg2ν
J2ν(hk)
h=/radicalBig
ae1
2c+be−1
2c,k =/radicalBig
ae−1
2c+be1
2c[a,b > 0,Imc=0 ] ET II 380(9)
2./integraldisplay∞
−∞e−cxi[Jν−ix(a)Jν+ix(b)−Yν−ix(a)Yν+ix(b)]dx=2/parenleftbiggh
k/parenrightbigg2ν
Y2ν(hk)
h=/radicalBig
ae1
2c+be−1
2c,k =/radicalBig
ae−1
2c+be1
2c[a,b > 0,Imc=0 ] ET II 380(10)
3.10/integraldisplay∞
−∞eiγxsech(πx)[J−ix(α)Jix(β)−Jix(α)J−ix(β)]dx=2iH(σ)sign( β−α)J0/parenleftBig
σ1/2/parenrightBig
/bracketleftbig
α,β,γ ∈R,α , β > 0,σ=α2+β2−2αβcoshγ,H(σ) the Heaviside step function/bracketrightbig
6.794
1./integraldisplay∞
0Kix(a)Kix(b)cosh[( π−ϕ)x]dx=π
2K0/parenleftBig/radicalbig
a2+b2−2abcosϕ/parenrightBig
EH II 55(42)
2./integraldisplay∞
0cosh/parenleftBigπ
2x/parenrightBig
Kix(a)dx=π
2[a>0] ET II 382(19)
6.794 Integration of Bessel functions 751
3./integraldisplay∞
0cosh(/rho1x)Kix+ν(a)K−ix+ν(a)dx=π
2K2ν/bracketleftBig
2acos/parenleftBig/rho1
2/parenrightBig/bracketrightBig
[2|arga|+|Re/rho1|<π] ET II 383(28)
4./integraldisplay∞
−∞sech/parenleftBigπ
2x/parenrightBig
Jix(a)dx=2s i n a [a>0] ET II 380(6)
5./integraldisplay∞
−∞cosech/parenleftBigπ
2x/parenrightBig
Jix(a)dx=−2icosa [a>0] ET II 380(7)
6./integraldisplay∞
0sech(πx)/braceleftBig
[Jix(a)]2+[Yix(a)]2/bracerightBig
dx=−Y0(2a)−E0(2a)
[a>0] ET II 380(12)
7./integraldisplay∞
0xsinh/parenleftBigπ
2x/parenrightBig
Kix(a)dx=πa
2[a>0] ET II 382(20)
8./integraldisplay∞
0xtanh(πx)Kix(β)Kix(α)dx=π
2/radicalbig
αβexp(−β−α)
α+β
[|argβ|<π , |argα|<π]ET II 175(4)
9./integraldisplay∞
0xsinh(πx)K2ix(α)Kix(β)dx=π3/2α
25/2√βexp/parenleftbigg
−β−α2
8β/parenrightbigg
/bracketleftBig
β>0,|argα|<π
4/bracketrightBig
ET II 175(5)
10./integraldisplay∞
0xsinh(πx)
x2+n2Kix(α)Kix(β)dx=π2
2In(β)Kn(α)[ 0 <β<α ;n=0,1,2,...]
=π2
2In(α)Kn(β)[ 0 <α<β ;n=0,1,2,...]
ET II 176(8)
11./integraldisplay∞
0xsinh(πx)Kix(α)Kix(β)Kix(γ)dx=π2
4exp/bracketleftbigg
−γ
2/parenleftbiggα
β+β
α+αβ
γ2/parenrightbigg/bracketrightbigg
/bracketleftBig
|argα|+|argβ|<π
2,γ > 0/bracketrightBig
ET II 176(9)
12./integraldisplay∞
0xsinh/parenleftBigπ
2x/parenrightBig
K1
2ix(α)K1
2ix(β)Kix(γ)dx=π2γ
2/radicalbig
γ2+4αβexp/bracketleftBigg
−(α+β)/radicalbig
γ2+4αβ
2√αβ/bracketrightBigg
[|argα|+|argβ|<π , γ> 0]
ET II 176(10)
13./integraldisplay∞
0xsinh(πx)K1
2ix+λ(α)K1
2ix−λ(α)Kix(γ)dx=0 [ 0 <γ< 2α]
=π2γ
22λ+1α2λz/bracketleftBig
(γ+z)2λ+(γ−z)2λ/bracketrightBig
z=/radicalbig
γ2−4α2 [0<2α<γ ]ET II 176(11)
752 Bessel Functions 6.795
6.795
1./integraldisplay∞
0cos(bx)Kix(a)dx=π
2e−acoshb/bracketleftBig
|Imb|<π
2,a > 0/bracketrightBig
EH II 55(46), ET II 175(2)
2./integraldisplay∞
0Jx(ax)J−x(ax)cos(πx)dx=1
4/parenleftbig
1−a2/parenrightbig−1/2[|a|<1] ET II 380(4)
3./integraldisplay∞
0xsin(ax)Kix(b)dx=πb
2sinhaexp (−bcosha)/bracketleftBig
|Ima|<π
2,b > 0/bracketrightBig
ET II 175(1)
4./integraldisplay−∞
−∞sin[(ν+ix)π]
n+ν+ixKν+ix(a)Kν−ix(b)dx=π2In(a)Kn+2ν(b)[ 0 <a<b ;n=0,1,...]
=π2Kn+2ν(a)In(b)[ 0 <b<a ;n=0,1,...]
ET II 382(25)
5./integraldisplay∞
0xsin/parenleftbigg1
2πx/parenrightbigg
K1
2ix(a)Kix(b)dx=π3/2b√
2aexp/parenleftbigg
−a−b2
8a/parenrightbigg
/bracketleftBig
|arga|<π
2,b > 0/bracketrightBig
ET II 175(6)
6.796
1./integraldisplay∞
−∞e1
2πxcos(bx)
sinh(πx)Jix(a)dx=−iexp (iacoshb)[ a>0,b > 0] ET II 380(8)
2./integraldisplay∞
0cos(bx)cosh/parenleftbigg1
2πx/parenrightbigg
Kix(a)dx=π
2cos(asinhb) EH II 55(47)
3./integraldisplay∞
0sin(bx)sin h/parenleftbigg1
2πx/parenrightbigg
Kix(a)dx=π
2sin (asinhb) EH II 55(48)
4./integraldisplay∞
0cos(bx)cosh( πx)[Kix(a)]2dx=−π2
4Y0/bracketleftbigg
2asinh/parenleftbiggb
2/parenrightbigg/bracketrightbigg
[a>0,b > 0] ET II 383(27)
5./integraldisplay∞
0sin(bx)sin h( πx)[Kix(a)]2dx=π2
4J0/bracketleftbigg
2asinh/parenleftbiggb
2/parenrightbigg/bracketrightbigg
[a>0,b > 0] ET II 382(26)
6.797
1./integraldisplay∞
0xeπxsinh(πx)Γ(ν+ix)Γ(ν−ix)H(2)
ix(a)H(2)
ix(b)dx
=i2ν√πΓ/parenleftbig1
2+ν/parenrightbig
(ab)ν(a+b)−νKν(a+b)
[a>0,b > 0,Reν>0]ET II 381(14)
2./integraldisplay∞
0xeπxsinh(πx)cosh( πx)Γ(ν+ix)Γ(ν−ix)H(2)
ix(a)H(2)
ix(b)dx=iπ3/22ν
Γ/parenleftbig1
2−ν/parenrightbig(b−a)−νH(2)
ν(b−a)
/bracketleftbig
0<a<b , 0<Reν<1
2/bracketrightbig
ET II 381(15)
6.812 Struve functions 753
3./integraldisplay∞
0xeπxsinh(πx)Γ/parenleftbiggν+ix
2/parenrightbigg
Γ/parenleftbiggν−ix
2/parenrightbigg
H(2)
ix(a)H(2)
ix(b)dx
=iπ22−ν(ab)ν/parenleftbig
a2+b2/parenrightbig−1
2νH(2)
ν/parenleftBig/radicalbig
a2+b2/parenrightBig
[a>0,b > 0,Reν>0]ET II 381(16)
4.11/integraldisplay∞
0xsinh(πx)Γ(λ+ix)Γ(λ−ix)Kix(a)Kix(b)dx=2λ−1π3/2(ab)λ(a+b)−λΓ/parenleftbig
λ+1
2/parenrightbig
Kλ(a+b)
[|arga|<π , Reλ>0,b > 0]
ET II 176(12)
5./integraldisplay∞
0xsinh(2 πx)Γ(λ+ix)Γ(λ−ix)Kix(a)Kix(b)dx=2λπ5
2
Γ/parenleftbig1
2−λ/parenrightbig/parenleftbiggab
|b−a|/parenrightbiggλ
Kλ(|b−a|)
/bracketleftbig
a>0,0<Reλ<1
2,b > 0/bracketrightbig
ET II 176(13)
6./integraldisplay∞
0xsinh(πx)Γ/parenleftbig
λ+1
2ix/parenrightbig
Γ/parenleftbig
λ−1
2ix/parenrightbig
Kix(a)Kix(b)dx=2π2/parenleftbiggab
2√
a2+b2/parenrightbigg
K2λ/parenleftBig/radicalbig
a2+b2/parenrightBig
/bracketleftBig
|arga|<π
2,Reλ>0,b > 0/bracketrightBig
ET II 177(14)
7./integraldisplay∞
0xtanh(πx)Kix(a)Kix(b)
Γ/parenleftbig3
4+1
2ix/parenrightbig
Γ/parenleftbig3
4−1
2ix/parenrightbigdx=1
2/radicalbigg
πab
a2+b2exp/parenleftBig
−/radicalbig
a2+b2/parenrightBig
/bracketleftBig
|arga|<π
2,b > 0/bracketrightBig
, ET II 177(15)
6.8 Functions Generated by Bessel Functions
6.81 Struve functions
6.811
1./integraldisplay∞
0Hν(bx)dx=−cot/parenleftbigνπ
2/parenrightbig
b[−2<Reν<0,b > 0] ET II 158(1)
2./integraldisplay∞
0Hν/parenleftbigga2
x/parenrightbigg
Hν(bx)dx=−J2ν/parenleftBig
2a√
b/parenrightBig
b/bracketleftbig
a>0,b > 0,Reν>−3
2/bracketrightbig
ET II 170(37)
3./integraldisplay∞
0Hν−1/parenleftbigga2
x/parenrightbigg
Hν(bx)dx
x=−1
a√
bJ2ν−1/parenleftBig
2a√
b/parenrightBig/bracketleftbig
a>0,b > 0,Reν>−1
2/bracketrightbig
ET II 170(38)
6.812
1./integraldisplay∞
0H1(bx)dx
x2+a2=π
2a[I1(ab)−L1(ab)] [Re a>0,b > 0] ET II 158(6)
754 Functions Generated by Bessel Functions 6.813
2./integraldisplay∞
0Hν(bx)
x2+a2dx=−π
2asin/parenleftBigνπ
2/parenrightBigLν(ab)+bcot/parenleftbigνπ
2/parenrightbig
1−ν21F2/parenleftbigg
1;3−ν
2;3+ν
2;a2b2
2/parenrightbigg
[Rea>0,b > 0,|Reν|<2]
ET II 159(7)
6.813
1./integraldisplay∞
0xs−1Hν(ax)dx=2s−1Γ/parenleftbigs+ν
2/parenrightbig
asΓ/parenleftbig1
2ν−1
2s+1/parenrightbigtan/parenleftbiggs+ν
2π/parenrightbigg
/bracketleftbigg
a>0,−1−Reν<Res<min/parenleftbigg3
2,1−Reν/parenrightbigg/bracketrightbigg
WA 429(2), ET I 335(52)
2./integraldisplay∞
0x−ν−1Hν(x)dx=2−ν−1π
Γ(ν+1 )/bracketleftbig
Reν>−3
2/bracketrightbig
ET II 383(2)
3./integraldisplay∞
0x−μ−νHμ(x)Hν(x)dx=2−μ−ν√πΓ(μ+ν)
Γ/parenleftbig
μ+1
2/parenrightbig
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig
μ+ν+1
2/parenrightbig
[Re(μ+ν)>0]WA 435(2), ET II 384(8)
4./integraldisplay1
0xν+1Hν(ax)dx=1
aHν+1(a)/bracketleftbig
a>0,Reν>−3
2/bracketrightbig
ET II 158(2)a
5./integraldisplay1
0x1−νHν(ax)dx=aν−1
2ν−1√πΓ/parenleftbig
ν+1
2/parenrightbig−1
aHν−1(a)
[a>0] ET II 158(3)a
6.814
1./integraldisplay∞
0xν+1Hν(bx)
(x2+a2)1−μdx=2μ−1πaμ+νb−μ
Γ(1−μ)cos[( μ+ν)π][I−μ−ν(ab)−Lμ+ν(ab)]
/bracketleftbig
Rea>0,b > 0,Reν>−3
2,Re(μ+ν)<1
2,Re(2μ+ν)<3
2/bracketrightbig
ET II 159(8)
6.815
1./integraldisplay1
0x1
2ν(1−x)μ−1Hν/parenleftbig
a√x/parenrightbig
dx=2μa−μΓ(μ)Hμ+ν(a)
/bracketleftbig
Reν>−3
2,Reμ>0/bracketrightbig
ET II 199(88)a
2./integraldisplay1
0xλ−1
2ν−3
2(1−x)μ−1Hν/parenleftbig
a√x/parenrightbig
dx=B(λ,μ)aν+1
2ν√πΓ/parenleftbig
ν+3
2/parenrightbig2F3/parenleftbigg
1,λ;3
2,ν+3
2,λ+μ;−a2
4/parenrightbigg
[Reλ>0,Reμ>0] ET II 199(89)a
6.82 Combinations of Struve functions, exponentials, and powers
6.821
1.6/integraldisplay∞
0e−αxH−n−1
2(βx)dx=(−1)nβn+1
2/parenleftBig
α+/radicalbig
α2+β2/parenrightBig−n−1
2 1/radicalbig
α2+β2
[Reα>|Imβ|] ET I 206(6)
6.831 Struve and trigonometric functions 755
2.6/integraldisplay∞
0e−αxL−n−1
2(βx)dx=βn+1
2/parenleftBig
α+/radicalbig
α2−β2/parenrightBig−n−1
2 1/radicalbig
α2−β2
[Reα>|Reβ|] ET I 208(26)
3./integraldisplay∞
0e−αxH0(βx)dx=2
πln/parenleftbigg√
α2+β2+β
α/parenrightbigg
/radicalbig
α2+β2[Reα>|Imβ|] ET II 205(1)
4./integraldisplay∞
0e−αxL0(βx)dx=2
πarcsin/parenleftBig
β
α/parenrightBig
/radicalbig
α2+β2[Reα>|Reβ|] ET II 207(18)
6.822/integraldisplay∞
0e(ν+1)xHν(asinhx)dx=/radicalbiggπ
acosec( νπ)/bracketleftBig
sinh/parenleftBiga
2/parenrightBig
Iν+1
2/parenleftBiga
2/parenrightBig
−cosh/parenleftBiga
2/parenrightBig
I−ν−1
2/parenleftBiga
2/parenrightBig/bracketrightBig
[Rea>0,−2<Reν<0]
ET II 385(11)
6.823
1./integraldisplay∞
0xλe−αxHν(bx)dx=bν+1Γ(λ+ν+2 )
2νaλ+ν+2√πΓ/parenleftbigg
ν+3
2/parenrightbigg3F2/parenleftbigg
1,λ+ν
2+1,λ+ν+3
2;3
2,ν+3
2;−b2
a2/parenrightbigg
[Rea>0,b > 0,Re(λ+ν)>−2]
ET II 161(19)
2./integraldisplay∞
0xνe−αxLν(βx)dx=(2β)νΓ/parenleftbig
ν+1
2/parenrightbig
√π/parenleftBig/radicalbig
α2−β2/parenrightBig2ν+1−Γ(2ν+1 )/parenleftBig
β
α/parenrightBigν
/radicalbiggπ
2α/parenleftbig
β2−α2/parenrightbig1
2ν+1
4P−ν−1
2
−ν−1
2/parenleftbiggβ
α/parenrightbigg
/bracketleftbig
Reα>|Reβ|,Reν>−1
2/bracketrightbig
ET I 209(35)a
6.824
1./integraldisplay∞
0tνe−atL2ν/parenleftBig
2√
t/parenrightBig
dt=1
a2ν+1e1
aΦ/parenleftbigg1√a/parenrightbigg
MI 51
2./integraldisplay∞
0tνe−atL−2ν/parenleftBig√
t/parenrightBig
dt=1
Γ/parenleftbig1
2−2ν/parenrightbig
a2ν+1e1
aγ/parenleftbigg1
2−2ν,1
a/parenrightbigg
MI 51
6.825/integraldisplay∞
0xs−1e−α2x2Hν(βx)dx=βν+1Γ/parenleftbig1
2+s
2+ν
2/parenrightbig
2ν+1√παν+s+1Γ/parenleftbig
ν+3
2/parenrightbig2F2/parenleftbigg
1,ν+s+1
2;3
2,ν+3
2;−β2
4α2/parenrightbigg
/bracketleftBig
Res>−Reν−1,|argα|<π
4/bracketrightBig
ET I 335(51)a, ET II 162(20)
6.83 Combinations of Struve and trigonometric functions
6.831/integraldisplay∞
0x−νsin(ax)Hν(bx)dx=0/bracketleftbig
0<b<a , Reν>−1
2/bracketrightbig
=√π2−νb−ν/parenleftbig
b2−a2/parenrightbigν−1
2
Γ/parenleftbig
ν+1
2/parenrightbig/bracketleftbig
0<a<b , Reν>−1
2/bracketrightbig
ET II 162(21)
756 Functions Generated by Bessel Functions 6.832
6.832/integraldisplay∞
0√xsin(ax)H1
4/parenleftbig
b2x2/parenrightbig
dx=−2−3/2√π√a
b2Y1
4/parenleftbigga2
4b2/parenrightbigg
[a>0] ET I 109(14)
6.84–6.85 Combinations of Struve and Bessel functions
6.841/integraldisplay∞
0Hν−1(ax)Yν(bx)dx=−aν−1b−ν/bracketleftbig
0<b<a , |Reν|<1
2/bracketrightbig
=0/bracketleftbig
0<a<b , |Reν|<1
2/bracketrightbig
ET II 114(36)
6.842/integraldisplay∞
0[H0(ax)−Y0(ax)]J0(bx)dx=4
π(a+b)K/parenleftbigg|a−b|
a+b/parenrightbigg
[a>0,b > 0] ET II 15(22)
6.843
1./integraldisplay∞
0J2ν/parenleftbig
a√x/parenrightbig
Hν(bx)dx=−1
bYν/parenleftbigga2
4b/parenrightbigg/bracketleftbig
a>0,b > 0,−1<Reν<5
4/bracketrightbig
ET II 164(10)
2./integraldisplay∞
0K2ν/parenleftbig
2a√x/parenrightbig
Hν(bx)dx=2ν
πbΓ(ν+1 )S−ν−1,ν/parenleftbigga2
b/parenrightbigg
[Rea>0,b > 0,Reν>−1]
ET II 168(27)
6.844/integraldisplay∞
0/bracketleftbigg
cos/parenleftbiggμ−ν
2π/parenrightbigg
Jμ/parenleftbig
a√x/parenrightbig
−sin/parenleftbiggμ−ν
2π/parenrightbigg
Yμ/parenleftbig
a√x/parenrightbig/bracketrightbigg
Kμ/parenleftbig
a√x/parenrightbig
Hν(bx)dx
=1
a2W1
2ν,1
2μ/parenleftbigga2
2b/parenrightbigg
W−1
2ν,1
2μ/parenleftbigga2
2b/parenrightbigg
/bracketleftBig
|arga|<π
4,b > 0,Reν>|Reμ|−2/bracketrightBig
ET II 169(35)
6.845
1./integraldisplay∞
0/bracketleftBig
H−ν/parenleftBiga
x/parenrightBig
−Y−ν/parenleftBiga
x/parenrightBig/bracketrightBig
Jν(bx)dx=4
πbcos(νπ)K2ν/parenleftBig
2√
ab/parenrightBig
/bracketleftbig
|arga|<π , b> 0,|Reν|<1
2/bracketrightbig
ET II 73(7)
2./integraldisplay∞
0/bracketleftbigg
J−ν/parenleftbigga2
x/parenrightbigg
+s i n ( νπ)Hν/parenleftbigga2
x/parenrightbigg/bracketrightbigg
Hν(bx)dx=1
b/bracketleftbigg2
πK2ν/parenleftBig
2a√
b/parenrightBig
−Y2ν/parenleftBig
2a√
b/parenrightBig/bracketrightbigg
/bracketleftbig
a>0,b > 0,−3
2<Reν<0/bracketrightbig
ET II 170(39)
6.846/integraldisplay∞
0/bracketleftbigg2
πK2ν/parenleftbig
2a√x/parenrightbig
+Y2ν/parenleftbig
2a√x/parenrightbig/bracketrightbigg
Hν(bx)dx=1
bJν/parenleftbigga2
b/parenrightbigg
/bracketleftbig
a>0,b > 0,|Reν|<1
2/bracketrightbig
ET II 169(30)
6.847/integraldisplay∞
0/bracketleftBig
cosνπ
2Jν(ax)+s i nνπ
2Hν(ax)/bracketrightBigdx
x2+k2=π
2k[Iν(ak)−Lν(ak)]
/bracketleftbig
a>0,Rek>0,−1
2<Reν<2/bracketrightbig
ET II 384(5)a, WA 467(8)
6.852 Combinations of Struve and Bessel functions 757
6.848
1./integraldisplay∞
0x[Iν(ax)−L−ν(ax)]Jν(bx)dx=2
π/parenleftBiga
b/parenrightBigν−1
cos(νπ)1
a2+b2/bracketleftbig
Rea>0,b > 0,−1<Reν<−1
2/bracketrightbig
ET II 74(12)
2./integraldisplay∞
0x[H−ν(ax)−Y−ν(ax)]Jν(bx)dx=2cos(νπ)
aνπbν−11
a+b/bracketleftbig
|arga|<π , −1
2<Reν, b > 0/bracketrightbig
ET II 73(5)
6.849
1./integraldisplay∞
0xKν(ax)Hν(bx)dx=a−ν−1bν+11
a2+b2/bracketleftbig
Rea>0,b > 0,Reν>−3
2/bracketrightbig
ET II 164(12)
2./integraldisplay∞
0x[Kμ(ax)]2H0(bx)dx=−2−μ−1πa−2μ/bracketleftbig
(z+b)2μ+(z−b)2μ/bracketrightbig
bzsec(μπ),
z=/radicalbig
4a2+b2/bracketleftbig
Rea>0,b > 0,|Reμ|<3
2/bracketrightbig
ET II 166(18)
6.851
1./integraldisplay∞
0x/braceleftbigg/bracketleftBig
J1
2ν(ax)/bracketrightBig2
−/bracketleftBig
Y1
2ν(ax)/bracketrightBig2/bracerightbigg
Hν(bx)dx
=0/bracketleftbig
0<b< 2a,−3
2<Reν<0/bracketrightbig
=4
πb1√
b2−4a2/bracketleftbig
0<2a<b , −3
2<Reν<0/bracketrightbig
ET II 164(7)
2./integraldisplay∞
0xν+1/braceleftBig
[Jν(ax)]2−[Yν(ax)]2/bracerightBig
Hν(bx)dx
=0/bracketleftbig
0<b< 2a,−3
4<Reν<0/bracketrightbig
=23ν+2a2νb−ν−1
√πΓ/parenleftbig1
2−ν/parenrightbig/parenleftbig
b2−4a2/parenrightbig−ν−1
2/bracketleftbig
0<2a<b , −3
4<Reν<0/bracketrightbig
ET II 163(6)
6.852
1./integraldisplay∞
0x1−μ−νJν(x)Hμ(x)dx=(2ν−1)2−μ−ν
(μ+ν−1)Γ/parenleftbig
μ+1
2/parenrightbig
Γ/parenleftbig
ν+1
2/parenrightbig
/bracketleftbig
Reν>1
2,Re(μ+ν)>1/bracketrightbig
ET II 383(4)
2./integraldisplay∞
0xμ−ν+1Yμ(ax)Hν(bx)dx
=0/bracketleftbig
0<b<a , Re(ν−μ)>0,−3
2<Reμ<1
2/bracketrightbig
=21+μ−νaμb−ν
Γ(ν−μ)/parenleftbig
b2−a2/parenrightbigν−μ−1/bracketleftbig
0<a<b , Re(ν−μ)>0,−3
2<Reμ<1
2/bracketrightbig
ET II 163(3)
758 Functions Generated by Bessel Functions 6.853
3./integraldisplay∞
0xμ+ν+1Kμ(ax)Hν(bx)dx=2μ+ν+1bν+1
√πaμ+2ν+3Γ/parenleftbigg
μ+ν+3
2/parenrightbigg
F/parenleftbigg
1,μ+ν+3
2;3
2;−b2
a2/parenrightbigg
/bracketleftbig
Rea>0,b > 0,Reν>−3
2,Re(μ+ν)>−3
2/bracketrightbig
ET II 165(13)
6.853
1./integraldisplay∞
0x1−μ[sin (μπ)Jμ+ν(ax)+c o s ( μπ)Yμ+ν(ax)]Hν(bx)dx
=0/bracketleftbig
0<b<a , 1<Reμ<3
2,Reν>−3
2,Re(ν−μ)<1
2/bracketrightbig
=bν/parenleftbig
b2−a2/parenrightbigμ−1
2μ−1aμ+νΓ(μ)/bracketleftbig
0<a<b , 1<Reμ<3
2,Reν>−3
2,Re(ν−μ)<1
2/bracketrightbig
ET II 163(4)
2./integraldisplay∞
0xλ+1
2[Iμ(ax)−L−μ(ax)]Jν(bx)dx
=2λ+1
2cos(μπ)
πb−λ−3
2G22
33⎛
⎜⎝b2
a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+μ
2,1−μ
2,1+μ
2
3
4+λ+ν
2,1+μ
2,3
4+λ−ν
2⎞
⎟⎠
/bracketleftbig
Rea>0,b > 0,Re(μ+ν+λ)>−3
2,−Reν−5
2<Re(λ−μ)<1/bracketrightbig
ET II 76(21)
3./integraldisplay∞
0xλ+1
2[Hμ(ax)−Yμ(ax)]Jν(bx)dx
=2λ+1
2cos(μπ)
π2b−λ−3
2G23
33⎛
⎜⎝b2
a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−μ
2,1−μ
2,1+μ
2
3
4+λ+ν
2,1−μ
2,3
4+λ−ν
2⎞
⎟⎠
/bracketleftbig
b>0,|arga|<π , Re(λ+μ)<1,Re(λ+ν)+3
2>|Reμ|/bracketrightbig
ET II 73(6)
4./integraldisplay∞
0√x/bracketleftBig
Iν−1
2(ax)−Lν−1
2(ax)/bracketrightBig
Jν(bx)dx=/radicalbigg
2
πaν−1
2b−ν1√
a2+b2/bracketleftbig
Rea>0,b > 0,|Reν|<1
2/bracketrightbig
ET II 74(11)
5./integraldisplay∞
0xμ−ν+1[Iμ(ax)−Lμ(ax)]Jν(bx)dx=2μ−ν+1aμ−1bν−2μ−1
√πΓ/parenleftbig
ν−μ+1
2/parenrightbigF/parenleftbigg
1,1
2;ν−μ+1
2;−b2
a2/parenrightbigg
/bracketleftbig
−1<2R eμ+1<Reν+1
2,Rea>0,b > 0/bracketrightbig
ET II 74(13)
6./integraldisplay∞
0xμ−ν+1[Iμ(ax)−L−μ(ax)]Jν(bx)dx=2μ−ν+1a−μ−1bν−1
Γ/parenleftbig1
2−μ/parenrightbig
Γ/parenleftbig1
2+ν/parenrightbigF/parenleftbigg
1,1
2+μ;1
2+ν;−b2
a2/parenrightbigg
/bracketleftbig
Rea>0,Reν>−1
2,Reμ>−1,b > 0/bracketrightbig
ET II 75(18)
6.854
1./integraldisplay∞
0xH1
2ν/parenleftbig
ax2/parenrightbig
Kν(bx)dx=Γ/parenleftbig1
2ν+1/parenrightbig
21−1
2νaπS−1
2ν−1,1
2ν/parenleftbiggb2
4a/parenrightbigg
[a>0,Reb>0,Reν>−2]
ET II 150(75)
6.857 Combinations of Struve and Bessel functions 759
2./integraldisplay∞
0xH1
2ν/parenleftbig
ax2/parenrightbig
Jν(bx)dx=−1
2aY1
2ν/parenleftbiggb2
4a/parenrightbigg/bracketleftbig
a>0,b > 0,−2<Reν<3
2/bracketrightbig
ET II 73(3)
6.855
1./integraldisplay∞
0x2ν+1
2/bracketleftBig
Iν+1
2/parenleftBiga
x/parenrightBig
−Lν+1
2/parenleftBiga
x/parenrightBig/bracketrightBig
Jν(bx)dx=23
2aν+1
2√πbν+1J2ν+1/parenleftBig√
2ab/parenrightBig
K2ν+1/parenleftBig√
2ab/parenrightBig
/bracketleftbig
Rea>0,b > 0,−1<Reν<1
2/bracketrightbig
ET II 76(22)
2./integraldisplay∞
0/bracketleftBig
H−ν−1/parenleftBiga
x/parenrightBig
−Y−ν−1/parenleftBiga
x/parenrightBig/bracketrightBig
Jν(bx)dx
x=−4
π√
abcos(νπ)K−2ν−1/parenleftBig
2√
ab/parenrightBig
/bracketleftbig
|arga|<π , b> 0,|Reν|<1
2/bracketrightbig
ET II 74(8)
3./integraldisplay∞
0x2ν+1
2/bracketleftBig
Hν+1
2/parenleftBiga
x/parenrightBig
−Yν+1
2/parenleftBiga
x/parenrightBig/bracketrightBig
Jν(bx)dx
=−25/2π−3/2aν+1
2b−ν−1sin(νπ)K2ν+1/parenleftBig√
2abe1
4πi/parenrightBig
K2ν+1/parenleftBig√
2abe−1
4πi/parenrightBig
/bracketleftbig
|arga|<π , b> 0,−1<Reν<−1
6/bracketrightbig
ET II 74(9)
6.856/integraldisplay∞
0xYν/parenleftbig
a√x/parenrightbig
Kν/parenleftbig
a√x/parenrightbig
Hν(bx)dx=1
2b2exp/parenleftbigg
−a2
2b/parenrightbigg
/bracketleftBig
b>0,|arga|<π
4,Reν>−3
2/bracketrightBig
ET II 169(32)
6.857
1./integraldisplay∞
0xexp/parenleftbigga2x2
8/parenrightbigg
K1
2ν/parenleftbigga2x2
8/parenrightbigg
Hν(bx)dx
=2√πa−ν
2−1bν
2−1cos/parenleftBigνπ
2/parenrightBig
Γ/parenleftbigg
−1
2ν/parenrightbigg
exp/parenleftbiggb2
2a2/parenrightbigg
Wk,m/parenleftbiggb2
a2/parenrightbigg
k=1
4ν, m =1
2+1
4ν/bracketleftbig
|arga|<3
4π, b > 0,−3
2<Reν<0/bracketrightbig
ET II 167(24)
2./integraldisplay∞
0xσ−2exp/parenleftbigg
−1
2a2x2/parenrightbigg
Kμ/parenleftbigg1
2a2x2/parenrightbigg
Hν(bx)dx
=√π
2ν+2a−ν−σbν+1Γ/parenleftbigν+σ
2+μ/parenrightbig
Γ/parenleftbigν+σ
2−μ/parenrightbig
Γ/parenleftbig3
2/parenrightbig
Γ/parenleftbig
ν+3
2/parenrightbig
Γ/parenleftbigν+σ
2/parenrightbig
×3F3/parenleftbigg
1,ν+σ
2+μ,ν+σ
2−μ;3
2,ν+3
2,ν+σ
2;−b2
4a2/parenrightbigg
/bracketleftBig
b>0,|arga|<π
4,Re(σ+ν)>2|Reμ|/bracketrightBig
ET II 167(23)
760 Functions Generated by Bessel Functions 6.861
6.86 Lommel functions
6.861
1./integraldisplay∞
0xλ−1Sμ,ν(x)dx=Γ/bracketleftbig1
2(1 +λ+μ)/bracketrightbig
Γ/bracketleftbig1
2(1−λ−μ)/bracketrightbig
Γ/bracketleftbig1
2(1 +μ+ν)/bracketrightbig
Γ/bracketleftbig1
2(1 +μ−ν)/bracketrightbig
22−λ−μΓ/bracketleftbig1
2(ν−λ)+1/bracketrightbig
Γ/bracketleftbig
1−1
2(λ+ν)/bracketrightbig
/bracketleftbig
−Reμ<Reλ+1<5
2/bracketrightbig
ET II 385(17)
6.862
1./integraldisplayu
0xλ−1
2μ−1
2(u−x)σ−1sμ,ν/parenleftbig
a√x/parenrightbig
dx
=Γ (σ)aμ+1uλ+σΓ(λ+1 )
(μ−ν+1 )(μ+ν+1 )Γ ( λ+σ+1 )
×2F3/parenleftbigg
1,1+λ;μ−ν+3
2,μ+ν+3
2,λ+σ+1 ;−a2u
4/parenrightbigg
[Reλ>−1,Reσ>0] ET II 199(92)
2./integraldisplay∞
ux1
2ν(x−u)μ−1sλ,ν/parenleftbig
a√x/parenrightbig
dx=B/bracketleftbig
μ,1
2(1−λ−ν)−μ/bracketrightbig
u1
2μ+1
2ν
aμSλ+μ,μ+ν/parenleftbig
a√u/parenrightbig
/bracketleftbig/vextendsingle/vextendsinglearg/parenleftbig
a√u/parenrightbig/vextendsingle/vextendsingle<π , 0<2R eμ<1−Re(λ+ν)/bracketrightbig
ET II 211(71)
6.863/integraldisplay∞
0√xe−αxsμ,1
4/parenleftbiggx2
2/parenrightbigg
dx=2−2μ−1√αΓ/parenleftbigg
2μ+3
2/parenrightbigg
S−μ−1,1
4/parenleftbiggα2
2/parenrightbigg
/bracketleftbig
Reα>0,Reμ>−3
4/bracketrightbig
ET I 209(38)
6.864/integraldisplay∞
0exp[(μ+1 )x]sμ,ν(asinhx)dx=2μ−2πcosec( μπ)Γ(/rho1)Γ(σ)
×/bracketleftBig
I/rho1/parenleftBiga
2/parenrightBig
Iσ/parenleftBiga
2/parenrightBig
−I−/rho1/parenleftBiga
2/parenrightBig
I−σ/parenleftBiga
2/parenrightBig/bracketrightBig
2/rho1=μ+ν+1,2σ=μ−ν+1 [ a>0,−2<Reμ<0]ET II 386(22)
6.865/integraldisplay∞
0√
sinhxcosh(νx)Sμ,1
2(acoshx)dx=B/parenleftbig1
4−μ+ν
2,1
4−μ−ν
2/parenrightbig
√a2μ+3
2Sμ+1
2,ν(a)
/bracketleftbig
|arga|<π , Reμ+|Reν|<1
2/bracketrightbig
ET II 388(31)
6.866
1./integraldisplay∞
0x−μ−1cos(ax)sμ,ν(x)dx
=0 [ a>1]
=2μ−1
2√πΓ/parenleftbiggμ+ν+1
2/parenrightbigg
Γ/parenleftbiggμ−ν+1
2/parenrightbigg/parenleftbig
1−a2/parenrightbig1
2μ+1
4Pμ−1
2
ν−1
2(a)[ 0 <a< 1]
ET II 386(18)
2./integraldisplay∞
0x−μsin(ax)Sμ,ν(x)dx=2−μ−1
2√πΓ/parenleftbigg
1−μ+ν
2/parenrightbigg
Γ/parenleftbigg
1−μ−ν
2/parenrightbigg/parenleftbig
a2−1/parenrightbig1
2μ−1
4Pμ−1
2
ν−1
2(a)
[a>1,Reμ<1−|Reν|]
ET II 387(23)
6.871 Thomson functions 761
6.867
1./integraldisplayπ/2
0cos(2μx)S2μ−1,2ν(acosx)dx
=π22μ−3a2μcosec(2 νπ)
Γ(1−μ−ν)Γ( 1−μ+ν)/bracketleftBig
Jμ+ν/parenleftBiga
2/parenrightBig
Yμ−ν/parenleftBiga
2/parenrightBig
−Jμ−ν/parenleftBiga
2/parenrightBig
Yμ+ν/parenleftBiga
2/parenrightBig/bracketrightBig
[Reμ>−2,|Reν|<1]ET II 388(29)
2./integraldisplayπ/2
0cos[(μ+1 )x]sμ,ν(acosx)dx=2μ−2πΓ(/rho1)Γ(σ)J/rho1/parenleftBiga
2/parenrightBig
Jσ/parenleftBiga
2/parenrightBig
2/rho1=μ+ν+1,2σ=μ−ν+1 [ R e μ>−2]ET II 386(21)
6.868/integraldisplayπ/2
0cos(2μx)
cosxS2μ,2ν(asecx)dx=π22μ−1
aWμ,ν/parenleftbig
aeiπ
2/parenrightbig
Wμ,ν/parenleftbig
ae−iπ
2/parenrightbig
[|arga|<π , Reμ<1] ET II 388(30)
6.869
1./integraldisplay∞
0x1−μ−νJν(ax)Sμ,−μ−2ν(x)dx=√πaν−1Γ(1−μ−ν)
2μ+2νΓ/parenleftbig
ν+1
2/parenrightbig/parenleftbig
a2−1/parenrightbig1
2(μ+ν−1)Pμ+ν−1
μ+ν(a)
/bracketleftbig
a>1,Reν>−1
2,Re(μ+ν)<1/bracketrightbig
ET II 388(28)
2./integraldisplay∞
0x−μJν(ax)sν+μ,−ν+μ+1(x)dx
=2ν−1Γ(ν)a−ν/parenleftbig
1−a2/parenrightbigμ/bracketleftbig
0<a< 1,Reμ>−1,−1e<Reν<3
2/bracketrightbig
=0/bracketleftbig
1<a , Reμ>−1,−1<Reν<3
2/bracketrightbig
ET II 388(28)
3./integraldisplay∞
0xKν(bx)sμ,1
2ν/parenleftbig
ax2/parenrightbig
dx=1
4aΓ/parenleftbigg
μ+1
2ν+1/parenrightbigg
Γ/parenleftbigg
μ−1
2ν+1/parenrightbigg
S−μ−1,1
2ν/parenleftbiggb2
4a/parenrightbigg
/bracketleftbig
Reμ>1
2|Reν|−2,a > 0,Reb>0/bracketrightbig
ET II 151(78)
6.87 Thomson functions
6.871
1./integraldisplay∞
0e−βxberxdx=/parenleftBig/radicalbig
β4+1+ β2/parenrightBig1/2
/radicalbig
2(β4+1 )ME 40
2./integraldisplay∞
0e−βxbeixdx=/parenleftBig/radicalbig
β4+1−β2/parenrightBig1/2
/radicalbig
2(β4+1 )ME 40
762 Functions Generated by Bessel Functions 6.872
6.872
1./integraldisplay∞
0e−βxberν/parenleftbig
2√x/parenrightbig
dx=1
2β/radicalbiggπ
β⎡
⎣J1
2(ν−1)/parenleftbigg1
2β/parenrightbigg
cos/parenleftbigg1
2β+3νπ
4/parenrightbigg
−J1
2(ν+1)/parenleftbigg1
2β/parenrightbigg
cos/parenleftbigg1
2β+3ν+6
4π/parenrightbigg⎤
⎦
MI 49
2./integraldisplay∞
0e−βxbeiν/parenleftbig
2√x/parenrightbig
dx=1
2β/radicalbiggπ
β⎡
⎣J1
2(ν−1)/parenleftbigg1
2β/parenrightbigg
sin/parenleftbigg1
2β+3ν
4π/parenrightbigg
−J1
2(ν+1)/parenleftbigg1
2β/parenrightbigg
sin/parenleftbigg1
2β+3ν+6
4π/parenrightbigg⎤
⎦
MI 49
3./integraldisplay∞
0e−βxber/parenleftbig
2√x/parenrightbig
dx=1
βcos1
βME 40
4./integraldisplay∞
0e−βxbei/parenleftbig
2√x/parenrightbig
dx=1
βsin1
βME 40
5./integraldisplay∞
0e−βxker/parenleftbig
2√x/parenrightbig
dx=−1
2β/bracketleftbigg
cos1
βci1
β+s i n1
βsi1
β/bracketrightbigg
MI 50
6./integraldisplay∞
0e−βxkei/parenleftbig
2√x/parenrightbig
dx=−1
2β/bracketleftbigg
sin1
βci1
β−cos1
βsi1
β/bracketrightbigg
MI 50
7./integraldisplay∞
0e−βxberν/parenleftbig
2√x/parenrightbig
beiν/parenleftbig
2√x/parenrightbig
dx=1
2βJν/parenleftbigg2
β/parenrightbigg
sin/parenleftbigg2
β+3νπ
2/parenrightbigg
[Reν>−1] MI 49
6.873/integraldisplay∞
0/bracketleftbig
ber2
ν/parenleftbig
2√x/parenrightbig
+b e i2
ν/parenleftbig
2√x/parenrightbig/bracketrightbig
e−βxdx=1
βIν/parenleftbigg2
β/parenrightbigg
[Reν>−1] ME 40
6.874
1./integraldisplay∞
0e−βx
√xber2ν/parenleftBig
2√
2x/parenrightBig
dx=/radicalbiggπ
βJν/parenleftbigg1
β/parenrightbigg
cos/parenleftbigg1
β−3π
4+3νπ
2/parenrightbigg
/bracketleftbig
Reν>−1
2/bracketrightbig
MI 49
2./integraldisplay∞
0e−βx
√xbei2ν/parenleftBig
2√
2x/parenrightBig
dx=/radicalbiggπ
βJν/parenleftbigg1
β/parenrightbigg
sin/parenleftbigg1
β−3π
4+3νπ
2/parenrightbigg
/bracketleftbig
Reν>−1
2/bracketrightbig
MI 49
3./integraldisplay∞
0xν
2berν/parenleftbig√x/parenrightbig
e−βxdx=2−ν
β1+νcos/parenleftbigg1
4β+3νπ
4/parenrightbigg
[Reν>−1] ME 40
4./integraldisplay∞
0xν
2beiν/parenleftbig√x/parenrightbig
e−βxdx=2−ν
β1+νsin/parenleftbigg1
4β+3νπ
4/parenrightbigg
[Reν>−1] ME 40
6.921 Mathieu, hyperbolic, and trigonometric functions 763
6.875
1./integraldisplay∞
0e−βx/bracketleftbigg
ker/parenleftbig
2√x/parenrightbig
−1
2lnxber/parenleftbig
2√x/parenrightbig/bracketrightbigg
dx=1
β/bracketleftbigg
lnβcos1
β+π
4sin1
β/bracketrightbigg
MI 50
2./integraldisplay∞
0e−βx/bracketleftbigg
kei/parenleftbig
2√x/parenrightbig
−1
2lnxbei/parenleftbig
2√x/parenrightbig/bracketrightbigg
dx=1
β/bracketleftbigg
lnβsin1
β−π
4cos1
β/bracketrightbigg
MI 50
6.876
1./integraldisplay∞
0xkeixJ1(ax)dx=−1
2aarctan a2[a>0] ET II 21(32)
2./integraldisplay∞
0xkerxJ1(ax)dx=1
2aln/radicalbig
(1 +a4)[ a>0] ET II 21(33)
6.9 Mathieu Functions
Notation :k2=q. For definition of the coefficients A(m)
pandB(m)
p, see section 8.6.
6.91 Mathieu functions
6.911
1./integraldisplay2π
0cem(z,q)cep(z,q)dz=0 [ m/negationslash=p] MA
2./integraldisplay2π
0[ce2n(z,q)]2dz=2π/bracketleftBig
A(2n)
0/bracketrightBig2
+π∞/summationdisplay
r=1/bracketleftBig
A(2n)
2r/bracketrightBig2
=π MA
3./integraldisplay2π
0[ce2n+1(z,q)]2dz=π∞/summationdisplay
r=0/bracketleftBig
A(2n+1)
2r+1/bracketrightBig2
=π MA
4./integraldisplay2π
0sem(z,q)sep(z,q)dz=0 [ m/negationslash=p] MA
5./integraldisplay2π
0[se2n+1(z,q)]2dz=π∞/summationdisplay
r=0/bracketleftBig
B(2n+1)
2r+1/bracketrightBig2
=π MA
6./integraldisplay2π
0[se2n+2(z,q)]2dz=π∞/summationdisplay
r=0/bracketleftBig
B(2n+2)
2r+2/bracketrightBig2
=π MA
7./integraldisplay2π
0sem(z,q)cep(z,q)dz=0 [ m=1,2,...;p=1,2,...] MA
6.92 Combinations of Mathieu, hyperbolic, and trigonometric functions
6.921
1./integraldisplayπ
0cosh (2 kcosusinhz)c e2n(u,q)du=πA(2n)
0
ce2n/parenleftbigπ
2,q/parenrightbig(−1)nCe2n(z,−q)
[q>0] MA
764 Mathieu Functions 6.922
2./integraldisplayπ
0cosh (2 ksinucoshz)c e2n(u,q)du=πA(2n)
0
ce2n(0,q)(−1)nCe2n(z,−q)
[q>0] MA
3./integraldisplayπ
0sinh (2 ksinucoshz)s e2n+1(u,q)du=πkB(2n+1)
1
se/prime
2n+1(0,q)(−1)nCe2n+1(z,−q)
[q>0] MA
4./integraldisplayπ
0sinh (2 kcosusinhz)c e2n+1(u,q)du=πkA(2n+1)
1
ce/prime
2n+1/parenleftbigπ
2,q/parenrightbig(−1)n+1Se2n+1(z,−q)
[q>0] MA
5./integraldisplayπ
0sinh (2 ksinusinz)s e2n+1(u,q)du=πkB(2n+1)
1
se/prime
2n+1(0,q)se2n+1(z,q)
[q>0] MA
6.922
1./integraldisplayπ
0cosucoshzcos (2ksinusinhz)c e2n+1(u,q)du=πA(2n+1)
1
2c e2n+1(0,q)Ce2n+1(z,q)
[q>0] MA
2./integraldisplayπ
0sinusinhzcos (2kcosucoshz)s e2n+1(u,q)du=πB1(2n+1)
2s e2n+1/parenleftBigπ
2,q/parenrightBigSe2n+1(z,q)
[q>0] MA
3./integraldisplayπ
0sinusinhzsin (2kcosucoshz)s e2n+2(u,q)du=−πkB(2n+2)
2
2s e/prime
2n+2/parenleftbigπ
2,q/parenrightbigSe2n+2(z,q)
[q>0] MA
4./integraldisplayπ
0cosucoshzsin (2ksinusinhz)s e2n+2(u,q)du=πkB(2n+2)
2
2s e/prime
2n+2(0,q)Se2n+2(z,q)
[q>0] MA
5./integraldisplayπ
0sinucoshzcosh (2 kcosusinhz)s e2n+1(u,q)du=πB1(2n+1)
2s e2n+1/parenleftBigπ
2,q/parenrightBig(−1)nCe2n+1(z,−q)
[q>0] MA
6./integraldisplayπ
0cosusinhzcosh (2 ksinucoshz)c e2n+1(u,q)du=πA(2n+1)
1
2c e2n+1(0,q)(−1)nSe2n+1(z,−q)
[q>0] MA
7./integraldisplayπ
0sinucoshzsinh (2 kcosusinhz)s e2n+2(u,q)du=πkB(2n+2)
2
2s e/prime
2n+2/parenleftBigπ
2,q/parenrightBig(−1)n+1Se2n+2(z,−q)
[q>0] MA
6.924 Mathieu, hyperbolic, and trigonometric functions 765
8./integraldisplayπ
0cosusinhzsinh (2 ksinucoshz)s e2n+2(u,q)du=πkB(2n+2)
2
2s e/prime
2n+2(0,q)(−1)nSe2n+2(z,−q)
[q>0] MA
6.923
1./integraldisplay∞
0sin(2kcoshzcoshu)sin h zsinhuSe2n+1(u,q)du=−πB1(2n+1)
4s e2n+1/parenleftbig1
2π,q/parenrightbigSe2n+1(z,q)
[q>0] MA
2./integraldisplay∞
0cos(2kcoshzcoshu)s i n h zsinhuSe2n+1(u,q)du=−πB1(2n+1)
4s e2n+1/parenleftbig1
2π,q/parenrightbigGey2n+1(z,q)
[q>0] MA
3./integraldisplay∞
0sin(2kcoshzcoshu)sin h zsinhuSe2n+2(u,q)du=−kπB2(2n+2)
4s e/prime
2n+2/parenleftbig1
2π,q/parenrightbigGey2n+2(z,q)
[q>0] MA
4./integraldisplay∞
0cos(2kcoshzcoshu)s i n h zsinhuSe2n+2(u,q)du=−kπB2(2n+2)
4s e2n+2/parenleftbig1
2π,q/parenrightbigSe2n+2(z,q)
[q>0] MA
5./integraldisplay∞
0sin(2kcoshzcoshu)Ce2n(u,q)du=πA(2n)
0
2c e2n/parenleftbig1
2π,q/parenrightbigCe2n(z,q)
[q>0] MA
6./integraldisplay∞
0cos(2kcoshzcoshu)C e2n(u,q)du=−πA(2n)
0
2c e2n/parenleftbig1
2π,q/parenrightbigFey2n(z,q)
[q>0] MA
7./integraldisplay∞
0sin(2kcoshzcoshu)Ce2n+1(u,q)du=kπA(2n+1)
1
2c e/prime
2n+1/parenleftbig1
2π,q/parenrightbigFey2n+1(z,q)
[q>0] MA
8./integraldisplay∞
0cos(2kcoshzcoshu)C e2n+1(u,q)du=kπA(2n+1)
1
2c e/prime
2n+1/parenleftbig1
2π,q/parenrightbigCe2n+1(z,q)
[q>0] MA
6.924
1./integraldisplayπ
0cos(2kcosucosz)c e2n(u,q)du=πA(2n)
0
ce2n/parenleftbig1
2π,q/parenrightbigce2n(z,q)
[q>0] MA
2./integraldisplayπ
0sin (2kcosucosz)c e2n+1(u,q)du=−πkA(2n+1)
1
ce/prime
2n+1/parenleftbig1
2π,q/parenrightbigce2n+1(z,q)
[q>0] MA
766 Mathieu Functions 6.925
3./integraldisplayπ
0cos(2kcosucoshz)c e2n(u,q)du=πA(2n)
0
ce2n/parenleftbig1
2π,q/parenrightbigCe2n(z,q)
[q>0] MA
4./integraldisplayπ
0cos(2ksinusinhz)c e2n(u,q)du=πA(2n)
0
ce2n(0,q)Ce2n(z,q)
[q>0] MA
5./integraldisplayπ
0sin (2kcosucoshz)c e2n+1(u,q)du=−πkA(2n+1)
1
ce/prime
2n+1/parenleftbig1
2π,q/parenrightbigCe2n+1(z,q)
[q>0] MA
6./integraldisplayπ
0sin (2ksinusinhz)s e2n+1(u,q)du=πkB(2n+1)
1
se/prime/prime
2n+1(0,q)Se2n+1(z,q)
[q>0] MA
6.925 Notation :z1=2k/radicalbig
cosh2ξ−sin2η,a n dt a n α=t a n h ξtanη
1./integraldisplay2π
0sin [z1cos(θ−α)] ce2n(θ,q)dθ=0. MA
2./integraldisplay2π
0cos[z1cos(θ−α)] ce2n(θ,q)dθ=2πA(2n)
0
ce2n(0,q)ce2n/parenleftbig1
2π,q/parenrightbigCe2n(ξ,q)ce2n(η,q) MA
3./integraldisplay2π
0sin [z1cos(θ−α)] ce2n+1(θ,q)dθ=−2πkA(2n+1)
1
ce2n+1(0,q)ce/prime
2n+1/parenleftbig1
2π,q/parenrightbigCe2n+1(ξ,q)ce2n+1(η,q)
MA
4./integraldisplay2π
0cos[z1cos(θ−α)] ce2n+1(θ,q)dθ=0 MA
5./integraldisplay2π
0sin [z1cos(θ−α)] se2n+1(θ,q)dθ=2πkB(2n+1)
1
se2n+1(0,q)se2n+1/parenleftbig1
2π,q/parenrightbigSe2n+1(ξ,q)se2n+1(η,q)
MA
6./integraldisplay2π
0cos[z1cos(θ−α)] se2n+1(θ,q)dθ=0 MA
7./integraldisplay2π
0sin [z1cos(θ−α)] se2n+2(θ,q)dθ=0 MA
8./integraldisplay2π
0cos[z1cos(θ−α)] se2n+2(θ,q)dθ=2πk2B2(2n+2)
se/prime
2n+2(0,q)se/prime
2n+2/parenleftbig1
2π,q/parenrightbigSe2n+2(ξ,q)se2n+2(η,q)
MA
6.926/integraldisplayπ
0sinusinzsin (2kcosucosz)s e2n+2(u,q)du=−πkB(2n+2)
2
2s e/prime
2n+2/parenleftbigπ
2,q/parenrightbigse2n+2(z,q)
[q>0] MA
6.941 Eigenfunctions of Helmholtz equation 767
6.93 Combinations of Mathieu and Bessel functions
6.931
1./integraldisplayπ
0J0/braceleftBig
k[2 (cos2 u+c o s2 z)]1/2/bracerightBig
ce2n(u,q)du=π/bracketleftBig
A(2n)
0/bracketrightBig2
ce2n(0,q)ce2n/parenleftBigπ
2,q/parenrightBigce2n(z,q) MA
2./integraldisplay2π
0Y0/braceleftBig
k[2(cos2 u+c o s h2 z)]1/2/bracerightBig
ce2n(u,q)du=2π/bracketleftBig
A(2n)
0/bracketrightBig2
ce2n(0,q)ce2n/parenleftBigπ
2,q/parenrightBigFey2n(z,q) MA
6.94 Relationships between eigenfunctions of the Helmholtz equation in different
coordinate systems
Notation : Particular solutions of the Helmholtz equation in three-dimensional infinite space
∇2Ψ+k2Ψ=0
in Cartesian ( x, y, z), spherical ( r, θ, φ), and cylindrical ( ρ, z, φ ) coordinates are
Ψkxkykz(x, y, z)∝ei(kxx+kyy+kzz)with k2=k2
x+k2
y+k2
z
Ψlm(r, θ, φ)∝eimφ/radicalbigg
k
rZl+1/2(kr)Pm
l(cosθ)
Ψmkz(ρ, z, φ )∝ei(mφ+kzz)Zl+1/2/parenleftBig
ρ/radicalbig
k2−k2z/parenrightBig
withPm
l(cosθ) the associated Legendre function, Zis any Bessel function, m=0,1,...,l ;l∈N,
r2=ρ2+z2,ρ=rsinθ,z=rcosθ,φ= arccot( x/y), and k2
t=k2−k2
z.
6.941
1./integraldisplayk
−keiρzJm/parenleftBig
ρ/radicalbig
k2−ρ2/parenrightBig
Pm
l/parenleftBigp
k/parenrightBig
dp=il−m/radicalbigg
2πk
rJl+1/2(kr)Pm
l/parenleftBigz
r/parenrightBig
[ρ>0,l≥m≥0]
2./integraldisplay∞
−∞e−iρzJl+1/2(kr)Pm
l/parenleftBigz
r/parenrightBig
dz=im−l/radicalbigg
2πr
kJm/parenleftBig
ρ/radicalbig
k2−ρ2/parenrightBig
Pm
l/parenleftBigρ
k/parenrightBig
[ρ>0,l≥m≥0]
3./integraldisplay∞
0Jm(ρkt)cos/bracketleftbigg
kxx+marcsin/parenleftbiggx
ρ/parenrightbigg/bracketrightbigg
dx
=(−1)m
/radicalbig
k2
t−k2xcos/bracketleftbigg
y/radicalBig
k2
t−k2x+marccos/parenleftbiggkx
kt/parenrightbigg/bracketrightbigg/bracketleftbig
k2
x<k2
t/bracketrightbig
=0/bracketleftbig
k2
x>k2
t/bracketrightbig
768 Mathieu Functions 6.941
4./integraldisplay∞
0Ym(ρkt)c o s/bracketleftbigg
kxx+marcsin/parenleftbiggx
ρ/parenrightbigg/bracketrightbigg
dx
=(−1)m
/radicalbig
k2
t−k2xsin/bracketleftbigg
y/radicalBig
k2
t−k2x+marccos/parenleftbiggkx
kt/parenrightbigg/bracketrightbigg/bracketleftbig
k2
x<k2
t/bracketrightbig
=(−1)m
/radicalbig
k2x−k2
texp/bracketleftbigg
−y/radicalBig
k2x−k2
t−msign (kx) arccosh/parenleftbigg|kx|
kt/parenrightbigg/bracketrightbigg/bracketleftbig
k2
x>k2
t/bracketrightbig
5./integraldisplay∞
−∞H(j)
l+1/2(kr)Pm
l/parenleftBigz
r/parenrightBig
e−ikzzdx=im−l/radicalbigg
2πr
kH(j)
m/parenleftBig
ρ/radicalbig
k2−k2z/parenrightBig
Pm
l/parenleftbiggkz
k/parenrightbigg
[ρ>0]
T h er e s u l ti st r u ef o r j=1i f π>arg/radicalbig
k2−k2z≥0, for j=2i f −π<arg/radicalbig
k2−k2z≤0.
6./integraldisplay∞
−∞H(j)
m/parenleftBig
ρ/radicalbig
k2−k2z/parenrightBig
Pm
l/parenleftbiggkz
k/parenrightbigg
eikzzdkz=il−m/radicalbigg
2πk
rH(j)
l+1/2(kr)Pm
l/parenleftBigz
r/parenrightBig
T h er e s u l ti st r u ef o r j=1i f π>arg/radicalbig
k2−k2z≥0, for j=2i f −π<arg/radicalbig
k2−k2z≤0.
7./integraldisplay∞
−∞Jl+1/2(kr)Pm
l/parenleftBigz
r/parenrightBig
e−ikzzdz=im−l/radicalbigg
2πr
kJm/parenleftBig
ρ/radicalbig
k2−k2z/parenrightBig
Pm
l/parenleftbiggkz
k/parenrightbigg/bracketleftbig
k2
z<k2/bracketrightbig
=0/bracketleftbig
k2
z>k2/bracketrightbig
8./integraldisplayk
−kJm/parenleftBig
ρ/radicalbig
k2−k2z/parenrightBig
Pm
l/parenleftbiggkz
k/parenrightbigg
eikzzdkz=il−m/radicalbigg
2πk
rJl+1/2(kr)Pm
l/parenleftBigz
r/parenrightBig
9./integraldisplay∞
−∞Yl+1/2(kr)Pm
l/parenleftBigz
r/parenrightBig
e−ikzzdz=im−l/radicalbigg
2πr
kYm/parenleftBig
ρ/radicalbig
k2−k2z/parenrightBig
Pm
l/parenleftbiggkz
k/parenrightbigg/bracketleftbig
k2
z<k2/bracketrightbig
=−2im−l/radicalbigg
2r
kπKm/parenleftBig
ρ/radicalbig
k2z−k2/parenrightBig
Pm
l/parenleftbiggkz
k/parenrightbigg/bracketleftbig
k2
z>k2/bracketrightbig
10. il−m/integraldisplayk
−kYm/parenleftBig
ρ/radicalbig
k2−k2z/parenrightBig
Pm
l/parenleftbiggkz
k/parenrightbigg
eikzzdkz
−4
π/integraldisplay∞
kcos/bracketleftbig
kzz+1
2π(m−l)/bracketrightbig
Pm
l/parenleftbiggkz
k/parenrightbigg
Km/parenleftBig
ρ/radicalbig
k2z−k2/parenrightBig
eikzzdkz
=/radicalbigg
2πk
rYl+1/2(kr)Pm
l/parenleftBigz
r/parenrightBig
7.113 Associated Legendre functions 769
7.1–7.2 Associated Legendre Functions
7.11 Associated Legendre functions
7.111/integraldisplay1
cosϕPν(x)dx=s i nϕP−1
ν(cosϕ) MO 90
7.112
1./integraldisplay1
−1Pm
n(x)Pm
k(x)dx=0 [ n/negationslash=k]
=2
2n+1(n+m)!
(n−m)![n=k]
SM III 185, WH
2./integraldisplay1
−1Qm
n(x)Pm
k(x)dx=(−1)m1−(−1)n+k(n+m)!
(k−n)(k+n+1 ) (n−m)!EH I 171(18)
3./integraldisplay1
−1Pν(x)Pσ(x)dx
=2πsinπ(σ−ν)+4s i n ( πν)sin(πσ)[ψ(ν+1 )−ψ(σ+1 ) ]
π2(σ−ν)(σ+ν+1 )[σ+ν+1/negationslash=0 ] EH I 170(7)
=π2−2( s inπν)2ψ/prime(ν+1 )
π2/parenleftbigg
ν+1
2/parenrightbigg[σ=ν] EH I 170(9)a
4./integraldisplay1
−1Qν(x)Qσ(x)dx=[ψ(ν+1 )−ψ(σ+1 ) ][ 1+c o s ( πσ)cos(νπ)]−π
2sinπ(ν−σ)
(σ−ν)(σ+ν+1 )
[σ+ν+1/negationslash=0 ; ν, σ /negationslash=−1,−2,−3,...]
EH I 170(11)
=1
2π2−ψ/prime(ν+1 )/bracketleftBig
1+( c o s νπ)2/bracketrightBig
2ν+1
[ν=σ, ν /negationslash=−1,−2,−3,...]
EH I 170(12)
5./integraldisplay1
−1Pν(x)Qσ(x)dx=1−cosπ(σ−ν)−2π−1sin(πν)cos(πσ)[ψ(ν+1 )−ψ(σ+1 ) ]
(ν−σ)(ν+σ+1 )
[Reν>0,Reσ>0,σ/negationslash=ν]
EH I 170(13)
=−sin(2νπ)ψ/prime(ν+1 )
π(2ν+1 )
[Reν>0,σ=ν]
EH I 171(14)
7.113 Notation :A=Γ/parenleftbig1
2+ν
2/parenrightbig
Γ/parenleftbig
1+σ
2/parenrightbig
Γ/parenleftbig1
2+σ
2/parenrightbig
Γ/parenleftbig
1+ν
2/parenrightbig
1./integraldisplay1
0Pν(x)Pσ(x)dx=Asinπσ
2cosπν
2−A−1sinπν
2cosπσ
2
1
2π(σ−ν)(σ+ν+1 )EH I 171(15)
770 Associated Legendre Functions 7.114
2./integraldisplay1
0Qν(x)Qσ(x)dx=ψ(ν+1 )−ψ(σ+1 )−π
2/bracketleftBig/parenleftbig
A−A−1/parenrightbig
sinπ(σ+ν)
2/parenleftbig
A+A−1/parenrightbig
sinπ(σ−ν)
2/bracketrightBig
(σ−ν)(σ+ν+1 )
[Reν>0,Reσ>0] EH I 171(16)
3./integraldisplay1
0Pν(x)Qσ(x)dx=A−1cosπ(ν−σ)
2−1
(σ−ν)(σ+ν+1 )[Reν>0,Reσ>0] EH I 171(17)
7.114
1./integraldisplay∞
1Pν(x)Qσ(x)dx=1
(σ−ν)(σ+ν+1 )[Re(σ−ν)>0,Re(σ+ν)>−1]
ET II 324(19)
2./integraldisplay∞
1Qν(x)Qσ(x)dx=ψ(σ+1 )−ψ(ν+1 )
(σ−ν)(σ+ν+1 )
[Re(ν+σ)>−1;σ, ν/negationslash=−1,−2,−3,...]EH I 170(5)
3./integraldisplay∞
1[Qν(x)]2dx=ψ/prime(ν+1 )
2ν+1/bracketleftbig
Reν>−1
2/bracketrightbig
EH I 170(6)
7.115/integraldisplay∞
1Qν(x)dx=1
ν(ν+1 )[Reν>0] ET II 324(18)
7.12–7.13 Combinations of associated Legendre functions and powers
7.121/integraldisplay1
cosϕxPν(x)dx=−sinϕ
(ν−1)(ν+2 )/bracketleftbig
sinϕPν(cosϕ) + cos ϕP1
ν(cosϕ)/bracketrightbig
MO 90
7.122
1./integraldisplay1
0[Pm
n(x)]2
1−x2dx=1
2m(n+m)!
(n−m)![0<m≤n] MO 74
2./integraldisplay1
0[Pμ
ν(x)]2dx
1−x2=−Γ(1 + μ+ν)
2μΓ(1−μ+ν)[Reμ<0,ν+μis a positive integer]
EH I 172(26)
3./integraldisplay1
0/bracketleftbig
Pn−ν
ν(x)/bracketrightbig2dx
1−x2=−n!
2(n−ν)Γ(1−n+2ν)[n=0,1,2,...;R e ν>n]
ET II 315(9)
7.123/integraldisplay1
−1Pm
n(x)Pk
n(x)dx
1−x2=0 [ 0 ≤m≤n,0≤k≤n;m/negationslash=k]
MO 74
7.124/integraldisplay1
−1xk(z−x)−1/parenleftbig
1−x2/parenrightbig1
2mPm
n(x)dx=(−2)m/parenleftbig
z2−1/parenrightbig1
2mQm
n(z)·zk
[m≤n;k=0,1,...,n −m;
zis in the complex plane with a cut along the interval ( −1,1) on the real axis]
ET II 279(26)
7.128 Associated Legendre functions and powers 771
7.125/integraldisplay1
−1/parenleftbig
1−x2/parenrightbig1
2mPm
k(x)Pm
l(x)Pm
n(x)dx=(−1)mπ−3/2(k+m)!(l+m)! (n+m)!(s−m)!
(k−m)!(l−m)!(n−m)!(s−k)!
×Γ/parenleftbig
m+1
2/parenrightbig
Γ/parenleftbig
t−k+1
2/parenrightbig
Γ/parenleftbig
t−l+1
2/parenrightbig
Γ/parenleftbig
t−n+1
2/parenrightbig
(s−l)!(s−n)! Γ/parenleftbig
s+3
2/parenrightbig
[2s=k+l+n+mand 2t=k+l−n−mare both even
l≥m, m ≤k−l−m≤n≤k+l+m]
ET II 280(32)
7.126
1./integraldisplay1
0Pν(x)xσdx=√π2−σ−1Γ(1 + σ)
Γ/parenleftbig
1+1
2σ−1
2ν/parenrightbig
Γ/parenleftbig1
2σ+1
2ν+3
2/parenrightbig[Reσ>−1] EH I 171(23)
2./integraldisplay1
0xσPm
ν(x)dx=(−1)mπ1/22−2m−1Γ/parenleftbig1+σ
2/parenrightbig
Γ(1 + m+ν)
Γ/parenleftbig1
2+1
2m/parenrightbig
Γ/parenleftbig3
2+σ
2+m
2/parenrightbig
Γ(1−m+ν)
×3F2/parenleftbiggm+ν+1
2,m−ν
2,m
2+1 ;m+1,3+σ+m
2;1/parenrightbigg
[Reσ>−1;m=0,1,2,...]ET II 313(2)
3./integraldisplay1
0xσPμ
ν(x)dx=π1/222μ−1Γ/parenleftbig1+σ
2/parenrightbig
Γ/parenleftbig1−μ
2/parenrightbig
Γ/parenleftbig3+σ−μ
2/parenrightbig3F2/parenleftbiggν−μ+1
2,−μ+ν
2,1−μ
2;1−μ,3+σ−μ
2;1/parenrightbigg
[Reσ>−1,Reμ<2] ET II 313(3)
4./integraldisplay∞
1xμ−1Qν(ax)dx=eμπiΓ(μ)a−μ/parenleftbig
a2−1/parenrightbig1
2μQ−μ
ν(a)
[|arg(a−1)|<π , Reμ>0,Re(ν−μ)>−1]ET II 325(26)
7.127/integraldisplay1
−1(1 +x)σPν(x)dx=2σ+1[Γ(σ+1 ) ]2
Γ(σ+ν+2 )Γ ( 1+ σ−ν)[Reσ>−1] ET II 316(15)
7.128
1./integraldisplay1
−1(1−x)−1
2μ(1 +x)1
2μ−1
2(z+x)μ−3
2Pμ
ν(x)dx
=−Γ/parenleftbig
μ−1
2/parenrightbig
(z−1)μ−1
2(z+1 )−1/2
π1/2e2μπiΓ(μ+ν)Γ (μ−ν−1)
×/braceleftBigg
Qμ
ν/bracketleftBigg/parenleftbigg1+z
2/parenrightbigg1/2/bracketrightBigg
Qμ−1
−ν−1/bracketleftBigg/parenleftbigg1+z
2/parenrightbigg1/2/bracketrightBigg
+Qμ−1
ν/bracketleftBigg/parenleftbigg1+z
2/parenrightbigg1/2/bracketrightBigg
Qμ
−ν−1/bracketleftBigg/parenleftbigg1+z
2/parenrightbigg1/2/bracketrightBigg/bracerightBigg
[−1
2<Reμ<1,
zis in the complex plane with a cut along the interval ( −1,1) of the real axis]
ET II 317(20)
2./integraldisplay1
−1(1−x)−1
2μ(1 +x)1
2μ−1
2(z+x)μ−1
2Pμ
ν(x)dx
=2e−2μπiΓ/parenleftbig1
2+μ/parenrightbig
π1/2Γ(μ−ν)Γ(μ+ν+1 )(z−1)μQμ
ν/bracketleftBigg/parenleftbigg1+z
2/parenrightbigg1/2/bracketrightBigg
Qμ
−ν−1/bracketleftBigg/parenleftbigg1+z
2/parenrightbigg1/2/bracketrightBigg
[−1
2<Reμ<1,
zis in the complex plane with a cut along the interval ( −1,1) of the real axis]
ET II 316(18)
772 Associated Legendre Functions 7.129
7.129/integraldisplay1
−1Pν(x)Pλ(x)(1 + x)λ+νdx=2λ+ν+1[Γ(λ+ν+1 ) ]4
[Γ(λ+1 )Γ ( ν+1 ) ]2Γ(2λ+2ν+2 )
[Re(ν+λ+1 )>0] EH I 172(30)
7.131
1./integraldisplay∞
1(x−1)−1
2μ(x+1 )1
2μ−1
2(z+x)μ−1
2Pμ
ν(x)dx
=π1/2Γ(−μ−ν)Γ(1−μ+ν)
Γ/parenleftbig1
2−μ/parenrightbig (z−1)μ/braceleftBigg
Pμ
ν/bracketleftBigg/parenleftbigg1+z
2/parenrightbigg1/2/bracketrightBigg/bracerightBigg2
[Re(μ+ν)<0,Re(μ−ν)<1,|arg(z+1 )|<π]ET II 321(6)
2./integraldisplay∞
1(x−1)−1
2μ(x+1 )1
2μ−1
2(z+x)μ−3
2Pμ
ν(x)dx
=π1/2Γ(1−μ−ν)Γ( 2−μ+ν)(z−1)μ−1
2(z+1 )−1/2
Γ/parenleftbig3
2−μ/parenrightbig Pμ
ν/bracketleftBigg/parenleftbigg1+z
2/parenrightbigg1/2/bracketrightBigg
Pμ−1
ν/bracketleftBigg/parenleftbigg1+z
2/parenrightbigg1/2/bracketrightBigg
[Reμ<1,Re(μ+ν)<1,Re(μ−ν)<2,|arg(1 + z)|<π]ET II 321(7)
7.132
1./integraldisplay1
−1/parenleftbig
1−x2/parenrightbigλ−1Pμ
ν(x)dx=π2μΓ/parenleftbig
λ+1
2μ/parenrightbig
Γ/parenleftbig
λ−1
2μ/parenrightbig
Γ/parenleftbig
λ+1
2ν+1
2/parenrightbig
Γ/parenleftbig
λ−1
2ν/parenrightbig
Γ/parenleftbig
−1
2μ+1
2ν+1/parenrightbig
Γ/parenleftbig
−1
2μ−1
2ν+1
2/parenrightbig
[2 Reλ>|Reμ|] ET II 316(16)
2./integraldisplay∞
1/parenleftbig
x2−1/parenrightbigλ−1Pμ
n(x)dx=2μ−1Γ/parenleftbig
λ−1
2μ/parenrightbig
Γ/parenleftbig
1−λ+1
2ν/parenrightbig
Γ/parenleftbig1
2−λ−1
2ν/parenrightbig
Γ/parenleftbig
1−1
2μ+1
2ν/parenrightbig
Γ/parenleftbig1
2−1
2μ−1
2ν/parenrightbig
Γ/parenleftbig
1−λ−1
2μ/parenrightbig
[Reλ>Reμ,Re(1−2λ−ν)>0,Re(2−2λ+ν)>0]ET II 320(2)
3.9/integraldisplay∞
1/parenleftbig
x2−1/parenrightbigλ−1Qμ
ν(x)dx=eμπiΓ/parenleftbig1
2+1
2ν+1
2μ/parenrightbig
Γ/parenleftbig
1−λ+1
2ν/parenrightbig
Γ/parenleftbig
λ+1
2μ/parenrightbig
Γ/parenleftbig
λ−1
2μ/parenrightbig
22−μΓ/parenleftbig
1+1
2ν−1
2μ/parenrightbig
Γ/parenleftbig1
2+λ+1
2ν/parenrightbig
[|Reμ|<2R eλ<Reν+2 ]
ET II 324(23)
4./integraldisplay1
0xσ/parenleftbig
1−x2/parenrightbig−1
2μPμ
ν(x)dx=2μ−1Γ/parenleftbig1
2+1
2σ/parenrightbig
Γ/parenleftbig
1+1
2σ/parenrightbig
Γ/parenleftbig
1+1
2σ−1
2ν−1
2μ/parenrightbig
Γ/parenleftbig1
2σ+1
2ν−1
2μ+3
2/parenrightbig
[Reμ<1,Reσ>−1] EH I 172(24)
5./integraldisplay1
0xσ/parenleftbig
1−x2/parenrightbig1
2mPm
ν(x)dx=(−1)m2−m−1Γ/parenleftbig1
2+1
2σ/parenrightbig
Γ/parenleftbig
1+1
2σ/parenrightbig
Γ( 1+ m+ν)
Γ(1−m+ν)Γ/parenleftbig
1+1
2σ+1
2m−1
2ν/parenrightbig
Γ/parenleftbig3
2+1
2σ+1
2m+1
2ν/parenrightbig
[Reσ>−1,m is a positive integer] EH I 172(25), ET II 313(4)
6./integraldisplay1
0/parenleftbig
1−x2/parenrightbigηPμ
ν(x)dx=2μ−1Γ/parenleftbig
1+η−1
2μ/parenrightbig
Γ/parenleftbig1
2+1
2σ/parenrightbig
Γ(1−μ)Γ/parenleftbig3
2+η+1
2σ−1
2μ/parenrightbig
×3F2/parenleftbiggν−μ+1
2,−μ+ν
2,1+η−μ
2;1−μ,3+σ−μ
2+η;1/parenrightbigg
/bracketleftbig
Re/parenleftbig
η−1
2μ/parenrightbig
>−1,Reσ>−1/bracketrightbig
ET II 314(6)
7.135 Associated Legendre functions and powers 773
7./integraldisplay∞
1x−ρ/parenleftbig
x2−1/parenrightbig−1
2μPμ
ν(x)dx=2ρ+μ−2Γ/parenleftbigρ+μ+ν
2/parenrightbig
Γ/parenleftbigρ+μ−ν−1
2/parenrightbig
√πΓ(ρ)
[Reμ<1,Re(ρ+μ+ν)>0,Re(ρ+μ−ν)>1]ET II 320(3)
7.133
1./integraldisplay∞
uQν(x)(x−u)μ−1dx=Γ (μ)eμπi/parenleftbig
u2−1/parenrightbig1
2μQ−μ
ν(u)
[|arg(u−1)|<π , 0<Reμ<1+R e ν]MO 90a
2./integraldisplay∞
u/parenleftbig
x2−1/parenrightbig1
2λQ−λ
ν(x)(x−u)μ−1dx=Γ (μ)eμπi/parenleftbig
u2−1/parenrightbig1
2λ+1
2μQ−λ−μ
ν(u)
[|arg(u−1)|<π , 0<Reμ<1+R e ( ν−λ)]ET II 204(30)
7.134
1./integraldisplay∞
1(x−1)λ−1/parenleftbig
x2−1/parenrightbig1
2μPμ
ν(x)dx=2λ+μΓ(λ)Γ(−λ−μ−ν)Γ(1−λ−μ+ν)
Γ(1−μ+ν)Γ(−μ−ν)Γ(1−λ−μ)
[Reλ>0,Re(λ+μ+ν)<0,Re(λ+μ−ν)<1]ET II 321(4)
2./integraldisplay∞
1(x−1)λ−1/parenleftbig
x2−1/parenrightbig−1
2μPμ
ν(x)dx=−2λ−μsinπνΓ(λ−μ)Γ(−λ+μ−ν)Γ(1−λ+μ+ν)
πΓ(1−λ)
[Re(λ−μ)>0,Re(μ−λ−ν)>0,Re(μ−λ+ν)>−1]ET II 321(5)
7.135
1./integraldisplay1
−1/parenleftbig
1−x2/parenrightbig−1
2μ(z−x)−1Pμ
μ+n(x)dx=2e−iμπ/parenleftbig
z2−1/parenrightbig−1
2μQμ
μ+n(z)
[n=0,1,2,...,R e μ+n>−1,zis in the complex plane with a cut along the interval ( −1,1)
of the real axis.] ET II 316(17)
2./integraldisplay∞
1(x−1)λ−1/parenleftbig
x2−1/parenrightbigμ/2(x+z)−ρPμ
ν(x)dx
=2λ+μ−ρΓ(λ−ρ)Γ(ρ−λ−μ−ν)Γ(ρ−λ−μ+ν+1 )
Γ(1−μ+ν)Γ(−μ−ν)Γ(1+ ρ−λ−μ)
×3F2/parenleftbigg
ρ, ρ−λ−μ−ν,ρ−λ−μ+ν+1 ;ρ−λ+1,ρ−λ−μ+1 ;1+z
2/parenrightbigg
+Γ(ρ−λ)Γ(λ)
Γ(ρ)Γ(1−μ)2μ(z+1 )λ−ρ
3F2/parenleftbigg
λ,−μ−ν,1−μ+ν;1−μ,1−ρ+λ;1+z
2/parenrightbigg
[Reλ>0,Re(ρ−λ−μ−ν)>0,Re(ρ−λ−μ+ν+1 )>0,|arg(z+1 )|<π]
ET II 322(9)
774 Associated Legendre Functions 7.136
3./integraldisplay∞
1(x−1)λ−1/parenleftbig
x2−1/parenrightbig−μ/2(x+z)−ρPμ
ν(x)dx
=−sin(νπ)Γ(λ−μ−ρ)Γ(ρ−λ+μ−ν)Γ (ρ−λ+μ+ν+1 )
2ρ−λ+μπΓ(1+ ρ−λ)
×3F2/parenleftbigg
ρ, ρ−λ+μ−ν,ρ−λ+μ+ν+1 ;1+ ρ−λ,1+ρ−λ+μ;1+z
2/parenrightbigg
+Γ(λ−μ)Γ(ρ−λ+μ)
Γ(ρ)Γ(1−μ)(z+1 )λ−ρ−μ
×3F2/parenleftbigg
λ−μ,−ν,ν+1 ;1+ λ−μ−ρ,1−μ;1+z
2/parenrightbigg
[Re(λ−μ)>0,Re (ρ−λ+μ−ν)>0,Re(ρ−λ+μ+ν+1 )>0,|arg(z+1 )|<π]
ET II 322(10)
7.136
1./integraldisplay1
−1/parenleftbig
1−x2/parenrightbigλ−1/parenleftbig
1−a2x2/parenrightbigμ/2Pν(ax)dx
=π2μΓ(λ)
Γ/parenleftbig1
2+λ/parenrightbig
Γ/parenleftbig1
2−1
2μ−1
2ν/parenrightbig
Γ/parenleftbig
1−1
2μ+1
2ν/parenrightbig2F1/parenleftbigg
−μ+ν
2,1−μ+ν
2;1
2+λ;a2/parenrightbigg
[Reλ>0,−1<a< 1] ET II 318(31)
2./integraldisplay∞
1/parenleftbig
x2−1/parenrightbigλ−1/parenleftbig
a2x2−1/parenrightbigμ/2Pμ
ν(ax)dx
=Γ(λ)Γ/parenleftbig
1−λ−1
2μ+1
2ν/parenrightbig
Γ/parenleftbig1
2−λ−1
2μ−1
2ν/parenrightbig
Γ/parenleftbig
1−1
2μ+1
2ν/parenrightbig
Γ/parenleftbig1
2−1
2ν−1
2μ/parenrightbig
Γ(1−λ−μ)
×2μ−1aμ−ν−1
2F1/parenleftbigg1−μ+ν
2,1−λ−μ−ν
2;1−λ−μ;1−1
a2/parenrightbigg
[Rea>0,Reλ>0,Re(ν−μ−2λ)>−2,Re(2λ+μ+ν)<1]ET II 325(25)
3./integraldisplay∞
1/parenleftbig
x2−1/parenrightbigλ−1/parenleftbig
a2x2−1/parenrightbig−1
2μQμ
ν(ax)dx=Γ/parenleftbigμ+ν+1
2/parenrightbig
Γ(λ)Γ/parenleftbig
1−λ+μ+ν
2/parenrightbig
2μ−2eμπia−μ−ν−1
Γ/parenleftbig
ν+3
2/parenrightbig
×2F1/parenleftbiggμ+ν+1
2,1−λ+μ+ν
2;ν+3
2;a−2/parenrightbigg
[|arg(a−1)|<π , Reλ>0,Re(2λ−μ−ν)<2]ET II 325(27)
7.137
1./integraldisplay∞
1x−1
2μ−1
2(x−1)−μ−1
2(1 +ax)1
2μQμ
ν(1 + 2 ax)dx
=π−1/2e−μπiΓ/parenleftbig1
2−μ/parenrightbig
a1
2μ/braceleftBig
Qμ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig/bracerightBig2
/bracketleftbig
|arga|<π , Reμ<1
2,Re(μ+ν)>−1/bracketrightbig
ET II 325(28)
2./integraldisplay∞
1x−1
2μ−1
2(x−1)−μ−3
2(1 +ax)1
2μQμ
ν(1 + 2 ax)dx
=−π−1/2e−μπiΓ/parenleftbig
−μ−1
2/parenrightbig
a1
2μ+1
2/parenleftbig
1+a2/parenrightbig−1/2Qμ+1
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
Qμ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
/bracketleftbig
|arga|<π , Reμ<−1
2,Re(μ+ν+2 )>0/bracketrightbig
ET II 326(29)
7.137 Associated Legendre functions and powers 775
3./integraldisplay1
0x−1
2μ−1
2(1−x)−μ−1
2(1 +ax)1
2μPμ
ν(1 + 2 ax)dx=π1/2Γ/parenleftbig1
2−μ/parenrightbig
a1
2μ/braceleftBig
Pμ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig/bracerightBig2
/bracketleftbig
Reμ<1
2,|arga|<π/bracketrightbig
ET II 319(32)
4./integraldisplay1
0x−1
2μ−1
2(1−x)−μ−3
2(1 +ax)1
2μPμ
ν(1 + 2 ax)dx
=π1/2Γ/parenleftbig
−1
2−μ/parenrightbig
a1
2μ+1
2Pμ+1
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
Pμ
ν/bracketleftbig
(1 +a)2/bracketrightbig
/bracketleftbig
Reμ<−1
2,|arga|<π/bracketrightbig
ET II 319(33)
5./integraldisplay1
0x1
2μ−1
2(1−x)μ−1
2(1 +ax)−1
2μPμ
ν(1 + 2 ax)dx
=π1/2Γ/parenleftbig1
2+μ/parenrightbig
a−1
2μPμ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
P−μ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
/bracketleftbig
Reμ>−1
2,|arga|<π/bracketrightbig
ET II 319(34)
6./integraldisplay1
0x1
2μ−1
2(1−x)μ−3
2(1 +ax)−1
2μPμ
ν(1 + 2 ax)dx
=1
2π1/2Γ/parenleftbig
μ−1
2/parenrightbig
a1
2−1
2μ(1 +a)−1/2/braceleftBig
P1−μ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
Pμ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig/bracerightBig
+(μ+ν)(1−μ+ν)P−μ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
Pμ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
/bracketleftbig
Reμ>1
2,|arga|<π/bracketrightbig
ET II 319(35)
7./integraldisplay1
0x−μ
2−1
2(1−x)−μ−1
2(1 +ax)1
2μQμ
ν(1 + 2 ax)dx
=π1/2Γ/parenleftbig1
2−μ/parenrightbig
a1
2μPμ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
Qμ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
/bracketleftbig
Reμ<1
2,|arga|<π/bracketrightbig
ET II 320(38)
8./integraldisplay1
0x−μ
2−1
2(1−x)−μ−3
2(1 +ax)1
2μQμ
ν(1 + 2 ax)dx
=1
2π1/2Γ/parenleftbig
−μ−1
2/parenrightbig
(1 +a)−1/2a1
2μ+1
2
×/braceleftBig
Pμ+1
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
Qμ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
+Pμ
ν/bracketleftBig
(1 +a)1/2/bracketrightBig
Qμ+1
ν/bracketleftBig
(1 +a)1/2/bracketrightBig/bracerightBig
/bracketleftbig
Reμ<−1
2,|arga|<π/bracketrightbig
ET II 320(39)
9./integraldisplayy
0(y−x)μ−1/bracketleftbig
x/parenleftbig
1+1
2γx/parenrightbig/bracketrightbig−1
2λPλ
ν(1 +γx)dx
=Γ (μ)/parenleftbigg2
γ/parenrightbigg1
2μ/bracketleftbigg
y/parenleftbigg
1+1
2γy/parenrightbigg/bracketrightbigg1
2μ−1
2λ
Pλ−μ
ν(1 +γy)
[Reλ<1,Reμ>0,|argγy|<π]ET II 193(52)
10./integraldisplayy
0(y−x)μ−1xσ+1
2λ−1/parenleftbig
1+1
2γx/parenrightbig−1
2λPλ
ν(1 +γx)dx
=/parenleftbigγ
2/parenrightbig−1
2λΓ(σ)Γ(μ)yσ+μ−1
Γ(1−λ)Γ(σ+μ)3F2/parenleftbigg
−ν,1+ν,σ;1−λ,σ+μ;−1
2γy/parenrightbigg
[Reσ>0,Reμ>0,|γy|<1]ET II 193(53)
776 Associated Legendre Functions 7.138
11./integraldisplayy
0(y−x)μ−1[x(1−x)]−1
2λPλ
ν(1−2x)dx=Γ (μ)[y(1−y)]1
2μ−1
2λPλ−μ
ν(1−2y)
[Reλ<1,Reμ>0,0<y< 1]
ET II 193(54)
12./integraldisplayy
0(y−x)μ−1xσ+1
2λ−1(1−x)−1
2λPλ
ν(1−2x)dx
=Γ(μ)Γ(σ)yσ+μ−1
Γ(σ+μ)Γ ( 1−λ)3F2(−ν,1+ν,σ;1−λ,σ+μ;y)
[Reσ>0,Reμ>0,0<y< 1]ET II 193(155)
7.138/integraldisplay∞
0(a+x)−μ−ν−2Pμ/parenleftbigga−x
a+x/parenrightbigg
Pν/parenleftbigga−x
a+x/parenrightbigg
dx=a−μ−ν−1[Γ(μ+ν+1 ) ]4
[Γ(μ+1 )Γ ( ν+1 ) ]2Γ(2μ+2ν+2 )
[|arga|<π , Re(μ+ν)>−1]
ET II 326(3)
7.14 Combinations of associated Legendre functions, exponentials, and powers
7.141
1./integraldisplay∞
1e−ax(x−1)λ−1/parenleftbig
x2−1/parenrightbig1
2μPμ
ν(x)dx=a−λ−μe−a
Γ(1−μ+ν)Γ(−μ−ν)G31
23/parenleftbigg
2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+μ,1
λ+μ,−ν,1+ν/parenrightbigg
[Rea>0,Reλ>0]
ET II 323(13)
2./integraldisplay∞
1e−ax(x−1)λ−1/parenleftbig
x2−1/parenrightbig1
2μQμ
ν(x)dx
=Γ(ν+μ+1 )eμπi
2Γ (ν−μ+1 )a−λ−μe−aG22
23/parenleftbigg
2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+μ,1
λ+μ, ν+1,−ν/parenrightbigg
[Rea>0,Reλ>0,Re(λ+μ)>0]
ET II 325(24)
3./integraldisplay∞
1e−ax(x−1)λ−1/parenleftbig
x2−1/parenrightbig−1
2μPμ
ν(x)dx=−π−1sin(νπ)aμ−λe−aG31
23/parenleftbigg
2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle1,1−μ
λ−μ,1+ν,−ν/parenrightbigg
[Rea>0,Re(λ−μ)>0]
ET II 323(15)
4./integraldisplay∞
1e−ax(x−1)λ−1/parenleftbig
x2−1/parenrightbig−1
2μQμ
ν(x)dx=1
2eμπiaμ−λe−aG22
23/parenleftbigg
2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−μ,1
λ−μ, ν+1,−ν/parenrightbigg
[Rea>0,Reλ>0,Re(λ−μ)>0]
ET II 323(14)
5./integraldisplay∞
1e−ax/parenleftbig
x2−1/parenrightbig−1
2μPμ
ν(x)dx=21/2π−1/2aμ−1
2Kν+1
2(a)
[Rea>0,Reμ<1]
ET II 323(11), MO 90
7.142/integraldisplay∞
1e−1
2ax/parenleftbiggx+1
x−1/parenrightbigg1
2μ
Pμ
ν−1
2(x)dx=2
aWμ,ν(a)/bracketleftbig
Reμ<1,ν−1
2/negationslash=0,±1,±2,.../bracketrightbig
BU 79(34), MO 118
7.146 Associated Legendre functions, exponentials, and powers 777
7.143
1./integraldisplay∞
0[x(1 +x)]−1
2μe−βxPμ
ν(1 + 2 x)dx=βμ−1
2√πe1
2βKν+1
2/parenleftbiggβ
2/parenrightbigg
[Reμ<1,Reβ>0] ET I 179(1)
2./integraldisplay∞
0/parenleftbigg
1+1
x/parenrightbigg1
2μ
e−βxPμ
ν(1 + 2 x)dx=e1
2β
βWμ,ν+1
2(β)
[Reμ<1,Reβ>0] ET I 179(2)
7.144
1./integraldisplay∞
0e−βxxλ+1
2μ−1(x+2 )1
2μQμ
ν(1 +x)dx
=Γ(ν+μ+1 )
Γ(ν−μ+1 )/braceleftbiggsin(νπ)
2βλ+μsin(μπ)E(−ν,ν+1,λ+μ;μ+1:2 β)
−sin [(μ+ν)π]
21−μβλsin(μπ)E(ν−μ+1,−ν−μ, λ:1−μ:2β)/bracerightbigg
[Reβ>0,Reλ>0,Re (λ+μ)>0]ET I 181(16)
2./integraldisplay∞
0e−βxxλ−1
2μ−1(x+2 )1
2μQμ
ν(1 +x)dx=−sin(νπ)
2βλ−μsin(μπ)E(−ν,ν+1,λ−μ:1−μ:2β)
−sin[(μ−ν)π]
21+μβλsin(μπ)E(μ+ν+1,μ−ν,λ:1+μ:2β)
[Reβ>0,Reλ>0,Re(λ−μ)]>0ET I 181(17)
7.145
1./integraldisplay∞
0e−βx
1+xPν/bracketleftbigg1
(1 +x)2−1/bracketrightbigg
dx=eβ
βWν+1
2,0(β)W−ν−1
2,0(β)
[Reβ>0] ET I 180(6)
2./integraldisplay∞
0x−1e−βxQ−1
2/parenleftbig
1+2x−2/parenrightbig
dx=π2
8/braceleftBigg/bracketleftbigg
J0/parenleftbigg1
2β/parenrightbigg/bracketrightbigg2
+/bracketleftbigg
Y0/parenleftbigg1
2β/parenrightbigg/bracketrightbigg2/bracerightBigg
[Reβ>0] ET II 327(5)
3./integraldisplay∞
0x−1e−axQν/parenleftbig
1+2x−2/parenrightbig
dx=1
2[Γ(ν+1 ) ]2a−1W−ν−1
2,0(ai)W−ν−1
2,0(−ai)
[Rea>0,Reν>−1] ET II 327(6)
7.146
1./integraldisplay∞
0x−1
2μe−βxPμ
ν/parenleftbig√
1+x/parenrightbig
dx=2μβ1
2μ−5
4eβ
2W1
2μ+1
4,1
2ν+1
4(β)
[Reμ<1,Reβ>0] ET I 180(7)
2./integraldisplay∞
0x−1
2μe−βx
√1+xPμ
ν/parenleftbig√
1+x/parenrightbig
dx=2μβ1
2μ−3
4eβ
2W1
2μ+1
4,1
2ν+1
4(β)
[Reμ<1,Reβ>0] ET I 180(8)a
778 Associated Legendre Functions 7.147
3./integraldisplay∞
0√xe−βxP1/4
ν/parenleftBig/radicalbig
1+x2/parenrightBig
P−1/4
ν/parenleftBig/radicalbig
1+x2/parenrightBig
dx=1
2/radicalbiggπ
2βH(1)
ν+1
2/parenleftbigg1
2β/parenrightbigg
H(2)
ν+1
2/parenleftbigg1
2β/parenrightbigg
[Reβ>0] ET I 180(9)
7.147/integraldisplay∞
0xλ−1/parenleftbig
x2+a2/parenrightbig1
2νe−βxPμ
ν/bracketleftBigg
x
(x2+a2)1/2/bracketrightBigg
dx
=2−ν−2aλ+ν
πΓ(−μ−ν)G32
24/parenleftBigg
a2β2
4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−
λ
2,1−λ
2
0,1
2,−λ+μ+ν
2,−λ−μ+ν
2/parenrightBigg
[a>0,Reβ>0,Reλ>0]ET II 327(7)
7.148/integraldisplay1
−1(1−x)−1
2μ(1 +x)1
2μ+ν−1exp/parenleftbigg
−1−x
1+xy/parenrightbigg
Pμ
ν(x)dx=2νy1
2μ+ν−1
2e1
2yW1
2μ−ν−1
2,1
2μ(y)
[Rey>0] ET II 317(21)
7.149/integraldisplay∞
1/parenleftbig
α2+β2+2αβx/parenrightbig−1/2exp/bracketleftBig
−/parenleftbig
α2+β2+2αβx/parenrightbig1/2/bracketrightBig
Pν(x)dx
=2π−1(αβ)−1/2Kν+1
2(α)Kν+1
2(β)
[Reα>0,Reβ>0] ET II 323(16)
7.15 Combinations of associated Legendre and hyperbolic functions
7.151
1./integraldisplay∞
0(sinhx)α−1P−μ
ν(coshx)dx=2−1−μΓ/parenleftbig1
2α+1
2μ/parenrightbig
Γ/parenleftbig1
2ν−1
2α+1/parenrightbig
Γ/parenleftbig1
2−1
2α−1
2ν/parenrightbig
Γ/parenleftbig1
2μ+1
2ν+1/parenrightbig
Γ/parenleftbig1
2+1
2μ−1
2ν/parenrightbig
Γ/parenleftbig
1+1
2μ−1
2α/parenrightbig
[Re(α+μ)>0,Re(ν−α+2 )>0,Re(1−α−ν)>0]EH I 172(28)
2./integraldisplay∞
0(sinhx)α−1Qμ
ν(coshx)dx=eiμπ2μ−αΓ/parenleftbig1
2+1
2ν+1
2μ/parenrightbig
Γ/parenleftbig
1+1
2ν−1
2α/parenrightbig
Γ/parenleftbig
1+1
2ν−1
2μ/parenrightbig
Γ/parenleftbig1
2+1
2ν+1
2α/parenrightbig
×Γ/parenleftbig1
2α+1
2μ/parenrightbig
Γ/parenleftbig1
2α−1
2μ/parenrightbig
[Re (α±μ)>0,Re(ν−α+2 )>0]EH I 172(29)
7.152/integraldisplay∞
0e−αxsinh2μ/parenleftbig1
2x/parenrightbig
P−2μ
2n/bracketleftbig
cosh/parenleftbig1
2x/parenrightbig/bracketrightbig
dx=Γ/parenleftbig
2μ+1
2/parenrightbig
Γ(α−n−μ)Γ/parenleftbig
α+n−μ+1
2/parenrightbig
4μ√πΓ(α+n+μ+1 )Γ/parenleftbig
α−n+μ+1
2/parenrightbig
/bracketleftbig
Reα>n +R eμ,Reμ>−1
4/bracketrightbig
ET I 181(15)
7.162 Associated Legendre functions, powers, and trigonometric functions 779
7.16 Combinations of associated Legendre functions, powers, and trigonometric
functions
7.161
1./integraldisplay1
0xλ−1/parenleftbig
1−x2/parenrightbig−1
2μsin(ax)Pμ
ν(x)dx
=π1/22μ−λ−1Γ(λ+1 )a
Γ/parenleftBig
1+λ−μ−ν
2/parenrightBig
Γ/parenleftBig
3+λ−μ+ν
2/parenrightBig
×2F3/parenleftbigg1+λ
2,1+λ
2;3
2,1+λ−μ−ν
2,3+λ−μ+ν
2;−a2
4/parenrightbigg
[Reλ>−1,Reμ<1] ET II 314(7)
2./integraldisplay1
0xλ−1/parenleftbig
1−x2/parenrightbig−1
2μcos(ax)Pμ
ν(x)dx
=π1/22μ−λΓ(λ)
Γ/parenleftbigg
1+λ−μ+ν
2/parenrightbigg
Γ/parenleftbigg1+λ−μ−ν
2/parenrightbigg
×2F3/parenleftbiggλ
2,λ+1
2;1
2,1+λ−μ−ν
2,1+λ−μ+ν
2;−a2
4/parenrightbigg
[Reλ>0,Reμ<1] ET II 314(8)
3./integraldisplay∞
0/parenleftbig
x2−1/parenrightbig1
2μsin(ax)Pμ
ν(x)dx=2μπ1/2a−μ−1
2
Γ/parenleftbig1
2−1
2μ−1
2ν/parenrightbig
Γ/parenleftbig
1−1
2μ+1
2ν/parenrightbigSμ+1
2,ν+1
2(a)
/bracketleftbig
a>0,Reμ<3
2,Re(μ+ν)<1/bracketrightbig
ET II 320(1)
7.162
1./integraldisplay∞
aPν/parenleftbig
2x2a−2−1/parenrightbig
sin(bx)dx=−πa
4c os(νπ)/braceleftBigg/bracketleftbigg
Jν+1
2/parenleftbiggab
2/parenrightbigg/bracketrightbigg2
−/bracketleftbigg
J−ν−1
2/parenleftbiggab
2/parenrightbigg/bracketrightbigg2/bracerightBigg
[a>0,b > 0,−1<Reν<0]
ET II 326(1)
2./integraldisplay∞
aPν/parenleftbig
2x2a−2−1/parenrightbig
cos(bx)dx
=−π
4a/bracketleftbigg
Jν+1
2/parenleftbiggab
2/parenrightbigg
J−ν−1
2/parenleftbiggab
2/parenrightbigg
−Yν+1
2/parenleftbiggab
2/parenrightbigg
Y−ν−1
2/parenleftbiggab
2/parenrightbigg/bracketrightbigg
[a>0,b > 0,−1<Reν<0]ET II 326(2)
3./integraldisplay∞
0/parenleftbig
x2+2/parenrightbig−1/2sin(ax)P−1
ν/parenleftbig
x2+1/parenrightbig
dx=2−1/2π−1asin(νπ)/bracketleftBig
Kν+1
2/parenleftBig
2−1/2a/parenrightBig/bracketrightBig2
[a>0,−2<Reν<1] ET I 98(22)
4./integraldisplay∞
0/parenleftbig
x2+2/parenrightbig−1/2sin(ax)Q1
ν/parenleftbig
x2+1/parenrightbig
dx=−2−3/2πaKν+1
2/parenleftBig
2−1/2a/parenrightBig
Iν+1
2/parenleftBig
2−1/2a/parenrightBig
/bracketleftbig
a>0,Reν>−3
2/bracketrightbig
ET 98(23)
780 Associated Legendre Functions 7.163
5./integraldisplay∞
0cos(ax)Pν/parenleftbig
1+x2/parenrightbig
dx=−√
2
πsin(νπ)/bracketleftbigg
Kν+1
2/parenleftbigga√
2/parenrightbigg/bracketrightbigg2
[a>0,−1<Reν<0] ET I 42(23)
6./integraldisplay∞
0cos(ax)Qν/parenleftbig
1+x2/parenrightbig
dx=π√
2Kν+1
2/parenleftbigga√
2/parenrightbigg
Iν+1
2/parenleftbigga√
2/parenrightbigg
[a>0,Reν>−1] ET I 42(24)
7./integraldisplay1
0cos(ax)Pν/parenleftbig
2x2−1/parenrightbig
dx=π
2Jν+1
2/parenleftBiga
2/parenrightBig
J−ν−1
2/parenleftBiga
2/parenrightBig
[a>0] ET I 42(25)
7.163
1./integraldisplay∞
a/parenleftbig
x2−a2/parenrightbig1
2ν−1
4sin(bx)P1
2−ν
0/parenleftbig
ax−1/parenrightbig
dx=b−ν−1
2cos/parenleftBig
ab−νπ
2+π
4/parenrightBig
/bracketleftbig
a>0,|Reν|<1
2/bracketrightbig
ET I 98(24)
2./integraldisplay1
0x−1cos(ax)Pν/parenleftbig
2x−2−1/parenrightbig
dx=−1
2πcosec( νπ)1F1((ν+1 ;1 ; ai))1F1(ν+1 ;1 ; −ai)
[a>0,−1<Reν<0] ET II 327(4)
7.164
1./integraldisplay∞
0x1/2sin(bx)/bracketleftBig
P−1/4
ν/parenleftBig/radicalbig
1+a2x2/parenrightBig/bracketrightBig2
dx=/radicalBig
2
πa−1b−1/2
Γ/parenleftbig5
4+ν/parenrightbig
Γ/parenleftbig1
4−ν/parenrightbig/bracketleftbigg
Kν+1
2/parenleftbiggb
2a/parenrightbigg/bracketrightbigg2
/bracketleftbig
Rea>0,b > 0,−5
4<Reν<1
4/bracketrightbig
ET II 327(8)
2./integraldisplay∞
0x1/2sin(bx)P−1/4
ν/parenleftBig/radicalbig
1+a2x2/parenrightBig
Q−1/4
ν−1/parenleftBig/radicalbig
1+a2x2/parenrightBig
dx
=/radicalbigπ
2e−1
4πiΓ/parenleftbig
ν+5
4/parenrightbig
ab1
2Γ/parenleftbig
ν+3
4/parenrightbigIν+1
2/parenleftbiggb
2a/parenrightbigg
Kν+1
2/parenleftbiggb
2a/parenrightbigg
/bracketleftbig
Rea>0,b > 0,Reν>−5
4/bracketrightbig
ET II 328(9)
3./integraldisplay∞
0x1/2sin(bx)P−1/4
ν/parenleftBig/radicalbig
1+a2x2/parenrightBig
P−1/4
ν−1/parenleftBig/radicalbig
1+a2x2/parenrightBigdx√
1+a2x2
=a−2b1/2
√
2πΓ/parenleftbig5
4+ν/parenrightbig
Γ/parenleftbig5
4−ν/parenrightbigKν−1
2/parenleftbiggb
2a/parenrightbigg
Kν+1
2/parenleftbiggb
2a/parenrightbigg
/bracketleftbig
Rea>0,b > 0,−5
4<Reν<5
4/bracketrightbig
ET II 328(10)
4./integraldisplay∞
0x1/2sin(bx)P1/4
ν/parenleftBig/radicalbig
1+a2x2/parenrightBig
P−3/4
ν/parenleftBig/radicalbig
1+a2x2/parenrightBigdx√
1+a2x2
=a−2b1/2
√
2πΓ/parenleftbig7
4+ν/parenrightbig
Γ/parenleftbig3
4−ν/parenrightbig/bracketleftbigg
Kν+1
2/parenleftbiggb
2a/parenrightbigg/bracketrightbigg2
/bracketleftbig
Rea>0,b > 0,−7
4<Reν<3
4/bracketrightbig
ET II 328(11)
7.171 Associated Legendre function and probability integral 781
5./integraldisplay∞
0x1/2cos(bx)/bracketleftBig
P1/4
ν/parenleftBig/radicalbig
1+a2x2/parenrightBig/bracketrightBig2
dx=a−1/parenleftbigπb
2/parenrightbig−1/2
Γ/parenleftbig3
4+ν/parenrightbig
Γ/parenleftbig
−1
4−ν/parenrightbig/bracketleftbigg
Kν+1
2/parenleftbiggb
2a/parenrightbigg/bracketrightbigg2
/bracketleftbig
Rea>0,b > 0,−3
4<Reν<−1
4/bracketrightbig
ET II 328(12)
6./integraldisplay∞
0x1/2cos(bx)P1/4
ν/parenleftBig/radicalbig
1+a2x2/parenrightBig
Q1/4
ν/parenleftBig/radicalbig
1+a2x2/parenrightBig
dx
=/radicalbigπ
2e1
4πiΓ/parenleftbig
ν+3
4/parenrightbig
ab1/2Γ/parenleftbig
ν+5
4/parenrightbigIν+1
2/parenleftbiggb
2a/parenrightbigg
Kν+1
2/parenleftbiggb
2a/parenrightbigg
/bracketleftbig
Rea>0,b > 0,Reν>−3
4/bracketrightbig
ET II 328(13)
7./integraldisplay∞
0x1/2cos(bx)P−1/4
ν/parenleftBig/radicalbig
1+a2x2/parenrightBig
P3/4
ν/parenleftBig/radicalbig
1+a2x2/parenrightBigdx√
1+a2x2
=a−2b1/2
√
2πΓ/parenleftbig5
4+ν/parenrightbig
Γ/parenleftbig1
4−ν/parenrightbig/bracketleftbigg
Kν+1
2/parenleftbiggb
2a/parenrightbigg/bracketrightbigg2
/bracketleftbig
Rea>0,b > 0,−5
4<Reν<1
4/bracketrightbig
ET II 328(14)
8./integraldisplay∞
0x1/2cos(bx)P1/4
ν/parenleftBig/radicalbig
1+a2x2/parenrightBig
P1/4
ν−1/parenleftBig/radicalbig
1+a2x2/parenrightBigdx√
1+a2x2
=a−2b1/2
√
2πΓ/parenleftbig3
4+ν/parenrightbig
Γ/parenleftbig3
4−ν/parenrightbigKν−1
2/parenleftbiggb
2a/parenrightbigg
Kν+1
2/parenleftbiggb
2a/parenrightbigg
/bracketleftbig
Rea>0,b > 0,|Reν|<3
4/bracketrightbig
ET II 329(15)
7.165/integraldisplay∞
0cos(ax)Pν(coshx)dx
=−sin(νπ)
4π2Γ/parenleftbigg1+ν+iα
2/parenrightbigg
Γ/parenleftbigg1+ν−iα
2/parenrightbigg
Γ/parenleftbigg
−ν+iα
2/parenrightbigg
Γ/parenleftbigg
−ν−iα
2/parenrightbigg
[a>0,−1<Reν<0] ET II 329(18)
7.166/integraldisplayπ
0P−μ
ν(cosϕ)sinα−1ϕdϕ=2−μπΓ/parenleftbig1
2α+1
2μ/parenrightbig
Γ/parenleftbig1
2α−1
2μ/parenrightbig
Γ/parenleftbig1
2+1
2α+1
2ν/parenrightbig
Γ/parenleftbig1
2α−1
2ν/parenrightbig
Γ/parenleftbig1
2μ+1
2ν+1/parenrightbig
Γ/parenleftbig1
2μ−1
2ν+1
2/parenrightbig
[Re(α±μ)>0] MO 90, EH I 172(27)
7.167/integraldisplaya
0P−μ
ν(cosx)P−η
ν[cos(a−x)]/bracketleftbiggsin(a−x)
sinx/bracketrightbiggηdx
sinx=2ηΓ(μ−η)Γ/parenleftbig
η+1
2/parenrightbig
(sina)η
√πΓ(η+μ+1 )P−μ
ν(cosa)
/bracketleftbig
Reμ>Reη>−1
2/bracketrightbig
ET II 329(16)
7.17 A combination of an associated Legendre function and the probability integral
7.171/integraldisplay∞
1/parenleftbig
x2−1/parenrightbig−1
2μexp/parenleftbig
a2x2/parenrightbig
[1−Φ(ax)]Pμ
ν(x)dx
=π−12μ−1Γ/parenleftbigg1+μ+ν
2/parenrightbigg
Γ/parenleftbiggμ−ν
2/parenrightbigg
aμ−3
2ea2
2W1
4−1
2μ,1
4+1
2ν/parenleftbig
a2/parenrightbig
[Rea>0,Reμ<1,Re (μ+ν)>−1,Re(μ−ν)>0]
ET II 324(17)
782 Associated Legendre Functions 7.181
7.18 Combinations of associated Legendre and Bessel functions
7.181
1./integraldisplay∞
1Pν−1
2(x)x1/2Yν(ax)dx=2−1/2a−1/bracketleftbig
cos/parenleftbig1
2a/parenrightbig
Jν/parenleftbig1
2a/parenrightbig
−sin/parenleftbig1
2a/parenrightbig
Yν/parenleftbig1
2a/parenrightbig/bracketrightbig
/bracketleftbig
a>0,Reν<1
2/bracketrightbig
ET II 108(3)a
2./integraldisplay∞
1Pν−1
2(x)x1/2Jν(ax)dx=−1√
2a/bracketleftbig
cos/parenleftbig1
2a/parenrightbig
Yν/parenleftbig1
2a/parenrightbig
+s i n/parenleftbig1
2a/parenrightbig
Jν/parenleftbig1
2a/parenrightbig/bracketrightbig
/bracketleftbig
|Reν|<1
2/bracketrightbig
ET II 344(36)a
7.182
1./integraldisplay∞
1xν/parenleftbig
x2−1/parenrightbig1
2λ−1
2Pλ−1
λ(x)Jν(ax)dx=2λ+νa−λΓ/parenleftbig1
2+ν/parenrightbig
π1/2Γ(1−λ)Sλ−ν,λ+ν(a)
/bracketleftbig
a>0,Reν<5
2,Re(2λ+ν)<3
2/bracketrightbig
ET II 345(38)a
2./integraldisplay∞
1x1
2−μ/parenleftbig
x2−1/parenrightbig−1
2μPμ
ν−1
2(x)Jν(ax)dx
=−2−3/2π1/2aμ−1
2/bracketleftBig
Jμ−1
2/parenleftBiga
2/parenrightBig
Yν/parenleftBiga
2/parenrightBig
+Yμ−1
2/parenleftBiga
2/parenrightBig
Jν/parenleftBiga
2/parenrightBig/bracketrightBig
/bracketleftbig
−1
4<Reμ<1,a > 0,|Reν|<1
2+2R e μ/bracketrightbig
ET II 344(37)a
3./integraldisplay∞
1x1
2−μ/parenleftbig
x2−1/parenrightbig−1
2μPμ
ν−1
2(x)Yν(ax)dx
=2−3/2π1/2aμ−1
2/bracketleftBig
Jν/parenleftBiga
2/parenrightBig
Jμ−1
2/parenleftBiga
2/parenrightBig
−Yν/parenleftBiga
2/parenrightBig
Yμ−1
2/parenleftBiga
2/parenrightBig/bracketrightBig
/bracketleftbig
−1
4<Reμ<1,a > 0,Re(2μ−ν)>−1
2/bracketrightbig
ET II 349(67)a
4./integraldisplay1
0x1
2−μ/parenleftbig
1−x2/parenrightbig−1
2μPμ
ν(x)Jν+1
2(ax)dx=/radicalbiggπ
2aμ−1
2J1
2−μ/parenleftbig1
2a/parenrightbig
Jν+1
2/parenleftbig1
2a/parenrightbig
[Reμ<1,Re(μ−ν)<2]
ET II 337(33)a
5./integraldisplay∞
1x1
2−μ/parenleftbig
x2−1/parenrightbig−1
2μPμ
ν−1
2(x)Kν(ax)dx=( 2π)−1/2aμ−1
2Kν/parenleftbig1
2a/parenrightbig
Kμ−1
2/parenleftbig1
2a/parenrightbig
[Reμ<1,Rea>0] ET II 135(5)a
6./integraldisplay∞
1xμ+1
2/parenleftbig
x2−1/parenrightbig−1
2μPμ
ν−1
2(x)Kν(ax)dx=/radicalbiggπ
2a−3/2e−1
2aWμ,ν(a)
[Reμ<1,Rea>0] ET II 135(3)a
7./integraldisplay∞
1xμ−3
2/parenleftbig
x2−1/parenrightbig−1
2μPμ
ν−1
2(x)Kν(ax)dx=/radicalbiggπ
2a−1/2e−1
2aWμ−1,ν(a)
[Reμ<1,Rea>0] ET II 135(4)a
8./integraldisplay∞
1xμ−1
2/parenleftbig
x2−1/parenrightbig−1
2μPμ
ν−3
2(x)Kν(ax)dx=/radicalbiggπ
2a−1e−1
2aWμ−1
2,ν−1
2(a)
[Reμ<1] ET II 135(6)a
7.182 Associated Legendre and Bessel functions 783
9./integraldisplay∞
1x1/2/parenleftbig
x2−1/parenrightbig1
2ν−1
4P1
2−ν
μ/parenleftbig
2x2−1/parenrightbig
Kν(ax)dx=π−1/2a−ν2ν−1/bracketleftBig
Kμ+1
2/parenleftBiga
2/parenrightBig/bracketrightBig2
/bracketleftbig
Reν>−1
2,Rea>0/bracketrightbig
ET II 136(11)a
10./integraldisplay∞
1x1/2/parenleftbig
x2−1/parenrightbig1
2ν−1
4P1
2−ν
μ/parenleftbig
2x2−1/parenrightbig
Yν(ax)dx
=π1/22ν−2a−ν/bracketleftBig
Jμ+1
2/parenleftBiga
2/parenrightBig
J−μ−1
2/parenleftBiga
2/parenrightBig
−Yμ+1
2/parenleftBiga
2/parenrightBig
Y−μ−1
2/parenleftBiga
2/parenrightBig/bracketrightBig
/bracketleftbig
Reν>−1
2,a > 0,Reν+|2R eμ+1|<3
2/bracketrightbig
ET II 108(5)a
11./integraldisplay∞
1x1/2/parenleftbig
x2−1/parenrightbig1
2ν−1
4P1
2−ν
μ/parenleftbig
2x2−1/parenrightbig
Jν(ax)dx
=−2ν−2a−νπ1/2sec(μπ)/braceleftbigg/bracketleftBig
Jμ+1
2/parenleftBiga
2/parenrightBig/bracketrightBig2
−/bracketleftBig
J−μ−1
2/parenleftBiga
2/parenrightBig/bracketrightBig2/bracerightbigg
/bracketleftbig
Reν>−1
2,a > 0,Reν−3
2<2R eμ<1
2−Reν/bracketrightbig
ET II 345(39)a
12./integraldisplay∞
1x/parenleftbig
x2−1/parenrightbig−1
2νPν
μ/parenleftbig
2x2−1/parenrightbig
Kν(ax)dx=2−νaν−1Kμ+1(a)
[Rea>0,Reν<1] ET II 136(10)a
13./integraldisplay∞
0x/parenleftbig
x2+a2/parenrightbig1
2νPν
μ/parenleftbig
1+2x2a−2/parenrightbig
Kν(xy)dx=2−νay−ν−1S2ν,2μ+1(ay)
[Rea>0,Rey>0,Reν<1]
ET II 135(7)
14./integraldisplay∞
0x/parenleftbig
x2+a2/parenrightbig1
2ν/bracketleftbig
(μ−ν)Pν
μ/parenleftbig
1+2x2a−2/parenrightbig
+(μ+ν)Pν
−μ/parenleftbig
1+2x2a−2/parenrightbig/bracketrightbig
Kν(xy)dx
=21−νμy−ν−2S2ν+1,2μ(ay)
[Rea>0,Rey>0,Reν<1]ET II 136(8)
15./integraldisplay∞
0x/parenleftbig
x2+a2/parenrightbig1
2ν−1/bracketleftbig
Pν
μ/parenleftbig
1+2x2a−2/parenrightbig
+Pν
−μ/parenleftbig
1+2x2a−2/parenrightbig/bracketrightbig
Kν(xy)dx=21−νy−νS2ν−1,2μ(ay)
[Rea>0,Rey>0,Reν<1]
ET II 136(9)
16./integraldisplay∞
0x1/2/parenleftbig
x2+2/parenrightbig−1
2ν−1
4P−ν−1
2μ/parenleftbig
x2+1/parenrightbig
Jν(xy)dx=y−1/221
2−νπ−1/2/bracketleftBig
Kμ+1
2/parenleftbig
2−1/2y/parenrightbig/bracketrightBig2
Γ/parenleftbig
ν+μ+3
2/parenrightbig
Γ/parenleftbig
ν−μ+1
2/parenrightbig
/bracketleftbig
−3
2−Reν<Reμ<Reν+1
2,y > 0/bracketrightbig
ET II 44(1)
17./integraldisplay∞
0x1/2/parenleftbig
x2+2/parenrightbig−1
2ν−1
4Qν+1
2μ/parenleftbig
x2+1/parenrightbig
Jν(xy)dx
=2−ν−1
2π1/2e(ν+1
2)πiyνKμ+1
2/parenleftBig
2−1/2y/parenrightBig
Iμ+1
2/parenleftBig
2−1/2y/parenrightBig
/bracketleftbig
Reν>−1,Re(2μ+ν)>−5
2,y > 0/bracketrightbig
ET II 46(12)
784 Associated Legendre Functions 7.183
7.183/integraldisplay∞
0x1−μ/parenleftbig
1+a2x2/parenrightbig−1
2μ−1
4Qμ+1
2
ν−1
2(±iax)Jν(xy)dx
=i(2π)1/2eiπ(μ∓1
2ν∓1
4)a−1yμ−1Iν/parenleftbig1
2a−1y/parenrightbig
Kμ/parenleftbig1
2a−1y/parenrightbig
/bracketleftbig
−3
4−1
2Reν<Reμ<1+R e ν, y > 0,Rea>0/bracketrightbig
ET II 46(11)
7.184
1./integraldisplay∞
1x1/2/parenleftbig
x2−1/parenrightbig1
2μ−1
4P−1
2−μ
−1
2+ν/parenleftbig
x−1/parenrightbig
Jν(xa)dx=21/2a−1−μπ−1/2cos/bracketleftbig
a+1
2(ν−μ)π/bracketrightbig
/bracketleftbig
|Reμ|<1
2,Reν>−1,a > 0/bracketrightbig
ET II 44(2)a
2./integraldisplay∞
1x−ν/parenleftbig
x2−1/parenrightbig1
4−1
2νPν−1
2μ/parenleftbig
2x−2−1/parenrightbig
Kν(ax)dx
=π1/22−νa−2+νWμ+1
2,ν−1
2(a)W−μ−1
2,ν−1
2(a)
/bracketleftbig
Reν<3
2,a > 0/bracketrightbig
ET II 370(45)a
3./integraldisplay∞
0xν/parenleftbig
1+x2/parenrightbig1
4+ν
2Qν+1
2μ/parenleftbigg
1+2
x2/parenrightbigg
Jν(ax)dx
=−ieiπνπ−1
22νa−ν−2/bracketleftbig
Γ/parenleftbig3
2+μ+ν/parenrightbig/bracketrightbig2Γ/parenleftbig1
2+ν−μ/parenrightbig
×W−μ−1
2,ν+1
2(a)/bracketleftBigg
cos(μπ)
Γ(2 + 2 ν)Mμ+1
2,ν+1
2(a)+sin(μπ)
Γ/parenleftbig
ν+μ+3
2/parenrightbigWμ+1
2,ν+1
2(a)/bracketrightBigg
/bracketleftbig
a>0,Re(μ+ν)>−3
2,Re(μ−ν)<1
2/bracketrightbig
ET II 46(14)
4./integraldisplay1
0xν/parenleftbig
1−x2/parenrightbig1
2ν+1
4P−ν−1
2μ/parenleftbig
2x−2−1/parenrightbig
Jν(xy)dx
=2ν+1
2yνΓ/parenleftbig3
2+μ+ν/parenrightbig
Γ/parenleftbig1
2+ν−μ/parenrightbig
(2π)1/2/bracketleftbig
Γ/parenleftbig3
2+ν/parenrightbig/bracketrightbig2
×1F1/parenleftbigg
ν+μ+3
2;2ν+2 ;iy/parenrightbigg
1F1/parenleftbigg
ν+μ+3
2;2ν+2 ;−iy/parenrightbigg
/bracketleftbig
y>0,−3
2−Reν<Reμ<Reν+1
2/bracketrightbig
ET II 45(3)
5./integraldisplay∞
0x−ν/parenleftbig
x2+a2/parenrightbig1
4−1
2νQ1
2−ν
μ/parenleftbig
1+2a2x−2/parenrightbig
Kν(xy)dx
=ie−iπνπ1/22−ν−1a−ν−1
2yν−2/bracketleftbig
Γ/parenleftbig3
2+μ−ν/parenrightbig/bracketrightbig2W−μ−1
2,ν−1
2(iay)W−μ−1
2,ν−1
2(−iay)
/bracketleftbig
Rea>0,Rey>0,Reμ>−3
2,Re(μ−ν)>−3
2/bracketrightbig
ET II 137(13)
6./integraldisplay∞
0x−ν/parenleftbig
x2+1/parenrightbig1
4−1
2νQ1
2−ν
μ/parenleftbig
1+2x−2/parenrightbig
Jν(ax)dx
=2−νa−ν−2ie−iνππ1/2Γ/parenleftbig3
2+μ−ν/parenrightbig
Γ(2ν)Mμ+1
2,ν−1
2(a)W−μ−1
2,ν−1
2(a)
/bracketleftbig
a>0,0<Reν<Reμ+3
2/bracketrightbig
ET II 47(15)a
7.187 Associated Legendre and Bessel functions 785
7./integraldisplay∞
0x−ν/parenleftbig
x2+a2/parenrightbig1
4−1
2νQ1
2−ν
−1
2/parenleftbig
1+2a2x−2/parenrightbig
Kν(xy)dx
=ie−iπνπ3/22−ν−3a1
2−νyν−1[Γ(1−ν)]2×/braceleftbigg/bracketleftBig
Jν−1
2/parenleftBigay
2/parenrightBig/bracketrightBig2
+/bracketleftBig
Yν−1
2/parenleftBigay
2/parenrightBig/bracketrightBig2/bracerightbigg
[Rea>0,Rey>0,Reν<1]ET II 136(12)
7.185/integraldisplay∞
0x1/2Qν−1
2/bracketleftbig/parenleftbig
a2+x2/parenrightbig
x−1/bracketrightbig
Jν(xy)dx=2−1/2πy−1exp/bracketleftBig
−/parenleftbig
a2−1
4/parenrightbig1/2y/bracketrightBig
Jν/parenleftbig1
2y/parenrightbig
/bracketleftbig
Reν>−1
2,y > 0/bracketrightbig
ET II 46(10)
7.186/integraldisplay∞
0x/parenleftbig
1+x2/parenrightbig−ν−1Pν/parenleftbigg1−x2
1+x2/parenrightbigg
J0(xy)dx=y2ν[2νΓ(ν+1 ) ]−2K0(y)
[Reν>0] ET II 13(10)
7.187
1./integraldisplay∞
0xPν
μ/parenleftBig/radicalbig
1+x2/parenrightBig
Kν(xy)dx=y−3/2Sν+1
2,μ+1
2(y)
[Reν<1,Rey>0] ET II 137(14)
2./integraldisplay∞
0x/bracketleftBig
Pλ−1
2/parenleftBig/radicalbig
1+a2x2/parenrightBig/bracketrightBig2
J0(xy)dx=2π−2y−1a−1cos(λπ)/bracketleftBig
Kλ/parenleftBigy
2a/parenrightBig/bracketrightBig2
/bracketleftbig
Rea>0,|Reλ|<1
4,y > 0/bracketrightbig
ET II 13(11)
3./integraldisplay∞
0x/parenleftbig
1+x2/parenrightbig−1/2Pν
μ/parenleftBig/radicalbig
1+x2/parenrightBig
Kν(xy)dx=y−1/2Sν−1
2,μ+1
2(y)
[Reν<1,Rey>0] ET II 137(15)
4./integraldisplay∞
0xP−1
2ν
μ/parenleftBig/radicalbig
1+a2x2/parenrightBig
Q−1
2ν
μ/parenleftBig/radicalbig
1+a2x2/parenrightBig
Jν(xy)dx
=y−1e−1
2νπiΓ/parenleftbig
1+μ+1
2ν/parenrightbig
aΓ/parenleftbig
1+μ−1
2ν/parenrightbig Iμ+1
2/parenleftBigy
2a/parenrightBig
Kμ+1
2/parenleftBigy
2a/parenrightBig
/bracketleftbig
Rea>0,y > 0,Reμ>−3
4,Reν>−1/bracketrightbig
ET II 47(16)
5./integraldisplay∞
0xPμ
σ−1
2/parenleftBig/radicalbig
1+a2x2/parenrightBig
Qμ
σ−1
2/parenleftBig/radicalbig
1+a2x2/parenrightBig
J0(xy)dx
=y−2eμπiΓ/parenleftbig1
2+σ−μ/parenrightbig
Γ(1 + 2 σ)Wμ,σ/parenleftBigy
a/parenrightBig
M−μ,σ/parenleftBigy
a/parenrightBig
/bracketleftbig
Rea>0,y > 0,Reσ>−1
4,Reμ<1/bracketrightbig
ET II 14(15)
6./integraldisplay∞
0xPμ
σ−1
2/parenleftBig/radicalbig
1+a2x2/parenrightBig
P−μ
σ−1
2/parenleftBig/radicalbig
1+a2x2/parenrightBig
J0(xy)dx
=2π−1y−2cos(σπ)Wμ,σ/parenleftBigy
a/parenrightBig
W−μ,σ/parenleftBigy
a/parenrightBig
/bracketleftbig
Rea>0,y > 0,|Reσ|<1
4/bracketrightbig
ET II 14(14)
7./integraldisplay∞
0x/braceleftBig
Pμ
σ−1
2/parenleftBig/radicalbig
1+a2x2/parenrightBig/bracerightBig2
J0(xy)dx=−iπ−1y−2Wμ,σ/parenleftBigy
a/parenrightBig/bracketleftBig
Wμ,σ/parenleftBig
eπiy
a/parenrightBig
−Wμ,σ/parenleftBig
e−πiy
a/parenrightBig/bracketrightBig
/bracketleftbig
Rea>0,y > 0,|Reσ|<1
4,Reμ<1/bracketrightbig
ET II 14(13)
786 Associated Legendre Functions 7.188
8./integraldisplay∞
0x/parenleftbig
1+a2x2/parenrightbig−1/2P−1
2−1
2ν
μ/parenleftBig/radicalbig
1+a2x2/parenrightBig
P1
2−1
2ν
μ/parenleftBig/radicalbig
1+a2x2/parenrightBig
Jν(xy)dx
=/bracketleftBig
Kμ+1
2/parenleftbigy
2a/parenrightbig/bracketrightBig2
πa2Γ/parenleftbigν
2+μ+3
2/parenrightbig
Γ/parenleftbigν
2−μ+1
2/parenrightbig
/bracketleftbig
Rea>0,y > 0,−5
4<Reμ<1
4/bracketrightbig
ET II 46(9)
9./integraldisplay∞
0x/braceleftBig
P−1
2ν
μ/parenleftBig/radicalbig
1+a2x2/parenrightBig/bracerightBig2
Jν(xy)dx=2/bracketleftBig
Kμ+1
2/parenleftbigy
2a/parenrightbig/bracketrightBig2
y−1
πaΓ/parenleftbig
1+μ+1
2ν/parenrightbig
Γ/parenleftbig1
2ν−μ/parenrightbig
/bracketleftbig
Rea>0,y > 0,−3
4<Reμ<−1
4,Reν>−1/bracketrightbig
ET II 45(7)
10./integraldisplay∞
0x/parenleftbig
1+a2x2/parenrightbig−1/2P−1
2ν
μ/parenleftBig/radicalbig
1+a2x2/parenrightBig
P−1
2ν
μ+1/parenleftBig/radicalbig
1+a2x2/parenrightBig
Jν(xy)dx
=Kμ+1
2/parenleftbigy
2a/parenrightbig
Kμ+3
2/parenleftbigy
2a/parenrightbig
πa2Γ/parenleftbig
2+1
2ν+μ/parenrightbig
Γ/parenleftbig1
2ν−μ/parenrightbig
/bracketleftbig
Rea>0,y > 0,−7
4<Reμ<−1
4/bracketrightbig
ET II 45(8)
7.188
1./integraldisplay∞
0x/parenleftbig
a2+x2/parenrightbig−1
2μP−ν
μ−1/bracketleftbigga√
a2+x2/bracketrightbigg
Jν(xy)dx=yμ−2e−ay
Γ(μ+ν)/bracketleftbig
Rea>0,y > 0,Reν>−1,Reμ>1
2/bracketrightbig
ET II 45(4)
2./integraldisplay∞
0xν+1/parenleftbig
x2+a2/parenrightbig1
2νPν/parenleftbiggx2+2a2
2a√
x2+a2/parenrightbigg
Jν(xy)dx=(2a)ν+1y−ν−1
πΓ(−ν)/bracketleftBig
Kν+1
2/parenleftBigya
2/parenrightBig/bracketrightBig2
[Rea>0,−1<Reν<0,y > 0]
ET II 45(5)
3./integraldisplay∞
0x1−ν/parenleftbig
x2+a2/parenrightbig−1
2νPν−1/parenleftbiggx2+2a2
2a√
x2+a2/parenrightbigg
Jν(xy)dx=(2a)1−νyν−1
Γ(ν)Iν−1
2/parenleftBigay
2/parenrightBig
Kν−1
2/parenleftBigay
2/parenrightBig
[Rea>0,y > 0,0<Reν<1]
ET II 45(6)
7.189
1./integraldisplay∞
0(a+x)μe−xP−2μ
ν/parenleftbigg
1+2x
a/parenrightbigg
Iμ(x)dx=0
/bracketleftbig
−1
2<Reμ<0,−1
2+R eμ<Reν<−1
2−Reμ/bracketrightbig
ET II 366(18)
2./integraldisplay∞
0(x+a)−μe−xP−2μ
ν/parenleftbigg
1+2x
a/parenrightbigg
Iμ(x)dx
=2μ−1Γ/parenleftbig
μ+ν+1
2/parenrightbig
Γ/parenleftbig
μ−ν−1
2/parenrightbig
ea
π1/2Γ( 2μ+ν+1 )Γ ( 2 μ−ν)W1
2−μ,1
2+ν(2a)
/bracketleftbig
|arga|<π , Reμ>/vextendsingle/vextendsingleReν+1
2/vextendsingle/vextendsingle/bracketrightbig
ET II 367(19)
7.192 Associated Legendre functions and functions generated by Bessel functions 787
3./integraldisplay∞
0x−μexP2μ
ν/parenleftbigg
1+2x
a/parenrightbigg
Kμ(x+a)dx
=π−1/22μ−1cos(μπ)Γ/parenleftbig
μ+ν+1
2/parenrightbig
Γ/parenleftbig
μ−ν+1
2/parenrightbig
W1
2−μ,1
2+ν(2a)
/bracketleftbig
|arga|<π , Reμ>/vextendsingle/vextendsingleReν+1
2/vextendsingle/vextendsingle/bracketrightbig
ET II 373(11)
4./integraldisplay∞
0x−1
2μ(x+a)−1/2e−xPμ
ν−1
2/parenleftbigga−x
a+x/parenrightbigg
Kν(a+x)dx=/radicalbiggπ
2a−1
2μΓ(μ,2a)
[a>0,Reμ<1] ET II 374(12)
5./integraldisplay∞
0(sinhx)μ+1(coshx)−2μ−3
2P−μ
ν[cosh(2 x)]Iμ−1
2(asechx)dx
=2μ−1
2Γ(μ−ν)Γ(μ+ν+1 )
π1/2aμ+3
2[Γ(μ+1 ) ]2Mν+1
2,μ(a)M−ν−1
2,μ(a)
[Reμ>Reν,Reμ>−Reν−1]ET II 378(44)
7.19 Combinations of associated Legendre functions and functions generated by
Bessel functions
7.191
1./integraldisplay∞
ax1/2/parenleftbig
x2−a2/parenrightbig−1
4−1
2νPν+1
2μ/parenleftbig
2x2a−2−1/parenrightbig
[Hν(x)−Yν(x)]dx
=2−ν−2π1/2acosec( μπ)cos(νπ)/braceleftBig/bracketleftbig
Yν/parenleftbig1
2a/parenrightbig/bracketrightbig2−/bracketleftbig
Jν/parenleftbig1
2a/parenrightbig/bracketrightbig2/bracerightBig
/bracketleftbig
−1<Reμ<0,Reν<1
2/bracketrightbig
ET II 384(6)
2./integraldisplay∞
0x1/2/parenleftbig
x2−a2/parenrightbig−1/4−ν/2Pν+1/2
μ/parenleftbig
2x2a−2−1/parenrightbig
[I−ν(x)−Lν(x)]dx
=2−ν−1π1/2acosec(2 μπ)cos(νπ)/braceleftBig/bracketleftbig
Iν/parenleftbig1
2a/parenrightbig/bracketrightbig2−/bracketleftbig
I−ν/parenleftbig1
2a/parenrightbig/bracketrightbig2/bracerightBig
/bracketleftbig
−1<Reμ<0,Reν<1
2/bracketrightbig
ET II 385(15)
7.192
1./integraldisplay1
0x(ν−μ−1)/2/parenleftbig
1−x2/parenrightbig(ν−μ−2)/4P(μ−ν+2)/2
ν−1/2(x)Sμ,ν(ax)dx
=2μ−3/2π1/2a−(ν−μ−1)/2Γ/parenleftbiggμ+ν+3
4/parenrightbigg
Γ/parenleftbiggμ−3ν+3
4/parenrightbigg
cos/parenleftbiggμ−ν
2π/parenrightbigg
×/bracketleftbig
Jν/parenleftbig1
2a/parenrightbig
Y−(μ−ν+1)/2/parenleftbig1
2a/parenrightbig
−Yν/parenleftbig1
2a/parenrightbig
J−(μ−ν+1)/2/parenleftbig1
2a/parenrightbig/bracketrightbig
[Re(μ−ν)<0,a > 0,|Re(μ+ν)|<1,Re(μ−3ν)<1]ET II 387(24)a
788 Associated Legendre Functions 7.193
2./integraldisplay∞
1x1/2/parenleftbig
x2−1/parenrightbig−β/2Pβ
ν(x)Sμ,1/2(ax)dx
=2−3/2+β−μaβ−1Γ/parenleftBig
β−μ+ν
2+1
4/parenrightBig
Γ/parenleftBig
β−μ−ν
2−1
4/parenrightBig
π1/2Γ/parenleftbig1
2−μ/parenrightbig Sμ−β+1,ν+1/2(a)
/bracketleftbig
Reβ<1,a > 0,Re(μ+ν−β)<−1
2,Re(μ−ν−β)<1
2/bracketrightbig
ET II 387(25)a
7.193
1./integraldisplay∞
1x−ν/parenleftbig
x2−1/parenrightbig1/4−ν/2Pν−1/2
μ/2−ν/2/parenleftbig
2x−2−1/parenrightbig
Sμ,ν(ax)dx
=2μ−νaν−2π1/2Γ/parenleftbig3ν−μ−1
2/parenrightbig
Γ/parenleftbig1+ν−μ
2/parenrightbig Wρ,σ/parenleftBig
aeiπ/2/parenrightBig
Wρ,σ/parenleftBig
ae−iπ/2/parenrightBig
ρ=1
2(μ+1−ν),σ=ν−1
2,/bracketleftbig
Re(μ−ν)<0,a > 0,Reν<3
2,Re(3ν−μ)>1/bracketrightbig
ET II 387(27)a
2./integraldisplay∞
1x/parenleftbig
x2−1/parenrightbig−ν/2Pν
λ/parenleftbig
2x2−1/parenrightbig
Sμ,ν(ax)dx
=aν−1Γ/parenleftbigν−μ+1
2+λ/parenrightbig
Γ/parenleftbigν−μ−1
2−λ/parenrightbig
2Γ/parenleftbig1−μ−ν
2/parenrightbig
Γ/parenleftbig1−μ+ν
2/parenrightbig Sμ−ν+1,2λ+1(a)
[Reν<1,a > 0,Re(μ−ν+λ)<−1,Re(μ−ν+λ)<0]ET II 387(26)a
7.21 Integration of associated Legendre functions with respect to the order
7.211
1./integraldisplay∞
0P−x−1
2(cosθ)dx=1
2cosec/parenleftbigg1
2θ/parenrightbigg
[0<θ<π ] ET II 329(19)
2./integraldisplay∞
−∞Px(cosθ)dx=c o s e c/parenleftbigg1
2θ/parenrightbigg
[0<θ<π ] ET II 329(20)
7.212/integraldisplay∞
0x−1tanh(πx)P−1
2+ix(cosha)dx=2e−1
2aK/parenleftbig
e−a/parenrightbig
[a>0] ET II 330(22)
7.213/integraldisplay∞
0xtanh(πx)
a2+x2P−1
2+ix(coshb)dx=Qa−1
2(coshb)[ R e a>0] ET II 387(23)
7.214/integraldisplay∞
0sinh(πx)cos(ax)P−1
2+ix(b)dx=1/radicalbig
2(b+c o s h a)
[a>0,|b|<1] ET I 42(27)
7.215/integraldisplay∞
0cos(bx)Pμ
−1
2+ix(cosha)dx=0 [ 0 <a<b ]
=/radicalbigπ
2(sinha)μ
Γ/parenleftbig1
2−μ/parenrightbig
(cosha−coshb)μ+1
2[0<b<a ]
ET II 330(21)
7.221 Integration of associated Legendre functions 789
7.216/integraldisplay∞
0cos(bx)Γ(μ+ix)Γ(μ−ix)P1
2−μ
−1
2+ix(cosha)dx=/radicalbigπ
2Γ(μ)(sin h a)μ−1
2
(cosha+c o s h b)μ
[a>0,b > 0,Reμ>0]
ET II 330(24)
7.217
1./integraldisplay∞
−∞/parenleftbigg
ν−1
2+ix/parenrightbigg
Γ/parenleftbigg1
2−ix/parenrightbigg
Γ/parenleftbigg
2ν−1
2+ix/parenrightbigg
P1
2−ν
ν+ix−1(cosθ)Iν−1
2+ix(a)Kν−1
2+ix(b)dx
=√
2π(sinθ)ν−1
2/parenleftbiggab
ω/parenrightbiggν
Kν(ω)
/bracketleftBig
ω=/parenleftbig
a2+b2+2abcosθ/parenrightbig1/2/bracketrightBig
ET II 383(29)
2./integraldisplay∞
0xeπxtanh(πx)P−1
2+ix(−cosθ)H(2)
ix(ka)H(2)
ix(kb)dx=−2(ab)1/2
πRe−ikR;
R=/parenleftbig
a2+b2−2abcosθ/parenrightbig1/2[a>0,b > 0,0<θ<π , Imk≤0]ET II 381(17)
3./integraldisplay∞
0xeπxsinh(πx)Γ(ν+ix)Γ (ν−ix)P1
2−ν
−1
2+ix(−cosθ)H(2)
ix(a)H(2)
ix(b)dx
=i(2π)1/2(sinθ)ν−1
2/parenleftbiggab
R/parenrightbiggν
H(2)
ν(R)
R=/parenleftbig
a2+b2−2abcosθ/parenrightbig1/2[a>0,b > 0,0<θ<π , Reν>0]ET II 381 (18)
4./integraldisplay∞
0xsinh(πx)Γ(λ+ix)Γ(λ−ix)Kix(a)Kix(b)P1
2−λ
−1
2+ix(β)dx=π1/2
√
2/parenleftbiggab
z/parenrightbiggλ/parenleftbig
β2−1/parenrightbig1
2λ−1
4Kλ(z)
z=/radicalbig
a2+b2+2abβ/bracketleftBig
|arga|<π
2,|arg(β−1)|<π , Reλ>0/bracketrightBig
ET II 177(16)
7.22 Combinations of Legendre polynomials, rational functions, and algebraic
functions
7.221
1./integraldisplay1
−1Pn(x)Pm(x)dx=0 [m/negationslash=n]
=2
2n+1[m=n] WH, EH I 170(8, 10)
2.6/integraldisplay1
0Pn(x)Pm(x)dx=1
2n+1[m=n]
=0 [ n−mis even ,m/negationslash=n]
=(−1)1
2(m+n−1)m!n!
2m+n−1(m−n)(n+m+1 )/bracketleftbig/parenleftbign
2/parenrightbig
!/parenleftbigm−1
2/parenrightbig
!/bracketrightbig2[nis even, mis odd]
WH
3./integraldisplay2π
0P2n(cosϕ)dϕ=2π/bracketleftbigg/parenleftbigg2n
n/parenrightbigg
2−2n/bracketrightbigg2
. MO 70, EH II 183(50)
790 Associated Legendre Functions 7.222
7.222
1./integraldisplay1
−1xmPn(x)dx=0 [ m<n ]
2./integraldisplay1
−1(1 +x)m+nPm(x)Pn(x)dx=2m+n+1[(m+n)!]4
(m!n!)2(2m+2n+1 ) !ET II 277(15)
3./integraldisplay1
−1(1 +x)m−n−1Pm(x)Pn(x)dx=0 [ m>n ] ET II 278(16)
4./integraldisplay1
−1/parenleftbig
1−x2/parenrightbignP2m(x)dx=2n2
(n−m)(2m+2n+1 )/integraldisplay1
−1/parenleftbig
1−x2/parenrightbign−1P2m(x)dx
[m<n ] WH
5./integraldisplay1
0x2Pn+1(x)Pn−1(x)dx=n(n+1 )
(2n−1)(2n+ 1)(2 n+3 )WH
7.223/integraldisplay1
−11
z−x{Pn(x)Pn−1(x)−Pn−1(x)Pn(z)}dx=−2
nWH
7.224 [zbelongs to the complex plane with a discontinuity along the interval from −1 to +1.]
1./integraldisplay1
−1(z−x)−1Pn(x)dx=2Qn(z) ET II 277(7)
2./integraldisplay1
−1x(z−x)−1P0(x)dx=2Q1(z) ET II 277(8)
3./integraldisplay1
−1xn+1(z−x)−1Pn(x)dx=2zn+1Qn(z)−2n+1(n!)2
(2n+1 ) !ET II 277(9)
4./integraldisplay1
−1xm(z−x)−1Pn(x)dx=2zmQn(z)[ m≤n] ET II 277(10)a
5./integraldisplay1
−1(z−x)−1Pm(x)Pn(x)dx=2Pm(z)Qn(z)[ m≤n] ET II 278(18)a
6./integraldisplay1
−1(z−x)−1Pn(x)Pn+1(x)dx=2Pn+1(z)Qn(z)−2
n+1ET II 278(19)
7./integraldisplay1
−1x(z−x)−1Pm(x)Pn(x)dx=2zPm(z)Qn(z)[ m<n ] ET II 278(21)
8./integraldisplay1
−1x(z−x)−1[Pn(x)]2dx=2zPn(z)Qn(z)−2
2n+1ET II 278(20)
7.225
1./integraldisplayx
−1(x−t)−1/2Pn(t)dt=/parenleftbigg
n+1
2/parenrightbigg−1
(1 +x)−1/2[Tn(x)+Tn+1(x)] EH II 187(43)
2./integraldisplay1
x(t−x)−1/2P−1/2Pn(t)dt=/parenleftbigg
n+1
2/parenrightbigg−1
(1−x)−1/2[Tn(x)−Tn+1(x)] EH II 187(44)
7.232 Legendre polynomials and powers 791
3./integraldisplay1
−1(1−x)−1/2Pn(x)dx=23/2
2n+1EH II 183(49)
4./integraldisplay1
−1(cosh 2 p−x)−1/2Pn(x)dx=2√
2
2n+1exp[−(2n+1 )p]
[p>0] WH
5.101
2/integraldisplay1
−1P/lscript(z)dz/radicalbig
(xy−z)2−(x2−1)(y2−1)=P/lscript(x)Q/lscript(y)( 1 <x≤y)
=P/lscript(y)Q/lscript(x)( 1 <y≤x)
7.226
1./integraldisplay1
−1/parenleftbig
1−x2/parenrightbig−1/2P2m(x)dx=/bracketleftBigg
Γ/parenleftbig1
2+m/parenrightbig
m!/bracketrightBigg2
ET II 276(4)
2./integraldisplay1
−1x/parenleftbig
1−x2/parenrightbig−1/2P2m+1(x)dx=Γ/parenleftbig1
2+m/parenrightbig
Γ/parenleftbig3
2+m/parenrightbig
m!(m+1 ) !ET II 276(5)
3./integraldisplay1
−1/parenleftbig
1+px2/parenrightbig−m−3/2P2m(x)dx=2
2m+1(−p)m(1 +p)−m−1/2
[|p|<1] MO 71
7.227/integraldisplay1
0x/parenleftbig
a2+x2/parenrightbig−1/2Pn/parenleftbig
1−2x2/parenrightbig
dx=/bracketleftBig
a+/parenleftbig
a2+1/parenrightbig1/2/bracketrightBig−2n−1
2n+1
[Rea>0] ET II 278(23)
7.22861
2Γ(1 + μ)/integraldisplay1
−1Pl(x)(z−x)−μ−1dx=/parenleftbig
z2−1/parenrightbig−μ/2e−iπμQμ
l(z)
[l=0,1,2,..., |arg(z−1)|<π]
7.23 Combinations of Legendre polynomials and powers
7.231
1./integraldisplay1
0xλP2m(x)dx=(−1)mΓ/parenleftbig
m−1
2λ/parenrightbig
Γ/parenleftbig1
2+1
2λ/parenrightbig
2Γ/parenleftbig
−1
2λ/parenrightbig
Γ/parenleftbig
m+3
2+1
2λ/parenrightbig [Reλ>−1] EH II 183(51)
2.6/integraldisplay1
0xλP2m+1(x)dx=(−1)mΓ/parenleftbig
m+1
2−1
2λ/parenrightbig
Γ/parenleftbig
1+1
2λ/parenrightbig
2Γ/parenleftbig1
2−1
2λ/parenrightbig
Γ/parenleftbig
m+2+1
2λ/parenrightbig
[Reλ>−2] EH II 183(52)
7.232
1./integraldisplay1
−1(1−x)a−1Pm(x)Pn(x)dx
=2aΓ(a)Γ(n−a+1 )
Γ(1−a)Γ(n+a+1 )4F3(−m, m+1,a,a;1,a+n+1,a−n;1)
[Rea>0] ET II 278(17)
792 Associated Legendre Functions 7.233
2./integraldisplay1
−1(1−x)a−1(1 +x)b−1Pn(x)dx=2a+b−1Γ(a)Γ(b)
Γ(a+b)3F2(−n,1+n, a;1,a+b;1)
[Rea>0,Reb>0] ET II 276(6)
3./integraldisplay1
0(1−x)μ−1Pn(1−γx)dx=Γ(μ)n!
Γ(μ+n+1 )P(μ,−μ)
n (1−γ)
[Reμ>0] ET II 190(37)a
4./integraldisplay1
0(1−x)μ−1xν−1Pn(1−γx)dx=Γ(μ)Γ(ν)
Γ(μ+ν)3F2/parenleftbigg
−n, n+1,ν;1,μ+ν;1
2γ/parenrightbigg
[Reμ>0,Reν>0] ET II 190(38)
7.233/integraldisplay1
0x2μ−1Pn/parenleftbig
1−2x2/parenrightbig
dx=(−1)n[Γ(μ)]2
2Γ (μ+n+1 )Γ ( μ−n)
[Reμ>0] ET II 278(22)
7.24 Combinations of Legendre polynomials and other elementary functions
7.241/integraldisplay∞
0Pn(1−x)e−axdx=e−aan/parenleftbigg1
ad
da/parenrightbiggn/parenleftbiggea
a/parenrightbigg
=an/parenleftbigg
1+1
2d
da/parenrightbiggn/parenleftbigg1
an+1/parenrightbigg
[Rea>0] ET I 171(2)
7.242/integraldisplay∞
0Pn/parenleftbig
e−x/parenrightbig
e−axdx=(a−1)(a−2)···(a−n+1 )
(a+n)(a+n−2)···(a−n+2 )
[n≥2,Rea>0] ET I 171(3)
7.243
1./integraldisplay∞
0P2n(coshx)e−axdx=/parenleftbig
a2−12/parenrightbig/parenleftbig
a2−32/parenrightbig
···/bracketleftbig
a2−(2n−1)2/bracketrightbig
a(a2−22)(a2−42)···[a2−(2n)2]
[Rea>2n] ET I 171(6)
2./integraldisplay∞
0P2n+1(coshx)e−axdx=a/parenleftbig
a2−22/parenrightbig/parenleftbig
a2−42/parenrightbig
···/bracketleftbig
a2−(2n)2/bracketrightbig
(a2−1)(a2−32)···[a2−(2n+1 )2]
[Rea>2n+1 ] ET I 171(7)
3./integraldisplay∞
0P2n(cosx)e−axdx=/parenleftbig
a2+12/parenrightbig/parenleftbig
a2+32/parenrightbig
···/bracketleftbig
a2+( 2n−1)2/bracketrightbig
a(a2+22)(a2+42)···[a2+( 2n)2]
[Rea>0] ET I 171(4)
4./integraldisplay∞
0P2n+1(cosx)e−axdx=a/parenleftbig
a2+22/parenrightbig/parenleftbig
a2+42/parenrightbig
···/bracketleftbig
a2+( 2n)2/bracketrightbig
(a2+12)(a2+32)···[a2+( 2n+1 )2]
[Rea>0] ET I 171(5)
5.11/integraldisplay1
−1eixαPn(x)dx=in/radicalbigg
2π
αJn+1
2(α)[ n=0,1,2,..., a> 0]
GH2 24 (171.10)
7.249 Legendre polynomials and elementary functions 793
7.244
1./integraldisplay1
0Pn/parenleftbig
1−2x2/parenrightbig
sinaxdx =π
2/bracketleftBig
Jn+1
2/parenleftBiga
2/parenrightBig/bracketrightBig2
[a>0] ET I 94(2)
2./integraldisplay1
0Pn/parenleftbig
1−2x2/parenrightbig
cosaxdx =π
2(−1)nJn+1
2/parenleftBiga
2/parenrightBig
J−n−1
2/parenleftBiga
2/parenrightBig
[a>0] ET I 38(1)
7.245
1./integraldisplay2π
0P2m+1(cosθ)c o sθd θ=π
24m+1/parenleftbigg2m
m/parenrightbigg/parenleftbigg2m+2
m+1/parenrightbigg
MO 70, EH II 183(5)
2./integraldisplayπ
0Pm(cosθ)sinnθ dθ=2(n−m+1 ) (n−m+3 )···(n+m−1)
(n−m)(n−m+2 )···(n+m)[n>m andn+mis odd]
=0 [ n≤morn+mis even]
MO 71
3.10/integraldisplay2π
0P2n+1(sinαsinφ)sinφd φ=(−1)n+12√πΓ/parenleftbig
n+3
2/parenrightbig
(2n+1 )Γ( n+2 )P1
2n+1(cosα)
/bracketleftbig
α/negationslash=1
2(2n+1 )π, n an integer/bracketrightbig
4./integraldisplay1
−1cos(αx)Pn(x)dx=0 [ nis odd]
=(−1)v/radicalbigg
2π
αJ2v+1
2(α)[ n=2vis even]
GH2 24 (171.10a)
7.246/integraldisplayπ
0Pn/parenleftbig
1−2s in2xsin2θ/parenrightbig
sinxdx=2s in ( 2 n+1 )θ
(2n+1 )s i n θMO 71
7.247/integraldisplay1
0P2n+1(x)sinaxdx√x=(−1)n+1/radicalbiggπ
2aJ2n+3
2(a)[ a>0] ET I 94(1)
7.248
1./integraldisplay1
−1/parenleftbig
a2+b2−2abx/parenrightbig−1/2sin/bracketleftBig
λ/parenleftbig
a2+b2−2abx/parenrightbig1/2/bracketrightBig
Pn(x)dx=π(ab)−1/2Jn+1
2(aλ)Jn+1
2(bλ)
[a>0,b > 0] ET II 277(11)
2./integraldisplay1
−1/parenleftbig
a2+b2−2abx/parenrightbig−1/2cos/bracketleftBig
λ/parenleftbig
a2+b2−2abx/parenrightbig1/2/bracketrightBig
Pn(x)dx=−π(ab)−1/2Jn+1
2(aλ)Yn+1
2(bλ)
[0≤a≤b] ET II 277(12)
7.249
1./integraldisplay1
−1Pn(x)arcsin xdx=0 [ nis even]
=π⎧
⎪⎪⎨
⎪⎪⎩(n−2)!!
21
2(n+1)/parenleftbiggn+1
2/parenrightbigg
!⎫
⎪⎪⎬
⎪⎪⎭2
[nis odd]
WH
794 Associated Legendre Functions 7.251
2. Pn(x)=1
tt−1/summationdisplay
t=0/parenleftbigg
x+/radicalbig
x2−1c os2πr
t/parenrightbiggn
[t>n]
7.25 Combinations of Legendre polynomials and Bessel functions
7.251
1./integraldisplay1
0xPn/parenleftbig
1−2x2/parenrightbig
Yν(xy)dx=π−1y−1[S2n+1(y)+πY2n+1(y)]
[n=0,1,...;y>0,ν > 0]
ET II 108(1)
2./integraldisplay1
0xPn/parenleftbig
1−2x2/parenrightbig
K0(xy)dx=y−1/bracketleftbigg
(−1)n+1K2n+1(y)+i
2S2n+1(iy)/bracketrightbigg
[y>0] ET II 134(1)
3./integraldisplay1
0xPn/parenleftbig
1−2x2/parenrightbig
J0(xy)dx=y−1J2n+1(y)[ y>0] ET II 13(1)
4./integraldisplay1
0xPn/parenleftbig
1−2x2/parenrightbig
[J0(ax)]2dx=1
2(2n+1 )/braceleftBig
[Jn(a)]2+[Jn+1(a)]2/bracerightBig
ET II 338(39)a
5./integraldisplay1
0xPn/parenleftbig
1−2x2/parenrightbig
J0(ax)Y0(ax)dx=1
2(2n+1 )[Jn(a)Yn(a)+Jn+1(a)Yn+1(a)]
ET II 339(48)a
6./integraldisplay1
0x2Pn/parenleftbig
1−2x2/parenrightbig
J1(xy)dx=y−1(2n+1 )−1[(n+1 )J2n+2(y)−nJ2n(y)]
[y>0] ET II 20(23)
7./integraldisplay1
0xμ−1Pn/parenleftbig
2x2−1/parenrightbig
Jν(ax)dx=2−ν−1aν/bracketleftbig
Γ/parenleftbig1
2μ+1
2ν/parenrightbig/bracketrightbig2
Γ(ν+1 )Γ/parenleftbig1
2μ+1
2ν+n+1/parenrightbig
Γ/parenleftbig1
2+1
2ν−n/parenrightbig
×2F3/parenleftbiggμ+ν
2,μ+ν
2;ν+1,μ+ν
2+n+1,μ+ν
2−n;−a2
4/parenrightbigg
[a>0,Re(μ+ν)>0]ET II 337(32)a
7.252/integraldisplay1
0e−axPn(1−2x)I0(ax)dx=e−a
2n+1[In(a)+In+1(a)]
[a>0] ET II 366(11)a
7.253/integraldisplayπ/2
0sin(2x)Pn(cos2x)J0(asinx)dx=a−1J2n+1(a) ET II 361(20)
7.254/integraldisplay1
0xPn/parenleftbig
1−2x2/parenrightbig
[I0(ax)−L0(ax)]dx=(−1)n[I2n+1(a)−L2n+1(a)]
[a>0] ET II 385(14)a
7.313 Gegenbauer polynomials Cν
n(x)and powers 795
7.3–7.4 Orthogonal Polynomials
7.31 Combinations of Gegenbauer polynomials Cν
n(x)and powers
7.311
1./integraldisplay1
−1/parenleftbig
1−x2/parenrightbigν−1
2Cν
n(x)dx=0/bracketleftbig
n>0,Reν>−1
2/bracketrightbig
ET II 280(1)
2./integraldisplay1
0xn+2ρ/parenleftbig
1−x2/parenrightbigν−1
2Cν
n(x)dx=Γ(2ν+n)Γ(2ρ+n+1 )Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig
ρ+1
2/parenrightbig
2n+1Γ(2ν)Γ(2ρ+1 )n!Γ (n+ν+ρ+1 )/bracketleftbig
Reρ>−1
2,Reν>−1
2/bracketrightbig
ET II 280(2)
3./integraldisplay1
−1(1−x)ν−1
2(1 +x)βCν
n(x)dx=2β+ν+1
2Γ(β+1 )Γ/parenleftbig
ν+1
2/parenrightbig
Γ(2ν+n)Γ/parenleftbig
β−ν+3
2/parenrightbig
n!Γ ( 2ν)Γ/parenleftbig
β−ν−n+3
2/parenrightbig
Γ/parenleftbig
β+ν+n+3
2/parenrightbig
/bracketleftbig
Reβ>−1,Reν>−1
2/bracketrightbig
ET II 280(3)
4./integraldisplay1
−1(1−x)α(1 +x)βCν
n(x)dx=2α+β+1Γ(α+1 )Γ ( β+1 )Γ( n+2ν)
n!Γ ( 2ν)Γ(α+β+2 )
×3F2/parenleftbigg
−n, n+2ν,α+1 ;ν+1
2,α+β+2 ;1/parenrightbigg
[Reα>−1,Reβ>−1] ET II 281(4)
7.312 In the following integrals, zbelongs to the complex plane with a cut along the interval of the real
axis from −1t o1 .
1./integraldisplay1
−1xm(z−x)−1/parenleftbig
1−x2/parenrightbigν−1
2Cν
n(x)dx=π1/223
2−ν
Γ(ν)e−(ν−1
2)πizm/parenleftbig
z2−1/parenrightbig1
2ν−1
4Qν−1
2
n+ν−1
2(z)
/bracketleftbig
m≤n,Reν>−1
2/bracketrightbig
ET II 281(5)
2./integraldisplay1
−1xn+1(z−x)−1/parenleftbig
1−x2/parenrightbigν−1
2Cν
n(x)dx=π1/223
2−ν
Γ(ν)e−(ν−1
2)πizn+1/parenleftbig
z2−1/parenrightbig1
2ν−1
4Qν−1
2
n+ν−1
2(z)
−π21−2ν−nn!
Γ(ν)Γ(ν+n+1 )/bracketleftbig
Reν>−1
2/bracketrightbig
ET II 281(6)
3.6/integraldisplay1
−1(z−x)−1/parenleftbig
1−x2/parenrightbigν−1
2Cν
m(x)Cν
n(x)dx=π1/223
2−ν
Γ(ν)e−(ν−1
2)πi/parenleftbig
z2−1/parenrightbig1
2ν−1
4Cν
m(z)Qν−1
2
n+ν−1
2(z)
/bracketleftbig
m≤n,Reν>−1
2/bracketrightbig
ET II 283(17)
7.313
1./integraldisplay1
−1/parenleftbig
1−x2/parenrightbigν−1
2Cν
m(x)Cν
n(x)dx=0/bracketleftbig
m/negationslash=n,Reν>−1
2/bracketrightbig
ET II 282(12), MO 98a, EH I 177(16)
2./integraldisplay1
−1/parenleftbig
1−x2/parenrightbigν−1
2[Cν
n(x)]2dx=π21−2νΓ(2ν+n)
n!(n+ν)[Γ (ν)]2/bracketleftbig
Reν>−1
2/bracketrightbig
ET II 281(8), MO 98a, EH I 177(17)
796 Orthogonal Polynomials 7.314
7.314
1./integraldisplay1
−1(1−x)ν−3
2(1 +x)ν−1
2[Cν
n(x)]2dx=π1/2Γ/parenleftbig
ν−1
2/parenrightbig
Γ(2ν+n)
n!Γ (ν)Γ(2ν)
/bracketleftbig
Reν>1
2/bracketrightbig
ET II 281(9)
2./integraldisplay1
−1(1−x)ν−1
2(1 +x)2ν−1[Cν
n(x)]2dx=23ν−1
2[Γ(2ν+n)]2Γ/parenleftbig
2n+ν+1
2/parenrightbig
(n!)2Γ(2ν)Γ/parenleftbig
3ν+2n+1
2/parenrightbig
[Reν>0] ET II 282(10)
3./integraldisplay1
−1(1−x)3ν+2n−3
2(1 +x)ν−1
2[Cν
n(x)]2dx
=π1/2/bracketleftbig
Γ/parenleftbig
ν+1
2/parenrightbig/bracketrightbig2Γ/parenleftbig
ν+2n+1
2/parenrightbig
Γ( 2ν+2n)Γ/parenleftbig
3ν+2n−1
2/parenrightbig
22ν+2n/bracketleftbig
n!Γ/parenleftbig
ν+n+1
2/parenrightbig
Γ(2ν)/bracketrightbig2Γ/parenleftbig
2ν+2n+1
2/parenrightbig
/bracketleftbig
Reν>1
6/bracketrightbig
ET II 282(11)
4./integraldisplay1
−1(1−x)ν−1
2(1 +x)ν+m−n−3
2Cν
m(x)Cν
n(x)dx
=(−1)m 22−2ν−m+nπ3/2Γ(2ν+n)
m!(n−m)! [Γ(ν)]2Γ/parenleftbig1
2+ν+m/parenrightbigΓ/parenleftbig
ν−1
2+m−n/parenrightbig
Γ/parenleftbig1
2−ν+m−n/parenrightbig
Γ/parenleftbig1
2−ν−n/parenrightbig
Γ/parenleftbig1
2+m−n/parenrightbig
/bracketleftbig
Reν>−1
2;n≥m/bracketrightbig
ET II 282(13)a
5./integraldisplay1
−1(1−x)2ν−1(1 +x)ν−1
2Cν
m(x)Cν
n(x)dx
=23ν−1
2Γ/parenleftbig
ν+1
2/parenrightbig
Γ(2ν+m)Γ ( 2ν+n)
m!n!Γ ( 2ν)Γ/parenleftbig1
2−ν/parenrightbigΓ/parenleftbig
ν+1
2+m+n/parenrightbig
Γ/parenleftbig1
2−ν+n−m/parenrightbig
Γ/parenleftbig
ν+1
2+n−m/parenrightbig
Γ/parenleftbig
3ν+1
2+m+n/parenrightbig
[Reν>0] ET II 282(14)
6./integraldisplay1
−1(1−x)ν−1
2(1 +x)3ν+m+n−3
2Cν
m(x)Cν
n(x)dx
=24ν+m+n−1/bracketleftbig
Γ/parenleftbig
ν+1
2/parenrightbig
Γ(2ν+m+n)/bracketrightbig2
Γ/parenleftbig
ν+m+1
2/parenrightbig
Γ/parenleftbig
ν+n+1
2/parenrightbig
Γ( 2ν+m)Γ/parenleftbig
ν+m+n+1
2/parenrightbig
Γ/parenleftbig
3ν+m+n−1
2/parenrightbig
Γ(2ν+n)Γ(4ν+2m+2n)/bracketleftbig
Reν>1
6/bracketrightbig
ET II 282(15)
7./integraldisplay1
−1(1−x)α(1 +x)ν−1
2Cμ
m(x)Cν
n(x)dx
=2α+ν+1
2Γ(α+1 )Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig
ν−α+n−1
2/parenrightbig
m!n!Γ/parenleftbig
ν−α−1
2/parenrightbig
Γ/parenleftbig
ν−α+n+3
2/parenrightbigΓ(2μ+m)Γ( 2ν+n)
Γ(2μ)Γ(2ν)
×4F3/parenleftbigg
−m, m+2μ, α+1,α−ν+3
2;μ+1
2,ν+α+n+3
2,α−ν−n+3
2;1/parenrightbigg
/bracketleftbig
Reα>−1,Reν>−1
2/bracketrightbig
ET II 283(16)
7.315/integraldisplay1
−1/parenleftbig
1−x2/parenrightbig1
2ν−1Cν
2n(ax)dx=π1/2Γ/parenleftbig1
2ν/parenrightbig
Γ/parenleftbig1
2ν+1
2/parenrightbigC1
2ν
n/parenleftbig
2a2−1/parenrightbig
[Reν>0] ET II 283(19)
7.323 Gegenbauer polynomials Cν
n(x)and elementary functions 797
7.316/integraldisplay1
−1/parenleftbig
1−x2/parenrightbigν−1Cν
n(cosαcosβ+xsinαsinβ)dx=22ν−1n![Γ (ν)]2
Γ(2ν+n)Cν
n(cosα)Cν
n(cosβ)
[Reν>0] ET II 283(20)
7.317
1./integraldisplay1
0(1−x)μ−1xλ−1
2Cλ
n(1−γx)dx=Γ(2λ+n)Γ/parenleftbig
λ+1
2/parenrightbig
Γ(μ)
Γ(2λ)Γ/parenleftbig
λ+μ+n+1
2/parenrightbigP(α,β)
n(1−γ)
α=λ+μ−1
2,β =λ−μ−1
2/bracketleftbig
Reλ>−1,λ/negationslash=0,−1
2,Reμ>0/bracketrightbig
ET II 190(39)a
2./integraldisplay1
0(1−x)μ−1xν−1Cλ
n(1−γx)dx=Γ(2λ+n)Γ(μ)Γ(ν)
n!Γ ( 2λ)Γ(μ+ν)
×3F2/parenleftbigg
−n, n+2λ,ν;λ+1
2,μ+ν;γ
2/parenrightbigg
[2λ/negationslash=0,−1,−2,..., Reμ>0,Reν>0]ET II 191(40)a
7.318/integraldisplay1
0x2ν/parenleftbig
1−x2/parenrightbigσ−1Cν
n/parenleftbig
1−x2y/parenrightbig
dx=Γ(2ν+n)Γ/parenleftbig
ν+1
2/parenrightbig
Γ(σ)
2Γ ( 2ν)Γ/parenleftbig
n+ν+σ+1
2/parenrightbigP(α,β)
n(1−y),
α=ν+σ−1
2,β =ν−σ−1
2/bracketleftbig
Reν>−1
2,Reσ>0/bracketrightbig
ET II 283(21)
7.319
1./integraldisplay1
0(1−x)μ−1xν−1Cλ
2n/parenleftBig
γx1/2/parenrightBig
dx=(−1)nΓ(λ+n)Γ(μ)Γ(ν)
n!Γ (λ)Γ(μ+ν)3F2/parenleftbigg
−n, n+λ,ν;1
2,μ+ν;γ2/parenrightbigg
[Reμ>0,Reν>0] ET II 191(41)a
2./integraldisplay1
0(1−x)μ−1xν−1Cλ
2n+1/parenleftBig
γx1/2/parenrightBig
dx=(−1)n2γΓ(μ)Γ(λ+n+1 )Γ/parenleftbig
ν+1
2/parenrightbig
n!Γ(λ)Γ/parenleftbig
μ+ν+1
2/parenrightbig
×3F2/parenleftbigg
−n, n+λ+1,ν+1
2;3
2,μ+ν+1
2;γ2/parenrightbigg
/bracketleftbig
Reμ>0,Reν>−1
2/bracketrightbig
ET II 191(42)
7.32 Combinations of Gegenbauer polynomials Cν
n(x)and elementary functions
7.321/integraldisplay1
−1/parenleftbig
1−x2/parenrightbigν−1
2eiaxCν
n(x)dx=π21−νinΓ(2ν+n)
n!Γ(ν)a−νJν+n(a)
/bracketleftbig
Reν>−1
2/bracketrightbig
ET II 281(7), MO 99a
7.322/integraldisplay2a
0[x(2a−x)]ν−1
2Cν
n/parenleftBigx
a−1/parenrightBig
e−bxdx=(−1)nπΓ(2ν+n)
n!Γ (ν)/parenleftBiga
2b/parenrightBigν
e−abIν+n(ab)
/bracketleftbig
Reν>−1
2/bracketrightbig
ET I 171(9)
7.323
1./integraldisplayπ
0Cν
n(cosϕ)(sinϕ)2νdϕ=0 [ n=1,2,3,...]
=2−2νπΓ(2ν+1 )[ Γ ( 1+ ν)]−2[n=0 ]
EH I 177(18)
798 Complete System of Orthogonal Step Functions 7.324
2.11/integraldisplayπ
0Cν
n(cosψcosψ/prime+s i nψsinψ/primecosϕ)( s i nϕ)2ν−1dϕ
=22ν−1n![Γ (ν)]2Cν
n(cosψ)Cν
n(cosψ/prime)[Γ (2ν+n)]−1
[Reν>0] EH I 177(20)
7.324
1./integraldisplay1
0/parenleftbig
1−x2/parenrightbigν−1
2Cν
2n+1(x)sinaxdx =(−1)nπΓ(2n+2ν+1 )J2n+ν+1(a)
(2n+1 ) !Γ ( ν)(2a)ν
/bracketleftbig
Reν>−1
2,a > 0/bracketrightbig
ET I 94(4)
2./integraldisplay1
0/parenleftbig
1−x2/parenrightbigν−1
2Cν
2n(x)cosaxdx =(−1)nπΓ(2n+2ν)Jν+2n(a)
(2n)! Γ(ν)(2a)ν
/bracketleftbig
Reν>−1
2,a > 0/bracketrightbig
ET I 38(3)a
7.325∗Complete System of Orthogonal Step Functions
Letsj(x)=(−1)⌊2jx⌋forj∈Nandcj(x)=(−1)⌊2jx+1/2⌋forj∈0+Nwhere ⌊z⌋denotes the integer
part of z.T h u s , cj(z)a n d sj(z) have minimal period j−1and manifest even and odd symmetry about
x=1/2, respectively, and so are the discrete analogues of cos2 πjxand sin 2 πjx. Furthermore, for j∈N
letjdenote its odd part: the quotient of jby its highest power-of-two factor. Then for all jandk∈N,
if (j,k) denotes their highest common factor and [ j,k] denotes their lowest common multiple:
1./integraldisplay1
0sj(x)sk(x)dx=/braceleftBigg(j,k)
[j,k]ifj/j=k/k
0 otherwise
2./integraldisplay1
0cj(x)ck(x)dx=/braceleftBigg
(−1)(j+k)/2+1(j,k)
[j,k]ifj/j=k/k
0 otherwise
7.33 Combinations of the polynomials Cν
n(x)and Bessel functions; Integration of
Gegenbauer functions with respect to the index
7.331
1./integraldisplay∞
1x2n+1−ν/parenleftbig
x2−1/parenrightbigν−2n−1
2Cν−2n
2n/parenleftbigg1
x/parenrightbigg
Jν(xy)dx
=(−1)n22n−ν+1y−ν+2n−1[(2n)!]−1Γ(2ν−2n)[Γ(ν−2n)]−1cosy
/bracketleftbig
y>0,2n−1
2<Reν<2n+1
2/bracketrightbig
ET II 44(10)a
7.334 Gegenbauer functions and Bessel functions 799
7.332
1./integraldisplay∞
0xν+1/parenleftbig
x2+β2/parenrightbig−1
2ν−3
4Cν+1
2
2n+1/bracketleftBig/parenleftbig
x2+β2/parenrightbig−1/2β/bracketrightBig
Jν+3
2+2n/bracketleftBig/parenleftbig
x2+β2/parenrightbig1/2a/bracketrightBig
Jν(xy)dx
=(−1)n21/2π−1/2a1
2−νyν/parenleftbig
a2−y2/parenrightbig−1/2sin/bracketleftBig
β/parenleftbig
a2−y2/parenrightbig1/2/bracketrightBig
Cν+1
2
2n+1/bracketleftBigg/parenleftbigg
1−y2
a2/parenrightbigg1/2/bracketrightBigg
[0<y<a ]
=0
[a<y< ∞][ a>0,Reβ>0,Reν>−1]
ET II 59(23)
2./integraldisplay∞
0xν+1/parenleftbig
x2+β2/parenrightbig−1
2ν−3
4Cν+1
2
2n/bracketleftBig
β/parenleftbig
x2+β2/parenrightbig−1/2/bracketrightBig
Jν+1
2+2n/bracketleftBig/parenleftbig
x2+β2/parenrightbig1/2a/bracketrightBig
Jν(xy)dx
=(−1)n21/2π−1/2a1
2−νyν/parenleftbig
a2−y2/parenrightbig−1/2cos/bracketleftBig
β/parenleftbig
a2−y2/parenrightbig1/2/bracketrightBig
Cν+1
2
2n/bracketleftBigg/parenleftbigg
1−y2
a2/parenrightbigg1/2/bracketrightBigg
[0<y<a ]
=0
[a<y< ∞][ a>0,Reβ>0,Reν>−1]
ET II 59(24)
7.333
1./integraldisplayπ
0(sinx)ν+1cos(acosθcosx)Cν+1
2n(cosx)Jν(asinθsinx)dx
=(−1)n
2/parenleftbigg2π
a/parenrightbigg1/2
(sinθ)νCν+1
2n(cosθ)Jν+1
2+n(a)[n=0,2,4,...]
=0 [ n=1,3,5,...]
[Reν>−1] WA 414(2)a
2./integraldisplayπ
0(sinx)ν+1sin (acosθcosx)Cν+1
2n(cosx)Jν(asinθsinx)dx
=0 [ n=0,2,4,...]
=(−1)n−1
2/parenleftbigg2π
a/parenrightbigg1/2
(sinθ)νCν+1
2n(cosθ)Jν+1
2+n(a)[n=1,3,5,...]
[Reν>−1] WA 414(3)a
7.334
1./integraldisplayπ
0(sinx)2νCν
n(cosx)Jν(ω)
ωνdx=πΓ(2ν+n)
2ν−1n!Γ(ν)Jν+n(α)
ανJν+n(β)
βν,
ω=/parenleftbig
α2+β2−2αβcosx/parenrightbig1/2/bracketleftbig
n=0,1,2,...;R e ν>−1
2/bracketrightbig
ET II 362(29)
2./integraldisplayπ
0(sinx)2νCν
n(cosx)Yν(ω)
ωνdx=πΓ(2ν+n)
2ν−1n!Γ (ν)Jν+n(α)
ανYν+n(β)
βν,
ω=/parenleftbig
α2+β2−2αβcosx/parenrightbig1/2/bracketleftbig
|α|<|β|,Reν−1
2/bracketrightbig
ET II 362(30)
800 Complete System of Orthogonal Step Functions 7.335
Integration of Gegenbauer functions with respect to the index
7.335/integraldisplayc+i∞
c−i∞[sin(απ)]−1tαCν
α(z)dα=−2i/parenleftbig
1+2tz+t2/parenrightbig−ν
[−2<Reν<c< 0,|arg (z±1)|<π]
EH I 178(25)
7.336/integraldisplay∞
−∞sech(πx)/parenleftbigg
ν−1
2+ix/parenrightbigg
Kν−1
2+ix(a)Iν−1
2+ix(b)Cν
−1
2+ix(−cosϕ)dx
=2−ν+1(ab)ν
Γ(ν)ω−νKν(ω)
ω=/radicalbig
a2+b2−2abcosϕ EH II 55(45)
7.34 Combinations of Chebyshev polynomials and powers
7.341/integraldisplay1
−1[Tn(x)]2dx=1−/parenleftbig
4n2−1/parenrightbig−1ET II 271(6)
7.342/integraldisplay1
−1Un/bracketleftBig
x/parenleftbig
1−y2/parenrightbig1/2/parenleftbig
1−z2/parenrightbig1/2+yz/bracketrightBig
dx=2
n+1Un(y)Un(z)
[|y|<1,|z|<1] ET II 275(34)
7.343
1./integraldisplay1
−1Tn(x)Tm(x)dx√
1−x2=0 [ m/negationslash=n]
=π
2[m=n/negationslash=0 ]
=π [m=n=0 ]
MO 104
2./integraldisplay1
−1/radicalbig
1−x2Un(x)Um(x)dx=0 [m/negationslash=n] ET II 274(28)
=π
2[m=n] ET II 274(27), MO 105a
7.344
1./integraldisplay1
−1(y−x)−1/parenleftbig
1−y2/parenrightbig−1/2Tn(y)dy=πUn−1(x)[ n=1,2,...] EH II 187(47)
2./integraldisplay1
−1(y−x)−1/parenleftbig
1−y2/parenrightbig1/2Un−1(y)dy=−πTn(x)[ n=1,2,...] EH II 187(48)
7.345
1./integraldisplay1
−1(1−x)−1/2(1 +x)m−n−3
2Tm(x)Tn(x)dx=0 [ m>n ] ET II 272(10)
2./integraldisplay1
−1(1−x)−1/2(1 +x)m+n−3
2Tm(x)Tn(x)dx=π(2m+2n−2)!
2m+n(2m−1)!(2n−1)!
[m+n/negationslash=0 ] ET II 272(11)
7.349 Chebyshev polynomials and powers 801
3./integraldisplay1
−1(1−x)1/2(1 +x)m+n+3
2Um(x)Un(x)dx=π(2m+2n+2 ) !
2m+n+2(2m+ 1)!(2 n+1 ) !ET II 274(31)
4./integraldisplay1
−1(1−x)1/2(1 +x)m−n−1
2Um(x)Un(x)dx=0 [ m>n ] ET II 274(30)
5./integraldisplay1
−1(1−x)(1 + x)1/2Um(x)Un(x)dx=25/2(m+1 ) (n+1 )/parenleftbig
m+n+3
2/parenrightbig/parenleftbig
m+n+5
2/parenrightbig
[1−4(m−n)2]
ET II 274(29)
6./integraldisplay1
−1(1 +x)−1/2(1−x)α−1Tm(x)Tn(x)dx
=π1/22α−1
2Γ(α)Γ/parenleftbig
n−α+1
2/parenrightbig
Γ/parenleftbig1
2−α/parenrightbig
Γ/parenleftbig
α+n+1
2/parenrightbig4F3/parenleftbigg
−m,m,α,α +1
2;1
2,α+n+1
2,α−n+1
2;1/parenrightbigg
[Reα>0] ET II 272(12)
7./integraldisplay1
−1(1 +x)1/2(1−x)α−1Um(x)Un(x)dx
=π1/22α−1
2(m+1 )(n+1 )Γ ( α)Γ/parenleftbig
n−α+3
2/parenrightbig
Γ/parenleftbig3
2−α/parenrightbig
Γ/parenleftbig3
2+α+n/parenrightbig
×4F3/parenleftbigg
−m, m+2,α,α−1
2;3
2,α+n+3
2,α−n−1
2;1/parenrightbigg
[Reα>0] ET II 275(32)
7.346/integraldisplay1
0xs−1Tn(x)dx√
1−x2=π
s2sB/parenleftbig1
2+1
2s+1
2n,1
2+1
2s−1
2n/parenrightbig
[Res>0] ET II 324(2)
7.347
1./integraldisplay1
−1(1−x)α(1 +x)βTn(x)dx=2α+β+2n+1(n!)2Γ(α+1 )Γ( β+1 )
(2n)! Γ(α+β+2 )
×3F2/parenleftbigg
−n, n, α +1 ;1
2,α+β+2 ;1/parenrightbigg
[Reα>−1,Reβ>−1] ET II 271(2)
2./integraldisplay1
−1(1−x)α(1 +x)βUn(x)dx=2α+β+2n+2[(n+1 ) ! ]2Γ(α+1 )Γ( β+1 )
(2n+2 ) !Γ ( α+β+2 )
×3F2/parenleftbigg
−n, n+1,α+1 ;3
2,α+β+2 ;1/parenrightbigg
ET II 273(22)
7.348/integraldisplay1
−1/parenleftbig
1−x2/parenrightbig−1/2U2n(xz)dx=πPn/parenleftbig
2z2−1/parenrightbig
[|z|<1] ET II 275(33)
7.349/integraldisplay1
−1/parenleftbig
1−x2/parenrightbig−1/2Tn/parenleftbig
1−x2y/parenrightbig
dx=1
2π[Pn(1−y)+Pn−1(1−y)] ET II 222(14)
802 Complete System of Orthogonal Step Functions 7.351
7.35 Combinations of Chebyshev polynomials and elementary functions
7.351/integraldisplay1
0x−1/2/parenleftbig
1−x2/parenrightbig−1
2e−2a
xTn(x)dx=π1/2Dn−1
2/parenleftBig
2a1/2/parenrightBig
D−n−1
2/parenleftBig
2a1/2/parenrightBig
[Rea>0] ET II 272(13)
7.352
1./integraldisplay∞
0xUn/bracketleftBig
a/parenleftbig
a2+x2/parenrightbig−1/2/bracketrightBig
(a2+x2)1
2n+1(eπx+1 )dx=a−n
2n−2−n−1ζ/parenleftbigg
n+1,a+1
2/parenrightbigg
[Rea>0] ET II 275(39)
2./integraldisplay∞
0xUn/bracketleftBig
a/parenleftbig
a2+x2/parenrightbig−1/2/bracketrightBig
(a2+x2)1
2n+1(e2πx−1)dx=1
2ζ(n+1,a)−a−n−1
4−a−n
2n
[Rea>0] ET II 276(40)
7.353
1./integraldisplay∞
0/parenleftbig
a2+x2/parenrightbig−1
2nsech/parenleftbigg1
2πx/parenrightbigg
Tn/bracketleftBig
a/parenleftbig
a2+x2/parenrightbig−1/2/bracketrightBig
dx=21−2n/bracketleftbigg
ζ/parenleftbigg
n,a+1
4/parenrightbigg
−ζ/parenleftbigg
n,a+3
4/parenrightbigg/bracketrightbigg
=21−nΦ/parenleftbigg
−1,n ,a+1
2/parenrightbigg
[Rea>0] ET II 273(19)
2./integraldisplay∞
0/parenleftbig
a2+x2/parenrightbig−1
2n/bracketleftbigg
cosh/parenleftbigg1
2πx/parenrightbigg/bracketrightbigg−2
Tn/bracketleftBig
a/parenleftbig
a2+x2/parenrightbig−1/2/bracketrightBig
dx=π−1n21−nζ/parenleftbigg
n+1,a+1
2/parenrightbigg
[Rea>0] ET II 273(20)
7.354
1./integraldisplay1
−1sin(xyz)cos/bracketleftBig/parenleftbig
1−x2/parenrightbig1/2/parenleftbig
1−y2/parenrightbig1/2z/bracketrightBig
T2n+1(x)dx=(−1)nπT2n+1(y)J2n+1(x)
ET II 271(4)
2./integraldisplay1
−1sin(xyz)sin/bracketleftBig/parenleftbig
1−x2/parenrightbig1/2/parenleftbig
1−y2/parenrightbig1/2z/bracketrightBig
U2n+1(x)dx=(−1)nπ/parenleftbig
1−y2/parenrightbig1/2U2n+1(y)J2n+2(z)
ET II 274(25)
3./integraldisplay1
−1cos(xyz)cos/bracketleftBig/parenleftbig
1−x2/parenrightbig1/2/parenleftbig
1−y2/parenrightbig1/2z/bracketrightBig
T2n(x)dx=(−1)nπT2n(y)J2n(z) ET II 271(5)
4./integraldisplay1
−1cos(xyz)sin/bracketleftBig/parenleftbig
1−x2/parenrightbig1/2/parenleftbig
1−y2/parenrightbig1/2z/bracketrightBig
U2n(x)dx=(−1)nπ/parenleftbig
1−y2/parenrightbig1/2U2n(y)J2n+1(z)
ET II 274(24)
7.355
1./integraldisplay1
0T2n+1(x)sinaxdx√
1−x2=(−1)nπ
2J2n+1(a)[ a>0] ET I 94(3)a
2./integraldisplay1
0T2n(x)cosaxdx√
1−x2=(−1)nπ
2J2n(a)[ a>0] ET I 38(2)a
7.374 Hermite polynomials 803
7.36 Combinations of Chebyshev polynomials and Bessel functions
7.361/integraldisplay1
0/parenleftbig
1−x2/parenrightbig−1/2Tn(x)Jν(xy)dx=1
2πJ1
2(ν+n)/parenleftbigg1
2y/parenrightbigg
J1
2(ν−n)/parenleftbigg1
2y/parenrightbigg
[y>0,Reν>−n−1] ET II 42(1)
7.362/integraldisplay∞
1/parenleftbig
x2−1/parenrightbig−1
2Tn/parenleftbigg1
x/parenrightbigg
K2μ(ax)dx=π
2aW1
2n,μ(a)W−1
2n,μ(a)
[Rea>0] ET II 366(17)a
7.37–7.38 Hermite polynomials
7.371/integraldisplayx
0Hn(y)dy= [2(n+1 ) ]−1[Hn+1(x)−Hn+1(0)] EH II 194(27)
7.372/integraldisplay1
−1/parenleftbig
1−t2/parenrightbigα−1
2H2n/parenleftbig√xt/parenrightbig
dx=(−1)nπ1/2(2n)! Γ/parenleftbig
α+1
2/parenrightbig
Lα
n(x)
Γ(n+α+1 )/bracketleftbig
Rea>−1
2/bracketrightbig
EH II 195(34)
7.373
1./integraldisplayx
0e−y2Hn(y)dy=Hn−1(0)−e−x2Hn−1(x)[ s e e 8.956 ] EH II 194(26)
2./integraldisplay∞
−∞e−x2H2m(xy)dx=√π(2m)!
m!/parenleftbig
y2−1/parenrightbigmEH II 195(28)
7.374
1./integraldisplay∞
−∞e−x2Hn(x)Hm(x)dx=0 [m/negationslash=n] SM II 567
=2n·n!√π[m=n]
SM II 568
2.11/integraldisplay∞
−∞e−2x2Hm(x)Hn(x)dx=(−1)⌊m
2⌋+⌊n
2⌋2m+n−1
2Γ/parenleftbiggm+n+1
2/parenrightbigg
[m+nis even]
=0 [ m+nis odd]
ET II 289(10)a
3./integraldisplay∞
−∞e−x2Hm(ax)Hn(x)dx=0 [ m<n ] ET II 290(20)a
4./integraldisplay∞
−∞e−x2H2m+n(ax)Hn(x)dx=√π2n(2m+n)!
m!/parenleftbig
a2−1/parenrightbigmanET II 291(21)a
5./integraldisplay∞
−∞e−2α2x2Hm(x)Hn(x)dx=2m+n−1
2α−m−n−1/parenleftbig
1−2α2/parenrightbigm+n
2Γ/parenleftbiggm+n+1
2/parenrightbigg
×2F1/parenleftbigg
−m, n;1−m−n
2;α2
2α2−1/parenrightbigg
/bracketleftbig
Reα2>0,α2/negationslash=1
2,m +nis even/bracketrightbig
ET II 289(12)a
6./integraldisplay∞
−∞e−(x−y)2Hn(x)dx=π1/2yn2nET II 288(2)a, EH II 195(31)
804 Complete System of Orthogonal Step Functions 7.375
7./integraldisplay∞
−∞e−(x−y)2Hm(x)Hn(x)dx=2nπ1/2m!yn−mLn−m
m/parenleftbig
−2y2/parenrightbig
[m≤n] BU 148(15), ET II 289(13)a
8./integraldisplay∞
−∞e−(x−y)2Hn(αx)dx=π1/2/parenleftbig
1−α2/parenrightbign
2Hn/bracketleftBigg
αy
(1−α2)1/2/bracketrightBigg
ET II 290(17)a
9./integraldisplay∞
−∞e−(x−y)2Hm(αx)Hn(αx)dx
=π1/2min(m,n)/summationdisplay
k=02kk!/parenleftbiggm
k/parenrightbigg/parenleftbiggn
k/parenrightbigg/parenleftbig
1−α2/parenrightbigm+n
2−kHm+n−2k/bracketleftBigg
αy
(1−α2)1/2/bracketrightBigg
ET II 291(26)a
10./integraldisplay∞
−∞e−(x−y)2
2uHn(x)dx=( 2πu)1/2(1−2u)n
2Hn/bracketleftBig
y(1−2u)−1/2/bracketrightBig
/bracketleftbig
0≤u<1
2/bracketrightbig
EH II 195(30)
7.375
1./integraldisplay∞
−∞e−2x2Hk(x)Hm(x)Hn(x)dx=π−121
2(m+n+k−1)Γ(s−k)Γ(s−m)Γ(s−n)
2s=k+m+n+1 [ k+m+nis even] ET II 290(14)a
2./integraldisplay∞
−∞e−x2Hk(x)Hm(x)Hn(x)dx=2m+n+k
2π1/2k!m!n!
(s−k)!(s−m)!(s−n)!,
2s=m+n+k [k+m+nis even]
ET II 290(15)a
7.376
1./integraldisplay∞
−∞eixye−x2
2Hn(x)dx=( 2π)1/2e−y2
2Hn(y)inMO 165a
2./integraldisplay∞
0e−2αx2xνH2n(x)dx=(−1)n22n−3
2−1
2νΓ/parenleftbigν+1
2/parenrightbig
Γ/parenleftbig
n+1
2/parenrightbig
√πα1
2(ν+1)F/parenleftbigg
−n,ν+1
2;1
2;1
2α/parenrightbigg
[Reα>0,Reν>−1] BU 150(18a)
3./integraldisplay∞
0e−2αx2xνH2n+1(x)dx=(−1)n22n−1
2νΓ/parenleftbigν
2+1/parenrightbig
Γ/parenleftbig
n+3
2/parenrightbig
√πα1
2ν+1F/parenleftbigg
−n,ν
2+1 ;3
2;1
2α/parenrightbigg
[Reα>0,Reν>−2] BU 150(18b)
7.3778/integraldisplay∞
−∞e−x2Hm(x+y)Hn(x+z)dx=2nπ1/2m!zn−mLn−m
m(−2yz)
[m≤n] ET II 292(30)a
7.378/integraldisplay∞
0xα−1e−βxHn(x)dx=2n⌊n
2⌋/summationdisplay
m=0n!Γ (α+n−2m)
m!(n−2m)!(−1)m2−2mβ2m−α−n
[Reα>0,ifnis even; Re α>−1,ifnis odd; Re β>0]ET I 172(11)a
7.385 Hermite polynomials 805
7.379
1./integraldisplay∞
−∞xe−x2H2m+1(xy)dx=π1/2(2m+1 ) !
m!y/parenleftbig
y2−1/parenrightbigmEH II 195(28)
2./integraldisplay∞
−∞xne−x2Hn(xy)dx=π1/2n!Pn(y) EH II 195(29)
7.381/integraldisplay∞
−∞(x±ic)νe−x2Hn(x)dx=2n−1−νπ1/2Γ/parenleftbign−ν
2/parenrightbig
Γ(−ν)exp/bracketleftbig
±1
2π(ν+n)i/bracketrightbig
[c>0] ET II 288(3)a
7.382/integraldisplay∞
0x−1/parenleftbig
x2+a2/parenrightbig−1e−x2H2n+1(x)dx=(−2)nπ1/2a−2/bracketleftBig
2νn!−(2n+1 ) !e1
2a2D−2n−2/parenleftBig
a√
2/parenrightBig/bracketrightBig
ET II 288(4)a
7.383
1./integraldisplay∞
0e−xpH2n+1/parenleftbig√x/parenrightbig
dx=(−1)n2n(2n+1 ) ! !π1/2(p−1)np−n−3
2
[Rep>0] EF 151(261)a, ET I 172(12)a
2./integraldisplay∞
0e−(b−βx)H2n+1/parenleftBig/radicalbig
(α−β)x/parenrightBig
dx=(−1)n√π/radicalbig
α−β(2n+1 ) !
n!(b−α)n
(b−β)n+3
2
[Re(b−β)>0] ET I 172(15)a
3./integraldisplay∞
01√xe−(b−β)xH2n/parenleftBig/radicalbig
(α−β)x/parenrightBig
dx=(−1)n√π(2n)!
n!(b−α)n
(b−β)n+1
2
[Re(b−β)>0] ET I 172(16)a
4./integraldisplay∞
0xa−1
2n−1e−bxHn/parenleftbig√x/parenrightbig
dx=2nΓ(a)b−a
2F1/parenleftbig
−1
2n,1
2−1
2n;1−a;b/parenrightbig
/bracketleftbigg
Rea>1
2n,ifnis even ,Rea>1
2n−1
2,ifnis odd ,Reb>0,
Ifais even, only the first 1 +/floorleftBign
2/floorrightBig
terms are kept in the series for 2F1⎤
⎦
ET I 172(14)a
5./integraldisplay∞
0x−1/2e−pxH2n/parenleftbig√x/parenrightbig
dx=(−1)n2n(2n−1)!!π1/2(p−1)np−n−1
2 MO 177a
7.384/integraldisplay∞
01√xe−bx/bracketleftbigg
Hn/parenleftbiggα+√x
λ/parenrightbigg
+Hn/parenleftbigga−√x
λ/parenrightbigg/bracketrightbigg
dx=/radicalbigg
2π
b/parenleftbig
1−λ−2b−1/parenrightbign
2Hn⎛
⎝α/radicalBig
λ2−1
b⎞
⎠
[Reb>0] ET I 173(17)a
7.385
1./integraldisplay∞
0e−bx
√ex−1H2n/bracketleftBig/radicalbig
s(1−e−x)/bracketrightBig
dx=(−1)n22n√π(2n)! Γ/parenleftbig
b+1
2/parenrightbig
Γ(n+b+1 )Ln
n(s)
/bracketleftbig
Reb>−1
2/bracketrightbig
ET I 174(23)a
806 Complete System of Orthogonal Step Functions 7.386
2./integraldisplay∞
0e−bxH2n+1/bracketleftBig√s√
1−e−x/bracketrightBig
dx=(−1)n22n√πs(2n+1 ) !Γ ( b)
Γ/parenleftbig
n+b+3
2/parenrightbigLb
n(s)
[Reb>0] ET I 174(24)a
7.386/integraldisplay∞
0x−n+1
2e−q2
4xHn/parenleftbiggq
2√x/parenrightbigg
e−pxdx=2nπ1/2pn−1
2e−q√pEF 129(117)
7.387
1./integraldisplay∞
0e−x2sinh/parenleftBig√
2βx/parenrightBig
H2n+1(x)dx=2n−1
2π1/2β2n+1e1
2β2ET II 289(7)a
2./integraldisplay∞
0e−x2cosh/parenleftBig√
2βx/parenrightBig
H2n(x)dx=2n−1π1/2β2ne1
2β2ET II 289(8)a
7.388
1./integraldisplay∞
0e−x2sin/parenleftBig√
2βx/parenrightBig
H2n+1(x)dx=(−1)n2n−1
2π1/2β2n+1e−1
2β2ET II 288(5)a
2./integraldisplay∞
0e−x2sin/parenleftBig√
2βx/parenrightBig
H2n+1(ax)dx=(−1)n2−1π1/2/parenleftbig
a2−1/parenrightbign+1
2e−1
2β2H2n+1/parenleftBigg
aβ√
2(a2−1)1/2/parenrightBigg
ET II 290(18)a
3./integraldisplay∞
0e−x2cos/parenleftBig√
2βx/parenrightBig
H2n(x)dx=(−1)n2n−1π1/2β2ne−1
2β2ET II 289(6)a
4./integraldisplay∞
0e−x2cos/parenleftBig√
2βx/parenrightBig
H2n(ax)dx=2−1π1/2/parenleftbig
1−a2/parenrightbigne−1
2β2H2n/bracketleftBigg
aβ√
2(a2−1)1/2/bracketrightBigg
ET II 290(19)a
5./integraldisplay∞
0e−y2[Hn(y)]2cos/parenleftBig√
2βy/parenrightBig
dy=π1/22n−1n!e−β2
2Ln/parenleftbig
β2/parenrightbig
EH II 195(33)
6.11/integraldisplay∞
0e−x2sin(bx)Hn(x)Hn+2m+1(x)dx=2n−1(−1)m√πn!b2m+1e−b2
4L2m+1
n/parenleftbiggb2
2/parenrightbigg
[b>0] ET I 39(11)a
7./integraldisplay∞
0e−x2cos(bx)Hn(x)Hn+2m(x)dx=2n−1
2/radicalbiggπ
2n!(−1)mb2me−b2
4L2m
n/parenleftbiggb2
2/parenrightbigg
[b>0] ET I 39(11)a
7.389/integraldisplayπ
0(cosx)nH2n/bracketleftBig
a(1−secx)1/2/bracketrightBig
dx=2−n(−1)nπ(2n)!
(n!)2[Hn(a)]2ET II 292(31)
7.39 Jacobi polynomials
7.391
1./integraldisplay1
−1(1−x)α(1 +x)βP(α,β)
n(x)P(α,β)
m(x)dx
=0 [ m/negationslash=n,Reα>−1,Reβ>−1]
=2α+β+1Γ(α+n+1 )Γ( β+n+1 )
n!(α+β+1+2 n)Γ (α+β+n+1 )[m=n,Reα>−1,Reβ>−1]
ET II 285(5, 9)
7.391 Jacobi polynomials 807
2./integraldisplay1
−1(1−x)ρ(1 +x)σP(α,β)
n(x)dx=2ρ+σ+1Γ(ρ+1 )Γ ( σ+1 )Γ ( n+1+ α)
n!Γ (ρ+σ+2 )Γ ( 1+ α)
×3F2(−n, α+β+n+1,ρ+1 ;α+1,ρ+σ+2 ;1 )
[Reρ>−1,Reσ>−1] ET II 284(3)
3.6/integraldisplay1
−1(1−x)α(1 +x)σP(α,β)
n(x)dx=2α+σ+1Γ(σ+1 )Γ ( α+1 )Γ ( σ−β+1 )
n!Γ (σ−β−n+1 )Γ ( α+σ+n+2 )
[Reα>−1,Reσ>−1] ET II 284(1)
4./integraldisplay1
−1(1−x)ρ(1 +x)βP(α,β)
n(x)dx=2β+ρ+1Γ(ρ+1 )Γ ( β+n+1 )Γ ( α−ρ+n)
n!Γ (α−ρ)Γ(β+ρ+n+2 )
[Reρ>−1,Reβ>−1] ET II 284(2)
5./integraldisplay1
−1(1−x)α−1(1 +x)β/bracketleftBig
P(α,β)
n(x)/bracketrightBig2
dx=2α+βΓ(α+n+1 )Γ ( β+n+1 )
n!αΓ(α+β+n+1 )
[Reα>0,Reβ>−1] ET II 285(6)
6./integraldisplay1
−1(1−x)2α(1 +x)β/bracketleftBig
P(α,β)
n(x)/bracketrightBig2
dx=24α+β+1Γ/parenleftbig
α+1
2/parenrightbig
[Γ(α+n+1 ) ]2Γ(β+2n+1 )
√π(n!)2Γ(α+1 )Γ ( 2 α+β+2n+2 )/bracketleftbig
Reα>−1
2,Reβ>−1/bracketrightbig
ET II 285(7)
7./integraldisplay1
−1(1−x)ρ(1 +x)βP(α,β)
n(x)P(ρ,β)
n(x)dx
=2ρ+β+1Γ(ρ+n+1 )Γ ( β+n+1 )Γ( α+β+2n+1 )
n!Γ (β+ρ+2n+2 )Γ ( α+β+n+1 )
[Reρ>−1,Reβ>−1] ET II285(10)
8./integraldisplay1
−1(1−x)ρ−1(1 +x)βP(α,β)
n(x)P(ρ,β)
n(x)dx=2ρ+βΓ(α+n+1 )Γ ( β+n+1 )Γ ( ρ)
n!Γ (α+1 )Γ ( ρ+β+n+1 )
[Reβ>−1,Reρ>0] ET II 286(11)
9.7/integraldisplay1
−1(1−x)α(1 +x)σP(α,β)
n(x)P(α,σ)
m(x)dx
=2α+σ+1Γ(α+n+1 )Γ( α+β+m+n+1 )Γ ( σ+m+1 )Γ ( σ−β+1 )
m!(n−m)! Γ (α+β+n+1 )Γ ( α+σ+m+n+2 )Γ( α−β+m−n+1 )
[Reα>−1,Reσ>−1]ET II 286(12)
10.6/integraldisplay1
−1(1−x)ρ(1 +x)βP(α,β)
n(x)P(ρ,β)
m(x)dx
=2β+ρ+1Γ(α+β+m+n+1 )Γ( β+n+1 )Γ ( ρ+m+1 )
m!(n−m)! Γ(α+β+n+1 )Γ( β+ρ+m+n+2 )Γ(α−ρ−m+n)
Γ(α−ρ)
[Reβ>−1,Reρ>−1]ET II 287(16)
11./integraldisplayx
0(1−y)α(1 +y)βP(α,β)
n(y)dy=1
2n/bracketleftBig
P(α+1,β+1)
n−1 (0)−(1−x)α+1(1 +x)β+1P(α+1,β+1)
n−1 (x)/bracketrightBig
EH II 173(38)
808 Complete System of Orthogonal Step Functions 7.392
7.392
1./integraldisplay1
0xλ−1(1−x)μ−1P(α,β)
n(1−γx)dx
=Γ(α+n+1 )Γ ( λ)Γ(μ)
n!Γ(α+1 )Γ ( λ+μ)3F2/parenleftbigg
−n, n+α+β+1,λ;α+1,λ+μ;1
2γ/parenrightbigg
[Reλ>0,Reμ>0] ET II 192(46)a
2./integraldisplay1
0xλ−1(1−x)μ−1P(α,β)
n(γx−1)dx
=(−1)nΓ(β+n+1 )Γ ( λ)Γ(μ)
n!Γ(β+1 )Γ ( λ+μ)3F2/parenleftbigg
−n, n+α+β+1,λ;β+1,λ+μ;1
2γ/parenrightbigg
a
[Reλ>0,Reμ>0] ET II 192(47)a
3./integraldisplay1
0xα(1−x)μ−1P(α,β)
n(1−γx)dx=Γ(α+n+1 )Γ ( μ)
Γ(α+μ+n+1 )P(α+μ,β−μ)
n (1−γ)
[Rea>−1,Reμ>0] ET II 191(43)a
4./integraldisplay1
0xβ(1−x)μ−1P(α,β)
n(γx−1)dx=Γ(β+n+1 )Γ ( μ)
Γ(β+μ+n+1 )P(α−μ,β+μ)
n (γ−1)
[Reβ>−1,Reμ>0]ET II 191(44)a
7.393
1./integraldisplay1
0/parenleftbig
1−x2/parenrightbigνsinbxP(ν,ν)
2n+1(x)dx=(−1)n√πΓ(2n+ν+2 )J2n+ν+3
2(b)
21
2−ν(2n+1 ) !bν+1
2
[b>0,Reν>−1] ET I 94(5)
2./integraldisplay1
0/parenleftbig
1−x2/parenrightbigνcosbxP(ν,ν)
2n(x)dx=(−1)n2ν−1
2√πΓ(2n+ν+1 )J2n+ν+1
2(b)
(2n)!bν+1
2
[b>0,Reν>−1] ET I 38(4)
7.41–7.42 Laguerre polynomials
7.411
1./integraldisplayt
0Ln(x)dx=Ln(t)−Ln+1(t)/(n+1 ) MO 110
2./integraldisplayt
0Lα
n(x)dx=Lα
n(t)−Lα
n+1(t)−/parenleftbiggn+α
n/parenrightbigg
+/parenleftbiggn+1+ α
n+1/parenrightbigg
EH II 189(16)a
3./integraldisplayt
0Lα+1
n−1(x)dx=−Lα
n(t)+/parenleftbiggn+α
n/parenrightbigg
EH II 189(15)a
4./integraldisplayt
0Lm(x)Ln(t−x)dx=Lm+n(t)−Lm+n+1(t) EH II 191(31)
5.∞/summationdisplay
k=0/bracketleftbigg/integraldisplayt
0Lk(x)
k!dx/bracketrightbigg2
=et−1[ t≥0] MO 110
7.414 Laguerre polynomials 809
7.412
1./integraldisplay1
0(1−x)μ−1xαLα
n(ax)dx=Γ(α+n+1 )Γ ( μ)
Γ(α+μ+n+1 )Lα+μ
n(a)
[Reα>−1,Reμ>0]
EH II 191(30)a, BU 129(14c)
2./integraldisplay1
0(1−x)μ−1xλ−1Lα
n(βx)dx=Γ(α+n+1 )Γ ( λ)Γ(μ)
n!Γ(α+1 )Γ ( λ+μ)2F2(−n, λ;α+1,λ+μ:β)
[Reλ>0,Reμ>0] ET II 192(50)a
7.413/integraldisplay1
0xα(1−x)βLα
m(xy)Lβ
n[(1−x)y]dx=(m+n)! Γ(α+m+1 )Γ ( β+n+1 )
m!n!Γ (α+β+m+n+2 )Lα+β+1
m+n(y)
[Reα>−1,Reβ>−1] ET II 293(7)
7.414
1.11/integraldisplay∞
ye−xLα
n(x)dx=e−y/bracketleftbig
Lα
n(y)−Lα
n−1(y)/bracketrightbig
EH II 191(29)
2./integraldisplay∞
0e−bxLn(λx)Ln(μx)dx=(b−λ−μ)n
bn+1Pn/bracketleftbiggb2−(λ+μ)b+2λμ
b(b−λ−μ)/bracketrightbigg
[Reb>0] ET I 175(34)
3.8/integraldisplay∞
0e−xxαLα
n(x)Lα
m(x)dx=0 [m/negationslash=n,Reα>−1] BU 115(8), ET II 293(3)
=Γ(α+n+1 )
n![m=n,Reα>0] BU 115(8), ET II 292(2)
4./integraldisplay∞
0e−bxxαLα
n(λx)Lα
m(μx)dx=Γ(m+n+α+1 )
m!n!(b−λ)n(b−μ)m
bm+n+α+1
×F/bracketleftbigg
−m,−n;−m−n−α,b(b−λ−μ)
(b−λ)(b−μ)/bracketrightbigg
[Reα>−1,Reb>0] ET I 175(35)
4(1)9./integraldisplay∞
0e−xxα+1/2Lα
n(x)Lα
m(x)dx=Γ(α+n+1 )2Γ(α+m+1 )Γ/parenleftbig
α+3
2/parenrightbig
Γ/parenleftbig
m−1
2/parenrightbig
n!m!Γ (α+1 )Γ/parenleftbig
−1
2/parenrightbig
×3F2/parenleftbig
−n, α+3
2,3
2;α+1,3
2−m;1/parenrightbig
5./integraldisplay∞
0e−bxLa
n(x)dx=n/summationdisplay
m=0/parenleftbigga+m−1
m/parenrightbigg(b−1)n−m
bn−m+1[Reb>0] ET I 174(27)
6./integraldisplay∞
0e−bxLn(x)dx=(b−1)nb−n−1[Reb>0] ET I 174(25)
7./integraldisplay∞
0e−sttβLα
n(t)dt=Γ(β+1 )Γ ( α+n+1 )
n!Γ (α+1 )s−β−1F/parenleftbigg
−n, β+1 ;α+1 ;1
s/parenrightbigg
[Reβ>−1,Res>0]
BU 119(4b), EH II 191(133)
8./integraldisplay∞
0e−sttαLα
n(t)dt=Γ(α+n+1 ) (s−1)n
n!sα+n+1[Reα>−1,Res>0]
EH II 191(32), MO 176a
810 Complete System of Orthogonal Step Functions 7.415
9./integraldisplay∞
0e−xxα+βLα
m(x)Lβ
n(x)dx=(−1)m+n(α+β)!/parenleftbiggα+m
n/parenrightbigg/parenleftbiggβ+n
m/parenrightbigg
[Re(α+β)>−1] ET II 293(4)
10.6/integraldisplay∞
0e−bxx2a[La
n(x)]2dx=22aΓ/parenleftbig
a+1
2/parenrightbig
Γ/parenleftbig
n+1
2/parenrightbig
π(n!)2b2a+1
×F/parenleftBigg
−n, a+1
2;1
2−n;/parenleftbigg
1−2
b/parenrightbigg2/parenrightBigg
Γ(a+n+1 )
/bracketleftbigg
Rea>−1
2,Reb>0/bracketrightbigg
ET I 174(30)
11./integraldisplay∞
0e−xxγ−1Lμ
n(x)dx=Γ(γ)Γ(1+ μ+n−γ)
n!Γ ( 1+ μ−γ)[Reγ>0] BU 120(4b)
12./integraldisplay∞
0e−x(s+a1+a2
2)xμ+βLμ
k(a1x)Lμ
k(a2x)dx
=Γ(1 + μ+β)Γ(1+ μ+k)
k!k!Γ(1+ μ)⎧
⎨
⎩dk
dhk⎡
⎣F/parenleftBig
1+μ+β
2,1+μ+β
2;1+μ;A2
B2/parenrightBig
(1−h)1+μB1+μ+β⎤
⎦⎫
⎬
⎭
h=0
A2=4a1a2h
(1−h)2;B=s+a1+a2
21+h
1−h/bracketleftbigg
Re/parenleftbigg
s+a1+a2
2/parenrightbigg
>0,a1>0,a2>0,Re(μ+β)>−1/bracketrightbigg
BU 142(19)
13./integraldisplay∞
0exp/bracketleftbigg
−x/parenleftbigg
s+a1+a2
2/parenrightbigg/bracketrightbigg
xμLμ
k(a1x)Lμ
k(a2x)dx=Γ(1 + μ+k)
b1+μ+k
0·bk
0
k!·P(μ,0)
k/parenleftbiggb2
1
b0b2/parenrightbigg
b0=s+a1+a2
2,b2
1=b0b2+2a1a2,b2=s−a1+a2
2/bracketleftbigg
Reμ>−1,Re/parenleftbigg
s+a1+a2
2/parenrightbigg
>0/bracketrightbigg
BU 144(22)
7.415/integraldisplay1
0(1−x)μ−1xλ−1e−βxLα
n(βx)dx=Γ(α+n+1 )
n!Γ (α+1 )B(λ,μ)2F2(α+n+1,λ;α+1,λ+μ;−β)
[Reλ>0,Reμ>0] ET II 193(51)a
7.416/integraldisplay∞
−∞xm−nexp/bracketleftbigg
−1
2(x−y)2/bracketrightbigg
Lm−n
n/parenleftbig
x2/parenrightbig
dx=(2π)1/2
n!in−m2−n+m
2Hn/parenleftbiggiy√
2/parenrightbigg
Hm/parenleftbiggiy√
2/parenrightbigg
BU 149(15b), ET II 293(8)a
7.417
1./integraldisplay∞
0xν−2n−1e−axsin(bx)Lν−2n−1
2n (ax)dx=(−1)niΓ(ν)b2n[(a−ib)−ν−(a+ib)−ν]
2(2n)!
[b>0,Rea>0,Reν>2n]
ET I 95(12)
2./integraldisplay∞
0xν−2n−2e−axsin(bx)Lν−2n−2
2n+1(ax)dx=(−1)n+1Γ(ν)b2n+1[(a+ib)−ν+(a−ib)−ν]
2(2n+1 ) !
[b>0,Rea>0,Reν>2n+1 ]
ET I 95(13)
7.421 Laguerre polynomials 811
3./integraldisplay∞
0xν−2ne−axcos(bx)L2n−1
ν−2n(ax)dx=i(−1)n+1Γ(ν)b2n−1[(a−ib)−ν−(a+ib)−ν]
2(2n−1)!
[b>0,Rea>0,Reν>2n−1]
ET I 39(12)
4./integraldisplay∞
0xν−2n−1e−axcos(bx)Lν−2n−1
2n (ax)dx=(−1)nΓ(ν)b2n[(a+ib)−ν+(a−ib)−ν]
2(2n)!
[b>0,Reν>2n,Rea>0]
ET I 39(13)
7.418
1./integraldisplay∞
0e−1
2x2sin(bx)Ln/parenleftbig
x2/parenrightbig
dx=(−1)ni
2n!1√
2π/braceleftBig
[D−n−1(ib)]2−[D−n−1(−ib)]2/bracerightBig
[b>0] ET I 95(14)
2./integraldisplay∞
0e−1
2x2cos(bx)Ln/parenleftbig
x2/parenrightbig
dx=/radicalbiggπ
2(n!)−1e−1
2b22−n/bracketleftbigg
Hn/parenleftbiggb√
2/parenrightbigg/bracketrightbigg2
[b>0] ET I 39(14)
3./integraldisplay∞
0x2n+1e−1
2x2sin(bx)Ln+1
2n/parenleftbigg1
2x2/parenrightbigg
dx=/radicalbiggπ
2b2n+1e−1
2b2Ln+1
2n/parenleftbiggb2
2/parenrightbigg
[b>0] ET I 95(15)
4./integraldisplay∞
0x2ne−1
2x2cos(bx)Ln−1
2n/parenleftbigg1
2x2/parenrightbigg
dx=/radicalbiggπ
2b2ne−1
2b2Ln+1
2n/parenleftbigg1
2b2/parenrightbigg
[b>0] ET I 39(16)
5./integraldisplay∞
0xe−1
2x2Lα
n/parenleftbigg1
2x2/parenrightbigg
L1
2−α
n/parenleftbigg1
2x2/parenrightbigg
sin(xy)dx=/parenleftBigπ
2/parenrightBig1/2
ye−1
2y2Lα
n/parenleftbigg1
2y2/parenrightbigg
L1
2−α
n/parenleftbigg1
2y2/parenrightbigg
ET II 294(11)
6./integraldisplay∞
0e−1
2x2Lα
n/parenleftbigg1
2x2/parenrightbigg
L−1
2−α
n/parenleftbigg1
2x2/parenrightbigg
cos(xy)dx=/parenleftBigπ
2/parenrightBig1/2
e−1
2y2Lα
n/parenleftbigg1
2y2/parenrightbigg
L−α−1
2n/parenleftbigg1
2y2/parenrightbigg
ET II 294(12)
7.419/integraldisplay∞
0xn+2ν−1
2exp[−(1 +a)x]L2ν
n(ax)Kν(x)dx
=π1/2Γ/parenleftbig
n+ν+1
2/parenrightbig
Γ/parenleftbig
n+3ν+1
2/parenrightbig
2n+2ν+1
2n!Γ( 2ν+1 )F/parenleftbigg
n+ν+1
2,n+3ν+1
2;2ν+1 ;−1
2a/parenrightbigg
/bracketleftbig
Rea>−2,Re(n+ν)>−1
2,Re(n+3ν)>−1
2/bracketrightbig
ET II 370(44)
7.421
1./integraldisplay∞
0xe−1
2αx2Ln/parenleftbigg1
2βx2/parenrightbigg
J0(xy)dx=(α−β)n
αn+1e−1
2αy2Ln/bracketleftbiggβy2
2α(β−α)/bracketrightbigg
[y>0,Reα>0] ET II 13(4)a
2./integraldisplay∞
0xe−x2Ln/parenleftbig
x2/parenrightbig
J0(xy)dx=2−2n−1
n!y2ne−1
4y2ET II 13(5)
812 Hypergeometric Functions 7.422
3./integraldisplay∞
0x2n+ν+1e−1
2x2Lν+n
n/parenleftbigg1
2x2/parenrightbigg
Jν(xy)dx=y2n+νe−1
2y2Lν+n
n/parenleftbigg1
2y2/parenrightbigg
[y>0,Reν>−1] MO 183
4./integraldisplay∞
0xν+1e−βx2Lν
n/parenleftbig
αx2/parenrightbig
Jν(xy)dx=2−ν−1β−ν−n−1(β−α)nyνe−y2
4βLν
n/bracketleftbiggαy2
4β(α−β)/bracketrightbigg
ET II 43(5)
5./integraldisplay∞
0e−1
2qx2xν+1Lν
n/bracketleftbiggx2
2q(1−q)/bracketrightbigg
Jν(xy)dx=qn+ν+1
(q−1)ne−qy2
2yνLν
n/parenleftbiggy2
2/parenrightbigg
[ν>0] MO 183
6.∗/integraldisplay∞
0xν+1e−x2Lν
n/parenleftbig
x2/parenrightbig
Jν(xy)dx=1
2n!/parenleftBigy
2/parenrightBig2n+ν
e−1
4y2
7.422
1./integraldisplay∞
0xν+1e−βx2/bracketleftBig
L1
2ν
n/parenleftbig
αx2/parenrightbig/bracketrightBig2
Jν(xy)dx
=yν
πn!Γ/parenleftbig
n+1+1
2ν/parenrightbig
(2β)−ν−1e−y2
4β
×n/summationdisplay
l=0(−1)lΓ/parenleftbig
n−l+1
2/parenrightbig
Γ/parenleftbig
l+1
2/parenrightbig
Γ/parenleftbig
l+1+1
2ν/parenrightbig
(n−l)!/parenleftbigg2α−β
β/parenrightbigg2l
Lν
2l/bracketleftbiggαy2
2β(2α−β)/bracketrightbigg
[y>0,Reβ>0,Reν>−1]ET II 43(7)
2.9/integraldisplay∞
0xν+1e−αx2Lν−σ
m/parenleftbig
αx2/parenrightbig
Lσ
n/parenleftbig
αx2/parenrightbig
Jν(xy)dx
=(−1)m+n(2α)−ν−1yνe−y2
4αLm−n−σ
n/parenleftbiggy2
4α/parenrightbigg
Ln−m+σ−ν
m/parenleftbiggy2
4α/parenrightbigg
[y>0,Reα>0,Reν>−1,n/negationslash=0,σ/negationslash=0,α/negationslash=1 ] ET II 43(8)
7.423
1./integraldisplay∞
0e−1
2x2Ln/parenleftbigg1
2x2/parenrightbigg
H2n+1/parenleftbiggx
2√
2/parenrightbigg
sin(xy)dx=/parenleftBigπ
2/parenrightBig1/2
e−1
2y2Ln/parenleftbigg1
2y2/parenrightbigg
H2n+1/parenleftbiggy
2√
2/parenrightbigg
ET II 294(13)a
2./integraldisplay∞
0e−1
2x2Ln/parenleftbigg1
2x2/parenrightbigg
H2n/parenleftbiggx
2√
2/parenrightbigg
cos(xy)dx=/parenleftBigπ
2/parenrightBig1/2
e−1
2y2Ln/parenleftbigg1
2y2/parenrightbigg
H2n/parenleftbiggy
2√
2/parenrightbigg
ET II 294(14)a
7.5 Hypergeometric Functions
7.51 Combinations of hypergeometric functions and powers
7.511/integraldisplay∞
0F(a,b;c;−z)z−s−1dx=Γ(a+s)Γ(b+s)Γ(c)Γ(−s)
Γ(a)Γ(b)Γ(c+s)
[c/negationslash=0,−1,−2,..., Res<0,Re(a+s)>0,Re(b+s)>0]EH I 79(4)
7.512 Hypergeometric functions and powers 813
7.512
1./integraldisplay1
0xα−γ(1−x)γ−β−1F(α,β;γ;x)dx=Γ/parenleftBig
1+α
2/parenrightBig
Γ(γ)Γ(α−γ+1 )Γ/parenleftBig
γ−α
2−β/parenrightBig
Γ(1+ α)Γ/parenleftBig
1+α
2−β/parenrightBig
Γ/parenleftBig
γ−α
2/parenrightBig
/bracketleftBig
Reα+1>Reγ>Reβ,Re/parenleftBig
γ−α
2−β/parenrightBig
>0/bracketrightBig
ET II 398(1)
2./integraldisplay1
0xρ−1(1−x)β−γ−nF(−n, β;γ;x)dx=Γ(γ)Γ (ρ)Γ(β−γ+1 )Γ ( γ−ρ+n)
Γ(γ+n)Γ(γ−ρ)Γ(β−γ+ρ+1 )
[n=0,1,2...;R e ρ>0,Re(β−γ)>n−1]ET II 398(2)
3./integraldisplay1
0xρ−1(1−x)β−ρ−1F(α,β;γ;x)dx=Γ(γ)Γ(ρ)Γ(β−ρ)Γ(γ−α−ρ)
Γ(β)Γ(γ−α)Γ(γ−ρ)
[Reρ>0,Re(β−ρ)>0,Re(γ−α−ρ)>0]ET II 399(3)
4./integraldisplay1
0xγ−1(1−x)ρ−1F(α,β;γ;x)dx=Γ(γ)Γ(ρ)Γ(γ+ρ−α−β)
Γ(γ+ρ−α)Γ(γ+ρ−β)
[Reγ>0,Reρ>0,Re(γ+ρ−α−β)>0]ET II 399(4)
5./integraldisplay1
0xρ−1(1−x)σ−1F(α,β;γ;x)dx=Γ(ρ)Γ(σ)
Γ(ρ+σ)3F2(α,β,ρ ;γ,ρ+σ;1)
[Reρ>0,Reσ>0,Re(γ+σ−α−β)>0]ET II 399(5)
6.10/integraldisplay1
0xλ−1(1−x)β−λ−1F/parenleftBig
α,β;λ;zx
b/parenrightBig
dx=B (λ,β−λ)(1−z/b)−αBU 9
7.11/integraldisplay1
0xγ−1(1−x)δ−γ−1F(α,β;γ;xz)F(δ−α,δ−β;δ−γ;( 1−x)ζ)dx
=Γ(γ)Γ(δ−γ)
Γ(δ)(1−ζ)α+β−δF(α,β;δ;z+ζ−zζ)
[0<Reγ<Reδ,|arg(1−z)|<π , |arg(1−ζ)|<π]ET II 400(11)
8./integraldisplay1
0xγ−1(1−x)/epsilon1−1(1−xz)−δF(α,β;γ;xz)F/bracketleftbigg
δ, β−γ;/epsilon1;(1−x)z
(1−xz)/bracketrightbigg
dx
=Γ(γ)Γ(/epsilon1)
Γ(γ+/epsilon1)F(α+δ, β;γ+/epsilon1;z)
[Reγ>0,Re/epsilon1>0,|arg(z−1)|<π]ET II 400(12), Eh I 78(3)
9./integraldisplay1
0xγ−1(1−x)ρ−1(1−zx)−σF(α,β;γ;x)dx
=Γ(γ)Γ(ρ)Γ(γ+ρ−α−β)
Γ(γ+ρ−α)Γ(γ+ρ−β)(1−z)−σ
×3F2/parenleftbigg
ρ, σ, γ +ρ−α−β;γ+ρ−α,γ+ρ−β;z
z−1/parenrightbigg
[Reγ>0,Reρ>0,Re (γ+ρ−α−β)>0,|arg(1−z)|<π]ET II 399(6)
814 Hypergeometric Functions 7.513
10./integraldisplay∞
0xγ−1(x+z)−σF(α,β;γ;−x)dx=Γ(γ)Γ(α−γ+σ)Γ(β−γ+σ)
Γ(σ)Γ(α+β−γ+σ)
×F(α−γ+σ, β−γ+σ;α+β−γ+σ;1−z)
[Reγ>0,Re(α−γ+σ)>0,Re (β−γ+σ)>0,|argz|<π]ET II 400(10)
11./integraldisplay1
0(1−x)μ−1xν−1
pFq(a1,...,a p;ν,b2,...,b q;ax)dx
=Γ(μ)Γ(ν)
Γ(μ+ν)pFq(a1,...,a p;μ+ν,b2,...,b q;a)
[Reμ>0,Reν>0,p≤q+1 ; i f p=q+1 ,t h e n |a|<1]ET II 200(94)
12./integraldisplay1
0(1−x)μ−1xν−1
pFq(a1,...,a p;b1,...,b q;ax)dx
=Γ(μ)Γ(ν)
Γ(μ+ν)p+1Fq+1(ν,a1,...,a p;μ+ν,b1,...,b q;a)
[Reμ>0,Reν>0,p≤q+1,ifp=q+1 ,t h e n |a|<1]ET II 200(95)
7.513/integraldisplay1
0xs−1/parenleftbig
1−x2/parenrightbigνF/parenleftbig
−n, a;b;x2/parenrightbig
dx=1
2B/parenleftBig
ν+1,s
2/parenrightBig
3F2/parenleftBig
−n, a,s
2;b,ν+1+s
2;1/parenrightBig
[Res>0,Reν>−1] ET I 336(4)
7.52 Combinations of hypergeometric functions and exponentials
7.521/integraldisplay∞
0e−st
pFq(a1,...,a p;b1,...,b q,t)dt=1
sp+1Fq/parenleftbig
1,a1,...,a p;b1,...,b q,s−1/parenrightbig
[p≤q] EH I 192
7.522
1.11/integraldisplay∞
0e−λxxγ−1
2F1(α,β;δ;−x)dx=Γ(δ)λ−γ
Γ(α)Γ(β)E(α,β,γ :δ:λ)
[Reλ>0,Reγ>0] EH I 205(10)
2.6/integraldisplay∞
0e−bxxa−1F/parenleftbigg1
2+ν,1
2−ν;a;−x
2/parenrightbigg
dx=2aeb1√πΓ(a)(2b)1
2−aKν(b)
[Rea>0,Reb>0] ET I 212(1)
3./integraldisplay∞
0e−bxxγ−1F(2α,2β;γ;−λx)dx=Γ (γ)b−γ/parenleftbiggb
λ/parenrightbiggα+β−1
2
eb
2λW1
2−α−β,α−β/parenleftbiggb
λ/parenrightbigg
[Reb>0,Reγ>0,|argλ|<π]
BU 78(30), ET I 212(4)
4.6/integraldisplay∞
0e−xttb−1F(a,a−c+1 ;b;−t)dt=xa−bΓ(b)Ψ(a,c;x)
[Reb>0,Rex>0] EH I 273(11)
5./integraldisplay∞
0e−xxs−1
pFq(a1,...,a p,b1,...,b q;ax)dx=Γ (s)p+1Fq(s, a1,...,a p;b1,...,b q;a)
[p<q , Res>0] ET I 337(11)
7.525 Hypergeometric functions and exponentials 815
6./integraldisplay∞
0xβ−1e−μx
2F2(−n, n+1 ;1,β;x)dx=Γ (β)μ−βPn/parenleftbigg
1−2
μ/parenrightbigg
[Reμ>0,Reβ>0] ET I 218(6)
7./integraldisplay∞
0xβ−1e−μx
2F2/parenleftbigg
−n, n;β,1
2;x/parenrightbigg
dx=Γ (β)μ−βcos/bracketleftbigg
2narcsin/parenleftbigg1√μ/parenrightbigg/bracketrightbigg
[Reμ>0,Reβ>0] ET I 218(7)
8./integraldisplay∞
0xρn−1e−μx
mFn(a1,...,a m;ρ1,...,ρ n;λx)dx
=Γ(ρn)μ−ρnmFn−1/parenleftbigg
a1,...,a m;ρ1,...,ρ n−1;λ
μ/parenrightbigg
[m≤n;R e ρn>0,Reμ>0,ifm<n ;R eμ>Reλ,ifm=n]ET I 219(16)a
9./integraldisplay∞
0xσ−1e−μx
mFn(a1,...,a m;ρ1,...,ρ n;λx)dx
=Γ (σ)μ−σ
m+1Fn/parenleftbigg
a1,...,a m,σ;ρ1,...,ρ n;λ
μ/parenrightbigg
[m≤n,Reσ>0,Reμ>0,ifm<n ;R eμ>Reλ,ifm=n]ET I 219(17)
7.523/integraldisplay∞
1(x−1)μ−1x−μ−1
2e−1
2axW2μ+1
2,λ(ax)dx=Γ (μ)e−1
2aWμ+1
2,λ(a)
[Reμ>0,Rea>0]
7.524
1./integraldisplay∞
0e−λxF/parenleftbigg
α,β;1
2;−x2/parenrightbigg
dx=λα+β−1S1−α−β,α−β(λ)
[Reλ>0] ET II 401(13)
2./integraldisplay∞
0e−st
pFq/parenleftbig
a1,...,a p;b1,...,b q;t2/parenrightbig
dx=s−1
p+2Fq/parenleftbigg
a1,...,a p,1,1
2;b1,...,b q;4
s2/parenrightbigg
[p<q] MO 176
3./integraldisplay∞
0e−st
0Fq/parenleftbigg1
q,2
q,...,q−1
q,1;tq
qq/parenrightbigg
dt=s−1exp/parenleftbig
s−q/parenrightbig
MO 176
7.525
1./integraldisplay∞
0xσ−1e−μx
mFn/parenleftbig
a1,...,a m;ρ1,...,ρ n;(λx)k/parenrightbig
dx
=Γ (σ)μ−σ
m+kFn/parenleftBigg
a1,...,a m,σ
k,σ+1
k,...,σ+k−1
k;ρ1,...,ρ n;/parenleftbiggkλ
μ/parenrightbiggk/parenrightBigg
/bracketleftbigg
m+k≤n+1,Reσ>0; Re μ>0,ifm+k≤n;
Re/parenleftBig
μ+kλe2πi
k/parenrightBig
>0;r=0,1,...,k −1f o rm+k=n+1/bracketrightbigg
ET I 220(19)
816 Hypergeometric Functions 7.526
2./integraldisplay∞
0xe−λxF/parenleftbig
α,β;3
2;−x2/parenrightbig
dx=λα+β−2S1−α−β,α−β(λ)
[Reλ>0] ET II 401(14)
7.526
1./integraldisplayγ+i∞
γ−i∞ests−bF/parenleftbigg
a,b;a+b−c+1 ;1 −1
s/parenrightbigg
dx=2πiΓ(a+b−c+1 )
Γ(b)Γ(b−c+1 )tb−1Ψ(a;c;t)
/bracketleftbigg
Reb>0,Re(b−c)>−1,γ >1
2/bracketrightbigg
EH I 273(12)
2./integraldisplay∞
0e−ttγ−1(x+t)−α(y+t)−a/primeF/bracketleftbigg
a,a/prime;γ;t(x+y+t)
(x+t)(y+t)/bracketrightbigg
dt=Γ (γ)Ψ(a,c;x)Ψ(a/prime,c;y),
γ=a+a/prime−c+1 [ R e γ>0,x y /negationslash=0 ] EH I 287(21)
3./integraldisplay∞
0xγ−1(x+y)−α(x+z)−βe−xF/bracketleftbigg
α,β;γ;x(x+y+z)
(x+y)(x+z)/bracketrightbigg
dx
=Γ (γ)(zy)−1
2−μey+z
2Wν,μ(y)Wλ,μ(z)
2ν=1−α+β−γ;2λ=1+ α−β−γ;2μ=α+β−γ
[Reγ>0,|argy|<π , |argz|<π]
ET II 401(15)
7.527
1./integraldisplay∞
0/parenleftbig
1−e−x/parenrightbigλ−1e−μxF/parenleftbig
α,β;γ;δe−x/parenrightbig
dx=B (μ, λ)3F2(α,β,μ ;γ,μ+λ;δ)
[Reλ>0,Reμ>0,|arg(1−δ)|<π]ET I 213(9)
2./integraldisplay∞
0/parenleftbig
1−e−x/parenrightbigμe−αxF/parenleftbig
−n, μ+β+n;β;e−x/parenrightbig
dx=B(α,μ+n+1 )B ( α,β+n−α)
B(α,β−α)
[Reα>0,Reμ>−1] ET I 213(10)
3./integraldisplay∞
0/parenleftbig
1−e−x/parenrightbigγ−1e−μxF/parenleftbig
α,β;γ;1−e−x/parenrightbig
dx=Γ(μ)Γ(γ−α−β+μ)Γ(γ)
Γ(γ−α+μ)Γ(γ−β+μ)
[Reμ>0,Reμ>Re(α+β−γ),Reγ>0]ET I 213(11)
4./integraldisplay∞
0/parenleftbig
1−e−x/parenrightbigγ−1e−μxF/bracketleftbig
α,β;γ;δ/parenleftbig
1−e−x/parenrightbig/bracketrightbig
dx=B (μ, γ)F(α,β;μ+γ;δ)
[Reμ>0,Reγ>0,|arg(1−δ)|<π]ET I 213(12)
7.542 Hypergeometric and Bessel functions 817
7.53 Hypergeometric and trigonometric functions
7.531
1./integraldisplay∞
0xsinμxF/parenleftbigg
α,β;3
2;−c2x2/parenrightbigg
dx=2−α−β+1πc−α−βμα+β−2Kα−β/parenleftbigμ
c/parenrightbig
Γ(α)Γ(β)/bracketleftbig
μ>0,Reα>1
2,Reβ>1
2/bracketrightbig
ET I 115(6)
2./integraldisplay∞
0cosμxF/parenleftbigg
α,β;1
2;−c2x2/parenrightbigg
dx=2−α−β+1πc−α−βμα+β−1Kα−β/parenleftbigμ
c/parenrightbig
Γ(α)Γ(β)
[μ>0,Reα>0,Reβ>0,c > 0]
ET I 61(9)
7.54 Combinations of hypergeometric and Bessel functions
7.541/integraldisplay∞
0xα+β−2ν−1(x+1 )−νexzKν[(x+1 )z]F(α,β;α+β−2ν;−x)dx
=π−1
2cos(νπ)Γ/parenleftbig1
2−α+ν/parenrightbig
Γ/parenleftbig1
2−β+ν/parenrightbig
Γ(γ)(2z)−1
2−1
2γW1
2γ,1
2(β−α)(2z)
γ=α+β−2ν/bracketleftbig
Re(α+β−2ν)>0,Re/parenleftbig1
2−α+ν/parenrightbig
>0,Re/parenleftbig1
2−β+ν/parenrightbig
>0,|argz|<3
2π/bracketrightbig
ET II 401(16)
7.542
1./integraldisplay∞
0xσ−1
pFp−1/parenleftbig
a1,...,a p;b1,...,b p−1;−λx2/parenrightbig
Yν(xy)dx
=Γ(b1)...Γ(bp−1)
2λ1
2σΓ(a1)...Γ(ap)Gp+2,1
p+2,p+3/parenleftbiggy2
4λ/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
∗
0,...,b∗p+1,
h, k, a∗
1,...,a∗p,l/parenrightbigg
a∗
j=aj−σ
2,j=1,...,p ;b∗
0=1−σ
2;b∗
j=bj−σ
2,
j=1,...,p −1;h=ν
2,k=−ν
2,l=−1+ν
2/bracketleftbig
|argλ|<π , Reσ>|Reν|,Reaj>1
2Reσ−3
4,y > 0/bracketrightbig
ET II 118(53)
2./integraldisplay∞
0xσ−1
pFp/parenleftbig
a1,...,a p;b1,...,b p;−λx2/parenrightbig
Yν(xy)dx
=Γ(b1)...Γ(bp)
2λ1
2σΓ(a1)...Γ(ap)Gp+2,1
p+2,p+3/parenleftbiggy2
4λ/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
∗
0,...,b∗p,l
h, k, a∗
1,...,a∗p,l/parenrightbigg
b∗
0=1−σ
2;a∗
j=aj−σ
2,b j∗=bj−σ
2;j=1,...,p ;h=ν
2,k=−ν
2,l=−1+ν
2/bracketleftbig
Reλ>0,Reσ>|Reν|,Reaj>1
2Reσ−3
4,y > 0/bracketrightbig
ET II 119(54)
818 Hypergeometric Functions 7.542
3./integraldisplay∞
0xσ−1
pFq/parenleftbig
a1,...,a p;b1,...,b q;−λx2/parenrightbig
Yν(xy)dx
=−π−12σ−1y−σcos/bracketleftBigπ
2(σ−ν)/bracketrightBig
Γ/parenleftbiggσ+ν
2/parenrightbigg
Γ/parenleftbiggσ−ν
2/parenrightbigg
×p+2Fq/parenleftbigg
a1,...,a p,σ+ν
2,σ−ν
2;b1,...,b q;−4λ
y2/parenrightbigg
[y>0,p≤q−1,Reσ>|Reν|]ET II 119(55)
4./integraldisplay∞
0xσ−1
pFq/parenleftbig
a1,...,a p;b1,...,b q;−λx2/parenrightbig
Kν(xy)dx
=2σ−2y−σΓ/parenleftbiggσ+ν
2/parenrightbigg
Γ/parenleftbiggσ−ν
2/parenrightbigg
p+2Fq/parenleftbigg
a1,...,a p,σ+ν
2,σ−ν
2;b1,...,b q;4λ
y2/parenrightbigg
[Rey>0,p≤q−1,Reσ>|Reν|]ET II 153(88)
5./integraldisplay∞
0x2ρ
pFp/parenleftbig
a1,...,a p;b1,...,b p;−λx2/parenrightbig
Jν(xy)dx
=22ρΓ(b1)...Γ(bp)
y2ρ+1Γ(a1)...Γ(ap)Gp+1,1
p+1,p+2/parenleftbiggy2
4λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle1,b
1,...,b p
h, a 1,...,a p,k/parenrightbigg
h=1
2+ρ+1
2ν, k =1
2+ρ−1
2ν/bracketleftbig
y>0,Reλ>0,−1−Reν<2R eρ<1
2+2R e ar,r=1,...,p/bracketrightbig
ET II 91(18)
6./integraldisplay∞
0x2ρ
m+1Fm/parenleftbig
a1,...,a m+1;b1,...,b m;−λ2x2/parenrightbig
Jν(xy)dx
=22ρΓ(b1)...Γ(bm)y−2ρ−1
Γ(a1)...Γ(am+1)Gm+2,1
m+1,m+3/parenleftbiggy2
4λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1,b
1,...,b m
h, a 1,...,a m+1,k/parenrightbigg
h=1
2+ρ+1
2ν, k =1
2+ρ−1
2ν,/bracketleftbig
y>0,Reλ>0,Re(2ρ+ν)>−1,Re (ρ−ar)<1
4;r=1,...,m +1/bracketrightbig
ET II 91(19)
7./integraldisplay∞
0xδF/parenleftbig
α,β;γ;−λ2x2/parenrightbig
Jν(xy)dx
=2δΓ(γ)
Γ(α)Γ(β)y−δ−1G22
24⎛
⎝y2
4λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−α,1−β
1+δ+ν
2,0,1−γ,1+δ−ν
2⎞
⎠
/bracketleftbig
y>0,Reλ>0,−1−Reν−2m in(R e α,Reβ)<Reδ<−1
2/bracketrightbig
ET II 82(9)
8./integraldisplay∞
0xδF/parenleftbig
α,β;γ;−λ2x2/parenrightbig
Jν(xy)dx=2δy−δ−1Γ(γ)
Γ(α)Γ(β)G31
24⎛
⎝y2
4λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1,γ
1+δ+ν
2,α,β,1+δ−ν
2⎞
⎠
/bracketleftbig
y>0,Reλ>0,−Reν−1<Reδ<2m ax( R e α,Reβ)−1
2/bracketrightbig
ET II 81(6)
9./integraldisplay∞
0xν+1F/parenleftbig
α,β;γ;−λ2x2/parenrightbig
Jν(xy)dx=2ν+1Γ(γ)
Γ(α)Γ(β)y−ν−2G30
13/parenleftbiggy2
4λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingleγ
ν+1,α ,β/parenrightbigg
/bracketleftbig
y>0,Reλ>0,−1<Reν<2m ax( R e α,Reβ)−
3
2/bracketrightbig
ET II 81(5)
7.542 Hypergeometric and Bessel functions 819
10./integraldisplay∞
0xν+1F/parenleftbig
α,β;ν+1 ;−λ2x2/parenrightbig
Jν(xy)dx=2ν−α−β+2Γ(ν+1 )
λα+βΓ(α)Γ(β)yα+β−ν−2Kα−β/parenleftBigy
λ/parenrightBig
/bracketleftbig
y>0,Reλ>0,−1<Reν<2m ax( R e α,Reβ)−3
2/bracketrightbig
ET II 81(3)
11./integraldisplay∞
0xν+1F/parenleftbig
α,β;ν+1 ;−λ2x2/parenrightbig
Kν(xy)dx=2ν+1λ−α−βyα+β−ν−2Γ(ν+1 )S1−α−β,α−β/parenleftBigy
λ/parenrightBig
[Rey>0,Reλ>0,Reν>−1]
ET II 152(86)
12./integraldisplay∞
0xν+1F/parenleftbigg
α,β;β+ν
2+1 ;−λ2x2/parenrightbigg
Jν(xy)dx=Γ/parenleftBig
β+ν+2
2/parenrightBig
yβ−1λ−ν−β−1
π1
2Γ(α)Γ(β)2β−1K1
2(ν−β+1)/parenleftBigy
2λ/parenrightBig2
/bracketleftbig
y>0,−1<Reν</parenleftbig
2m ax( R e α,Reβ)−3
2/parenrightbig/bracketrightbig
ET II 81(4)
13./integraldisplay∞
0xσ+1
2F/parenleftbig
α,β;γ;−λ2x2/parenrightbig
Yν(xy)dx=λ−σ−1y−1
2Γ(γ)√
2Γ (α)Γ(β)G41
35/parenleftbiggy2
4λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−p, γ−p, l
h, k, α −p, β−p, l/parenrightbigg
h=
1
4+1
2ν, k =1
4−1
2ν, l =−1
4−1
2ν, p =1
2+1
2σ/bracketleftbig
y>0,Reλ>0,Reσ>|Reν|−3
2,Reσ<2R eα,Reσ<2R eβ/bracketrightbig
ET II 118(52)
14./integraldisplay∞
0xν+2F/parenleftbigg1
2,1
2−ν;3
2;−λ2x2/parenrightbigg
Yν(xy)dx=2νy−ν−1
π1
2λ2Γ/parenleftbig1
2−ν/parenrightbigKν/parenleftBigy
2λ/parenrightBig
Kν+1/parenleftBigy
2λ/parenrightBig
/bracketleftbig
y>0,Reλ>0,−3
2<Reν<−1
2/bracketrightbig
ET II 117(49)
15./integraldisplay∞
0xν+2F/parenleftbigg
1,2ν+3
2;ν+2 ;−λ2x2/parenrightbigg
Yν(xy)dx=π−1
22−νλ−2ν−3Γ(ν+2 )
Γ/parenleftbig
2ν+3
2/parenrightbig/bracketleftBig
Kν/parenleftBigy
2λ/parenrightBig/bracketrightBig2
/bracketleftbig
y>0,Reλ>0,−1
2<Reν<1
2/bracketrightbig
ET II 117(50)
16./integraldisplay∞
0xν+2F/parenleftbigg
1,μ+ν+3
2;3
2;−λ2x2/parenrightbigg
Yν(xy)dx=π1
22−μ−ν−1λ−μ−2ν−3yμ+ν
Γ/parenleftbig
μ+ν+3
2/parenrightbig Kμ/parenleftBigy
λ/parenrightBig
/bracketleftbig
y>0,Reλ>0,−3
2<Reν<1
2,Re(2μ+ν)>−3
2/bracketrightbig
ET II 118(51)
17./integraldisplay∞
0x2α+νF/parenleftbigg
α−ν−1
2,α;2α;−λ2x2/parenrightbigg
Jν(xy)dx
=iΓ/parenleftbig1
2+α/parenrightbig
Γ/parenleftbig1
2+α+ν/parenrightbig
π21−ν−2αλ2α−1yν+2W1
2−α,−1
2−ν/parenleftBigy
λ/parenrightBig/bracketleftBig
W1
2−α,−1
2−ν/parenleftBig
e−iπy
λ/parenrightBig
−W1
2−α,−1
2−ν/parenleftBig
eiπy
λ/parenrightBig/bracketrightBig
/bracketleftbig
y>0,Reλ>0,Reν<−1
2,Re(α+ν)>−1
2/bracketrightbig
ET II 80(1)
18./integraldisplay∞
0x2α−νF/parenleftbigg
ν+α−1
2,α;2α;−λ2x2/parenrightbigg
Jν(xy)dx
=22α−νΓ/parenleftbig1
2+α/parenrightbig
yν−2
λ2α−1Γ(2ν)Mα−1
2,ν−1
2/parenleftBigy
λ/parenrightBig
W1
2−α,ν−1
2/parenleftBigy
λ/parenrightBig
ET II 80(2)
820 Confluent Hypergeometric Functions 7.543
7.543
1./integraldisplay∞
0x−2α−1F/parenleftbigg1
2+α,1+α;1+2 α;−4λ2
x2/parenrightbigg
Jν(xy)dx=λ−2αI1
2ν+α(λy)K1
2ν−α(λy)
/bracketleftbig
y>0,Reλ>0,Reν>−1,Reα>−1
2/bracketrightbig
ET II 81(7)
2./integraldisplay∞
0xν+1−4αF/parenleftbigg
α,α+1
2;ν+1 ;−λ2
x2/parenrightbigg
Jν(xy)dx
=Γ(ν)
Γ(2α)2νλ1−2αy2α−ν−1Iν/parenleftbigg1
2λy/parenrightbigg
K2α−ν−1/parenleftbigg1
2λy/parenrightbigg
/bracketleftbig
y>0,Reλ>0,Reα−1<Reν<4R eα−3
2/bracketrightbig
ET II 81(8)
7.544/integraldisplay∞
0xν+1(1 +x)−2αF/bracketleftbigg
α,ν+1
2;2ν+1 ;4x
(1 +x)2/bracketrightbigg
Jν(xy)dx
=Γ(ν+1 )Γ ( ν−α+1 )
Γ(α)22ν−2α+1y2(α−ν−1)Jν(y)
/bracketleftbig
y>0,−1<Reν<2R eα−3
2/bracketrightbig
ET II 82(10)
7.6 Confluent Hypergeometric Functions
7.61 Combinations of confluent hypergeometric functions and powers
7.611
1./integraldisplay∞
0x−1Wk,μ(x)dx=π3
22ksec(μπ)
Γ/parenleftbig3
4−1
2k+1
2μ/parenrightbig
Γ/parenleftbig3
4−1
2k−1
2μ/parenrightbig
/bracketleftbig
|Reμ|<1
2/bracketrightbig
ET II 406(22)
2./integraldisplay∞
0x−1Mk,μ(x)Wλ,μ(x)dx=Γ(2μ+1 )
(k−λ)Γ/parenleftbig1
2+μ−λ/parenrightbig
/bracketleftbig
Reμ>−1
2,Re(k−λ)>0/bracketrightbig
BU 116(11), ET II 409(39)
3./integraldisplay∞
0x−1Wk,μ(x)Wλ,μ(x)dx
=1
(k−λ)sin(2 μπ)/bracketleftBigg
1
Γ/parenleftbig1
2−k+μ/parenrightbig
Γ/parenleftbig1
2−λ−μ/parenrightbig−1
Γ/parenleftbig1
2−k−μ/parenrightbig
Γ/parenleftbig1
2−λ+μ/parenrightbig/bracketrightBigg
/bracketleftbig
|Reμ|<1
2/bracketrightbig
BU 116(12), ET II 409(40)
4./integraldisplay∞
0{Wκ,μ(z)}2dz
z=π
sin 2πμψ/parenleftbig1
2+μ−κ/parenrightbig
−ψ/parenleftbig1
2−μ−κ/parenrightbig
Γ/parenleftbig1
2+μ−κ/parenrightbig
Γ/parenleftbig1
2−μ−κ/parenrightbig
/bracketleftbig
|Reμ|<1
2/bracketrightbig
BU 117(12a)
5./integraldisplay∞
01
z[Wκ,0(z)]2dx=ψ/prime/parenleftbig1
2−κ/parenrightbig
/bracketleftbig
Γ/parenleftbig1
2−κ/parenrightbig/bracketrightbig2BU 117(12b)
7.613 Confluent hypergeometric functions and powers 821
6./integraldisplay∞
0xρ−1Wk,μ(x)W−k,μ(x)dx=Γ(ρ+1 )Γ/parenleftbig1
2ρ+1
2+μ/parenrightbig
Γ/parenleftbig1
2ρ+1
2−μ/parenrightbig
2Γ/parenleftbig
1+1
2ρ+k/parenrightbig
Γ/parenleftbig
1+1
2ρ−k/parenrightbig
[Reρ>2|Reμ|−1] ET II 409(41)
7.11/integraldisplay∞
0xρ−1Wk,μ(x)Wλ,ν(x)dx
=Γ(1−μ+ν+ρ)Γ(1+ μ+ν+ρ)Γ(−2ν)
Γ/parenleftbig1
2−λ−ν/parenrightbig
Γ/parenleftbig3
2−k+ν+ρ/parenrightbig
×3F2/parenleftbigg
1−μ+ν+ρ,1+μ+ν+ρ,1
2−λ+ν;1+2 ν,3
2−k+ν+ρ;1/parenrightbigg
+Γ(1 + μ−ν+ρ)Γ(1−μ−ν+ρ)Γ(2ν)
Γ/parenleftbig1
2−λ+ν/parenrightbig
Γ/parenleftbig3
2−k−ν+ρ/parenrightbig
×3F2/parenleftbigg
1+μ−ν+ρ,1−μ−ν+ρ,1
2−λ−ν;1−2ν,3
2−k−ν+ρ;1/parenrightbigg
[|Reμ|+|Reν|<Reρ+1 ] ET II 410(42)
7.612
1./integraldisplay∞
0tb−1
1F1(a;c;−t)dt=Γ(b)Γ(c)Γ(a−b)
Γ(a)Γ(c−b)[0<Reb<Rea] EH I 285(10)
2./integraldisplay∞
0tb−1Ψ(a,c;t)dt=Γ(b)Γ(a−b)Γ(b−c+1 )
Γ(a)Γ(a−c+1 )[0<Reb<ReaRec<Reb+1 ]
EH I 285(11)
7.613
1./integraldisplayt
0xγ−1(t−x)c−γ−1
1F1(a;γ;x)dx=tc−1Γ(γ)Γ(c−γ)
Γ(c)1F1(a;c;t)
[Rec>Reγ>0]
BU 9(16)a, EH I 271(16)
2./integraldisplayt
0xβ−1(t−x)γ−1
1F1(t;β;x)dx=Γ(β)Γ(γ)
Γ(β+γ)tβ+γ−1
1F1(t;β+γ;t)
[Reβ>0,Reγ>0] ET II 401(1)
3./integraldisplay1
0xλ−1(1−x)2μ−λ
1F1/parenleftbigg1
2+μ−ν;λ;xz/parenrightbigg
dx=B (λ,1+2μ−λ)e1
2zz−1
2−μMν,μ(z)
[Reλ>0,Re(2μ−λ)>−1]
BU 14(14)
4./integraldisplayt
0xβ−1(t−x)δ−1
1F1(t;β;x)1F1(γ;δ;t−x)dx=Γ(β)Γ(δ)
Γ(β+δ)tβ+δ−1
1F1(t+γ;β+δ;t)
[Reβ>0,Reδ>0]
ET II 402(2), EH I 271(15)
5./integraldisplayt
0xμ−1
2(t−x)ν−1
2Mk,μ(x)Mλ,ν(t−x)dx=Γ(2μ+1 )Γ ( 2 ν+1 )
Γ(2μ+2ν+2 )tμ+νMk+λ,μ+ν+1
2(t)
/bracketleftbigg
Reμ>−1
2,Reν>−1
2/bracketrightbigg
BU 128(14), ET II 402(7)
822 Confluent Hypergeometric Functions 7.621
6./integraldisplay1
0xβ−1(1−x)σ−β−1
1F1(α;β;λx)1F1[σ−α;σ−β;μ(1−x)]dx
=Γ(β)Γ(σ−β)
Γ(σ)eλ
1F1(α;σ;μ−λ)
[0<Reβ<Reσ] ET II 402(3)
7.62–7.63 Combinations of confluent hypergeometric functions and exponentials
7.621
1./integraldisplay∞
0e−sttαMμ,ν(t)dt=Γ/parenleftbig
α+ν+3
2/parenrightbig
/parenleftbig1
2+s/parenrightbigα+ν+3
2F/parenleftbigg
α+ν+3
2,−μ+ν+1
2;2ν+1 ;2
2s+1/parenrightbigg
/bracketleftbig
Re/parenleftbig
α+μ+3
2/parenrightbig
>0,Res>1
2/bracketrightbig
BU 118(1), MO 176a, EH I 270(12)a
2./integraldisplay∞
0e−sttμ−1
2Mλ,μ(qt)dt=qμ+1
2Γ(2μ+1 )/parenleftbig
s−1
2q/parenrightbigλ−μ−1
2/parenleftbig
s+1
2q/parenrightbig−λ−μ−1
2
/bracketleftbigg
Reμ>−1
2,Res>|Req|
2/bracketrightbigg
BU 119(4c), MO 176a, EH I 271(13)a
3./integraldisplay∞
0e−sttαWλ,μ(qt)dt=Γ/parenleftbig
α+μ+3
2/parenrightbig
Γ/parenleftbig
α−μ+3
2/parenrightbig
qμ+1
2
Γ(α−λ+2 )/parenleftbigg
s+1
2q/parenrightbigg−α−μ−3
2
×F/parenleftbigg
α+μ+3
2,μ−λ+1
2;α−λ+2 ;2s−q
2s+q/parenrightbigg
/bracketleftbigg
Re/parenleftbigg
α±μ+3
2/parenrightbigg
>0,Res>−q
2,q > 0/bracketrightbigg
EH I 271(14)a, BU 121(6), MO 176
4./integraldisplay∞
0e−sttb−1
1F1(a;c;kt)dt=Γ (b)s−bF/parenleftbig
a,b;c;ks−1/parenrightbig
[|s|>|k|]
=Γ (b)(s−k)−bF/parenleftbigg
c−a,b;c;k
k−s/parenrightbigg
[|s−k>|k||]
[Reb>0,Res>max(0 ,Rek)]EH I 269(5)
5./integraldisplay∞
0tc−1
1F1(a;c;t)e−stdt=Γ (c)s−c/parenleftbig
1−s−1/parenrightbig−a[Rec>0,Res>1] EH I 270(6)
6./integraldisplay∞
0tb−1Ψ(a,c;t)e−stdt=Γ(b)Γ(b−c+1 )
Γ(a+b−c+1 )F(b,b−c+1 ;a+b−c+1 ;1 −s)
[Reb>0,Rec<Reb+1,|1−s|<1]
=Γ(b)Γ(b−c+1 )
Γ(a+b−c+1 )s−bF/parenleftbig
a,b;a+b−c+1 ;1 −s−1/parenrightbig
/bracketleftbig
Res>1
2/bracketrightbig
EH I 270(7)
7./integraldisplay∞
0e−b
2xxν−1Mκ,μ(bx)dx=Γ(1 + 2 μ)Γ(κ−ν)Γ/parenleftbig1
2+μ+ν/parenrightbig
Γ/parenleftbig1
2+μ+κ/parenrightbig
Γ/parenleftbig1
2+μ−ν/parenrightbigbν
/bracketleftbig
Re/parenleftbig
ν+1
2+μ/parenrightbig
>0,Re (κ−ν)>0/bracketrightbig
BU 119(3)a, ET I 215(11)a
7.622 Confluent hypergeometric functions and exponentials 823
8./integraldisplay∞
0e−sxMκ,μ(x)dx
x=2Γ ( 1+2 μ)e−iπκ
Γ/parenleftbig1
2+μ+κ/parenrightbig/parenleftbiggs−1
2
s+1
2/parenrightbiggκ
2
Qκ
μ−1
2(2s)
/bracketleftbig
Re/parenleftbig1
2+μ/parenrightbig
>0,Res>1
2/bracketrightbig
BU 119(4a)
9./integraldisplay∞
0e−sxWκ,μ(x)dx
x=π
cos/parenleftBigπμ
2/parenrightBig/parenleftbiggs−1
2
s+1
2/parenrightbiggκ
2
Pκ
μ−1
2(2s)
/bracketleftbig
Re/parenleftbig1
2±μ/parenrightbig
>0,Res>−1
2/bracketrightbig
BU 121(7)
10./integraldisplay∞
0xk+2μ−1e−3
2xWk,μ(x)dx=Γ/parenleftbig
k+μ+1
2/parenrightbig
Γ/bracketleftbig1
4(2k+6μ+5 )/bracketrightbig
/parenleftbig
k+3μ+1
2/parenrightbig
Γ/bracketleftbig1
4(2μ−2k+3 )/bracketrightbig
/bracketleftbig
Re(k+μ)>−1
2,Re(k+3μ)>−1
2/bracketrightbig
BU 122(8a), ET II 406(23)
11./integraldisplay∞
0e−1
2xxν−1Wκ,μ(x)dx=Γ/parenleftbig
ν+1
2−μ/parenrightbig
Γ/parenleftbig
ν+1
2+μ/parenrightbig
Γ(ν−κ+1 )/bracketleftbig
Re/parenleftbig
ν+1
2±μ/parenrightbig
>0/bracketrightbig
BU 122(8b)
12./integraldisplay∞
0e1
2xxν−1Wκ,μ(x)dx=Γ(−κ−μ)Γ/parenleftbig1
2+μ+ν/parenrightbig
Γ/parenleftbig1
2−μ+ν/parenrightbig
Γ/parenleftbig1
2−μ−κ/parenrightbig
Γ/parenleftbig1
2+μ−κ/parenrightbig
/bracketleftbig
Re/parenleftbig
ν+1
2±μ/parenrightbig
>0,Re (κ+ν)<0/bracketrightbig
BU 122(8c)a
7.622
1./integraldisplay∞
0e−sttc−1
1F1(a;c;t)1F1(α;c;λt)dt
=Γ (c)(s−1)−a(s−λ)−αsa+α−cF/bracketleftbig
a,α;c;λ(s−1)−1(s−λ)−1/bracketrightbig
[Rec>0,Res>Reλ+1 ] EH I 287(22)
2./integraldisplay∞
0e−ttρ
1F1(a;c;t)Ψ(a/prime;c/prime;λt)dt
=CΓ(c)Γ(β)
Γ(γ)λσF/parenleftbig
c−a,β;γ;1−λ−1/parenrightbig
,
ρ=c−1,σ=−c, β =c−c/prime+1,γ=c−a+a/prime−c/prime+1,C =Γ(a/prime−a)
Γ(a/prime),or
ρ=c+c/prime−2,σ=1−c−c/prime,β=c+c/prime−1,γ=a/prime−a+c, C =Γ(a/prime−a−c/prime+1 )
Γ(a/prime−c/prime+1 )
EH I 287(24)
824 Confluent Hypergeometric Functions 7.623
3./integraldisplay∞
0xν−1e−bxMλ1,μ1−1
2(a1x)...Mλn,μn−1
2(anx)dx
=aμ1
1...aμn
n(b+A)−ν−MΓ(ν+M)
×FA/parenleftbigg
ν+M;μ1−λ1,...,μ n−λn;2μ1,...,2μn;a1
b+A,...,an
b+A/parenrightbigg
,
M=μ1+···+μn,A =1
2(a1+···+an)/bracketleftbig
Re(ν+M)>0,Re/parenleftbig
b±1
2a1±···+1
2an/parenrightbig
>0/bracketrightbig
ET I 216(14)
7.623
1./integraldisplay∞
0e−xxc+n−1(x+y)−1
1F1(a;c;x)dx=(−1)nΓ(c)Γ(1−a)yc+n−1Ψ(c−a,c;y)
[−Rec<n< 1−Rea, n =0,1,2,..., |argy|<π]EH I 285(16)
2./integraldisplayt
0x−1(t−x)k−1e1
2(t−x)Mk,μ(x)dx=Γ(k)Γ(2μ+1 )
Γ/parenleftbig
k+μ+1
2/parenrightbigπ1
2tk−1
2lμ/parenleftbigg1
2t/parenrightbigg
/bracketleftbig
Rek>0,Reμ>−1
2/bracketrightbig
ET II 402(5)
3./integraldisplayt
0xk−1(t−x)λ−1e1
2(t−x)Mk+λ,μ(x)dx=Γ(λ)Γ/parenleftbig
k+μ+1
2/parenrightbig
tk+λ−1
Γ/parenleftbig
k+λ+μ+1
2/parenrightbigMk,μ(t)
/bracketleftbig
Re(k+μ)>−1
2,Reλ>0/bracketrightbig
ET II 402(6)
4./integraldisplayt
0x−k−λ−1(t−x)λ−1e1
2xWk,μ(x)dx=Γ(λ)Γ/parenleftbig1
2−k−λ+μ/parenrightbig
Γ/parenleftbig1
2−k−λ−μ/parenrightbig
tk+1Γ/parenleftbig1
2−k+μ/parenrightbig
Γ/parenleftbig1
2−k−μ/parenrightbigWk+λ,μ(t)
/bracketleftbig
Reλ>0,Re(k+λ)<1
2−|Reμ|/bracketrightbig
ET II 405(21)
5./integraldisplay∞
1(x−1)μ−1xλ−1
2e1
2axWk,λ(ax)dx=Γ(μ)Γ/parenleftbig1
2−k−λ−μ/parenrightbig
Γ/parenleftbig1
2−k−λ/parenrightbig a−1
2μe1
2aWk+1
2μ,λ+1
2μ(a)
/bracketleftbig
|arg(a)|<3
2π,0<Reμ<1
2−Re(k+λ)/bracketrightbig
ET II 211(72)a
6.11/integraldisplay∞
1(x−1)μ−1xμ−1
2e−1
2axW2μ+1
2,λ(ax)dx=Γ (μ)e−1
2aWμ+1
2,λ(a)
[Reμ>0,Rea>0] ET II 211(74)a
7./integraldisplay∞
1(x−1)μ−1xk−μ−1e−1
2axWk,λ(ax)dx=Γ (μ)e−1
2aWk−μ,λ(a)
[Reμ>0,Rea>0] ET II 211(73)a
7.624 Confluent hypergeometric functions and exponentials 825
8./integraldisplay1
0(1−x)μ−1xk−μ−1e−1
2axWk,λ(ax)dx
=Γ (μ)e−1
2asec[(k−μ−λ)π]
×/braceleftBigg
sin(μπ)Γ/parenleftbig
k−μ+λ+1
2/parenrightbig
Γ(2λ+1 )Mk−μ,λ(a) + cos[( k−λ)π]Wk−μ,λ(a)/bracerightBigg
/bracketleftbig
0<Reμ<Rek−|Reλ|+1
2/bracketrightbig
ET II 200(93)a
7.624
1./integraldisplay∞
0xρ−1/bracketleftBig
x1
2+(a+x)1
2/bracketrightBig2σ
e−1
2xMk,μ(x)dx
=−σΓ(2μ+1 )aσ
π1
2Γ/parenleftbig1
2+k+μ/parenrightbigG23
34/parenleftBigg
a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2,1,1−k+ρ
1
2+μ+ρ,−σ, σ,1
2−μ+ρ/parenrightBigg
/bracketleftbig
|arga|<π , Re(μ+ρ)>−1
2,Re(k−ρ−σ)>0/bracketrightbig
ET II 403(8)
2./integraldisplay∞
0xρ−1/bracketleftBig
x1
2+(a+x)1
2/bracketrightBig2σ
e−1
2xWk,μ(x)dx=−π−1
2σaσG32
34/parenleftBigg
a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2,1,1−k+ρ
1
2+μ+ρ,1
2−μ+ρ,−σ, σ/parenrightBigg
/bracketleftbig
|arga|<π , Reρ>|Reμ|−1
2/bracketrightbig
ET II 406(24)
3./integraldisplay∞
0xρ−1/bracketleftBig
x1
2+(a+x)1
2/bracketrightBig2σ
e−1
2xWk,μ(x)dx
=−σπ−1
2aσ
Γ/parenleftbig1
2−k+μ/parenrightbig
Γ/parenleftbig1
2−k−μ/parenrightbigG33
34/parenleftBigg
a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2,1,1+k+ρ
1
2+μ+ρ,1
2−μ+ρ,−σ, σ/parenrightBigg
/bracketleftbig
|arga|<π , Reρ>|Reμ|−1
2,Re(k+ρ+σ)<0/bracketrightbig
ET II 406(25)
4./integraldisplay∞
0xρ−1(a+x)−1
2/bracketleftBig
x1
2+(a+x)1
2/bracketrightBig2σ
e−1
2xMk,μ(x)dx
=Γ(2μ+1 )aσ
π1
2Γ/parenleftbig1
2+k+μ/parenrightbigG23
34/parenleftbigg
a/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,1
2,1
2−k−ρ
−σ, ρ+μ, ρ−μ, σ/parenrightbigg
/bracketleftbig
|arga|<π , Re(ρ+μ)>−1
2,Re(k−ρ−σ)>−1
2/bracketrightbig
ET II 403(9)
5./integraldisplay∞
0xρ−1(a+x)−1
2/bracketleftBig
x1
2+(a+x)1
2/bracketrightBig2σ
e−1
2xWk,μ(x)dx
=π−1
2aσ
Γ/parenleftbig1
2−k+μ/parenrightbig
Γ/parenleftbig1
2−k−μ/parenrightbigG33
34/parenleftbigg
a/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,
1
2,1
2+k+ρ
−σ, ρ+μ, ρ−μ, σ/parenrightbigg
/bracketleftbig
|arga|<π , Reρ>|Reμ|−1
2,Re(k+ρ+σ)<1
2/bracketrightbig
ET II 406(26)
6./integraldisplay∞
0xρ−1(a+x)−1
2/bracketleftBig
x1
2+(a+x)1
2/bracketrightBig2σ
e−1
2xWk,μ(x)dx=π−1
2aσG32
34/parenleftbigg
a/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,
1
2,1
2−k+ρ
−σ, ρ+μ, ρ−μ, σ/parenrightbigg
/bracketleftbig
|arga|<π , Reρ>|Reμ|−1
2/bracketrightbig
ET II 406(27)
826 Confluent Hypergeometric Functions 7.625
7.625
1./integraldisplay∞
0xρ−1exp/bracketleftbig
−1
2(α+β)x/bracketrightbig
Mk,μ(αx)Wλ,ν(βx)dx
=Γ(1 + μ+ν+ρ)Γ(1+ μ−ν+ρ)
Γ/parenleftbig3
2−λ+μ+ρ/parenrightbig αμ+1
2β−μ−ρ−1
2
×3F2/parenleftbigg1
2+k+μ,1+μ+ν+ρ,1+μ−ν+ρ;2μ+1,3
2−λ+μ+ρ;−α
β/parenrightbigg
[Reα>0,Reβ>0,Re (ρ+μ)>|Reν|−1]ET II 410(43)
2./integraldisplay∞
0xρ−1exp/bracketleftbigg
−1
2(α+β)x/bracketrightbigg
Wk,μ(αx)Wλ,ν(βx)dx
=β−ρ/bracketleftbig
Γ/parenleftbig1
2−k+μ/parenrightbig
Γ/parenleftbig1
2−k−μ/parenrightbig
Γ/parenleftbig1
2−λ+ν/parenrightbig
Γ/parenleftbig1
2−λ−ν/parenrightbig/bracketrightbig−1
×G33
33/parenleftBigg
β
α/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2+μ,1
2−μ,1+λ+ρ
1
2+ν+ρ,1
2−ν+ρ,−k/parenrightBigg
[|Reμ|+|Reν|<Reρ+1,Re(k+λ+ρ)<0]ET II 410(44)a
3./integraldisplay∞
0xρ−1exp/bracketleftbigg
−1
2(α+β)x/bracketrightbigg
Wk,μ(αx)Wλ,ν(βx)dx=β−ρG22
33/parenleftBigg
β
α/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2+μ,1
2−ν,1−λ+ρ
1
2+ν+ρ,1
2−ν+ρ, k/parenrightBigg
ET II 411(46)
4./integraldisplay∞
0xρ−1exp/bracketleftbigg
−1
2(α−β)x/bracketrightbigg
Wk,μ(αx)Wλ,ν(βx)dx
=β−ρ/bracketleftbig
Γ/parenleftbig1
2−λ+ν/parenrightbig
Γ/parenleftbig1
2−λ−ν/parenrightbig/bracketrightbig−1G23
33/parenleftBigg
β
α/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2+μ,1
2−μ,1+λ+ρ
1
2+ν+ρ,1
2−ν+ρ, k/parenrightBigg
[Reα>0,|Reμ|+|Reν|<Reρ+1 ] ET II 411(45)
7.626
1./integraldisplay1
0/bracketleftbiggk
x−1
4(ξ+η)exp/bracketleftbigg
−1
2(ξ+η)x/bracketrightbigg
xc/bracketrightbigg
1F1(a;c;ξx)1F1(a;c;ηx)dx
=0 [ ξ/negationslash=η,Rec>0]
=a
ξe−ξ[1F1(a+1 ;c;ξ)]2[ξ=η,Rec>0]
[where ξandηare any two zeros of the function 1F1(a;c;x)]EH I 285
2./integraldisplay∞
1/bracketleftbiggk
x−1
4(ξ+η)/bracketrightbigg
e−1
2(ξ+η)xxcΨ(a,c;ξx)Ψ (a,c;ηx)dx=0 [ ξ/negationslash=η];
=−ξ−1e−ξ[Ψ(a−1,c;ξ)]2[ξ=η]
[where ξandηare any two zeros of the function Ψ( a,c;x)]EH I 286
7.627 Confluent hypergeometric functions and exponentials 827
7.627
1./integraldisplay∞
0x2λ−1(a+x)−μ−1
2e1
2xWk,μ(a+x)dx=Γ(2λ)Γ/parenleftbig1
2−k+μ−2λ/parenrightbig
Γ/parenleftbigg1
2−k+μ/parenrightbigg aλ−μ−1
2Wk+λ,μ−λ(a)
/bracketleftbigg
|arga|<π , 0<2R eλ<1
2−Re(k+μ)/bracketrightbigg
ET II 411(50)
2./integraldisplay∞
0x2λ−1(a+x)−μ−1
2e−1
2xM−1
2x
k,μ(a+x)dx
=Γ(2λ)Γ(2μ+1 )Γ/parenleftbig
k+μ−2λ+1
2/parenrightbig
Γ/parenleftbig
k+μ+1
2/parenrightbig
Γ(1−2λ+2μ)aλ−μ−1
2Mk−λ,μ−λ(a)
/bracketleftbig
Reλ>0,Re(k+μ−2λ)>−1
2/bracketrightbig
ET II 405(20)
3./integraldisplay∞
0x2λ−1(a+x)−μ−1
2e−1
2xWk,μ(a+x)dx=Γ ( 2 λ)aλ−μ−1
2Wk−λ,μ−λ(a)
[|arga|<π , Reλ>0] ET II 411(47)
4./integraldisplay∞
0xλ−1(a+x)k−λ−1e−1
2xWk,μ(a+x)dx=Γ (λ)ak−1Wk−λ,μ(a)
[|arga|<π , Reλ>0] ET II 411(48)
5./integraldisplay∞
0xρ−1(a+x)−σe−1
2xWk,μ(a+x)dx=Γ (ρ)aρe1
2aG30
23/parenleftbigg
a/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,1−k−σ
−ρ,
1
2+μ−σ,1
2−μ−σ/parenrightbigg
[|arga|<π , Reρ>0] ET II 411(49)
6./integraldisplay∞
0xρ−1(a+x)−σe1
2xWk,μ(a+x)dx
=Γ(ρ)aρe−1
2a
Γ/parenleftbig1
2−k+μ/parenrightbig
Γ/parenleftbig1
2−k−μ/parenrightbigG31
23/parenleftbigg
a/vextendsingle/vextendsingle/vextendsingle/vextendsinglek−σ+1,0
−ρ,1
2+μ−σ,1
2−μ−σ/parenrightbigg
[|arga|<π , 0<Reρ<Re(σ−k)]ET II 412(51)
7./integraldisplay∞
0e−1
2(a+x)(a+x)2κ−1
(ax)κWκ,μ(x)dx
x=Γ/parenleftbig1
2−μ−κ/parenrightbig
Γ/parenleftbig1
2+μ−κ/parenrightbig
aΓ(1−2κ)Wκ,μ(a)
/bracketleftbig
Re/parenleftbig1
2±μ−κ/parenrightbig
>0/bracketrightbig
BU 126(7a)
8./integraldisplay∞
0e−1
2xxγ+α−1Mκ,μ(x)dx
(x+a)α
=Γ(1 + 2 μ)Γ/parenleftbig1
2+μ+γ/parenrightbig
Γ(κ−γ)
Γ/parenleftbig1
2+μ−γ/parenrightbig
Γ/parenleftbig1
2+μ+κ/parenrightbig2F2/parenleftbigg
α,κ−γ;1
2+μ−γ,1
2−μ−γ;a/parenrightbigg
+Γ/parenleftbig
α+γ+1
2+μ/parenrightbig
Γ/parenleftbig
−γ−1
2−μ/parenrightbig
Γ(α)aγ+1
2+μ
×2F2/parenleftbigg
α+γ+μ+1
2,κ+μ+1
2;1+2 μ,3
2+μ+γ;a/parenrightbigg
/bracketleftbig
Re/parenleftbig
γ+α+1
2+μ/parenrightbig
>0,Re (γ−κ)<0/bracketrightbig
BU 126(8)a
828 Confluent Hypergeometric Functions 7.628
9./integraldisplay∞
0e−1
2xxn+μ+1
2Mκ,μ(x)dx
x+a=(−1)n+1an+μ+1
2e1
2aΓ(1 + 2 μ)Γ/parenleftbigg1
2−μ+κ/parenrightbigg
W−κ,μ(a)
/bracketleftbigg
n=0,1,2,..., Re/parenleftBig
μ+1+n
2/parenrightBig
>0,Re/parenleftbigg
κ−μ−1
2/parenrightbigg
<n , |arga|<π/bracketrightbigg
BU 127(10a)a
7.628
1./integraldisplay∞
0e−ste−t2t2c−2
1F1/parenleftbig
a;c;t2/parenrightbig
dt=21−2cΓ(2c−1)Ψ/parenleftbigg
c−1
2,a+1
2;1
4s2/parenrightbigg
/bracketleftbig
Rec>1
2,Res>0/bracketrightbig
EH I 270(11)
2./integraldisplay∞
0t2ν−1e−1
2at2e−stM−3ν,ν/parenleftbiggt2
a/parenrightbigg
dt=1
2√πΓ(4ν+1 )a−νs−4νeas2/8K2ν/parenleftbiggas2
8/parenrightbigg
/bracketleftbig
Rea>0,Reν>−1
4,Res>0/bracketrightbig
ET I 215(12)
3./integraldisplay∞
0t2μ−1e−1
2at2e−stMλ,μ/parenleftbiggt2
a/parenrightbigg
dt
=2−3μ−λΓ(4μ+1 )a1
2(λ+μ−1)sλ−μ−1eas2
8W−1
2(λ+3μ),1
2(λ−μ)/parenleftbiggas2
4/parenrightbigg
/bracketleftbig
Rea>0,Reμ>−1
4,Res>0/bracketrightbig
ET I 215(13)
7.629
1.8/integraldisplay∞
0tkexp/parenleftBiga
2t/parenrightBig
e−stWk,μ/parenleftBiga
t/parenrightBig
dt=21−2k√as−k−1
2S2k,2μ/parenleftbig
2√as/parenrightbig
/bracketleftbig
|arga|<π , Re (k±μ)>−1
2,Res>0/bracketrightbig
ET I 217(21)
2./integraldisplay∞
0t−kexp/parenleftBig
−a
2t/parenrightBig
e−stWk,μ/parenleftBiga
t/parenrightBig
dt=2√ask−1
2K2μ/parenleftbig
2√as/parenrightbig
[Rea>0,Res>0] ET I 217(22)
7.631
1./integraldisplay∞
0xρ−1exp/bracketleftbigg1
2/parenleftbig
α−1x−βx−1/parenrightbig/bracketrightbigg
Wk,μ/parenleftbig
α−1x/parenrightbig
Wλ,ν/parenleftbig
βx−1/parenrightbig
dx
=βρ/bracketleftbig
Γ/parenleftbig1
2−k+μ/parenrightbig
Γ/parenleftbig1
2−k−μ/parenrightbig/bracketrightbig−1
×G41
24/parenleftbiggβ
α/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+k,1−λ−ρ
1
2+μ,1
2−μ,1
2+ν−ρ,1
2−ν−ρ/parenrightbigg
/bracketleftbig
|argα|<3
2π,Reβ>0,Re(k+ρ)<−|Reν|−1
2/bracketrightbig
ET II 412(55)
2./integraldisplay∞
0xρ−1exp/bracketleftbigg1
2/parenleftbig
α−1x−βx−1/parenrightbig/bracketrightbigg
Wk,μ/parenleftbig
α−1x/parenrightbig
Wλ,ν/parenleftbig
βx−1/parenrightbig
dx
=βρ/bracketleftbig
Γ/parenleftbig1
2−k+μ/parenrightbig
Γ/parenleftbig1
2−k−μ/parenrightbig
Γ/parenleftbig1
2−λ+ν/parenrightbig
Γ/parenleftbig1
2−λ−ν/parenrightbig/bracketrightbig−1
×G42
24/parenleftbiggβ
α/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+k,1+λ−ρ
1
2+μ,1
2−μ,1
2+ν−ρ,1
2−ν−ρ/parenrightbigg
/bracketleftbig
|argα|<3
2π,|argβ|<3
2π,Re(λ−ρ)<1
2−|Reμ|,Re(k+ρ)<1
2−|Reν|/bracketrightbig
ET II 412(57)
7.644 Confluent hypergeometric and trigonometric functions 829
3./integraldisplay∞
0xρ−1exp/bracketleftbigg1
2/parenleftbig
α−1x+βx−1/parenrightbig/bracketrightbigg
Wk,μ/parenleftbig
α−1x/parenrightbig
Wλ,ν/parenleftbig
βx−1/parenrightbig
dx
=βρG40
24/parenleftbiggβ
α/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−k,1−λ−ρ
1
2+μ,1
2−μ,1
2+ν−ρ,1
2−ν−ρ/parenrightbigg
[Reα>0,Reβ>0] ET II 412(54)
7.632/integraldisplay∞
0e−st/parenleftbig
et−1/parenrightbigμ−1
2exp/parenleftbigg
−1
2λet/parenrightbigg
Mk,μ/parenleftbig
λet−λ/parenrightbig
dt
=Γ(2μ+1 )Γ/parenleftbig1
2+k−μ+s/parenrightbig
Γ(s+1 )W−k−1
2s,μ−1
2s(λ)
/bracketleftbig
Reμ>−1
2,Res>Re(μ−k)−1
2/bracketrightbig
ET I 216(15)
7.64 Combinations of confluent hypergeometric and trigonometric functions
7.641/integraldisplay∞
0cos(ax)1F1(ν+1 ;1 ; ix)1F1(ν+1 ;1 ; −ix)dx
=−a−1sin(νπ)Pν/parenleftbig
2a−2−1/parenrightbig
[0<a< 1];
=0 [ 1 <a< ∞]
[−1<Reν<0] ET II 402(4)
7.64211/integraldisplay∞
0cos(2xy)1F1/parenleftbig
a;c;−x2/parenrightbig
dx=1
2π1
2Γ(c)
Γ(a)|y|2α−1e−y2Ψ/parenleftbig
c−1
2,a+1
2;y2/parenrightbig
EH I 285(12)
7.643
1./integraldisplay∞
0x4νe−1
2x2sin(bx)1F1/parenleftbigg1
2−2ν;2ν+1 ;1
2x2/parenrightbigg
dx=/radicalbiggπ
2b4νc−1
2b2
1F1/parenleftbigg1
2−2ν;1+2 ν;1
2b2/parenrightbigg
/bracketleftbig
b>0,Reν>−1
4/bracketrightbig
ET I 115(5)
2./integraldisplay∞
0x2ν−1e−1
4x2sin(bx)M3ν,ν/parenleftbigg1
2x2/parenrightbigg
dx=/radicalbiggπ
2b2ν−1e−1
4b2M3ν,ν/parenleftbigg1
2b2/parenrightbigg
/bracketleftbig
b>0,Reν>−1
4/bracketrightbig
ET I 116(10)
3./integraldisplay∞
0x−2ν−1e1
4x2cos(bx)W3ν,ν/parenleftbigg1
2x2/parenrightbigg
dx=/radicalbiggπ
2b−2ν−1e1
4b2W3ν,ν/parenleftbigg1
2b2/parenrightbigg
/bracketleftbig
Reν<1
4,b > 0/bracketrightbig
ET I 61(7)
4./integraldisplay∞
0x−2νe1
4x2sin(bx)W3ν−1,ν/parenleftbigg1
2x2/parenrightbigg
dx=/radicalbiggπ
2b−2νe1
4b2W3ν−1,ν/parenleftbigg1
2b2/parenrightbigg
/bracketleftbig
Reν<1
2,b > 0/bracketrightbig
ET I 116(9)
7.644
1.11/integraldisplay∞
0x−μ−1
2e−1
2xsin/parenleftBig
2ax1
2/parenrightBig
Mk,μ(x)dx=π1
2ak+μ−1Γ(3−2μ)
Γ/parenleftbig1
2+k+μ/parenrightbigexp/parenleftbigg
−a2
2/parenrightbigg
Wρ,σ/parenleftbig
a2/parenrightbig
,
2ρ=k−3μ+1,2σ=k+μ−1[ a>0,Re(k+μ)>0]ET II 403(10)
830 Confluent Hypergeometric Functions 7.651
2./integraldisplay∞
0xρ−1sin/parenleftBig
cx1
2/parenrightBig
e−1
2xWk,μ(x)dx=cΓ(1 + μ+ρ)Γ( 1−μ+ρ)
Γ/parenleftbig3
2−k+ρ/parenrightbig
×2F2/parenleftbigg
1+μ+ρ,1−μ+ρ;3
2,3
2−k+ρ;−c2
4/parenrightbigg
[Reρ>|Reμ|−1] ET II 407(28)
3./integraldisplay∞
0xρ−1sin/parenleftBig
cx1
2/parenrightBig
e1
2xWk,μ(x)dx
=π1
2
Γ/parenleftbig1
2−k+μ/parenrightbig
Γ/parenleftbig1
2−k−μ/parenrightbigG22
23/parenleftBigg
c2
4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2+μ−ρ,1
2−μ−ρ
1
2,−k−ρ,0/parenrightBigg
/bracketleftbig
c>0,Reρ>|Reμ|−1,Re(k+ρ)<1
2/bracketrightbig
ET II 407(29)
4./integraldisplay∞
0xρ−1cos/parenleftBig
cx1
2/parenrightBig
e−1
2xWk,μ(x)dx=Γ/parenleftbig1
2+μ+ρ/parenrightbig
Γ/parenleftbig1
2−μ+ρ/parenrightbig
Γ(1−k+ρ)
×2F2/parenleftbigg1
2+μ+ρ,1
2−μ+ρ;1
2,1−k+ρ;−c2
4/parenrightbigg
/bracketleftbig
Reρ>|Reμ|−1
2/bracketrightbig
ET II 407(30)
5./integraldisplay∞
0xρ−1cos/parenleftBig
cx1
2/parenrightBig
e1
2xWk,μ(x)dx
=π1
2
Γ/parenleftbig1
2−k+μ/parenrightbig
Γ/parenleftbig1
2−k−μ/parenrightbigG22
23/parenleftBigg
c2
4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2+μ−ρ,1
2−μ−ρ
0,−k−ρ,1
2/parenrightBigg
/bracketleftbig
c>0,Reρ>|Reμ|−1
2,Re(k+ρ)<1
2/bracketrightbig
ET II 407(31)
7.65 Combinations of confluent hypergeometric functions and Bessel functions
7.651
1./integraldisplay∞
0Jν(xy)M−1
2μ,1
2ν(ax)W1
2μ,1
2ν(ax)dx
=ay−μ−1Γ(ν+1 )
Γ/parenleftbig1
2−1
2μ+1
2ν/parenrightbig/bracketleftBig
a+/parenleftbig
a2+y2/parenrightbig1
2/bracketrightBigμ/parenleftbig
a2+y2/parenrightbig−1
2
/bracketleftbig
y>0,Reν>−1,Reμ<1
2,Rea>0/bracketrightbig
ET II 85(19)
2./integraldisplay∞
0Mk,1
2ν(−iax)M−k,1
2ν(−iax)Jν(xy)dx
=ae−1
2(ν+1)πi[Γ(1 + ν)]2
Γ/parenleftbig1
2+k+1
2ν/parenrightbig
Γ/parenleftbig1
2−k+1
2ν/parenrightbigy−1−2k
×/parenleftbig
a2−y2/parenrightbig−1
2/braceleftbigg/bracketleftBig
a+/parenleftbig
a2−y2/parenrightbig1
2/bracketrightBig2k
+/bracketleftBig
a−/parenleftbig
a2−y2/parenrightbig1
2/bracketrightBig2k/bracerightbigg
[0<y<a ];
=0 [ a<y< ∞]
/bracketleftbig
a>0,Reν>−1,|Rek|<1
4/bracketrightbig
ET II 85(18)
7.661 Confluent hypergeometric functions, Bessel functions, and powers 831
7.652/integraldisplay∞
0M−μ,1
2ν/braceleftBig
a/bracketleftBig/parenleftbig
b2+x2/parenrightbig1
2−b/bracketrightBig/bracerightBig
Wμ,1
2ν/braceleftBig
a/bracketleftBig/parenleftbig
b2+x2/parenrightbig1
2+b/bracketrightBig/bracerightBig
Jν(xy)dx
=ay−2μ−1Γ(1 + ν)/bracketleftBig/parenleftbig
a2+y2/parenrightbig1
2+a/bracketrightBig2μ
Γ/parenleftbig1
2+1
2ν−μ/parenrightbig
(A2+Y2)1
2exp/bracketleftBig
−b/parenleftbig
a2+y2/parenrightbig1
2/bracketrightBig
/bracketleftbig
y>0,Reν>−1,Reμ<1
4,Rea>0,Reb>0/bracketrightbig
ET II 87(29)
7.66 Combinations of confluent hypergeometric functions, Bessel functions, and
powers
7.661
1./integraldisplay∞
0x−1Wk,μ(ax)M−k,μ(ax)J0(xy)dx
=e−ikπΓ(1 + 2 μ)
Γ/parenleftbig1
2+μ+k/parenrightbigPk
μ−1
2/bracketleftBigg/parenleftbigg
1+y2
a2/parenrightbigg1
2/bracketrightBigg
Qk
μ−1
2/bracketleftBigg/parenleftbigg
1+y2
a2/parenrightbigg1
2/bracketrightBigg
/bracketleftbig
y>0,Rea>0,Reμ>−1
2,Rek<3
4/bracketrightbig
ET II 18(44)
2./integraldisplay∞
0x−1Wk,μ(ax)W−k,μ(ax)J0(xy)dx=1
2πcos(μπ)Pk
μ−1
2/bracketleftBigg/parenleftbigg
1+y2
a2/parenrightbigg1
2/bracketrightBigg
P−k
μ−1
2/bracketleftBigg/parenleftbigg
1+y2
a2/parenrightbigg1
2/bracketrightBigg
/bracketleftbig
y>0,Rea>0,|Reμ|<1
2/bracketrightbig
ET II 18(45)
3./integraldisplay∞
0x2μ−νWk,μ(ax)M−k,μ(ax)Jν(xy)dx
=22μ−ν+2ka2kyν−2μ−2k−1Γ(2μ+1 )
Γ/parenleftbig
ν−k−μ+1
2/parenrightbig
×3F2/parenleftbigg1
2−k,1−k,1
2−k+μ;1−2k,1
2−k−μ+ν;−y2
a2/parenrightbigg
/bracketleftbig
y>0,Reμ>−1
2,Rea>0,Re(2μ+2k−ν)<1
2/bracketrightbig
ET II 85(20)
4./integraldisplay∞
0x2ρ−νWk,μ(iax)Wk,μ(−iax)Jν(xy)dx
=22ρ−νyν−2ρ−1π−1
2/bracketleftbigg
Γ/parenleftbigg1
2−k+μ/parenrightbigg
Γ/parenleftbigg1
2−k−μ/parenrightbigg/bracketrightbigg−1
G24
44/parenleftBigg
y2
a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2,0,1
2−μ,1
2+μ
ρ+1
2,−k,k,ρ −ν+1
2/parenrightBigg
/bracketleftbig
y>0,Rea>0,Reρ>|Reμ|−1,Re(2ρ+2k−ν)<1
2/bracketrightbig
ET II 86(23)a
5./integraldisplay∞
0x2ρ−νWk,μ(ax)M−k,μ(ax)Jν(xy)dx
=22ρ−νΓ(2μ+1 )
π1
2Γ/parenleftbig1
2−k+μ/parenrightbigyν−2ρ−1G23
44/parenleftBigg
y2
a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2,0,1
2−μ,1
2+μ
ρ+1
2,−k,k,ρ −ν+1
2/parenrightBigg
/bracketleftbig
y>0,Rea>0,Reρ>−1,Re(ρ+μ)>−1,Re(2e+2k+ν)<1
2/bracketrightbig
ET II 86(21)a
832 Confluent Hypergeometric Functions 7.662
6./integraldisplay∞
0x2ρ−νWk,μ(ax)W−k,μ(ax)Jν(xy)dx
=Γ(ρ+1+ μ)Γ(ρ+1−μ)Γ(2ρ+2 )
Γ/parenleftbig3
2+k+ρ/parenrightbig
Γ/parenleftbig3
2−k+ρ/parenrightbig
Γ(1 + ν)yν2−ν−1a−2ρ−1
×4F3/parenleftbigg
ρ+1,ρ+3
2,ρ+1+ μ, ρ+1−μ;3
2+k+ρ,3
2−k+ρ,1+ν;−y2
a2/parenrightbigg
[y>0,Reρ>|Reμ|−1,Rea>0]ET II 86(22)a
7.662
1./integraldisplay∞
0x−1M−μ,1
4ν/parenleftbigg1
2x2/parenrightbigg
Wμ,1
4ν/parenleftbigg1
2x2/parenrightbigg
Jν(xy)dx=Γ/parenleftbig
1+1
2ν/parenrightbig
Γ/parenleftbig1
2+1
4ν−μ/parenrightbigI1
4ν−μ/parenleftbigg1
4y2/parenrightbigg
K1
4ν+μ/parenleftbigg1
4y2/parenrightbigg
[y>0,Reν>−1] ET II 86(24)
2./integraldisplay∞
0x−1Mα−β,1
4ν−γ/parenleftbigg1
2x2/parenrightbigg
Wα+β,1
4ν+γ/parenleftbigg1
2x2/parenrightbigg
Jν(xy)dx
=Γ/parenleftbig
1+1
2ν−2γ/parenrightbig
Γ/parenleftbig
1+1
2ν−2β/parenrightbigy−2Mα−γ,1
4ν−β/parenleftbigg1
2y2/parenrightbigg
Wα+γ,1
4ν+β/parenleftbigg1
2y2/parenrightbigg
/bracketleftbig
y>0,Reβ<1
8,Reν>−1,Re(ν−4γ)>−2/bracketrightbig
ET II 86(25)
3./integraldisplay∞
0x−1Mk,0/parenleftbig
iax2/parenrightbig
Mk,0/parenleftbig
−iax2/parenrightbig
K0(xy)dx=π
16/braceleftBigg/bracketleftbigg
Jk/parenleftbiggy2
8a/parenrightbigg/bracketrightbigg2
+/bracketleftbigg
Yk/parenleftbiggy2
8a/parenrightbigg/bracketrightbigg2/bracerightBigg
[a>0] ET II 152(83)
4./integraldisplay∞
0x−1Mk,μ/parenleftbig
iax2/parenrightbig
Mk,μ/parenleftbig
−iax2/parenrightbig
K0(xy)dx=ay−2[Γ(2μ+1 ) ]2W−μ,k/parenleftbiggiy2
4a/parenrightbigg
W−μ,k/parenleftbigg
−iy2
4a/parenrightbigg
/bracketleftbig
a>0,Rey>0,Reμ>−1
2/bracketrightbig
ET II 152(84)
7.663
1./integraldisplay∞
0x2ρ
1F1/parenleftbig
a;b;−λx2/parenrightbig
Jν(xy)dx=22ρΓ(b)
Γ(a)y2ρ+1G21
23/parenleftbiggy2
4λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle1,b
1
2+ρ+1
2ν,a,1
2+ρ−1
2ν/parenrightbigg
/bracketleftbig
y>0,−1−Reν<2R eρ<1
2+2R e a,Reλ>0/bracketrightbig
ET II 88(6)
2./integraldisplay∞
0xν+1
1F1/parenleftbigg
2a−ν;a+1 ;−1
2x2/parenrightbigg
Jν(xy)dx=2ν−a+1
2Γ(a+1 )
π1
2Γ(2a−ν)y2a−ν−1e−1
4y2Ka−ν−1
2/parenleftbigg1
4y2/parenrightbigg
/bracketleftbig
y>0,Reν>−1,Re(4a−3ν)>1
2/bracketrightbig
ET II 87(1)
3./integraldisplay∞
0xa
1F1/parenleftbigg
a;1+a+ν
2;−1
2x2/parenrightbigg
Jν(xy)dx=ya−1
1F1/parenleftbigg
a;1+a+ν
2;−y2
2/parenrightbigg
/bracketleftbig
y>0,Rea>−1
2,Re(a+ν)>−1/bracketrightbig
ET II 87(2)
7.664 Confluent hypergeometric functions, Bessel functions, and powers 833
4./integraldisplay∞
0xν+1−2a
1F1/parenleftbigg
a;1+ν−a;−1
2x2/parenrightbigg
Jν(xy)dx
=π1
2Γ(1 + ν−a)
Γ(a)2−2a+ν+1
2y2a−ν−1e−1
4y2Ia−1
2/parenleftbigg1
4y2/parenrightbigg
/bracketleftbig
y>0,Rea−1<Reν<4R ea−1
2/bracketrightbig
ET II 87(3)
5./integraldisplay∞
0x1F1/parenleftbig
λ;1;−x2/parenrightbig
J0(xy)dx=/bracketleftbig
22λ−1Γ(λ)/bracketrightbig−1y2λ−2e−1
4y2
[y>0,Reλ>0] ET II 18(46)
6./integraldisplay∞
0xν+1
1F1/parenleftbig
a;b;−λx2/parenrightbig
Jν(xy)dx
=21−aΓ(b)
Γ(a)λ1
2a+1
2νya−2e−y2
8λWk,μ/parenleftbiggy2
4λ/parenrightbigg
,2k=a−2b+ν+2,2μ=a−ν−1
/bracketleftbig
y>0,−1<Reν<2R ea−1
2,Reλ>0/bracketrightbig
ET II 88(4)
7./integraldisplay∞
0x2b−ν−1
1F1/parenleftbig
a;b;−λx2/parenrightbig
Jν(xy)dx=22b−2a−ν−1Γ(b)
Γ(a−b+ν+1 )λ−ay2a−2b+ν
×1F1/parenleftbigg
a;1+a−b+ν;−y2
4λ/parenrightbigg
/bracketleftbig
y>0,0<Reb<3
4+R e/parenleftbig
a+1
2ν/parenrightbig
,Reλ>0/bracketrightbig
ET II 88(5)
7.664
1./integraldisplay∞
0xW1
2ν,μ/parenleftBiga
x/parenrightBig
W−1
2ν,μ/parenleftBiga
x/parenrightBig
Kν(xy)dx=2ay−1K2μ/bracketleftBig
(2ay)1
2e1
4iπ/bracketrightBig
K2μ/bracketleftBig
(2ay)1
2e−1
4iπ/bracketrightBig
[Rey>0,Rea>0] ET II 152(85)
2./integraldisplay∞
0xW1
2ν,μ/parenleftbigg2
x/parenrightbigg
W−1
2ν,μ/parenleftbigg2
x/parenrightbigg
Jν(xy)dx
=−4y−1/braceleftBig
sin/bracketleftbig/parenleftbig
μ−1
2ν/parenrightbig
π/bracketrightbig
J2μ/parenleftBig
2y1
2/parenrightBig
+c o s/bracketleftbig/parenleftbig
μ−1
2ν/parenrightbig
π/bracketrightbig
Y2μ/parenleftBig
2y1
2/parenrightBig/bracerightBig
K2μ/parenleftBig
2y1
2/parenrightBig
[y>0,Re (ν±2μ)>−1]ET II 87(27)
3./integraldisplay∞
0xW1
2ν,μ/parenleftbigg2
x/parenrightbigg
W−1
2ν,μ/parenleftbigg2
x/parenrightbigg
Yν(xy)dx
=4y−1/braceleftBig/braceleftBig
cos/bracketleftbig/parenleftbig
μ−1
2ν/parenrightbig
π/bracketrightbig
J2μ/parenleftBig
2y1
2/parenrightBig
−sin/bracketleftbig/parenleftbig
μ−1
2ν/parenrightbig
π/bracketrightbig
Y2μ/parenleftBig
2y1
2/parenrightBig/bracerightBig
K2μ/parenleftBig
2y1
2/parenrightBig/bracerightBig
/bracketleftbig
y>0,|Reμ|<1
4/bracketrightbig
ET II 117(48)
4./integraldisplay∞
0xW−1
2ν,μ/parenleftbigg2
x/parenrightbigg
M1
2ν,μ/parenleftbigg2
x/parenrightbigg
Jν(xy)dx=4Γ ( 1+2 μ)y−1
Γ/parenleftbig1
2+1
2ν+μ/parenrightbigJ2μ/parenleftBig
2y1
2/parenrightBig
K2μ/parenleftBig
2y1
2/parenrightBig
/bracketleftbig
y>0,Reν>−1,Reμ>−1
4/bracketrightbig
ET II 86(26)
834 Confluent Hypergeometric Functions 7.665
5./integraldisplay∞
0xW−1
2ν,μ/parenleftbiggia
x/parenrightbigg
W−1
2ν,μ/parenleftbigg
−ia
x/parenrightbigg
Jν(xy)dx
=4ay−1/bracketleftbig
Γ/parenleftbig1
2+μ+1
2ν/parenrightbig
Γ/parenleftbig1
2−μ+1
2ν/parenrightbig/bracketrightbig−1Kμ/bracketleftBig
(2iay)1
2/bracketrightBig
Kμ/bracketleftBig
(−2iay)1
2/bracketrightBig
/bracketleftbig
y>0,Rea>0,|Reμ|<1
2,Reν>−1/bracketrightbig
ET II 87(28)
7.665
1./integraldisplay∞
0x−1
2Jν/parenleftBig
ax1
2/parenrightBig
K1
2ν−μ/parenleftbigg1
2x/parenrightbigg
Mk,μ(x)dx
=Γ(2μ+1 )
aΓ/parenleftbig
k+1
2ν+1/parenrightbigW1
2(k−μ),1
2k−1
4ν/parenleftbigga2
2/parenrightbigg
M1
2(k+μ),1
2k+1
4ν/parenleftbigga2
2/parenrightbigg
/bracketleftbig
a>0,Rek>−1
4,Reμ>−1
2,Reν>−1/bracketrightbig
ET II 405(18)
2./integraldisplay∞
0x1
2c+1
2c/prime−1Ψ(a,c;x)1F1(a/prime;c/prime;−x)Jc+c/prime−2/bracketleftBig
2(xy)1
2/bracketrightBig
dx
=Γ(c/prime)
Γ(a+a/prime)y1
2c+1
2c/prime−1Ψ(c/prime−a/prime,c+c/prime−a−a/prime;y)1F1(a/prime;a+a/prime;−y)
/bracketleftbig
Rec/prime>0,1<Re (c+c/prime)<2R e(a+a/prime)+1
2/bracketrightbig
EH I 287(23)
7.666/integraldisplay∞
0x1
2c−1
21F1/parenleftBig
a;c;−2x1
2/parenrightBig
Ψ/parenleftBig
a,c;2x1
2/parenrightBig
Jc−1/bracketleftBig
2(xy)1
2/bracketrightBig
dx
=2−cΓ(c)
Γ(a)ya−1
2c−1
2/bracketleftBig
1+( 1+ y)1
2/bracketrightBigc−2a
(1 +y)−1
2
/bracketleftbig
Rec>2,Re(c−2a)<1
2/bracketrightbig
EH I 285(13)
7.67 Combinations of confluent hypergeometric functions, Bessel functions, expo-
nentials, and powers
7.671
1./integraldisplay∞
0xk−3
2exp/bracketleftbigg
−1
2(a+1 )x/bracketrightbigg
Kν/parenleftbigg1
2ax/parenrightbigg
Mk,ν(x)dx
=π1
2Γ(k)Γ(k+2ν)
ak+νΓ/parenleftbig
k+ν+1
2/parenrightbig2F1/parenleftbig
k,k+2ν;2ν+1 ;−a−1/parenrightbig
[Rea>0,Rek>0,Re(k+2ν)>0]ET II 405(17)
2./integraldisplay∞
0x−k−3
2exp/bracketleftbigg
−1
2(a−1)x/bracketrightbigg
Kμ/parenleftbigg1
2ax/parenrightbigg
Wk,μ(x)dx
=πΓ(−k)Γ(2μ−k)Γ(−2μ−k)
Γ/parenleftbig1
2−k/parenrightbig
Γ/parenleftbig1
2+μ−k/parenrightbig
Γ/parenleftbig1
2−μ−k/parenrightbig22k+1ak−ν
2F1/parenleftbig
−k,2μ−k;−2k;1−a−1/parenrightbig
[Rea>0,Rek<2R eμ<−Rek]ET II 408(36)
7.672 Confluent hypergeometric functions, Bessel functions, exponentials, and powers 835
7.672
1./integraldisplay∞
0x2ρe−1
2ax2Mk,μ/parenleftbig
ax2/parenrightbig
Jν(xy)dx
=Γ(2μ+1 )
Γ/parenleftbig
μ+k+1
2/parenrightbig22ρy−2ρ−1G21
23/parenleftBigg
y2
4a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2−μ,1
2+μ
1
2+ρ+1
2ν,k,1
2+ρ−1
2ν/parenrightBigg
/bracketleftbig
y>0,−1−Re/parenleftbig1
2ν+μ/parenrightbig
<Reρ<Rek−1
4,Rea>0/bracketrightbig
ET II 83(10)
2./integraldisplay∞
0x2ρe−1
2ax2Wk,μ/parenleftbig
ax2/parenrightbig
Jν(xy)dx
=Γ/parenleftbig
1+μ+1
2ν+ρ/parenrightbig
Γ/parenleftbig
1−μ+1
2ν+ρ/parenrightbig
2−ν−1
Γ(ν+1 )Γ/parenleftbig3
2−k+1
2ν+ρ/parenrightbig a−1
2ν−ρ−1
2yν
×2F2/parenleftbigg
λ+μ, λ−μ;ν+1,1
2−k+λ;−y2
4a/parenrightbigg
,
λ=1+1
2ν+ρ/bracketleftbig
y>0,Rea>0,Re/parenleftbig
ρ±μ+1
2ν/parenrightbig
>−1/bracketrightbig
ET II 85(16)
3./integraldisplay∞
0x2ρe1
2ax2Wk,μ/parenleftbig
ax2/parenrightbig
Jν(xy)dx=22ρy−2ρ−1
Γ/parenleftbig1
2+μ−k/parenrightbig
Γ/parenleftbig1
2−μ−k/parenrightbig
×G22
23/parenleftBigg
y2
4a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2−μ,1
2+μ
1
2+ρ+1
2ν,−k,1
2+ρ−1
2ν/parenrightBigg
/bracketleftbig
y>0,|arga|<π , −1−Re/parenleftbig1
2ν±μ/parenrightbig
<Reρ<−1
4−Rek/bracketrightbig
ET II 85(17)
4./integraldisplay∞
0x2λ+1
2e−1
4x2Mk,μ/parenleftbigg1
2x2/parenrightbigg
Yν(xy)dx=2λy−1/2Γ(2μ+1 )
Γ/parenleftbig1
2+k+μ/parenrightbigG31
34/parenleftbiggy2
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle−μ−λ, μ −λ,
h, κ, −λ−
1
2,l/parenrightbigg
h=1
4+1
2ν, κ =1
4−1
2ν, l =−1
4−1
2ν/bracketleftbig
y>0,Re(k−λ)>0,Re (2λ+2μ±ν)>−5
2/bracketrightbig
ET II 116(45)
5./integraldisplay∞
0x2λ+1
2e1
4x2Wk,μ/parenleftbigg1
2x2/parenrightbigg
Yν(xy)dx
=2λ/bracketleftbigg
Γ/parenleftbigg1
2−k+μ/parenrightbigg
Γ/parenleftbigg1
2−k−μ/parenrightbigg/bracketrightbigg−1
G32
34/parenleftbiggy2
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle−μ−λ, μ −λ, l
h, κ, −1
2−k−λ, l/parenrightbigg
y−1/2,
h=1
4+1
2ν, κ =1
4−1
2ν, l =−1
4−1
2ν/bracketleftbig
y>0,Re(k+λ)<0,Re (2λ±2μ±ν)>−5
2/bracketrightbig
ET II 117(47)
6./integraldisplay∞
0x−1/2e−1
2x2M1
2ν−1
4,1
2ν+1
4/parenleftbig
x2/parenrightbig
Jν(xy)dx=( 2ν+1 ) 2−νyν−1/bracketleftbigg
1−Φ/parenleftbigg1
2y/parenrightbigg/bracketrightbigg
/bracketleftbig
y>0,Reν>−1
2/bracketrightbig
ET II 82(1)
7./integraldisplay∞
0x−1e−1
2x2M1
2ν+1
2,1
2ν+1
2/parenleftbig
x2/parenrightbig
Jν(xy)dx=Γ(ν+2 )yν
Γ/parenleftbig
ν+3
2/parenrightbig
2ν/bracketleftbigg
1−Φ/parenleftbigg1
2y/parenrightbigg/bracketrightbigg
[y>0,Reν>−1] ET II 82(2)
836 Confluent Hypergeometric Functions 7.672
8./integraldisplay∞
0e−1
4x2Mk,1
2ν/parenleftbigg1
2/parenrightbigg
x2Jν(xy)dx=2−kΓ(ν+1 )
Γ/parenleftbig
k+1
2ν+1
2/parenrightbigy2k−1e−1
2y2
/bracketleftbig
y>0,Reν>−1,Rek<1
2/bracketrightbig
ET II 83(7)
9./integraldisplay∞
0xν−2μe−1
4x2Mk,μ/parenleftbigg1
2/parenrightbigg
x2Jν(xy)dx
=21
2(1
2−k−3μ+ν)Γ(2μ+1 )
Γ/parenleftbig
μ+k+1
2/parenrightbigyk+μ−3
2e−1
4y2Wα,β/parenleftbigg1
2y2/parenrightbigg
,
2α=k−3μ+ν+1
2,2β=k+μ−ν−1
2 /bracketleftbig
y>0,−1<Reν<2R e(k+μ)−1
2/bracketrightbig
ET II 83(9)
10./integraldisplay∞
0xν−2μe1
4x2Wk,±μ/parenleftbigg1
2x2/parenrightbigg
Jν(xy)dx=Γ(1 + ν−2μ)
Γ(1 + 2 β)2β−μyk+μ−3
2e−1
4y2Mα,β/parenleftbigg1
2y2/parenrightbigg
2α=1
2+k+ν−3μ,2β=1
2−k+ν−μ
[y>0,Reν>−1,Re(ν−2μ)>−1]
ET II 84(14)
11./integraldisplay∞
0xν−2μe−1
4x2Wk,±μ/parenleftbigg1
2x2/parenrightbigg
Jν(xy)dx
=Γ(1 + ν−2μ)
Γ/parenleftbig1
2+μ−k/parenrightbig21
2(1
2+k−3μ+ν)yμ−k−3
2e1
4y2Wα,β/parenleftbigg1
2y2/parenrightbigg
,
2α=k+3μ−ν−1
2,2β=k−μ+ν+1
2 /bracketleftbig
y>0,Reν>−1,Re(ν−2μ)>−1,Re/parenleftbig
k−μ+1
2ν/parenrightbig
<−1
4/bracketrightbig
ET II 84(15)
12./integraldisplay∞
0x2μ−νe−1
4x2Mk,μ/parenleftbigg1
2x2/parenrightbigg
Jν(xy)dx
=Γ(2μ+1 )
Γ/parenleftbig1
2+k−μ+ν/parenrightbig21
2(1
2−k+3μ−ν)yk−μ−3
2e−1
4y2Mα,β/parenleftbigg1
2y2/parenrightbigg
2α=1
2+k+3μ−ν, 2β=−1
2+k−μ+ν/bracketleftbig
y>0,−1
2<Reμ<Re/parenleftbig
k+1
2ν/parenrightbig
−1
4/bracketrightbig
ET II 83(8)
13./integraldisplay∞
0x2μ−νe−1
4x2Mk,μ/parenleftbigg1
2x2/parenrightbigg
Yν(xy)dx
=π−12μ+βyk−μ−3
2e−1
4y2Γ( 2μ+1 )
×Γ/parenleftbig1
2−k−μ/parenrightbig/braceleftbigg
cos[(ν−2μ)π]Γ(2μ−ν−1)
Γ(2β+1 )Mα,β/parenleftbig1
2y2/parenrightbig
−sin[(ν+k−μ)π]Wα,β/parenleftbig1
2y2/parenrightbig/bracerightbigg
2α=3μ−ν+k+1
2,2β=μ−ν−k+1
2 /bracketleftbig
y>0,−1<2R eμ<Re(2k+ν)+1
2,Re(2μ−ν)>−1/bracketrightbig
ET II 116(44)
7.673 Confluent hypergeometric functions, Bessel functions, exponentials, and powers 837
14./integraldisplay∞
0x2μ+νe−1
4x2Mk,μ/parenleftbigg1
2x2/parenrightbigg
Yν(xy)dx
=π−12μ+βyk−μ−3
2Γ(2μ+1 )
×Γ/parenleftbig1
2−μ−k/parenrightbig
e−1
4y2/braceleftBigg
cos(2μπ)Γ(2μ+ν+1 )
Γ/parenleftbig
μ+ν−k+3
2/parenrightbigMα,β/parenleftbig1
2y2/parenrightbig
+ sin[( μ−k)π]Wα,β/parenleftbig1
2y2/parenrightbig/bracerightbigg
2α=3μ+ν+k+1
2,2β=μ+ν−k+1
2 /bracketleftbig
y>0,−1<2R eμ<Re(2k−ν)+1
2,Re(2μ+ν)>−1/bracketrightbig
ET II 116(43)
15./integraldisplay∞
0x2μ+νe−1
2ax2Mk,μ/parenleftbig
ax2/parenrightbig
Kν(xy)dx=2μ−k−1
2a1
4−1
2(μ+ν+k)yk−μ−3
2
×Γ(2μ+1 )Γ ( 2 μ+ν+1 )e x p/parenleftbiggy2
8a/parenrightbigg
Wκ,m/parenleftbiggy2
4a/parenrightbigg
,
2κ=−3μ−ν−k−1
2,2m=μ+ν−k+1
2 /bracketleftbig
Rey>0,Rea>0,Reμ>−1
2,Re(2μ+ν)>−1/bracketrightbig
ET II 152(82)
7.673
1.10/integraldisplay∞
0e−1
2axx1
2(μ−ν−1)Mκ,1
2μ(ax)Jν/parenleftBig
2√
bx/parenrightBig
dx
=/parenleftbiggb
a/parenrightbiggκ−1
2−1+μ
4
a−1
2(μ+1−ν)Γ(1 + μ)e−b
2a1
Γ/parenleftbigg
1+κ+ν
2−1+μ
4/parenrightbigg
×M1
2(κ−ν−1)+3
4(1+μ),κ+ν
2−1+μ
4/parenleftbiggb
a/parenrightbigg
/bracketleftbigg
Re(1 + μ)>0,Re/parenleftbigg
κ+ν−μ
2/parenrightbigg
>−3
4,Imb=0/bracketrightbigg
BU 128(12)a
2./integraldisplay∞
0e1
2axx1
2(ν−1∓μ)Wκ,1
2μ(ax)Jν/parenleftBig
2√
bx/parenrightBig
dx=a−1
2(ν+1∓μ)Γ(ν+1∓μ)eb
2a
Γ/parenleftbig1±μ
2−κ/parenrightbig/parenleftBiga
b/parenrightBig1
2(κ+1)+1
4(1∓ν)
×W1
2(κ+1−ν)−3
4(1∓μ),1
2(κ+ν)+1
4(1∓μ)/parenleftbiggb
a/parenrightbigg
/bracketleftbigg
Re/parenleftbiggν∓μ
2+κ/parenrightbigg
<3
4,Reν>−1/bracketrightbigg
BU 128(13)
838 Confluent Hypergeometric Functions 7.674
7.674
1./integraldisplay∞
0xρ−1e−1
2κJλ+ν/parenleftBig
ax1/2/parenrightBig
Jλ−ν/parenleftBig
ax1/2/parenrightBig
Wk,μ(x)dx
=/parenleftbig1
2a/parenrightbig2λΓ/parenleftbig1
2+λ+μ+ρ/parenrightbig
Γ/parenleftbig1
2+λ−μ+ρ/parenrightbig
Γ( 1+ λ+ν)Γ ( 1+ λ−ν)Γ(1+ λ−k+ρ)
×4F4/parenleftbigg
1+λ,1
2+λ,1
2+λ+μ+ρ,1
2+λ−μ+ρ;1+λ+ν,
1+λ−ν,1+2λ,1+λ−k+ρ;−a2/parenrightbigg
/bracketleftbig
|Reμ|<Re(λ+ρ)+1
2/bracketrightbig
ET II 409(37)
2./integraldisplay∞
0xρ−1e−1
2κIλ+ν/parenleftBig
ax1/2/parenrightBig
Kλ−ν/parenleftBig
ax1/2/parenrightBig
Wk,μ(x)dx
=π−1/2
2G24
45/parenleftbigg
a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,
1
2,1
2+μ−ρ,1
2−μ−ρ
λ,ν,−λ,−ν,k−ρ/parenrightbigg
/bracketleftbig
|Reμ|<Re(λ+ρ)+1
2,|Reμ|<Re(ν+ρ)+1
2/bracketrightbig
ET II 409(38)
Combinations of Struve functions and confluent hypergeometric functions
7.675
1./integraldisplay∞
0x2λ+1
2e−1
4x2Mk,μ/parenleftbigg1
2x2/parenrightbigg
Hν(xy)dx=2−λΓ(2μ+1 )
y1/2Γ/parenleftbig1
2+k+μ/parenrightbigG22
34/parenleftbiggy2
2/vextendsingle/vextendsingle/vextendsingle/vextendsinglel,−μ−λ, mu −λ
l,k−λ−
1
2,h ,κ/parenrightbigg
h=1
4+1
2ν, κ =1
4−1
2ν, l =3
4+1
2ν/bracketleftbig
Re(2λ+2μ+ν)>−7
2,Re(k−λ)>0,y > 0,Re(2λ−2k+ν)<−1
2/bracketrightbig
ET II 171(42)
2./integraldisplay∞
0x2λ+1
2e−1
4x2Wk,μ/parenleftbigg1
2x2/parenrightbigg
Hν(xy)dx
=21
4−λ−1
2νπ−1/2yν+1Γ/parenleftbig7
4+1
2ν+λ+μ/parenrightbig
Γ/parenleftbig7
4+1
2ν+λ−μ/parenrightbig
Γ/parenleftbig
ν+3
2/parenrightbig
Γ/parenleftbig9
4+λ−k−1
2ν/parenrightbig
×3F3/parenleftbigg
1,7
4+ν
2+λ+μ,7
4+ν
2+λ−μ;3
2,ν+3
2,9
4+λ−k+ν
2;−y2
2/parenrightbigg
/bracketleftbig
Re(2λ+ν)>2|Reμ|−7
4,y > 0/bracketrightbig
ET II 171(43)
3./integraldisplay∞
0x2λ+1
2e1
4x2Wk,μ/parenleftbigg1
2x2/parenrightbigg
Hν(xy)dx
=/bracketleftbigg
2λΓ/parenleftbigg1
2−k+μ/parenrightbigg
Γ/parenleftbigg1
2−k−μ/parenrightbigg/bracketrightbigg−1
y−1/2G23
34/parenleftbiggy2
2/vextendsingle/vextendsingle/vextendsingle/vextendsinglel,−μ−λ,μ−λ
l,−k−λ−1
2,h ,κ/parenrightbigg
h=1
4+1
2ν, κ =1
4−1
2ν, l =3
4+1
2ν/bracketleftbig
y>0,Re(2λ+ν)>2|Reμ|−7
2,Re (2k+2λ+ν)<−1
2,Re(k+λ)<0/bracketrightbig
ET II 172(46)a
4./integraldisplay∞
0e1
2x2W−1
2ν−1
2,1
2ν/parenleftbig
x2/parenrightbig
Hν(xy)dx=2−ν−1yνπe1
4y2/bracketleftBig
1−Φ/parenleftBigy
2/parenrightBig/bracketrightBig
[y>0,Reν>−1] ET II 171(44)
7.681 Confluent hypergeometric functions and other special functions 839
7.68 Combinations of confluent hypergeometric functions and other special functions
Combinations of confluent hypergeometric functions and associated Legendre functions
7.681
1./integraldisplay∞
0x−1/2(a+x)μe−1
2xP−2μ
ν/parenleftBig
1+2x
a/parenrightBig
Mk,μ(x)dx
=−sin(νπ)
πΓ(k)Γ(2μ+1 )Γ/parenleftbig
k−μ+ν+1
2/parenrightbig
Γ/parenleftbig
k−μ−ν−1
2/parenrightbig
e1
2aWρ,σ(a),
ρ=1
2−k+μ, σ =1
2+ν/bracketleftbig
|arga|<π , Reμ>−1
2,Re(k−μ)>/vextendsingle/vextendsingleReν+1
2/vextendsingle/vextendsingle/bracketrightbig
ET II 403(11)
2./integraldisplay∞
0x−1/2(a+x)−μe−1
2xP−2μ
ν/parenleftBig
1+2x
a/parenrightBig
Mk,μ(x)dx
=Γ(2μ+1 )Γ/parenleftbig
k+μ+ν+1
2/parenrightbig
Γ/parenleftbig
k+μ−ν−1
2/parenrightbig
e1
2a
Γ/parenleftbig
k+μ+1
2/parenrightbig
Γ(2μ+ν+1 )Γ ( 2 μ−ν)W1
2−k−μ,1
2+ν(a)
/bracketleftbig
|arga|<π , Reμ>−1
2,Re(k+μ)>/vextendsingle/vextendsingleReν+1
2/vextendsingle/vextendsingle/bracketrightbig
ET II 403(12)
3./integraldisplay∞
0x−1
2−1
2μ−ν(a+x)1
2μe−1
2xPμ
k+ν−3
2/parenleftBig
1+2x
a/parenrightBig
Wk,ν(x)dx
=Γ(1−μ−2ν)
Γ/parenleftbig3
2−k−μ−ν/parenrightbiga−1
4+1
2k−1
2νe1
2aWρ,σ(a)
2ρ=1
2+2μ+ν−k,2σ=k+3ν−3
2
[|arga|<π , Reμ<1,Re(μ+2ν)<1]
ET II 407(32)
4./integraldisplay∞
0x−1
2−1
2μ−ν(a+x)−1
2μe−1
2xPμ
k+μ+ν−3
2/parenleftBig
1+2x
a/parenrightBig
Wk,ν(x)dx
=Γ(1−μ−2ν)
Γ/parenleftbig3
2−k−μ−ν/parenrightbiga−1
2+1
2k−1
2νe1
2aWρ,σ(a)
2ρ=1
2−k+ν, 2σ=k+2μ+3ν−3
2
[|arga|<π , Reμ<1,Re(μ+2ν)<1]
ET II 408(33)
5./integraldisplay∞
0xμ−1
4k−1
2ν−1
2(a+x)1
2νe−1
2xQν
μ−k+3
2/parenleftBig
1+2x
a/parenrightBig
Mk,ν(x)dx
=eνπiΓ(1 + 2 μ−ν)Γ(1+2 μ)Γ/parenleftbig5
2−k+μ+ν/parenrightbig
2Γ/parenleftbig1
2+k+μ/parenrightbig a1
4(κ+2μ−2ν+5)e1
2aWρ,σ(a)
2ρ=1
2−k−μ+2ν, 2σ=k−3μ−3
2/bracketleftbig
|arga|<π , Reμ>−1
2,Re(2μ−ν)>−1/bracketrightbig
ET II 404(14)
840 Confluent Hypergeometric Functions 7.682
7.682
1./integraldisplay∞
0x−1/2e−1
2xP−2μ
ν/bracketleftbigg/parenleftBig
1+x
a/parenrightBig1/2/bracketrightbigg
Mk,μ(x)dx
=Γ(2μ+1 )Γ/parenleftbig
k+1
2ν/parenrightbig
Γ/parenleftbig
k−1
2ν−1
2/parenrightbig
e1
2a
22μa1/4Γ/parenleftbig
k+μ+1
2/parenrightbig
Γ/parenleftbig
μ+1
2ν+1
2/parenrightbig
Γ/parenleftbig
μ−1
2ν/parenrightbigW3
4−k,1
4+1
2ν(a)
/bracketleftbig
|arga|<π , Rek>1
2Reν−1
2,Rek>−1
2Reν/bracketrightbig
ET II 404(13)
2./integraldisplay∞
0x1
2(k+μ+ν)−1(a+x)−1/2e−1
2xQ1−k+μ−ν
k−μ−ν−1/bracketleftbigg/parenleftBig
1+x
a/parenrightBig1/2/bracketrightbigg
Mk,μ(x)dx
=e(1−k+μ−ν)πi2μ−k−νa1
2(k+μ−1)Γ/parenleftbig1
2−ν/parenrightbig
Γ(1 + 2 μ)Γ(k+μ+ν)
Γ/parenleftbig
k+μ+1
2/parenrightbig e1
2aWρ,σ(a),
ρ=1
2−k−1
2ν, σ =μ+1
2ν/bracketleftbig
|arga|<π , Reμ>−1
2,Re(k+μ+ν)>0/bracketrightbig
ET II 404(15)
3./integraldisplay∞
0xν−1
2e−1
2xQ2μ−2ν
2k−2ν−3/bracketleftbigg/parenleftBig
1+x
a/parenrightBig1/2/bracketrightbigg
Mk,μ(x)dx
=e2(μ−ν)πi22μ−2ν−1a1
2(k+μ−1)e1
2aΓ(2μ+1 )Γ ( ν+1 )Γ/parenleftbig
k+μ−2ν−1
2/parenrightbig
Γ/parenleftbig
k+μ+1
2/parenrightbig Wρ,σ(a),
2ρ=1−k+μ−2ν, 2σ=k−μ−2ν−2/bracketleftbig
|arga|<π , Reμ>−1
2,Reν>−1,Re(k+μ−2ν)>1
2/bracketrightbig
ET II 404(16)
4./integraldisplay∞
0x−1
2−1
2μ−νe−1
2xPμ
2k+μ+2ν−3/bracketleftbigg/parenleftBig
1+x
a/parenrightBig1
2/bracketrightbigg
Wk,ν(x)dx
=2μΓ(1−μ−2ν)
Γ/parenleftbig3
2−k−μ−ν/parenrightbiga−1
2+1
2k−1
2νe1
2aWρ,σ(a),
2ρ=1−k+μ+ν, 2σ=k+μ+3ν−2
[|arga|<π , Reμ<1,Re(μ+2ν)<1]
ET II 408(34)
5.8/integraldisplay∞
0x−1
2−1
2μ−ν(a+x)−1/2e−1
2xPμ
2k+μ+2ν−2/bracketleftbigg/parenleftBig
1+x
a/parenrightBig1/2/bracketrightbigg
Wk,ν(x)dx
=2μΓ(1−μ−2ν)
Γ/parenleftbig3
2−k−μ−ν/parenrightbiga−1
2+1
2k−1
2νe1
2aWρ,σ(a),2ρ=μ+ν−k, 2σ=k+μ+3ν−1
[|arga|<π , Reμ>0,Reν>0]ET II 408(35)
A combination of confluent hypergeometric functions and orthogonal polynomials
7.6838/integraldisplay1
0e−1
2axxα(1−x)μ−α
2−1Lα
n(ax)Mα−1+α
2,μ−α−1
1[a(1−x)]dx
=Γ(μ−α)
Γ(1 + μ)Γ(1 + n+α)
n!a−1+α
2Mα+n,μ
2(a)
[Rea>−1,Re(μ−α)>0,n=0,1,2,...]BU 129(14b)
7.711 Parabolic cylinder functions 841
A combination of hypergeometric and confluent hypergeometric functions
7.684/integraldisplay∞
0xρ−1e−1
2xMγ+ρ,β+ρ+1
2(x)2F1/parenleftbigg
α,β;γ;−λ
x/parenrightbigg
dx
=Γ(α+β+2ρ)Γ(2β+2ρ)Γ(γ)
Γ(β)Γ(β+γ+2ρ)λ1
2β+ρ−1
2e1
2λWk,μ(λ);
k=1
2−α−1
2β−ρ, μ =1
2β+ρ
[|argλ|<π , Re(β+ρ)>0,Re (α+β+2ρ)>0,Reγ>0]
ET II 405(19)
7.69 Integration of confluent hypergeometric functions with respect to the index
7.691/integraldisplay∞
−∞sech(πx)Wix,0(α)W−ix,0(β)dx=2(aβ)1/2
α+βexp/bracketleftbigg
−1
2(α+β)/bracketrightbigg
ET II 414(61)
7.692/integraldisplayi∞
−i∞Γ(−a)Γ(c−a)Ψ(a,c;x)Ψ(c−a,c;y)da=2πiΓ(c)Ψ(c,2c;x+y) EH I 285(15)
7.693
1./integraldisplay∞
−∞Γ(ix)Γ(2k+ix)Wk+ix,k−1
2(α)W−k−ix,k−1
2(β)dx
=2π1/2Γ(2k)(aβ)k(α+β)1
2−2kK2k−1
2/parenleftbigga+β
2/parenrightbigg
ET II 414(62)
2./integraldisplayi∞
−i∞Γ/parenleftbig1
2+ν+μ+x/parenrightbig
Γ/parenleftbig1
2+ν+μ−x/parenrightbig
Γ/parenleftbig1
2+ν−μ+x/parenrightbig
Γ/parenleftbig1
2+ν−μ−x/parenrightbig
×Mμ+ix,ν(α)Mμ−ix,ν(β)dx
=2π(aβ)ν+1
2[Γ(2ν+1 ) ]2Γ(2ν+2μ+1 )Γ ( 2 ν−2μ+1 )
(α+β)2ν+1Γ(4ν+2 )M2μ,2ν+1
2(α+β)
/bracketleftbig
Reν>|Reμ|−1
2/bracketrightbig
ET II 413(59)
7.69411/integraldisplay∞
−∞e−2ρxiΓ/parenleftbig1
2+ν+ix/parenrightbig
Γ/parenleftbig1
2+ν−ix/parenrightbig
Mix,ν(α)Mix,ν(β)dx
=π/radicalbig
αβ[Γ(2ν+1 ) ]2sechρexp/bracketleftbigg
−1
2(α+β)tan h ρ/bracketrightbigg
J2ν/parenleftBig/radicalbig
αβsechρ/parenrightBig
/bracketleftbig
|Imρ|<1
2π,Reν>−1
2/bracketrightbig
7.7 Parabolic Cylinder Functions
7.71 Parabolic cylinder functions
7.711
1./integraldisplay∞
−∞Dn(x)Dm(x)dx=0 [ m/negationslash=n]
=n!(2π)1/2[m=n]
WH
842 Parabolic Cylinder Functions 7.721
2./integraldisplay∞
0Dμ(±t)Dν(t)dt=π21
2(μ+ν+1)
μ−ν/bracketleftBigg
1
Γ/parenleftbig1
2−1
2μ/parenrightbig
Γ/parenleftbig
−1
2ν/parenrightbig∓1
Γ/parenleftbig1
2−1
2ν/parenrightbig
Γ/parenleftbig
−1
2μ/parenrightbig/bracketrightBigg
[when the lower sign is taken, Re μ>Reν]BU 11 117(13a), EH II 122(21)
3./integraldisplay∞
0[Dν(t)]2dt=π1/22−3/2ψ/parenleftbig1
2−1
2ν/parenrightbig
−ψ/parenleftbig
−1
2ν/parenrightbig
Γ(−ν)BU 117(13b)a, EH II 122(22)a
7.72 Combinations of parabolic cylinder functions, powers, and exponentials
7.721
1./integraldisplay∞
−∞e−1
4x2(x−z)−1Dn(x)dx=±ie∓nπi(2π)1/2n!e−1
4z2D−n−1(∓iz)
[The upper or lower sign is taken accordingly as the imaginary part of zis positive or negative.]
WH
2./integraldisplay∞
1xν(x−1)1
2μ−1
2ν−1exp/bracketleftbigg
−(x−1)2a2
4/bracketrightbigg
Dμ(ax)dx=2μ−ν−2aμ
2−ν
2−1Γ/parenleftbiggμ−ν
2/parenrightbigg
Dν(a)
[Re(μ−ν)>0] ET II 395(4)a
7.722
1./integraldisplay∞
0e−3
4x2xνDν+1(x)dx=2−1
2−1
2νΓ(ν+1 )s i n1
4(1−ν)π
[Reν>−1] WH
2./integraldisplay∞
0e−1
4x2xμ−1D−ν(x)dx=π1/22−1
2μ−1
2νΓ(μ)
Γ/parenleftbig1
2μ+1
2ν+1
2/parenrightbig [Reμ>0] EH II 122(20)
3.11/integraldisplay∞
0e−3
4x2xνDν−1(x)dx=2−1
2νΓ(ν)sin/parenleftbigg1
4πν/parenrightbigg
[Reν>−1] ET II 395(2)
7.723
1./integraldisplay∞
0e−1
4x2xν/parenleftbig
x2+y2/parenrightbig−1Dν(x)dx=/parenleftBigπ
2/parenrightBig1/2
Γ(ν+1 )yν−1e1
4y2D−ν−1(y)
[Rey>0,Reν>−1]
EH II 121(18)a, ET II 396(6)a
2./integraldisplay∞
0e−1
4x2xν−1/parenleftbig
x2+y2/parenrightbig−1/2Dν(x)dx=yν−1Γ(ν)e1
4y2D−ν(y)
[Rey>0,Reν>0] ET II 396(7)
3./integraldisplay1
0x2ν−1/parenleftbig
1−x2/parenrightbigλ−1ea2x2
4D−2λ−2ν(ax)dx=Γ(λ)Γ(2ν)
Γ(2λ+2ν)2λ−1ea2
4D−2ν(a)
[Reλ>0,Reν>0] ET II 395(3)a
7.724/integraldisplay∞
−∞e−(x−y)2
2μe1
4x2Dν(x)dx=( 2πμ)1/2(1−μ)1
2νey2
4−4μDν/bracketleftBig
y(1−μ)−1/2/bracketrightBig
[0<Reμ<1]
EH II 121(15)
7.731 Parabolic cylinder and hyperbolic functions 843
7.725
1./integraldisplay∞
0e−pt(2t)ν−1
2e−t
2D−ν−2/parenleftBig√
2t/parenrightBig
dt=/parenleftBigπ
2/parenrightBig1/2/parenleftbig√p+1−1/parenrightbigν+1
(ν+1 )pν+1
[Reν>−1] MO 175
2./integraldisplay∞
0e−pt(2t)ν−1
2e−t
2D−ν/parenleftBig√
2t/parenrightBig
dt=/parenleftBigπ
2/parenrightBig1/2/parenleftbig√p+1−1/parenrightbigν
pν√p+1
[Reν>−1] MO 175
3./integraldisplay∞
0e−bxD2n+1/parenleftBig√
2x/parenrightBig
dx=(−2)nΓ/parenleftbig
n+3
2/parenrightbig/parenleftbig
b−1
2/parenrightbign/parenleftbig
b+1
2/parenrightbig−n−3
2
/bracketleftbig
Reb>−1
2/bracketrightbig
ET I 210(3)
4./integraldisplay∞
0/parenleftbig√x/parenrightbig−1e−bxD2n/parenleftBig√
2x/parenrightBig
dx=(−2)nΓ/parenleftbigg
n+1
2/parenrightbigg/parenleftbigg
b−1
2/parenrightbiggn/parenleftbigg
b+1
2/parenrightbigg−n−1
2
/bracketleftbig
Reb>−1
2/bracketrightbig
ET I 210(5)
5./integraldisplay∞
0x−1
2(ν+1)e−sxDν/parenleftbig√x/parenrightbig
dx=√π/parenleftbigg
1+/radicalBig
1
2+2s/parenrightbiggν1/radicalBig
1
4+s
/bracketleftbig
Res>−1
4,Reν<1/bracketrightbig
ET I 210(7)
6./integraldisplay∞
0e−ztt−1+β
2D−ν/bracketleftBig
2(kt)1/2/bracketrightBig
dt=21−β−ν
2π1/2Γ(β)
Γ/parenleftbig1
2ν+1
2β+1
2/parenrightbig(z+k)−β
2F/parenleftbiggν
2,β
2;ν+β+1
2;z−k
z+k/parenrightbigg
/bracketleftBig
Re(z+k)>0,Rez
k>0/bracketrightBig
EH II 121(11)
7.726/integraldisplay∞
−∞eixy−(1+λ)x2
4Dν/bracketleftBig
x(1−λ)1/2/bracketrightBig
dx=( 2π)1/2λ1
2νe−(1+λ)y2
4λDν/bracketleftBig
i/parenleftbig
λ−1−1/parenrightbig1/2y/bracketrightBig
[Reλ>0]
EH II 121(16)
7.727/integraldisplay∞
0e1
2xe−bx
(ex−1)μ+1
2exp/parenleftbigg
−a
1−e−x/parenrightbigg
D2μ/parenleftbigg2√a√
1−e−x/parenrightbigg
dx=e−a2b+μΓ(b+μ)D−2b/parenleftbig
2√a/parenrightbig
[Rea>0,Reb>−Reμ]
ET I 211(13)
7.728/integraldisplay∞
0(2t)−ν
2e−pte−q2
8tDν−1/parenleftbiggq√
2t/parenrightbigg
dt=/parenleftBigπ
2/parenrightBig1
2p1
2ν−1e−q√pMO 175
7.73 Combinations of parabolic cylinder and hyperbolic functions
7.731
1./integraldisplay∞
0cosh(2 μx)exp/bracketleftBig
−(asinhx)2/bracketrightBig
D2k(2acoshx)dx=2k−3
2π1/2a−1Wk,μ/parenleftbig
2a2/parenrightbig
/bracketleftbig
Re2a>0/bracketrightbig
ET II 398(20)
844 Parabolic Cylinder Functions 7.741
2./integraldisplay∞
0cosh(2 μx)exp/bracketleftBig
(asinhx)2/bracketrightBig
D2k(2acoshx)dx=Γ(μ−k)Γ(−μ−k)
2k+5
2aΓ(−2k)Wk+1
2,μ/parenleftbig
2a2/parenrightbig
/bracketleftbigg
|arga|<3π
4,Rek+|Reμ|<0/bracketrightbigg
ET II 398(21)
7.74 Combinations of parabolic cylinder and trigonometric functions
7.741
1./integraldisplay∞
0sin(bx)/braceleftBig
[D−n−1(ix)]2−[D−n−1(−ix)]2/bracerightBig
dx=(−1)n+1i
n!π√
2πe−1
2b2Ln/parenleftbig
b2/parenrightbig
[b>0] ET I 115(3)
2./integraldisplay∞
0e−1
4x2sin(bx)D2n+1(x)dx=(−1)n/radicalbiggπ
2b2n+1e−1
2b2
[b>0] ET I 115(1)
3./integraldisplay∞
0e−1
4x2cos(bx)D2n(x)dx=(−1)n/radicalbiggπ
2b2ne−1
2b2[b>0] ET I 60(2)
4./integraldisplay∞
0e−1
4x2sin(bx)/bracketleftBig
D2ν−1
2(x)−D2ν−1
2(−x)/bracketrightBig
dx=√
2πsin/bracketleftbig/parenleftbig
ν−1
4/parenrightbig
π/bracketrightbig
b2ν−1
2e−1
2b2
/bracketleftbig
Reν>1
4,b > 0/bracketrightbig
ET I 115(2)
5./integraldisplay∞
0e−1
2x2cos(bx)/bracketleftBig
D2ν−1
2(x)+D2ν−1
2(−x)/bracketrightBig
dx=21
4−2ν√πb2ν−1
2e−1
4b2
cosec/bracketleftbig/parenleftbig
ν+1
4/parenrightbig
π/bracketrightbig
/bracketleftbig
Reν>1
4,b > 0/bracketrightbig
ET I 61(4)
7.742
1./integraldisplay∞
0x2ρ−1sin(ax)e−x2
4D2ν(x)dx=2ν−ρ−1
2π1/2aΓ( 2ρ+1 )
Γ(ρ−ν+1 )
×2F2/parenleftbigg
ρ+1
2,ρ+1 ;3
2,ρ−ν+1 ;−a2
2/parenrightbigg
/bracketleftbig
Reρ>−1
2/bracketrightbig
ET II 396(8)
2./integraldisplay∞
0x2ρ−1sin(ax)ex2
4D2ν(x)dx=2ρ−ν−2
Γ(−2ν)G22
23/parenleftBigg
a2
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2−ρ,1−ρ
−ρ−ν,1
2,0/parenrightBigg
/bracketleftbig
a>0,Reρ>−1
2,Re(ρ+ν)<1
2/bracketrightbig
ET II 396(9)
3./integraldisplay∞
0x2ρ−1cos(ax)e−x2
4D2ν(x)dx=2ν−ρΓ(2ρ)π1/2
Γ/parenleftbig
ρ−ν+1
2/parenrightbig2F2/parenleftbigg
ρ, ρ+1
2;1
2,ρ−ν+1
2;−a2
2/parenrightbigg
[Reρ>0] ET II 396(10)a
7.752 Parabolic cylinder and Bessel functions 845
4./integraldisplay∞
0x2ρ−1cos(ax)ex2
4D2ν(x)dx=2ρ−ν−2
Γ(−2ν)G22
23/parenleftBigg
a2
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2−ρ,1−ρ
−ρ−ν,0,1
2/parenrightBigg
/bracketleftbig
a>0,Reρ>0,Re(ρ+ν)<1
2/bracketrightbig
ET II 396(11)
7.743/integraldisplayπ/2
0(cosx)−μ−2(sinx)−νDν(asinx)Dμ(acosx)dx=−/parenleftbig1
2π/parenrightbig1/2(1 +μ)−1Dμ+ν+1(a)
[Reν<1,Reμ<−1] ET II 397(19)
7.744
1./integraldisplay∞
0sin(bx)/bracketleftBig
D−ν−1
2/parenleftBig√
2x/parenrightBig
−D−ν−1
2/parenleftBig
−√
2x/parenrightBig/bracketrightBig
Dν−1
2/parenleftBig√
2x/parenrightBig
dx
=−√
2πsin/bracketleftbig/parenleftbig1
4+1
2ν/parenrightbig
π/bracketrightbig
b−ν−1
2/parenleftbig
1+√
1+b2/parenrightbigν
√
1+b2
[b>0] ET I 115(4)
2./integraldisplay∞
0cos(bx)/bracketleftBig
D−2ν−1
2/parenleftBig√
2x/parenrightBig
+D−2ν−1
2/parenleftBig
−√
2x/parenrightBig/bracketrightBig
D2ν−1
2/parenleftBig√
2x/parenrightBig
dx
=−√πsin/bracketleftbig/parenleftbig
ν−1
4/parenrightbig
π/bracketrightbig/parenleftbig
1+√
1+b2/parenrightbig2ν
√
1+b2b2ν+1
2
[b>0] ET I 60(3)
7.75 Combinations of parabolic cylinder and Bessel functions
7.751
1./integraldisplay∞
0[Dn(ax)]2J1(xy)dx=(−1)n−1y−1/bracketleftBig
Dn/parenleftBigy
a/parenrightBig/bracketrightBig2
[y>0] ET II 20(24)
2./integraldisplay∞
0J0(xy)Dn(ax)Dn+1(ax)dx=(−1)ny−1Dn/parenleftBigy
a/parenrightBig
Dn+1/parenleftBigy
a/parenrightBig
/bracketleftbig
y>0,|arga|<1
4π/bracketrightbig
ET II 17(42)
3./integraldisplay∞
0J0(xy)Dν(x)Dν+1(x)dx=2−1y−1[Dν(−y)Dν+1(y)−Dν+1(−y)Dν(y)] ET II 397(17)a
7.752
1./integraldisplay∞
0xνe−1
4x2D2ν−1(x)Jν(xy)dx=−1
2sec(νπ)yν−1e−1
4y2[D2ν−1(y)−D2ν−1(−y)]
/bracketleftbig
y>0,Reν>−1
2/bracketrightbig
ET II 76(1), MO 183
2./integraldisplay∞
0xνe1
4x2D2ν−1(x)Jν(xy)dx=21
2−νπsin(νπ)y−νΓ(2ν)e1
4y2Kν/parenleftbig1
4y2/parenrightbig
/bracketleftbig
y>0,−1
2<Reν<1
2/bracketrightbig
ET II 77(4)
3./integraldisplay∞
0xν+1e−1
4x2D2ν(x)Jν(xy)dx=1
2sec(νπ)yν−1e−1
4y2[D2ν+1(y)−D2ν+1(−y)]
[y>0,Reν>−1] ET II 78(13)
846 Parabolic Cylinder Functions 7.752
4./integraldisplay∞
0xνe−1
4x2D2ν+1(x)Jν(xy)dx=1
2sec(νπ)e−1
4y2yν[D2ν(y)+D2ν(−y)]
/bracketleftbig
y>0,Reν>−1
2/bracketrightbig
ET II 77(5)
5./integraldisplay∞
0xν+1e−1
4x2D2ν+2(x)Jν(xy)dx=−1
2sec(νπ)yνe−1
4y2[D2ν+2(y)+D2ν+2(−y)]
[Reν>−1,y > 0] ET II 78(16)
6./integraldisplay∞
0xν+1e1
4x2D2ν+2(x)Jν(xy)dx=π−1sin(νπ)Γ(2ν+3 )y−ν−2e1
4y2Kν+1/parenleftbig1
4y2/parenrightbig
/bracketleftbig
y>0,−1<Reν<−5
6/bracketrightbig
ET II 78(19)
7./integraldisplay∞
0xνe−1
4x2D−2ν(x)Jν(xy)dx=2−1/2π1/2y−νe−1
4y2Iν/parenleftbig1
4y2/parenrightbig
/bracketleftbig
y>0,Reν>−1
2/bracketrightbig
ET II 77(8)
8./integraldisplay∞
0xνe1
4x2D−2ν(x)Jν(xy)dx=yν−1e1
4y2D−2ν(y)/bracketleftbig
Reν>−1
2,y > 0/bracketrightbig
ET II 77(9), EH II 121(17)
9./integraldisplay∞
0xνe1
4x2D−2ν−2(x)Jν(xy)dx=( 2ν+1 )−1yνe1
4y2D−2ν−1(y)
/bracketleftbig
y>0,Reν>−1
2/bracketrightbig
ET II 77(10)
10./integraldisplay∞
0xνe−1
4a2x2D2μ(ax)Jν(xy)dx=2μ−1
2Γ/parenleftbig
ν+1
2/parenrightbig
yν
Γ(ν−μ+1 )a1+2ν1F1/parenleftbigg
ν+1
2;ν−μ+1 ;−y2
2a2/parenrightbigg
/bracketleftbig
y>0,|arga|<1
4π,Reν>−1
2/bracketrightbig
ET II 77(11)
11./integraldisplay∞
0xνe1
4a2x2D2μ(ax)Jν(xy)dx=Γ/parenleftbig1
2+ν/parenrightbig
a2k2m+μ
Γ/parenleftbig1
2−μ/parenrightbig
yμ+3
2ey2
4a2Wk,m/parenleftbiggy2
4a2/parenrightbigg
2k=1
2+μ−ν,2m=1
2+μ+ν/bracketleftbig
y>0,|arga|<1
4π,−1
2<Reν<Re/parenleftbig1
2−2μ/parenrightbig/bracketrightbig
ET II 78(12)
12./integraldisplay∞
0xν+1e−1
4a2x2D2μ(ax)Jν(xy)dx=2μΓ/parenleftbig
ν+3
2/parenrightbig
yν
Γ/parenleftbig
ν−μ+3
2/parenrightbig
a2ν+21F1/parenleftbigg
ν+3
2;ν−μ+3
2;−y2
2a2/parenrightbigg
/bracketleftbig
y>0,|arga|<1
4π,Reν>−1/bracketrightbig
ET II 79(23)
13./integraldisplay∞
0xν+1e1
4a2x2D2μ(ax)Jν(xy)dx=Γ/parenleftbig3
2+ν/parenrightbig
21
2+m+μa2k+1
Γ(−μ)yμ+2ey2
4a2Wk,m/parenleftbiggy2
2a2/parenrightbigg
2k=μ−ν−1,2m=μ+ν+1/bracketleftbig
y>0,|arga|<3
4π,−1<Reν<−1
2−2R eμ/bracketrightbig
ET II 79(24)
7.754 Parabolic cylinder and Bessel functions 847
14./integraldisplay∞
0xλ+1
2e1
4a2x2Dμ(ax)Jν(xy)dx=2λ−1
2μπ−1
2
Γ(−μ)yλ+3
2G22
23/parenleftBigg
y2
2a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2,1
3
4+λ+ν
2,−μ
2,3
4+λ−ν
2/parenrightBigg
/bracketleftbig
y>0,|arga|<3
4π,Reμ<−Reλ<Reν+3
2/bracketrightbig
ET II 80(26)
15./integraldisplay∞
0xν+1e1
4x2D−2ν−1(x)Jν(xy)dx=( 2ν+1 )yν−1e1
4y2D−2ν−2(y)
/bracketleftbig
y>0,Reν>−1
2/bracketrightbig
ET II 79(20)
16./integraldisplay∞
0xν+1e−1
4x2D−2ν−3(x)Jν(xy)dx=2−1/2π1/2y−ν−2e−1
4y2Iν+1/parenleftbig1
4y2/parenrightbig
[y>0,Reν>−1] ET II 79(21)
17./integraldisplay∞
0xν+1e1
4x2D−2ν−3(x)Jν(xy)dx=yνe1
4y2D−2ν−3(y)
[y>0,Reν>−1] ET II 79(22)
18./integraldisplay∞
0xνe1
4a2x2D1
2ν−1
2(ax)Yν(xy)dx=−π−123
4ν+3
4a−νy−1Γ(ν+1 )ey2
4a2W−1
2ν−1
2,1
2ν/parenleftbiggy2
2a2/parenrightbigg
/bracketleftbig
y>0,|arga|<3
4π,−1
2<Reν<2
3/bracketrightbig
ET II 115(39)
7.753
1./integraldisplay∞
0xν−1
2e−(x+a)2Iν−1
2(2ax)Dν(2x)dx=1
2π−1/2Γ(ν)aν−1
2D−ν(2a)
[Rea>0,Reν>0] ET II 397(12)
2./integraldisplay∞
0xν−3
2e−(x+a)2Iν−3
2(2ax)Dν(2x)dx=1
2π−1/2Γ(ν)aν−3
2D−ν(2a)
[Rea>0,Reν>1] ET II 397(13)
7.754
1./integraldisplay∞
0xνe−1
4x2{[1∓2c os (νπ)]D2ν−1(x)−D2ν−1(−x)}Jν(xy)dx
=±yν−1e−1
4y2{[1∓2c os (νπ)]D2ν−1(y)−D2ν−1(−y)}
/bracketleftbig
y>0,Reν>−1
2/bracketrightbig
ET II 76(2, 3)
2./integraldisplay∞
0xνe−1
4x2{[1∓2c os (νπ)]D2ν+1(x)−D2ν+1(−x)}Jν(xy)dx
=∓yνe−1
4y2{[1∓2c os(νπ)]D2ν(y)+D2ν(−y)}
/bracketleftbig
y>0,Reν>−1
2/bracketrightbig
ET II 77(6, 7)
3./integraldisplay∞
0xν+1e−1
4x2{[1±2c os (νπ)]D2ν(x)+D2ν(−x)}Jν(xy)dx
=±yν−1e−1
4y2{[1±2c os (νπ)]D2ν+1(y)−D2ν+1(−y)}
[y>0,Reν>−1] ET II 78(14, 15)
848 Parabolic Cylinder Functions 7.755
4./integraldisplay∞
0xν+1e−1
4x2{[1∓2c os (νπ)]D2ν+2(x)+D2ν+2(−x)}Jν(xy)dx
=±yνe−1
4y2{[1∓2c os (νπ)]D2ν+2(y)+D2ν+2(−y)}
[y>0,Reν>−1] ET II 78(17, 18)
7.755
1./integraldisplay∞
0x−1/2Dν/parenleftbig√ax/parenrightbig
D−ν−1/parenleftbig√ax/parenrightbig
J0(xy)dx
=2−3/2πa−1/2P1
2ν+1
4
−1
4/bracketleftBigg/parenleftbigg
1+4y2
a2/parenrightbigg1/2/bracketrightBigg
P1
2ν−1
4
1
4/bracketleftBigg/parenleftbigg
1+4y2
a2/parenrightbigg1/2/bracketrightBigg
[y>0,Rea>0] ET II 17(43)
2./integraldisplay∞
0x1/2D−1
2−ν/parenleftBig
ae1
4πix1/2/parenrightBig
D−1
2−ν/parenleftBig
ae−1
4πix1/2/parenrightBig
Jν(xy)dx
=2−νπ1/2y−ν−1/parenleftbig
a2+2y/parenrightbig−1/2/bracketleftbig
Γ/parenleftbig
ν+1
2/parenrightbig/bracketrightbig−1/bracketleftBig/parenleftbig
a2+2y/parenrightbig1/2−a/bracketrightBig2ν
/bracketleftbig
y>0,Rea>0,Reν>−1
2/bracketrightbig
ET II 80(27)
3. a/integraldisplay∞
0D−1
2−ν/parenleftBig
ae1
4πix−1/2/parenrightBig
D−1
2−ν/parenleftBig
ae−1
4πix−1/2/parenrightBig
Jν(xy)dx
=21/2π1/2y−1/bracketleftbig
Γ/parenleftbig
ν+1
2/parenrightbig/bracketrightbig−1exp/bracketleftBig
−a(2y)1/2/bracketrightBig
/bracketleftbig
y>0,Rea>0,eReν>−1
2/bracketrightbig
ET II 80(28)a
4./integraldisplay∞
0x1/2Dν−1
2/parenleftBig
ax−1/2/parenrightBig
D−ν−1
2/parenleftBig
ax−1/2/parenrightBig
Yν(xy)dx
=y−3/2exp/parenleftBig
−ay1/2/parenrightBig
sin/bracketleftBig
ay1/2−1
2/parenleftbig
ν−1
2/parenrightbig
π/bracketrightBig
/bracketleftbig
y>0,|arga|<1
4π/bracketrightbig
ET II 115(40)
5./integraldisplay∞
0x1/2Dν−1
2/parenleftBig
ax−1/2/parenrightBig
D−ν−1
2/parenleftBig
ax−1/2/parenrightBig
Kν(xy)dx=2−1y−3/2πexp/bracketleftBig
−a(2y)1/2/bracketrightBig
/bracketleftbig
Rey>0,|arga|<1
4π/bracketrightbig
ET II 151(81)
Combinations of parabolic cylinder and Struve functions
7.756/integraldisplay∞
0x−νe−1
4x2[Dμ(x)−Dμ(−x)]Hν(xy)dx
=23/2Γ/parenleftbig1
2μ+1
2/parenrightbig
Γ/parenleftbig1
2μ+ν+1/parenrightbigyμ+νsin/parenleftbigg1
2μπ/parenrightbigg
1F1/parenleftbigg1
2μ+1
2;1
2μ+ν+1 ;−1
2y2/parenrightbigg
/bracketleftbig
y>0,Re(μ+ν)>−3
2,Reμ>−1/bracketrightbig
ET II 171(41)
7.773 Integration of a parabolic cylinder function 849
7.76 Combinations of parabolic cylinder functions and confluent hypergeometric
functions
7.761
1./integraldisplay∞
0e1
4t2t2c−1D−ν(t)1F1/parenleftbigg
a;c;−1
2pt2/parenrightbigg
dt
=π1/2
2c+1
2νΓ(2c)Γ/parenleftbig1
2ν−c+a/parenrightbig
Γ/parenleftbig1
2ν/parenrightbig
Γ/parenleftbig
a+1
2+1
2ν/parenrightbigF/parenleftbigg
a,c+1
2;a+1
2+1
2ν;1−p/parenrightbigg
[|1−p|<1,Rec>0,Reν>2R e (c−a)]EH II 121(12)
2./integraldisplay∞
0e1
4t2t2c−2D−ν(t)1F1/parenleftbigg
a;c;−1
2pt2/parenrightbigg
dt
=π1/2
2c+1
2ν−1
2Γ(2c−1)Γ/parenleftbig1
2ν+1
2−c+a/parenrightbig
Γ/parenleftbig1
2+1
2ν/parenrightbig
Γ/parenleftbig
a+1
2ν/parenrightbigF/parenleftbigg
a,c−1
2;a+1
2ν;1−p/parenrightbigg
/bracketleftbig
|1−p|<1,Rec>1
2,Reν>2R e (c−a)−1/bracketrightbig
EH II 121(13)
7.77 Integration of a parabolic cylinder function with respect to the index
7.771/integraldisplay∞
0cos(ax)Dx−1
2(β)D−x−1
2(β)dx=1
2/parenleftBigπ
cosa/parenrightBig1/2
exp/parenleftbigg
−β2cosa
2/parenrightbigg/bracketleftbig
|a|<1
2π/bracketrightbig
=0/bracketleftbig
|a|>1
2π/bracketrightbig
ET II 298(22)
7.772
1./integraldisplay−1
2+i∞
−1
2−i∞⎡
⎣/parenleftbig
tan1
2ϕ/parenrightbigν
cos1
2ϕDν/parenleftBig
−e1
4iπξ/parenrightBig
D−ν−1/parenleftBig
e1
4iπη/parenrightBig
+/parenleftbig
cot1
2ϕ/parenrightbigν
sin1
2ϕD−ν−1/parenleftBig
e1
4iπξ/parenrightBig
Dν/parenleftBig
−e1
4iπη/parenrightBig⎤
⎦dν
sinνπ
=−2i(2π)1/2exp/bracketleftbig
−1
4i/parenleftbig
ξ2−η2/parenrightbig
cosϕ−1
2iξηsinϕ/bracketrightbig
EH II 125(7)
2./integraldisplay−1
2+i∞
−1
2−i∞/parenleftbig
tan1
2ϕ/parenrightbigν
cos1
2ϕDν/parenleftBig
−e1
4iπζ/parenrightBig
D−ν−1/parenleftBig
e1
4iπη/parenrightBigdν
sinνπ
=−2iD0/bracketleftBig
e1
4iπ/parenleftbig
ζcos1
2ϕ+ηsin1
2ϕ/parenrightbig/bracketrightBig
D−1/bracketleftBig
e1
4iπ/parenleftbig
ηcos1
2ϕ−ζsin1
2ϕ/parenrightbig/bracketrightBig
EH II 125(8)
7.773
1./integraldisplayc+i∞
c−i∞Dν(z)tνΓ(−ν)dν=2πie−1
4z2−zt−1
2t2/bracketleftBig
c<0,|argt|<π
4/bracketrightBig
EH II 126(10)
850 Meijer’s and MacRobert’s Functions ( GandE) 7.774
2./integraldisplayc+i∞
c−i∞[Dν(x)D−ν−1(iy)+Dν(−x)D−ν−1(iy)]t−ν−1dν
sin(−νπ)
=2πi
/parenleftBigπ
2/parenrightBig1/2/parenleftbig
1+t2/parenrightbig−1
2exp/bracketleftbigg1
41−t2
1+t2/parenleftbig
x2+y2/parenrightbig
+itxy
1+t2/bracketrightbigg
/bracketleftbigg
−1<c< 0,|argt|<1
2π/bracketrightbigg
EH II 126(11)
7.774/integraldisplayc+i∞
c−i∞Dν/bracketleftBig
k1
2(1 +i)ξ/bracketrightBig
D−ν−1/bracketleftBig
k1
2(1 +i)η/bracketrightBig
Γ/parenleftbig
−1
2ν/parenrightbig
Γ/parenleftbig1
2+1
2ν/parenrightbig
dν=21/2π2H(2)
0/bracketleftbig1
2k/parenleftbig
ξ2+η2/parenrightbig/bracketrightbig
[−1<c< 0,Reik≥0] EH II 125(9)
7.8 Meijer’s and MacRobert’s Functions ( GandE)
7.81 Combinations of the functions GandEand the elementary functions
7.811
1./integraldisplay∞
0Gm,n
p,q/parenleftbigg
ηx/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
Gμ,ν
σ,τ/parenleftbigg
ωx/vextendsingle/vextendsingle/vextendsingle/vextendsinglec
1,...,c σ
d ,...,d τ/parenrightbigg
dx
=1
ηGn+μ,m+ν
q+σ,p+τ/parenleftbiggω
η/vextendsingle/vextendsingle/vextendsingle/vextendsingle−b
1,...,−bm,c1,...,c σ,−bm+1,...,−bq
−a1,...,−an,d1,...,d τ,−an+1,...,−ap/parenrightbigg
subject to the following constraints
•m, n, p, q, μ, ν, σ, τ are integers;
•1≤n≤p<q<p +τ−σ
•1
2p+1
2q−n<m ≤q,0≤ν≤σ,1
2σ+1
2τ−ν<μ ≤τ
•Re (bj+dk)>−1( j=1,...,m ;k=1,...,μ )
•Re (aj+ck)<1( j=1,...,n ;k=1,...,τ )
•ω/negationslash=0,η/negationslash=0,|argη|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,|argω|</parenleftbig
μ+ν−1
2σ−1
2τ/parenrightbig
π
•The following must not be integers:
bj−bk(j=1,...,m ;k=1,...,m ;j/negationslash=k),
aj−ak(j=1,...,n ;k=1,...,n ;j/negationslash=k),
dj−dk(j=1,...,μ ;k=1,...,μ ;j/negationslash=k),
aj+dk(j=1,...,n ;k=1,...,n );
•The following must not be positive integers:
aj−bk(j=1,...,n ;k=1,...,m )
cj−dk(j=1,...,ν ;k=1,...,μ )
Formula 7.811 1 also holds for four sets of restrictions. See C. S. Meijer, Neue Integraldarstel-
lungen f¨ ur Whittakersche Funktionen, Nederl. Akad. Wetensch. Proc. 44(1941), 82–92.
ET II 422(14)
Hereafter, Gm,n
p,qwill be written as Gmn
pq, and commas will only be inserted in entries like
Gm,n+1
p+1,q+1, where their omission could cause ambiguity.
7.811 GandEand the elementary functions 851
2./integraldisplay1
0xρ−1(1−x)σ−1Gmn
pq/parenleftbigg
αx/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx=Γ (σ)Gm,n+1
p+1,q+1/parenleftbigg
α/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−ρ, a
1,...,a p
b1,...,b q,1−ρ−σ/parenrightbigg
where
•(p+q)<2(m+n)
•|arga|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π
•Re (ρ+bj)>0,j=1,...,m
•Reσ>0
•either
p+q≤2(m+n),|argα|≤/parenleftbig
m+n−1
2ρ−1
2q/parenrightbig
π,
Re (ρ+bj)>0;j=1,...,m ;R e σ>0,
Re⎡
⎣p/summationdisplay
j=1aj−q/summationdisplay
j=1bj+(p−q)/parenleftbigg
ρ−1
2/parenrightbigg⎤
⎦>−1
2,
or
p<q (orp≤qfor|α|<1),Re (p+bj)>0;j=1,...,m ;R e σ>0
ET II 417(1)
3./integraldisplay∞
1x−ρ(x−1)σ−1Gmn
pq/parenleftbigg
αx/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx=Γ (σ)Gm+1,n
p+1,q+1/parenleftbigg
α/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p,ρ
ρ−σ, b1,...,b q/parenrightbigg
where
•p+q<2(m+n)
•|argα|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π
•Re (ρ−σ−aj)>−1;j=1,...,n
•Reσ>0
•either
p+q≤2(m+n),|argα|≤/parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,
Re (ρ−σ−aj)>−1;j=1,...,n ;R e σ>0,
Re⎡
⎣p/summationdisplay
j=1aj−q/summationdisplay
j=1bj+(q−p)/parenleftbigg
ρ−σ+1
2/parenrightbigg⎤
⎦>−1
2,
or
q<p (orq≤pfor|α|>1),Re (ρ−σ−aj)>−1;j=1,...,n ;R e σ>0
ET II 417(2)
4./integraldisplay∞
0xρ−1Gmn
pq/parenleftbigg
αx/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx=/producttextm
j=1Γ(bj+ρ)/producttextn
j=1Γ(1−aj−ρ)/producttextq
j=m+1Γ( 1−bj−ρ)/producttextp
j=n+1Γ(aj+ρ)α−ρ
p+q<2(m+n),|argα|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,−min
1≤j≤mRebj<Reρ<1−max
1≤j≤nReaj
ET II 418(3)a, ET I 337(14)
852 Meijer’s and MacRobert’s Functions ( GandE) 7.812
5./integraldisplay∞
0xρ−1(x+β)−σGmn
pq/parenleftbigg
αx/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx=βρ−σ
Γ(σ)Gm+1,n+1
p+1,q+1/parenleftbigg
αβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−ρ, a
1,...,a p
σ−ρ, b1,...,b q/parenrightbigg
where
•p+q<2(m+n)
•|argα|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π
•|argβ|<π
•Re (ρ+bj)>0,j=1,...,m
•Re (ρ−σ+aj)<1,j=1,...,n
•either
p≤q, p +q≤2(m+n),|argα|≤/parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,|argβ|<π
Re (ρ=bj)>0,j=1,...,m , Re (ρ−σ+aj)<1,j=1,...,n ,
Re⎡
⎣p/summationdisplay
j=1aj−q/summationdisplay
j=1bj−(q−p)/parenleftbigg
ρ−σ−1
2/parenrightbigg⎤
⎦>1,
or
p≥q, p +q≤2(m+n),|argα|≤/parenleftbigg
m+n−1
2p−1
2q/parenrightbigg
π,|argβ|<π ,
Re (ρ+bj)>0,j=1,...,m, Re (ρ−σ+aj)<1,j=1,...,n ,
Re⎡
⎣p/summationdisplay
j=1aj−q/summationdisplay
j=1bj+(p−q)/parenleftbigg
ρ−1
2/parenrightbigg⎤
⎦>1
ET II 418(4)
7.812
1./integraldisplay1
0xβ−1(1−x)γ−β−1E/parenleftBig
a1,...,a p:ρ1,...,ρ q;z
xm/parenrightBig
dx
=Γ (γ−β)mβ−γE(a1,...,a p+m:ρ1,...,ρ q+m:z)
ap+k=β+k−1
m,ρ q+k=γ+k−1
m,k =1,...,m
[Reγ>Reβ>0,m =1,2,...]ET II 414(2)
2./integraldisplay∞
0xρ−1(1 +x)−σE[a1,...,a p:ρ1,...,ρ q:( 1+ x)z]dx
=Γ (ρ)E(a1,...,a p,σ−ρ;ρ1,...,ρ q,σ;z)
[Reσ>Reρ>0] ET II 415(3)
7.815 GandEand the elementary functions 853
3./integraldisplay∞
0(1 +x)−βxs−1Gmn
pq/parenleftbiggax
1+x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx=Γ (β−s)Gm,n+1
p+1,q+1/parenleftbigg
a/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−s, a
1,...,a p
b1,...,b q,1−β/parenrightbigg
/bracketleftbigg
−min Re bk<Res<Reβ,1≤k≤m;(p+q)<2(m+n),
|arga|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π/bracketrightbigg
ET I 338(19)
7.813
1./integraldisplay∞
0x−ρe−βxGmn
pq/parenleftbigg
αx/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx=βρ−1Gm,n+1
p+1,q/parenleftbiggα
β/vextendsingle/vextendsingle/vextendsingle/vextendsingleρ, a
1,...,a p
b1,...,b q/parenrightbigg
/bracketleftbigg
p+q<2(m+n),|argα|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,
|argβ|<1
2π,Re (bj−ρ)>−1,j=1,...,m/bracketrightbigg
ET II 419(5)
2./integraldisplay∞
0e−βxGmn
pq/parenleftbigg
αx2/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx=π−1/2β−1Gm,n+2
p+2,q/parenleftbigg4α
β2/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,
1
2,a1,...,a p
b1,...,b q/parenrightbigg
/bracketleftbigg
p+q<2(m+n),|argα|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,
|argβ|<1
2π,Rebj>−1
2;j=1,...,m/bracketrightbigg
ET II 419(6)
7.814
1./integraldisplay∞
0xβ−1e−xE(a1,...,a p:ρ1,...,ρ q:xz)dx
=πcosec( βπ)/bracketleftbigg
E/parenleftbig
a1,...,a p:1−β,ρ1,...,ρ q:e±iπz/parenrightbig
−z−βE/parenleftbig
a1+β,...,a p+β:1+β,ρ1+β,...,ρ l+β:e±iπz/parenrightbig/bracketrightbigg
[p≥q+1 ,R e( ar+β)>0,r=1,...,p ,|argz|<π. The formula holds also for p<q +1 ,
provided the integral converges.] ET II 415(4)
2./integraldisplay∞
0xβ−1e−xE/parenleftbig
a1,...,a p:ρ1,...,ρ q:x−mz/parenrightbig
dx
=( 2π)1
2−1
2mmβ−1
2E/parenleftbig
a1,...,a p+m:ρ1,...,ρ q:m−mz/parenrightbig
/bracketleftbigg
Reβ>0,a p+k=β+k−1
m,k=1,...,m ;m=1,2,.../bracketrightbigg
ET II 415(5)
7.815
1./integraldisplay∞
0sin(cx)Gmn
pq/parenleftbigg
αx2/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx=√πc−1Gm,n+1
p+2,q/parenleftbigg4α
c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,a
1,...,a p,1
2
b1,...,b q/parenrightbigg
/bracketleftbig
p+q<2(m+n),|argα|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,
c>0,Rebj>−1,j=1,2,...,m, Reaj<1
2,j=1,...,n/bracketrightbig
ET II 420(7)
854 Meijer’s and MacRobert’s Functions ( GandE) 7.821
2./integraldisplay∞
0cos(cx)Gmn
pq/parenleftbigg
αx2/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx=π1/2c−1Gm,n+1
p+2,q/parenleftbigg4α
c2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2,a1,...,a p,0
b1,...,b q/parenrightbigg
/bracketleftbig
p+q<2(m+n),|argα|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,
c>0,Rebj>−1
2,j=1,...,m , Reaj<1
2,j=1,...,n/bracketrightbig
ET II 420(8)
7.82 Combinations of the functions GandEand Bessel functions
7.821
1./integraldisplay∞
0x−ρJν/parenleftbig
2√x/parenrightbig
Gmn
pq/parenleftbigg
αx/vextendsingle/vextendsingle/vextendsingle/vextendsinglea1,...,a p
b1,...,b q/parenrightbigg
dx=Gm,n+1
p+2,q/parenleftbigg
α/vextendsingle/vextendsingle/vextendsingle/vextendsingleρ−1
2ν,a1,...,a p,ρ+1
2ν
b1,...,b q/parenrightbigg
/bracketleftbigg
p+q<2(m+n),|argα|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π
−3
4+m a x
1≤j≤nReaj<Reρ<1+1
2Reν+m i n
1≤j≤mRebj/bracketrightbigg
ET II 420(9)
2./integraldisplay∞
0x−ρYν/parenleftbig
2√x/parenrightbig
Gmn
pq/parenleftbigg
αx/vextendsingle/vextendsingle/vextendsingle/vextendsinglea1,...,a p
b1,...,b q/parenrightbigg
dx
=Gm,n+2
p+3,q+1/parenleftBigg
α/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleρ−1
2ν,ρ+1
2ν,a1,...,a p,ρ+1
2+1
2ν
b1,...,b q,ρ+1
2+1
2ν/parenrightBigg
/bracketleftbigg
p+q<2(m+n),|argα|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,
−3
4+m a x
1≤j≤nReaj<Reρ<min
1≤j≤mRebj+1
2|Reν|+1/bracketrightbigg
ET II 420(10)
3./integraldisplay∞
0x−ρKν/parenleftbig
2√x/parenrightbig
Gmn
pq/parenleftbigg
αx/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx=1
2Gm,n+2
p+2,q/parenleftbigg
α/vextendsingle/vextendsingle/vextendsingle/vextendsingleρ−
1
2ν,ρ+1
2ν,a1,...,a p
b1,...,b q/parenrightbigg
/bracketleftbigg
p+q<2(m+n),|argα|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,
Reρ<1−1
2|Reν|+m i n
1≤j≤mRebj/bracketrightbigg
ET II 421(11)
7.822
1./integraldisplay∞
0x2ρJν(xy)Gmn
pq/parenleftbigg
λx2/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx=22ρ
y2ρ+1Gm,n+1
p+2,q/parenleftbigg4λ
y2/vextendsingle/vextendsingle/vextendsingle/vextendsingleh, a
1,...,a p,k
b1,...,b q/parenrightbigg
h=1
2−ρ−1
2ν, k =1
2−ρ+1
2ν
/bracketleftbigg
p+q<2(m+n),|argλ|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,Re/parenleftbig
bj+ρ+1
2ν/parenrightbig
>−1
2,
j=1,2,...,m, Re (aj+ρ)<3
4,j=1,...,n , y> 0/bracketrightbigg
ET II 91(20)
7.823 GandEand Bessel functions 855
2./integraldisplay∞
0x1/2Yν(xy)Gmn
pq/parenleftbigg
λx2/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx
=( 2λ)−1/2y−1/2Gn+2,m
q+1,p+3/parenleftBigg
y2
4λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2−b1,...,1
2−bq,l
h, k,1
2−a1,...,1
2−ap,l/parenrightBigg
h=1
4+1
2ν, k =1
4−1
2ν, l =−1
4−1
2ν
/bracketleftbigg
p+q<2(m+n),|argλ|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π, y > 0,
Reaj<1,j=1,...,n , Re/parenleftbig
bj±1
2ν/parenrightbig
>−3
4,j=1,...,m/bracketrightbigg
ET II 119(56)
3./integraldisplay∞
0x1/2Kν(xy)Gmn
pq/parenleftbigg
λx2/vextendsingle/vextendsingle/vextendsingle/vextendsinglea1,...,a p
b1,...,b q/parenrightbigg
dx
=2−3/2λ−1/2y−1/2Gn+2,m
q,p+2/parenleftBigg
y2
4λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2−b1,...,1
2−bq
h, k,1
2−a1,...,1
2−ap/parenrightBigg
h=1
4+1
2ν, k =1
4−1
2ν
/bracketleftbigg
Rey>0,p+q<2(m+n),|argλ|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,
Rebj>1
2|Reν|−3
4,j=1,...,m/bracketrightbigg
ET II 153(90)
7.823
1./integraldisplay∞
0xβ−1Jν(x)E/parenleftbig
a1,...,a p:ρ1,...,ρ q:x−2mz/parenrightbig
dx
=( 2π)−m(2m)β−1/braceleftbig
exp/bracketleftbig1
2π(β−ν−1)i/bracketrightbig
E/bracketleftbig
a1,...,a p+2m:ρ1,...,ρ q:( 2m)−2mze−mπi/bracketrightbig
+e x p/bracketleftbig
−1
2π(β−ν−1)i/bracketrightbig
E/bracketleftbig
a1,...,a p+2m:ρ1,...,ρ q:( 2m)−2mzemπi/bracketrightbig/bracerightbig
,
ap+k=β+ν+2k−2
2m,a p+m+k=β−ν+2k−2
2m,m =1,2,...,;k=1,...,m
/bracketleftbig
Re(β+ν)>0,Re(2arm−β)>−3
2,r=1,...,p/bracketrightbig
ET II 415(7)
2./integraldisplay∞
0xβ−1Kν(x)E/parenleftbig
a1,...,a p:ρ1,...,ρ q:x−2mz/parenrightbig
dx
=( 2π)1−m2β−2mβ−1
×E/bracketleftbig
a1,...,a p+2m:ρ1,...,ρ q:( 2m)−2mz/bracketrightbig
,
ap+k=β+ν+2k−2
2m,a p+m+k=β−ν+2k−2
2m,k =1,2,...,m
[Reβ>|Reν|,m =1,2,...]
ET II 416(8)
856 Meijer’s and MacRobert’s Functions ( GandE) 7.824
7.824
1./integraldisplay∞
0x1/2Hν(xy)Gmn
pq/parenleftbigg
λx2/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx
=( 2λy)−1/2Gn+1,m+1
q+1,p+3/parenleftBigg
y2
4λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglel,
1
2−b1,...,1
2−bq
l,1
2−a1,...,1
2−ap,h ,k/parenrightBigg
h=1
4+ν
2,k=1
4−ν
2,l=3
4+ν
2/bracketleftbigg
p+q<2(m+n),|argλ|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π, y > 0,
Reaj<min/parenleftbig
1,3
4−1
2ν/parenrightbig
,j=1,...,n , Re (2bj+ν)>−5
2,j=1,...,m/bracketrightbigg
ET II 172(47)
2./integraldisplay∞
0x−ρHν/parenleftbig
2√x/parenrightbig
Gmn
pq/parenleftbigg
αx/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx
=Gm+1,n+1
p+3,q+1/parenleftBigg
α/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleρ−
1
2−1
2ν,a1,...,a p,ρ+1
2ν,ρ−1
2ν
ρ−1
2−1
2ν,b1,...,b q/parenrightBigg
/bracketleftbigg
p+q<2(m+n),|argα|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,
max/parenleftbigg
−3
4,Reν−1
2/parenrightbigg
+m a x
1≤j≤nReaj<Reρ<min
1≤j≤mRebj+1
2Reν+3
2/bracketrightbigg
ET II 421(12)
7.83 Combinations of the functions GandEand other special functions
7.831/integraldisplay∞
1x−ρ(x−1)σ−1F(k+σ−ρ, λ+σ−ρ;σ;1−x)Gmn
pq/parenleftbigg
αx/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
dx
=Γ (σ)Gm+2,n
p+2,q+2/parenleftbigg
α/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p,k+λ+σ−ρ, ρ
k,λ,b 1,...,b q/parenrightbigg
where ET II 421(13)
•Re⎡
⎣p/summationdisplay
j=1aj−q/summationdisplay
j=1bj+(q−p)/parenleftbigg
k+1
2/parenrightbigg⎤
⎦>−1
2
•Re⎡
⎣p/summationdisplay
j=1aj−q/summationdisplay
j=1bj+(q−p)/parenleftbigg
λ+1
2/parenrightbigg⎤
⎦>−1
2
•either
p+q<2(m+n),|argα|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,
Reσ>0,Rek≥Reλ>Reaj−1,j=1,...,n ,
7.832 GandEand other special functions 857
or
p+q≤2(m+n),|argα|≤/parenleftbig
m+n−1
2p−1
2q/parenrightbig
π,
Reσ>0,Rek≥Reλ>Reaj−1,j=1,...,n ,
7.832/integraldisplay∞
0xβ−1e−1
2xWκ,μ(x)E/parenleftbig
a1,...,a p:ρ1,...,ρ q:x−mz/parenrightbig
dx
=( 2π)1
2−1
2mmβ+κ−1
2E/parenleftbig
a1,...,a p+2m:ρ1,...,ρ q+m:m−mz/parenrightbig
,
ap+k=β+k+μ−1
2
m,a p+m+k=β−μ+k−1
2
m,ρ q+k=β−κ+k
m,k =1,...,m
/bracketleftbig
Reβ>|Reμ|−1
2,m =1,2,.../bracketrightbig
ET II 416(10)
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8–9 Special Functions
8.1 Elliptic Integrals and Functions
8.11 Elliptic integrals
8.110
1. Every integral of the form/integraldisplay
R/parenleftBig
x,/radicalbig
P(x)/parenrightBig
dx,w h e r e P(x) is a third- or fourth-degree polyno-
mial, can be reduced to a linear combination of integrals leading to elementary functions and the
following three integrals:
/integraldisplaydx/radicalbig
(1−x2)(1−k2x2),/integraldisplay√
1−k2x2
√
1−x2dx,/integraldisplaydx
(1−nx2)/radicalbig
(1−x2)(1−k2x2),
which are called respectively elliptic integrals of the first, second, and third kind in the Legendre
normal form . The results of this reduction for the more frequently encountered integrals are
given in formulas 3.13–3.17 .T h en u m b e r kis called the modulus∗of these integrals; the number
k/prime=√
1−k2is called the complementary modulus, and the number nis called the parameter of
the integral of the third kind. BY (110.04)
2. By means of the substitution x=s i nϕ, elliptic integrals can be reduced to the normal trigono-
metric forms/integraldisplaydϕ/radicalbig
1−k2sin2ϕ,/integraldisplay/radicalBig
1−k2sin2ϕdϕ,/integraldisplaydϕ/parenleftbig
1−nsin2ϕ/parenrightbig/radicalbig
1−k2sin2ϕ.BY (110.04)
The results of reducing integrals of trigonometric functions to normal form are given in 2.58–
2.62.
3.11Elliptic integrals from 0 to 1 in the 8.110 1 formulation (or from 0 toπ
2in the 8.110 2 formu-
lation) are called complete elliptic integrals .
4.∗Take note that in mathematical software, and elsewhere, the notation for elliptic integrals is often
modified by replacing the parameter k2that is used here with k.
8.111
Notations:
1. Δ ϕ=/radicalBig
1−k2sin2ϕ;k/prime=/radicalbig
1−k2;k2<1
∗The quantity kis sometimes called the module of the functions.
859
860 Elliptic Integrals and Functions 8.112
2. The elliptic integral of the first kind:
F(ϕ, k)=/integraldisplayϕ
0dα/radicalbig
1−k2sin2α=/integraldisplaysinϕ
0dx/radicalbig
(1−x2)( 1−k2x2)
3. The elliptic integral of the second kind:
E(ϕ, k)=/integraldisplayϕ
0/radicalbig
1−k2sin2αdα=/integraldisplaysinϕ
0√
1−k2x2
√
1−x2dx FI II 135
4.11The elliptic integral of the third kind:
Π(ϕ, n, k )=/integraldisplayϕ
0dα
/parenleftbig
1−nsin2α/parenrightbig/radicalbig
1−k2sin2α=/integraldisplaysinϕ
0dx
(1−nx2)/radicalbig
(1−x2)(1−k2x2)
BY (110.04)
5. D(ϕ, k)=F(ϕ, k)−E(ϕ, k)
k2=/integraldisplayϕ
0sin2αd α/radicalbig
1−k2sin2α=/integraldisplaysinϕ
0x2dx/radicalbig
(1−x2)(1−k2x2)
6.∗/integraldisplayπ/2
0dx/radicalbig
a2+s i n2xarctan/parenleftBigg
b/radicalbig
a2+s i n2x/parenrightBigg
=π
2|a|F/parenleftbigg
arcsin/parenleftbiggb√
a2+b2+1/parenrightbigg
,i
a/parenrightbigg
[aandbare real]
8.112 Complete elliptic integrals
1. K(k)=F/parenleftBigπ
2,k/parenrightBig
=K/prime(k/prime)
2. E(k)=E/parenleftBigπ
2,k/parenrightBig
=E/prime(k/prime)
3. K/prime(k)=F/parenleftBigπ
2,k/prime/parenrightBig
=K(k/prime)
4. E/prime(k)=E/parenleftBigπ
2,k/prime/parenrightBig
=E(k/prime)
5.D=D/parenleftBigπ
2,k/parenrightBig
=K−E
k2
In writing complete elliptic integrals, the modulus k, which acts as an independent variable, is often
omitted, and we write
K(≡K(k)),K/prime/parenleftbig
≡K/prime(k)/parenrightbig
,E(≡E(k)),E/prime/parenleftbig
≡E/prime(k)/parenrightbig
.
Series representations
8.113
1.K=π
2/braceleftBigg
1+/parenleftbigg1
2/parenrightbigg2
k2+/parenleftbigg1·3
2·4/parenrightbigg2
k4+···+/parenleftbigg(2n−1)!!
2nn!/parenrightbigg2
k2n+.../bracerightBigg
=π
2F/parenleftbigg1
2,1
2;1;k2/parenrightbigg
FI II 487, WH 499
8.116 Elliptic integrals 861
2.K=π
1+k/prime/braceleftBigg
1+/parenleftbigg1
2/parenrightbigg2/parenleftbigg1−k/prime
1+k/prime/parenrightbigg2
+/parenleftbigg1·3
2·4/parenrightbigg2/parenleftbigg1−k/prime
1+k/prime/parenrightbigg4
+···+/parenleftbigg(2n−1)!!
2nn!/parenrightbigg2/parenleftbigg1−k/prime
1+k/prime/parenrightbigg2n
+.../bracerightBigg
DW
3.K=l n4
k/prime+/parenleftbigg1
2/parenrightbigg2/parenleftbigg
ln4
k/prime−2
1·2/parenrightbigg
k/prime2+/parenleftbigg1·3
2·4/parenrightbigg2/parenleftbigg
ln4
k/prime−2
1·2−2
3·4/parenrightbigg
k/prime4
+/parenleftbigg1·3·5
2·4·6/parenrightbigg2/parenleftbigg
ln4
k/prime−2
1·2−2
3·4−2
5·6/parenrightbigg
k/prime6+...
DW
See also 8.197 1a n d8.197 2.
8.114
1.6E=π
2/braceleftBigg
1−1
22k2−12·3
22·42k4−···−/parenleftbigg(2n−1)!!
2nn!/parenrightbigg2k2n
2n−1−.../bracerightBigg
=π
2F/parenleftbigg
−1
2,1
2;1;k2/parenrightbigg
WH 518, FI II 487
2.E=(1 +k/prime)π
4/braceleftBigg
1+1
22/parenleftbigg1−k/prime
1+k/prime/parenrightbigg2
+12
22·42/parenleftbigg1−k/prime
1+k/prime/parenrightbigg4
+···+/parenleftbigg(2n−3)!!
2nn!/parenrightbigg2/parenleftbigg1−k/prime
1+k/prime/parenrightbigg2n
+.../bracerightBigg
DW
3.E=1+1
2/parenleftbigg
ln4
k/prime−1
1·2/parenrightbigg
k/prime2+12·3
22·4/parenleftbigg
ln4
k/prime−2
1·2−1
3·4/parenrightbigg
k/prime4
+12·32·5
22·42·6/parenleftbigg
ln4
k/prime−2
1·2−2
3·4−1
5·6/parenrightbigg
k/prime6+...
DW
8.115 D=π/braceleftBigg
1
1/parenleftbigg1
2/parenrightbigg2
+2
3/parenleftbigg1·3
2·4/parenrightbigg2
k2+···+n
2n−1/bracketleftbigg(2n−1)!!
2nn!/bracketrightbigg2
k2(n−1)+.../bracerightBigg
ZH 43(158)
8.116/integraldisplayπ
2
0/radicalbig
1−k2sin2ϕ
1−n2sin2ϕdϕ=/radicalbig
n/prime2−k/prime2/parenleftBigg
arccos1
n/prime
n/prime/radicalbig
n/prime2−1+R/parenrightBigg
,where ZH 44(163)
R=k/prime2
2/parenleftbigg
p+1
2/parenrightbigg1
n/prime3+k/prime4
16/bracketleftbigg
−1+/parenleftbigg
p+1
4/parenrightbigg1
n/prime3/parenleftbigg
1+6
n/prime2/parenrightbigg/bracketrightbigg
+k/prime6
16/bracketleftbigg
−7
16−1
n/prime2+/parenleftbigg
p+1
6/parenrightbigg1
n/prime3/parenleftbigg3
8+1
n/prime2+5
n/prime4/parenrightbigg/bracketrightbigg
+15k/prime8
256/bracketleftbigg
−37
144−21
40n/prime2−1
n/prime4+/parenleftbigg
p+1
8/parenrightbigg1
n/prime3/parenleftbigg5
24+9
20n/prime2+1
n/prime4+14
3n/prime6/parenrightbigg/bracketrightbigg
+...,
p=l n4
k/prime,k/prime=4e−p,k/prime2=1−k2,n/prime2=1−n2ZH 44(163)
862 Elliptic Integrals and Functions 8.117
Trigonometric series
8.117 Forsmall values of kandϕ, we may use the series
1. F(ϕ, k)=2
πKϕ−sinϕcosϕ/parenleftbigg
a0+2
3a1sin2ϕ+2·4
3·5a2sin4ϕ+.../parenrightbigg
,where
a0=2
πK−1;an=an−1−/bracketleftbigg(2n−1)!!
2nn!/bracketrightbigg2
k2nZH 10(19)
2. E(ϕ, k)=2
πEϕ+s i nϕcosϕ/parenleftbigg
b0+2
3b1sin2ϕ+2·4
3·5b2sin4ϕ+.../parenrightbigg
,where
b0=1−2
πE,b n=bn−1−/bracketleftbigg(2n−1)!!
2nn!/bracketrightbigg2k2n
2n−1ZH 27(86)
8.118 Forkc l o s et o1 ,w em a yu s et h es e r i e s
1. F(ϕ, k)=2
πK/primeln tan/parenleftBigϕ
2+π
4/parenrightBig
−tanϕ
cosϕ/parenleftbigg
a/prime
0−2
3a/prime
1tan2ϕ+2·4
3·5a/prime
2tan4ϕ−.../parenrightbigg
,where
a/prime
0=2
πK/prime−1;a/prime
n=an−1−/bracketleftbigg(2n−1)!!
2nn!/bracketrightbigg2
k/prime2nZH 10(23)
2. E(ϕ, k)=2
π(K/prime−E/prime)lnt an/parenleftBigϕ
2+π
2/parenrightBig
+tanϕ
cosϕ/parenleftbigg
b/prime
1−2
3b/prime
2tan2ϕ+2·4
3·5b/prime
3tan4ϕ−.../parenrightbigg
+1
sinϕ/bracketleftBig
1−cosϕ/radicalbig
1−k2sinϕ/bracketrightBig
,
where
b/prime
0=2
π(K/prime−E/prime),b/prime
n=b/prime
n−1−/bracketleftbigg(2n−3)!!
2n−1(n−1)!/bracketrightbigg2/parenleftbigg2n−1
2n/parenrightbigg
k/prime2nZH 27(90)
For the expansion of complete elliptic integrals in Legendre polynomials, see 8.928 .
8.119 Representation in the form of an infinite product:
1. K(k)=π
2∞/productdisplay
n=1(1 +kn),where
kn=1−/radicalBig
1−k2
n−1
1+/radicalBig
1−k2
n−1;k0=k FI II 166
See also 8.197 .
8.125 Functional relations between elliptic integrals 863
8.12 Functional relations between elliptic integrals
8.121
1. F(−ϕ, k)=−F(ϕ, k) JA
2. E(−ϕ, k)=−E(ϕ, k) JA
3. F(nπ±ϕ, k)=2nK(k)±F(ϕ, k) JA
4. E(nπ±ϕ, k)=2nE(k)±E(ϕ, k) JA
8.122 E(k)K/prime(k)+E/prime(k)K(k)−K(k)K/prime(k)=π
2FI II 691, 791
8.123
1.∂F
∂k=1
k/prime2/parenleftBigg
E−k/prime2F
k−ksinϕcosϕ/radicalbig
1−k2sin2ϕ/parenrightBigg
MO 138, BY (710.07)
2.dK(k)
dk=E(k)
kk/prime2−K(k)
kFI II 691
3.∂E
∂k=E−F
kMO 138
4.dE(k)
dk=E(k)−K(k)
kFI II 690
8.124
1. The functions KandK/primesatisfy the equation
d
dk/braceleftbigg
kk/prime2du
dk/bracerightbigg
−ku=0. WH 499, WH 502
2. The functions EandE/prime−K/primesatisfy the equation
k/prime2d
dk/parenleftbigg
kdu
dk/parenrightbigg
+ku=0. WH
8.125
1. F/parenleftbigg
ψ,1−k/prime
1+k/prime/parenrightbigg
=( 1+ k/prime)F(ϕ, k)[ t a n ( ψ−ϕ)=k/primetanϕ] MO 130
2. E/parenleftbigg
ψ,1−k/prime
1+k/prime/parenrightbigg
=2
1+k/prime[E(ϕ, k)+k/primeF(ϕ, k)]−1−k/prime
1+k/primesinψ
[tan(ψ−ϕ)=k/primetanϕ] MO 131
3. F/parenleftBigg
ψ,2√
k
1+k/parenrightBigg
=( 1+ k)F(ϕ, k)/bracketleftbigg
sinψ=(1 +k)sinϕ
1+ksin2ϕ/bracketrightbigg
4. E/parenleftBigg
ψ,2√
k
1+k/parenrightBigg
=1
1+k/bracketleftbigg
2E(ϕ, k)−k/prime2F(ϕ, k)+2ksinϕcosϕ
1+ksin2ϕ/radicalBig
1−k2sin2ϕ/bracketrightbigg
/bracketleftbigg
sinψ=(1 +k)sinϕ
1+ksin2ϕ/bracketrightbigg
MO 131
864 Elliptic Integrals and Functions 8.126
8.126 In particular,
1. K/parenleftbigg1−k/prime
1+k/prime/parenrightbigg
=1+k/prime
2K(k) MO 130
2. E/parenleftbigg1−k/prime
1+k/prime/parenrightbigg
=1
1+k/prime[E(k)+k/primeK(k)] MO 130
3. K/parenleftBigg
2√
k
1+k/parenrightBigg
=( 1+ k)K(k) MO 130
4. E/parenleftBigg
2√
k
1+k/parenrightBigg
=1
1+k/bracketleftbig
2E(k)−k/prime2K(k)/bracketrightbig
MO 130
8.12711
k1 sinϕ1 cosϕ1F(ϕ1,k1) E(ϕ1,k1)
ik
k/primek/primesinϕ
Δϕcosϕ
Δϕk/primeF(ϕ, k)1
k/prime/bracketleftBig
E(ϕ, k)−k2sinϕcosϕ
Δϕ/bracketrightBig
k/prime−itanϕ secϕ−iF(ϕ, k) ia[E(ϕ, k)−F(ϕ, k)−Δϕtanϕ]
1
kksinϕ Δϕ kF(ϕ, k)1
k/bracketleftbig
E(ϕ, k)−k/prime2F(ϕ, k)/bracketrightbig
1
k/prime−ik/primetanϕΔϕ
cosϕ−ik/primeF(ϕ, k)i
k/prime/bracketleftbig
E(ϕ, k)−k/prime2F(ϕ, k)−Δϕtanϕ/bracketrightbig
k/prime
ik−iksinϕ
Δϕ1
Δϕ−ikF(ϕ, k)i
k/bracketleftBig
E(ϕ, k)−F(ϕ, k)−k2sinϕcosϕ
Δϕ/bracketrightBig
(see8.111 1) MO 131
8.128 In particular,
1. K/parenleftbigg
ik
k/prime/parenrightbigg
=k/primeK(k)[ I m ( k)<0] MO 130
2. K/parenleftbigg
ik
k/prime/parenrightbigg
=k/prime/bracketleftbig
K/prime(k/prime)−iK(k)/bracketrightbig
[Im(k)<0] MO 130
3. K/parenleftbigg1
k/parenrightbigg
=k/bracketleftbig
K(k)+iK/prime(k)/bracketrightbig
[Im(k)<0] MO 130
For integrals of elliptic integrals, see 6.11–6.15 . For indefinite integrals of complete elliptic integrals,
see5.11.
8.129 Special values:
1. K/parenleftBig
sinπ
4/parenrightBig
=K/parenleftBigg√
2
2/parenrightBigg
=K/prime/parenleftBigg√
2
2/parenrightBigg
=√
2/integraldisplay1
0dt√
1−t4=1
4√π/bracketleftbigg
Γ/parenleftbigg1
4/parenrightbigg/bracketrightbigg2
MO 130
2. K/prime/parenleftBig√
2−1/parenrightBig
=√
2K/parenleftBig√
2−1/parenrightBig
MO 130
8.130 Elliptic functions 865
3. K/prime/parenleftBig
sinπ
12/parenrightBig
=√
3K/parenleftBig
sinπ
12/parenrightBig
MO 130
4. K/prime/parenleftBig
tan2π
8/parenrightBig
=K/prime/parenleftBigg
2−√
2
2+√
2/parenrightBigg
=2K/parenleftBig
tan2π
8/parenrightBig
MO 130
5.∗K/parenleftBig
sinπ
12/parenrightBig
=√
3−1
2√
2
6.∗E=π√
3
12K+/radicalbigg
2
3k/primeK
7.∗E/prime=π√
3
4K/prime+/radicalbigg
2
3kK/prime
8.13 Elliptic functions
8.130 Definition and general properties.
1. A single-valued function f(z) of a complex variable, which is not a constant, is said to be elliptic
if it has two periods 2 ω1and 2ω2,t h a ti s
f(z+2mω1+2nω2)=f(z)[ m, nintegers] .
The ratio of the periods of an analytic function cannot be a real number . For an elliptic function
f(z), the z-plane can be partitioned into parallelograms—the period parallelograms—the vertices
of which are the points z0+2mω1+2nω2. At corresponding points of these parallelograms, the
function f(z) has the same value. ZH 117, SI 299
2. Suppose that αis the angle between the sides aandbof one of the period parallelograms. Then,
τ=ω1
ω2=a
beiα,q=eiπτ=e−a
bπsinα/bracketleftBig
cos/parenleftBiga
bπcosα/parenrightBig
+isin/parenleftBiga
bπcosα/parenrightBig/bracketrightBig
.
3. The derivative of an elliptic function is also an elliptic function with the same periods.
SM III 598
4. A non-constant elliptic function has a finite number of poles in a period parallelogram: it can
have no more than two simple and one second-order pole in such a parallelogram. Suppose that
these poles lie at the points a1,a2,...,anand that their orders are α1,α2,...,αn. Suppose
that the zeros of an analytic function that occur in a single parallelogram are b1,b2,...,bmand
that the orders of the zeros are β1,β2,...,βm, respectively. Then,
γ=α1+α2+···+αn=β1+β2+···+βm. ZH 118
The number γrepresenting this sum is called the order of the elliptic function.
5. The sum of the residues of an elliptic function with respect to all the poles belonging to a period
parallelogram is equal to zero.
6. The difference between the sum of all the zeros and the sum of all the poles of an elliptic function
that are located in a period parallelogram is equal to one of its periods.
7. Every two elliptic functions with the same periods are related by an algebraic relationship.
GO II 151
866 Elliptic Integrals and Functions 8.141
8.7A non-constant single-valued function which is not constant cannot have more than two periods.
GO II 147
9. An elliptic function of order γassumes an arbitrary value γtimes in a period parallelogram.
SM 601, SI 301
8.14 Jacobian elliptic functions
8.141 Consider the upper limit ϕof the integral
u=/integraldisplayϕ
0dα/radicalbig
1−k2sin2α
as a function of u. Using the notation
ϕ=a m u
we call this upper limit the amplitude .T h eq u a n t i t y uis called the argument , and its dependence on ϕ
is written
u=a r g ϕ.
8.142 The amplitude is an infinitely many-valued function of uand has a period of 4 Ki.T h e branch
points of the amplitude correspond to the values of the argument
u=2mK+( 2n+1 )K/primei, ZH 67–69
where mandnare arbitrary integers (see also 8.151 ).
8.143 The first two of the following functions
snu=s i nϕ= sin am u, cnu=c o s ϕ= cosam u,
dnu=Δϕ=/radicalBig
1−k2sin2ϕ=dϕ
du
are called, respectively, the sine-amplitude and the cosine-amplitude while the third may be called the
delta amplitude . All these elliptic functions were exhibited by Jacobi and they bear his name. SI 16
The Jacobian elliptic functions are doubly periodic functions and have two simple poles in a period
parallelogram. ZH 69
8.144
1. u=/integraldisplaysnu
0dt/radicalbig
(1−t2)(1−k2t2)SI 21(23)
2. u=/integraldisplaycnu
1dt/radicalbig
(1−t2)(k/prime2+k2t2)SI 21(23)
3. u=/integraldisplaydnu
1dt/radicalbig
(1−t2)(t2−k/prime2)SI 21(23)
8.145 Power series representations:
1.11snu=u−1+k2
3!u3+1+1 4 k2+k4
5!u5−1 + 135 k2+ 135 k4+k6
7!u7
+1 + 1228 k2+ 5478 k4+ 1228 k6+k8
9!u9−...
[|u|<|K/prime|] ZH 81(97)
8.146 Jacobian elliptic functions 867
2. cn u=1−1
2!u2+1+4k2
4!u4−1+4 4 k2+1 6k4
6!u6+1 + 408 k2+ 912 k4+6 4k6
8!u8−...
[|u|<|K/prime|] ZH 81(98)
3. dn u=
1−k2
2!u2+k2/parenleftbig
4+k2/parenrightbig
4!u4−k2/parenleftbig
16 + 44 k2+k4/parenrightbig
6!u6+k2/parenleftbig
64 + 912 k2+ 408 k4+k6/parenrightbig
8!u8−...
[|u|<|K/prime|] ZH 81(99)
4. am u
=u−k2
3!u3+k2/parenleftbig
4+k2/parenrightbig
5!u5−k2/parenleftbig
16 + 44 k2+k4/parenrightbig
7!u7+k2/parenleftbig
64 + 912 k2+ 408 k4+k6/parenrightbig
9!u9−...
[|u|<|K/prime|] LA 380(4)
8.146 Representation as a trigonometric series or a product/parenleftBig
q=e−πK/prime
K=eπiτ/parenrightBig
∗
1.11snu=2π
kK∞/summationdisplay
n=1qn−1
2
1−q2n−1sin(2n−1)πu
2KWH 511a, ZH 84(108)
2.11cnu=2π
kK∞/summationdisplay
n=1qn−1
2
1+q2n−1cos(2n−1)πu
2KWH 511a, ZH 84(109)
3. dn u=π
2K+2π
K∞/summationdisplay
n=1qn
1+q2ncosnπu
KWH 511a, ZH 84(110)
4.11amu=πu
2K+2∞/summationdisplay
n=11
nqn
1+q2nsinnπu
KWH 511a
5.1
snu=π
2K/bracketleftBigg
1
sinπu
2K+4∞/summationdisplay
n=1q2n−1
1−q2n−1sin(2n−1)πu
2K/bracketrightBigg
LA 369(3)
6.1
cnu=π
2k/primeK/bracketleftBigg
1
cosπu
2K+4∞/summationdisplay
n=1(−1)nq2n−1
1+q2n−1cos(2n−1)πu
2K/bracketrightBigg
LA 369(3)
7.1
dnu=π
2k/primeK/bracketleftBigg
1+4∞/summationdisplay
n=1(−1)nqn
1+q2ncosnπu
K/bracketrightBigg
LA 369(3)
8.snu
cnu=π
2k/primeK/bracketleftBigg
tanπu
2K+4∞/summationdisplay
n=1(−1)nq2n
1+q2nsinnπu
K/bracketrightBigg
LA 369(4)
9.11snu
dnu=−2π
kk/primeK∞/summationdisplay
n=1(−1)nqn−1
2
1+q2n−1sin(2n−1)πu
2KLA 369(4)
10.cnu
snu=π
2K/bracketleftBigg
cotπu
2K−4∞/summationdisplay
n=1q2n
1+q2nsinπnu
K/bracketrightBigg
LA 369(5)
∗The expansions 1–22 are valid in every strip of the form/vextendsingle/vextendsingle/vextendsingleImπu
2K/vextendsingle/vextendsingle/vextendsingle<1
2πImτ. The expansions 23–25 are valid in an
arbitrary bounded portion of u.
868 Elliptic Integrals and Functions 8.146
11.cnu
dnu=−2π
kK∞/summationdisplay
n=1(−1)nqn−1
2
1−q2n−1cos(2n−1)πu
2KLA 369(5)
12.dnu
snu=π
2K/bracketleftBigg
1
sinπu
2K−4∞/summationdisplay
n=1q2n−1
1+q2n−1sin(2n−1)πu
2K/bracketrightBigg
LA 369(6)
13.dnu
cnu=π
2K/bracketleftBigg
1
cosπu
2K−4∞/summationdisplay
n=1(−1)nq2n−1
1−q2n−1cos(2n−1)πu
2K/bracketrightBigg
LA 369(6)
14.cnudnu
snu=π
2K/bracketleftBigg
cotπu
2K−4∞/summationdisplay
n=1qn
1+qnsinnπu
K/bracketrightBigg
LA 369(7)
15.snudnu
cnu=π
2K/braceleftBigg
tanπu
2K+4∞/summationdisplay
n=1qn
1+(−1)nqnsinnπu
K/bracerightBigg
LA 369(7)
16.snucnu
dnu=4π2
k2K∞/summationdisplay
n=1q2n−1
1−q2(2n−1)sin(2n−1)πu
KLA 369(7)
17.snu
cnudnu=π
2( 1−k2)K/bracketleftBigg
tanπu
2K+4∞/summationdisplay
n=1(−1)nqn
1−qnsinnπu
K/bracketrightBigg
LA 369(8)
18.cnu
snudnu=π
2K/bracketleftBigg
cotπu
2K−4∞/summationdisplay
n=1(−1)nqn
1+(−1)nqnsinnπu
K/bracketrightBigg
LA 369(8)
19.dnu
snucnu=π
K/bracketleftBigg
1
sinπu
K+4∞/summationdisplay
n=1q2(2n−1)
1−q2(2n−1)sin(2n−1)πu
K/bracketrightBigg
LA 369(8)
20.11lnsnu=l n2K
π+l ns i nπu
2K−4∞/summationdisplay
n=11
nqn
1+qnsin2nπu
2KLA 369(2)
21. lncn u=l nc o sπu
2K−4∞/summationdisplay
n=11
nqn
1+(−1)nqnsin2nπu
2KLA 369(2)
22. lndn u=−8∞/summationdisplay
n=11
2n−1q2n−1
1−q2(2n−1)sin2(2n−1)πu
2KLA 369(2)
23.11snu=24√q√
ksinπu
2K∞/productdisplay
n=11−2q2ncosπu
K+q4n
1−2q2n−1cosπu
K+q4n−2WH 508a, ZH 86(145)
24. cn u=2√
k/prime4√q√
kcosπu
2K∞/productdisplay
n=11+2q2ncosπu
K+q4n
1−2q2n−1cosπu
K+q4n−2WH 508a, ZH 86(146)
25. dn u=√
k/prime∞/productdisplay
n=11+2q2n−1cosπu
K+q4n−2
1−2q2n−1cosπu
K+q4n−2WH 508a, ZH 86(147)
26. sn3u=∞/summationdisplay
n=0/bracketleftBigg
1+k2
2k3−(2n+1 )2
2k3π2
4K2/bracketrightBigg
2πqn+1
2sin(2n+1 )πu
2K
K(1−q2n+1)
/bracketleftBig/vextendsingle/vextendsingle/vextendsingleImu
2K/vextendsingle/vextendsingle/vextendsingle<Imτ/bracketrightBig
MO 147
8.148 Jacobian elliptic functions 869
27.1
sn2u=π2
4K2cosec2πu
2K+K−E
K−2π2
K2∞/summationdisplay
n=1nq2ncosnπu
K
1−q2n
/bracketleftbigg/vextendsingle/vextendsingle/vextendsingleImu
2K/vextendsingle/vextendsingle/vextendsingle<1
2Imτ/bracketrightbigg
MO 148
8.147
1. sn u=π
2kK∞/summationdisplay
n=−∞1
sinπ
2K[u−(2n−1)iK/prime]MO 149
2. cn u=πi
2kK∞/summationdisplay
n=−∞(−1)n
sinπ
2K[u−(2n−1)iK/prime]MO 150
3. dn u=πi
2K∞/summationdisplay
n=−∞(−1)n
tanπ
2K[u−(2n−1)iK/prime]MO150
8.148 The Weierstrass expansions of the functions sn u,c nu,d nu:
snu=B
A,cnu=C
A,dnu=D
A, ZH 82–83(105,106,107)
where
A=1−∞/summationdisplay
n=1(−1)n+1an+1u2n+2
(2n+2 ) !B=∞/summationdisplay
n=0(−1)nbnu2n+1
(2n+1 ) !
C=∞/summationdisplay
n=0(−1)ncnu2n
(2n)!D=∞/summationdisplay
n=0(−1)ndnu2n
(2n)!
and
a2=2k2,a3=8/parenleftbig
k2+k4/parenrightbig
,a4=3 2/parenleftbig
k2+k6/parenrightbig
+6 8k4,a5= 128/parenleftbig
k2+k8/parenrightbig
+ 480/parenleftbig
k4+k6/parenrightbig
,
a6= 512/parenleftbig
k2+k10/parenrightbig
+ 3008/parenleftbig
k4+k8/parenrightbig
+ 5400 k6, ...
b0=1,b1=1+ k2,b2=1+ k4+4k2,b3=1+ k6+9/parenleftbig
k2+k4/parenrightbig
,
b4=1+ k8+1 6/parenleftbig
k2+k6/parenrightbig
−6k4,b5=1+ k10+2 5/parenleftbig
k2+k8/parenrightbig
−494/parenleftbig
k4+k6/parenrightbig
,
b6=1+ k12+3 6/parenleftbig
k2+k10/parenrightbig
−5781/parenleftbig
k4+k8/parenrightbig
−12184 k6, ...
c0=1,c1=1,c2=1+2 k2,c3=1+6 k2+8k4,c4=1+1 2 k2+6 0k4+3 2k6,
c5=1+2 0 k2+ 348 k4+ 448 k6+ 128 k8,c6=1+3 0 k2+ 2372 k4+ 4600 k6+ 2880 k8+ 512 k10, ...
d0=1,d1=k2,d2=2k2+k4,d3=8k2+6k4+k6,d4=3 2k2+6 0k4+1 2k4+k8,
d5= 128 k2+ 448 k4+ 348 k6+2 0k8+k10,
d6= 512 k2+ 2880 k4+ 4600 k6+ 2372 k8+3 0k10+k12, ...
870 Elliptic Integrals and Functions 8.151
8.15 Properties of Jacobian elliptic functions and functional relationships between
them
8.151 The periods, zeros, poles, and residues of Jacobian elliptic functions:
1.
Periods Zeros Poles Residues
snu 4mK+2nK/primei 2mK+2nK/primei 2mK+( 2n+1 )K/primei(−1)m1
k
cnu4mK+2n(K+K/primei) (2m+1 )K+2nK/primei 2mK+( 2n+1 )K/primei(−1)m−1i
k
dnu 2mK+4nK/primei (2m+1 )K+( 2n+1 )K/primei2mK+( 2n+1 )K/primei(−1)n−1i
SM 630, ZH 69–72
2.
u∗=u+K u+iKu+K+iK/primeu+2Ku+2iK/primeu+2K+2iK/prime
snu∗=cnu
dnu1
ksnu1
kdnu
cnu−snu snu −snu
cnu∗=−k/primesnu
dnu−i
kdnu
snu−ik/prime
kcnu−cnu−cnu cnu
dnu∗=k/prime1
dnu−icnu
snuik/primesnu
cnudnu −dnu −dnu
SM 630
3.
u∗=0 −u1
2K1
2(K+iK/prime)1
2iK/primeu+2mK+2nK/primei
snu∗=0 −snu1√
1+k/prime√
1+k+i√
1−k√
2ki√
k(−1)msnu
cnu∗=1 cnu√
k/prime
√
1+k/prime(1−i)√
k/prime
√
2k√
1+k√
k(−1)m+ncnu
dnu∗=1 dnu√
k/prime√
k/prime/parenleftbig√
1+k/prime−i√
1−k/prime/parenrightbig
√
2√
1+k (−1)ndnu
SI 19, SI 18(13), WH, WH WH WH
8.152 Jacobian elliptic functions 871
8.152 Transformation formulasu1 l1 sn(u1,k1) cn (u1,k1) dn (u1,k1)
ku1
kksn(u,k) dn(u,k) cn(u,k)
iu k/primeisn(u,k)
cn(u,k)1
cn(u,k)dn(u,k)
cn(u,k)
k/primeu ik
k/primek/primesn(u,k)
dn(u,k)cn(u,k)
dn(u,k)1
dn(u,k)
iku ik/prime
kiksn(u,k)
dn(u,k)1
dn(u,k)cn(u,k)
dn(u,k)
ik/primeu1
k/primeik/primesn(u,k)
cn(u,k)dn(u,k)
cn(u,k)1
cn(u,k)
(1 +k)u2√
k
1+k(1 +k)sn(u,k)
1+ksn2(u,k)cn(u,k)d n(u,k)
1+ksn2(u,k)1−ksn2(u,k)
1+ksn2(u,k)
(1 +k/prime)u1−k/prime
1+k/prime(1 +k/prime)sn(u,k)cn(u,k)
dn(u,k)1−(1 +k/prime)s n2(u,k)
dn(u,k)1−(1−k/prime)s n2(u,k)
dn(u,k)/parenleftBig
1+√
k/prime/parenrightBig2
2u/parenleftBigg
1−√
k/prime
1+√
k/prime/parenrightBigg2
k2sn(u,k)dcn(u,k)√k1[1 + dn( u,k)] [k/prime+d n ( u,k)]dn(u,k)−√
k/prime
1−√
k/prime√1+k1/parenleftBig
dn(u,k)+√
k/prime/parenrightBig
/radicalbig
[1 + dn( u,k)] [k/prime+d n ( u,k)]
×/radicalBig
2(1+k/prime)
[1+dn( u,k)][k/prime+dn( u,k)]
JA
872 Elliptic Integrals and Functions 8.153
8.153
1. sn( iu,k)=isn (u,k/prime)
cn (u,k/prime)SI 50(64)
2. cn( iu,k)=1
cn (u,k/prime)SI 50(65)
3. dn( iu,k)=dn (u,k/prime)
cn(u,k/prime)SI 50(65)
4. sn( u,k)=k−1sn/parenleftbig
ku,k−1/parenrightbig
5. cn( u,k)=d n/parenleftbig
ku,k−1/parenrightbig
6. dn( u,k)=c n/parenleftbig
ku,k−1/parenrightbig
7.11sn(u,ik)=1√
1+k2sn/parenleftBig
u√
1+k2,k/parenleftbig
1+k2/parenrightbig−1/2/parenrightBig
dn/parenleftBig
u√
1+k2,k(1 +k2)−1/2/parenrightBig
8.11cn(u,ik)=sn/parenleftBig
u/parenleftbig
1+k2/parenrightbig1/2,k/parenleftbig
1+k2/parenrightbig−1/2/parenrightBig
dn/parenleftBig
u(1 +k2)1/2,k(1 +k2)−1/2/parenrightBig
9.11dn(u,ik)=1
dn/parenleftBig
u(1 +k2)1/2,k(1 +k2)−1/2/parenrightBig
Functional relations
8.154
1. sn2u=1−cn2u
1+d n2 uMO 146
2. cn2u=cn 2u+d n2 u
1+d n2 uMO 146
3. dn2u=dn 2u+k2cn 2u+k/prime2
1+d n2 uMO 146
4. sn2u+c n2u=1 SI 16(9)
5. dn2u+k2sn2u=1 SI 16(9)
8.155
1.1−dn 2u
1+d n2 u=k2sn2ucn2u
dn2uMO 146
2.1−cn2u
1 + cn2 u=sn2udn2u
cn2uMO 146
8.156
1. sn ( u±v)=snucnvdnv±snvcnudnu
1−k2sn2usn2vSI 46(56)
8.160 The Weierstrass function ℘(u) 873
2. cn ( u±v)=cnucnv∓snusnvdnudnv
1−k2sn2usn2vSI 46(57)
3. dn ( u±v)=dnudnv∓k2snusnvcnucnv
1−k2sn2usn2vSI 46(58)
8.157
1. snu
2=±1
k/radicalbigg
1−dnu
1+c n u=±/radicalbigg
1−cnu
1+d n uSI 47(61), SU 67(15)
2. cnu
2=±/radicalbigg
cnu+d nu
1+d n u=±k/prime
k/radicalbigg
1−dnu
dnu−cnuSI 48(62), SI 67(16)
3. dnu
2=±/radicalbigg
cnu+d nu
1+c n u=±k/prime/radicalbigg
1−cnu
dnu+c nuSI 48(63), SI 67(17)
8.158
1.d
dusnu=c nudnu SI 21(21)
2.d
ducnu=−snudnu SI 21(21)
3.8d
dudnu=−k2dnucnu SI 21(21)
8.159 Jacobian elliptic functions are solutions of the following differential equations:
1.d
dusnu=/radicalbig
(1−sn2u)(1−k2sn2u) SI 21(22)
2.d
ducnu=−/radicalbig
(1−cn2u)(k/prime2+k2cn2u), SI 21(22)
3.d
dudnu=−/radicalBig
(1−dn2u)(d n2u−k/prime2) SI 21(22)
For the indefinite integrals of Jacobi’s elliptic functions, see 5.13.
8.16 The Weierstrass function ℘(u)
8.160 The Weierstrass elliptic function ℘(u) is defined by
1. ℘(u)=1
u2+/summationdisplay/prime
m,n/braceleftBigg
1
(u−2mω1−2nω2)2−1
(2mω1+2nω2)2/bracerightBigg
, SI 307(6)
where the symbol/summationtext/primemeans that the summation is made over all combinations of integers m
andnexcept for the combination m=n=0 ;2 ω1and 2ω2are the periods of the function ℘(u).
Obviously,
2. ℘(u+2mω1+2nω2)=℘(u)a n d I m/parenleftbiggω1
ω2/parenrightbigg
/negationslash=0,
874 Elliptic Integrals and Functions 8.161
3.d
du℘(u)=−2/summationdisplay
m,n1
(u−2mω1−2nω2)3,
where the summation is made over all integral values of mandn.
The series 8.160 1a n d8.160 3 converge everywhere except at the poles, that is, at the points
2mω1+2nω2(where mandnare integers).
4. The function ℘(u)i sadoubly periodic function and has one second-order pole in a period paral-
lelogram. SI 306
8.161 The function ℘(u) satisfies the differential equation
1./bracketleftbiggd℘(u)
du/bracketrightbigg2
=4℘3(u)−g2℘(u)−g3, SI 142, 310, WH
where
2. g2=6 0/summationdisplay/prime
m,n(mω1+nω2)−4;g3= 140/summationdisplay/prime
m,n(mω1+nω2)−6WH, SI 310
The functions g2andg3are called the invariants of the function ℘(u).
8.162 u=/integraldisplay∞
℘(u)dz/radicalbig
4z3−g2z−g3=/integraldisplay∞
℘(u)dz/radicalbig
4(z−e1)(z−e2)(z−e3),
where e1,e2,a n d e3are the roots of the equation 4 z3−g2z−g3=0 ;t h a ti s ,
e1+e2+e3=0,e1e2+e2e3+e3e1=−g2
4,e1e2e3=g3
4SI 142, 143, 144
8.163 ℘(ω1)=e1,℘(ω1)+ω2=e2,℘(ω2)=e3. Here, it is assumed that if e1,e2,a n d e3lie on a
straight line in the complex plane, e2lies between e1ande3.
8.164 The number Δ = g3
2−27g2
3is called the discriminant of the function ℘(u). If Δ >0, all roots
e1,e2,a n d e3of the equation 4 z3−g2z−g3=0( w h e r e g2andg3are real numbers) are real.I n t h i s
case, the roots e1,e2,a n d e3are numbered in such a way that e1>e2>e3.
1. If Δ >0, then
ω1=/integraldisplay∞
e1dz/radicalbig
4z3−g2z−g3,ω 2=i/integraldisplaye3
−∞dz/radicalbig
g3+g2z−4z3,
where ω1is real and ω2is a purely imaginary number. Here, the values of the radical in the
integrand are chosen in such a way that ω1andω2
iwill be positive.
2. If Δ <0, the root e2of the equation 4 z3−g2z−g3=0i s real, and the remaining two roots ( e1
ande3)a r ecomplex conjugates . Suppose that e1=α+iβ,a n d e3=α−iβ. In this case, it is
convenient to take
ω/prime=/integraldisplay∞
e1dz/radicalbig
4z3−g2z−g3and ω/prime/prime=/integraldisplay∞
e3dz/radicalbig
4z3−g2z−g3
as basic semiperiods.
In the first integral, the integration is taken over a path lying entirely in the upper half-plane and in
the second over a path lying entirely in the lower half-plane. SI 151(21, 22)
8.169 The Weierstrass function ℘(u) 875
8.165 Series representation:
1. ℘(u)=1
u2+g2u2
4·5+g3u4
4·7+g2
2u6
24·3·52+3g2g3u8
24·5·7·11+... WH
8.166 Functional relations
1. ℘(u)=℘(−u),℘/prime(u)=−℘/prime(−u)
2. ℘(u+v)=−℘(u)−℘(v)+1
4/bracketleftbigg℘/prime(u)−℘/prime(v)
℘(u)−℘(v)/bracketrightbigg2
SI 163(32)
8.167 ℘(u;g2,g3)=μ2℘/parenleftbigg
μu;g2
μ4,g3
μ6/parenrightbigg
(the formula for homogeneity)
SI 149(13)
The special case: μ=i.
1. ℘(u;g2,g3)=−℘(iu;g2,−g3)
8.168 An arbitrary elliptic function can be expressed in terms of the elliptic function ℘(u)h a v i n gt h e
same periods as the original function and its derivative ℘/prime(u). This expression is rational with respect to
℘(u) and linear with respect to ℘/prime(u).
8.169 A connection with the Jacobian elliptic functions. For Δ >0( s e e8.164 1).
1. ℘/parenleftbiggu√e1−e2/parenrightbigg
=e1+(e1−e3)cn2(u;k)
sn2(u;k)
=e2+(e1−e3)dn2(u;k)
sn2(u;k)
=e3+(e1−e3)1
sn2(u;k)
SI 145(5), ZH 120(197–199)a
2. ω1=K√e1−e3,ω 2=iK/prime
√e1−e3, SI 154(29)
where
3. k=/radicalbigge2−e3
e1−e3,k/prime=/radicalbigge1−e2
e1−e3SI 145(7)
For Δ <0( s e e8.164 2)
4. ℘/parenleftBigg
u
4/radicalbig
9α2+β2/parenrightBigg
=e2+/radicalbig
9α2+β21+c n ( 2 u;k)
1−cn(2u;k); SI 147(12)
5. ω/prime=K−iK/prime
2/radicalbig
9α2+β2,ω/prime/prime=K+iK/prime
4/radicalbig
9α2+β2, SI 153(28)
where
6.11k=/radicalBigg
1
2−3e2
4/radicalbig
9α2+β2;k/prime=/radicalBigg
1
2+3e2
4/radicalbig
9α2+β2SI 147
For Δ = 0, all the roots e1,e2,a n d e3are real, and if g2g3/negationslash= 0, two of them are equal to each
other. If e1=e2/negationslash=e3,t h e n
876 Elliptic Integrals and Functions 8.171
7. ℘(u)=3g3
g2−9g3
2g2coth2/parenleftbigg
u/radicalbigg
−9g3
2g2/parenrightbigg
SI 148
Ife1/negationslash=e2=e3,t h e n
8. ℘(u)=−3g3
2g2+9g3
2g21
sin2/parenleftBig
u/radicalBig
9g3
2g2/parenrightBig SI 149
Ifg2=g3=0 ,t h e n e1=e2=e3=0 ,a n d
9. ℘(u)=1
u2SI 149
8.17 The functions ζ(u) and σ(u)
8.171 Definitions:
1. ζ(u)=1
u−/integraldisplayu
0/parenleftbigg
℘(z)−1
z2/parenrightbigg
dz SI 181(45)
2. σ(u)=uexp/braceleftbigg/integraldisplayu
0/parenleftbigg
℘(z)−1
z2/parenrightbigg
dz/bracerightbigg
SI 181(46)
8.172 Series and infinite-product representation
1. ζ(u)=1
u+/summationdisplay/prime
m,n/parenleftBigg
1
u−2mω1−2nω2+1
2mω1+2nω2+u
(2mω1−2nω2)2/parenrightBigg
SI 307(8)
2. σ(u)=u/productdisplay/prime
mn,/parenleftbigg
1−u
2mω1+2nω2/parenrightbigg
exp/braceleftBigg
u
2mω1+2nω2+u2
2(2mω1+2nω2)2/bracerightBigg
SI 308(9)
8.173
1. ζ(u)=u−g2u3
22·3·5−g3u5
22·5·7−g2
2u7
24·3·52·7−3g2g3u9
24·5·7·9·11−··· SI 181(49)
2. σ(u)=u−g2u5
24·3·5−g3u7
23·3·5·7−g2
2u9
29·32·5·7−3g2g3u11
27·32·52·7·11−··· SI 181(49)
8.174 ζ(u)=ζ(ω1)
ω1u+π
2ω1cotπu
2ω1+π
2ω1∞/summationdisplay
n=1/braceleftbigg
cot/parenleftbiggπu
2ω1+nπω2
ω1/parenrightbigg
+c o t/parenleftbiggπu
2ω1−nπω2
ω1/parenrightbigg/bracerightbigg
MO 154
=ζ(ω1)
ω1u+π
2ω1cotπu
2ω1+2π
ω1∞/summationdisplay
n=1q2n
1−q2nsinπnu
ω1MO 155
Functional relations and properties
8.175 ζ(u)=−ζ(−u),σ(u)=−σ(−u) SI 181
8.176
1. ζ(u+2ω1)=ζ(u)+2ζ(ω1) SI 184(57)
8.181 Theta functions 877
2. ζ(u+2ω2)=ζ(u)+2ζ(ω2) SI 184(57)
3. σ(u+2ω1)=−σ(u)exp{2(u+ω1)ζ(ω1)}. SI 185(60)
4. σ(u+2ω2)=−σ(u)exp{2(u+ω2)ζ(ω2)}. SI 185(60)
5. ω2ζ(ω1)−ω1ζ(ω2)=π
2i SI 186(62)
8.177
1. ζ(u+v)−ζ(u)−ζ(v)=1
2℘/prime(u)−℘/prime(v)
℘(u)−℘(v)SI 182(53)
2. ℘(u)−℘(v)=−σ(u−v)σ(u+v)
σ2(u)σ2(v)SI 183(54)
3. ζ(u−v)+ζ(u+v)−2ζ(u)=℘/prime(u)
℘(u)−℘(v)SI 182(51)
8.178
1. ζ(u;ω1,ω2)=tζ(tu;tω1,t ω2) MO 154
2.8σ(u;ω1,ω2)=t−1σ(tu;tω1,t ω2) MO 156
For the indefinite integrals of Weierstrass elliptic functions, see 5.14.
8.18–8.19 Theta functions
8.180 Theta functions are defined as the sums (for |q|<1) of the following series:
1. ϑ4(u)=∞/summationdisplay
n=−∞(−1)nqn2e2nui=1+2∞/summationdisplay
n=1(−1)nqn2cos2nu WH
2. ϑ1(u)=1
i∞/summationdisplay
n=−∞(−1)nq(n+1
2)2
e(2n+1)ui=2∞/summationdisplay
n=1(−1)n+1q(n−1
2)2
sin(2n−1)u WH
3.11ϑ2(u)=∞/summationdisplay
n=−∞q(n+1
2)2
e(2n+1)ui=2∞/summationdisplay
n=1q(n−1
2)2
cos(2n−1)u WH
4. ϑ3(u)=∞/summationdisplay
n=−∞qn2e2nui=1+2∞/summationdisplay
n=1qn2cos2nu WH
The notations ϑ(u,q)a n d ϑ(u|τ), where τandqare related by q=eiπτ, are also used. Here, qis called
thenome of the theta function and τitsparameter .
8.181 Representation of theta functions in terms of infinite products
1. ϑ4(u)=∞/productdisplay
n=1/parenleftBig
1−2q2n−1cos 2u+q2(2n−1)/parenrightBig/parenleftbig
1−q2n/parenrightbig
SI 200(9), ZH 90(9)
2. ϑ3(u)=∞/productdisplay
n=1/parenleftBig
1+2q2n−1cos 2u+q2(2n−1)/parenrightBig/parenleftbig
1−q2n/parenrightbig
SI 200(9), ZH 90(9)
878 Elliptic Integrals and Functions 8.182
3. ϑ1(u)=24√qsinu∞/productdisplay
n=1/parenleftbig
1−2q2ncos2u+q4n/parenrightbig/parenleftbig
1−q2n/parenrightbig
SI 200(9), ZH 90(9)
4.8ϑ2(u)=24√qcosu∞/productdisplay
n=1/parenleftbig
1+2q2ncos2u+q4n/parenrightbig/parenleftbig
1−q2n/parenrightbig
SI 200(0), ZH 90(9)
Functional relations and properties
8.182 Quasiperiodicity. Suppose that q=eπτi(Imτ>0). Then, theta functions that are periodic
functions of uare called quasiperiodic functions ofτandu. This property follows from the equations
1. ϑ4(u+π)=ϑ4(u) SI 200(10)
2. ϑ4(u+τπ)=−1
qe−2iuϑ4(u) SI 200(10)
3. ϑ1(u+π)=−ϑ1(u) SI 200(10)
4. ϑ1(u+τπ)=−1
qe−2iuϑ1(u) SI 200(10)
5. ϑ2(u+π)=−ϑ2(u) SI 200(10)
6. ϑ2(u+τπ)=1
qe−2iuϑ2(u) SI 200(10)
7. ϑ3(u+π)=ϑ3(u) SI 200(10)
8. ϑ3(u+τπ)=1
qe−2iuϑ3(u) SI 200(10)
8.183
1. ϑ4/parenleftbig
u+1
2π/parenrightbig
=ϑ3(u) WH
2. ϑ1/parenleftbig
u+1
2π/parenrightbig
=ϑ2(u) WH
3. ϑ2/parenleftbig
u+1
2π/parenrightbig
=−ϑ1(u) WH
4. ϑ3/parenleftbig
u+1
2π/parenrightbig
=ϑ4(u) WH
5. ϑ4/parenleftbig
u+1
2πτ/parenrightbig
=iq−1/4e−iuϑ1(u) WH
6. ϑ1/parenleftbig
u+1
2πτ/parenrightbig
=iq−1/4e−iuϑ4(u) WH
7. ϑ2/parenleftbig
u+1
2πτ/parenrightbig
=q−1/4e−iuϑ3(u) WH
8. ϑ3/parenleftbig
u+1
2πτ/parenrightbig
=q−1/4e−iuϑ2(u) WH
8.184 Even and odd theta functions
1. ϑ1(−u)=−ϑ1(u) WH
2. ϑ2(−u)=ϑ2(u) WH
3. ϑ3(−u)=ϑ3(u) WH
4. ϑ4(−u)=ϑ4(u) WH
8.185 ϑ4
4(u)+ϑ4
2(u)=ϑ4
1(u)+ϑ4
3(u) WH
8.1867Considering the theta functions as functions of two independent variables uandτ,w eh a v e
8.192 Theta functions 879
πi∂2ϑk(u|τ)
∂u2+4∂ϑk(u|τ)
∂τ=0 [ k=1,2,3,4] WH
8.187 We denote the partial derivatives of the theta functions with respect to uby a prime and consider
them as functions of the single argument u. Then,
1. ϑ/prime
1(0) = ϑ2(0)ϑ3(0)ϑ4(0) WH
2.ϑ/prime/prime/prime
1(0)
ϑ/prime
1(0)=ϑ/prime/prime
2(0)
ϑ2(0)+ϑ/prime/prime
3(0)
ϑ3(0)+ϑ/prime/prime
4(0)
ϑ4(0)WH
8.188 ϑ1(u)ϑ2(u)ϑ3(u)ϑ4(0) =1
2ϑ1(2u)ϑ2(0)ϑ3(0)ϑ4(0) WH
8.189 The zeros of the theta functions:
1.8ϑ4(u)=0f o r u=2mπ
2+( 2n−1)πτ
2SI 201
2.10ϑ1(u)=0f o r u=2mπ
2+2nπτ
2SI 201
3. ϑ2(u)=0f o r u=( 2m−1)π
2+2nπτ
2SI 201
4. ϑ3(u)=0f o r u=( 2m−1)π
2+( 2n−1)πτ
2[mandnare integers or zero] SI 201
For integrals of theta functions, see 6.16.
8.191 Connections with the Jacobian elliptic functions:
Forτ=iK/prime
K, i.e. for q=e x p/parenleftBig
−πK/prime
K/parenrightBig
,
1. sn u=1√
kϑ1/parenleftBigπu
2K/parenrightBig
ϑ4/parenleftBigπu
2K/parenrightBig=1√
kH(u)
Θ(u)SI 206(22), SI 209(35)
2. cn u=/radicalbigg
k/prime
kϑ2/parenleftBigπu
2K/parenrightBig
ϑ4/parenleftBigπu
2K/parenrightBig=/radicalbigg
k/prime
kH1(u)
Θ(u)SI 207(23), SI 209(35)
3. dn u=√
k/primeϑ3/parenleftBigπu
2K/parenrightBig
ϑ4/parenleftBigπu
2K/parenrightBig=√
k/primeΘ1(u)
Θ(u)SI 207(24), SI 209(35)
8.192 Series representation of the functions H,H1,Θ ,Θ 1.
In these formulas, q=e x p/parenleftBig
−πK/prime
K/parenrightBig
.
1. Θ( u)=ϑ4/parenleftBigπu
2K/parenrightBig
=1+2∞/summationdisplay
n=1(−1)nqn2cosnπu
KSI 207(25), SI 212(42)
2. H(u)=ϑ1/parenleftBigπu
2K/parenrightBig
=2∞/summationdisplay
n=1(−1)n+14/radicalBig
q(2n+1)2sin(2n−1)πu
2KSI 207(25), SI 212(43)
3. Θ 1(u)=ϑ3/parenleftBigπu
2K/parenrightBig
=1+2∞/summationdisplay
n=1qn2cosnπu
KSI 207(25), SI 212(45)
880 Elliptic Integrals and Functions 8.193
4. H1(u)=ϑ2/parenleftBigπu
2K/parenrightBig
=2∞/summationdisplay
n=14/radicalbig
q(2n−1)2cos(2n−1)πu
2KSI 207(25), SI 212(44)
8.193 Connections with the Weierstrass elliptic functions
1. ℘(u)=e1+⎡
⎣H1/parenleftBig
u√
λ/parenrightBig
H/prime(0)
H1(0)H/parenleftBig
u√
λ/parenrightBig⎤
⎦2
λ=e2+⎡
⎣Θ1/parenleftBig
u√
λ/parenrightBig
H/prime(0)
Θ1(0)H/prime/parenleftBig
u√
λ/parenrightBig⎤
⎦2
λ=e3+⎡
⎣Θ/parenleftBig
u√
λ/parenrightBig
H/prime(0)
Θ(0)H/prime/parenleftBig
u√
λ/parenrightBig⎤
⎦2
λ
SI 235(77,78)
2. ζ(u)=η1u
ω1+√
λH/prime/parenleftBig
u√
λ/parenrightBig
H/parenleftBig
u√
λ/parenrightBig SI 234(73)
3. σ(u)=1√
λexp/parenleftbiggη1u2
2ω1/parenrightbiggH/parenleftBig
u√
λ/parenrightBig
H/prime(0)SI 234(72)
where
λ=e1−e3;η1=ζ(ω1)=−ω1λ
3H/prime/prime/prime(0)
H/prime(0)SI 236
8.194 The connection with elliptic integrals:
1. E(u,k)=u−uΘ/prime/prime(0)
Θ(0)+Θ/prime(u)
Θ(u)SI 228(65)
2.11Π/parenleftbig
u,−k2sin2a,k/parenrightbig
=/integraldisplayu
0dϕ
1−k2sin2asn2ϕ=u+sna
cnadna/bracketleftbiggΘ/prime(a)
Θ(a)u+1
2lnΘ(u−a)
Θ(u+a)/bracketrightbigg
SI 228(65)
q-series and products, q=e x p/parenleftBig
−πK/prime
K/parenrightBig
8.195π
2/bracketleftBigg
1+2∞/summationdisplay
n=1qn2/bracketrightBigg2
=K=π
2Θ2(K) (cf. 8.197 1) SI 219
8.196 E=K−KΘ/prime/prime(0)
Θ(0)=K−2π2
K∞/summationdisplay
n=1(−1)n+1n2qn2
1+2∞/summationdisplay
n=1(−1)nqn2SI 230(67)
8.197
1. 1 + 2∞/summationdisplay
n=1qn2=/radicalbigg
2K
π=ϑ3(0) (cf. 8.195 ) WH
2.∞/summationdisplay
n=1q(2n−1
2)2
=/radicalbigg
kK
2π=1
2ϑ2(0) WH
8.199 Theta functions 881
3. 4√q∞/productdisplay
n=1/parenleftbigg1+q2n
1+q2n−1/parenrightbigg4
=k SI 206(17, 18)
4.∞/productdisplay
n=1/parenleftbigg1−q2n−1
1+q2n−1/parenrightbigg4
=k/primeSI 206(19, 20)
5. 24√q∞/productdisplay
n=1/parenleftbigg1−q2n
1−q2n−1/parenrightbigg2
=2√
kK
πWH
6.∞/productdisplay
n=1/parenleftbigg1−q2n
1+q2n/parenrightbigg2
=2√
k/primeK
πWH
8.198
1. λ=1
21−√
k/prime
1+√
k/prime=∞/summationdisplay
n=0q(2n+1)2
1+2∞/summationdisplay
n=1q4n2[for 0 <k< 1, we have 0 <λ<1
2]WH
The series
2. q=λ+2λ5+1 5λ9+ 150 λ13+ 1707 λ17+... WH
is used to determine qfrom the given modulus k.
8.19910Identities involving products of theta functions
1. ϑ1(x, q)ϑ1(y,q)=ϑ3/parenleftbig
x+y,q2/parenrightbig
ϑ2/parenleftbig
x−y,q2/parenrightbig
−ϑ2/parenleftbig
x+y,q2/parenrightbig
ϑ3/parenleftbig
x−y,q2/parenrightbig
LW 7(1.4.7)
2. ϑ1(x, q)ϑ2(y,q)=ϑ1/parenleftbig
x+y,q2/parenrightbig
ϑ4/parenleftbig
x−y,q2/parenrightbig
+ϑ4/parenleftbig
x+y,q2/parenrightbig
ϑ1/parenleftbig
x−y,q2/parenrightbig
LW 8(1.4.8)
3. ϑ2(x, q)ϑ2(y,q)=ϑ2/parenleftbig
x+y,q2/parenrightbig
ϑ3/parenleftbig
x−y,q2/parenrightbig
+ϑ3/parenleftbig
x+y,q2/parenrightbig
ϑ2/parenleftbig
x−y,q2/parenrightbig
LW 8(1.4.9)
4. ϑ3(x, q)ϑ3(y,q)=ϑ3/parenleftbig
x+y,q2/parenrightbig
ϑ3/parenleftbig
x−y,q2/parenrightbig
+ϑ2/parenleftbig
x+y,q2/parenrightbig
ϑ2/parenleftbig
x−y,q2/parenrightbig
LW 8(1.4.10)
5. ϑ3(x, q)ϑ4(y,q)=ϑ4/parenleftbig
x+y,q2/parenrightbig
ϑ4/parenleftbig
x−y,q2/parenrightbig
−ϑ1/parenleftbig
x+y,q2/parenrightbig
ϑ1/parenleftbig
x−y,q2/parenrightbig
LW 8(1.4.11)
6. ϑ4(x, q)ϑ4(y,q)=ϑ3/parenleftbig
x+y,q2/parenrightbig
ϑ3/parenleftbig
x−y,q2/parenrightbig
−ϑ2/parenleftbig
x+y,q2/parenrightbig
ϑ2/parenleftbig
x−y,q2/parenrightbig
LW 8(1.4.12)
7. ϑ1(x+y)ϑ1(x−y)ϑ2
4(0) = ϑ2
3(x)ϑ2
2(y)−ϑ2
2(x)ϑ2
3(y)=ϑ2
1(x)ϑ2
4(y)−ϑ2
4(x)ϑ2
1(y) LW 8(1.4.16)
8. ϑ2(x+y)ϑ2(x−y)ϑ2
4(0) = ϑ2
4(x)ϑ2
2(y)−ϑ2
1(x)ϑ2
3(y)=ϑ2
2(x)ϑ2
4(y)−ϑ2
3(x)ϑ2
1(y) LW 8(1.4.17)
9. ϑ3(x+y)ϑ3(x−y)ϑ2
4(0) = ϑ2
4(x)ϑ2
3(y)−ϑ2
1(x)ϑ2
2(y)=ϑ2
3(x)ϑ2
4(y)−ϑ2
2(x)ϑ2
1(y) LW 8(1.4.18)
10. ϑ4(x+y)ϑ4(x−y)ϑ2
4(0) = ϑ2
4(x)ϑ2
4(y)−ϑ2
1(x)ϑ2
1(y) LW 8(1.4.15)
11. ϑ4(x+y)ϑ4(x−y)ϑ2
4(0) = ϑ2
3(x)ϑ2
3(y)−ϑ2
2(x)ϑ2
2(y)=ϑ2
4(x)ϑ2
4(y)−ϑ2
1(x)ϑ2
1(y) LW 9(1.4.19)
12. ϑ1(x+y)ϑ1(x−y)ϑ2
3(0) = ϑ2
1(x)ϑ2
3(y)−ϑ2
3(x)ϑ2
1(y)=ϑ2
4(x)ϑ2
2(y)−ϑ2
2(x)ϑ2
4(y) LW 9(1.4.23)
13. ϑ2(x+y)ϑ2(x−y)ϑ2
3(0) = ϑ2
2(x)ϑ2
3(y)−ϑ2
4(x)ϑ2
1(y)=ϑ2
3(x)ϑ2
2(y)−ϑ2
1(x)ϑ2
4(y) LW 9(1.4.24)
14. ϑ3(x+y)ϑ3(x−y)ϑ2
3(0) = ϑ2
1(x)ϑ2
1(y)+ϑ2
3(x)ϑ2
3(y)=ϑ2
2(x)ϑ2
2(y)+ϑ2
4(x)ϑ2
4(y) LW 9(1.4.25)
15. ϑ4(x+y)ϑ4(x−y)ϑ2
3(0) = ϑ2
1(x)ϑ2
2(y)+ϑ2
3(x)ϑ2
4(y)=ϑ2
2(x)ϑ2
1(y)+ϑ2
4(x)ϑ2
3(y) LW 9(1.4.26)
882 Elliptic Integrals and Functions 8.199(2)
16. ϑ1(x+y)ϑ1(x−y)ϑ2
2(0) = ϑ2
1(x)ϑ2
2(y)−ϑ2
2(x)ϑ2
1(y)=ϑ2
4(x)ϑ2
3(y)−ϑ2
3(x)ϑ2
4(y) LW 9(1.4.30)
17. ϑ2(x+y)ϑ2(x−y)ϑ2
2(0) = ϑ2
2(x)ϑ2
2(y)−ϑ2
1(x)ϑ2
1(y)=ϑ2
3(x)ϑ2
3(y)−ϑ2
4(x)ϑ2
4(y) LW 10(1.4.31)
18. ϑ3(x+y)ϑ3(x−y)ϑ2
2(0) = ϑ2
3(x)ϑ2
2(y)+ϑ2
4(x)ϑ2
1(y)=ϑ2
2(x)ϑ2
3(y)+ϑ2
1(x)ϑ2
4(y) LW 10(1.4.32)
19. ϑ4(x+y)ϑ4(x−y)ϑ2
2(0) = ϑ2
4(x)ϑ2
2(y)+ϑ2
3(x)ϑ2
1(y)=ϑ2
1(x)ϑ2
3(y)+ϑ2
2(x)ϑ2
4(y) LW 10(1.4.33)
20. ϑ2
3(x)ϑ2
3(0) = ϑ2
4(x)ϑ2
4(0) + ϑ2
2(x)ϑ2
2(0) LW 11(1.4.49)
21. ϑ2
4(x)ϑ2
3(0) = ϑ2
1(x)ϑ2
2(0) + ϑ2
3(x)ϑ2
4(0) LW 11(1.4.50)
22. ϑ2
4(x)ϑ2
2(0) = ϑ2
1(x)ϑ2
3(0) + ϑ2
2(x)ϑ2
4(0) LW 11(1.4.51)
23. ϑ2
3(x)ϑ2
2(0) = ϑ2
1(x)ϑ2
4(0) + ϑ2
2(x)ϑ2
3(0) LW 11(1.4.52)
24.8ϑ4
3(x)=ϑ4
2(0) + ϑ4
4(0) LW 11(1.4.53)
8.199(2)10Derivatives of ratios of theta functions
1.d
dx(ϑ1/ϑ4)=ϑ2
4(0)ϑ2(x)ϑ3(x)/ϑ2
4(x) LW 19(1.9.3)
2.d
dx(ϑ2/ϑ4)=−ϑ2
3(0)ϑ1(x)ϑ3(x)/ϑ2
4(x) LW 19(1.9.6)
3.d
dx(ϑ3/ϑ4)=−ϑ2
2(0)ϑ1(x)ϑ2(x)/ϑ2
4(x) LW 19(1.9.7)
4.d
dx(ϑ1/ϑ3)=ϑ2
3(0)ϑ2(x)ϑ4(x)/ϑ2
3(x) LW 19(1.9.8)
5.d
dx(ϑ2/ϑ3)=−ϑ2
4(0)ϑ1(x)ϑ4(x)/ϑ2
3(x) LW 19(1.9.9)
6.d
dx(ϑ1/ϑ2)=ϑ2
2(0)ϑ3(x)ϑ4(x)/ϑ2
2(x) LW 19(1.9.10)
7.d
dx(ϑ4/ϑ1)=−ϑ2
4(0)ϑ2(x)ϑ3(x)/ϑ2
1(x) LW 19(1.9.11)
8.d
dx(ϑ4/ϑ2)=ϑ2
3(0)ϑ1(x)ϑ3(x)/ϑ2
2(x) LW 20(1.9.12)
9.d
dx(ϑ4/ϑ3)=ϑ2
2(0)ϑ1(x)ϑ2(x)/ϑ2
3(x) LW 20(1.9.13)
10.d
dx(ϑ3/ϑ1)=−ϑ2
3(0)ϑ2(x)ϑ4(x)/ϑ2
1(x) LW 20(1.9.14)
11.d
dx(ϑ3/ϑ2)=ϑ2
4(0)ϑ1(x)ϑ4(x)/ϑ2
2(x) LW 20(1.9.15)
12.d
dx(ϑ2/ϑ1)=−ϑ2
2(0)ϑ3(x)ϑ4(x)/ϑ2
1(x) LW 20(1.9.16)
8.199(3)10Derivatives of theta functions
1.d
dulnϑ1(u) = cot u+4s i n2 u∞/summationdisplay
n=1q2n
1−2q2ncos 2u+q4n
8.212 The exponential integral function Ei (x) 883
2.d
dulnϑ2(u)=−tanu−4s in2 u∞/summationdisplay
n=1q2n
1+2q2ncos2u+q4n
3.d
dulnϑ3(u)=−4s in2 u∞/summationdisplay
n=1q2n−1
1+2q2ncos 2u+q4n−2
4.d
dulnϑ4(u)=4s i n2 u∞/summationdisplay
n=1q2n−1
1−2q2ncos2u+q4n−2
5.d2
du2lnϑ2(u)=−∞/summationdisplay
n=−∞sech2{i(u+nπτ)}
8.2 The Exponential Integral Function and Functions Generated by It
8.21 The exponential integral function Ei (x)
8.211
1. Ei( x)=−/integraldisplay∞
−xe−t
tdt=/integraldisplayx
−∞et
tdt=l i(ex)[ x<0]
2.11Ei(x)=−lim
ε→0+/bracketleftbigg/integraldisplay−ε
−xe−t
tdt+/integraldisplay∞
εe−t
tdt/bracketrightbigg
=P V/integraldisplayx
−∞et
tdt
[x>0]
3.7Ei(x)=1
2{Ei(x+i0) + Ei( x−i0)} [x>0] ET I 386
8.212
1.8Ei(−x)=C+l nx+/integraldisplayx
0e−t−1
tdt [x>0] NT 11(1)
=C+e−xlnx+/integraldisplayx
0e−tlntd t [x>0] NT 11(10)
2.7Ei(x)=ex/bracketleftbigg1
x+/integraldisplay∞
0e−tdt
(x−t)2/bracketrightbigg
[x>0] (cf. 8.211 1)
3. Ei( −x)=e−x/bracketleftbigg
−1
x+/integraldisplay∞
0e−tdt
(x+t)2/bracketrightbigg
[x>0] (cf. 8.211 1) LA 281(28)
4. Ei( ±x)=±e±x/integraldisplay1
0dt
x±lnt[x>0] (cf. 8.211 1)
5. Ei( ±xy)=±e±xy/integraldisplay∞
0e−xt
y∓tdt [Rey>0,x > 0] NT 19(11)
6. Ei( ±x)=−e±x/integraldisplay∞
0e−it
t±ixdt [x>0] NT 23(2, 3)
7.8Ei(xy)=exy/integraldisplay1
0ty−1
x+l ntdt LA 282(44)a
884 The Exponential Integral Function and Functions Generated by It 8.213
8. Ei( −xy)=−e−xy/integraldisplay1
0ty−1
x−lntdt LA 282(45)a
=x−1e−xy/bracketleftBigg/integraldisplay1
0tx−1
(y−lnt)2dt−y−1/bracketrightBigg
[x>0,y > 0] LA 283(47)a
9. Ei( x)=ex/integraldisplay∞
11
x−lntdt
t2[x>0] LA 283(48)
10. Ei( −x)=−e−x/integraldisplay∞
11
x+l ntdt
t2[x>0] LA 283(48)
11. Ei( −x)=−e−x/integraldisplay∞
0tcost+xsint
t2+x2dt [x>0] NT 23(6)
12. Ei( −x)=−e−x/integraldisplay∞
0tcost−xsint
t2+x2dt [x<0] NT 23(6)
13. Ei( −x)=2
π/integraldisplay∞
0cost
tarctant
xdt [Rex>0] NT 25(13)
14. Ei( −x)=2e−x
π/integraldisplay∞
0xcost−tsint
t2+x2lntd t [x>0] NT 26(7)
15. Ei( x)=2l n x−2ex
π/integraldisplay∞
0xcost+tsint
t2+x2lntd t [x>0] NT 27(8)
16. Ei( −x)=−x/integraldisplay∞
1e−txlntd t [x>0] NT 32(12)
See also 3.327 ,3.881 8,3.916 2a n d3 , 4.326 1,4.326 2,4.331 2,4.351 3,4.425 3,4.581 .F o r
integrals of the exponential integral function, see 6.22–6.23,6.78.
Series and asymptotic representations
8.213
1. li( x)=C+l n(−lnx)+∞/summationdisplay
k=1(lnx)k
k·k![0<x< 1] NT 3(9)
2. li( x)=C+l nl n x+∞/summationdisplay
k=1(lnx)k
k·k![x>1] NT 3(10)
8.214
1. Ei( x)=C+l n (−x)+∞/summationdisplay
k=1xk
k·k![x<0]
2. Ei( x)=C+l nx+∞/summationdisplay
k=1xk
k·k![x>0]
3. Ei( x)−Ei(−x)=2x∞/summationdisplay
k=0x2k
(2k+ 1)(2 k+1 ) ![x>0] NT 39(13)
8.219 The exponential integral function Ei (x) 885
8.2157Ei(z)=ez
z/bracketleftBiggn/summationdisplay
k=0k!
zk+Rn(z)/bracketrightBigg
|Rn(z)|=O/parenleftBig
|z|−n−1/parenrightBig
[z→∞,|arg(−z)|≤π−δ;δ>0 small] ,|Rn(z)|≤(n+1 ) !|z|−n−1[Rez≤0]
8.2167Ei(nx)−Ei(−nx)=enx/prime/parenleftbigg1
nx+1
n2x2+kn
n3x3/parenrightbigg
,
where x/prime=xsign Re( x),k n=O(1),andn→∞ NT 39(15)
8.217 Functional relations:
1. ex/primeEi(−x/prime)−e−x/primeEi(x/prime)=−2/integraldisplay∞
0x/primesint
t2+x2dt NT 24(11)
=4
π/integraldisplay∞
0x/primecost
t2+x2lntd t−2e−x/primelnx/prime[x/prime=xsign Re x] NT 27(9)
2. ex/primeEi(−x/prime)+e−x/primeEi(x/prime)=−2/integraldisplay∞
0tcost
t2+x2dt=2e−x/primelnx/prime−4
π/integraldisplay∞
0tsint
t2+x2lntd t
[x/prime=xsign Re x]NT 24(10), NT 27(10)
3. Ei( −x)−Ei/parenleftbigg
−1
x/parenrightbigg
=2
π/integraldisplay∞
0cost
tarctant/parenleftbig
x−1
x/parenrightbig
1+t2dt
[Rex>0] NT 25(14)
4. Ei( −αx)Ei(−βx)−ln(αβ)Ei[−(α+β)x]=e−(α+β)x/integraldisplay∞
0e−txln[(α+t)(β+t)]
t+α+βdt NT 32(9)
See also 3.723 1a n d5 , 3.742 2a n d4 , 3.824 4,4.573 2.
•For a connection with a confluent hypergeometric function, see 9.237 .
•For integrals of the exponential integral function, see 5.21,5.22,5.23,6.22,a n d6.23.
8.218 Two numerical values:
1. Ei( −1) =−0.219 383 934 395 520 273 665 ... NT 89
2. Ei(1) = 1 .895 117 816 355 936 755 478 ... NT 89
8.219∗Definite integrals of exponential functions
1.∗/integraldisplay∞
0Ei2(x)e−2xdx=π2
4
2.∗/integraldisplay∞
0Ei2(−x)e2xdx=π2
4
3.∗/integraldisplay∞
0Ei(x)Ei(−x)dx=0
886 The Exponential Integral Function and Functions Generated by It 8.221
8.22 The hyperbolic sine integral shixand the hyperbolic cosine integral chix
8.221
1. shi x=/integraldisplayx
0sinht
tdt=−i/bracketleftBigπ
2+s i (ix)/bracketrightBig
(see8.230 1) EH II 146(17)
2.11chix=C+l nx+/integraldisplayx
0cosht−1
tdt EH II 146(18)
8 . 2 3T h es i n ei n t e g r a la n dt h ec o s i n ei n t e g r a l : sixandcix
8.230
1.10si(x)=−/integraldisplay∞
xsint
tdt=−π
2+S i (x),where Si( x)=/integraldisplayx
0sint
tdt NT 11(3)
2.10ci(x)=−/integraldisplay∞
xcost
tdt=C+l nx+/integraldisplayx
0cost−1
tdt [ci(x) is also written Ci( x)] NT 11(2)
8.231
1. si( xy)=−/integraldisplay∞
xsinty
tdt NT 18(7)
2. ci( xy)=−/integraldisplay∞
xcosty
tdt NT 18(6)
3. si( x)=−/integraldisplayπ/2
0e−xcostcos (xsint)dt NT 13(26)
8.232
1. si( x)=−π
2+∞/summationdisplay
k=1(−1)k+1x2k−1
(2k−1)(2k−1)!NT 7(4)
2.7ci(x)=C+l n (x)+∞/summationdisplay
k=1(−1)kx2k
2k(2k)!NT 7(3)
8.233
1. ci( x)±isi(x)=E i( ±ix) NT 6a
2. ci( x)−ci/parenleftbig
xe±πi/parenrightbig
=∓πi NT 7(5)
3. si( x)+s i ( −x)=−π NT 7(7)
8.234
1.7Ei(−x)−ci(x)=/integraldisplayπ/2
0e−xcosϕsin(ssinϕ)dϕ NT 13(27)
2. [ci( x)]2+[ s i (x)]2=−2/integraldisplayπ/2
0exp (−xtanϕ)lnc os ϕ
sinϕcosϕdϕ
[Rex>0] (see also 4.366 )
NT 32(11)
See also 3.341 ,3.351 1a n d2 , 3.354 1a n d2 , 3.721 2a n d3 , 3.722 1, 3, 5 and 7, 3.723 8 and 11,
4.338 1,4.366 1.
8.250 The probability integral, Fresnel integrals and error functions 887
8.235
1. lim
x→+∞(x/rho1si(x)) = 0 ,lim
x→+∞(x/rho1ci(x)) = 0 [ /rho1<1] NT 38(5)
2. lim
x→−∞si(x)=−π, lim
x→−∞ci(x)=±πi NT 38(6)
•For integrals of the sine integral and cosine integral, see 6.24–6.26,6.781 ,6.782 ,a n d6.783 .
•For indefinite integrals of the sine integral and cosine integral, see 5.3.
8.24 The logarithm integral li(x)
8.240
1. li( x)=/integraldisplayx
0dt
lnt=E i( l n x)[ x<1] JA
2. li( x) = lim
ε→0/bracketleftbigg/integraldisplay1−ε
0dt
lnt+/integraldisplayx
1+εdt
lnt/bracketrightbigg
=E i( l n x)[ x>1] JA
3. li/braceleftbig
exp/parenleftbig
−xe±πi/parenrightbig/bracerightbig
=E i/parenleftbig
−xe±iπ/parenrightbig
=E i( x∓i0) = Ei( x)±iπ=l i(ex)±iπ
[x>0] JA, NT 2(6)
Integral representations
8.241
1. li( x)=/integraldisplaylnx
−∞et
tdt=xln ln1
x−/integraldisplay∞
−lnxe−tlntd t [x<1] LA 281(33)
2. li( x)=x/integraldisplay1
0dt
lnx+l ntLA 280(22)
=x
lnx+x/integraldisplay1
0dt
(lnx+l nt)2LA 280(29)
=x/integraldisplay∞
11
lnx−lntdt
t2[x<1] LA 280(30)
3. li( ax)=1
lna/integraldisplayx
−∞at
tdt [x>0]
For integrals of the logarithm integral, see 6.21
8.25 The probability integral Φ(x), the Fresnel integrals S(x)andC(x), the error
function erf(x), and the complementary error function erfc(x)
8.250 Definition:
1.11Φ(x)=e r f ( x)=2√π/integraldisplayx
0e−t2dt (called the error function)
2. S(x)=2√
2π/integraldisplayx
0sint2dt
888 The Exponential Integral Function and Functions Generated by It 8.251
3. C(x)=2√
2π/integraldisplayx
0cost2dt
4.11erfc(x)=1−erf(x) (called the complementary error function)
5.∗/integraldisplay∞
0e−(p+x)y
π(p+x)sin/parenleftbig
a√x/parenrightbig
dx
=−sinh (a√p)+1
2e−a√pΦ/parenleftbigga
2√y−√py/parenrightbigg
+1
2ea√pΦ/parenleftbigga
2√y+√py/parenrightbigg
6.∗/integraldisplay∞
0e−(p+x)y
π(p+x)cos/parenleftbig
a√x/parenrightbig
dx=1√πyexp/parenleftbigg
−a2
4y−py/parenrightbigg
−√p
2e−a√pΦ/parenleftbigga
a√y−√py/parenrightbigg
+√p
2e√pΦ/parenleftbigga
2√y+√py/parenrightbigg
−√pcosh (a√p)
[Rep>0,a , b are real]
7.∗/integraldisplayp
0exp/parenleftbig
−x2/parenrightbig
Φ(p−x)dx=/integraldisplayp
0exp/parenleftbig
−x2/parenrightbig
erf(p−x)dx=√π
2/bracketleftbigg
Φ/parenleftbiggp√
2/parenrightbigg/bracketrightbigg2
8.∗/integraldisplayp
0x2exp/parenleftbig
−x2/parenrightbig
Φ(p−x)dx=/integraldisplayp
0x2exp/parenleftbig
−x2/parenrightbig
erf(p−x)dx
=√π
4/bracketleftbigg
Φ/parenleftbiggp√
2/parenrightbigg/bracketrightbigg2
−p
2√
2Φ/parenleftbigg
−x2
2/parenrightbigg
erf/parenleftbiggp√
2/parenrightbigg
9.∗/integraldisplay(b+a)/√
2
(b−a)/√
2exp/parenleftbig
−x2/parenrightbig
Φ/parenleftBig
b√
2−x/parenrightBig
dx+/integraldisplay(a+b)/√
2
(a−b)/√
2exp/parenleftbig
−x2/parenrightbig
Φ/parenleftBig
a√
2−x/parenrightBig
dx=√πΦ(a)Φ(b)
Integral representations
8.251
1. Φ( x)=1√π/integraldisplayx2
0e−t
√
tdt (see also 3.361 1)
2. S(x)=1√
2π/integraldisplayx2
0sint√
tdt
3. C(x)=1√
2π/integraldisplayx2
0cost√
tdt
8.252
1. Φ( xy)=2y√π/integraldisplayx
0e−t2y2dt/bracketleftbig
Rey2>0/bracketrightbig
2. S(xy)=2y√
2π/integraldisplayx
0sin/parenleftbig
t2y2/parenrightbig
dt
3. C(xy)=2y√
2π/integraldisplayx
0cos/parenleftbig
t2y2/parenrightbig
dt
8.255 The probability integral, Fresnel integrals and error functions 889
4. Φ( xy)=1−2√πe−x2y2/integraldisplay∞
0e−t2y2ty dt√
t2+x2/bracketleftbig
Rey2>0/bracketrightbig
NT 19(11)a
=1−2x
πe−x2y2/integraldisplay∞
0e−t2y2dt
t2+x2/bracketleftbig
Rey2>0/bracketrightbig
NT 19(13)a
5.7Φ/parenleftbigg−y
2xi/parenrightbigg
−Φ/parenleftBigy
2xi/parenrightBig
=4xiey2
4x2√π/integraldisplay∞
0e−t2y2sin(ty)dt/bracketleftbig
Rex2>0/bracketrightbig
NT 28(3)a
6.8Φ/parenleftBigy
2x/parenrightBig
=1−2√πxe−y2
4/integraldisplay∞
0e−t2x2−tydt/bracketleftbig
Rex2>0/bracketrightbig
NT 27(1)a
See also 3.322 ,3.362 2,3.363 ,3.468 ,3.897 ,6.511 4a n d5 .
8.2538Series representations:
1.11erf(x)=2√πe−x2xF1/parenleftbigg
1;3
2;x2/parenrightbigg
=2√π∞/summationdisplay
k=1(−1)k+1 x2k−1
(2k−1)(k−1)!NT 7(9)a
=2√πe−x2∞/summationdisplay
k=02kx2k+1
(2k+1 ) ! !NT 10(11)a
2. S(x)=2√
2π/parenleftbigg
xsinx2F/parenleftbigg
1;5
4,3
4;−1
4x2/parenrightbigg
−2
3x3cosx2F/parenleftbigg
1;7
4,5
4;−1
4x2/parenrightbigg/parenrightbigg
=2√
2π∞/summationdisplay
k=0(−1)kx4k+3
(2k+ 1)!(4 k+3 )NT 8(14)a
=2√
2π/braceleftBigg
sin2x∞/summationdisplay
k=0(−1)k22kx4k+1
(4k+1 ) ! !−cosx2∞/summationdisplay
k=0(−1)k22k+1x4k+3
(4k+3 ) ! !/bracerightBigg
NT 10(13)a
3. C(x)=2√
2π/parenleftbigg2
3x3sinx2F/parenleftbigg
1;7
4,5
4;−1
4x2/parenrightbigg
−xcosx2F/parenleftbigg
1;5
4,3
4;−1
4x2/parenrightbigg/parenrightbigg
=2√
2π∞/summationdisplay
k=0(−1)kx4k+1
(2k)!(4k+1 )NT 8(13)a
=2√
2π/braceleftBigg
sin2x∞/summationdisplay
k=0(−1)k22k+1x4k+3
(4k+3 ) ! !+c o s x2∞/summationdisplay
k=0(−1)k22kx4k+1
(4k+1 ) ! !/bracerightBigg
NT 10(12)a
For the expansions in Bessel functions, see 8.515 2,8.515 3.
Asymptotic representations
8.2548Φ(z)=1−e−z2
√πz/bracketleftBiggn/summationdisplay
k=0(−1)k(2k−1)!!
(2z2)k+O/parenleftBig
|z|−2n−z/parenrightBig/bracketrightBigg
,
[z→∞,|arg(−z)|≤π−δ;δ>0 small]
where
|Rn|<Γ/parenleftbig
n+1
2/parenrightbig
|x|n+1
2cosϕ
2,x=|x|eiϕandϕ2<π2NT 37(10)a
8.255
1. S(x)=1
2−1√
2πxcosx2+O/parenleftbigg1
x2/parenrightbigg
[x→∞] MO 127a
890 The Exponential Integral Function and Functions Generated by It 8.256
2. C(x)=1
2+1√
2πxsinx2+O/parenleftbigg1
x2/parenrightbigg
[x→∞] MO 127a
8.256 Functional relations:
1. C(z)+iS(z)=/radicalbigg
i
2Φ/parenleftbiggz√
i/parenrightbigg
=2√
2π/integraldisplayz
0eit2dt
2. C(z)−iS(z)=1√
2iΦ/parenleftBig
z√
i/parenrightBig
=2√
2π/integraldisplayz
0e−it2dt
3./bracketleftbig
cos2uC(u)+s i n u2S(u)/bracketrightbig
=1
2/bracketleftbig
cos2u+s i nu2/bracketrightbig
+/radicalbigg
2
π/integraldisplay∞
0e−2utsint2dt
[Reu≥0] NT 28(6)a
4./bracketleftbig
cos2uS(u)−sinu2C(u)/bracketrightbig
=1
2/bracketleftbig
cos2u−sinu2/bracketrightbig
−/radicalbigg
2
π/integraldisplay∞
0e−2utcost2dt
[Reu≥0] NT 28(5)a
5.11/bracketleftbigg
C(x)−1
2/bracketrightbigg2
+/bracketleftbigg
S(x)−1
2/bracketrightbigg2
=2
π/integraldisplayπ/2
0exp/parenleftbig
−x2tanϕ/parenrightbig
sinϕ
2√cosϕ
sin 2ϕdϕ
(see also 6.322 ) NT 33(18)a
•For a connection with a confluent hypergeometric function, see 9.236 .
•For a connection with a parabolic cylinder function, see 9.254 .
8.257
1. lim
x→+∞/parenleftbig
x/rho1/bracketleftbig
S(x)−1
2/bracketrightbig/parenrightbig
=0 [ /rho1<1] NT 38(11)
2. lim
x→+∞/parenleftbig
x/rho1/bracketleftbig
C(x)−1
2/bracketrightbig/parenrightbig
=0 [ /rho1<1] NT 38(11)
3. lim
x→+∞S(x)=1
2NT 38(12)a
4. lim
x→+∞C(x)=1
2NT 38(12)a
•For integrals of the probability integral, see 6.28–6.31.
•For integrals of Fresnel’s sine integral and cosine integral, see 6.32.
8.25810Integrals involving the complementary error function
1./integraldisplay∞
0erfc2(x)e−βx2dx=1√βπ/parenleftbigg
−arccos/parenleftbigg1
1+β/parenrightbigg
+ 2arctan/parenleftBig/radicalbig
β/parenrightBig/parenrightbigg
[β>0]
2./integraldisplay∞
0xerfc2(x)e−βx2dx=1
2β/parenleftBigg
1−4
πarctan/parenleftbig√1+β/parenrightbig
√1+β/parenrightBigg
[β>0]
8.262 Lobachevskiy’s function Lfunction]Lobachevskiyfunction(ZdddZLZdddZ)( x) 891
3./integraldisplay∞
0x3erfc2(x)e−βx2dx=1
2β2/parenleftBigg
1−4
πarctan/parenleftbig√1+β/parenrightbig
√1+β/parenrightBigg
+1
βπ/parenleftBigg
1
(1 +β)(β2+2β+2 )−arctan/parenleftbig√1+β/parenrightbig
(1 +β)3
2/parenrightBigg
[β>0]
4./integraldisplay∞
0xerfc/parenleftbig√x/parenrightbig
e−βxdx=1
β2/bracketleftbigg
1−1+3
2β
(1 +β)3
2/bracketrightbigg
[β>0]
5.11/integraldisplay∞
0√xerfc/parenleftbig√x/parenrightbig
e−βxdx=1√π/parenleftBigg
1
2arctan/parenleftbig√β/parenrightbig
β3
2−1
2β(1 +β)/parenrightBigg
[β>0]
8.259∗Integrals involving the error function and an exponential function
1./integraldisplay∞
−∞e−px2Φ(a+bx)dx=/radicalbiggπ
pΦ/parenleftBigg
a√p/radicalbig
b2+p/parenrightBigg
[Rep>0],a , b real
2./integraldisplay∞
−∞x2e−px2Φ(a+bx)dx=1
2p/radicalbiggπ
pΦ/parenleftBigg
a√p/radicalbig
b2+p/parenrightBigg
−ab2
p(b2+p)3/2exp/parenleftbigg
−a2p
b2+p/parenrightbigg
[Rep>0,a , b are real]
3./integraldisplay∞
−∞x2ne−px2Φ(a+bx)dx=(−1)n∂n
∂pn/bracketleftBigg/radicalbiggπ
pΦ/parenleftBigg
a√p/radicalbig
b2+p/parenrightBigg/bracketrightBigg
[n=0,1,..., Rep>0,a , b are real]
8.26 Lobachevskiy’s function L(x)
8.260 Definition:
L(x)=−/integraldisplayx
0lncos td t LO III 184(10)
For integral representations of the function L(x), see also 3.531 8,3.532 2,3.533 ,a n d4.224 .
8.261 Representation in the form of a series:
L(x)=xln 2−1
2∞/summationdisplay
k=1(−1)k−1sin 2kx
k2LO III 185(11)
8.262 Functional relationships:
1. L(−x)=−L(x)/bracketleftBig
−π
2≤x≤π
2/bracketrightBig
LO III 185(13)
2. L(π−x)=πln 2−L(x) LO III 286
3. L(π+x)=πln 2 + L(x) LO III 286
4. L(x)−L/parenleftBigπ
2−x/parenrightBig
=/parenleftBig
x−π
4/parenrightBig
ln 2−1
2L/parenleftBigπ
2−2x/parenrightBig/bracketleftBig
0≤x<π
4/bracketrightBig
LO III 186(14)
892 Euler’s Integrals of the First and Second Kinds 8.310
8.3 Euler’s Integrals of the First and Second Kinds and Functions
Generated by Them
8.31 The gamma function (Euler’s integral of the second kind): Γ(z)
8.310 Definition:
1. Γ( z)=/integraldisplay∞
0e−ttz−1dt [Rez>0] (Euler) FI II 777(6)
Generalization:
2. Γ( z)=−1
2isinπz/integraldisplay
C(−t)z−1e−tdt
forznot an integer. The contour Cis shown in the drawing: WH
Γ(z) is an analytic function zwith simple poles at the points z=−l(forl=0 ,1 ,2 , ...)t ow h i c h
correspond to residues(−1)l
l!.Γ (z) satisfies the relation Γ(1) = 1. W H ,M O1
Integral representations
8.311 Γ(z)=1
e2πiz−1/integraldisplay(0+)
∞e−ttz−1dt MO 2
8.312
1. Γ( z)=/integraldisplay1
0/parenleftbigg
ln1
t/parenrightbiggz−1
dt [Rez>0] FI II 778
2. Γ( z)=xz/integraldisplay∞
0e−xttz−1dt [Rez>0,Rex>0] FI II 779(8)
3. Γ( z)=2azea
sinπz/integraldisplay∞
0e−at2/parenleftbig
1+t2/parenrightbigz−1
2cos[2at+( 2z−1)arctan t]dt
[a>0] WH
4. Γ( z)=1
2s inπz/integraldisplay∞
0e−t2tz−1/parenleftbig
1+t2/parenrightbigz
2{3s in[t+zarccot( −t)] + sin [ t+(z−2)arccot( −t)]}dt
[arccot denotes an obtuse angle] WH
5. Γ( y)=xye−iβy/integraldisplay∞
0ty−1exp/parenleftbig
−xte−iβ/parenrightbig
dt
/bracketleftBig
x, y, β real,x > 0,y > 0,|β|<π
2/bracketrightBig
MO 8
6. Γ( z)=bz
2s inπz/integraldisplay∞
−∞ebti(it)z−1dt [b>0,0<Rez<1] NH 154(3)
8.315 The gamma function (Euler’s integral of the second kind): Γ(z) 893
7. Γ( z)=/parenleftbig√
a2+b2/parenrightbigz
cos/parenleftbig
zarctanb
a/parenrightbig/integraldisplay∞
0e−atcos(bt)tz−1dt NH 152(1)a
=/parenleftbig√
a2+b2/parenrightbigz
sin/parenleftbig
zarctanb
a/parenrightbig/integraldisplay∞
0e−atsin(bt)tz−1dt NH 152(2)
[a>0,b≥0,Rez>0]
8. Γ( z)=bz
cosπz
2/integraldisplay∞
0cos(bt)tz−1dt
=bz
sinπz
2/integraldisplay∞
0sin(bt)tz−1dt
[b>0,0<Rez<1] NH 152(5)
9. Γ( z)=/integraldisplay∞
0e−t(t−z)tz−1lntd t
[Rez>0] NH 173(7)
10. Γ( z)=/integraldisplay∞
−∞exp/parenleftbig
zt−et/parenrightbig
dt
[Rez>0] NH 145(14)
11.11Γ(x)cosαx=λx/integraldisplay∞
0tx−1e−λtcosαcos(λtsinα)dt
/bracketleftBig
λ>0,x > 0,−π
2<α<π
2/bracketrightBig
WH
12. Γ( x)sinαx=λx/integraldisplay∞
0tx−1e−λtcosαsin(λtsinα)dt
/bracketleftBig
λ>0,x > 0,−π
2<α<π
2/bracketrightBig
WH
13. Γ( −z)=/integraldisplay∞
0⎡
⎢⎢⎢⎢⎣e
−t−n/summationdisplay
k=0(−1)ktk
k!
tz+1⎤
⎥⎥⎥⎥⎦dt [n=⌊Rez⌋]
MO 2
8.313 Γ/parenleftbiggz+1
v/parenrightbigg
=vuz+1
v/integraldisplay∞
0exp (−utv)tzdt [Reu>0,Rev>0,Rez>−1]
J A ,M O7 a
8.314∗Γ(z)=/integraldisplay∞
1e−ttz−1dt+∞/summationdisplay
n=0(−1)k
k!(z+k)[z→0, in|argz|<π]
8.315
1.111
Γ(z)=i
2π/integraldisplay
C(−t)−ze−tdt [for the contour C,s e e8.310 2]
2.8/integraldisplay∞
−∞ebti
(a+it)2dt=2πe−abbz−1
Γ(z)
/integraldisplay∞
−∞e−bti
(a+it)zdt=0/bracketleftbig
Rea>0,b > 0,Rez>0,|arg(a+it)|<1
2π/bracketrightbig
894 Euler’s Integrals of the First and Second Kinds 8.321
3.1
Γ(z)=a1−zea
π/integraldisplayπ/2
0cos(atanθ−zθ)c o sz−2θd θ [Rez>1] NH 157(14)
See also 3.324 2,3.326 ,3.328 ,3.381 4,3.382 2,3.389 2,3.433 ,3.434 ,3.478 1,3.551 1, 2,3.827 1,
4.267 7,4.272 ,4.353 1,4.369 1,6.214 ,6.223 ,6.246 ,6.281 .
8.32 Representation of the gamma function as series and products
8.321 Representation in the form of a series:
1.6Γ(z+1 )=∞/summationdisplay
k=0ckzk
/bracketleftbigg
c0=1,c n+1=/summationtextn
k=0(−1)k+1sk+1cn−k
n+1;s1=C,s n=ζ(n)f o rn≥2,|z|<1/bracketrightbigg
NH 40(1, 3)
2.111
Γ(z+1 )=∞/summationdisplay
k=0dkzk
/bracketleftbigg
d0=1,d n+1=/summationtextn
k=0(−1)ksk+1dn−k
n+1;s1=C,s n=ζ(n)f o rn≥2/bracketrightbigg
NH 41(4, 6)
Infinite-product representation
8.32211Γ(z)=e−Cz1
z∞/productdisplay
k=1ez/k
1+z
k[Rez>0] SM 269
=1
z∞/productdisplay
k=1/parenleftbig
1+1
k/parenrightbigz
1+z
k[Rez>0] WH
= lim
n→∞nz
zn/productdisplay
k=1k
z+k[Rez>0] SM 267(130)
8.3237Γ(z)=2zze−z∞/productdisplay
k=12k/radicalBig
B/parenleftbig
2k−1z,1
2/parenrightbig
NH 98(12)
8.3247Γ(1 + z)=4z∞/productdisplay
k=1Γ/parenleftbigg1
2+z
2k/parenrightbigg
√πMO 3
8.325
1.Γ(α)Γ(β)
Γ(α+γ)Γ(β−γ)=∞/productdisplay
k=0/bracketleftbigg/parenleftbigg
1+γ
α+k/parenrightbigg/parenleftbigg
1−γ
β+k/parenrightbigg/bracketrightbigg
NH 62(2)
2.11eCxΓ(z+1 )
Γ(z−x+1 )=∞/productdisplay
k=1/bracketleftbigg/parenleftbigg
1−x
z+k/parenrightbigg
ex/k/bracketrightbigg
[z/negationslash=0,−1,−2,...;R e z>0,Re(z−x)>0]
3.7√π
Γ/parenleftbig
1+z
2/parenrightbig
Γ/parenleftbig1
2−z
2/parenrightbig=∞/productdisplay
k=1/parenleftbigg
1−z
2k−1/parenrightbigg/parenleftBig
1+z
2k/parenrightBig
MO 2
8.331 Functional relations involving the gamma function 895
8.326
1.[Γ(x)]2
Γ(2x)
B(x+iy,x−iy)=/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓ(x)
Γ(x−iy)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
=∞/productdisplay
k=0/parenleftbigg
1+y2
(x+k)2/parenrightbigg
[x, yare real ,x/negationslash=0,−1,−2,...]
LO V, NH 63(4)
2.11Γ(x+iy)
Γ(x)=xe−iCy
x+iy∞/productdisplay
n=1exp/parenleftbigiy
n/parenrightbig
1+iy
x+n[x, yare real ,x/negationslash=0,−1,−2,...]
MO 2
8.327 Asymptotic representation for large arguments:
1.∗Γ(z)∼zz−1
2e−z√
2π/braceleftbigg
1+1
12z+1
288z2−139
51840 z3−571
2488320 z4+O/parenleftbig
z−5/parenrightbig/bracerightbigg
[|argz|<π] WH
Forzreal and positive, the remainder of the series is less than the last term that is retained.
2.∗n!∼√
2πn/parenleftBign
e/parenrightBign
or equivalently Γ( n+1 )∼√
2πn/parenleftBign
e/parenrightBign
[Stirling’s asymptotic formula for n/greatermuch0]AS 6.1.38
3.∗lnΓ(z)∼/parenleftbigg
z−1
2/parenrightbigg
lnz−z+1
2ln(2π)+1
12z−1
360z3+1
1260z5−1
1680z7+...
[z→∞,|argz|<π] AS 6.1.38
8.328
1. lim
|y|→∞|Γ(x+iy)|eπ
2|y||y|1
2−x=√
2π [xandyare real] MO 6
2. lim
|z|→∞Γ(z+a)
Γ(z)e−alnz=1 MO 6
8.33 Functional relations involving the gamma function
8.331
1. Γ( x+1 )= xΓ(x)
2.∗Γ(x+a)=(x+a−1)Γ(x+a−1)
=Γ(x+a+1 )
(x+a)
3.∗Γ(x−a)=(x−a−1)Γ(x−a−1)
=Γ(x−a+1 )
(x−a)
896 Euler’s Integrals of the First and Second Kinds 8.332
8.332
1. |Γ(iy)|2=π
ysinhπy[yis real] MO 3
2./vextendsingle/vextendsingleΓ/parenleftbig1
2+iy/parenrightbig/vextendsingle/vextendsingle2=π
coshπy[yis real]
3. Γ(1 + ix)Γ(1−ix)=πx
sinhxπ[xis real] LO V
4. Γ(1 + x+iy)Γ(1−x+iy)Γ(1+ x−iy)Γ(1−x−iy)=2π2/parenleftbig
x2+y2/parenrightbig
cosh2 yπ−cos 2xπ
[xandyare real] LO V
8.333 [Γ(n+1 ) ]n=G(n+1 )n/productdisplay
k=1kk,
where nis a natural number and
G(z+1 )=( 2 π)z
2exp/bracketleftbigg
−z(z+1 )
2−C
2z2/bracketrightbigg∞/productdisplay
n=1/braceleftbigg/parenleftBig
1+z
n/parenrightBign
exp/parenleftbigg
−z+z2
2n/parenrightbigg/bracerightbigg
WH
8.334
1.n/productdisplay
k=11
Γ/parenleftbig
−zexp2πki
n/parenrightbig=−zn∞/productdisplay
k=1/bracketleftBig
1−/parenleftBigz
k/parenrightBign/bracketrightBig
[n=2,3,3...] MO 2
2. Γ/parenleftbig1
2+x/parenrightbig
Γ/parenleftbig1
2−x/parenrightbig
=π
cosπx
3. Γ(1 −x)Γ(x)=π
sinπxFI II 430
Special cases
8.3357Γ(nx)=( 2 π)1−n
2nnx−1
2n−1/productdisplay
k=0Γ/parenleftbigg
x+k
n/parenrightbigg
[product theorem] FI II 782a, WH
1. Γ(2 x)=22x−1
√πΓ(x)Γ/parenleftbig
x+1
2/parenrightbig
[doubling formula]
2. Γ(3 x)=33x−1
2
2πΓ(x)Γ/parenleftbig
x+1
3/parenrightbig
Γ/parenleftbig
x+2
3/parenrightbig
3.n−1/productdisplay
k=1Γ/parenleftbiggk
n/parenrightbigg
Γ/parenleftbigg
1−k
n/parenrightbigg
=(2π)n−1
nWH
4.10∞/summationdisplay
n=0Γ2/parenleftbig
n−1
2/parenrightbig
4(n!)2Γ2/parenleftbig
−1
2/parenrightbig=1
4+1
16+1
256+1
1024+25
65536+···=1
π
8.336 Γ/parenleftbigg
−yz+xi
2y/parenrightbigg
Γ(1−z)=( 2 i)z+1yΓ/parenleftbigg
1+yz−xi
2y/parenrightbigg/integraldisplay∞
0e−txsinz(ty)dt
[Re(yi)>0,Re(x−yzi)>0]
NH 133(10)
8.339 Functional relations involving the gamma function 897
•For a connection with the psi function, see 8.361 1.
•For a connection with the beta function, see 8.384 1.
•For integrals of the gamma function, see 8.412 4,8.414 ,9.223 ,9.242 3,9.242 4.
8.337
1./bracketleftbig
Γ/prime(x)/bracketrightbig2<Γ(x)Γ/prime/prime(x)[ x>0] MO 1
2. For x>0, min Γ(1 + x)=0.88560 ...is attained when x=0.46163 ... JA
Particular values
8.338
1. Γ(1) = Γ(2) = 1
2. Γ/parenleftbig1
2/parenrightbig
=√π
3. Γ/parenleftbig
−1
2/parenrightbig
=−2√π
4./bracketleftbigg
Γ/parenleftbigg1
4/parenrightbigg/bracketrightbigg4
=1 6π2∞/productdisplay
k=1(4k−1)2/bracketleftbig
(4k+1 )2−1/bracketrightbig
[(4k−1)2−1](4k+1 )2MO 1a
5.8/productdisplay
k=1Γ/parenleftbiggk
3/parenrightbigg
=640
36/parenleftbiggπ√
3/parenrightbigg3
WH
8.339 Forna natural number
1. Γ( n)=(n−1)!
2. Γ/parenleftbig
n+1
2/parenrightbig
=√π
2n(2n−1)!!
3. Γ/parenleftbig1
2−n/parenrightbig
=(−1)n2n√π
(2n−1)!!
4.Γ/parenleftbig
p+n+1
2/parenrightbig
Γ/parenleftbig
p−n+1
2/parenrightbig=/parenleftbig
4p2−12/parenrightbig/parenleftbig
4p2−32/parenrightbig
.../bracketleftbig
4p2−(2n−1)2/bracketrightbig
22nWA 221
5.∗Γ(n+k)=(n+k−1)!
=Γ(n+k+1 )
(n+k)[n+k≥0,1,...]
6.∗Γ(n−k)=(n−k−1)!
=Γ(n−k+1 )
(n−k)[n−k≥0,1,...]
898 Euler’s Integrals of the First and Second Kinds 8.341
8.34 The logarithm of the gamma function
8.341 Integral representation:
1. lnΓ( z)=/parenleftbigg
z−1
2/parenrightbigg
lnz−z+1
2ln 2π+/integraldisplay∞
0/parenleftbigg1
2−1
t+1
et−1/parenrightbigge−tz
tdt
[Rez>0] WH
2.11lnΓ(z)=zlnz−z−1
2lnz+l n√
2π+2/integraldisplay∞
0arctant
z
e2πt−1dt
/bracketleftbigg
Rez>0 and arctan w=/integraldisplayw
0du
1+u2is taken over a rectangular path in the w-plane/bracketrightbigg
WH
3. lnΓ( z)=/integraldisplay∞
0/braceleftbigge−zt−e−t
1−e−t+(z−1)e−t/bracerightbiggdt
t[Rez>0] WH
4. lnΓ( z)=/integraldisplay∞
0/braceleftbigg
(z−1)e−t+(1 +t)−z−(1 +t)−1
ln(1 + t)/bracerightbiggdt
t
[Rez>0] WH
5. lnΓ( x)=lnπ−lnsinπx
2+1
2/integraldisplay∞
0/braceleftBigg
sinh/parenleftbig1
2−x/parenrightbig
t
sinht
2−(1−2x)e−t/bracerightBigg
dt
t
[0<x< 1] WH
6. lnΓ( z)=/integraldisplay1
0/braceleftbiggtz−t
t−1−t(z−1)/bracerightbiggdt
tlnt[Rez>0] WH
7. lnΓ( z)=/integraldisplay∞
0/bracketleftbigg
(z−1)e−t+e−tz−e−t
1−e−t/bracketrightbiggdt
t[Rez>0] NH 187(7)
See also 3.427 9,3.554 5.
8.342 Series representations:
1.11lnΓ(z+1 )
=1
2/bracketleftbigg
ln/parenleftBigπz
sinπz/parenrightBig
−ln1+z
1−z/bracketrightbigg
+( 1−C)z+∞/summationdisplay
k=11−ζ(2k+1 )
2k+1z2k+1
=−Cz+∞/summationdisplay
k=2(−1)kzk
kζ(k) [|z|<1]NH 38(16, 12)
2. lnΓ(1 + x)=1
2lnπx
sinπx−Cx−∞/summationdisplay
n=1x2n+1
2n+1ζ(2n+1 )
[|x|<1] NH 38(14)
8.343
1. lnΓ( x)=l n√
2π+∞/summationdisplay
n=1/braceleftbigg1
2ncos2nπx+1
nπ(C+l n2 nπ)sin2nπx/bracerightbigg
[0<x< 1] FI III 558
8.352 The incomplete gamma function 899
2. lnΓ( z)=zlnz−z−1
2lnz+l n√
2π+1
2∞/summationdisplay
m=1m
(m+1 ) (m+2 )∞/summationdisplay
n=11
(z+n)m+1
[|argz|<π] MO 9
8.3447Asymptotic expansion for large values of |z|:
ln Γ(z)=zlnz−z−1
2lnz+l n√
2π+n−1/summationdisplay
k=1B2k
2k(2k−1)z2k−1+Rn(z),
where
|Rn(z)|<|B2n|
2n(2n−1)|z|2n−1cos2n−1/parenleftbig1
2argz/parenrightbig MO5
For integrals of lnΓ( x), see6.44.
8.35 The incomplete gamma function
8.350 Definition:
1. γ(α,x)=/integraldisplayx
0e−ttα−1dt [Reα>0] EH II 133(1), NH 1(1)
2.11Γ(α,x)=/integraldisplay∞
xe−ttα−1dt EH II 133(2), NH 2(2), LE 339
3.∗Γ(z,0) = Γ( z)
4.∗Γ(a,∞)=0
5.∗γ(a,0) = 0
8.351
1. γ∗(α,x)=x−α
Γ(α)γ(α,x) is an analytic function with respect to αandx EH II 133(5)
2. Another definition of Γ( α,x) that is also suitable for the case Re α≤0:
γ(α,x)=xα
αe−xΦ( 1,1+α;x)=xα
αΦ(a,1+a;−x) EH II 133(3)
3. For fixed x,Γ (α,x) is an entire function of α. For non-integral α,Γ (α,x) is a multiple-valued
function of xwith a branch point at x=0 .
4. A second definition of Γ( α,x):
Γ(α,x)=xαe−xΨ(1,1+α;x)=e−xΨ(1−α,1−α;x) EH II 133(4)
8.352 Special cases:
1. γ(1 +n, x)=n!/bracketleftBigg
1−e−x/parenleftBiggn/summationdisplay
m=0xm
m!/parenrightBigg/bracketrightBigg
[n=0,1,...]
EH II 136(17, 16), NH 6(11)
2. Γ(1 + n, x)=n!e−xn/summationdisplay
m=0xm
m![n=0,1,...] EH II 136(16, 18)
900 Euler’s Integrals of the First and Second Kinds 8.353
3.11Γ(−n, x)=(−1)n
n!/bracketleftbigg
Ei(−z)−1
2ln(−z)+1
2ln/parenleftbigg
−1
z/parenrightbigg
−lnz/bracketrightbigg
−e−zn/summationdisplay
k=1zk−n−1
(−n)k
[n=1,2,...]
4.∗Γ(n, x)=(n−1)!e−xn−1/summationdisplay
m=0xm
m!
5.∗Γ(−n+1,x)=(−1)n+1
(n−1)!/bracketleftBigg
Γ(0,x)−e−zn−2/summationdisplay
m=0(−1)mm!
xm+1/bracketrightBigg
[n=2,3,...]
6.∗γ(n, x)=(n−1)!/bracketleftBigg
1−e−xn−1/summationdisplay
m=0xm
m!/bracketrightBigg
[n=1,2,...]
7.∗Γ(n, x)=(n−1)!e−xn−1/summationdisplay
m=0xm
m![n=1,2,...]
8.∗Γ(−n+k,x)=(−1)n−k
(n−k)!/bracketleftBigg
Γ(0,x)−e−xn−k−1/summationdisplay
m=0(−1)mm!
xm+1/bracketrightBigg
[n−k≥1,k=0,1,...]
8.353 Integral representations:
1. γ(α,x)=xαcosecπα/integraldisplayπ
0excosθcos(αθ+xsinθ)dθ [x/negationslash=0,Reα>0,α/negationslash=1,2,...]
EH II 137(2)
2. γ(α,x)=x1
2α/integraldisplay∞
0e−tt1
2α−1Jα/parenleftBig
2√
xt/parenrightBig
dt [Reα>0] EH II 138(4)
3. Γ( α,x)=ρ−xxα
Γ(1−α)/integraldisplay∞
0e−tt−α
x+tdt [Reα<1,x > 0]
EH II 137(3), NH 19(12)
4. Γ( α,x)=2x1
2αe−x
Γ(1−α)/integraldisplay∞
0e−tt−1
2αKα/bracketleftBig
2√
xt/bracketrightBig
dt [Reα<1] EH II 138(5)
5. Γ( α,xy)=yαe−xy/integraldisplay∞
0e−ty(t+x)α−1dt
[Rey>0,x > 0,Reα>1] (See also 3.936 5,3.944 1–4) NH 19(10)
For integrals of the gamma function, see 6.45.
8.354 Series representations:
1. γ(α,x)=∞/summationdisplay
n=0(−1)nxα+n
n!(α+n)EH II 135(4)
8.356 The incomplete gamma function 901
2. Γ( α,x)=Γ ( α)−∞/summationdisplay
n=0(−1)nxα+n
n!(α+n)[α/negationslash=0,−1,−2,...]
EH II 135(5), LE 340(2)
3. Γ( α,x)−Γ(α,x+y)=γ(α,x+y)−γ(α,x)
=e−xxα−1∞/summationdisplay
k=0(−1)k[1−e−yek(y)] Γ(1−α+k)
xkΓ(1−α)
ek(x)=k/summationdisplay
m=0xm
m![|y|<|x|]EH II 139(2)
4. γ(α,x)=Γ ( α)e−xx1
2α∞/summationdisplay
n=0x1
2nIn+α/parenleftbig
2√x/parenrightbign/summationdisplay
m=0(−1)m
m![x/negationslash=0,α/negationslash=0,−1,−2,...]
EH II 139(3)
5. Γ( α,x)=e−xxα∞/summationdisplay
n=0Lα
n(x)
n+1[x>0] EH II 140(5)
8.355 Γ(α,x)γ(α,y)=e−x−y(xy)α∞/summationdisplay
n=0n!Γ (α)
(n+1 )Γ ( α+n+1 )Lα
n(x)Lα
n(y)
[y>0,x≥y, α /negationslash=0,−1,...]
EH II 139(4)
8.356 Functional relations:
1.11γ(α+1,x)=αγ(α,x)−xαe−xEH II 134(2)
2. Γ( α+1,x)=αΓ(α,x)+xαe−xEH II 134(3)
3. Γ( α,x)+γ(α,x)=Γ ( α) EH II 134(1)
4.dγ(α,x)
dx=−dΓ(α,x)
dx=xα−1e−xEH II 135(8)
5.Γ(α+n, x)
Γ(α+n)=Γ(α,x)
Γ(α)+e−xn−1/summationdisplay
s=0xα+s
Γ(α+s+1 )NH 4(3)
6.11Γ(α)Γ(α+n, x)−Γ(α+n)Γ(α,x)=Γ ( α+n)γ(α,x)−Γ(α)γ(α+n, x) NH 5
7.∗Γ(a+k,x)=(a+k−1)Γ(a+k−1,x)+xa+k−1e−x
=1
a+k/bracketleftbig
Γ(a+k+1,x)−xa+ke−x/bracketrightbig
8.∗Γ(a−k,x)=(a−k−1)Γ(a−k−1,x)+xa−k−1e−x
=1
a−k/bracketleftbig
Γ(a−k+1,x)−xa−ke−x/bracketrightbig
9.∗γ(a+k,x)=(a+k−1)γ(a+k−1,x)−xa+k−1e−x
=1
a+k/bracketleftbig
Γ(a+k+1,x)+xa+ke−x/bracketrightbig
902 Euler’s Integrals of the First and Second Kinds 8.357
10.∗γ(a−k,x)=(a−k−1)γ(a−k−1,x)−xa−k−1e−x
=1
a−k/bracketleftbig
γ(a−k+1,x)+xa−ke−x/bracketrightbig
8.357 Asymptotic representation for large values of |x|:
1. Γ( α,x)=xα−1e−x/bracketleftBiggM−1/summationdisplay
m=0(−1)mΓ(1−α+m)
xmΓ(1−α)+O/parenleftBig
|x|−M/parenrightBig/bracketrightBigg
/bracketleftbigg
|x|→∞ ,−3π
2<argx<3π
2,M =1,2,.../bracketrightbigg
EH II 135(6), NH 37(7), LE 340(3)
8.358 Representation as a continued fraction:
Γ(α,x)=e−xxα
x+1−α
1+1
x+2−α
1+2
x+3−α
1+...EH II 136(13), NH 42(9)
8.359 Relationships with other functions:
1. Γ(0 ,x)=−Ei(−x) EH II 143(1)
2. Γ/parenleftbigg
0,ln1
x/parenrightbigg
=−li(x) EH II 143(2)
3. Γ/parenleftbig1
2,x2/parenrightbig
=√π−√πΦ(x) EH II 147(2)
4.11γ/parenleftbig1
2,x2/parenrightbig
=√πΦ(x) EH II 147(1)
8.36 The psi function ψ(x)
8.360 Definition:
1. ψ(x)=d
dxlnΓ(x)
8.361 Integral representations:
1.8ψ(z)=dln Γ(z)
dz=/integraldisplay∞
0/parenleftbigge−t
t−e−zt
1−e−t/parenrightbigg
dt [Rez>0] NH 183(1), WH
2. ψ(z)=/integraldisplay∞
0/braceleftbigg
e−t−1
(1 +t)z/bracerightbiggdt
t[Rez>0] NH 184(7), WH
3. ψ(z)=l n z−1
2z−2/integraldisplay∞
0td t
(t2+z2)(e2πt−1)[Rez>0] WH
4. ψ(z)=/integraldisplay1
0/parenleftbigg1
−lnt−tz−1
1−t/parenrightbigg
dt [Rez>0] WH
8.363 The psi function ψ(x) 903
5. ψ(z)=/integraldisplay∞
0e−t−e−zt
1−e−tdt−C, WH
6. ψ(z)=/integraldisplay∞
0/braceleftbig
(1 +t)−1−(1 +t)−z/bracerightbigdt
t−C, [Rez>0] WH
7. ψ(z)=/integraldisplay1
0tz−1−1
t−1dt−C FI II 796, WH
8. ψ(z)=l n z+/integraldisplay∞
0e−tz/bracketleftbigg1
t−1
1−e−t/bracketrightbigg
dt [Rez>0] MO 4
See also 3.244 3,3.311 6,3.317 1,3.457 ,3.458 2,3.471 14,4.253 1a n d6 , 4.275 2,4.281 4,4.482 5.
For integrals of the psi function, see 6.46,6.47.
Series representation
8.362
1. ψ(x)=−C−∞/summationdisplay
k=0/parenleftbigg1
x+k−1
k+1/parenrightbigg
FI II 799(26), KU 26(1)
=−C−1
x+x∞/summationdisplay
k=11
k(x+k)FI II 495
2. ψ(x)=l n x−∞/summationdisplay
k=0/bracketleftbigg1
x+k−ln/parenleftbigg
1+1
x+k/parenrightbigg/bracketrightbigg
MO 4
3. ψ(x)=−C+π2
6(x−1)−(x−1)∞/summationdisplay
k=1/parenleftbigg1
k+1−1
x+k/parenrightbiggk−1/summationdisplay
n=01
x+nNH 54(12)
8.363
1. ψ(x+1 )= −C+∞/summationdisplay
k=2(−1)kζ(k)xk−1NH 37(5)
2. ψ(x+1 )=1
2x−π
2cotπx−x2
1−x2−C+∞/summationdisplay
k=1[1−ζ(2k+1 ) ]x2kNH 38(10)
3. ψ(x)−ψ(y)=∞/summationdisplay
k=0/parenleftbigg1
y+k−1
x+k/parenrightbigg
(see also 3.219 ,3.231 5,3.311 7,3.688 20,4.253 1,4.295 37) NH 99(3)
4. ψ(x+iy)−ψ(x−iy)=∞/summationdisplay
k=02yi
y2+(x+k)2
5. ψ/parenleftbiggp
q/parenrightbigg
=−C+∞/summationdisplay
k=0/parenleftbigg1
k+1−q
p+kq/parenrightbigg
(see also 3.244 3) NH 29(1)
904 Euler’s Integrals of the First and Second Kinds 8.364
6.8ψ/parenleftbiggp
q/parenrightbigg
=−C−ln(2q)−π
2cotpπ
q+2[q+1
2]−1/summationdisplay
k=1/bracketleftbigg
cos2kpπ
qln sinkπ
q/bracketrightbigg
[q=2,3,...,p =1,2,...,q −1]
MO 4, EH I 19(29)
7. ψ/parenleftbiggp
q/parenrightbigg
−ψ/parenleftbiggp−1
q/parenrightbigg
=q∞/summationdisplay
n=2∞/summationdisplay
k=01
(p+kq)n−1NH 59(3)
8. ψ(n)(x)=(−1)n+1n!∞/summationdisplay
k=01
(x+k)n+1=(−1)n+1n!ζ(n+1,x) NH 37(1)
Infinite-product representation
8.364
1. eψ(x)=x∞/productdisplay
k=0/parenleftbigg
1+1
x+k/parenrightbigg
e−1
x+k NH 65(12)
2. eyψ(x)=Γ(x+y)
Γ(x)∞/productdisplay
k=0/parenleftbigg
1+y
x+k/parenrightbigg
e−y
x+k NH 65(11)
See also 8.37.
•For a connection with Riemann’s zeta function, see 9.533 2.
•For a connection with the gamma function, see 4.325 12 and 4.352 1.
•For a connection with the beta function, see 4.253 1.
•For series of psi functions, see 8.403 2,8.446 ,a n d8.447 3 (Bessel functions), 8.761 (derivatives
of associated Legendre functions with respect to the degree), 9.153 ,9.154 (hypergeometric
function), 9.237 (confluent hypergeometric function).
•For integrals containing psi functions, see 6.46–6.47 .
8.365 Functional relations:
1. ψ(x+1 )= ψ(x)+1
xJA
2. ψ/parenleftbiggx+1
2/parenrightbigg
−ψ/parenleftBigx
2/parenrightBig
=2β(x) (cf. 8.370)
3. ψ(x+n)=ψ(x)+n−1/summationdisplay
k=01
x+kGA 154(64)a
4. ψ(n+1 )= −C+n/summationdisplay
k=11
kMO 4
5. lim
n→∞[ψ(z+n)−lnn]=0 MO 3
6. ψ(nz)=1
nn−1/summationdisplay
k=0ψ/parenleftbigg
z+k
n/parenrightbigg
+l nn [n=2,3,4,...] MO 3
8.367 The psi function ψ(x) 905
7. ψ(x−n)=ψ(x)−n/summationdisplay
k=11
x−k
8. ψ(1−z)=ψ(z)+πcotπz GA 155(68)a
9. ψ/parenleftbig1
2+z/parenrightbig
=ψ/parenleftbig1
2−z/parenrightbig
+πtanπz JA
10. ψ/parenleftbig3
4−n/parenrightbig
=ψ/parenleftbig1
4+n/parenrightbig
+π [n=0,±1,±2,...]
8.366 Particular values
1. ψ(1) =−C (cf.8.367 1)
2. ψ/parenleftbig1
2/parenrightbig
=−C−2ln2= −1.963510026 ... GA 155a
3. ψ/parenleftbig1
2±n/parenrightbig
=−C+2/bracketleftBiggn/summationdisplay
k=11
2k−1−ln2/bracketrightBigg
JA
4. ψ/parenleftbig1
4/parenrightbig
=−C−π
2−3ln2 GA 157a
5. ψ/parenleftbig3
4/parenrightbig
=−C+π
2−3ln2 GA 157a
6. ψ/parenleftbig1
3/parenrightbig
=−C−π
2/radicalBig
1
3−3
2ln 3 GA 157a
7. ψ/parenleftbig2
3/parenrightbig
=−C+π
2/radicalBig
1
3−3
2ln 3 GA 157a
8. ψ/prime(1) =π2
6=1.644934066848 ... JA
9. ψ/prime/parenleftbig1
2/parenrightbig
=π2
2=4.9348022005 ... JA
10. ψ/prime(−n)=∞ [nis a natural number] JA
11. ψ/prime(n)=π2
6−n−1/summationdisplay
k=11
k2[nis a natural number] JA
12. ψ/prime/parenleftbig1
2+n/parenrightbig
=π2
2−4n/summationdisplay
k=11
(2k−1)2[nis a natural number] JA
13. ψ/prime/parenleftbig1
2−n/parenrightbig
=π2
2+4n/summationdisplay
k=11
(2k−1)2[nis a natural number] JA
8.367 Euler’s constant (also denoted by γ):
1.C=−ψ(1) = 0 .577 215 664 90 ... FI II 319, 795
2.C= lim
n→∞/bracketleftBiggn−1/summationdisplay
k=11
k−lnn/bracketrightBigg
FI II 801a
3.C= lim
x→1+0/bracketleftbigg
ζ(x)−1
x−1/bracketrightbigg
FI II 804
906 Euler’s Integrals of the First and Second Kinds 8.370
Integral representations:
4.C=−/integraldisplay∞
0e−tlntd t FI II 807
5.C=−/integraldisplay1
0ln/parenleftbigg
ln1
t/parenrightbigg
dt FI II 807
6.C=/integraldisplay1
0/bracketleftbigg1
lnt+1
1−t/bracketrightbigg
dt DW
7.C=−/integraldisplay∞
0/bracketleftbigg
cost−1
1+t/bracketrightbiggdt
tMO 10
8.C=1−/integraldisplay∞
0/bracketleftbiggsint
t−1
1+t/bracketrightbiggdt
tMO 10
9.C=−/integraldisplay∞
0/bracketleftbigg
e−t−1
1+t/bracketrightbiggdt
tFI II 795, 802
10.C=−/integraldisplay∞
0/bracketleftbigg
e−t−1
1+t2/bracketrightbiggdt
tD W ,M O1 0
11.C=/integraldisplay∞
0/bracketleftbigg1
et−1−1
tet/bracketrightbigg
dt DW
12.C=/integraldisplay1
0/parenleftbig
1−e−t/parenrightbigdt
t−/integraldisplay∞
1e−t
tdt FI II 802
See also 8.361 5–8.361 7,3.311 6,3.435 3a n d4 , 3.476 2,3.481 1a n d2 , 3.951 10,4.283 9,
4.331 1,4.421 1,4.424 1,4.553 ,4.572 ,6.234 ,6.264 1,6.468 .
13. Asymptotic expansions
C=n−1/summationdisplay
k=11
k−lnn+1
2n+1
12n2−1
120n4+1
252n6−1
240n8+...
···+B2r
2r1
n2r+B2r+2
2(r+1 )θ
n2r+2[0<θ< 1] FI II 827
8.37 The function β(x)
8.370 Definition:
β(x)=1
2/bracketleftbigg
ψ/parenleftbiggx+1
2/parenrightbigg
−ψ/parenleftBigx
2/parenrightBig/bracketrightbigg
NH 16(13)
8.371 Integral representations:
1.3β(x)=/integraldisplay1
0tx−1
1+tdt [Rex>0] WH
2. β(x)=/integraldisplay∞
0e−xt
1+e−tdt [Rex>0] MO 4
3. β/parenleftbiggx+1
2/parenrightbigg
=/integraldisplay∞
0e−xt
coshtdt [Rex>−1]
See also 3.241 1,3.251 7,3.522 2a n d4 , 3.623 2a n d3 , 4.282 2,4.389 3,4.532 1a n d3 .
8.377 The function β(x) 907
Series representation
8.372
1.7β(x)=∞/summationdisplay
k=0(−1)k
x+k[−x/negationslash∈N] NH 37, 101(1)
2.7β(x)=∞/summationdisplay
k=01
(x+2k)(x+2k+1 )[−x/negationslash∈N] NH 101(2)
3.8β(x)=1
2∞/summationdisplay
k=0k!
x(x+1 )...(x+k)1
2k[−x/negationslash∈N]
[βhas simple poles at x=−nwith residue ( −1)n]NH 246(7)
8.373
1.6β(x+1 )=l n2+∞/summationdisplay
k=1(−1)k/parenleftbig
1−2−k/parenrightbig
ζ(k+1 )xk[|x|<1] NH 37(5)
2.6β(x+1 )=l n2 −1+1
2x−π
2s inπx+1
1−x2−∞/summationdisplay
k=1/bracketleftbig
1−/parenleftbig
1−2−2k/parenrightbig
ζ(2k+1 )/bracketrightbig
x2k
[0<|x|<2;x/negationslash=±1] NH 38(11)
8.374dn
dxnβ(x)=(−1)nn!∞/summationdisplay
k=0(−1)k
(x+k)n+1[−x∈N] NH 37(2)
8.375 Representation in the form of a finite sum:
1.6β/parenleftbiggp
q/parenrightbigg
=π
2s inpπ
q−⌊q−1
2⌋/summationdisplay
k=0cosp(2k+1 )π
qln sin(2k+1 )π
2q
[q=2,3,...,p =1,2,3,...,q −1] (see also 8.362 5–7) NH 23(9)
2. β(n)=(−1)n+1ln2 +n−1/summationdisplay
k=1(−1)k+n+1
k
Functional relations
8.3762n/summationdisplay
k=0(−1)kβ/parenleftbiggx+k
2n+1/parenrightbigg
=( 2n+1 )β(x) NH 19
8.377n/summationdisplay
k=1β/parenleftbig
2kx/parenrightbig
=ψ(2nx)−ψ(x)−nln 2 NH 20(10)
908 Euler’s Integrals of the First and Second Kinds 8.380
8.38 The beta function (Euler’s integral of the first kind): B(x, y)
Integral representation
8.380
1. B( x, y)=/integraldisplay1
0tx−1(1−t)y−1dt∗
=2/integraldisplay1
0t2x−1/parenleftbig
1−t2/parenrightbigy−1dt [Rex>0,Rey>0] FI II 774(1)
2. B( x, y)=2/integraldisplayπ/2
0sin2x−1ϕcos2y−1ϕdϕ [Rex>0,Rey>0] KU 10
3. B( x, y)=/integraldisplay∞
0tx−1
(1 +t)x+ydt=2/integraldisplay∞
0t2x−1
(1 +t2)x+ydt [Rex>0,Rey>0] FI II 775
4. B( x, y)=22−y−x/integraldisplay1
−1(1 +t)2x−1(1−t)2y−1
(1 +t2)x+ydt [Rex>0,Rey>0] MO 7
5. B( x, y)=/integraldisplay1
0tx−1+ty−1
(1 +t)x+ydt=/integraldisplay∞
1tx−1+ty−1
(1 +t)x+ydt [Rex>0,Rey>0] BI (1)(15)
6. B( x, y)=1
2x+y−1/integraldisplay1
0/bracketleftBig
(1 +t)x−1(1−t)y−1+( 1+ t)y−1(1−t)x−1/bracketrightBig
dt
[Rex>0,Rey>0] BI (1)(15)
7. B( x, y)=zy(1 +z)x/integraldisplay1
0tx−1(1−t)y−1
(t+z)x+ydt
[Rex>0,Rey>0,0>z> −1,Re(x+y)<1]NH 163(8)
8. B( x, y)=zy(1 +z)x/integraldisplayπ/2
0cos2x−1ϕsin2y−1ϕ
(z+c o s2ϕ)x+ydϕ
[Rex>0,Rey>0,0>z> −1,Re(x+y)<1]NH 163(8)
See also 3.196 3,3.198 ,3.199 ,3.215 ,3.238 3,3.251 1–3, 11, 3.253 ,3.312 1,3.512 1a n d
2,3.541 1,3.542 1,3.621 5,3.623 1,3.631 1, 8, 9, 3.632 2,3.633 1, 4,3.634 1, 2,3.637 ,
3.642 1,3.667 8,3.681 2.
9. B( x, x)=1
22x−2/integraldisplay1
0/parenleftbig
1−t2/parenrightbigx−1dt=1
22x−1/integraldisplay1
0(1−t)x−1
√
tdt
See8.384 4,8.382 3, and also 3.621 1,3.642 2,3.665 1,3.821 6,3.839 6.
10. B( x+y,x−y)=41−x/integraldisplay∞
0cosh 2 yt
cosh2xtdt [Rex>|Rey|,Rex>0] MO 9
11. B/parenleftBig
x,y
z/parenrightBig
=z/integraldisplay1
0(1−tz)x−1ty−1dt/bracketleftBig
Rez>0,Rey
z>0,Rex>0/bracketrightBig
FI II 787a
∗This equation is used as the definition of the function B( x, y).
8.384 The beta function (Euler’s integral of the first kind): B(x, y) 909
8.381
1./integraldisplay∞
−∞dt
(a+it)x(b−it)y=2π(a+b)1−x−y
(x+y−1)B(x, y)
[a>0,b > 0;xandyare real ,x+y>1]MO 7
2./integraldisplay∞
−∞dt
(a−it)x(b−it)y=0
[a>0,b > 0;xandyare real ,x+y>1]MO 7
3. B( x+iy,x−iy)=21−2xαe−2iγy/integraldisplay∞
−∞e2iαytdt
cosh2x(αt−γ)
[y,α,γ are real ,α > 0; Re x>0]
MI 8a
For an integral representation of lnB( x, y), see3.428 7.
4.1
B(x, y)=2x+y−1(x+y−1)
π/integraldisplayπ/2
0cos[(x−y)t]c osx+y−2td t NH 158(5)a
=2x+y−2(x+y−1)
πcos/bracketleftbig
(x−y)π
2/bracketrightbig/integraldisplayπ
0cos[(x−y)t]s inx+y−2td t NH 159(8)a
=2x+y−2(x+y−1)
πsin/bracketleftbig
(x−y)π
2/bracketrightbig/integraldisplayπ
0sin[(x−y)t]s inx+y−2td t NH 159(9)a
Series representation
8.382
1. B( x, y)=1
y∞/summationdisplay
n=0(−1)ny(y−1)...(y−n)
n!(x+n)[y>0] WH
2. lnB/parenleftbigg1+x
2,1
2/parenrightbigg
ln√
2π+1
2/bracketleftbigg
ln/parenleftbiggtanπx
2
x/parenrightbigg
−ln/parenleftbigg1+x
1−x/parenrightbigg/bracketrightbigg
+∞/summationdisplay
k=01−/parenleftbig
1−2−2k/parenrightbig
ζ(2k+1 )
2k+1x2k+1
[|x|<2] NH 39(17)
3. B/parenleftbigg
z,1
2/parenrightbigg
=∞/summationdisplay
k=1(2k−1)!!
2kk!1
z+k+1
z(see also 8.384 and8.380 9) WH
8.383 Infinite-product representation:
(x+y+1 )B ( x+1,y+1 )=∞/productdisplay
k=1k(x+y+k)
(x+k)(y+k)[x, y /negationslash=−1,−2,...] MO 2
8.384 Functional relations involving the beta function:
1. B( x, y)=Γ(x)Γ(y)
Γ(x+y)=B (y,x) FI II 779
2. B( x, y)B(x+y,z)=B ( y,z)B(y+z,x) MO 6
910 Bessel Functions and Functions Associated with Them 8.391
3.∞/summationdisplay
k=0B(x, y+k)=B ( x−1,y) WH
4. B( x, x)=21−2xB/parenleftbig1
2,x/parenrightbig
(see also 8.380 9a n d8.382 3)
FI II 784
5. B( x, x)B/parenleftbig
x+1
2,x+1
2/parenrightbig
=π
24x−1xWH
6.1
B(n, m)=m/parenleftbiggn+m−1
n−1/parenrightbigg
=n/parenleftbiggn+m−1
m−1/parenrightbigg
[mandnare natural numbers]
For a connection with the psi function, see 4.253 1.
8.39 The incomplete beta function Bx(p, q)
8.3917Bx(p, q)=/integraldisplayx
0tp−1(1−t)q−1dt=xp
p2F1(p,1−q;p+1 ;x) ET I 373
8.392 Ix(p, q)=Bx(p, q)
B(p, q)ET II 429
8.4–8.5 Bessel Functions and Functions Associated with Them
8.40 Definitions
8.401 Bessel functions Zν(z) are solutions of the differential equation
d2Zν
dz2+1
zdZν
dz+/parenleftbigg
1−ν2
z2/parenrightbigg
Zν=0 KU 37(1)
Special types of Bessel functions are what are called Bessel functions of the first kind Jν(z), Bessel
functions of the second kind Yν(z) (also called Neumann functions and often written Nν(z)), and Bessel
functions of the third kind H(1)
ν(z)a n dH(2)
ν(z) (also called Hankel’s functions).
8.402 Jν(z)=zν
2ν∞/summationdisplay
k=0(−1)k z2k
22kk!Γ(ν+k+1 )[|argz|<π] KU 55(1)
8.403
1. Yν(z)=1
sinνπ[cosνπJν(z)−J−ν(z)] [for non-integer ν,|argz|<π]
KU 41(3)
8.407 Definitions 911
2. πYn(z)=2Jn(z)lnz
2−n−1/summationdisplay
k=0(n−k−1)!
k!/parenleftBigz
2/parenrightBig2k−n
−∞/summationdisplay
k=0(−1)k 1
k!(k+n)!/parenleftBigz
2/parenrightBign+2k
[ψ(k+1 )+ ψ(k+n+1 ) ]
KU 43(10)
=2Jn(z)/parenleftBig
lnz
2+C/parenrightBig
−n−1/summationdisplay
k=0(n−k−1)!
k!/parenleftBigz
2/parenrightBig2k−n
−/parenleftBigz
2/parenrightBign1
n!n/summationdisplay
k=11
k−∞/summationdisplay
k=1(−1)k/parenleftbigz
2/parenrightbign+2k
k!(k+n)!/bracketleftBiggn+k/summationdisplay
m=11
m+k/summationdisplay
m=11
m/bracketrightBigg
[n+ 1 a natural number ,|argz|<π]
KU 44, WA 75(3)a
8.404
1. Y−n(z)=(−1)nYn(z)[ nis a natural number] KU 41(2)
2. J−n(z)=(−1)nJn(z)[ nis a natural number] KU 41(2)
8.4057
1. H(1)
ν(z)=Jν(z)+iYν(z) KU 44(1)
2. H(2)
ν(z)=Jν(z)−iYν(z) KU 44(1)
In all relationships that hold for an arbitrary Bessel function Zν(z), that is, for the functions Jν(z),
Yν(z), and linear combinations of them, for example, H(1)
ν(z)a n d H(2)
ν(z), we shall write simply the
letter Zinstead of the letters J,Y,H(1),a n d H(2).
Modified Bessel functions of imaginary argument Iν(z)andKν(z)
8.406
1. Iν(z)=e−π
2νiJν/parenleftbig
eπ
2iz/parenrightbig /bracketleftBig
−π<argz≤π
2/bracketrightBig
WA 92
2. Iν(z)=e3
2πνiJν/parenleftBig
e−3
2πiz/parenrightBig/bracketleftBigπ
2<argz≤π/bracketrightBig
WA 92
For integer ν,
3. In(z)=i−nJn(iz) KU 46(1)
8.407
1.8Kν(z)=πi
2eπ
2νiH(1)
ν/parenleftBig
ze1
2πi/parenrightBig /bracketleftbig
−π<argz≤1
2π/bracketrightbig
2.8Kν(z)=−πi
2e−π
2νiH(2)
−ν/parenleftBig
ze−1
2πi/parenrightBig /bracketleftbig
−1
2π<argz≤π/bracketrightbig
WA 92(8)
For the differential equation defining these functions, see 8.494 .
912 Bessel Functions and Functions Associated with Them 8.411
8.41 Integral representations of the functions Jν(z)andNν(z)
8.411
1.11Jn(z)=1
2π/integraldisplayπ
−πe−niθ+izsinθdθ
=1
π/integraldisplayπ
0cos(nθ−zsinθ)dθ [n=0,1,2,...] WH
2. J2n(z)=1
π/integraldisplayπ
0cos2nθcos (zsinθ)dθ=2
π/integraldisplayπ/2
0cos2nθcos (zsinθ)dθ
[nan integer] WA 30(7)
3.11J2n+1(z)=1
π/integraldisplayπ
0sin(2n+1 )θsin (zsinθ)dθ
=2
π/integraldisplayπ/2
0sin(2n+1 )θsin(zsinθ)dθ [nan integer] WA 30(6)
4. Jν(z)=2/parenleftbigz
2/parenrightbigν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplayπ/2
0sin2νθcos(zcosθ)dθ
/bracketleftbig
Reν>−1
2/bracketrightbig
WH
5. Jν(z)=/parenleftbigz
2/parenrightbigν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbigg1
2/parenrightbigg/integraldisplayπ
0sin2νθcos(zcosθ)dθ/bracketleftbig
Reν>−1
2/bracketrightbig
6. Jν(z)=/parenleftbigz
2/parenrightbigν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbigg1
2/parenrightbigg/integraldisplayπ/2
−π/2cos(zsinθ)c o s2νθd θ
/bracketleftbig
Reν>−1
2/bracketrightbig
KU 65(5), WA 35(4)a
7. Jν(z)=/parenleftbigz
2/parenrightbigν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplayπ
0e±izcosϕsin2νϕdϕ/bracketleftbig
Re/parenleftbig
ν+1
2/parenrightbig
>0/bracketrightbig
WH
8. Jν(z)=/parenleftbigz
2/parenrightbigν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplay1
−1/parenleftbig
1−t2/parenrightbigν−1
2cosztdt/bracketleftbig
Reν>−1
2/bracketrightbig
KU 65(6), WH
9. Jν(x)=2/parenleftbigx
2/parenrightbig−ν
Γ/parenleftbig1
2−ν/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplay∞
1sinxt
(t2−1)ν+1
2dt/bracketleftbig
−1
2<Reν<1
2,x > 0/bracketrightbig
MO 37
10. Jν(z)=/parenleftbigz
2/parenrightbigν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplay1
−1eizt/parenleftbig
1−t2/parenrightbigν−1
2dt/bracketleftbig
Reν>−1
2/bracketrightbig
WA 34(3)
11. Jν(x)=2
π/integraldisplay∞
0sin/parenleftBig
xcosht−νπ
2/parenrightBig
coshνtdt WA 199(12)
12. Jν(z)=2ν+1zν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplayπ/2
0/parenleftBig
cosν−1
2θ/parenrightBig
sin/parenleftbig
z−νθ+1
2θ/parenrightbig
sin2ν+1θe−2zcotθdθ
/bracketleftBig
|argz|<π
2,Re/parenleftbig
ν+1
2/parenrightbig
>0/bracketrightBig
WH
8.414 Integral representations of the functions Jν(z)andNν(z) 913
13.10Jν(z)=1
π/integraldisplayπ
0cos(νθ−zsinθ)dθ−sinνπ
π/integraldisplay∞
0e−νθ−zsinhθdθ
[Rez>0] WA 195(4)
14. Jν(z)=e±νπi
π/bracketleftbigg/integraldisplayπ
0cos(νθ+zsinθ)dθ−sinνπ/integraldisplay∞
0e−νθ+zsinhθdθ/bracketrightbigg
/bracketleftbigg
forπ
2<|argz|<π ,with the upper sign taken for |argz|>π
2
and the lower sign taken for |argz|<−π
2/bracketrightbigg
WH
8.412
1. Jν(z)=1
2πi/integraldisplay(0+)
−∞t−ν−1exp/bracketleftbiggz
2/parenleftbigg
t−1
t/parenrightbigg/bracketrightbigg
dt/bracketleftBig
|argz|<π
2/bracketrightBig
WH, WA 195(2)
2. Jν(z)=zν
2ν+1πi/integraldisplay(0+)
−∞t−ν−1exp/parenleftbigg
t−z2
4t/parenrightbigg
dt WA 195(1)
3.8Jν(z)=zν
2ν+1πi∞/summationdisplay
k=1(−1)kz2k
22kk!/integraldisplay(0+)
−∞ett−ν−k−1dt WA 195(1)
4. Jν(x)=1
2πi/integraldisplayi∞
−i∞Γ(−t)
Γ(ν+t+1 )/parenleftBigx
2/parenrightBigν+2t
dt [Reν>0,x > 0] WA 214(7)
5.7Jν(z)=Γ/parenleftbig1
2−ν/parenrightbig/parenleftBigz
2/parenrightBigν
2πiΓ/parenleftbig1
2/parenrightbig/integraldisplay(1+,−1−)
A/parenleftbig
t2−1/parenrightbigν−1
2cos(zt)dt
/bracketleftbigg
ν/negationslash=1
2,3
2,...; The point Afalls to the right of the point t=1 ,
and arg( t−1) = arg( t+ 1) = 0 at the point A/bracketrightbigg
WH
6.8Jν(z)=1
2π/integraldisplayπ+∞i
−π+∞ie−izsinθ+iνθdθ [Rez>0]
The path of integration being taken around the
semi-infinite strip y≥0,−π≤x≤π.
8.4138Jν/parenleftBig/radicalbig
z2+ζ2/parenrightBig
(z2−ζ2)ν
2=1
π(z+ζ)ν/braceleftbigg/integraldisplay∞
0eζcostcos(zsint−νt)dt
−sinνπ/integraldisplay∞
0exp (−zsinht−ζcosht−νt)dt/bracerightbigg
[Re(z+ζ)>0] MO 40
8.414/integraldisplay∞
2xJ0(t)
tdt=1
4π/integraldisplay−1
2+i∞
−1
2−i∞Γ(−t)
tΓ(1 + t)x2tdt [x>0] MO 41
See3.715 2, 9, 10, 13, 14, 19–21, 3.865 1, 2, 4, 3.996 4.
914 Bessel Functions and Functions Associated with Them 8.415
•For an integral representation of J0(z), see3.714 2,3.753 2, 3, and 4.124 .
•For an integral representation of J1(z), see3.697 ,3.711 ,3.752 2, and 3.753 5.
8.415
1. Y0(x)=4
π2/integraldisplay1
0arcsin t√
1−t2sin(xt)dt−4
π2/integraldisplay∞
1ln/parenleftbig
t+√
t2−1/parenrightbig
√
t2−1sin(xt)dt
[x>0] MO 37
2. Yν(x)=−2/parenleftbigx
2/parenrightbig−ν
Γ/parenleftbig1
2−ν/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplay∞
1cosxt
(t2−1)ν+1
2dt/bracketleftbig
−1
2<Reν<1
2,x > 0/bracketrightbig
KU 89(28)a, MO 38
3. Yν(x)=−2
π/integraldisplay∞
0cos/parenleftBig
xcosht−νπ
2/parenrightBig
coshνtdt [−1<Reν<1,x > 0] WA 199(13)
4.8Yν(z)=1
π/integraldisplayπ
0sin(zsinθ−νθ)dθ−1
π/integraldisplay∞
0/parenleftbig
eνt+e−νtcosνπ/parenrightbig
e−zsinhtdt
[Rez>0] WA 197(1)
5. Yν(z)=2/parenleftbigz
2/parenrightbigν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/bracketleftBigg/integraldisplayπ/2
0sin (zsinθ)c o s2νθd θ−/integraldisplay∞
0e−zsinhθcosh2νθd θ/bracketrightBigg
/bracketleftbig
Reν>−1
2,Rez>0/bracketrightbig
WA 181(5)a
6. Yν(z)=−2ν+1zν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplayπ
2
0cosν−1
2θcos/parenleftbig
z−νθ+1
2θ/parenrightbig
sin2ν+1θe−2zcotθdθ
/bracketleftBig
|argz|<π
2,Re/parenleftbig
ν+1
2/parenrightbig
>0/bracketrightBig
WA 186(8)
For an integral representation of Y0(z), see3.714 3,3.753 4,3.864 . See also 3.865 3.
8.42 Integral representations of the functions H(1)
ν(z)andH(2)
ν(z)
8.421
1. H(1)
ν(x)=e−νπi
2
πi/integraldisplay∞
−∞eixcosht−νtdt
=2e−νπi
2
πi/integraldisplay∞
0eixcoshtcoshνtdt
[−1<Reν<1,x > 0] WA 199(10)
2. H(2)
ν(x)=−eνπi
2
πi/integraldisplay∞
−∞e−ixcosht−νtdt
=−2eνπi
2
πi/integraldisplay∞
0e−ixcoshtcoshνtdt
[−1<Reν<1,x > 0] WA 199(11)
8.422 Integral representations of the functions H(1)
ν(z)andH(2)
ν(z) 915
3. H(1)
ν(z)=−2ν+1izν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplayπ/2
0cosν−1
2tei(z−νt+t
2)
sin2ν+1texp (−2zcott)dt
/bracketleftbig
Reν>−1
2,Rez>0/bracketrightbig
WA 186(5)
4. H(2)
ν(z)=2ν+1izν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplayπ/2
0cosν−1
2te−i(z−νt+t
2)
sin2ν+1texp (−2zcott)dt
/bracketleftbig
Reν>−1
2,Rez>0/bracketrightbig
WA 186(6)
5. H(1)
ν(x)=−2i/parenleftbigx
2/parenrightbig−ν
√πΓ/parenleftbig1
2−ν/parenrightbig/integraldisplay∞
1eixt
(t2−1)ν+1
2dt/bracketleftbig
−1
2<Reν<1
2,x > 0/bracketrightbig
WA 87(1)
6. H(2)
ν(x)=2i/parenleftbigx
2/parenrightbig−ν
√πΓ/parenleftbig1
2−ν/parenrightbig/integraldisplay∞
1e−ixt
(t2−1)ν+1
2dt/bracketleftbig
−1
2<Reν<1
2,x > 0/bracketrightbig
WA 187(2)
7. H(1)
ν(z)=−i
πe−1
2iνπ/integraldisplay∞
0exp/bracketleftbigg1
2iz/parenleftbigg
t+1
t/parenrightbigg/bracketrightbigg
t−ν−1dt
[0<argz<π;o ra r g z=0a n d −1<Reν<1]MO 38
8. H(1)
ν(xz)=−i
πe−1
2iνπzν/integraldisplay∞
0exp/bracketleftbigg1
2ix/parenleftbigg
t+z2
t/parenrightbigg/bracketrightbigg
t−ν−1dt
/bracketleftBig
0<argz<π
2,x > 0,Reν>−1; or arg z=π
2,x > 0a n d −1<Reν<1/bracketrightBig
MO 38
9. H(1)
ν(xz)=/radicalbigg
2
πzxνexp/bracketleftBig
i/parenleftBig
xz−π
2ν−π
4/parenrightBig/bracketrightBig
Γ/parenleftbig
ν+1
2/parenrightbig/integraldisplay∞
0/parenleftbigg
1+it
2z/parenrightbiggν−1
2
tν−1
2e−xtdt
/bracketleftbig
Reν>−1
2,−1
2π<argz<3
2π, x > 0/bracketrightbig
MO 39
10. H(1)
ν(z)=−2ie−iνπ/parenleftBigz
2/parenrightBigν
√πΓ/parenleftbig
ν+1
2/parenrightbig/integraldisplay∞
0eizcoshtsinh2νtd t
/bracketleftbig
0<argz<π , Reν>−1
2or arg z=0a n d −1
2<Reν<1
2/bracketrightbig
MO 38
11. H(1)
0(x)=−i
π/integraldisplay∞
−∞exp/parenleftbig
i√
x2+t2/parenrightbig
√
x2+t2dt [x>0] MO 38
8.422
1. H(1)
ν(z)=Γ/parenleftbig1
2−ν/parenrightbig/parenleftBigz
2/parenrightBigν
πiΓ/parenleftbig1
2/parenrightbig/integraldisplay(1+)
1+∞ieizt/parenleftbig
t2−1/parenrightbigν−1
2dt [−π<argz<2π] WA 183(4)
2. H(2)
ν(z)=Γ/parenleftbig1
2−ν/parenrightbig/parenleftBigz
2/parenrightBigν
πiΓ/parenleftbig1
2/parenrightbig/integraldisplay(−1−)
−1+∞ieizt/parenleftbig
t2−1/parenrightbigν−1
2dt
[−2π<argz<π]
The paths of integration are shown in the drawing.
916 Bessel Functions and Functions Associated with Them 8.423
8.423
1. H(1)
ν(z)=−1
π/integraldisplay−π+∞i
−∞ie−izsinθ+iνθdθ [Rez>0] WA 197(2)a
2. H(2)
ν(z)=−1
π/integraldisplay−∞i
π+∞ie−izsinθ+iνθdθ [Rez>0] WA 197(3)a
The path of integration for 8.423 1 is shown in the left-hand drawing and for 8.423 2 in the right-hand
drawing.
8.424
1. H(1)
ν(z)Jν(ζ)=1
πi/integraldisplayγ+i∞
0exp/bracketleftbigg1
2/parenleftbigg
t−z2+ζ2
t/parenrightbigg/bracketrightbigg
Iν/parenleftbiggzζ
t/parenrightbiggdt
t
[γ>0,Reν>−1,|ζ|<|z|]MO 45
2. H(2)
ν(z)Jν(ζ)=i
π/integraldisplayγ−i∞
0exp/bracketleftbigg1
2/parenleftbigg
t−z2+ζ2
t/parenrightbigg/bracketrightbigg
Iν/parenleftbiggzζ
t/parenrightbiggdt
t
[γ>0,Reν>−1,|ζ|<|z|]MO 45
8.43 Integral representations of the functions Iν(z)andKν(z)
The function Iν(z)
8.431
1. Iν(z)=/parenleftbigz
2/parenrightbigν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplay1
−1/parenleftbig
1−t2/parenrightbigν−1
2e±ztdt/bracketleftbig
Re/parenleftbig
ν+1
2/parenrightbig
>0/bracketrightbig
WA 94(9)
2. Iν(z)=/parenleftbigz
2/parenrightbigν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplay1
−1/parenleftbig
1−t2/parenrightbigν−1
2coshztdt/bracketleftbig
Re/parenleftbig
ν+1
2/parenrightbig
>0/bracketrightbig
WA 94(9)
3. Iν(z)=/parenleftbigz
2/parenrightbigν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplayπ
0e±zcosθsin2νθd θ/bracketleftbig
Re/parenleftbig
ν+1
2/parenrightbig
>0/bracketrightbig
WA 94(9)
4. Iν(z)=/parenleftbigz
2/parenrightbigν
Γ/parenleftbig
ν+1
2/parenrightbig
Γ/parenleftbig1
2/parenrightbig/integraldisplayπ
0cosh (zcosθ)s i n2νθd θ/bracketleftbig
Re/parenleftbig
ν+1
2/parenrightbig
>0/bracketrightbig
WA 94(9)
8.432 Integral representations of the functions Iν(z)andKν(z) 917
5. Iν(z)=1
π/integraldisplayπ
0ezcosθcosνθdθ−sinνπ
π/integraldisplay∞
0e−zcosht−νtdt
/bracketleftBig
|argz|≤π
2,Reν>0/bracketrightBig
WA 201(4)
See also 3.383 2,3.387 1,3.471 6,3.714 5.
For an integral representation of I0(z)a n dI1(z), see3.366 1,3.534 3.856 6.
The function Kν(z)
8.432
1. Kν(z)=/integraldisplay∞
0e−zcoshtcoshνtdt/bracketleftBig
|argz|<π
2or Re z=0a n d ν=0/bracketrightBig
MO 39
2. Kν(z)=/parenleftbigz
2/parenrightbigνΓ/parenleftbig1
2/parenrightbig
Γ/parenleftbig
ν+1
2/parenrightbig/integraldisplay∞
0e−zcoshtsinh2νtd t
/bracketleftbig
Reν>−1
2,Rez>0; or Re z=0a n d −1
2<Reν<1
2/bracketrightbig
WA 190(5), WH
3. Kν(z)=/parenleftbigz
2/parenrightbigνΓ/parenleftbig1
2/parenrightbig
Γ/parenleftbig
ν+1
2/parenrightbig/integraldisplay∞
1e−zt/parenleftbig
t2−1/parenrightbigν−1
2dt
/bracketleftBig
Re/parenleftbig
ν+1
2/parenrightbig
>0,|argz|<π
2;o rR e z=0a n d ν=0/bracketrightBig
WA 190(4)
4. Kν(x)=1
cosνπ
2/integraldisplay∞
0cos(xsinht)cosh νtdt [x>0,−1<Reν<1] WA 202(13)
5. Kν(xz)=Γ/parenleftbig
ν+1
2/parenrightbig
(2z)ν
xνΓ/parenleftbig1
2/parenrightbig/integraldisplay∞
0cosxt dt
(t2+z2)ν+1
2/bracketleftBig
Re/parenleftbig
ν+1
2/parenrightbig
≥0,x > 0,|argz|<π
2/bracketrightBig
WA 191(1)
6.11Kν(z)=1
2/parenleftBigz
2/parenrightBigν/integraldisplay∞
0e−t−z2/4tdt
tν+1/bracketleftBig
|argz|<π
2,Rez2>0/bracketrightBig
WA 203(15)
7.7Kν(xz)=zν
2/integraldisplay∞
0exp/bracketleftbigg
−x
2/parenleftbigg
t+z2
t/parenrightbigg/bracketrightbigg
t−ν−1dt
/bracketleftBig
|argz|<π
4or|argz|=π
4and Re ν<1/bracketrightBig
MO 39
8. Kν(xz)=/radicalbiggπ
2zxνe−xz
Γ/parenleftbig
ν+1
2/parenrightbig/integraldisplay∞
0e−xttν−1
2/parenleftbigg
1+t
2z/parenrightbiggν−1
2
dt
/bracketleftbig
|argz|<π , Reν>−1
2,x>0/bracketrightbig
MO 39
9. Kν(xz)=√π
Γ/parenleftbig
ν+1
2/parenrightbig/parenleftBigx
2z/parenrightBigν/integraldisplay∞
0exp/parenleftbig
−x√
t2+z2/parenrightbig
√
t2+z2t2νdt
/bracketleftBig
Reν>−1
2,Rez>0,Re/radicalbig
t2+z2>0,x > 0/bracketrightBig
MO 39
918 Bessel Functions and Functions Associated with Them 8.433
See also 3.383 3,3.387 3, 6,3.388 2,3.389 4,3.391 ,3.395 1,3.471 9,3.483 ,3.547 2,3.856 ,3.871 3,
4,7.141 5.
8.433 K1
3/parenleftbigg2x√x
3√
3/parenrightbigg
=3√x/integraldisplay∞
0cos/parenleftbig
t3+xt/parenrightbig
dt KU 98(31), WA 211(2)
For an integral representation of K0(z), see3.754 2,3.864 ,4.343 ,4.356 ,4.367 .
8.44 Series representation
The function Jν(z)
8.440 Jν(z)=/parenleftBigz
2/parenrightBigν∞/summationdisplay
k=0(−1)k
k!Γ (ν+k+1 )/parenleftBigz
2/parenrightBig2k
[|argz|<π] WH 358 a
8.441 Special cases:
1. J0(z)=∞/summationdisplay
k=0(−1)kz2k
22k(k!)2
2. J1(z)=−J/prime
0(z)=z
2∞/summationdisplay
k=0(−1)kz2k
22kk!(k+1 ) !
3. J1
3(z)=1
Γ/parenleftbig4
3/parenrightbig3/radicalbiggz
2∞/summationdisplay
k=0(−1)k/parenleftbig
z√
3/parenrightbig2k
22kk!·1·4·7·····(3k+1 )
4. J−1
3(z)=1
Γ/parenleftbig2
3/parenrightbig3/radicalbigg
2
z/braceleftBigg
1+∞/summationdisplay
k=1(−1)k/parenleftbig
z√
3/parenrightbig2k
22kk!·2·5·8·····(3k−1)/bracerightBigg
For the expansion of Jν(z) in Laguerre polynomials, see 8.975 3.
8.442
1.7Jν(z)Jμ(z)=∞/summationdisplay
m=0(−1)m/parenleftbig1
2z/parenrightbigμ+ν+2m(μ+ν+m+1 ) m
m!Γ (μ+m+1 )Γ ( ν+m+1 )
2.8Jν(az)Jμ(bz)=/parenleftbigaz
2/parenrightbigν/parenleftbiggbz
2/parenrightbiggμ
Γ(μ+1 )∞/summationdisplay
k=0(−1)k/parenleftBigaz
2/parenrightBig2k
F/parenleftbigg
−k,−ν−k;μ−1;b2
a2/parenrightbigg
k!Γ (ν+k+1 )MO 28
The function Yν(z)
8.44311Yν(z)=1
sinνπ/braceleftBigg
cosνπ/parenleftBigz
2/parenrightBigν∞/summationdisplay
k=0(−1)k z2k
22kk!Γ(ν+k+1 )
−/parenleftBigz
2/parenrightBig−ν∞/summationdisplay
k=0(−1)k z2k
22kk!Γ (k−ν+1 )/bracerightBigg
[ν/negationslash= an integer] (cf. 8.403 1)
Forν+ 1 a natural number, see 8.403 2.; for νa negative integer, see 8.404 1
8.447 Series representation 919
8.444 Special cases,
1. πY0(z)=2J0(z)/parenleftBig
lnz
2+C/parenrightBig
−2∞/summationdisplay
k=1(−1)k
(k!)2/parenleftBigz
2/parenrightBig2kk/summationdisplay
m=11
mKU 44
2.11πY1(z)=2J1(z)/parenleftBig
lnz
2+C/parenrightBig
−2
z−z
2−∞/summationdisplay
k=2(−1)k+1/parenleftBigz
2/parenrightBig2k−1
k!(k−1)!/parenleftBigg
1
k+2k−1/summationdisplay
m=11
m/parenrightBigg
The functions Iν(z)andKn(z)
8.445 Iν(z)=∞/summationdisplay
k=01
k!Γ(ν+k+1 )/parenleftBigz
2/parenrightBigν+2k
WH 372a
8.4468Kn(z)=1
2n−1/summationdisplay
k=0(−1)k(n−k−1)!
k!/parenleftbigz
2/parenrightbign−2k
+(−1)n+1∞/summationdisplay
k=0/parenleftbigz
2/parenrightbign+2k
k!(n+k)!/bracketleftbigg
lnz
2−1
2ψ(k+1 )−1
2ψ(n+k+1 )/bracketrightbigg
WA 95(15)
=(−1)n+1In(z)/parenleftbigg
ln1
2z+C/parenrightbigg
+1
2(−1)n∞/summationdisplay
l=0/parenleftbigz
2/parenrightbign+2l
l!(n+l)!/parenleftBiggl/summationdisplay
k=11
k+n+l/summationdisplay
k=11
k/parenrightBigg
+1
2n−1/summationdisplay
l=0(−1)l(n−l−1)!
l!/parenleftBigz
2/parenrightBig2l−n
[n+ 1 is a natural number]
MO 29
8.447 Special cases:
1. I0(z)=∞/summationdisplay
k=0/parenleftbigz
2/parenrightbig2k
(k!)2
2. I1(z)=I/prime
0(z)=∞/summationdisplay
k=0/parenleftbigz
2/parenrightbig2k+1
k!(k+1 ) !
3. K0(z)=−lnz
2I0(z)+∞/summationdisplay
k=0z2k
22k(k!)2ψ(k+1 ) WA 95(14)
920 Bessel Functions and Functions Associated with Them 8.451
8.45 Asymptotic expansions of Bessel functions
8.451 For large values of |z|∗
1. J±ν(z)=/radicalbigg
2
πz/braceleftBigg
cos/parenleftBig
z∓π
2ν−π
4/parenrightBig/bracketleftBiggn−1/summationdisplay
k=0(−1)k
(2z)2kΓ/parenleftbig
ν+2k+1
2/parenrightbig
(2k)! Γ/parenleftbig
ν−2k+1
2/parenrightbig+R1/bracketrightBigg
−sin/parenleftBig
z∓π
2ν−π
4/parenrightBig/bracketleftBiggn−1/summationdisplay
k=0(−1)k
(2z)2k+1Γ/parenleftbig
ν+2k+3
2/parenrightbig
(2k+1 ) !Γ/parenleftbig
ν−2k−1
2/parenrightbig+R2/bracketrightBigg/bracerightBigg
[|argz|<π]( s e e 8.339 4)WA 222(1, 3)
2.11Y±ν(z)=/radicalbigg
2
πz/braceleftBigg
sin/parenleftBig
z∓π
2ν−π
4/parenrightBig/bracketleftBiggn−1/summationdisplay
k=0(−1)k
(2z)2kΓ/parenleftbig
ν+2k+1
2/parenrightbig
(2k)! Γ/parenleftbig
ν−2k+1
2/parenrightbig+R1/bracketrightBigg
+c o s/parenleftBig
z∓π
2ν−π
4/parenrightBig/bracketleftBiggn−1/summationdisplay
k=0(−1)k
(2z)2k+1Γ/parenleftbig
ν+2k+3
2/parenrightbig
(2k+1 ) !Γ/parenleftbig
ν−2k−1
2/parenrightbig+R2/bracketrightBigg/bracerightBigg
[|argz|<π]( s e e 8.339 4)WA 222(2, 4, 5)
3.11H(1)
ν(z)=/radicalbigg
2
πzei(z−π
2ν−π
4)/bracketleftBiggn−1/summationdisplay
k=0(−1)k
(2iz)kΓ/parenleftbig
ν+k+1
2/parenrightbig
k!Γ/parenleftbig
ν−k+1
2/parenrightbig+θ1(−1)n
(2iz)nΓ/parenleftbig
ν+n+1
2/parenrightbig
k!Γ/parenleftbig
ν−n+1
2/parenrightbig/bracketrightBigg
/bracketleftbig
Reν>−1
2,|argz|<π/bracketrightbig
(see8.339 4)WA 221(5)
4.11H(2)
ν(z)=/radicalbigg
2
πze−i(z−π
2ν−π
4)/bracketleftBiggn−1/summationdisplay
k=01
(2iz)kΓ/parenleftbig
ν+k+1
2/parenrightbig
k!Γ/parenleftbig
ν−k+1
2/parenrightbig+θ21
(2iz)nΓ/parenleftbig
ν+n+1
2/parenrightbig
k!Γ/parenleftbig
ν−n+1
2/parenrightbig/bracketrightBigg
/bracketleftbig
Reν>−1
2,|argz|<π/bracketrightbig
(see8.339 4)WA 221(6)
For indices of the form ν=2n−1
2(where nis a natural number), the series 8.451 terminate. In
this case, the closed formulas 8.46are valid for all values.
5. Iν(z)∼ez
√
2πz∞/summationdisplay
k=0(−1)k
(2z)kΓ/parenleftbig
ν+k+1
2/parenrightbig
k!Γ/parenleftbig
ν−k+1
2/parenrightbig
+exp/bracketleftbig
−z±/parenleftbig
ν+1
2/parenrightbig
πi/bracketrightbig
√
2πz∞/summationdisplay
k=01
(2z)kΓ/parenleftbig
ν+k+1
2/parenrightbig
k!Γ/parenleftbig
ν−k+1
2/parenrightbig
[The + sign is taken for −1
2π<argz<3
2π,t h e−sign for −3
2π<argz<1
2π]∗(see8.339 4)]
WA 226(2,3)
6.11Kν(z)=/radicalbiggπ
2ze−z/bracketleftBiggn−1/summationdisplay
k=01
(2z)kΓ/parenleftbig
ν+k+1
2/parenrightbig
k!Γ/parenleftbig
ν−k+1
2/parenrightbig+θ3Γ/parenleftbig
ν+n+1
2/parenrightbig
(2z)nn!Γ/parenleftbig
ν−n+1
2/parenrightbig/bracketrightBigg
(see8.339 4) WA 231, 245(9)
An estimate of the remainders of the asymptotic series in formulas 8.451 :
∗An estimate of the remainders in formulas 8.451 is given in 8.451 7a n d8.451 8.
∗The contradiction that this condition contains at first glance is explained by the so-called Stokes phenomenon (see
Watson, G.N., A Treatise on the Theory of Bessel Functions , 2nd Edition, Cambridge Univ. Press, 1944, page 201).
8.452 Asymptotic expansions of Bessel functions 921
7. |R1|</vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓ/parenleftbig
ν+2n+
1
2/parenrightbig
(2z)2n(2n)! Γ/parenleftbig
ν−2n+1
2/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg
n>ν
2−1
4/bracketrightbigg
WA 231
8. |R2|</vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓ/parenleftbig
ν+2n+
3
2/parenrightbig
(2z)2n+1(2n+1 ) !Γ/parenleftbig
ν−2n−1
2/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg
n≥ν
2−3
4/bracketrightbigg
WA 231
For−π
2<argz<3
2π,νreal, and n+1
2>|ν| WA 245
|θ1|</braceleftBigg
1, if Imz≥0
|sec(arg z)|,if Imz≤0
For−3
2π<argz<π
2,νreal, and n+1
2>|ν| WA 246
|θ2|</braceleftBigg
1, if Imz≤0
|sec(arg z)|,if Imz≥0
Forνreal, WA 245
|θ3|</braceleftBigg
1i f R e z≥0
|cosec (arg z)|,if Rez<0
Reθ3≥0,if Rez≥0
Forνandzreal and n≥ν−1
2, WA 231
0≤|θ3|≤1
In particular, it follows from 8.451 7a n d8.451 8 that for real positive values of zandν, the errors
|R1|and|R2|are less than the absolute value of the first discarded term. For values of |argz|close to π,
the series 8.451 1a n d8.451 2 may not be suitable for calculations. In particular, the error for |argz|>π
can be greater in absolute value than the first discarded term.
“Approximation by tangents”
8.45211For large values of the index (where the argument is less than the index).
Suppose that x>0a n d ν>0. Let us set ν/x=c o s h α. Then, for large values of ν, the following
expansions are valid:
1. Jν/parenleftBigν
coshα/parenrightBig
∼exp(νtanhα−να)√
2νπtanhα/braceleftbigg
1+1
ν/parenleftbigg1
8cothα−5
24coth3α/parenrightbigg
+1
ν2/parenleftbigg9
128coth2α−231
576coth4α+1155
3456coth6α/parenrightbigg
+.../bracerightbigg
WA 269(3)
922 Bessel Functions and Functions Associated with Them 8.453
2. Yν/parenleftBigν
coshα/parenrightBig
∼−exp (να−νtanhα)/radicalbigπ
2νtanhα/braceleftbigg
1−1
ν/parenleftbigg1
8cothα−5
24coth3α/parenrightbigg
+1
ν2/parenleftbigg9
128coth2α−231
576coth4α+1155
3456coth6α/parenrightbigg
+.../bracerightbigg
WA 270(5)
8.453 For large values of the index (where the argument is greater than the index).
Suppose that x>0a n d ν>0. Let us set ν/x=c o s β. Then, for large values of ν, the following
expansions are valid:
1. Jν(νsecβ)∼/radicalbigg2
νπtanβ⎧
⎨
⎩⎡
⎣1−1
ν2⎛
⎝9
128cot2β+231
576cot4β
+1155
3456cot6β⎞
⎠+...⎤
⎦cos/parenleftBig
νtanβ−νβ−π
4/parenrightBig
+/bracketleftbigg1
ν/parenleftbigg1
8cotβ+5
24cot3β/parenrightbigg
−.../bracketrightbigg
sin/parenleftBig
νtanβ−νβ−π
4/parenrightBig⎤
⎦
WA 271(4)
2. Yν(νsecβ)∼/radicalbigg2
νπtanβ⎧
⎨
⎩⎡
⎣1−1
ν2⎛
⎝9
128cot2β+231
576cot4β
+1155
3456cot6β⎞
⎠+...⎤
⎦sin/parenleftBig
νtanβ−νβ−π
4/parenrightBig
−1
ν/parenleftbigg1
8cotβ+5
24cot3β/parenrightbigg
−...⎤
⎦cos/parenleftBig
νtanβ−νβ−π
4/parenrightBig⎤
⎦
WA 271(5)
3. H(1)
ν(νsecβ)∼exp/bracketleftbig
νi(tanβ−β)−π
4i/bracketrightbig
/radicalbigπ
2νtanβ/braceleftbigg
1−i
ν/parenleftbigg1
8cotβ+5
24cot3β/parenrightbigg
−1
ν2/parenleftbigg9
128cot2β+231
576cot4β+1155
3456cot6β/parenrightbigg
+.../bracerightbigg
WA 271(1)
4. H(2)
ν(νsecβ)∼exp/bracketleftbig
−νi(tanβ−β)+π
4i/bracketrightbig
/radicalbigπ
2νtanβ/braceleftbigg
1+i
ν/parenleftbigg1
8cotβ+5
24cot3β/parenrightbigg
−1
ν2/parenleftbigg9
128cot2β+231
576cot4β+1155
3456cot6β/parenrightbigg
+.../bracerightbigg
WA 271(2)
Formulas 8.453 are not valid when |x−ν|is of a size comparable to x1
3. For arbitrary small (and
also large) values of |x−ν|, we may use the following formulas:
8.457 Asymptotic expansions of Bessel functions 923
8.454 Suppose that x>0a n d ν>0, we set
w=/radicalbigg
x2
ν2−1;
Then,
1. H(1)
ν(x)=w√
3exp/braceleftbigg/bracketleftbiggπ
6+ν/parenleftbigg
w−w3
3−arctan w/parenrightbigg/bracketrightbigg
i/bracerightbigg
H(1)
1
3/parenleftBigν
3w3/parenrightBig
+O/parenleftbigg1
|ν|/parenrightbigg
2. H(2)
ν(x)=w√
3exp/braceleftbigg/bracketleftbigg
−π
6−ν/parenleftbigg
w−w3
3−arctan w/parenrightbigg/bracketrightbigg
i/bracerightbigg
H(2)
1
3/parenleftBigν
3w3/parenrightBig
+O/parenleftbigg1
|ν|/parenrightbigg
MO 34
The absolute value of the error O/parenleftbigg1
|ν|/parenrightbigg
is then less than 24√
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
8.455 Forxreal and νa natural number ( ν=n), ifn/greatermuch1, the following approximations are valid:
1.7Jn(x)≈1
π/radicalbigg
2(n−x)
3xK1
3/braceleftBigg
[2(n−x)]3
2
3√x/bracerightBigg
[n>x ] (see also 8.433 )
WA 276(1)
≈1
2e2
3πi/radicalbigg
2(n−x)
3xH(1)
1
3/braceleftBigg
i
3[2(n−x)]3
2√x/bracerightBigg
[n>x ]
MO 34
≈1√
3/radicalbigg
2(x−n)
3x/braceleftBigg
J1
3/bracketleftBigg
{2(x−n)}3
2
3√x/bracketrightBigg
+J−1
3/bracketleftBigg
{2(x−n)}3
2
3√x/bracketrightBigg/bracerightBigg
(see also 8.441 3,8.441 4)
WA 276(2)
2. Yn(x)≈/radicalbigg
2(x−n)
3x/braceleftBigg
J−1
3/bracketleftBigg
{2(x−n)}3
2
3√x/bracketrightBigg
−J1
3/bracketleftBigg
{2(x−n)}3
2
3√x/bracketrightBigg/bracerightBigg
[x>n ] WA 276(3)
An estimate of the error in formulas 8.455 has not yet been achieved.
8.45611J2
ν(z)+Y2
ν(z)≈2
πz∞/summationdisplay
k=0(2k−1)!!
2kz2kΓ/parenleftbig
ν+k+1
2/parenrightbig
k!Γ/parenleftbig
ν−k+1
2/parenrightbig[|argz|<π] (see also 8.479 1)
WA 250(5)
8.457 J2
ν(x)+J2
ν+1(x)≈2
πx[x/greatermuch|ν|] WA 223
924 Bessel Functions and Functions Associated with Them 8.461
8.46 Bessel functions of order equal to an integer plus one-half
The function Jν(z)
8.461
1.11Jn+1
2(z)=/radicalbigg
2
πz⎧
⎪⎨
⎪⎩sin/parenleftBig
z−π
2n/parenrightBig⌊n
2⌋/summationdisplay
k=0(−1)k(n+2k)!
(2k)!(n−2k)!(2z)−2k
+c o s/parenleftBig
z−π
2n/parenrightBig⌊n−1
2⌋/summationdisplay
k=0(−1)k(n+2k+1 ) !
(2k+1 ) ! ( n−2k−1)!(2z)−(2k+1)⎫
⎪⎬
⎪⎭
[n+ 1 is a natural number] (cf. 8.451 1)KU 59(6), WA 66(2)
2. J−n−1
2(z)=/radicalbigg
2
πz⎧
⎪⎨
⎪⎩cos/parenleftBig
z+π
2n/parenrightBig⌊n
2⌋/summationdisplay
k=0(−1)k(n+2k)!
(2k)!(n−2k)!(2z)2k
−sin/parenleftBig
z+π
2n/parenrightBig⌊n−1
2⌋/summationdisplay
k=0(−1)k(n+2k+1 ) !
(2k+1 ) ! ( n−2k−1)!(2z)2k+1⎫
⎪⎬
⎪⎭
[n+ 1 is a natural number] (cf. 8.451 1)KU 58(7), WA 67(5)
8.462
1. Jn+1
2(z)=1√
2πz/braceleftBigg
eizn/summationdisplay
k=0i−n+k−1(n+k)!
k!(n−k)!(2z)k+e−izn/summationdisplay
k=0(−i)−n+k−1(n+k)!
k!(n−k)!(2z)k/bracerightBigg
[n+ 1 is a natural number]
KU 59(6), WA 66(1)
2. J−n−1
2(z)=1√
2πz/braceleftBigg
eizn/summationdisplay
k=0in+k(n+k)!
k!(n−k)!(2z)k+e−izn/summationdisplay
k=0(−i)n+k(n+k)!
k!(n−k)!(2z)k/bracerightBigg
[n+ 1 is a natural number]
KU 59(7), WA 67(4)
8.463
1. Jn+1
2(z)=(−1)nzn+1
2/radicalbigg
2
πdn
(zd z)n/parenleftbiggsinz
z/parenrightbigg
KU 58(4)
2. J−n−1
2(z)=zn+1
2/radicalbigg
2
πdn
(zd z)n/parenleftBigcosz
z/parenrightBig
KU 58(5)
8.464 Special cases:
1. J1
2(z)=/radicalbigg
2
πzsinz DW
2. J−1
2(z)=/radicalbigg
2
πzcosz DW
8.469 Bessel functions of order equal to an integer plus one-half 925
3. J3
2(z)=/radicalbigg
2
πz/parenleftbiggsinz
z−cosz/parenrightbigg
DW
4. J−3
2(z)=/radicalbigg
2
πz/parenleftBig
−sinz−cosz
z/parenrightBig
DW
5.8J5
2(z)=/radicalbigg
2
πz/braceleftbigg/parenleftbigg3
z2−1/parenrightbigg
sinz−3
zcosz/bracerightbigg
DW
6. J−5
2(z)=/radicalbigg
2
πz/braceleftbigg3
zsinz+/parenleftbigg3
z2−1/parenrightbigg
cosz/bracerightbigg
DW
The function Yn+1
2(z)
8.465
1. Yn+1
2(z)=(−1)n−1J−n−1
2(z) JA
2. Y−n−1
2(z)=(−1)nJn+1
2(z) JA
The functions H(1,2)
n+1
2(z),In+1
2(z),Kn+1
2(z)
8.466
1. H(1)
n−1
2(z)=/radicalbigg
2
πzi−neizn−1/summationdisplay
k=0(−1)k(n+k−1)!
k!(n−k−1)!1
(2iz)k
(cf.8.451 3)
2. H(2)
n−1
2(z)=/radicalbigg
2
πzine−izn−1/summationdisplay
k=0(n+k−1)!
k!(n−k−1)!1
(2iz)k(cf.8.451 4)
8.467 I±(n+1
2)(z)=1√
2πz/bracketleftBigg
ezn/summationdisplay
k=0(−1)k(n+k)!
k!(n−k)!(2z)k±(−1)n+1e−zn/summationdisplay
k=0(n+k)!
k!(n−k)!(2z)k/bracketrightBigg
(cf.8.451 5) KU 60a
8.468 Kn+1
2(z)=/radicalbiggπ
2ze−zn/summationdisplay
k=0(n+k)!
k!(n−k)!(2z)k(cf.8.451 6) KU 60
8.469 Special cases:
1. Y1
2(z)=−/radicalbigg
2
πzcosz
2. Y−1
2(z)=/radicalbigg
2
πzsinz
3. K±1
2(z)=/radicalbiggπ
2ze−zWA 95(13)
4. H(1)
1
2(z)=/radicalbigg
2
πzeiz
iMO 27
926 Bessel Functions and Functions Associated with Them 8.471
5. H(2)
1
2(z)=/radicalbigg
2
πze−iz
−iMO 27
6. H(1)
−1
2(z)=/radicalbigg
2
πzeizMO 27
7. H(2)
−1
2(z)=/radicalbigg
2
πze−izMO 27
8.47–8.48 Functional relations
8.4718Recursion formulas:
1. zZν−1(z)+zZν+1(z)=2νZν(z) KU 56(13), WA 56(1), WA 79(1), WA 88(3)
2. Zν−1(z)−Zν+1(z)=2d
dzZν(z) KU 56(12), WA 56(2), WA 79(2), We 88(4)
Sonin and Nielsen, in their construction of the theory of Bessel functions, defined Bessel functions as
analytic functions of zthat satisfy the recursion relations 8.471 .Zdenotes J,N,H(1),H(2)or any
linear combination of these functions, the coefficients of which are independent of zandν.
8.472 Consequences of the recursion formulas for Zdefined as above:
1. zd
dzZν(z)+νZν(z)=zZν−1(z) KU 56(11), WA 56(3), WA 79(3), WA 88(5)
2. zd
dzZν(z)−νZν(z)=−zZν+1(z) KU 56(10), WA 56(4), WA 79(4), WA 88(6)
3./parenleftbiggd
zd z/parenrightbiggm
(zνZν(z)) =zν−mZν−m(z) KU 56(8), WA 57(5), WA 89(9)
4./parenleftbiggd
zd z/parenrightbiggm/parenleftbig
z−νZν(z)/parenrightbig
=(−1)mz−ν−mZν+m(z) WA 89(10), Ku 55(5), WA 57(6)
5. Z−n(z)=(−1)nZn(z)[ nis a natural number] (cf. 8.404 )
8.473 Special cases:
1. J2(z)=2
zJ1(z)−J0(z)
2. Y2(z)=2
zY1(z)−Y0(z)
3. H(1,2)
2(z)=2
zH(1,2)
1(z)−H(1,2)
0(z)
4.d
dzJ0(z)=−J1(z)
5.d
dzY0(z)=−Y1(z)
6.d
dzH(1,2)
0(z)=−H(1,2)
1(z)
8.4748Each of the pairs of functions Jν(z)a n dJ−ν(z)( f o r ν/negationslash=0 ,±1,±2,...),Jν(z)a n dYν(z), and
H(1)
ν(z)a n dH(2)
ν(z), which are solutions of equation 8.401 , and also the pair Iν(z)a n dKν(z) is a pair
of linearly independent functions. The Wronskians of these pairs are, respectively,
8.478 Functional relations 927
−2
πzsinνπ,2
πz,−4i
πz,−1
zKU 52(10, 11, 12), WA 90(1, 4)
8.4756The functions Jν(z), and Yν(z),H(1,2)
ν(z),Iν(z),Kν(z), with the exception of Jn(z)a n dIn(z),
fornan integer are non-single-valued :z= 0 is a branch point for these functions. The branches of these
functions that lie on opposite sides of the cut ( −∞, 0) are connected by the relations
8.476
1. Jν/parenleftbig
emπiz/parenrightbig
=emνπiJν(z) WA 90(1)
2. Yν/parenleftbig
emπiz/parenrightbig
=e−mνπiYν(z)+2isinmνπcotνπJν(z) WA 90(3)
3. Y−ν/parenleftbig
emπiz/parenrightbig
=e−mνπiY−ν(z)+2isinmνπcosecνπJν(z) WA 90(4)
4. Iν/parenleftbig
emπiz/parenrightbig
=emνπiIν(z) WA 95(17)
5. Kν/parenleftbig
emπiz/parenrightbig
=e−mνπiKν(z)−iπsinmνπ
sinνπIν(z)[ νnot an integer] WA 95(18)
6. H(1)
ν/parenleftbig
emπiz/parenrightbig
=e−mνπiH(1)
ν(z)−2e−νπisinmνπ
sinνπJν(z)
=sin(1−m)νπ
sinνπH(1)
ν(z)−e−νπisinmνπ
sinνπH(2)
ν(z)
WA 95(5)
7. H(2)
ν/parenleftbig
emπiz/parenrightbig
=e−mνπiH(2)
ν(z)+2eνπisinmνπ
sinνπJν(z)
=sin(1 + m)νπ
sinνπH(2)
ν(z)+eνπisinmνπ
sinνπH(1)
ν(z)
[man integer] WA 90(6)
8. H(1)
ν/parenleftbig
eiπz/parenrightbig
=−H(2)
−ν(z)=−e−iπνH(2)
ν(z) MO 26
9. H(2)
ν/parenleftbig
e−iπz/parenrightbig
=−H(1)
−ν(z)=−eiπνH(1)
ν(z) MO 26
10.8H(2)
ν(z)=H(1)
ν(z) MO 26
8.477
1. Jν(z)Yν+1(z)−Jν+1(z)Yν(z)=−2
πzWA 91(12)
2. Iν(z)Kν+1(z)+Iν+1(z)Kν(z)=1
zWA 95(20)
See also 3.863 .
•For a connection with Legendre functions, see 8.722 .
•For a connection with the polynomials Cλ
n(t), see8.936 4.
•For a connection with a confluent hypergeometric function, see 9.235 .
8.478 Forν>0a n d x>0, the product
x/bracketleftbig
J2
ν(x)+Y2
ν(x)/bracketrightbig
,
considered as a function of x, decreases monotonically, if ν>1
2and increases monotonically if 0 <ν<1
2.
MO 35
928 Bessel Functions and Functions Associated with Them 8.479
8.479
1.111√
x2−ν2>π
2/bracketleftbig
J2
ν(x)+Y2
ν(x)/bracketrightbig
≥1
x/bracketleftbig
x≥ν≥1
2/bracketrightbig
MO 35
2. |Jn(nz)|≤1/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglezexp√
1−z2
1+√
1−z2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<1,na natural number/bracketrightBigg
MO 35
Relations between Bessel functions of the first, second, and third kinds
8.481 Jν(z)=Y−ν(z)−Yν(z)cosνπ
sinνπ=H(1)
ν(z)−iYν(z)
=H(2)
ν(z)+iYν(z)=1
2/parenleftBig
H(1)
ν(z)+H(2)
ν(z)/parenrightBig
(cf.8.403 1,8.405 ) WA 89(1), JA
8.482 Yν(z)=Jν(z)cosνπ−J−ν(z)
sinνπ=iJν(z)−iH(1)
ν(z)
=iH(2)
ν(z)−iJν(z)=i
2/parenleftBig
H(2)
ν(z)−H(1)
ν(z)/parenrightBig
(cf.8.403 1,8.405 ) WA 89(3), JA
8.483
1. H(1)
ν(z)=J−ν(z)−e−νπiJν(z)
isinνπ=Y−ν(z)−e−νπiYν(z)
sinνπ=Jν(z)+iYν(z) WA 89(5)
2. H(2)
ν(z)=eνπiJν(z)−J−ν(z)
isinνπ=Y−ν(z)−eνπiYν(z)
sinνπ=Jν(z)−iYν(z)
(cf.8.405 ) WA 89(6)
8.484
1. H(1)
−ν(z)=eνπiH(1)
ν(z) WA 89(7)
2. H(2)
−ν(z)=e−νπiH(2)
ν(z) WA 89(7)
8.4857Kν(z)=π
2I−ν(z)−Iν(z)
sinνπ[νnot an integer] (see also 8.407 )
WA 92(6)
8.486 Recursion formulas for the functions Iν(z)a n dKν(z) and their consequences:
1. zIν−1(z)−zIν+1(z)=2νIν(z) WA 93(1)
2. Iν−1(z)+Iν+1(z)=2d
dzIν(z) WA 93(2)
3. zd
dzIν(z)+νIν(z)=zIν−1(z) WA 93(3)
4. zd
dzIν(z)−νIν(z)=zIν+1(z) WA 93(4)
5./parenleftbiggd
zd z/parenrightbiggm
{zνIν(z)}=zν−mIν−m(z) WA 93(5)
8.486(1) Functional relations 929
6./parenleftbiggd
zd z/parenrightbiggm/braceleftbig
z−νIν(z)/bracerightbig
=z−ν−mIν+m(z) WA 93(6)
7. I−n(z)=ln(z)[ na natural number] WA 93(8)
8. I2(z)=−2
zl1(z)+I0(z)
9.d
dzI0(z)=I1(z) WA 93(7)
10. zKν−1(z)−zKν+1(z)=−2νKν(z) WA 93(1)
11. Kν−1(z)+Kν+1(z)=−2d
dzKν(z) WA 93(2)
12. zd
dzKν(z)+νKν(z)=−zKν−1(z) WA 93(3)
13. zd
dzKν(z)−νKν(z)=−zKν+1(z) WA 93(4)
14./parenleftbiggd
zd z/parenrightbiggm
{zνKν(z)}=(−1)mzν−mKν−m(z) WA 93(5)
15./parenleftbiggd
zd z/parenrightbiggm/braceleftbig
z−νKν(z)/bracerightbig
=(−1)mz−ν−mKν+m(z) WA 93(6)
16. K−ν(z)=Kν(z) WA 93(8)
17. K2(z)=2
zK1(z)+K0(z)
18.d
dzK0(z)=−K1(z) WA 93(7)
19.∂Jν(z)
∂ν=/bracketleftBig
lnz
2−ψ(ν+1 )/bracketrightBig
Jν(z)+(z/2)ν+1
Γ(ν+1 )∞/summationdisplay
n=0(z/2)nJn+1(z)
n!(ν+n+1 )2LUKE 360
8.486(1)7Differentiation with respect to order
1.∂Jν(z)
∂ν=Jν(z)ln/parenleftbigg1
2z/parenrightbigg
−∞/summationdisplay
k=0(−1)k/parenleftbigg1
2z/parenrightbiggν+2kψ(ν+k+1 )
k!Γ(ν+k+1 )
/bracketleftbig
ν/negationslash=norn+1
2,ninteger/bracketrightbig
MS 3.1.3
2.∂J−ν(z)
∂ν=−J−ν(z)ln/parenleftbigg1
2z/parenrightbigg
+∞/summationdisplay
k=0(−1)k/parenleftbigg1
2z/parenrightbigg−ν+2kψ(−ν+k+1 )
k!Γ (−ν+k+1 )
/bracketleftbig
ν/negationslash=norn+1
2,ninteger/bracketrightbig
MS 3.1.3
3.∂Yν(z)
∂ν=c o t πν∂Jν(z)
∂ν−cosecπν∂J−ν(z)
∂ν−πcosecπνYν(z)
/bracketleftbig
ν/negationslash=norn+1
2,ninteger/bracketrightbig
MS 3.1.3
4.∂Iν(z)
∂ν=Iν(z)ln/parenleftbigg1
2z/parenrightbigg
−∞/summationdisplay
k=0/parenleftbigg1
2z/parenrightbiggν+2kψ(ν+k+1 )
k!Γ (ν+k+1 )/bracketleftbig
ν/negationslash=norn+1
2,ninteger/bracketrightbig
MS 3.1.3
930 Bessel Functions and Functions Associated with Them 8.486(1)
5.∂Kν(z)
∂ν=−πcotπνKν(z)+1
2πcosecπν/bracketleftbigg∂I−ν(z)
∂ν−∂Iν(z)
∂ν/bracketrightbigg
/bracketleftbig
ν/negationslash=norn+1
2,ninteger/bracketrightbig
MS 3.1.3
6./bracketleftbigg∂Jν(z)
∂ν/bracketrightbigg
ν=±n=1
2π(±1)nYn(z)±(±1)n1
2n!n−1/summationdisplay
k=0/parenleftbig1
2z/parenrightbigk−nJk(z)
k!(n−k)[n=0,1,...]MS 3.2.3
7./bracketleftbigg∂Yν(z)
∂ν/bracketrightbigg
ν=±n=−1
2π(±1)nJn(z)±(±1)n1
2n!n−1/summationdisplay
k=0/parenleftbig1
2z/parenrightbigk−nYk(z)
k!(n−k)[n=0,1,...]
MS 3.2.3
8./bracketleftbigg∂Iν(z)
∂ν/bracketrightbigg
ν=±n=(−1)n+1Kn(z)±(−1)n1
2n!n−1/summationdisplay
k=0(−1)k/parenleftbig1
2z/parenrightbigk−nIk(z)
k!(n−k)[n=0,1,...]
MS 3.2.3
9./bracketleftbigg∂Kν(z)
∂ν/bracketrightbigg
ν=±n=±1
2n!n−1/summationdisplay
k=0/parenleftbig1
2z/parenrightbigk−nKk(z)
k!(n−k)[n=0,1,...] MS 3.2.3
10. ( −1)n/bracketleftbigg∂
∂νIν(z)/bracketrightbigg
ν=n=−Kn(z)+1
2n!n−1/summationdisplay
k=0(−1)k/parenleftbigg1
2z/parenrightbiggk−n
Ik(z)
k!(n−k)
[n=0,1,...] AS 9.6.44
11.11/bracketleftbigg∂Kν(z)
∂ν/bracketrightbigg
ν=n=1
2n!n−1/summationdisplay
k=0/parenleftbig1
2z/parenrightbigk−nKk(z)
k!(n−k)[n=0,1,...] AS 9.6.45
Special cases
12./bracketleftbigg∂Jν(z)
∂ν/bracketrightbigg
ν=0=1
2πY0(z) MS 3.2.3
13./bracketleftbigg∂Yν(z)
∂ν/bracketrightbigg
ν=0=−1
2πJ0(z) MS 3.2.3
14./bracketleftbigg∂Iν(z)
∂ν/bracketrightbigg
ν=0=−K0(z) MS 3.2.3
15./bracketleftbigg∂Kν(z)
∂ν/bracketrightbigg
ν=0=0 MS 3.2.3
16./bracketleftbigg∂Jν(x)
∂ν/bracketrightbigg
ν=1
2=/parenleftbig1
2πx/parenrightbig−1/2[sinxCi(3x)−cosxSi(2x)] MS 3.3.3
17./bracketleftbigg∂Jν(x)
∂ν/bracketrightbigg
ν=−1
2=/parenleftbig1
2πx/parenrightbig−1/2[cosxCi(2x)+s i n xSi(2x)] MS 3.3.3
18./bracketleftbigg∂Yν(x)
∂ν/bracketrightbigg
ν=1
2=/parenleftbig1
2πx/parenrightbig−1/2{cosxCi(2x)+s i n x[Si(2x)−π]} MS 3.3.3
19./bracketleftbigg∂Yν(x)
∂ν/bracketrightbigg
ν=−1
2=−/parenleftbig1
2πx/parenrightbig−1/2{sinxCi(2x)−cosx[Si(2x)−π]} MS 3.3.3
8.491 Differential equations leading to Bessel functions 931
20./bracketleftbigg∂Iν(x)
∂ν/bracketrightbigg
ν=±1
2=( 2πx)−1/2/bracketleftbig
exEi(−2x)∓e−xEi(2x)/bracketrightbig
MS 3.3.3
21./bracketleftbigg∂Kν(x)
∂ν/bracketrightbigg
ν=±1
2=∓/parenleftBigπ
2x/parenrightBig1
2exEi(−2x) MS 3.3.3
8.487 Continuity with respect to the order∗:
1. lim
ν→nYν(z)=Yn(z)[ nan integer] WA 76
2. lim
ν→nH(1,2)
ν(z)=H(1,2)
n(z)[ nan integer] WA 183
3. lim
ν→nKν(z)=Kn(z)[ nan integer] WA 92
8.49 Differential equations leading to Bessel functions
See also 8.401
8.491
1.1
zd
dz(zu/prime)+/parenleftbigg
β2−ν2
z2/parenrightbigg
u=0 u=Zν(βz) JA
2.1
zd
dz(zu/prime)+/bracketleftbigg/parenleftbig
βγzγ−1/parenrightbig2−/parenleftBigνγ
z/parenrightBig2/bracketrightbigg
u=0 u=Zν(βzγ) JA
3. u/prime/prime+1−2α
zu/prime+/bracketleftbigg/parenleftbig
βγzγ−1/parenrightbig2+α2−ν2γ2
z2/bracketrightbigg
u=0 u=zαZν(βzγ) JA
4. u/prime/prime+/bracketleftbigg/parenleftbig
βγzγ−1/parenrightbig2−4ν2γ2−1
4z2/bracketrightbigg
u=0 u=√zZν(βzγ) JA
5. u/prime/prime+/parenleftbigg
β2−4ν2−1
4z2/parenrightbigg
u=0 u=√zZν(βz) JA
6. u/prime/prime+1−2α
zu/prime+/parenleftbigg
β2+α2−ν2
z2/parenrightbigg
u=0 u=zαZν(βz) JA
7. u/prime/prime+bzmu=0 u=√zZ1
m+2/parenleftBigg
2√
b
m+2zm+2
2/parenrightBigg
JA 111(5)
8. u/prime/prime+1
zu/prime+4/parenleftbigg
z2−ν2
z2/parenrightbigg
u=0 u=Zν/parenleftbig
z2/parenrightbig
WA 111(6)
9. u/prime/prime+1
zu/prime+1
4z/parenleftbigg
1−ν2
z/parenrightbigg
u=0 u=Zν/parenleftbig√z/parenrightbig
WA 111(7)
10. u/prime/prime+1−ν
zu/prime+1
4u
z=0 u=zν
2Zν/parenleftbig√z/parenrightbig
WA 111(9)a
11. u/prime/prime+β2γ2z2β−2u=0 u=z1/2Z1
2β/parenleftbig
γzβ/parenrightbig
WA 110(3)
∗The continuity of the functions Jν(z)a n dIν(z) follows directly from the series representations of these functions.
932 Bessel Functions and Functions Associated with Them 8.492
12. z2u/prime/prime+( 2α−2βν+1 )zu/prime+/bracketleftbig
β2γ2z2β+α(α−2βν)/bracketrightbig
u=0
u=zβν−αZν/parenleftbig
γzβ/parenrightbig
WA 112(21)
8.492
1. u/prime/prime+/parenleftbig
e2z−ν2/parenrightbig
u=0 u=Zν(ez) WA 112(22)
2. u/prime/prime+e2/z−ν2
z4u=0 u=zZν/parenleftBig
e1/z/parenrightBig
WA 112(22)
8.493
1. u/prime/prime+/parenleftbigg1
z−2t anz/parenrightbigg
u/prime−/parenleftbiggν2
z2+tanz
z/parenrightbigg
u=0 u=s e c zZν(z) JA
2. u/prime/prime+/parenleftbigg1
z+ 2cot z/parenrightbigg
u/prime−/parenleftbiggν2
z2−cotz
z/parenrightbigg
u=0 u=c o s e c zZν(z) JA
8.494
1. u/prime/prime+1
zu/prime−/parenleftbigg
1+ν2
z2/parenrightbigg
u=0 u=Zν(iz)=C1Iν(z)+C2Kν(z)JA
2. u/prime/prime+1
zu/prime−/bracketleftbigg1
z+/parenleftBigν
2z/parenrightBig2/bracketrightbigg
u=0 u=Zν/parenleftbig
2i√z/parenrightbig
JA
3. u/prime/prime+u/prime+1
z2/parenleftbigg1
4−ν2/parenrightbigg
u=0 u=√ze−z
2Zν/parenleftbiggiz
2/parenrightbigg
JA
4.10u/prime/prime+/parenleftbigg2ν+1
z−k/parenrightbigg
u/prime−2ν+1
2zku=0 u=z−νe1
2kxZν/parenleftbiggikz
2/parenrightbigg
JA
5. u/prime/prime+1−ν
zu/prime−1
4u
z=0 u=zν
2Zν/parenleftbig
i√z/parenrightbig
WA 111(8)
6. u/prime/prime±u√z=0
u=√zZ2
3/parenleftBig
4
3z3
4/parenrightBig
,u =√zZ2
3/parenleftBig
4
3iz3
4/parenrightBig
WA 111(10)
7. u/prime/prime±zu=0
u=√zZ1
3/parenleftBig
2
3z3
2/parenrightBig
,u =√zZ1
3/parenleftBig
2
3iz3
2/parenrightBig
WA 111(10)
8. u/prime/prime−/parenleftbigg
c2+ν(ν+1 )
z2/parenrightbigg
u=0 u=√zZν+1
2(icz) WA 108(1)
9. u/prime/prime−2ν
zu/prime−c2u=0 u=zν+1
2Zν+1
2(icz) WA 109(3, 4)
10. u/prime/prime−c2z2ν−2u=0 u=√zZ1
2ν/parenleftBig
ic
νzν/parenrightBig
WA 109(5, 6)
8.495
1. u/prime/prime+1
zu/prime+/parenleftbigg
i−ν2
z2/parenrightbigg
u=0 u=Zν/parenleftBig
z√
i/parenrightBig
JA
8.511 Series of Bessel functions 933
2. u/prime/prime+/parenleftbigg1
z∓2i/parenrightbigg
u/prime−/parenleftbiggν2
z2±i
z/parenrightbigg
u=0 u=e±izZν(z) JA
3. u/prime/prime+1
zu/prime+seiαu=0 u=Z0/parenleftBig√szei
2α/parenrightBig
JA
4. u/prime/prime+/parenleftbigg
seiα+1
4z2/parenrightbigg
u=0 u=√zZ0/parenleftBig√szei
2α/parenrightBig
JA
8.496
1.d2
dz2/parenleftbigg
z4d2u
dz2/parenrightbigg
−z2u=0 u=1
z/braceleftBig
Z2/parenleftbig
2√z/parenrightbig
+Z2/parenleftbig
2i√z/parenrightbig/bracerightBig
WA 122(7)
2.d2
dz2/parenleftbigg
z16
5d2u
dz2/parenrightbigg
−z8
5u=0 u=z−7/10/braceleftbigg
Z5
6/parenleftBig
5
3z3
5/parenrightBig
+Z5
6/parenleftBig
5
3iz3
5/parenrightBig/bracerightbigg
WA 122(8)
3.d2
dz2/parenleftbigg
z12d2u
dz2/parenrightbigg
−z6u=0
u=z−4/braceleftBig
Z10/parenleftBig
2z−1/2/parenrightBig
+Z10/parenleftbig
2iz−1/2/parenrightbig/bracerightBig
WA 122(9)
4.d4u
dz4+2
zd3u
dz3−2ν2+1
z2d2u
dz2+2ν2+1
z3du
dz+/parenleftbiggν4−4ν2
z4−1/parenrightbigg
u=0,
u=A1Jν(z)+A2Yν(z)+A3Iν(z)+A4Kν(z),where A1,A2,A3,A4are constants MO 29
8.51–8.52 Series of Bessel functions
8.511 Generating functions for Bessel functions:
1. exp1
2/parenleftbigg
t−1
t/parenrightbigg
z=J0(z)+∞/summationdisplay
k=1/bracketleftbig
tk+(−t)−k/bracketrightbig
Jk(z)=∞/summationdisplay
k=−∞Jk(z)tk
[|z|<|t|] KU 119(12)
2. exp/parenleftbigg
t−1
t/parenrightbigg
z=/braceleftBigg∞/summationdisplay
k=−∞tkJk(z)/bracerightBigg/braceleftBigg∞/summationdisplay
m=−∞tmJm(z)/bracerightBigg
WA 40
3. exp ( ±izsinϕ)=J0(z)+2∞/summationdisplay
k=1J2k(z)cos2 kϕ±2i∞/summationdisplay
k=0J2k+1(z)sin(2 k+1 )ϕ KU 120(13)
4. exp ( izcosϕ)=/radicalbiggπ
2z∞/summationdisplay
k=0(2k+1 )ikJk+1
2(z)Pk(cosϕ) WA 401(1)
=∞/summationdisplay
k=−∞ikJk(z)eikϕMO 27
=J0(z)+2∞/summationdisplay
k=1ikJk(z)coskϕ MO 27
934 Bessel Functions and Functions Associated with Them 8.512
5./radicalbigg
i
πeizcos2ϕ/integraldisplay√
2zcosϕ
−∞e−it2dt=1
2J0(z)+∞/summationdisplay
k=1e1
4kπiJk
2(z)coskϕ MO 28
The series/summationtextJk(z)
8.512
1. J0(z)+2∞/summationdisplay
k=1J2k(z)=1 WA 44
2.∞/summationdisplay
k=0(n+2k)(n+k−1)!
k!Jn+2k(z)=/parenleftBigz
2/parenrightBign
[n=1,2,...] WA 45
3.∞/summationdisplay
k=0(4k+ 1)(2 k−1)!!
2kk!J2k+1
2(z)=/radicalbigg
2z
π
8.513
Notation : In formulas 8.513 Q(p)
k=⌊k−1
2⌋/summationdisplay
m=0(−1)m/parenleftBig
k
m/parenrightBig
(k−2m)p
2kk!
1.∞/summationdisplay
k=1(2k)2pJ2k(z)=p/summationdisplay
k=0Q(2p)
2kz2k[p=1,2,3,...] WA 46(1)
2.∞/summationdisplay
k=0(2k+1 )2p+1J2k+1(z)=p/summationdisplay
k=0Q(2p+1)
2k+1z2k+1[p=0,1,2,3,...] WA 46(2)
In particular:
3.∞/summationdisplay
k=0(2k+1 )3J2k+1(z)=1
2/parenleftbig
z+z3/parenrightbig
WA 47(4)
4.∞/summationdisplay
k=1(2k)2J2k(z)=1
2z2WA 47(4)
5.∞/summationdisplay
k=12k(2k+ 1)(2 k+2 )J2k+1(z)=1
2z3WA 47(4)
8.514
1.∞/summationdisplay
k=0(−1)kJ2k+1(z)=sinz
2WH
2. J0(z)+2∞/summationdisplay
k=1(−1)kJ2k(z) = cos z WH
3.∞/summationdisplay
k=1(−1)k+1(2k)2J2k(z)=zsinz
2WA 32(9)
8.518 Series of Bessel functions 935
4.∞/summationdisplay
k=0(−1)k(2k+1 )2J2k+1(z)=zcosz
2WA 32(10)
5. J0(z)+2∞/summationdisplay
k=1J2k(z)cos2 kθ=c o s( zsinθ) KU 120(14), WA 32
6.∞/summationdisplay
k=0J2k+1(z)sin(2 k+1 )θ=sin (zsinθ)
2KU 120(15), WA 32
7.∞/summationdisplay
k=0J2k+1(x)=1
2/integraldisplayx
0J0(t)dt [xis real] WA 638
8.515
1.∞/summationdisplay
k=0(−1)ktk
k!/parenleftbigg2z+t
2z/parenrightbiggk
Jν+k(z)=/parenleftbiggz
z+t/parenrightbiggν
Jν(z+t) AD (9140)
2.∞/summationdisplay
k=1J2k−1
2/parenleftbig
x2/parenrightbig
=S(x) MO 127a
3.∞/summationdisplay
k=0J2k+1
2/parenleftbig
x2/parenrightbig
=C(x) MO 127a
8.516∞/summationdisplay
k=0(2n+2k)(2n+k−1)!
k!J2n+2k(2zsinθ)=(zsinθ)2nWA 47
The series/summationtextAkJk(kx)and/summationtextAkJ/prime
k(kx)
8.517
1.∞/summationdisplay
k=1Jk(kz)=z
2(1−z)/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglezexp√
1−z2
1+√
1−z2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<1/bracketrightBigg
WA 615(1)
2.∞/summationdisplay
k=1(−1)kJk(kz)=−z
2(1 + z)/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglezexp√
1−z2
1+√
1−z2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<1/bracketrightBigg
WA 622(1)
3.∞/summationdisplay
k=1J2k(2kz)=z2
2(1−z2)/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglezexp√
1−z2
1+√
1−z2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<1/bracketrightBigg
MO 58
8.518
1.11∞/summationdisplay
k=1J/prime
k(kx)
k=1
2+x
4[0≤x<1] MO 58
2.11∞/summationdisplay
k=1(−1)k−1J/prime
k(kx)
k=1
2−x
4[0≤x<1] MO 58
3.∞/summationdisplay
k=1kJ/prime
k(kx)=1
2(1−x)2[0≤x<1] MO 58
936 Bessel Functions and Functions Associated with Them 8.519
4.∞/summationdisplay
k=1(−1)k−1J/prime
k(kx)k=1
2(1 + x)2[0≤x<1] MO 58
The series/summationtextAkJ0(kx)
8.519 If, on the interval [0 ≤x≤π], a function f(x) possesses a continuous derivative with respect to
xthat is of bounded variation, then
1. f(x)=a0
2+∞/summationdisplay
k=1akJ0(kx)[ 0 <x<π ]
where
2. a0=2f(0) +2
π/integraldisplayπ
0du/integraldisplayπ/2
0uf/prime(usinϕ)dϕ
3. an=2
π/integraldisplayπ
0du/integraldisplayπ/2
0uf/prime(usinϕ)c o snu dϕ WH
8.521 Examples:
1.∞/summationdisplay
k=1J0(kx)=−1
2+1
x+2n/summationdisplay
m=11√
x2−4m2π2[2nπ < x < 2(n+1 )π] MO 59
2.∞/summationdisplay
k=1(−1)k+1J0(kx)=1
2[0<x<π ] KU 124(12)
3.∞/summationdisplay
k=11
(2k−1)2J0{(2k−1)x}π2
8−|x|
2[−π<x<π ] KU 124
=π2
8+/radicalbig
x2−π2−x
2−πarccosπ
x[π<x< 2π] MO 59
4.∞/summationdisplay
k=1e−kzJ0/parenleftBig
k/radicalbig
x2+y2/parenrightBig
=1
r−1
2+∞/summationdisplay
k=1/braceleftBigg
1/radicalbig
(2kiπ+z)2+x2+y2−1/radicalbig
(2kiπ−z)2+x2+y2/bracerightBigg
=1
r−1
2+∞/summationdisplay
k=11
(2k)!B2kr2k−1P2k−1/parenleftBigz
r/parenrightBig
[0<r< 2π]MO 59
where r=/radicalbig
x2+y2+z2and where the radical indicates the square root with a positive real part. In
formula 8.521 4, the first equation holds when xandyare real and Re z>0; the second equation holds
when x,y,a n d zare all real.
8.523 Series of Bessel functions 937
The series/summationtextAkZ0(kx)s i nkxand/summationtextAkZ0(kx)coskx
8.522
1.∞/summationdisplay
k=1J0(kx)coskxt=−1
2+m/summationdisplay
l=11/radicalbig
x2−(2πl+tx)2+1
x√
1−t2+n/summationdisplay
l=11/radicalbig
x2−(2πl−tx)2
MO 59
2.∞/summationdisplay
k=1J0(kx)sinkxt=1
2π/braceleftBiggn/summationdisplay
l=11
l−m/summationdisplay
l=11
l/bracerightBigg
+∞/summationdisplay
l=m+1/braceleftBigg
1/radicalbig
(2πl+tx)2−x2−1
2πl/bracerightBigg
−∞/summationdisplay
l=n+1/braceleftBigg
1/radicalbig
(2πl−tx)2−x2−1
2πl/bracerightBigg
MO 59
3.∞/summationdisplay
k=1Y0(kx)coskxt=−1
π/parenleftBig
C+l nx
4π/parenrightBig
+1
2π/braceleftBiggm/summationdisplay
l=11
l+n/summationdisplay
l=11
l/bracerightBigg
−∞/summationdisplay
l=m+1/braceleftBigg
1/radicalbig
(2πl+tx)2−x2−1
2πl/bracerightBigg
−∞/summationdisplay
l=n+1⎧
⎨
⎩1/radicalBig
(2πl−tx)2−x2−1
2πl⎫
⎬
⎭
MO 60
In formulas 8.522 ,x>0,0≤t<1,2πm < x (1−t)<2(m+1 )π,2nπ < x (1 +t)<2(n+1 )π,m+1a n d
n+ 1 are natural numbers.
8.523
1.∞/summationdisplay
k=1(−1)kJ0(kx)coskxt=−1
2+m/summationdisplay
l=11/radicalBig
x2−[(2l−1)π+tx]2+n/summationdisplay
l=11/radicalBig
x2−[(2l−1)π−tx]2
MO 60
2.∞/summationdisplay
k=1(−1)kJ0(kx)sinkxt1
2π/braceleftBiggn/summationdisplay
l=11
l−m/summationdisplay
l=11
l/bracerightBigg
+∞/summationdisplay
l=m+1⎧
⎨
⎩1/radicalBig
[(2l−1)π+tx]2−x2−1
2lπ⎫
⎬
⎭
−∞/summationdisplay
l=n+1⎧
⎨
⎩1/radicalBig
[(2l−1)π−tx]2−x2−1
2lπ⎫
⎬
⎭
MO 60
938 Bessel Functions and Functions Associated with Them 8.524
3.∞/summationdisplay
k=1(−1)kY0(kx)coskxt−1
π/parenleftBig
C+l nx
4π/parenrightBig
+1
2π/braceleftBiggm/summationdisplay
l=11
l+n/summationdisplay
l=11
l/bracerightBigg
−∞/summationdisplay
l=m+1⎧
⎨
⎩1/radicalBig
[(2l−1)π+tx]2−x2−1
2lπ⎫
⎬
⎭
−∞/summationdisplay
l=n+1⎧
⎨
⎩1/radicalBig
[(2l−1)π−tx]2−x2−1
2lπ⎫
⎬
⎭
MO 60
In formulas 8.523 ,x>0,0≤t<1, (2m−1)π<x (1−t)<(2m+1 )π,( 2n−1)π<x (1 +t)<(2n+1 )π,
mandnare natural numbers.
8.524
1.∞/summationdisplay
k=1J0(kx)coskxt=−1
2+n/summationdisplay
l=m+11/radicalbig
x2−(2lπ−tx)2MO 60
2.∞/summationdisplay
k=1J0(kx)sinkxtm/summationdisplay
l=01/radicalbig
(2lπ−tx)2−x2+∞/summationdisplay
l=1/braceleftBigg
1/radicalbig
(2lπ+tx)2−x2−1
2lπ/bracerightBigg
−∞/summationdisplay
l=n+1/braceleftBigg
1/radicalbig
(2lπ−tx)2−x2−1
2lπ/bracerightBigg
+1
2πn/summationdisplay
l=11
l
MO 60
3.6∞/summationdisplay
k=1Y0(kx)coskxt−1
π/parenleftBig
C+l nx
4π/parenrightBig
−m/summationdisplay
l=01/radicalbig
(2πl−tx)2−x2+1
2πn/summationdisplay
l=11
l
−∞/summationdisplay
l=1/braceleftBigg
1/radicalbig
(2lπ+tx)2−x2−1
2lπ/bracerightBigg
−∞/summationdisplay
l=n+1/braceleftBigg
1/radicalbig
(2lπ−tx)2−x2−1
2lπ/bracerightBigg
MO 61
In formulas 8.524 ,x>0,t >1,2mπ < x (t−1)<2(m+1 )π,2nπ < x (t+1 )<2(n+1 )π,m+1a n d
n+ 1 are natural numbers.
8.525
1.∞/summationdisplay
k=1(−1)kJ0(kx)coskxt=−1
2+n/summationdisplay
l=m+11/radicalBig
x2−[(2l−1)π−tx]2MO 61
8.526 Series of Bessel functions 939
2.∞/summationdisplay
k=1(−1)kJ0(kx)sinkxt=m/summationdisplay
l=11/radicalBig
[(2l−1)π−tx]2−x2+1
2πn/summationdisplay
l=11
l
+∞/summationdisplay
l=1⎧
⎨
⎩1/radicalBig
[(2l−1)π+tx]2−x2−1
2lπ⎫
⎬
⎭
−∞/summationdisplay
l=n+1⎧
⎨
⎩1/radicalBig
[(2l−1)π−tx]2−x2−1
2lπ⎫
⎬
⎭
MO 61
3.∞/summationdisplay
k=1(−1)kY0(kx)coskxt=−1
π/parenleftBig
C+l nx
4π/parenrightBig
+1
2πn/summationdisplay
l=11
l
−m/summationdisplay
l=11/radicalBig
[(2l−1)π−tx]2−x2
−∞/summationdisplay
l=1⎧
⎨
⎩1/radicalBig
[(2l−1)π+tx]2−x2−1
2lπ⎫
⎬
⎭
−∞/summationdisplay
l=n+1⎧
⎨
⎩1/radicalBig
[(2l−1)π−tx]2−x2−1
2lπ⎫
⎬
⎭
MO 61
In formulas 8.525 ,x>0,t>1, (2m−1)π<x (t−1)<(2m+1 )π,( 2n−1)π<x (t+1 )<(2n+1 )π,m
andnare natural numbers.
8.526
1.∞/summationdisplay
k=1K0(kx)coskxt=1
2/parenleftBig
C+l nx
4π/parenrightBig
+π
2x√
1+t2+π
2∞/summationdisplay
l=1/braceleftBigg
1/radicalbig
x2+( 2lπ−tx)2−1
2lπ/bracerightBigg
+π
2∞/summationdisplay
l=1/braceleftBigg
1/radicalbig
x2+( 2lπ+tx)2−1
2lπ/bracerightBigg
MO 61
2.∞/summationdisplay
k=1(−1)kK0(kx)coskxt=1
2/parenleftBig
C+l nx
4π/parenrightBig
+π
2∞/summationdisplay
l=1⎧
⎨
⎩1/radicalBig
x2+[ ( 2l−1)π−xt]2−1
2lπ⎫
⎬
⎭
+π
2∞/summationdisplay
l=1⎧
⎨
⎩1/radicalBig
x2+[ ( 2l−1)π+xt]2−1
2lπ⎫
⎬
⎭
[x>0,treal] (see also 8.66)MO 62
940 Bessel Functions and Functions Associated with Them 8.530
8.53 Expansion in products of Bessel functions
“Summation theorems”
8.530 Suppose that r>0,/rho1>0,ϕ>0, and R=/radicalbig
r2+/rho12−2r/rho1cosϕ; that is, suppose that r, /rho1,a n d
Rare the sides of a triangle such that the angle between the sides rand/rho1is equal to ϕ. Suppose also
that/rho1<ra n dt h a t ψis the angle opposite the side /rho1,s ot h a t
1. 0 <ψ<π
2,e2iψ=r−/rho1e−iϕ
r−/rho1eiϕ
When these conditions are satisfied, we have the “summation theorem” for Bessel functions:
1. eiνψZν(mR)=∞/summationdisplay
k=−∞Jk(m/rho1)Zν+k(mr)eikϕ[mis an arbitrary complex number]
WA 394(6)
ForZν=Jνandνan integer, the restriction /rho1<r is superfluous. MO 31
8.531 Special cases:
1. J0(mR)=J0(m/rho1)J0(mr)+2∞/summationdisplay
k=1Jk(m/rho1)Jk(mr)coskϕ WA 391(1)
2. H(1,2)
0(mR)=J0(m/rho1)H(1,2)
0(mr)+2∞/summationdisplay
k=1Jk(m/rho1)H(1,2)
k(mr)coskϕ MO 31
3. J0(zsinα)=J2
0/parenleftBigz
2/parenrightBig
+2∞/summationdisplay
k=1J2
k/parenleftBigz
2/parenrightBig
cos 2kα
=/radicalbigg
2π
z∞/summationdisplay
k=0/parenleftbigg
2k+1
2/parenrightbigg(2k−1)!!
2kk!J2k+1
2(z)P2k(cosα) MO 31
8.532 The term “summation theorem” is also applied to the formula
1.Zν(mR)
Rν=2νm−νΓ(ν)∞/summationdisplay
k=0(ν+k)Jν+k(m/rho1)
/rho1νZν+k(mr)
rνCν
k(cosϕ)
[ν/negationslash=−1,−2,−3,...; the conditions on r,/rho1,R,ϕ,a n d mare the same as in formula 8.530 ;f o rZν=Jν
andνan integer, formula 8.532 1 is valid for arbitrary r, /rho1,a n d ϕ]. WA 398(4)
8.533 Special cases:
1.eimR
R=πi
2√r/rho1∞/summationdisplay
k=0(2k+1 )Jk+1
2(m/rho1)H(1)
k+1
2(mr)Pk(cosϕ) MO 31
2.e−imR
R=−πi
2√r/rho1∞/summationdisplay
k=0(2k+1 )Jk+1
2(m/rho1)H(2)
k+1
2(mr)Pk(cosϕ) MO 31
8.543 The zeros of Bessel functions 941
8.534 A degenerate addition theorem ( r→∞):
eim/rho1cosϕ=/radicalbiggπ
2m/rho1∞/summationdisplay
k=0ik(2k+1 )Jk+1
2(m/rho1)Pk(cosϕ) WA 401(1)
=2νΓ(ν)∞/summationdisplay
k=0(ν+k)ik(m/rho1)−νJν+k(m/rho1)Cν
k(cosϕ)[ν/negationslash=0,−1,−2,...]WA 401(2)
8.535 The term “product theorem” is also applied to the formula
Zν(λz)=λν∞/summationdisplay
k=01
k!Zν+k(z)/parenleftbigg1−λ2
2z/parenrightbiggk/bracketleftBig
|1−λ|2<1/bracketrightBig
ForZν=Jν, it is valid for all values of λandz. MO 32
8.536
1.∞/summationdisplay
k=0(2n+2k)(2n+k−1)!
k!J2
n+k(z)=(2n)!
(n!)2/parenleftBigz
2/parenrightBig2n
[n>0] WA 47(1)
2. 2∞/summationdisplay
k=nkΓ(n+k)
Γ(k−n+1 )J2
k(z)=(2n)!
(n!)2/parenleftBigz
2/parenrightBig2n
[n>0] WA 47(2)
3. J2
0(z)+2∞/summationdisplay
k=1J2
k(z)=1 WA 41(3)
8.537
1.∞/summationdisplay
k=−∞Zν−k(t)Jk(z)=Zν(z+t)[ |z|<|t|] WA 158(2)
2.∞/summationdisplay
k=−∞Jk(z)Jn−k(z)=Jn(2z) WA 41
8.538
1.∞/summationdisplay
k=−∞(−1)kJ−ν+k(t)Jk(z)=J−ν(z+t)[ |z|<|t|] WA 159
2.∞/summationdisplay
k=−∞Zν+k(t)Jk(z)=Zν(t−z)[ |z|<|t|] WA 159(5)
8.54 The zeros of Bessel functions
8.541 For arbitrary real ν, the function Jν(z) has infinitely many real zeros. For ν>−1, all its zeros
are real. WA 526, 530
A Bessel function Zν(z) has no multiple zeros except possibly the coordinate origin. WA 528
8.542 All zeros of the function Y0(z) with positive real parts are real. WA 531
8.543 If−(2s+2 )<ν< −(2s+ 1), where sis a natural number or 0, then Jν(z) has exactly 4 s+2
complex roots, two of which are purely imaginary. If −(2s+1 )<ν< −2s,w h e r e sis a natural number,
then the function Jν(z) has exactly 4 scomplex zeros, none of which are purely imaginary. WA 532
942 Bessel Functions and Functions Associated with Them 8.544
8.544 Ifxνandx/prime
νare, respectively, the smallest positive zeros of the functions Jν(z)a n d J/prime
ν(z)f o r
ν>0, then xν>νandx/prime
ν>ν. Suppose also that yνis the smallest positive zero of the function Yν(z).
Then, xν<yν<x/prime
ν. WA 534, 536
Suppose that zν,m(form=1,2,3,...) are the zeros of the function z−νJν(z), numbered in order of
the absolute value of their real parts. Here, we assume that ν/negationslash=−1,−2,−3,.... Then, for arbitrary z
Jν(z)=/parenleftbigz
2/parenrightbigν
Γ(ν+1 )∞/productdisplay
m=1/parenleftbigg
1−z2
z2ν,m/parenrightbigg
. WA 550
8.5458The number of zeros of the function z−νJν(z) that occur between the imaginary axis and the
line on which
Rez=/parenleftbig
m+1
2Reν+1
4/parenrightbig
π, WA 497
is exactly m.
8.546 Forν≥0, the number of zeros of the function Kν(z) that occur in the region Re z<0,|argz|<π
is equal to the even number closest to ν−1
2. WA 562
8.547 Large zeros of the functions Jν(z)cosα−Yν(z)sinα,w h e r e νandαare real numbers, are given
by the asymptotic expansion
xν,m∼/parenleftbigg
m+1
2ν−1
4/parenrightbigg
π−α−4ν2−1
8/bracketleftbig/parenleftbig
m+1
2ν−1
4/parenrightbig
π−α/bracketrightbig
−/parenleftbig
4ν2−1/parenrightbig/parenleftbig
28ν2−31/parenrightbig
384/bracketleftbig/parenleftbig
m+1
2ν−1
4/parenrightbig
π−α/bracketrightbig3−...KU 109(24), WA 558
8.548 In particular, large zeros of the function J0(z) are given by the expansion
x0,m∼π
4(4m−1) +1
2π(4m−1)−31
6π3(4m−1)3+3779
15π5(4m−1)5−... KU 109(25), WA 556
This series is suitable for calculating all (except the smallest x01) zeros of the function J0(z) correctly
to at least five digits.
8.549 To calculate the roots xν,mof the function Jν(z) of smallest absolute value, we may use the
identity
∞/summationdisplay
m=11
x16ν,m=429ν5+ 7640 ν4+ 53752 ν3+ 185430 ν2+ 311387 ν+ 202738
216(ν+1 )8(ν+2 )4(ν+3 )2(ν+4 )2(ν+5 ) (ν+6 )(ν+7 )(ν+8 ).KU 112(27)a, WA 554
8.55 Struve functions
8.550 Definitions:
1. Hν(z)=∞/summationdisplay
m=0(−1)m/parenleftbigz
2/parenrightbig2m+ν+1
Γ/parenleftbig
m+3
2/parenrightbig
Γ/parenleftbig
ν+m+3
2/parenrightbig WA 358(2)
2. Lν(z)=−ie−iνπ
2Hν/parenleftbig
zeiπ
2/parenrightbig
=∞/summationdisplay
m=0/parenleftbigz
2/parenrightbig2m+ν+1
Γ/parenleftbig
m+3
2/parenrightbig
Γ/parenleftbig
ν+m+3
2/parenrightbig WA 360(11)
8.551 Integral representations:
1. Hν(z)=2/parenleftbigz
2/parenrightbigν
√πΓ/parenleftbig
ν+1
2/parenrightbig/integraldisplay1
0/parenleftbig
1−t2/parenrightbigν−1
2sinztdt=2/parenleftbigz
2/parenrightbigν
√πΓ/parenleftbig
ν+1
2/parenrightbig/integraldisplayπ/2
0sin (zcosϕ)( s i nϕ)2νdϕ
/bracketleftbig
Reν>−1
2/bracketrightbig
WA 358(1)
8.554 Struve functions 943
2. Lν(z)=2/parenleftbigz
2/parenrightbigν
√πΓ/parenleftbig
ν+1
2/parenrightbig/integraldisplayπ/2
0sinh (zcosϕ)(sinϕ)2νdϕ
/bracketleftbig
Reν>−1
2/bracketrightbig
WA 360(11)
8.552 Special cases:
1.6Hn(z)=1
π⌊n−1
2⌋/summationdisplay
m=0Γ/parenleftbig
m+1
2/parenrightbig/parenleftBigz
2/parenrightBign−2m−1
Γ/parenleftbig
n+1
2−m/parenrightbig−En(z)[ n=1,2,...] EH II 40(66), WA 337(1)
2.6H−n(z)=(−1)n+11
π⌊n−1
2⌋/summationdisplay
m=0Γ/parenleftbig
n−m−1
2/parenrightbig/parenleftBigz
2/parenrightBig−n+2m+1
Γ/parenleftbig
m+3
2/parenrightbig −E−n(z)
[n=1,2,...] EH II 40(67), WA 337(2)
3. Hn+1
2(z)=Yn+1
2(z)+1
πn/summationdisplay
m=0Γ/parenleftbig
m+1
2/parenrightbig/parenleftBigz
2/parenrightBig−2m+n−1
2
Γ(n+1−m)
[n=0,1,...] EH II 39(64)
4. H−(n+1
2)(z)=(−1)nJn+1
2(z)[ n=0,1,...] EH II 39(65)
5. L−(n+1
2)(z)=In+1
2(z)[ n=0,1,...] EH II 39(65)
6. H1
2(z)=√
2√πz(1−cosz) EH II 39, WA 364(3)
7. H3
2(z)=/parenleftBigz
2π/parenrightBig1/2/parenleftbigg
1+2
z2/parenrightbigg
−/parenleftbigg2
πz/parenrightbigg1/2/parenleftBig
sinz+cosz
z/parenrightBig
WA 364(3)
8.553 Functional relations:
1. Hν/parenleftbig
zeimπ/parenrightbig
=eiπ(ν+1)mHν(z)[ m=1,2,3,...] WA 362(5)
2.d
dz[zνHν(z)] =zνHν−1(z) WA 358
3.d
dz/bracketleftbig
z−νHν(z)/bracketrightbig
=2−νπ−1/2/bracketleftbig
Γ/parenleftbig
ν+3
2/parenrightbig/bracketrightbig−1−z−νHν+1(z) WA 359
4. Hν−1(z)+Hν+1(z)=2νz−1Hν(z)+π−1/2/parenleftBigz
2/parenrightBigν/bracketleftbig
Γ/parenleftbig
ν+3
2/parenrightbig/bracketrightbig−1WA 359(5)
5. Hν−1(z)−Hν+1(z)=2H/prime
ν(z)−π−1/2/parenleftBigz
2/parenrightBigν/bracketleftbig
Γ/parenleftbig
ν+3
2/parenrightbig/bracketrightbig−1WA 359(6)
8.554 Asymptotic representations:
Hν(ξ)=Yν(ξ)+1
πp−1/summationdisplay
m=0Γ/parenleftbig
m+1
2/parenrightbig/parenleftbiggξ
2/parenrightbigg−2m+ν−1
Γ/parenleftbig
ν+1
2−m/parenrightbig +O/parenleftBig
|ξ|ν−2p−1/parenrightBig
[|argξ|<π] EH II 39(63), WA 363(2)
For the asymptotic representation of Yν(ξ), see8.451 2.
944 Bessel Functions and Functions Associated with Them 8.555
8.555 The differential equation for Struve functions:
z2y/prime/prime+zy/prime+/parenleftbig
z2−ν2/parenrightbig
y=1√π4/parenleftbigz
2/parenrightbigν+1
Γ/parenleftbig
ν+1
2/parenrightbig WA 359(10)
8.56 Thomson functions and their generalizations
berν(z), bei ν(z), her ν(z), hei ν(z), ker ν(z), kei ν(z)
8.561
1. ber ν(z)+ibeiν(z)=Jν/parenleftBig
ze3
4πi/parenrightBig
WA 96(6)
2. ber ν(z)−ibeiν(z)=Jν/parenleftBig
ze−3
4πi/parenrightBig
. WA 96(6)
8.562
1. her ν(z)+iheiν(z)=Hν
(1)/parenleftBig
ze3
4πi/parenrightBig
(see also 8.567 ) WA 96(7)
2. her ν(z)−iheiν(z)=Hν
(1)/parenleftBig
ze−3
4πi/parenrightBig
(see also 8.567 ) WA 96(7)
8.563
1. ber 0(z)≡ber(z); bei 0(z)≡bei(z) WA 96(8)
2. ker( z)≡−π
2hei0(z); kei( z)≡π
2hei0(z) WA 96(8)
For integral representations, see 6.251 ,6.536 ,6.537 ,6.772 4,6.777 .
Series representation
8.564
1. ber( z)=∞/summationdisplay
k=0(−1)kz4k
24k[(2k)!]2WA 96(3)
2. bei( z)=∞/summationdisplay
k=0(−1)kz4k+2
24k+2[(2k+1 ) ! ]2WA 96(4)
3. ker( z)=/parenleftbigg
ln2
z−C/parenrightbigg
ber(z)+π
4bei(z)+∞/summationdisplay
k=1(−1)kz4k
24k[(2k)!]22k/summationdisplay
m=11
mWA 96(9)a, DW
4. kei( z)=/parenleftbigg
ln2
z−C/parenrightbigg
bei(z)−π
4ber(z)+∞/summationdisplay
k=0(−1)k z4k+2
24k+2[(2k+1 ) ! ]22k+1/summationdisplay
m=11
mWA 96(10)a, DW
8.565 ber2
ν(z)+b e i2
ν(z)=∞/summationdisplay
k=0(z/2)2ν+4k
k!Γ (ν+k+1 )Γ ( ν+2k+1 )WA 163(6)
8.570 Lommel functions 945
Asymptotic representation
8.566
1. ber( z)=eα(z)
√
2πzcosβ(z)/bracketleftBig
|argz|<π
4/bracketrightBig
WA 227(1)
2. bei( z)=eα(z)
√
2πzsinβ(z)/bracketleftBig
|argz|<π
4/bracketrightBig
WA 227(1)
3. ker( z)=/radicalbiggπ
2zeα(−z)cosβ(−z)/bracketleftbigg
|argz|<5
4π/bracketrightbigg
WA 227(2)
4. kei( z)=/radicalbiggπ
2zeα(−z)sinβ(−z)/bracketleftbigg
|argz|<5
4π/bracketrightbigg
, WA 227(2)
where
α(z)∼z√
2+1
8z√
2−25
384z3√
2−13
128z4−...,
β(z)∼z√
2−π
8−1
8z√
2−1
16z2−25
384z3√
2+...
8.567 Functional relations
1. ker( z)+ikei(z)=K0/parenleftBig
z√
i/parenrightBig
(see8.562 ) WA 96(5), DW
2. ker( z)−ikei(z)=K0/parenleftBig
z√
−i/parenrightBig
(see8.562 ) WA 96(5), DW
For integrals of Thomson’s functions, see 6.87.
8.57 Lommel functions
8.570 Definitions of the Lommel functions sμ,ν(z)a n dSμ,ν(z):
1. sμ,ν(z)=(−1)mzμ+1+2 m
[(μ+1 )2−ν2][(μ+3 )2−ν2].../bracketleftBig
(μ+2m+1 )2−ν2/bracketrightBig
=zμ−1∞/summationdisplay
m=0(−1)m/parenleftbigz
2/parenrightbig2m+2Γ/parenleftbig1
2μ−1
2ν+1
2/parenrightbig
Γ/parenleftbig1
2μ+1
2ν+1
2/parenrightbig
Γ/parenleftbig1
2μ−1
2ν+m+3
2/parenrightbig
Γ/parenleftbig1
2μ+1
2ν+m+3
2/parenrightbig
[μ±νis not a negative odd integer] EH II 40(69), WA 377(2)
2.11Sμ,ν(z)=sμ,ν(z)+2μ−1Γ/parenleftbig1
2μ−1
2ν+1
2/parenrightbig
Γ/parenleftbig1
2μ+1
2ν+1
2/parenrightbig
×cos/bracketleftbig1
2(μ−ν)π/bracketrightbig
J−ν(z)−cos/bracketleftbig1
2(μ+ν)π/bracketrightbig
Jν(z)
sinνπEH II 40(71), WA 379(2)
=sμ,ν(z)+2μ−1Γ/parenleftbig1
2μ−1
2ν+1
2/parenrightbig
Γ/parenleftbig1
2μ+1
2ν+1
2/parenrightbig
×/braceleftbig
sin/bracketleftbig1
2(μ−ν)π/bracketrightbig
Jν(z)−cos/bracketleftbig1
2(μ−ν)π/bracketrightbig
Yν(z)/bracerightbig
EH II 41(71), WA 379(3)
946 Bessel Functions and Functions Associated with Them 8.571
Integral representations
8.571 sμ,ν(z)=π
2/bracketleftbigg
Yν(z)/integraldisplayz
0zμJν(z)dz−Jν(z)/integraldisplayz
0zμYν(z)dz/bracketrightbigg
WA 378(9)
8.572 sμ,ν(z)
=2μ/parenleftBigz
2/parenrightBig1
2(1+ν+μ)
Γ/parenleftbigg1
2+1
2μ−1
2ν/parenrightbigg/integraldisplayπ/2
0J1
2(1+μ−ν)(zsinθ)(sinθ)1
2(1+ν−μ)(cosθ)ν+μdθ
[Re(ν+μ+1 )>0] EH II 42(86)
8.573 Special cases:
1. S1,2n(z)=zO2n(z) WA 382(1)
2. S0,2n+1(z)=z
2n+1O2n+1(z) WA 382(1)
3. S−1,2n(z)=1
4nS2n(z) WA 382(2)
4. S0,2n+1(z)=1
2S2n+1(z) WA 382(2)
5. Sν,ν(z)=Γ/parenleftbigg
ν+1
2/parenrightbigg√π2ν−1Hν(z) EH II 42(84)
6. Sν,ν(z)=[Hν(z)−Yν(z)] 2ν−1√πΓ/parenleftbig
ν+1
2/parenrightbig
EH II 42(84)
8.574 Connections with other special functions:
1. Jν(z)=1
πsin(νπ)[s0,ν(z)−νs−1,ν(z)] EH II 41(82)
2. Eν(z)=−1
π[(1 + cos νπ)s0,ν(z)+ν(1−cosνπ)s−1,ν(z)] EH II 42(83)
A connection with a hypergeometric function
3. sμ,ν(z)=zμ+1
(μ−ν+1 ) (μ+ν+1 )1F2/parenleftbigg
1;μ−ν+3
2,μ+ν+3
2;−z2
4/parenrightbigg
EH II 40(69), WA 378(10)
8.575 Functional relations:
1. sμ+2,ν(z)=zμ+1−/bracketleftbig
(μ+1 )2−ν2/bracketrightbig
sμ,ν(z) EH II 41(73), WA 380(1)
2.8s/prime
μ,ν(z)+/parenleftBigν
z/parenrightBig
sμ,ν(z)=(μ+ν−1)sμ−1,ν−1(z) EH II 41(74), WA 380(2)
3. s/prime
μ,ν(z)−/parenleftBigν
z/parenrightBig
sμ,ν(z)=(μ−ν−1)sμ−1,ν+1(z) EH II 41(75), WA 380(3)
4./parenleftBig
2ν
z/parenrightBig
sμ,ν(z)=(μ+ν−1)sμ−1,ν−1(z)−(μ−ν−1)sμ−1,ν+1(z) EH II 41(76), WA 380(4)
5.82s/prime
μ,ν(z)=(μ+ν−1)sμ−1,ν−1(z)+(μ−ν−1)sμ−1,ν+1(z) EH II 41(77), WA 380(5)
In formulas 8.575 1–5,sμ,ν(z) can be replaced with Sμ,ν(z).
8.579 Lommel functions 947
8.576 Asymptotic expansion of Sμ,ν(z).
In the case in which μ±νis not a positive odd integer, Sμ,ν(z) has the following asymptotic expansion:
Sμ,ν(z)∼zμ−1∞/summationdisplay
m=0(−1)m/parenleftbigg1−μ+ν
2/parenrightbigg
m/parenleftbigg1−μ−ν
2/parenrightbigg
m/parenleftBigz
2/parenrightBig−2m
[|z|→∞ ,|argz|<π] WA 347, 352
The series terminates and is equal to Sμ,ν(z)w h e n μ±νis a positive odd integer.
8.577 Lommel functions satisfy the following differential equation:
z2w/prime/prime+zw/prime+/parenleftbig
z2−ν2/parenrightbig
w=zμ+1WA 377(1), EH II 40(68)
8.578 Lommel functions of two variables Uν(w,z)a n dVν(w,z):
Definition
1. Uν(w,z)=∞/summationdisplay
m=0(−1)m/parenleftBigw
z/parenrightBigν+2m
Jν+2m(z) EH II 42(87), WA 591(5)
2. Vν(w,z) = cos/bracketleftbigg1
2/parenleftbigg
w+z2
w+νπ/parenrightbigg/bracketrightbigg
+U−ν+2(w,z) EH II 42(88), WA 591(6)
Particular values:
3. U0(z,z)=V0(z,z)=1
2{J0(z) + cos z} WA 591(9)
4. U1(z,z)=−V1(z,z)=1
2sinz WA 591(10)
5. U2n(z,z)=(−1)n
2/braceleftBigg
cosz−n−1/summationdisplay
m=0(−1)mε2mJ2m(z)/bracerightBigg
[n≥1],ε m=/braceleftBigg
2,m > 0,
1,m=0WA 591(11)
6. U2n+1(z,z)=(−1)n
2/braceleftBigg
sinz−n−1/summationdisplay
m=0(−1)mε2m+1J2m+1(z)/bracerightBigg
[n≥0],ε m=/braceleftBigg
2,m > 0,
1,m=0WA 591(12)
7. Vn(w,z)=(−1)nUn/parenleftbiggz2
w,z/parenrightbigg
8. Uν(w,0) =/parenleftbigw
2/parenrightbig1/2
Γ(ν−1)Sν−3
2,1
2/parenleftBigw
2/parenrightBig
WA 593(9)
9. V−ν+2(w,0) =/parenleftbigw
2/parenrightbig1/2
Γ(ν−1)Sν−3
2,1
2/parenleftBigw
2/parenrightBig
WA 593(10)
8.579 Functional relations:
1. 2∂
∂wUν(w,z)=Uν−1(w,z)+/parenleftBigz
w/parenrightBig2
Uν+1(w,z) WA 593(2)
2. 2∂
∂wVν(w,z)=Vν+1(w,z)+/parenleftBigz
w/parenrightBig2
Vν−1(w,z) WA 593(4)
3. The function Uν(w,z) is a particular solution of the differential equation
948 Bessel Functions and Functions Associated with Them 8.580
∂2U
∂z2−1
z∂U
∂z+z2U
w2=/parenleftBigw
z/parenrightBigν−2
Jν(z) WA 592(2)
4. The function Vν(w,z) is a particular solution of the differential equation
∂2V
∂z2−1
z∂V
∂z+z2V
w2=/parenleftBigw
z/parenrightBig−ν
J−ν+2(z) WA 592(3)
8.58 Anger and Weber functions J ν(z)and E ν(z)
8.580 Definitions:
1. The Anger function Jν(z):
Jν(z)=1
π/integraldisplayπ
0cos(νθ−zsinθ)dθ WA 336(1), EH II 35(32)
2. The Weber function Eν(z):
Eν(z)=1
π/integraldisplayπ
0sin (νθ−zsinθ)dθ WA 336(2), EH II 35(32)
8.581 Series representations:
1. Jν(z)= cosνπ
2∞/summationdisplay
n=0(−1)n/parenleftbigz
2/parenrightbig2n
Γ/parenleftbig
n+1+1
2ν/parenrightbig
Γ/parenleftbig
n+1−1
2ν/parenrightbig
+sinνπ
2∞/summationdisplay
n=0(−1)n/parenleftbigz
2/parenrightbig2n+1
Γ/parenleftbig
n+3
2+1
2ν/parenrightbig
Γ/parenleftbig
n+3
2−1
2ν/parenrightbig
EH II 36(36), WA 337(3)
2. Eν(z)=sinνπ
2∞/summationdisplay
n=0(−1)n/parenleftbigz
2/parenrightbig2n
Γ/parenleftbig
n+1+1
2ν/parenrightbig
Γ/parenleftbig
n+1−1
2ν/parenrightbig
−cosνπ
2∞/summationdisplay
n=0(−1)n/parenleftbigz
2/parenrightbig2n+1
Γ/parenleftbig
n+3
2+1
2ν/parenrightbig
Γ/parenleftbig
n+3
2−1
2ν/parenrightbig
EH II 36(37), WA 338(4)
8.582 Functional relations:
1.62J/prime
ν(z)=Jν−1(z)−Jν+1(z) EH II 36(40), WA 340(2)
2.62E/prime
ν(z)=Eν−1(z)−Eν+1(z) EH II 36(41), WA 340(6)
3.6Jν−1(z)+Jν+1(z)=2νz−1Jν(z)−2(πz)−1sin(νπ) EH II 36(42), WA 340(1)
4.6Eν−1(z)+Eν+1(z)=2νz−1Eν(z)−2(πz)−1(1−cosνπ) EH II 36(43), WA 340(5)
8.591 Neumann’s and Schl¨ afli’s polynomials 949
8.583 Asymptotic expansions:
1.6Jν(z)=Jν(z)+sinνπ
πz⎡
⎣p−1/summationdisplay
n=0(−1)n22nΓ/parenleftbig
n+1+ν
2/parenrightbig
Γ/parenleftbig1+ν
2/parenrightbigΓ/parenleftbig
n+1−ν
2/parenrightbig
Γ/parenleftbig1−ν
2/parenrightbigz−2n
+O/parenleftBig
|z|−2p/parenrightBig
−νp−1/summationdisplay
n=0(−1)n22nΓ/parenleftbig
n+1+1
2ν/parenrightbig
Γ/parenleftbig
n+1−1
2ν/parenrightbig
Γ/parenleftbig
1+1
2ν/parenrightbig
Γ/parenleftbig
1−1
2ν/parenrightbigz−2n−1+νO/parenleftBig
|z|−2p−1/parenrightBig⎤
⎦
[|argz|<π] EH II 37(47), WA 344(1)
2. Eν(z)=−Yν(z)
−1+c o s ( νπ)
πz/bracketleftBiggp−1/summationdisplay
n=0(−1)n22nΓ/parenleftbig
n+1+ν
2/parenrightbig
Γ/parenleftbig
n+1−ν
2/parenrightbig
Γ/parenleftbig1+ν
2/parenrightbig
Γ/parenleftbig1−ν
2/parenrightbigz−2n+O/parenleftBig
|z|−2p/parenrightBig/bracketrightBigg
−ν(1−cosνπ)
zπ/bracketleftBiggp−1/summationdisplay
n=0(−1)n22nΓ/parenleftbig
n+1+1
2ν/parenrightbig
Γ/parenleftbig
n+1−1
2ν/parenrightbig
Γ/parenleftbig
1+1
2ν/parenrightbig
Γ/parenleftbig
1−1
2ν/parenrightbigz−2n−1+O/parenleftBig
|z|−2p−1/parenrightBig/bracketrightBigg
WA344(2), EH II 37(48)
For the asymptotic expansion of Jν(z)a n dYν(z), see8.451 .
8.584 The Anger and Weber functions satisfy the differential equation
y/prime/prime+z−1y/prime+/parenleftbigg
1−ν2
z2/parenrightbigg
y=f(ν,z),
where f(ν,z)=z−ν
πz2sinνπforJν(z) WA 341(9), EH II 37(44)
andf(ν,z)=−1
πz2[z+ν+(z−ν)cosνπ]f o rEν(z) EH II 37(45), WA 341(10)
8.59 Neumann’s and Schl¨ afli’s polynomials: On(z)andSn(z)
8.590 Definition of Neumann’s polynomials
1. On(z)=1
4⌊n
2⌋/summationdisplay
m=0n(n−m−1)!
m!/parenleftBigz
2/parenrightBig2m−n−1
[n≥1] WA 299(2), EH II 33(6)
2. O−n(z)=(−1)nOn(z)[ n≥1] WA 303(8)
3. O0(z)=1
zWA 299(3), EH II 33(7)
4. O1(z)=1
z2EH II 33(7)
5. O2(z)=1
z+4
z3EH II 33(7)
In general, On(z) is a polynomial in z−1of degree n+1 .
8.591 Functional relations:
1. O/prime
0(z)=−O1(z) EH II 33(9), WA 301(3)
2. 2 O/prime
n(z)=On−1(z)−On+1(z)[ n≥1] EH II 33(10), WA 301(2)
950 Mathieu Functions 8.592
3. ( n−1)On+1(z)+(n+1 )On−1(z)−2z−1/parenleftbig
n2−1/parenrightbig
On(z)=2nz−1/parenleftBig
sinnπ
2/parenrightBig2
[n≥1] EH II 33(11), WA 301(1)
4. nzOn−2(z)−/parenleftbig
n2−1/parenrightbig
On(z)=(n−1)zO/prime
n(z)+n/parenleftBig
sinnπ
2/parenrightBig2
EH II 33(12), WA 303(4)
5. nzOn+1(z)−/parenleftbig
n2−1/parenrightbig
On(z)=−(n+1 )zO/prime
n(z)+n/parenleftBig
sinnπ
2/parenrightBig2
EH II 33(13), WA 303(5)a
8.592 The generating function:
1
z−ξ=J0(ξ)z−1+2∞/summationdisplay
n=1Jn(ξ)On(z)[ |ξ|<|z|] EH II 32(1), WA 298(1)
8.593 The integral representation:
On(z)=/integraldisplay∞
0/bracketleftbig
u+√
u2+z2/bracketrightbign+/bracketleftbig
u−√
u2+z2/bracketrightbign
2zn+1e−udu
See also 3.547 6, 8,3.549 1, 2. EH II 32(3), WA 305(1)
8.594 The inequality
|On(z)|≤2n−1n!|z|−n−1e1
4|z|2[n>1] EH II 33(8), WA 300(8)
8.595 Neumann’s polynomial On(z) satisfies the differential equation
z2d2y
dz2+3zdy
dz+/parenleftbig
z2+1−n2/parenrightbig
y=z/parenleftBig
cosnπ
2/parenrightBig2
+n/parenleftBig
sinnπ
2/parenrightBig2
EH II 33(14), WA 303(1)
8.596 Schl¨afli’s polynomials Sn(z). These are the functions that satisfy the formulas
1. S0(z)=0 EH II 34(18), WA 312(2)
2. Sn(z)=1
n/bracketleftbigg
2zOn(z)−2/parenleftBig
cosnπ
2/parenrightBig2/bracketrightbigg
[n≥1] EH II 34(19), WA 312(3)
=⌊n
2⌋/summationdisplay
m=0(n−m−1)!
m!/parenleftBigz
2/parenrightBig2m−n[n≥1] EH II 34(18)
3. S−n(z)=(−1)n+1Sn(z) WA 313(6)
8.597 Functional relations:
1. Sn−1(z)+Sn+1(z)=4On(z) WA 313(7)
Other functional relations may be obtained from 8.591 by replacing On(z) with the expression for Sn(z)
given by 8.596 2.
8.6 Mathieu Functions
8.60 Mathieu’s equation
d2y
dz2+/parenleftbig
a−2k2cos2z/parenrightbig
y=0,k2=q MA
8.621 Recursion relations for the coefficients A(2n)
2r,A(2n+1)
2r+1,B(2n+1)
2r+1,B(2n+2)
2r+2 951
8.61 Periodic Mathieu functions
8.610 In general, Mathieu’s equation 8.60does not have periodic solutions. If kis a real number, there
exist infinitely many eigenvalues a , not identically equal to zero, corresponding to the periodic solutions
y(z)=y(2π+z).
Ifkis nonzero, there are no other linearly independent periodic solutions. Periodic solutions of Mathieu’s
equations are called Mathieu’s periodic functions orMathieu functions of the first kind , or, more simply,
Mathieu functions .
8.611 Mathieu’s equation has four series of distinct periodic solutions:
1. ce 2n(z,q)=∞/summationdisplay
r=0A(2n)
2rcos2rz MA
2. ce 2n+1(z,q)=∞/summationdisplay
r=0A(2n+1)
2r+1cos(2r+1 )z MA
3. se 2n+1(z,q)=∞/summationdisplay
r=0B(2n+1)
2r+1sin(2r+1 )z MA
4. se 2n+2(z,q)=∞/summationdisplay
r=0B(2n+2)
2r+2sin(2r+2 )z MA
5. The coefficients AandBdepend on q. The eigenvalues aof the functions ce 2n,c e2n+1,s e2n,
se2n+1are denoted by a2n,a2n+1,b2n,b2n+1.
8.612 The solutions of Mathieu’s equation are normalized so that/integraldisplay2π
0y2dx=π MO 65
8.613
1. lim
q→0ce0(x)=1√
2
2. lim
q→0cen(x) = cos nx [n/negationslash=0 ]
3. lim
q→0sen(x)=s i n nx MO 65
8.62 Recursion relations for the coefficients A(2n)
2r,A(2n+1)
2r+1,B(2n+1)
2r+1,B(2n+2)
2r+2
8.621
1. aA(2n)
0−qA(2n)
2=0 MA
2. ( a−4)A(2n)
2−q/parenleftBig
A(2n)
4+2A(2n)
0/parenrightBig
=0 MA
3./parenleftbig
a−4r2/parenrightbig
A(2n)
2r−q/parenleftBig
A(2n)
2r+2+A(2n)
2r−2/parenrightBig
=0 [ r≥2] MA
952 Mathieu Functions 8.622
8.622
1. ( a−1−q)A(2n+1)
1 −qA(2n+1)
3 =0 MA
2./bracketleftbig
a−(2r+1 )2/bracketrightbig
A(2n+1)
2r+1−q/parenleftBig
A(2n+1)
2r+3+A(2n+1)
2r−1/parenrightBig
=0 [ r≥1] MA
8.623
1. ( a−1+q)B(2n+1)
1 −qB(2n+1)
3 =0 MA
2./bracketleftbig
a−(2r+1 )2/bracketrightbig
B(2n+1)
2r+1−q/parenleftBig
B(2n+1)
2r+3+B(2n+1)
2r−1/parenrightBig
=0
[r≥1] MA
8.624
1. ( a−4)B(2n+2)
2 −qB(2n+2)
4 =0 MA
2.11/parenleftbig
a−4r2/parenrightbig
B(2n+2)
2r−q/parenleftBig
B(2n+2)
2r+2+B(2n+2)
2r−2/parenrightBig
=0 [ r≥2] MA
8.625 We can determine the coefficients AandBfrom equations 8.612 ,8.613 and8.621 -8.624 pro-
vided ais known. Suppose, for example, that we need to determine the coefficients A(2n)
2rfor the function
ce2n(z,q). From the recursion formulas, we have
1./vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea−q 000 ...
−2qa−4−q 00 ...
0−qa −16 −q 0 ...
00 −qa −36 −q
00 0 −qa −64
............/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=0
ST
For given qin equation 8.625 1, we may determine the eigenvalues
2. a=A0,A2,A4,... [|A0|≤|A2|≤|A4|≤...]
If we now set a=A2n, we can determine the coefficients A(2n)
2rfrom the recursion formulas 8.621
up to a proportionality coefficient. This coefficient is determined from the formula
3. 2/bracketleftBig
A(2n)
0/bracketrightBig2
+∞/summationdisplay
r=1/bracketleftBig
A(2n)
2r/bracketrightBig2
=1, MA
which follows from the conditions of normalization.
8.63 Mathieu functions with a purely imaginary argument
8.630 If, in equation 8.60, we replace zwithiz, we arrive at the differential equation
1.11d2y
dz2+(−a+2qcosh2 z)y=0
We can find the solutions of this equation if we replace the argument zwithizin the functions ce n(z,q)
and se n(z,q). The functions obtained in this way are called associated Mathieu functions of the first kind
and are denoted as follows:
8.652 Mathieu functions for negative q 953
1. Ce 2n(z,q),Ce2n+1(z,q),Se2n+1(z,q),Se2n+2(z,q)
8.631
1. Ce 2n(z,q)=∞/summationdisplay
r=0A(2n)
2rcosh2 rz MA
2. Ce 2n+1(z,q)=∞/summationdisplay
r=0A(2n+1)
2r+1cosh(2 r+1 )z MA
3. Se 2n+1(z,q)=∞/summationdisplay
r=0B(2n+1)
2r+1sinh(2 r+1 )z MA
4. Se 2n+2(z,q)=∞/summationdisplay
r=0B(2n+2)
2r+2sinh(2 r+2 )z MA
8.64 Non-periodic solutions of Mathieu’s equation
Along with each periodic solution of equation 8.60, there exists a second non-periodic solution that is
linearly independent. The non-periodic solutions are denoted as follows:
fe2n(z,q),fe2n+1(z,q),ge2n+1(z,q),ge2n+2(z,q).
Analogously, the second solutions of equation 8.630 1 are denoted by
Fe2n(z,q),Fe2n+1(z,q),Ge2n+1(z,q),Ge2n+2(z,q).
8.65 Mathieu functions for negative q
8.651 If we replace the argument zin equation 8.60with±/parenleftBigπ
2±z/parenrightBig
, we get the equation
d2y
dz2+(a+2qcos 2z)y=0. MA
This equation has the following solutions:
8.652
1. ce 2n(z,−q)=(−1)nce2n/parenleftbig1
2π−z,q/parenrightbig
MA
2. ce 2n+1(z,−q)=(−1)nse2n+1/parenleftbig1
2π−z,q/parenrightbig
MA
3. se 2n+1(z,−q)=(−1)nce2n+1/parenleftbig1
2π−z,q/parenrightbig
MA
4. se 2n+2(z,−q)=(−1)nse2n+2/parenleftbig1
2π−z,q/parenrightbig
MA
5. fe 2n(z,−q)=(−1)n+1fe2n/parenleftbig1
2π−z,q/parenrightbig
MA
6. fe 2n+1(z,−q)=(−1)nge2n+1/parenleftbig1
2π−z,q/parenrightbig
MA
7. ge2n+1(z,−q)=(−1)nfe2n+1/parenleftbig1
2π−z,q/parenrightbig
MA
8. ge2n+2(z,−q)=(−1)nge2n+2/parenleftbig1
2π−z,q/parenrightbig
MA
954 Mathieu Functions 8.653
8.653 Analogously, if we replace zwithπ
2i+zin equation 8.630 1, we get the equation
d2y
dz2−(a+2qcoshz)y=0.
It has the following solutions:
8.654
1. Ce 2n(z,−q)=(−1)nCe2n/parenleftBigπ
2i+z,q/parenrightBig
MA
2. Ce 2n+1(z,−q)=(−1)n+1iSe2n+1/parenleftbig1
2πi+z,q/parenrightbig
MA
3. Se 2n+1(z,−q)=(−1)n+1iCe2n+1/parenleftbig1
2πi+z,q/parenrightbig
MA
4. Se 2n+2(z,−q)=(−1)n+1Se2n+2/parenleftbig1
2πi+z,q/parenrightbig
MA
5. Fe 2n(z,−q)=(−1)nFe2n/parenleftbig1
2πi+z,q/parenrightbig
MA
6.11Fe2n+1(z,−q)=(−1)n+1iGe2n+1/parenleftbig1
2πi+z,q/parenrightbig
MA
7.11Ge2n+1(z,−q)=(−1)n+1iFe2n+1/parenleftbig1
2πi+z,q/parenrightbig
MA
8.11Ge2n+2(z,−q)=(−1)n+1Ge2n+2/parenleftbig1
2πi+z,q/parenrightbig
MA
8.66 Representation of Mathieu functions as series of Bessel functions
8.661
1. ce 2n(z,q)=ce2n/parenleftbigπ
2,q/parenrightbig
A(2n)
0∞/summationdisplay
r=0(−1)rA(2n)
2rJ2r(2kcosz) MA
=ce2n(0,q)
A(2n)
0∞/summationdisplay
r=0(−1)rA(2n)
2rI2r(2ksinz) MA
2. ce 2n+1(z,q)=−ce/prime
2n+1/parenleftbigπ
2,q/parenrightbig
kA(2n+1)
1∞/summationdisplay
r=0(−1)rA(2n+1)
2r+1J2r+1(2kcosz) MA
=ce2n+1(0,q)
kA1(2n+1 )cotz∞/summationdisplay
r=0(−1)r(2r+1 )A(2n+1)
2r+1I2r+1(2ksinz) MA
3. se 2n+1(z,q)=se2n+1/parenleftbigπ
2,q/parenrightbig
kB(2n+1)
1tanz∞/summationdisplay
r=0(−1)r(2r+1 )B(2n+1)
2r+1J2r+1(2kcosz) MA
=se/prime
2n+1(0,q)
kB(2n+1)
1∞/summationdisplay
r=0(−1)rB(2n+1)
2r+1I2r+1(2ksinz) MA
4. se 2n+2(z,q)=−se/prime
2n+2/parenleftbigπ
2,q/parenrightbig
k2B(2n+2)
2tanz∞/summationdisplay
r=0(−1)r(2r+2 )B(2n+2)
2r+2J2r+2(2kcosz) MA
=se/prime
2n+2(0,q)
k2B(2n+2)
2cotz∞/summationdisplay
r=0(−1)r(2r+2 )B(2n+2)
2r+2I2r+2(2ksinz) MA
8.662
1. fe 2n(z,q)=−πfe/prime
2n(0,q)
2c e2n/parenleftbigπ
2,q/parenrightbig∞/summationdisplay
r=0(−1)rA(2n)
2rIm/bracketleftbig
Jr/parenleftbig
keiz/parenrightbig
Yr/parenleftbig
ke−iz/parenrightbig/bracketrightbig
MA
8.663 Representation of Mathieu functions 955
2. fe 2n+1(z,q)=πkfe/prime
2n+1(0,q)
2c e/prime
2n+1/parenleftbigπ
2,q/parenrightbig
×∞/summationdisplay
r=0(−1)rA(2n+1)
2r+1Im/bracketleftbig
Jr/parenleftbig
keiz/parenrightbig
Yr+1/parenleftbig
ke−iz/parenrightbig
+Jr+1/parenleftbig
keiz/parenrightbig
Yr/parenleftbig
ke−iz/parenrightbig/bracketrightbig
MA
3. ge2n+1(z,q)=−πkge2n+1(0,q)
2s e2n+1/parenleftbigπ
2,q/parenrightbig
×∞/summationdisplay
r=0(−1)rB(2n+1)
2r+1Re/bracketleftbig
Jr/parenleftbig
keiz/parenrightbig
Yr+1/parenleftbig
ke−iz/parenrightbig
−Jr+1/parenleftbig
keiz/parenrightbig
Yr/parenleftbig
ke−iz/parenrightbig/bracketrightbig
MA
4. ge2n+2(z,q)=−πk2ge2n+2(0,q)
2s e/prime
2n+2/parenleftbig1
2π,q/parenrightbig
×∞/summationdisplay
r=0(−1)rRe/bracketleftbig
Jk/parenleftbig
keiz/parenrightbig
Yr+2/parenleftbig
ke−iz/parenrightbig
−Jr+2/parenleftbig
keiz/parenrightbig
Yr/parenleftbig
ke−iz/parenrightbig/bracketrightbig
MA
The expansions of the functions Fe nand Ge nas series of the functions Yνare denoted, respectively,
by Feynand Geyn, and the expansions of these functions as series of the functions Kνare denoted,
respectively, by Fek nand Gek n.
8.663
1. Fey2n(z,q)=ce2n(0,q)
A(2n)
0∞/summationdisplay
r=0A(2n)
2rY2r(2ksinhz)
k2=q[|sinhz|>1,Rez>0]
MA
=ce2n/parenleftbigπ
2,q/parenrightbig
A(2n)
0∞/summationdisplay
r=0(−1)rA(2n)
2rY2r(2kcoshz)
[|coshz|>1]
MA
=ce2n(0,q)ce2n/parenleftbigπ
2,q/parenrightbig
/bracketleftBig
A(2n)
0/bracketrightBig2∞/summationdisplay
r=0(−1)rA(2n)
2rJr/parenleftbig
ke−z/parenrightbig
Yr(kez)
MA
956 Mathieu Functions 8.663
2. Fey2n+1(z,q)=ce2n+1(0,q)coth z
kA1(2n+1 )∞/summationdisplay
r=0(2r+1 )A(2n+1)
2r+1Y2r+1(2ksinhz),
k2=q,[|sinhz|>1,Rez>0]
MA
=−ce/prime
2n+1/parenleftbigπ
2,q/parenrightbig
kA(2n+1)
1∞/summationdisplay
r=0(−1)rA(2n+1)
2r+1Y2r+1(2kcoshz)
[|coshz|>1]
MA
=−ce2n+1(0,q)ce/prime
2n+1/parenleftbigπ
2,q/parenrightbig
k/bracketleftBig
A(2n+1)
1/bracketrightBig2
×∞/summationdisplay
r=0(−1)rA(2n+1)
2r+1/bracketleftbig
Jr/parenleftbig
ke−z/parenrightbig
Yr+1(kez)+Jr+1/parenleftbig
ke−z/parenrightbig
Yr(kez)/bracketrightbig
MA
3. Gey2n+1(z,q)=se/prime
2n+1(0,q)
kB(2n+1)
1∞/summationdisplay
r=0B(2n+1)
2r+1Y2r+1(2ksinhz)
[|sinhz|>1,Rez>0]
MA
=se2n+1/parenleftbigπ
2,q/parenrightbig
kB(2n+1)
1tanhz∞/summationdisplay
r=0(−1)r(2r+1 )B(2n+1)
2r+1Y2r+1(2kcoshz)
[|coshz|>1]
MA
=se2n+1(0,q)se2n+1/parenleftbigπ
2,q/parenrightbig
k/bracketleftBig
B(2n+1)
1/bracketrightBig2∞/summationdisplay
r=0(−1)rB(2n+1)
2r+1
×/bracketleftbig
Jr/parenleftbig
ke−z/parenrightbig
Yr+1(kez)/bracketrightbig
Jr+1/parenleftbig
ke−z/parenrightbig
Yr(kez)
MA
8.671 The general theory 957
4. Gey2n+2(z,q)=se/prime
2n+2(0,q)
k2B(2n+2)
2cothz∞/summationdisplay
r=0(2r+2 )B(2n+2)
2r+2Y2r+2(2ksinhz)
[|sinhz|>1,Rez>0]
MA
=−se/prime
2n+2/parenleftbigπ
2,q/parenrightbig
k2B(2n+2)
2tanhz∞/summationdisplay
r=0(−1)r(2r+2 )B(2n+2)
2r+2Y2r+2(2kcoshz)
[|coshz|>1]
MA
=se/prime
2n+2(0,q)se/prime
2n+2(π
2,q)
k2/bracketleftBig
B(2n+2)
2/bracketrightBig2∞/summationdisplay
r=0(−1)rB(2n+2)
2r+2
×/bracketleftbig
Jr/parenleftbig
ke−z/parenrightbig
Yr+2(kez)/bracketrightbig
−Jr+2/parenleftbig
ke−z/parenrightbig
Yr(kez)
MA
8.664
1. Fek 2n(z,q)=ce2n(0,q)
πA(2n)
0∞/summationdisplay
r=0(−1)rA(2n)
2rK2r(−2iksinhz)
k2=q, [|sinhz|>1,Rez>0]
MA
2. Fek 2n+1(z,q)=ce2n+1(0,q)
πkA(2n+1)
1cothz∞/summationdisplay
r=0(−1)r(2r+1 )A(2n+1)
2r+1K2r+1(−2iksinhz)
k2=q [|sinhz|>1,Rez>0]
MA
3. Gek 2n+1(z,q)=se2n+1/parenleftbigπ
2,q/parenrightbig
πkB(2n+1)
1tanhz∞/summationdisplay
r=0(2r+1 )B(2n+1)
2r+1K2r+1(−2ikcoshz) MA
4. Gek 2n+2(z,q)=se/prime
2n+2/parenleftbigπ
2,q/parenrightbig
πk2B(2n+2)
2tanhz∞/summationdisplay
r=0(2r+2 )B(2n+2)
2r+2K2r+2(−2ikcoshz) MA
8.67 The general theory
Ifiμis not an integer, the general solution of equation 8.60can be found in the form
8.671
1. y=Aeμz∞/summationdisplay
r=−∞c2re2rzi+Be−μz∞/summationdisplay
r=−∞c2re−2rziMA
The coefficients c2rcan be determined from the homogeneous system of linear algebraic equations
2.11c2r+ξ2r(c2r+2+c2r−2)=0,r =...,−2,−1,0,1,2,..., MA
where
958 Associated Legendre Functions 8.700
ξ2r=q
(2r−iμ)2−a
The condition that this system be compatible yields an equation that μmust satisfy:
3.7Δ(iμ)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle· ···· · · · ·
·ξ
−41ξ−400 0 0 ·
·0ξ−21ξ−200 0 ·
·00 ξ01ξ000 ·
·000 ξ21ξ20·
· ···· · · · ·/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=0
MA
This equation can also be written in the form
4. cosh μπ=1−2Δ(0)sin2/parenleftbiggπ√a
2/parenrightbigg
, where Δ(0) is the value that is assumed by the determinant
of the preceding article if we set μ= 0 in the expressions for ξ2r.
5. If the pair ( a,q) is such that |coshμπ|<1,thenμ=iβ,Imβ=0,and the solution 8.671 1i s
bounded on the real axis.
6. If |coshμπ|>1,μmay be real or complex, and the solution 8.671 1 will not be bounded on the
real axis.
7. If cosh μπ=±1, then iμwill be an integer. In this case, one of the solutions will be of period
πor 2π(depending on whether nis even or odd). The second solution is non-periodic (see 8.61
and8.64).
8.7–8.8 Associated Legendre Functions
8.70 Introduction
8.700 Anassociated Legendre function is a solution of the differential equation
1./parenleftbig
1−z2/parenrightbigd2u
dz2−2zdu
dz+/bracketleftbigg
ν(ν+1 )−μ2
1−z2/bracketrightbigg
u=0,
in which νandμare arbitrary complex constants.
This equation is a special case of (Riemann’s) hypergeometric equation (see 9.151 ). The points
+1,−1,∞
are, in general, its singular points , specifically, its ordinary branch points.
We are interested, on the one hand, in solutions of the equation that correspond to real values of the
independent variable zthat lie in the interval [ −1,1] and, on the other hand, in solutions corresponding
to an arbitrary complex number zsuch that Re z>1. These are multiple-valued in the z-plane. To
separate these functions into single-valued branches, we make a cut along the real axis from −∞to +1.
We are also interested in those solutions of equation 8.700 1 for which νorμor both are integers. Of
special significance is the case in which μ=0 .
8.701 In connection with this, we shall use the following notations:
The letter zwill denote an arbitrary complex variable ; the letter xwill denote a realvariable that
varies over the interval [ −1,+1]. We shall sometimes set x=c o s ϕ,where ϕis a real number.
We shall use the symbols Pμ
ν(z),Qμ
ν(z) to denote those solutions of equation 8.700 1 that are single-
valued and regular for |z|<1 and, in particular, uniquely determined for z=x.
8.706 Introduction 959
We shall use the symbols Pμ
ν(z),Qμ
ν(z) to denote those solutions of equation 8.700 1 that are single-
valued and regular forRe z>1. When these functions cannot be unrestrictedly extended without violating
their single-valuedness, we make a cut along the real axis to the left of the point z= 1. The values of the
functions Pμ
ν(z)a n dQμ
ν(z) on the upper and lower boundaries of that portion of the cuts lying between
the points −1 and +1 are denoted, respectively, by
Pμ
ν(x±i0),Qμ
ν(x±i0).
The letters nandmdenote natural numbers or zero. The letters νandμdenote arbitrary complex
numbers unless the contrary is stated.
The upper index will be omitted when it is equal to zero. That is, we set
P0
ν(z)=Pν(z),Q0
ν(z)=Qν(z)
Thelinearly independent functions
8.702 Pμ
ν(z)=1
Γ(1−μ)/parenleftbiggz+1
z−1/parenrightbiggμ
2
F/parenleftbigg
−ν,ν+1 ; 1 −μ;1−z
2/parenrightbigg
/bracketleftbigg
argz+1
z−1=0,ifzis real and greater than 1 and/bracketrightbigg
MO 80, WH
8.703 Qμ
ν(z)=eμπiΓ(ν+μ+1 )Γ/parenleftbig1
2/parenrightbig
2ν+1Γ/parenleftbig
ν+3
2/parenrightbig/parenleftbig
z2−1/parenrightbigμ
2z−ν−μ−1F/parenleftbiggν+μ+2
2,ν+μ+1
2;ν+3
2;1
z2/parenrightbigg
[arg/parenleftbig
z2−1/parenrightbig
=0w h e n zis real and greater than 1; arg z=0w h e n zis real and greater than zero] which
are solutions of the differential equation 8.700 1, are called associated Legendre functions (orspherical
functions )o fthe first andsecond kinds , respectively. They are uniquely defined, respectively, in the
intervals |1−z|<2a n d |z|>1, with the portion of the real axis that lies between −∞and +1 excluded.
They can be extended by means of hypergeometric series to the entire z-plane where the above-mentioned
cut was made. These expressions for Pμ
ν(z)a n dQμ
ν(z) lose their meaning when 1 −μandν+3
2are non-
positive integers, respectively. MO 80
When zi sar e a ln u m b e rl y i n go nt h ei n t e r v a l[ −1,+1], so that ( z=x=c o s ϕ), we take the following
functions as linearly independent solutions of the equation:
8.704 Pμ
ν(x)=1
2/bracketleftBig
e1
2μπiPμ
ν(cosϕ+i0) +e−1
2μπiPμ
ν(cosϕ−i0)/bracketrightBig
EH I 143(1)
=1
Γ(1−μ)/parenleftbigg1+x
1−x/parenrightbiggμ
2
F/parenleftbigg
−ν,ν+1 ;1 −μ;1−x
2/parenrightbigg
EH I 143(6)
8.705 Qμ
ν(x)=1
2e−μπi/bracketleftBig
e−1
2μπiQμ
ν(x+i0) +e1
2μπiQμ
ν(x−i0)/bracketrightBig
EH I 143(2)
=π
2s inμπ/bracketleftbigg
Pμ
ν(x)cosμπ−Γ(ν+μ+1 )
Γ(ν−μ+1 )P−μ
ν(x)/bracketrightbigg
(cf.8.732 5)
Ifμ=±mis an integer, the last equation loses its meaning. In this case, we get the following formulas
by passing to the limit:
8.706
1. Qm
ν(x)=(−1)m/parenleftbig
1−x2/parenrightbigm
2dm
dxmQν(x) (cf. 8.752 1) EH I 149(7)
2.11Q−m
ν(x)=Γ(ν−m+1 )
Γ(ν+m+1 )Qm
ν(x) EH I 144(18)
The functions Qμ
ν(z) are not defined when ν+μis equal to a negative integer. Therefore, we must
exclude the cases when ν+μ=−1,−2,−3,...for these formulas.
The functions
960 Associated Legendre Functions 8.707
P±μ
ν(±z),Q±μ
ν(±z),P±μ
−ν−1(±z),Q±μ
−ν−1(±z)
arelinearly independent solutions of the differential equation for ν+μ/negationslash=0,±1,±2,....
8.707 Nonetheless, two linearly independent solutions can always be found. Specifically, for ν±μnot
an integer, the differential equation 8.700 1 has the following solutions:
1. P±μ
ν(±z),Q±μ
ν(±z),P±μ
−ν−1(±z),Q±μ
−ν−1(±z)
respectively, for z=x=c o s ϕ,
2. P±μ
ν(±x),Q±μ
ν(±x),P±μ
−ν−1(±x),Q±μ
−ν−1(±x).
Ifν±μis not an integer, the solutions
3. Pμ
ν(z),Qμ
ν(z),respectively, and Pμ
ν(x),Qμ
ν(x)
are linearly independent. If ν±μis an integer but μitself is not an integer, the following functions
are linearly independent solutions of equation 8.700 1:
4. Pμ
ν(z),P−μ
ν(z),respectively, and Pμ
ν(x),P−μ
ν(x).
Ifμ=±m, ν=n,o rν=−n−1, the following functions are linearly independent solutions of
equation 8.700 1f o rn≥m:
5. Pm
n(z),Qm
n(z),respectively, and Pm
n(x),Qm
n(x),
and for n<m, the following functions will be linearly independent solutions
6. P−m
n(z),Qm
n(z),respectively, and P−m
n(x),Qm
n(x).
8.71 Integral representations
8.711
1. P−μ
ν(z)=/parenleftbig
z2−1/parenrightbigμ
2
2μ√πΓ/parenleftbig
μ+1
2/parenrightbig/integraldisplay1
−1/parenleftbig
1−t2/parenrightbigμ−1
2
/parenleftbig
z+t√
z2−1/parenrightbigμ−νdt/bracketleftbig
Reμ>−1
2,|arg (z±1)|<π/bracketrightbig
MO 88
2. Pm
ν(z)=(ν+1 ) (ν+2 )...(ν+m)
π/integraldisplayπ
0/bracketleftBig
z+/radicalbig
z2−1c osϕ/bracketrightBigν
cosmϕdϕ
=(−1)mν(ν−1)...(ν−m+1 )
π/integraldisplayπ
0cosmϕdϕ
/bracketleftbig
z+√
z2−1c osϕ/bracketrightbigν+1
/bracketleftBig
|argz|<π
2,arg/parenleftBig
z+/radicalbig
z2−1c osϕ/parenrightBig
=a r g zforϕ=π
2/bracketrightBig
(cf.8.822 1)SM 483(15), WH
3. Qμ
ν(z)=√πeμπiΓ(ν+μ+1 )
2μΓ/parenleftbig
μ+1
2/parenrightbig
Γ(ν−μ+1 )/parenleftbig
z2−1/parenrightbigμ
2/integraldisplay∞
0sinh2μtd t
/parenleftbig
z+√
z2−1c os h t/parenrightbigν+μ+1
[Re (ν±μ)>−1,|arg (z±1)|<π] (cf. 8.822 2)MO 88
4. Qμ
ν(z)=eμπiΓ(ν+1 )
Γ(ν−μ+1 )/integraldisplay∞
0coshμt dt
/parenleftbig
z+√
z2−1c os h t/parenrightbigν+1
[Re(ν+μ)>−1,ν/negationslash=−1,−2,−3,..., |arg (z±1)|<π]W H ,M O8 8
8.714 Integral representations 961
5./integraldisplay1
−1P2
l(x)P0
l(x)dx=−l!
(l−2)!1
2l+1=−l(l−1)
2l+1
8.712 Qμ
ν(z)=eμπiΓ(ν+μ+1 )
2ν+1Γ(ν+1 )/parenleftbig
z2−1/parenrightbig−μ
2/integraldisplay1
−1/parenleftbig
1−t2/parenrightbigν(z−t)−ν−μ−1dt
[Re(ν+μ)>−1,Reμ>−1,|arg (z±1)|<π] (cf. 8.821 2)MO 88a, EH I 155(5)a
8.713
1. Qμ
ν(z)=eμπiΓ/parenleftbigg
μ+1
2/parenrightbigg
√
2π/parenleftbig
z2−1/parenrightbigμ
2/braceleftBigg/integraldisplayπ
0cos/parenleftbig
ν+1
2/parenrightbig
td t
(z−cost)μ+1
2−cosνπ/integraldisplay∞
0e−(ν+1
2)tdt
(z+c o s h t)μ+1
2/bracerightBigg
/bracketleftbig
Reμ>−1
2,Re(ν+μ)>−1,|arg (z±1)|<π/bracketrightbig
MO 89
2. P−μ
ν(z)=/parenleftbig
z2−1/parenrightbigμ
2
2νΓ(μ−ν)Γ(ν+1 )/integraldisplay∞
0sinh2ν+1t
(z+c o s h t)ν+μ+1dt
[Rez>−1,|arg (z±1)|<π , Re(ν+1 )>0,Re(μ−ν)>0]MO 89
3. P−μ
ν(z)=/radicalbigg
2
πΓ/parenleftbig
μ+1
2/parenrightbig/parenleftbig
z2−1/parenrightbigμ
2
Γ(ν+μ+1 )Γ ( μ−ν)/integraldisplay∞
0cosh/parenleftbig
ν+1
2/parenrightbig
td t
(z+c o s h t)μ+1
2
[Rez>−1,|arg (z±1)|<π , Re(ν+μ)>−1,Re(μ−ν)>0]MO 89
8.714
1. Pμ
ν(cosϕ)=/radicalbigg
2
πsinμϕ
Γ/parenleftbig1
2−μ/parenrightbig/integraldisplayϕ
0cos/parenleftbig
ν+1
2/parenrightbig
td t
(cost−cosϕ)μ+1
2/bracketleftbig
0<ϕ<π , Reμ<1
2/bracketrightbig
; (cf. 8.823 )
MO 87
2. P−μ
ν(cosϕ)=Γ(2μ+1 )s i nμϕ
2μΓ(μ+1 )Γ ( ν+μ+1 )Γ ( μ−ν)/integraldisplay∞
0tν+μdt
(1 + 2 tcosϕ+t2)μ+1
2
[Re(ν+μ)>−1,Re(μ−ν)>0]
MO 89
3. Qμ
ν(cosϕ)=1
2μ+1Γ(ν+μ+1 )
Γ(ν−μ+1 )sinμϕ
Γ/parenleftbig
μ+1
2/parenrightbig
×/integraldisplay∞
0/bracketleftBigg
sinh2μt
(cosϕ+isinϕcosht)ν+μ+1+sinh2μt
(cosϕ−isinϕcosht)ν+μ+1/bracketrightBigg
dt
/bracketleftbig
Re(ν+μ+1 )>0,Re(ν−μ+1 )>0,Reμ>−1
2/bracketrightbig
MO 89
4. Pμ
ν(cosϕ)=i
2μΓ(ν+μ+1 )
Γ(ν−μ+1 )sinμϕ
Γ/parenleftbig
μ+1
2/parenrightbig
×/integraldisplay∞
0/bracketleftBigg
sinh2μt
(cosϕ+isinϕcosht)ν+μ+1−sinh2μt
(cosϕ−isinϕcosht)ν+μ+1/bracketrightBigg
dt
/bracketleftbig
Re (ν±μ+1 )>0,Reμ>−1
2/bracketrightbig
MO 89
962 Associated Legendre Functions 8.715
8.715
1. Pμ
ν(coshα)=√
2s in hμα√πΓ/parenleftbig1
2−μ/parenrightbig/integraldisplayα
0cosh/parenleftbig
ν+1
2/parenrightbig
td t
(coshα−cosht)μ+1
2
/bracketleftbig
α>0,Reμ<1
2/bracketrightbig
MO 87
2. Qμ
ν(coshα)=/radicalbiggπ
2eμπisinhμα
Γ/parenleftbig1
2−μ/parenrightbig/integraldisplay∞
αe−(ν+1
2)tdt
(cosht−coshα)μ+1
2/bracketleftbig
α>0,Reμ<1
2,Re(ν+μ)>−1/bracketrightbig
MO 87
See also 3.277 1, 4, 5, 7, 3.318 ,3.516 3,3.518 1, 2,3.542 2,3.663 1,3.894 ,3.988 3,6.622 3,6.628 1,
4–7, and also 8.742 .
8.72 Asymptotic series for large values of |ν|
8.7216For real values of μ,|ν|/greatermuch1,|ν|/greatermuch|μ|,|argν|<π,w eh a v e :
1. Pμ
ν(cosϕ)=2√πΓ(ν+μ+1 )∞/summationdisplay
k=0Γ/parenleftbig
μ+k+1
2/parenrightbig
Γ/parenleftbig
μ−k+1
2/parenrightbigcos/bracketleftBig/parenleftbig
ν+k+1
2/parenrightbig
ϕ+π
4(2k−1) +μπ
2/bracketrightBig
k!Γ/parenleftbig
ν+k+3
2/parenrightbig
(2 sinϕ)k+1
2/bracketleftbigg
ν+μ/negationslash=−1,−2,−3,...;ν/negationslash=−3
2,−5
2,7
2...;f o rπ
6<ϕ<5π
6
This series also converges for complex values of νandμ.
In the remaining cases, it is an asymptotic expansion for
|ν|/greatermuch|μ|,|ν|/greatermuch1,ifν>0,μ>0a n d0 <ε≤ϕ≤π−ε/bracketrightbigg
MO 92
2.6Qμ
ν(cosϕ)=√πΓ(ν+μ+1 )
×∞/summationdisplay
k=0(−1)kΓ/parenleftbig
μ+k+1
2/parenrightbig
Γ/parenleftbig
μ−k+1
2/parenrightbigcos/bracketleftBig/parenleftbig
ν+k+1
2/parenrightbig
ϕ−π
4(2k−1) +μπ
2/bracketrightBig
k!Γ/parenleftbig
ν+k+3
2/parenrightbig
(2sinϕ)k+1
2/bracketleftbigg
ν+μ/negationslash=−1,−2,−3,...;ν/negationslash=−3
2,−5
2,−7
2,...;f o rπ
6<ϕ<5
6π
This series also converges for complex values of νandμ.
In the remaining cases, it is an asymptotic expansion for
|ν|/greatermuch|μ|,|ν|/greatermuch1,ifν>0,μ > 0,0<ε≤ϕ≤π−ϕ/bracketrightbigg
EH I 147(6), MO 92
3. Pμ
ν(cosϕ)=2√πΓ(ν+μ+1 )
Γ/parenleftbig
ν+3
2/parenrightbigcos/bracketleftBig/parenleftbig
ν+1
2/parenrightbig
ϕ−π
4+μπ
2/bracketrightBig
√2s inϕ/bracketleftbigg
1+O/parenleftbigg1
ν/parenrightbigg/bracketrightbigg
/bracketleftbigg
0<ε≤ϕ≤π−ε,|ν|/greatermuch1
ε/bracketrightbigg
MO 92
Forν>0,μ>0a n d ν>μ, it follows from formulas 8.721 1a n d8.721 2t h a t
8.724 Asymptotic series 963
4. ν−μPμ
ν(cosϕ)=/radicalbigg2
νπsinϕcos/bracketleftbigg/parenleftbigg
ν+1
2/parenrightbigg
ϕ−π
4+μπ
2/bracketrightbigg
+O/parenleftbigg1√
ν3/parenrightbigg
5. ν−μQμ
ν(cosϕ)=/radicalbiggπ
2νsinϕcos/bracketleftbigg/parenleftbigg
ν+1
2/parenrightbigg
ϕ+π
4+μπ
2/bracketrightbigg
O/parenleftbigg1√
ν3/parenrightbigg
/bracketleftbigg
0<ε≤ϕ≤π−ε;ν/greatermuch1
ε/bracketrightbigg
MO 92
8.722 Ifϕis sufficiently close to 0 or πthatνϕorν(π−ϕ) is small in comparison with 1, the asymptotic
formulas 8.721 become unsuitable. In this case, the following asymptotic representation is applicable for
μ≤0,ν/greatermuch1, and small values of ϕ:
1./bracketleftbigg/parenleftbigg
ν+1
2/parenrightbigg
cosϕ
2/bracketrightbiggμ
P−μ
ν(cosϕ)=Jμ(η)+sin2ϕ
2/bracketleftbiggJμ+1(η)
2η−Jμ+2(η)+η
6Jμ+3(η)/bracketrightbigg
+O/parenleftBig
sin4ϕ
2/parenrightBig
where η=( 2ν+1 )s i nϕ
2. In particular, it follows that
1. lim
ν→∞νμP−μ
ν/parenleftBig
cosx
ν/parenrightBig
=Jμ(x)[ x≥0,μ≥0] MO 93
8.723 We can see how the functions Pμ
ν(z)a n dQμ
ν(z) behave for large |ν|and real values of z>3
2√
2:
1. Pμ
ν(coshα)=2μ
√π/braceleftBigg
Γ/parenleftbig
−ν−1
2/parenrightbig
Γ(−ν−μ)e(μ−ν)αsinhμα
(e2α−1)μ+1
2F/parenleftbigg
μ+1
2,−μ+1
2;ν+3
2;1
1−e2α/parenrightbigg
+Γ/parenleftbig
ν+1
2/parenrightbig
Γ(ν−μ+1 )e(ν+μ+1)αsinhμα
(e2α−1)μ+1
2F/parenleftbigg
μ+1
2,−μ+1
2;−ν+1
2;1
1−e2α/parenrightbigg/bracerightBigg
/bracketleftbig
ν/negationslash=±1
2,±3
2,±5
2,...;a>1
2ln2/bracketrightbig
MO 94
2. Qμ
ν(coshα)=eμπi2μ√πΓ(ν+μ+1 )
Γ/parenleftbig
ν+3
2/parenrightbige−(ν+μ+1)α
(1−e−2α)μ+1
2sinhμα
×F/parenleftbigg
μ+1
2,−μ+1
2;ν+3
2;1
1−e2α/parenrightbigg
/bracketleftbig
μ+ν+1/negationslash=0,−1,−2,...;α>1
2ln2/bracketrightbig
MO 94
See also 8.776 .
8.724 For the inequalities in 8.776 1–4,νandμare arbitrary real numbers satisfying the inequalities
ν≥1,ν−μ+1>0, and μ≥0:
1./vextendsingle/vextendsingleP±μ
ν(cosϕ)/vextendsingle/vextendsingle</radicalbigg
8
νπΓ(ν±μ+1 )
Γ(ν+1 )1
sinμ+1
2ϕMO 91-92
2./vextendsingle/vextendsingleQ±μ
ν(cosϕ)/vextendsingle/vextendsingle</radicalbigg
2π
νΓ(ν±μ+1 )
Γ(ν+1 )1
sinμ+1
2ϕMO 91-92
3./vextendsingle/vextendsingleP±μ
ν(cosϕ)/vextendsingle/vextendsingle<2√νπΓ(ν±μ+1 )
Γ(ν+1 )1
sinμ+1
2ϕMO 91-92
964 Associated Legendre Functions 8.725
4./vextendsingle/vextendsingleQ±μ
ν(cosϕ)/vextendsingle/vextendsingle</radicalbiggπ
νΓ(ν±μ+1 )
Γ(ν+1 )1
sinμ+1
2ϕMO 91-92
5.8/vextendsingle/vextendsingle/vextendsingle/radicalbig
sinϕPm
n(cosϕ)/vextendsingle/vextendsingle/vextendsingle<Γ/parenleftbig
n+1
2/parenrightbig
Γ(n−m+1 )2(m+n)2/nsup
0<t<∞/vextendsingle/vextendsingle/vextendsingle√
tJm(t)/vextendsingle/vextendsingle/vextendsingle
[uniformly 0 ≤m≤n]
8.725 10For fixed zandνand Re μ→∞, with znot on the real axis between −∞and−1a n d+ ∞
and +1, the following are asymptotic expansions in which the upper and lower signs are taken accordingto whether Im zis greater than or less than 0:
1. P
μ
ν(z)=Γ(ν+μ+1 )Γ ( μ−ν)
πΓ(μ+1 )/parenleftbiggz+1
z−1/parenrightbigg1
2μ
sinμπ⎡
⎣F/parenleftbigg
−ν,ν+1 ;1+ μ;1
2+1
2z/parenrightbigg
−sinνπ
sinμπe∓iμπ/parenleftbiggz−1
z+1/parenrightbiggμ
F/parenleftbigg
−ν,ν+1 ;1+ μ;1
2−1
2z/parenrightbigg⎤
⎦
AS 8.10.1
2. Qμ
ν(z)=1
2eiμπΓ(ν+μ+1 )
Γ(μ+1 )/parenleftbiggz+1
z−1/parenrightbigg1
2μ
Γ(μ−ν)⎡
⎣F/parenleftbigg
−ν,ν+1 ;1+ μ;1
2+1
2z/parenrightbigg
−e∓iνπ/parenleftbiggz−1
z+1/parenrightbiggμ
F/parenleftbigg
−ν,ν+1 ;1+ μ;1
2−1
2z/parenrightbigg⎤
⎦
AS 8.10.2
3. Q−μ
ν(z)=e−iμπcosec[ π(ν−μ)]
2πΓ(1 + μ)⎡
⎣e∓iνπ/parenleftbiggz+1
z−1/parenrightbigg−1
2μ
F/parenleftbigg
−ν, ν+1 ;1+ μ;1
2−1
2z/parenrightbigg
−/parenleftbiggz−1
z+1/parenrightbigg−1
2μ
F/parenleftbigg
−ν, ν+1 ;1+ μ;1
2+1
2z/parenrightbigg⎤
⎦
AS 8.10.3
8.73–8.74 Functional relations
8.731
1./parenleftbig
z2−1/parenrightbigdPμ
ν(z)
dz=(ν−μ+1 )Pμ
ν+1(z)−(ν+1 )zPμ
ν(z)
(cf.8.832 1,8.914 2)
E HI1 6 1 ( 1 0 ) ,M O8 1
1(1)9/parenleftbig
z2−1/parenrightbigdPμ
ν(z)
dz=νzPμ
ν(z)−(ν+μ)Pμ
ν−1(z) AS 8.5.4
1(2)/parenleftbig
z2−1/parenrightbigdPμ
ν(z)
dz=(ν+μ)(ν−μ+1 )/radicalbig
z2−1Pμ−1
ν(z)−μzPμ
ν(z) AS 8.5.2
8.733 Functional relations 965
2. (2 ν+1 )zPμ
ν(z)=(ν−μ+1 )Pμ
ν+1(z)+(ν+μ)Pμ
ν−1(z)
(cf.8.832 2,8.914 1)
EH I 160(2), MO 81
3. Pμ+2
ν(z)+2 ( μ+1 )z√
z2−1Pμ+1
ν(z)=(ν−μ)(ν+μ+1 )Pμ
ν(z) MO 82, EH I 160(1)
3(1)9Pμ+1
ν(z)=/parenleftbig
z2−1/parenrightbig−1/2/bracketleftbig
(ν−μ)zPμ
ν(z)−(ν+μ)Pμ
ν−1(z)/bracketrightbig
AS 8.5.1
4. Pμ
ν+1(z)−Pμ
ν−1(z)=( 2 ν+1 )/radicalbig
z2−1Pμ−1
ν(z) EH I 160(3), MO 82
4(1)9(ν−μ+1 )Pμ
ν+1(z)=( 2 ν+1 )zPμ
ν(z)−(ν+μ)Pμ
ν−1(z) AS 334(8.5.3)
4(2)9Pμ
ν+1(z)=Pμ
ν−1(z)+( 2 ν+1 )/parenleftbig
z2−1/parenrightbig1/2Pμ−1
ν(z) AS 334(8.5.5)
5. Pμ
−ν−1(z)=Pμ
ν(z) (cf. 8.820 ,8.832 4)
EH I 140(1), MO 82
8.732
1./parenleftbig
z2−1/parenrightbigdQμ
ν(z)
dz=(ν−μ+1 )Qμ
ν+1(z)−(ν+1 )zQμ
ν(z)
(cf.8.832 3) MO 82
2.10(2ν+1 )zQμ
ν(z)=(ν−μ+1 )Qμ
ν+1(z)+(ν+μ)Qμ
ν−1(z)
(cf.8.832 4) MO 82
3. Qμ+2
ν(z)+2 ( μ+1 )z√
z2−1Qμ+1
ν(z)=(ν−μ)(ν+μ+1 )Qμ
ν(z) MO 82
4. Qμ
ν−1(z)−Qμ
ν+1(z)=−(2ν+1 )/radicalbig
z2−1Qμ−1
ν(z) MO 82a
5. e−μπiQμ
ν(x±i0) =e±1
2μπi/bracketleftBig
Qμ
ν(x)∓iπ
2Pμ
ν(x)/bracketrightBig
MO 83
8.733
1./parenleftbig
1−x2/parenrightbigdPμ
ν(x)
dx=Pμ
ν(x)−(ν−μ+1 )Pμ
ν+1(x) (cf. 8.731 1)
=−νxPμ
ν(x)+(ν+μ)Pμ
ν−1(x)
=−/radicalbig
1−x2Pμ+1
ν(x)−μxPμ
ν(x);
=(ν−μ+1 ) (ν+μ)/radicalbig
1−x2Pμ−1
ν(x)+μxPμ
ν(x)
MO 82
2. (2 ν+1 )xPμ
ν(x)=(ν−μ+1 )Pμ
ν+1(x)+(ν+μ)Pμ
ν−1(x)
(cf.8.731 2) MO 82
3.11Pμ+2
ν(x)+2 ( μ+1 )x√
1−x2Pμ+1
ν(x)+(ν−μ)(ν+μ+1 )Pμ
ν(x)=0
(cf.8.731 3) MO 82
4. Pμ
ν−1(x)−Pμ
ν+1(x)=( 2 ν+1 )/radicalbig
1−x2Pμ−1
ν(x) (cf. 8.731 4) MO 82
5. Pμ
−ν−1(x)=Pμ
ν(x) (cf. 8.731 5)
966 Associated Legendre Functions 8.734
8.734
1. ( ν+μ+1 )zQν
μ(z)+/radicalbig
z2−1Qμ+1
ν(z)=(ν−μ+1 )Qμ
ν+1(z) MO 82
2. ( ν+μ)Qμ
ν−1(z)+/radicalbig
z2−1Qμ+1
ν(z)=(ν−μ)zQμ
ν(z) MO 82
3. Qμ
ν−1(z)−zQμ
ν(z)=−(ν−μ+1 )/radicalbig
z2−1Qμ−1
ν(z) MO 82
4. zQμ
ν(z)−Qμ
ν+1(z)=−(ν+μ)/radicalbig
z2−1Qμ−1
ν(z) MO 82
5. ( ν+μ)(ν+μ+1 )Qμ
ν−1(z)+( 2ν+1 )/radicalbig
z2−1Qμ+1
ν(z)=(ν−μ)(ν−μ+1 )Qμ
ν+1(z) MO 82
8.735
1. ( ν+μ+1 )xPμ
ν(x)+/radicalbig
1−x2Pμ+1
ν(x)=(ν−μ+1 )Pμ
ν+1(x) MO 83
2. ( ν−μ)xPμ
ν(x)−(ν+μ)Pμ
ν−1(x)=/radicalbig
1−x2Pμ+1
ν(x) MO 83
3. Pμ
ν−1(x)−xPμ
ν(x)=(ν−μ+1 )/radicalbig
1−x2Pμ−1
ν(x) MO 83
4. xPμ
ν(x)−Pμ
ν+1(x)=(ν+μ)/radicalbig
1−x2Pμ−1
ν(x) MO 83
5. ( ν−μ)(ν−μ+1 )Pμ
ν+1(x)=(ν+μ)(ν+μ+1 )Pμ
ν−1(x)+( 2ν+1 )/radicalbig
1−x2Pμ+1
ν(x) MO 83
8.736
1. P−μ
ν(z)=Γ(ν−μ+1 )
Γ(ν+μ+1 )/bracketleftbigg
Pμ
ν(z)−2
πe−μπisinμπQμ
ν(z)/bracketrightbigg
MO 83
2. Pμ
ν(−z)=eνπiPμ
ν(z)−2
πsin[(ν+μ)π]e−μπiQμ
ν(z)[ I m z<0] (cf. 8.833 1) MO 83
3. Pμ
ν(−z)=e−νπiPμ
ν(z)−2
πsin[(ν+μ)π]e−μπiQμ
ν(z)
[Imz>0] (cf. 8.833 2) MO 83
4. Q−μ
ν(z)=e−2μπiΓ(ν−μ+1 )
Γ(ν+μ+1 )Qμ
ν(z) MO 82
5. Qμ
ν(−z)=−e−νπiQμ
ν(z)[ I m z<0] MO 82
6. Qμ
ν(−z)=−eνπiQμ
ν(z)[ I m z>0] MO 82
7.6Qμ
ν(z)sin[(ν+μ)π]−Qμ
−ν−1(z)sin[(ν−μ)π]=πeμπicosνπPμ
ν(z) MO 83
8.737
1. P−μ
ν(x)=Γ(ν−μ+1 )
Γ(ν+μ+1 )/bracketleftbigg
cosμπPμ
ν(x)−2
πsin(μπ)Qμ
ν(x)/bracketrightbigg
MO 84
2. Pμ
ν(−x) = cos[( ν+μ)π]Pμ
ν(x)−2
πsin[(ν+μ)π]Qμ
ν(x) MO 84
3. Qμ
ν(−x)=−cos[(ν+μ)π]Qμ
ν(x)−π
2sin[(ν+μ)π]Pμ
ν(x) MO 83, EH I 144(15)
4. Qμ
−ν−1(x)=sin[(ν+μ)π]
sin[(ν−μ)π]Qμ
ν(x)−πcosνπcosμπ
sin[(ν−μ)π]Pμ
ν(x) MO 84
8.742 Functional relations 967
8.738
1.11Qμ
ν(icotϕ) = exp/bracketleftbigg
iπ/parenleftbigg
μ−ν+1
2/parenrightbigg/bracketrightbigg√πΓ(ν+μ+1 )/radicalbigg
1
2sinϕP−ν−1
2
−μ−1
2(cosϕ)
/bracketleftBig
0<ϕ<π
2/bracketrightBig
MO 83
2.6Pμ
ν(icotϕ)=/radicalbigg
2
πexp/bracketleftbigg
iπ/parenleftbigg
ν+1
4/parenrightbigg/bracketrightbigg√sinϕ
Γ(−ν−μ)Q−ν−1
2
−μ−1
2(cosϕ−i0)
/bracketleftBig
0<ϕ<π
2/bracketrightBig
MO 83
8.739 e−μπiQμ
ν(coshα)=√πΓ(ν+μ+1 )√
2s in h αP−ν−1
2
−μ−1
2(cothα)[ R e ( c o s h α)>0] MO 83
8.741
1. P−μ
ν(x)dPμ
ν(x)
dx−Pμ
ν(x)dP−μ
ν(x)
dx=2s inμπ
π(1−x2)MO 83
2. Pμ
ν(x)dQμ
ν(x)
dx−Qμ
ν(x)dPμ
ν(x)
dx=22μ
1−x2Γ/parenleftbigν+μ+1
2/parenrightbig
Γ/parenleftbigν+μ
2+1/parenrightbig
Γ/parenleftbigν−μ+1
2/parenrightbig
Γ/parenleftbigν−μ
2+1/parenrightbig MO 83
8.742
1.Γ(ν−μ−1)
Γ(ν+μ+1 )/braceleftbigg
cosμπPμ
ν(cosϕ)−2
πsinμπQμ
ν(cosϕ)/bracerightbigg
=/radicalbigg
2
πcosecμϕ
Γ/parenleftbig
μ+1
2/parenrightbig/integraldisplayϕ
0cos/parenleftbig
ν+1
2/parenrightbig
td t
(cost−cosϕ)1
2−μ
/bracketleftbig
Reμ>−1
2/bracketrightbig
MO 88
2.Γ(ν−μ+1 )
Γ(ν+μ+1 )/braceleftbigg
cosνπPμ
ν(cosϕ)−2
πsinνπQμ
ν(cosϕ)/bracerightbigg
=/radicalbigg
2
πcosecμϕ
Γ/parenleftbig
μ+1
2/parenrightbig/integraldisplayπ
ϕcos/bracketleftbig/parenleftbig
ν+1
2/parenrightbig
(t−π)/bracketrightbig
dt
(cosϕ−cost)1
2−μ
/bracketleftbig
Reμ>−1
2/bracketrightbig
MO 88
3. Pμ
ν(cosϕ)cos(ν+μ)π−2
πQμ
ν(cosϕ)s i n (ν+μ)π=/radicalbigg
2
πsinμϕ
Γ/parenleftbig1
2−μ/parenrightbig/integraldisplayπ
ϕcos/bracketleftbig/parenleftbig
ν+1
2/parenrightbig
(t−π)/bracketrightbig
dt
(cosϕ−cost)μ+1
2/bracketleftbig
Reμ<1
2/bracketrightbig
MO 88
4. cos μπPμ
ν(cosϕ)−2
πsinμπQμ
ν(cosϕ)
=1
2μ√πΓ(ν+μ+1 )
Γ(ν−μ+1 )sinμϕ
Γ/parenleftbig
μ+1
2/parenrightbig/integraldisplayπ
0sin2μtd t
(cosϕ±isinϕcost)ν−μ
/bracketleftbig
Reμ>−1
2,0<ϕ<π/bracketrightbig
MO 38
For integrals of Legendre functions, see 7.11–7.21.
968 Associated Legendre Functions 8.751
8.75 Special cases and particular values
8.751
1. Pm
ν(x)=(−1)mΓ(ν+m+1 )/parenleftbig
1−x2/parenrightbigm
2
2mΓ(ν−m+1 )m!F/parenleftbigg
m−ν,m+ν+1 ;m+1 ;1−x
2/parenrightbigg
MO 84
2. Pm
ν(z)=Γ(ν+m+1 )/parenleftbig
z2−1/parenrightbigm
2
2mm!Γ (ν−m+1 )F/parenleftbigg
m−ν,m+ν+1 ;m+1 ;1−z
2/parenrightbigg
MO 84
3.8Qμ
n+1
2(z)=eμπiΓ/parenleftbigg
μ+n+3
2/parenrightbigg
2n+3
2(n+1 ) !/parenleftbig
z2−1/parenrightbigμ
2π1/2z−n−μ−3/2F/parenleftbiggμ+n+5
2
2,μ+n+3
2
2;n+2 ;1
z2/parenrightbigg
MO 84
8.752
1. Pm
ν(x)=(−1)m/parenleftbig
1−x2/parenrightbigm
2dm
dxmPν(x) WH, MO 84, EH I 148(6)
2. P−m
ν(x)=(−1)mΓ(ν−m+1 )
Γ(ν+m+1 )Pm
ν(x)=/parenleftbig
1−x2/parenrightbig−m
2/integraldisplay1
x.../integraldisplay1
xPν(x)(dx)m
[m≥1] HO 99a, MO 85, EH I 149(10)a
3. P−m
ν(z)=/parenleftbig
z2−1/parenrightbig−m
2/integraldisplayz
1.../integraldisplayz
1Pν(z)(dz)m[m≥1] MO 85, EH I 149(8)
4. Qm
ν(z)=/parenleftbig
z2−1/parenrightbigm
2dm
dzmQν(z) WH, MO 85, EH I 148(5)
5. Q−m
ν(z)=(−1)m/parenleftbig
z2−1/parenrightbig−m
2/integraldisplay∞
z.../integraldisplay∞
zQν(z)(dz)m
[m≥1] MO 85, EH I 149(9)
Special values of the indices
8.753
1. Pμ
0(cosϕ)=1
Γ(1−μ)cotμϕ
2MO 84
2. P−1
ν(cosϕ)=−1
ν(ν+1 )dPν(cosϕ)
dϕMO 84
3. Pm
n(z)≡0,Pm
n(x)≡0f o r m>n MO 85
8.754
1. P1/2
ν−1
2(coshα)=/radicalbigg
2
πsinhαcoshνα MO 85
2. P1/2
ν−1
2(cosϕ)=/radicalbigg2
πsinϕcosνϕ MO 85
3. P−1/2
ν−1
2(cosϕ)=/radicalbigg2
πsinϕsinνϕ
νMO 85
8.762 Derivatives with respect to the order 969
4. Q1/2
ν−1
2(coshα)=i/radicalbiggπ
2s in h αe−ναMO 85
8.755
1. P−ν
ν(cosϕ)=1
Γ(1 + ν)/parenleftbiggsinϕ
2/parenrightbiggν
MO 85
2. P−ν
ν(coshα)=1
Γ(1 + ν)/parenleftbiggsinhα
2/parenrightbiggν
MO 85
Special values of Legendre functions
8.756
1. Pμ
ν(0) =2μ√π
Γ/parenleftbigν−μ
2+1/parenrightbig
Γ/parenleftbig−ν−μ+1
2/parenrightbig MO 84
2.dPμ
ν(0)
dx=2μ+1sin1
2(ν+μ)πΓ/parenleftbigν+μ
2+1/parenrightbig
√πΓ/parenleftbigν−μ+1
2/parenrightbig MO 84
3. Qμ
ν(0) =−2μ−1√πsin1
2(ν+μ)πΓ/parenleftbigν+μ+1
2/parenrightbig
Γ/parenleftbigν−μ
2+1/parenrightbig MO84
4.dQμ
ν(0)
dx=2μ√πcos1
2(ν+μ)πΓ/parenleftbigν+μ
2+1/parenrightbig
Γ/parenleftbigν−μ+1
2/parenrightbig MO 84
8.76 Derivatives with respect to the order
8.761∂P−μ
ν(x)
∂ν=1
Γ(μ+1 )/parenleftbigg1−x
1+x/parenrightbiggμ
2∞/summationdisplay
n=1(−ν)(1−ν)...(n−1−ν)(ν+1 ) (ν+2 )...(ν+n)
(μ+1 ) (μ+2 )...(μ+n)1·2...n
×[ψ(ν+n+1 )−ψ(ν−n+1 ) ]/parenleftbigg1−x
2/parenrightbiggn
[ν/negationslash=0,±1,±2,...;R e μ>−1]MO 94
8.762
1./bracketleftbigg∂Pν(cosϕ)
∂ν/bracketrightbigg
ν=0= 2ln cosϕ
2MO 94
2./bracketleftbigg∂P−1
ν(cosϕ)
∂ν/bracketrightbigg
ν=0=−tanϕ
2−2c otϕ
2ln cosϕ
2MO 94
3./bracketleftbigg∂P−1
ν(cosϕ)
∂ν/bracketrightbigg
ν=1=−1
2tanϕ
2sin2ϕ
2+s i nϕln cosϕ
2MO 94
•For a connection with the polynomials Cλ
n(x), see8.936 .
•For a connection with a hypergeometric function, see 8.77.
970 Associated Legendre Functions 8.771
8.77 Series representation
For a representation in the form of a series, see 8.721 . It is also possible to represent associated Legendre
functions in the form of a series by expressing them in terms of a hypergeometric function.
8.771
1. Pμ
ν(z)=/parenleftbiggz+1
z−1/parenrightbiggμ
21
Γ(1−μ)F/parenleftbigg
−ν,ν+1 ;1 −μ;1−z
2/parenrightbigg
MO 15
2.8Qμ
ν(z)=eμπi
2ν+1Γ(ν+μ+1 )
Γ/parenleftbig
ν+3
2/parenrightbigΓ/parenleftbig1
2/parenrightbig/parenleftbig
z2−1/parenrightbigμ
2
zν+μ+1F/parenleftbiggν+μ
2+1,ν+μ+1
2;ν+3
2;1
z2/parenrightbigg
MO 15
See also 8.702 ,8.703 ,8.704 ,8.723 ,8.751 ,8.772 .
The analytic continuation for |z|>1
The formulas are consequences of theorems on the analytic continuation of hypergeometric series (see
9.154 and9.155 ):
8.772
1. Pμ
ν(z)=sin(ν+μ)πΓ(ν+μ+1 )
2ν+1√πcosνπΓ/parenleftbig
ν+3
2/parenrightbig/parenleftbig
z2−1/parenrightbigμ
2z−ν−μ−1F/parenleftbiggν+μ
2+1,ν+μ+1
2;ν+3
2;1
z2/parenrightbigg
+2νΓ/parenleftbig
ν+1
2/parenrightbig
√πΓ(ν−μ+1 )/parenleftbig
z2−1/parenrightbigμ
2zν−μF/parenleftbiggμ−ν+1
2,μ−ν
2;1
2−ν;1
z2/parenrightbigg
[2ν/negationslash=±1,±3,±5,...;|z|>1;|arg (z±1)|<π]MO 85
2. Pμ
ν(z)=Γ/parenleftbig
−ν−1
2/parenrightbig/parenleftbig
z2−1/parenrightbig−ν+1
2
2ν+1√πΓ(−ν−μ)F/parenleftbiggν−μ+1
2,ν+μ+1
2;ν+3
2;1
1−z2/parenrightbigg
+2νΓ/parenleftbig
ν+1
2/parenrightbig
√πΓ(ν−μ+1 )/parenleftbig
z2−1/parenrightbigν
2F/parenleftbiggμ−ν
2,−μ+ν
2;1
2−ν;1
1−z2/parenrightbigg
/bracketleftbig
2ν/negationslash=±1,±3,±5;...;/vextendsingle/vextendsingle1−z2/vextendsingle/vextendsingle>1;|arg (z±1)|<π/bracketrightbig
MO 85
3. Pμ
ν(z)=1
Γ(1−μ)/parenleftbiggz−1
z+1/parenrightbigg−μ
2/parenleftbiggz+1
2/parenrightbiggν
F/parenleftbigg
−ν,−ν−μ;1−μ;z−1
z+1/parenrightbigg
/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglez−1
z+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle<1/bracketrightbigg
MO 86
8.773
1. Qμ
ν(z)=eμπi√πΓ(ν+μ+1 )
2ν+1Γ/parenleftbig
ν+3
2/parenrightbig/parenleftbig
z2−1/parenrightbig−ν+1
2F/parenleftbiggν+μ+1
2,ν−μ+1
2;ν+3
2;1
1−z2/parenrightbigg
/bracketleftbig
ν+μ/negationslash=−1,−2,−3,...;|arg (z±1)|<π;/vextendsingle/vextendsingle1−z2/vextendsingle/vextendsingle>1/bracketrightbig
MO 86
2. Qμ
ν(z)=1
2eμπi/braceleftBigg
Γ(μ)/parenleftbiggz+1
z−1/parenrightbiggμ
2
F/parenleftbigg
−ν,ν+1 ;1 −μ;1−z
2/parenrightbigg
+Γ(−μ)Γ(ν+μ+1 )
Γ(ν−μ+1 )/parenleftbiggz−1
z+1/parenrightbiggμ
2
F/parenleftbigg
−ν,ν+1 ; 1+ μ;1−z
2/parenrightbigg/bracerightBigg
[|arg (z±1)|<π , |1−z|<2]MO 86
8.777 Series representation 971
8.774 Pμ
ν(icotϕ)=/radicalbigg
sinϕ
2πΓ/parenleftbig
−ν−1
2/parenrightbig
Γ(−ν−μ)e−i(ν+1)π
2/parenleftBig
tanϕ
2/parenrightBigν+1
2F/parenleftbigg1
2+μ,1
2−μ;ν+3
2;s in2ϕ
2/parenrightbigg
+/radicalbigg
sinϕ
2πΓ/parenleftbig
ν+1
2/parenrightbig
Γ(ν−μ+1 )eiνπ
2/parenleftBig
cotϕ
2/parenrightBigν+1
2F/parenleftbigg1
2+μ,1
2−μ;1
2−ν;s in2ϕ
2/parenrightbigg
/bracketleftBig
2ν/negationslash=±1,±3,±5,..., 0<ϕ<π
2/bracketrightBig
MO 86
8.775
1.6Pμ
ν(x)=2μcos/parenleftbig1
2(ν+μ)π/parenrightbig
Γ/parenleftbigν+μ+1
2/parenrightbig
√πΓ/parenleftbigν−μ
2+1/parenrightbig/parenleftbig
1−x2/parenrightbigμ
2F/parenleftbiggν+μ+1
2,μ−ν
2;1
2;x2/parenrightbigg
+2μ+1
√πsin/parenleftbig1
2(ν+μ)π/parenrightbig
Γ/parenleftbigν+μ
2+1/parenrightbig
Γ/parenleftbigν−μ+1
2/parenrightbig x/parenleftbig
1−x2/parenrightbigμ
2F/parenleftbiggν+μ
2+1,−ν+μ+1
2;3
2;x2/parenrightbigg
MO 87
2.6Qμ
ν(x)=−√π
21−μsin/parenleftbig1
2(ν+μ)π/parenrightbig
Γ/parenleftbigν+μ+1
2/parenrightbig
Γ/parenleftbigν−μ
2+1/parenrightbig/parenleftbig
1−x2/parenrightbigμ
2F/parenleftbiggν+μ+1
2,μ−ν
2;1
2;x2/parenrightbigg
+2μ√πcos/parenleftbig1
2(ν+μ)π/parenrightbig
Γ/parenleftbigν+μ
2+1/parenrightbig
Γ/parenleftbigν−μ+1
2/parenrightbig x/parenleftbig
1−x2/parenrightbigμ
2F/parenleftbiggν+μ
2+1,μ−ν+1
2;3
2;x2/parenrightbigg
MO 87
8.776 For|z|/greatermuch1
1. Pμ
ν(z)=/braceleftBigg
2νΓ/parenleftbig
ν+1
2/parenrightbig
√πΓ(ν−μ+1 )zν+Γ/parenleftbig
−ν−1
2/parenrightbig
2ν+1√πΓ(−ν−μ)z−ν−1/bracerightBigg/parenleftbigg
1+O/parenleftbigg1
z2/parenrightbigg/parenrightbigg
[2ν/negationslash=±1,±3,±5,..., |argz|<π]
MO 87
2. Qμ
ν(z)=√πeμπi
2ν+1Γ(μ+ν+1 )
Γ/parenleftbig
ν+3
2/parenrightbigz−ν−1/parenleftbigg
1+O/parenleftbigg1
z2/parenrightbigg/parenrightbigg
[2ν/negationslash=−3,−5,−7,...;|argz|<π]
MO 87
8.777 Setζ=z+√
z2−1. The variable ζis uniquely defined by this equation on the entire z-plane
in which a cut is made from −∞to +1. Here, we are considering that branch of the variable ζfor which
values of ζexceeding 1 correspond to real values of zexceeding 1. In this case,
1. Pμ
ν(z)=2μΓ/parenleftbig
−ν−1
2/parenrightbig
√πΓ(−ν−μ)/parenleftbig
z2−1/parenrightbigμ
2
ζν+μ+1F/parenleftbigg1
2+μ, ν+μ+1 ;ν+3
2;1
ζ2/parenrightbigg
+2μ
√πΓ/parenleftbig
ν+1
2/parenrightbig
Γ(ν−μ+1 )/parenleftbig
z2−1/parenrightbigμ
2
ζμ−νF/parenleftbigg1
2+μ, μ−ν;1
2−ν;1
ζ2/parenrightbigg
[2ν/negationslash=±1,±3,±5,...;|arg(z−1)|<π]MO 86
2. Qμ
ν(z)=2μeμπi√πΓ(ν+μ+1 )
Γ/parenleftbig
ν+3
2/parenrightbig/parenleftbig
z2−1/parenrightbigμ
2
ζν+μ+1F/parenleftbigg1
2+μ, ν+μ+1 ;ν+3
2;1
ζ2/parenrightbigg
[|arg(z−1)|<π] MO 86
972 Associated Legendre Functions 8.781
8.78 The zeros of associated Legendre functions
8.781 The function P−μ
ν(cosϕ), considered as a function of ν, has infinitely many zeros for μ≥0. These
are all simple and real. If a number ν0is a zero of the function P−μ
ν(cosϕ), the number −ν0−1 is also
a zero of this function. MO 91
8.782 Ifνandμare both real and μ≤0, or if νandμare integers, the function Pμ
ν(t)h a sn o realzeros
exceeding 1. If νandμare both real with ν<μ< 0, the function Pμ
ν(t) has no real zeros exceeding 1
when sin μπsin(μ−ν)π>0, but does have one such zero when sin μπsin(μ−ν)π<0. Finally, if μ≤ν,
the function Pμ
ν(t) has no zeros exceeding 1 for ⌊μ⌋even but does have one zero for ⌊μ⌋odd.
8.783 Ifν>−3
2andν+μ+1>0, the function Qμ
ν(t) has no real zeros exceeding 1. MO 91
8.784 The function P−1
2+iλ(z) has infinitely many zeros for real λ. All these zeros are realandgreater
than unity .
8.785 Forna natural number, the function Pn(x) has exactly nreal zeros which lie in the closed
interval −1,+1.
8.786 The function Qn(z) has no zeros for which |arg(z−1)|<πifnis a natural number. The function
Qn(cosϕ) has exactly n+ 1 zeros in the interval 0 ≤ϕ≤π. MO 91
8.787 The following approximate formula can be used to calculate the values of νfor which the equation
P−μ
ν(cosϕ) = 0 holds for given small values of ϕ:
ν+1
2=−jμ
2s inϕ
2/braceleftBigg
1−sin2ϕ
2
6/parenleftbigg
1−4μ2−1
j2μ/parenrightbigg
+O/parenleftBig
sin4ϕ
2/parenrightBig/bracerightBigg
. MO 93
Here, jμdenotes an arbitrary nonzero root of the equation Jμ(z)=0( f o r μ≥0). If ϕis close to π
then, instead of this formula, we can use the following formulas:
1. ν≈μ+k+Γ(2μ+k+1 )
Γ(μ)Γ(μ+1 )Γ ( k+1 )/parenleftbiggπ−ϕ
3/parenrightbigg2μ
[μ>0,k=0,1,2,...] MO 93
2. ν≈k+1
2ln/parenleftBig
2
π−ϕ/parenrightBig [μ=0,k=0,1,2,...] MO 93
8.79 Series of associated Legendre functions
8.791
1.1
z−t=∞/summationdisplay
k=0(2k+1 )Pk(t)Qk(z)/bracketleftBig/vextendsingle/vextendsingle/vextendsinglet+/radicalbig
t2−1/vextendsingle/vextendsingle/vextendsingle</vextendsingle/vextendsingle/vextendsinglez+/radicalbig
z2−1/vextendsingle/vextendsingle/vextendsingle/bracketrightBig
Here, tmust lie inside an ellipse passing through the point zwith foci at the points ±1.
2.1
√
1−2tz+t2lnz−t+√
1−2tz+t2
√
z2−1=∞/summationdisplay
k=0tkQk(z)
[Rez>1,|t|<1] MO 78
8.792 P−α
ν(cosϕ)P−β
ν(cosψ)=sinνπ
π∞/summationdisplay
k=0(−1)k/bracketleftbigg1
ν−k−1
ν+k+1/bracketrightbigg
P−α
k(cosϕ)P−β
k(cosψ)
[a≥0,β≥0,νreal,−π<ϕ ±ψ<π ]MO 94
8.796 Series of associated Legendre functions 973
8.793 P−μ
ν(cosϕ)=sinνπ
π∞/summationdisplay
k=0(−1)k/parenleftbigg1
ν−k−1
ν+k+1/parenrightbigg
P−μ
k(cosϕ)[ μ≥0,0<ϕ<π ]
MO 94
Addition theorems
8.794
1.11Pν(cosψ1cosψ2+s i nψ1sinψ2cosϕ)
=Pν(cosψ1)Pν(cosψ2)+2∞/summationdisplay
k=1(−1)kP−k
ν(cosψ1)Pk
ν(cosψ2)c o skϕ
=Pν(cosψ1)Pν(cosψ2)+2∞/summationdisplay
k=1Γ(ν−k+1 )
Γ(ν+k+1 )Pk
ν(cosψ1)Pk
ν(cosψ2)coskϕ
[0≤ψ1<π , 0≤ψ2<π , ψ 1+ψ2<π , ϕ real] (cf. 8.814 ,8.844 1)MO 90
2. Qν(cosψ1)c o sψ2+s i nψ1sinψ2cosϕ
=Pν(cosψ1)Qν(cosψ2)+2∞/summationdisplay
k=1(−1)kP−k
ν(cosψ1)Qνk(cosψ2)coskϕ
/bracketleftBig
0<ψ1<π
2,0<ψ2<π , 0<ψ1+ψ2<π;ϕreal/bracketrightBig
(cf.8.844 3)MO 90
8.795
1. Pν/parenleftbigg
z1z2−/radicalBig
z2
1−1/radicalBig
z2
2−1c osϕ/parenrightbigg
=Pν(z1)Pν(z2)+2∞/summationdisplay
k=1(−1)kPk
ν(z1)P−k
ν(z2)coskϕ
[Rez1>0,Rez2>0,|arg (z1−1)|<π , |arg (z2−1)|<π]MO 91
2. Qν/parenleftbigg
x1x2−/radicalBig
x2
1−1/radicalBig
x2
2−1c osϕ/parenrightbigg
=Pν(x1)Qν(x2)+2∞/summationdisplay
k=1(−1)kP−k
ν(x1)Qk
ν(x2)coskϕ
[1<x1<x2,ν/negationslash=−1,−2,−3,..., ϕ real] MO 91
3. Qn/parenleftbigg
x1x2+/radicalBig
x2
1+1/radicalBig
x2
2+1c o s h α/parenrightbigg
=∞/summationdisplay
k=n+11
(k−n−1)!(k+n)!Qk
n(ix1)Qk
n(ix2)e−kα
[x1>0,x2>0,α > 0] MO 91
8.796 Pν(−cosψ1cosψ2−sinψ1sinψ2cosϕ)=Pν(−cosψ1)Pν(cosψ2)+2∞/summationdisplay
k=1(−1)kΓ(ν+k+1 )
Γ(ν−k+1 )
×P−k
ν(−cosψ1)P−k
ν(cosψ2)coskϕ
[0<ψ2<ψ1<π , ϕ real] (cf. 8.844 2)MO 91
See also 8.934 3.
974 Associated Legendre Functions 8.810
8.81 Associated Legendre functions with integer indices
8.810 Forinteger values of νandμ, the differential equation 8.700 1. (with |ν|>|μ|) has a simple
solution in the real domain, namely:
u=Pm
n(x)=(−1)m/parenleftbig
1−x2/parenrightbigm
2dm
dxmPn(x).
The functions Pm
n(x) are called associated Legendre functions (orspherical functions )of the first kind .
The number nis called the degree , and the number mis called the order of the function Pm
n(x). The
functions {cosmϑPm
n(cosϕ),sinmϑPm
n(cosϕ)}, which depend on the angles ϕandϑ, are also called
Legendre functions of the first kind, or, more specifically, tesseral harmonics form<n andsectoral
harmonics form=n. These last functions are periodic with respect to the angles ϕandϑ.T h e i rp e r i o d s
are, respectively, πand 2π. They are single-valued and continuous everywhere on the surface of the unit
sphere x2
1+x2
2+x2
3=1( w h e r e x1=s i nϕcosϑ,x2=s i nϕsinϑ,x3=c o s ϕ), and they are solutions of
the differential equation
1
sinϕ∂
∂ϕ/parenleftbigg
sinϕ∂Y
∂ϕ/parenrightbigg
+1
sin2ϕ∂2Y
∂ϑ2+n(n+1 )Y=0.
8.811 The integral representation
Pm
n(cosϕ)=(−1)m(n+m)!
Γ/parenleftbig
m+1
2/parenrightbig
(n−m)!/radicalbigg
2
πsin−mϕ/integraldisplayϕ
0(cost−cosϕ)m−1
2cos/parenleftbig
n+1
2/parenrightbig
td t MO 75
8.812 The series representation:
Pm
n(x)=(−1)m(n+m)!
2mm!(n−m)!/parenleftbig
1−x2/parenrightbigm
2/braceleftbigg
1−(n−m)(m+n+1 )
1!(m+1 )1−x
2
+(n−m)(n−m+1 ) (m+n+1 ) (m+n+2 )
2!(m+1 ) (m+2 )/parenleftbigg1−x
2/parenrightbigg2
−.../bracerightBigg
MO 73
=(−1)m(2n−1)!!
(n−m)!/parenleftbig
1−x2/parenrightbigm
2/braceleftbigg
xn−m−(n−m)(n−m−1)
2(2n−1)xn−m−2
+(n−m)(n−m−1)(n−m−2)(n−m−3)
2·4(2n−1)(2n−3)xn−m−4−.../bracerightbigg
MO 73
=(−1)m(2n−1)!!
(n−m)!/parenleftbig
1−x2/parenrightbigm
2xn−mF/parenleftbiggm−n
2,m−n+1
2;1
2−n;1
x2/parenrightbigg
MO 73
8.813 Special cases:
1. P1
1(x)=−/parenleftbig
1−x2/parenrightbig1/2=−sinϕ MO 73
2. P1
2(x)=−3/parenleftbig
1−x2/parenrightbig1/2x=−3
2sin 2ϕ MO 73
3. P2
2(x)=3/parenleftbig
1−x2/parenrightbig
=3
2(1−cos2ϕ) MO 73
4. P1
3(x)=−3
2/parenleftbig
1−x2/parenrightbig1/2/parenleftbig
5x2−1/parenrightbig
=−3
8(sinϕ+5s i n3 ϕ) MO 73
5. P2
3(x)=1 5/parenleftbig
1−x2/parenrightbig
x=15
4(cosϕ−cos3ϕ) MO 73
6. P3
3(x)=−15/parenleftbig
1−x2/parenrightbig3/2=−15
4(3 sinϕ−sin 3ϕ) MO 73
8.820 Legendre functions 975
Functional relations
For recursion formulas, see 8.731 .
8.814 Pn(cosϕ1cosϕ2+s i nϕ1sinϕ2cosΘ)
=Pn(cosϕ1)Pn(cosϕ2)+2n/summationdisplay
m=1(n−m)!
(n+m)!Pm
n(cosϕ1)Pm
n(cosϕ2)cosmΘ
[0≤ϕ1≤π,0≤ϕ2≤π] (“addition theorem”) MO 74
8.815 If
Yn1(ϕ, ϑ)=A0Pn1(cosϕ)+n1/summationdisplay
m=1(amcosmϑ+bmsinmϑ)Pm
n1(cosϕ),
Zn2(ϕ, ϑ)=α0Pn2(cosϕ)+n2/summationdisplay
m=1(αmcosmϑ+βmsinmϑ)Pm
n2(cosϕ),
then/integraldisplay2π
0dϑ/integraldisplayπ
0sinϕdϕYn1(ϕ, ϑ)Yn2(ϕ, ϑ)=0,
/integraldisplay2π
0dϑ/integraldisplayπ
0sinϕdϕYn(ϕ, ϑ)Pn[cosϕcosψ+s i nϕsinψcos(ϑ−θ)] =4π
2n+1Yn(ψ,θ) MO 75
8.816 (cosϕ+isinϕcosϑ)n=Pn(cosϕ)+2n/summationdisplay
m=1(−1)mn!
(n+m)!cosmϑPm
n(cosϕ) MO 75
For integrals of the functions, Pm
n(x), see7.112 1,7.122 1.
8.82–8.83 Legendre functions
8.820 The differential equation
d
dz/bracketleftbigg/parenleftbig
1−z2/parenrightbigdu
dz/bracketrightbigg
+ν(ν+1 )u=0 ( c f . 8.700 1),
where the parameter νcan be an arbitrary number, has the following two linearly independent solutions:
1. Pν(z)=F/parenleftbigg
−ν,ν+1 ;1 ;1−z
2/parenrightbigg
2. Qν(z)=Γ(ν+1 )Γ/parenleftbig1
2/parenrightbig
2ν+1Γ/parenleftbig
ν+3
2/parenrightbigz−ν−1F/parenleftbiggν+2
2,ν+1
2;2ν+3
2;1
z2/parenrightbigg
SM 518(137)
The functions Pν(z)a n dQν(z) are called Legendre functions of the first andsecond kind respec-
tively. If νis not an integer, the function Pν(z)h a ssingularities atz=−1a n d z=∞. However,
ifν=n=0,1,2,...,the function Pν(z) becomes the Legendre polynomial P n(z)( s e e8.91)F o r
ν=−n=−1,−2,...,we have
P−n−1(z)=Pn(z).
3. If ν/negationslash=0,1,2,...,the function Qν(z) has singularities at the points z=±1a n d z=∞.T h e s e
points are branch points of the function. On the other hand, if ν=n=0,1,2,...,the function
Qn(z) is single-valued for |z|>1 and regular for z=∞.
976 Associated Legendre Functions 8.821
4. In the right half-plane,
Pν(z)=/parenleftbigg1+z
2/parenrightbiggν
F/parenleftbigg
−ν,−ν;1;z−1
z+1/parenrightbigg
[Rez>0]
5. The function Pν(z) is uniquely determined by equations 8.820 1a n d8.820 4 within a circle of
radius 2 with its center at the point z= 1 in the right half-plane.
Forz=x=c o s ϕ, a solution of equation 8.820 is the function
6. Pν(x)=Pν(cosϕ)=F/parenleftBig
−ν,ν+1;1;s in2ϕ
2/parenrightBig
;
In general,
7. Pν(z)=P−ν−1(z)=Pν(x)=P−ν−1(x),forz=x
8. The function Qν(z)f o r|z|>1 is uniquely determined by equation 8.820 2e v e r y w h e r ei nt h e
z-plane in which a cut is made from the point z=−∞to the point z= 1. By means of a
hypergeometric series, the function can be continued analytically inside the unit circle. On the
cut (−1≤x≤+1) of the real axis, the function Qν(x) is determined by the equation
9. Qν(x)=1
2[Qν(x+i0) +Qν(x−i0)] HO 52(53), WH
Integral representations
8.821
1. Pν(z)=1
2πi/integraldisplay(1+,z+)
A/parenleftbig
t2−1/parenrightbigν
2ν(t−z)ν+1dt
Here, Ais a point on the real axis to the right of the point t= 1 and to the right of zifzis real.
At the point A,w es e t
arg(t−1) = arg( t+1 )=0a n d [ |arg(t−z)|<π] WH
2. Qν(z)=1
4isinνπ/integraldisplay(1−,1+)
A/parenleftbig
t2−1/parenrightbigν
2ν(z−t)ν+1dt
[νis not an integer; the point Ais at the end of the major axis of an ellipse to the right of t=1
drawn in the t-plane with foci at the points ±1 and with a minor axis sufficiently small that the
point zlies outside it. The contour begins at the point A, follows the path (1 −,−1+), and returns
toA;|argz|≤πand|arg(z−t)|→argzast→0 on the contour; arg( t+1 )=a r g ( t−1) = 0 at
the point A;zdoes not lie on the real axis between −1 and 1.]
Forν=nan integer,
3. Qn(z)=1
2n+1/integraldisplay1
−1/parenleftbig
1−t2/parenrightbign(z−t)−n−1dt SM 517(134), WH
8.822
1. Pν(z)=1
π/integraldisplayπ
0dϕ
/parenleftbig
z+√
z2−1c osϕ/parenrightbigν+1=1
π/integraldisplayπ
0/parenleftBig
z+/radicalbig
z2−1c osϕ/parenrightBigν
dϕ
/bracketleftBig
Rez>0a n d a r g/braceleftBig
z+/radicalbig
z2−1c osϕ/bracerightBig
=a r g zforϕ=π
2/bracketrightBig
WH
8.827 Legendre functions 977
2. Qν(z)=/integraldisplay∞
0dϕ
/parenleftbig
z+√
z2−1c os h ϕ/parenrightbigν+1,
/bracketleftBig
Reν>−1; if νis not an integer ,/braceleftBig/parenleftBig
z+/radicalbig
z2−1/parenrightBig
coshϕ/bracerightBig
forϕ= 0 has its principal value/bracketrightBig
WH
8.823 Pν(cosθ)=2
π/integraldisplayθ
0cos/parenleftbig
ν+1
2/parenrightbig
ϕ/radicalbig
2(cos ϕ−cosθ)dϕ WH
8.824 Qn(z)=2nn!/integraldisplay∞
z.../integraldisplay∞
z(dz)n+1
(z2−1)n+1=2n/integraldisplay∞
z(t−z)n
(t2−1)n+1dt
=(−1)n
(2n−1)!!dn
dzn/bracketleftBigg
/parenleftbig
z2−1/parenrightbign/integraldisplay∞
zdt
(t2−1)n+1/bracketrightBigg
[Rez>1]
W H ,M O7 8
8.825 Qn(z)=1
2/integraldisplay1
−1Pn(t)
z−tdt [|arg(z−1)|<π] W H ,M O7 8
See also 6.622 3,8.842 .
8.826 Fourier series:
1. Pn(cosϕ)=2n+2
πn!
(2n+1 ) ! !/bracketleftbigg
sin(n+1 )ϕ+1
1n+1
2n+3sin(n+3 )ϕ
+1·3(n+1 ) (n+2 )
1·2(2n+ 3)(2 n+5 )sin(n+5 )ϕ+.../bracketrightbigg
[0<ϕ<π ] MO 79
2. Qn(cosϕ)=2n+1 n!
(2n+1 ) ! !⎡
⎣cos(n+1 )ϕ+1
1n+1
2n+3cos(n+3 )ϕ
+1·3
1·2(n+1 ) (n+2 )
(2n+ 3)(2 n+5 )cos(n+5 )ϕ+...⎤
⎦
[0<ϕ<π ] MO 79
The expressions for Legendre functions in terms of a hypergeometric function (see 8.820 ) provide other
series representations of these functions.
Special cases and particular values
8.827
1. Q0(x)=1
2ln1+x
1−x=a r c t a n h x JA
2. Q1(x)=x
2ln1+x
1−x−1 JA
3. Q2(x)=1
4/parenleftbig
3x2−1/parenrightbig
ln1+x
1−x−3
2x JA
4. Q3(x)=1
4/parenleftbig
5x3−3x/parenrightbig
ln1+x
1−x−5
2x2+2
3JA
978 Associated Legendre Functions 8.828
5. Q4(x)=1
16/parenleftbig
35x4−30x2+3/parenrightbig
ln1+x
1−x−35
8x3+55
24x JA
6. Q5(x)=1
16/parenleftbig
63x5−70x3+1 5x/parenrightbig
ln1+x
1−x−63
8x4+49
8x2−8
15JA
8.828
1. Pν(1) = 1 MO 79
2. Pν(0) =−1
2sinνπ√
π3Γ/parenleftbiggν+1
2/parenrightbigg
Γ/parenleftBig
−ν
2/parenrightBig
MO 79
8.829 Qν(0) =1
4√π(1−cosνπ)Γ/parenleftbiggν+1
2/parenrightbigg
Γ/parenleftBig
−ν
2/parenrightBig
MO 79
Functional relationships
8.831
1. Qν(x)=π
2s inνπ[cosνπPν(x)−Pν(−x)] [ ν/negationslash=0,±1,±2,...] MO 76
2. Qn(x)=1
2Pn(x)ln1+x
1−x−Wn−1(x)[ n=0,1,2,...],
where
3. Wn−1(x)=⌊n−1
2⌋/summationdisplay
k=02(n−2k)−1
(2k+1 ) (n−k)Pn−2k−1(x)=n/summationdisplay
k=11
kPk−1(x)Pn−k(x)
and
4. W−1(x)≡0 (see also 8.839 ) SM 516(131), MO 76
5.∞/summationdisplay
k=0(−1)k/parenleftbigg1
ν−k−1
ν+k+1/parenrightbigg
Pk(cosϕ)=π
sinνπPν(cosϕ)
[νnot an integer; 0 ≤ϕ<π ]MO 77
6.∞/summationdisplay
k=0(−1)k/parenleftbigg1
ν−k−1
ν+k+1/parenrightbigg
Pk(cosϕ)Pk(cosψ)=π
sinνπPν(cosϕ)Pν(cosψ)
[νnot an integer ,−π<ϕ +ψ<π , −π<ϕ −ψ<π ]MO 77
See also 8.521 4.
8.832
1./parenleftbig
z2−1/parenrightbigd
dzPν(z)=(ν+1 )[Pν+1(z)−zPν(z)] WH
2. (2 ν+1 )zPν(z)=(ν+1 )Pν+1(z)+νPν−1(z) WH
3./parenleftbig
z2−1/parenrightbigd
dzQν(z)=(ν+1 )/bracketleftbig
Qν+1(z)−zQν(z)/bracketrightbig
WH
4. (2 ν+1 )zQν(z)Qν+1(z)+νQν−1(z) WH
8.838 Legendre functions 979
8.833
1. Pν(−z)=eνπiPν(z)−2
πsinνπQν(z)[ I m z<0] MO 77
2. Pν(−z)=e−νπiPν(z)−2
πsinνπQν(z)[ I m z>0] MO 77
3. Qν(−z)=−e−νπiQν(z)[ I m z<0] MO 77
4. Qν(−z)=−eνπiQν(z)[ I m z>0] MO 77
8.834
1. Qν(x±i0) =Qν(x)∓πi
2Pν(x) MO 77
2. Qn(z)=1
2Pn(z)lnz+1
z−1−Wn−1(z)( s e e 8.831 3) MO 77
8.835
1. Qν(z)−Q−ν−1(z)=πcotνπPν(z)[ s i n νπ/negationslash=0 ] MO 77
2. Q−ν−1(cosϕ)=Qν(cosϕ)−πcotνπPν(cosϕ)[ s i n νπ/negationslash=0 ] MO 77
3. Qν(−cosϕ)=−cosνπQν(cosϕ)−π
2sinνπPν(cosϕ) MO 77
8.836
1. Qn(z)=1
2nn!dn
dzn/bracketleftbigg/parenleftbig
z2−1/parenrightbignlnz+1
z−1/bracketrightbigg
−1
2Pn(z)lnz+1
z−1MO 79
2. Qn(x)=1
2nn!dn
dxn/bracketleftbigg/parenleftbig
x2−1/parenrightbignln1+x
1−x/bracketrightbigg
−1
2Pn(x)ln1+x
1−xMO 79
8.837
1. Pν(x)=Pν(cosϕ)=F/parenleftBig
−ν,ν+1;1;s in2ϕ
2/parenrightBig
(cf.8.820 6) MO 76
2. Pν(z)=tanνπ
2ν+1√πΓ(ν+1 )
Γ/parenleftbig
ν+3
2/parenrightbigz−ν−1F/parenleftbiggν
2+1,ν+1
2;ν+3
2;1
z2/parenrightbigg
+2ν
√πΓ/parenleftbig
ν+1
2/parenrightbig
Γ(ν+1 )zνF/parenleftbigg1−ν
2,−ν
2;1
2−ν;1
z2/parenrightbigg
MO 78
See also 8.820 .
For integrals of Legendre functions, see 7.1–7.2.
8.838 Inequalities (0 ≤ϕ≤π,ν>1, and C0is a number that does not depend on the values of ν
orϕ):
1. |Pν(cosϕ)−Pν+2(cosϕ)|≤2C0/radicalbigg
1
νπMO 78
2./vextendsingle/vextendsingleQν(cosϕ)−Qν+2(cosϕ)/vextendsingle/vextendsingle<C0/radicalbiggπ
νMO 78
With regard to the zeros of Legendre functions of the second kind, see 8.784 ,8.785 ,a n d8.786 .F o r
the expansion of Legendre functions in series of associated Legendre functions, see 8.794 ,8.795 ,a n d
8.796 .
980 Associated Legendre Functions 8.839
8.839 A differential equation leading to the functions Wn−1(see8.831 3):
/parenleftbig
1−x2/parenrightbigd2Wn−1
dx2−2xdWn−1
dx+(n+1 )nWn−1=2dPν
dxMO 76
8.84 Conical functions
8.840 Let us set
ν=−1
2+iλ,
where λis a real parameter, in the defining differential equation 8.700 1 for associated Legendre functions.
We then obtain the differential equation of the so-called conical functions. A conical function is a special
case of the associated Legendre function. However, the Legendre functions
P−1
2+iλ(x),Q−1
2+iλ(x)
have certain peculiarities that make us distinguish them as a special class—the class of conical functions.
The most important of these peculiarities is the following
8.841 The functions
P−1
2+iλ(cosϕ)=1+4λ2+12
22sin2ϕ
2+/parenleftbig
4λ2+12/parenrightbig/parenleftbig
4λ2+32/parenrightbig
2242sin4ϕ
2+...
are real for real values of ϕ. Also,
P−1
2+iλ(x)≡P−1
2−iλ(x) MO 95
8.842 Integral representations:
1. P−1
2+iλ(cosϕ)=2
π/integraldisplayϕ
0coshλudu/radicalbig
2(cos u−cosϕ)=2
πcoshλπ/integraldisplay∞
0cosλudu/radicalbig
2(cos ϕ+c o s h u)MO 95
2.6Q−1
2∓λi(cosϕ)=±isinhλπ/integraldisplay∞
0cosλudu/radicalbig
2(c os h u+c o s ϕ)+/integraldisplay∞
0cosλudu/radicalbig
2( c os h u−cosϕ)MO 95
Functional relations
(See also 8.73)
8.843 P−1
2+iλ(−cosϕ)=coshλπ
π/bracketleftBig
Q−1
2+iλ(cosϕ)+Q−1
2−iλ(cosϕ)/bracketrightBig
MO 95
8.844
1. P−1
2+iλ(cosψcosϑ+s i nψsinϑcosϕ)
=P−1
2+iλ(cosψ)P−1
2+iλ(cosϑ)+2∞/summationdisplay
k=1(−1)k22kPk
−1
2+iλ(cosψ)Pk
−1
2+iλ(cosϑ)c o skϕ
(4λ2+12)(4λ2+32)···[4λ2+( 2k−1)2]/bracketleftBig
0<ϑ<π
2,0<ψ<π , 0<ψ+ϑ<π/bracketrightBig
(cf.8.794 1)MO 95
2. P−1
2+iλ(−cosψcosϑ−sinψsinϑcosϕ)
=P−1
2+iλ(cosψ)P−1
2+iλ(−cosϑ)+2∞/summationdisplay
k=1(−1)k22kPk
−1
2+iλ(cosψ)Pk
−1
2+iλ(−cosϑ)c o skϕ
(4λ2+1 )( 4 λ2+32)···[4λ2+( 2k−1)2]/bracketleftBig
0<ψ<π
2<ϑ , ψ +ϑ<π/bracketrightBig
(cf.8.796 )MO 95
8.852 Toroidal functions 981
3. Q−1
2+iλ(cosψcosϑ+s i nψsinϑcosϕ)
=P−1
2+iλ(cosψ)Q−1
2+iλ(cosϑ)+2∞/summationdisplay
k=1(−1)k22kPk
−1
2+iλ(cosψ)Qk
−1
2+iλ(cosϑ)c o skϕ
(4λ2+1 )( 4 λ2+32)···/bracketleftBig
4λ2+( 2k−1)2/bracketrightBig
/bracketleftBig
0<ψ<π
2<ϑ , ψ +ϑ<π/bracketrightBig
(cf.8.794 2)MO 96
Regarding the zeros of conical functions, see 8.784 .
8.85 Toroidal functions
8.850 Solutions of the differential equation
1.d2u
dη2+coshη
sinhηdu
dη−/parenleftbigg
n2−1
4+m2
sinh2η/parenrightbigg
u=0,
are called toroidal functions. They are equivalent (under a coordinate transformation) to associated
Legendre functions. In particular, the functions
Pm
n−1
2(coshη),Qm
n−1
2(sinhη) MO 96
are solutions of equation 8.850 1.
The following formulas, obtained from the formulas obtained earlier for associated Legendre functions,
are valid for toroidal functions:
8.851 Integral representations:
1. Pm
n−1
2(coshη)=Γ/parenleftbig
n+m+1
2/parenrightbig
Γ/parenleftbig
n−m+1
2/parenrightbig(sinhη)m
2m√πΓ/parenleftbig
m+1
2/parenrightbig/integraldisplayπ
0sin2mϕd ϕ
(coshη+s i n h ηcosϕ)n+m+1
2
=(−1)m
2πΓ/parenleftbig
n+1
2/parenrightbig
Γ/parenleftbig
n−m+1
2/parenrightbig/integraldisplay2π
0cosmϕdϕ
(coshη+s i n h ηcosϕ)n+1
2
MO 96
2. Qm
n−1
2(coshη)=(−1)mΓ/parenleftbig
n+1
2/parenrightbig
Γ/parenleftbig
n−m+1
2/parenrightbig/integraldisplay∞
0coshmt dt
(coshη+s i n h ηcosht)n+1
2[n≥m]
=(−1)mΓ/parenleftbig
n+m+1
2/parenrightbig
Γ/parenleftbig
n+1
2/parenrightbig/integraldisplayln cothη
2
0(coshη−sinhηcosht)n−1
2coshmt dt
MO 96
8.852 Functional relations:
1. Qm
n−1
2(coshη)=(−1)m2mΓ/parenleftbig
n+m+1
2/parenrightbig√π
Γ(n+1 )sinhm/parenleftBig
ηe−(n+m+1
2)η/parenrightBig
×F/parenleftbig
m+1
2,n+m+1
2;n+1 ;e−2η/parenrightbig
MO 96
∗Sometimes called torus functions
982 Orthogonal Polynomials 8.853
2. P−m
n−1
2(coshη)=2−2m
Γ(m+1 )/parenleftbig
1−e−2η/parenrightbigme−(n+1
2)ηF/parenleftbig
m+1
2,n+m+1
2;2m+1 ;1 −e−2η/parenrightbig
MO 96
8.853 An asymptotic representation Pn−1
2(coshη) for large values of n:
Pn−1
2(coshη)=Γ(n)e(n−1
2)η
√πΓ/parenleftbig
n+1
2/parenrightbig
×/bracketleftBigg
2Γ2/parenleftbig
n+1
2/parenrightbig
πn!Γ(n)ln (4eη)e−2nηF/parenleftbigg1
2,n+1
2;n+1 ;e−2η/parenrightbigg
+A+B/bracketrightBigg
,
where
A=1+1
221·(2n−1)
1·(n−1)e−2η+1
241·3·(2n−1)(2n−3)
1·2·(n−1)(n−2)e−4η+···+1
22n−2/parenleftbigg(2n−1)!!
(n−1)!/parenrightbigg2
e−2(n−1)η
B=Γ/parenleftbig
n+1
2/parenrightbig
√
π3Γ(n)∞/summationdisplay
k=1Γ/parenleftbig
k+1
2/parenrightbig
Γ/parenleftbigg
n+k+1
2/parenrightbigg
Γ(n+k+1 )Γ ( k+1 )/parenleftBig
un+k+uk−vn+k−1
2−vk−1
2/parenrightBig
e−2(n+k)η
Here,
ur=r/summationdisplay
s=11
s,vr−1
2=r/summationdisplay
s=12
2s−1[ris a natural number] MO 97
8.9 Orthogonal Polynomials
8.90 Introduction
8.901 Suppose that w(x) is a nonnegative real function of a real variable x. Let ( a,b) be a fixed interval
on the x-axis. Let us suppose further that, for n=0,1,2,...,the integral
/integraldisplayb
axnw(x)dx
exists and that the integral
/integraldisplayb
aw(x)dx
is positive. In this case, there exists a sequence of polynomials p0(x),p1(x),...,p n(x),..., that is uniquely
determined by the following conditions:
1. pn(x) is a polynomial of degree nand the coefficient of xnin this polynomial is positive.
2. The polynomials p0(x),p1(x),...are orthonormal; that is,
/integraldisplayb
apn(x)pm(x)w(x)dx=/braceleftBigg
0f o r n/negationslash=m,
1f o r n=m.
We say that the polynomials pn(x) constitute a system of orthogonal polynomials on the interval
(a,b)with the weight function w(x).
8.910 Legendre polynomials 983
8.902 Ifqnis the coefficient of xnin the polynomial pn(x), then
1.n/summationdisplay
k=0pk(x)pk(y)=qn
qn+1pn+1(x)pn(y)−pn(x)pn+1(y)
x−y(Darboux–Christoffel formula)
EH II 159(10)
2.11n/summationdisplay
k=0[pk(x)]2=qn
qn+1/bracketleftbig
pn(x)p/prime
n+1(x)−p/prime
n(x)pn+1(x)/bracketrightbig
EH II 159(11)
8.903 Between any three consecutive orthogonal polynomials, there is a dependence
pn(x)=(Anx+Bn)pn−1(x)−Cnpn−2(x)[ n=2,3,4,...]
In this formula, An,Bn,a n d Cnare constants and
An=qn
qn−1,C n=qnqn−2
q2
n−1MO 102
8.904 Examples of normalized systems of orthogonal polynomials:
Notation and name Interval Weight/parenleftbig
n+1
2/parenrightbig1/2Pn(x)s e e 8.91 (−1,+1) 1
2λΓ(λ)/bracketleftbigg(n+λ)n!
2πΓ( 2λ+n)/bracketrightbigg1/2
Cλ
n(x)s e e 8.93 (−1,+1)/parenleftbig
1−x2/parenrightbigλ−1
2
/radicalbiggεn
πTn(x),ε0=1,εn=2f o r n=1,2,3,... see8.94 (−1,+1)/parenleftbig
1−x2/parenrightbig−1/2
2−n
2π−1/4(n!)−1/2Hn(x)s e e 8.95 (−∞,∞) e−x2
/bracketleftbiggΓ(n+1 )Γ ( α+β+1+ n)(α+β+1+2 n)
Γ(α+1+ n)Γ(β+1+ n)2α+β+1/bracketrightbigg1/2
P(α,β)
n(x)s e e 8.96 (−1,+1) (1−x)α(1 +x)β
/bracketleftbiggΓ(n+1 )
Γ(α+n+1 )/bracketrightbigg1/2
(−1)nLα
n(x)s e e 8.97 (0,∞) xαe−x
Cf.7.221 1,7.313 ,7.343 ,7.374 1,7.391 1,7.414 3.
8.91 Legendre polynomials
8.910 Definition. The Legendre polynomials Pn(z) are polynomials satisfying equation 8.700 1 with
μ=0a n d ν=n: that is, they satisfy the differential equation
1./parenleftbig
1−z2/parenrightbigd2u
dz2−2zdu
dz+n(n+1 )u=0
This equation has a polynomial solution if, and only if, nis an integer. Thus, Legendre polyno-
mials constitute a special type of associated Legendre function.
Legendre polynomials of degree nare of the form
2. Pn(z)=1
2nn!dn
dzn/parenleftbig
z2−1/parenrightbign
984 Orthogonal Polynomials 8.911
8.911 Legendre polynomials written in expanded form:
1. Pn(z)=1
2n⌊n
2⌋/summationdisplay
k=0(−1)k(2n−2k)!
k!(n−k)!(n−2k)!zn−2k
=(2n)!
2n(n!)2/parenleftbigg
zn−n(n−1)
2(2n−1)zn−2+n(n−1)(n−2)(n−3)
2·4(2n−1)(2n−3)zn−4−.../parenrightbigg
=(2n−1)!!
n!znF/parenleftbigg
−n
2,1−n
2;1
2−n;1
z2/parenrightbigg
HO 13, AD (9001), MO 69
2. P2n(z)=(−1)n(2n−1)!!
2nn!/parenleftbigg
1−2n(2n+1 )
2!z2+2n(2n−2)(2n+ 1)(2 n+3 )
4!z4−.../parenrightbigg
=(−1)n(2n−1)!!
2nn!F/parenleftbigg
−n, n+1
2;1
2;z2/parenrightbigg
AD (9002), MO 69
3. P2n+1(z)=(−1)n(2n+1 ) ! !
2nn!/parenleftbigg
z−2n(2n+3 )
3!z3+2n(2n−2)(2n+ 3)(2 n+5 )
5!z5−.../parenrightbigg
=(−1)n(2n+1 ) ! !
2nn!zF/parenleftbigg
−n, n+3
2;3
2;z2/parenrightbigg
AD (9002), MO 69
4. Pn(cosϕ)=(2n−1)!!
2nn!⎛
⎝cosnϕ+1
1n
2n−1cos(n−2)ϕ
+1·3
1·2n(n−1)
(2n−1)(2n−3)cos(n−4)ϕ
+1·3·5
1·2·3n(n−1)(n−2)
(2n−1)(2n−3)(2n−5)cos(n−6)ϕ−...⎞
⎠
WH
5. P2n(cosϕ)=(−1)n(2n−1)!!
2nn!
×/braceleftbigg
sin2nϕ−(2n)2
2!sin2n−2ϕcos2ϕ+···+(−1)n2nn!
(2n−1)!!cos2nϕ/bracerightbigg
AD (9011)
6. P2n+1(cosϕ)=(−1)n(2n+1 ) ! !
2nn!cosϕ
×/braceleftbigg
sin2nϕ−(2n)2
3!sin2n−2ϕcos2ϕ+···+(−1)n2nn!
(2n+1 ) ! !cos2nϕ/bracerightbigg
AD (9012)
7. Pn(z)=n/summationdisplay
k=0(−1)k(n+k)!
(n−k)! (k!)22k+1/bracketleftbig
(1−z)k+(−1)n(1 +z)k/bracketrightbig
WH
8.914 Legendre polynomials 985
8.912 Special cases:
1. P0(x)=1 JA
2. P1(x)=x=c o s ϕ JA
3. P2(x)=1
2/parenleftbig
3x2−1/parenrightbig
=1
4(3 cos2 ϕ+1 ) JA
4. P3(x)=1
2/parenleftbig
5x3−3x/parenrightbig
=1
8(5 cos 3 ϕ+ 3cos ϕ) JA
5. P4(x)=1
8/parenleftbig
35x4−30x2+3/parenrightbig
=1
64(35cos4 ϕ+2 0c o s2 ϕ+9 ) JA
6. P5(x)=1
8/parenleftbig
63x5−70x3+1 5x/parenrightbig
=1
128(63cos5 ϕ+3 5c o s3 ϕ+3 0c o s ϕ) JA
7.10P6(x)=1
16/parenleftbig
231x6−315x4+ 105 x2−5/parenrightbig
=1
512(231cos6 ϕ+ 126cos4 ϕ+ 105cos2 ϕ+ 50)
8. P7(x)=1
16/parenleftbig
429x7−693x5+ 315 x3−35x/parenrightbig
=1
1024(429cos7 ϕ+ 231cos5 ϕ+ 189cos3 ϕ+ 175cos ϕ)
9. P8(x)=1
128/parenleftbig
6435x8−12012 x6+ 6930 x4−1260x2+3 5/parenrightbig
=1
16384(6435cos8 ϕ−3432cos6 ϕ+ 2772cos4 ϕ−2520cos2 ϕ+ 1225)
8.913 Integral representations:
1. Pn(cosϕ)=2
π/integraldisplayπ
ϕsin/parenleftbig
n+1
2/parenrightbig
t/radicalbig
2(cos ϕ−cost)dt WH
See also 3.611 3,3.661 3, 4.
2.7Schl¨afli’s integral formula:
Pn(z)=1
2πi/integraldisplay
C/parenleftbig
t2−1/parenrightbign
2n(t−z)n+1dt,
withCa simple contour containing z. SA 175(9)
3.10Laplace integral formula:
Pn(z)=1
π/integraldisplayπ
0/bracketleftBig
x+/parenleftbig
x2−1/parenrightbig1/2cosϕ/bracketrightBign
dϕ [|x|≤1] SA 180(19)
Functional relations
8.914 Recurrence formulas:
1. ( n+1 )Pn+1(z)−(2n+1 )zPn(z)+nPn−1(z)=0 WH
986 Orthogonal Polynomials 8.915
2./parenleftbig
z2−1/parenrightbigdPn
dz=n[zPn(z)−Pn−1(z)] =n(n+1 )
2n+1[Pn+1(z)−Pn−1(z)] WH
8.915
1.10n/summationdisplay
k=0(2k+1 )Pk(x)Pk(y)=(n+1 )Pn(x)Pn+1(y)−Pn(y)Pn+1(x)
y−x
(Christoffel summation formula)
MO 70
1(1)10.(y−x)n/summationdisplay
k=0(2k+1 )Pk(x)Qk(y)=1−(n+1 )/bracketleftbig
Pn+1(x)Qn(y)−Pn(x)Qn+1(y)/bracketrightbig
AS 335(8.9.2)
2.7⌊n−1
2⌋/summationdisplay
k=0(2n−4k−1)Pn−2k−1(z)=P/prime
n(z) (summation theorem) MO 70
3.7⌊n−2
2⌋/summationdisplay
k=0(2n−4k−3)Pn−2k−2(z)=zP/prime
n(z)−nPn(z) SM 491(42), WH
4.10⌊n
2⌋/summationdisplay
k=1(2n−4k+1 ) [k(2n−2k+1 )−2]Pn−2k(z)=z2P/prime/prime
n(z)−n(n−1)Pn(z) WH
5.11m/summationdisplay
k=0am−kakan−k
an+m−k/parenleftbigg2n+2m−4k+1
2n+2m−2k+1/parenrightbigg
Pn+m−2k(z)=Pn(z)Pm(z)
/bracketleftbigg
ak=(2k−1)!!
k!,m≤n/bracketrightbigg
AD (9036)
8.916
1. Pn(cosϕ)=(2n−1)!!
2nn!e∓inϕF/parenleftbigg1
2,−n;1
2−n;e±2iϕ/parenrightbigg
MO 69
2. Pn(cosϕ)=F/parenleftBig
n+1,−n;1;s in2ϕ
2/parenrightBig
MO 69
3. Pn(cosϕ)=(−1)nF/parenleftBig
n+1,−n;1;c os2ϕ
2/parenrightBig
WH
4. Pn(cosϕ) = cosnϕF/parenleftbigg
−1
2n,1
2−1
2n;1;−tan2ϕ/parenrightbigg
HO 23
5. Pn(cosϕ) = cos2nϕ
2F/parenleftBig
−n,−n;1;−tan2ϕ
2/parenrightBig
HO 23, 29, WH
See also 8.911 1,8.911 2,8.911 3. For a connection with other functions, see 8.936 3,8.836 ,8.962 2.
•For integrals of Legendre polynomials, see 7.22–7.25 .
•For the zeros of Legendre polynomials, see 8.785 .
8.918 Legendre polynomials 987
8.917 Inequalities:
1. P0(x)<P1(x)<P2(x)<···<Pn(x)<... [x>1] MO 71
2. For x>−1,P0(x)+P1(x)+···+Pn(x)>0. MO 71
3. [ Pn(cosϕ)]2>sin(2n+1 )ϕ
(2n+1 )s i n ϕMO 71
4.√nsinϕ|Pn(cosϕ)|≤1. MO 71
5. |Pn(cosϕ)|≤1. WH
6.10Letn≥2. The successive relative maxima of |Pn(x)|,w h e n xdecreases from 1 to 0, form a
decreasing sequence. More precisely, if μ1,μ2,...,μ ⌊n/2⌋denote these maxima corresponding to
decreasing values of x,w eh a v e
1>μ1>μ2>···>μ⌊n/2⌋ SZ 162(7.3.1)
7.10Letn≥2. The successive relative maxima of (sin θ)1/2|Pn(cosθ)|when θincreases from 0 to
π/2, form an increasing sequence. SZ 163(7.3.2)
8.10We have
(sinθ)1/2|Pn(cosθ)|<(2/π)1/2n−1/2[0≤qθ≤qπ] SZ 163(7.3.8)
Here the constant (2 /π)1/2cannot be replaced by a smaller one.
9.10max
0≤qθ≤qπ(sinθ)1/2|Pn(cosθ)|∼=(2/π)1/2n−1
2 [n→∞] SZ 164(7.3.12)
10.10Stieltjes’ first theorem:
|Pn(cosθ)|≤/parenleftbigg2
π/parenrightbigg1/24√
nsinθ[n=1,2,...,0<θ<π ] SA 197(8)
11.10Stieltjes’ second theorem:
|Pn(x)−Pn+2(x)|<4√π√n+2[|x|≤1] SA 199(15)
12.10/vextendsingle/vextendsingle/vextendsingle/vextendsingledP
n(x)
dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle<2
√π√n
1−x2[|x|<1,n=1,2,...] SA 201(18)
13.10|Pn+1(x)+Pn(x)|<6/parenleftbigg2
πn/parenrightbigg1
2
(1−x)−1/2[|x|<1,n=0,1,...] SA 201(19)
8.91810Asymptotic approximations:
1. Pn(cosθ)=/parenleftbigg2
πnsinϕ/parenrightbigg1/2
cos/bracketleftbigg/parenleftbigg
n+1
2/parenrightbigg
θ−π
4/bracketrightbigg
+O/parenleftBig
n−3/2/parenrightBig
[ε≤θ≤π−ε,0<ε<π / 2m] (Laplace’s formula) SA 208(1)
988 Orthogonal Polynomials 8.921
2. Pn(cosθ)=/parenleftbigg2
πnsinθ/parenrightbigg1/2/braceleftbigg/parenleftbigg
1−1
4n/parenrightbigg
cos/bracketleftbigg/parenleftbigg
n+1
2/parenrightbigg
θ−π
4/bracketrightbigg
+1
8ncosθsin/bracketleftbigg/parenleftbigg
n+1
2/parenrightbigg
θ−π
4/bracketrightbigg/bracerightbigg
+O/parenleftBig
n−5/2/parenrightBig
[ε≤θ≤π−ε,0<ε<π / 2] (Bonnet–Heine formula) SA 208(2)
8.91910Series of products of Legendre and Chebyshev polynomials
1. 2/integraldisplay1
−1Tn(x)Pn(x)dx=i+j=n/summationdisplay
i,j=0/integraldisplay1
−1Pi(x)Pj(x)Pn(x)dx
8.92 Series of Legendre polynomials
8.921 The generating function:
1√
1−2tz+t2=∞/summationdisplay
k=0tkPk(z)/bracketleftBig
|t|<min/vextendsingle/vextendsingle/vextendsinglez±/radicalbig
z2−1/vextendsingle/vextendsingle/vextendsingle/bracketrightBig
SM 489(31), WH
=∞/summationdisplay
k=01
tk+1Pk(z)/bracketleftBig
|t|>max/vextendsingle/vextendsingle/vextendsinglez±/radicalbig
z2−1/vextendsingle/vextendsingle/vextendsingle/bracketrightBig
MO 70
8.922
1. z2n=1
2n+1P0(z)+∞/summationdisplay
k=1(4k+1 )2n(2n−2)...(2n−2k+2 )
(2n+ 1)(2 n+3 )...(2n+2k+1 )P2k(z) MO 72
2. z2n+1=3
2n+3P1(z)+∞/summationdisplay
k=1(4k+3 )2n(2n−2)...(2n−2k+2 )
(2n+ 3)(2 n+5 )...(2n+2k+3 )P2k+1(z) MO 72
3.1√
1−x2=π
2∞/summationdisplay
k=0(4k+1 )/braceleftbigg(2k−1)!!
2kk!/bracerightbigg2
P2k(x)[ |x|<1,(−1)!!≡1]
MO 72, LA 385(15)
4.x√
1−x2=π
2∞/summationdisplay
k=0(4k+3 )(2k−1)!!(2k+1 ) ! !
22k+1k!(k+1 ) !P2k+1(x)
[|x|<1,(−1)!!≡1] LA 385(17)
5./radicalbig
1−x2=π
2/braceleftBigg
1
2−∞/summationdisplay
k=1(4k+1 )(2k−3)!!(2k−1)!!
22k+1k!(k+1 ) !P2k(x)/bracerightBigg
[|x|<1,(−1)!!≡1] LA 385(18)
6.10/radicalbigg
1−x
2=2
3P0(x)−2∞/summationdisplay
n=11
(2n−1)(2n+3 )Pn(x)[ −1≤x≤1]
8.925 Series of Legendre polynomials 989
7.101−ρ2
(1−2ρx+ρ2)1/2=1+∞/summationdisplay
n=0(2n+1 )ρnPn(x), [|ρ|<1,|x|≤1] SA 170(4)
8.923 arcsin x=π
2∞/summationdisplay
k=1/braceleftbigg(2k−1)!!
2kk!/bracerightbigg2
[P2k+1(x)−P2k−1(x)] +πx/2
[|x|<1,(−1)!!≡1] WH
8.924
1. −1 + cos nπ
2(n2−1)P0(cosθ)−1 + cos nπ
2∞/summationdisplay
k=0(4k+5 )n2/parenleftbig
n2−22/parenrightbig
.../bracketleftbig
n2−(2k)2/bracketrightbig
(n2−12)(n2−32)...[n2−(2k+3 )2]P2k+2(cosθ)
−3(1−cosnπ)
2(n2−22)P1(cosθ)
−1−cosnπ
2∞/summationdisplay
k=1(4k+3 )/parenleftbig
n2−12/parenrightbig
.../bracketleftbig
n2−(2k−1)2/bracketrightbig
(n2−22)(n2−42)...[n2−(2k+2 )2]P2k+1(cosθ) = cos nθ
AD (9060.1)
2.−sinnπ
2(n2−1)P0(cosθ)−sinnπ
2∞/summationdisplay
k=0(4k+5 )n2/parenleftbig
n2−22/parenrightbig
.../bracketleftbig
n2−(2k)2/bracketrightbig
(n2−12)(n2−32)...[n2−(2k+3 )2]P2k+2(cosθ)
+3s innπ
2(n2−22)P1(cosθ)
+sinnπ
2∞/summationdisplay
k=1(4k+3 )/parenleftbig
n2−12/parenrightbig/parenleftbig
n2−32/parenrightbig
.../bracketleftbig
n2−(2k−1)2/bracketrightbig
(n2−22)(n2−42)...[n2−(2k+2 )2]P2k+1(cosθ)=s i n nθ
AD (9060.2)
3.32n−1n!
(2n−1)!!Pn(cosθ)−n⌊n/2⌋/summationdisplay
k=1(2n−4k+1 )2n−2k−1(n−k−1)!(2k−3)!!
(2n−2k+1 ) ! !k!Pn−2k(cosθ)
=c o s nθ
AD (9061.1)
4.(2n−1)!!Pn−1(cosθ)
2n−1(n−1)!−n
2n+1∞/summationdisplay
k=0(2n+2k−1)!!(2k−1)!! (2 n+4k+3 )
22k(n+k+1 ) ! ( k+1 ) !Pn+2k+1(cosθ)
=4s innθ
π
AD (9061.2)
8.925
1.∞/summationdisplay
k=14k−1
22k(2k−1)2/bracketleftbigg(2k−1)!!
k!/bracketrightbigg2
P2k−1(cosθ)=1−2θ
π
2.∞/summationdisplay
k=14k+1
22k+1(2k−1)(k+1 )/bracketleftbigg(2k−1)!!
k!/bracketrightbigg2
P2k(cosθ)=1
2−2s inθ
πAD (9062.2)
3.∞/summationdisplay
k=1k(4k−1)
22k−1(2k−1)/bracketleftbigg(2k−1)!!
k!/bracketrightbigg2
P2k−1(cosθ)=2c otθ
πAD (9062.3)
990 Orthogonal Polynomials 8.926
4.∞/summationdisplay
k=14k+1
22k/bracketleftbigg(2k−1)!!
k!/bracketrightbigg2
P2k(cosθ)=2
πsinθ−1 AD (9062.4)
8.926
1.∞/summationdisplay
n=11
nPn(cosθ)=l n2t anπ−θ
4
sinθ=−ln sinθ
2−ln/parenleftbigg
1+s i nθ
2/parenrightbigg
AD (9063.2)
2.∞/summationdisplay
n=11
n+1Pn(cosθ)=l n1+s i nθ
2
sinθ
2−1 AD (9063.1)
8.927∞/summationdisplay
k=0cos/parenleftbig
k+1
2/parenrightbig
βPk(cosϕ)=1/radicalbig
2(cos β−cosϕ)[0≤β<ϕ<π ]
=0 [ 0 <ϕ<β<π ]
MO 72
8.928
1.∞/summationdisplay
n=1(−1)n(4k+1 )[ ( 2 n−1)!!]3
23n(n!)3P2n(cosθ)=4K(sinθ)
π2−1 AD (9064.1)
2.∞/summationdisplay
n=1(−1)n+1(4n+1 )[ ( 2 n−1)!!]3
(2n−1)(2n+2 ) 23n(n!)3P2n(cosθ)=4E(sinθ)
π2−1
2AD (9064.2)
•For series of products of Bessel functions and Legendre polynomials, see 8.511 4,8.531 3,
8.533 1,8.533 2, and 8.534 .
•For series of products of Legendre and Chebyshev polynomials, see 8.919 .
8.93 Gegenbauer polynomials Cλ
n(t)
8.930 Definition. The polynomials Cλ
n(t) of degree nare the coefficients of αnin the power-series
expansion of the function
/parenleftbig
1−2tα+α2/parenrightbig−λ=∞/summationdisplay
n=0Cλ
n(t)αnWH
Thus, the polynomials Cλ
n(t)a r ea generalization of the Legendre polynomials .
1.10Cλ
0(t)=1
2.10Cλ
1(t)=2λt
3.10Cλ
2(t)=2λ(λ+1 )t2−λ
4.10Cλ
3(t)=1
3λ/parenleftbig
4λ2+1 2λ+8/parenrightbig
t3−2λ(λ+1 )t
5.11Cλ
4(t)=2
3λ/parenleftbig
λ3+6λ2+1 1λ+6/parenrightbig
t4−2λ/parenleftbig
λ2+3λ+2/parenrightbig
t2+1
2λ(λ+1 )
6.10Cλ
5(t)=1
15λ/parenleftbig
4λ4+4 0λ3+ 140 λ2+ 200 λ+9 6/parenrightbig
t5
−1
3λ/parenleftbig
4λ3+2 4λ2+4 4λ+2 4/parenrightbig
t3+λ/parenleftbig
λ2+3λ+2/parenrightbig
t
8.934 Gegenbauer polynomials Cλ
n(t) 991
7.10Cλ
6(t)=1
45λ/parenleftbig
λ5+6 0λ4+ 340 λ3+ 900 λ2+ 1096 λ+ 480/parenrightbig
t6
−1
3λ/parenleftbig
2λ4+2 0λ3+7 0λ2+ 100 λ+4 8/parenrightbig
t4
+λ/parenleftbig
λ3+6λ2+1 1λ+6/parenrightbig
t2+1
6λ/parenleftbig
λ2+3λ+2/parenrightbig
8.931 Integral representation:
Cλ
n(t)=1√πΓ(2λ+n)
n!Γ ( 2λ)Γ/parenleftbig2λ+1
2/parenrightbig
Γ(λ)/integraldisplayπ
0/parenleftBig
t+/radicalbig
t2−1c osϕ/parenrightBign
sin2λ−1ϕdϕ MO 99
See also 3.252 11,3.663 2,3.664 4.
Functional relations
8.932 Expressions in terms of hypergeometric functions:
1. Cλ
n(t)=Γ(2λ+n)
Γ(n+1 )Γ ( 2 λ)F/parenleftbigg
2λ+n,−n;λ+1
2;1−t
2/parenrightbigg∗
MO 97
=2nΓ(λ+n)
n!Γ (λ)tnF/parenleftbigg
−n
2,1−n
2;1−λ−n;1
t2/parenrightbigg
MO 99
2. Cλ
2n(t)=(−1)n
(λ+n)B(λ,n+1 )F/parenleftbigg
−n, n+λ;1
2;t2/parenrightbigg
MO 99
3. Cλ
2n+1(t)=(−1)n2t
B(λ,n+1 )F/parenleftbigg
−n, n+λ+1 ;3
2;t2/parenrightbigg
MO 99
8.933 Recursion formulas:
1. ( n+2 )Cλ
n+2(t)=2 ( λ+n+1 )tCλ
n+1(t)−(2λ+n)Cλ
n(t) Mo 98
2. nCλ
n(t)=2λ/bracketleftBig
tCλ+1
n−1(t)−Cλ+1
n−2(t)/bracketrightBig
WH
3. (2 λ+n)Cλ
n(t)=2λ/bracketleftBig
Cλ+1
n(t)−tCλ+1
n−1(t)/bracketrightBig
WH
4. nCλ
n(t)=( 2 λ+n−1)tCλ
n−1(t)−2λ/parenleftbig
1−t2/parenrightbig
Cλ+1
n−2(t) WH
8.934
1. Cλ
n(t)=(−1)n
2nΓ(2λ+n)Γ/parenleftbig2λ+1
2/parenrightbig
Γ(2λ)Γ/parenleftbig2λ+1
2+n/parenrightbig/parenleftbig
1−t2/parenrightbig1
2−λ
n!dn
dtn/bracketleftBig/parenleftbig
1−t2/parenrightbigλ+n−1
2/bracketrightBig
WH
2. Cλ
n(cosϕ)=n/summationdisplay
k,l=0
k+l=nΓ(λ+k)Γ(λ+l)
k!l![Γ (λ)]2cos(k−l)ϕ MO 99
∗Equation 8.932.1 defines the generalized functions Cλ
n(t), where the subscript nc a nb ea na r b i t r a r yn u m b e r .
992 Orthogonal Polynomials 8.935
3. Cλ
n(cosψcosϑ+s i nψsinϑcosϕ)
=Γ(2λ−1)
[Γ(λ)]2n/summationdisplay
k=022k(n−k)![Γ(λ+k)]2
Γ(2λ+n+k)(2λ+2k−1)sinkψsinkϑ
×Cλ+k
n−k(cosψ)Cλ+k
n−k(cosϑ)Cλ−1
2
k(cosϕ)
/bracketleftbig
ψ,ϑ,ϕ real; λ/negationslash=1
2/bracketrightbig
[“summation theorem”] (see also 8.794–8.796 )WH
4. lim
λ→0Γ(λ)Cλ
n(cosϕ)=2c osnϕ
nMO 98
For orthogonality, see 8.904, 7.313 .
8.935 Derivatives:
1.dk
dtkCλ
n(t)=2kΓ(λ+k)
Γ(λ)Cλ+k
n−k(t) MO 99
In particular,
2.11dCλ
n(t)
dt=2λCλ+1
n−1(t) WH
For integrals of the polynomials Cλ
n(x)s e e7.31–7.33 .
8.936 Connections with other functions:
1. Cλ
n(t)=Γ(2λ+n)Γ/parenleftbig
λ+1
2/parenrightbig
Γ(2λ)Γ(n+1 )/braceleftbigg1
4/parenleftbig
t2−1/parenrightbig/bracerightbigg1
4−λ
2
P1
2−λ
λ+n−1
2(t) MO 98
2. Cm+1
2
n−m(t)=1
(2m−1)!!dmPn(t)
dtm=(−1)m/parenleftbig
1−t2/parenrightbig−m
2m!2m
(2m)!Pm
n(t)
[m+ 1 a natural number] MO 98, WH
3. C1/2
n(t)=Pn(t)
4. Jλ−1
2(rsinϑsinα)(rsinϑsinα)−λ+1
2e−ircosϑcosα
=√
2Γ(λ)
Γ/parenleftbig
λ+1
2/parenrightbig∞/summationdisplay
k=0(λ+k)i−kJλ+k(r)Cλ
k(cosϑ)Cλ
k(cosα)
rλCλ
k(1)
MO 99
5. lim
λ→∞λ−n
2Cλ
2n/parenleftBigg
t/radicalbigg
2
λ/parenrightBigg
=2−n
2
n!Hn(t) MO 99a
See also 8.932 .
8.937 Special cases and particular values:
1. C1
n(cosϕ)=sin(n+1 )ϕ
sinϕMO 99
2. C0
0(cosϕ)=1 MO 98
3. Cλ
0(t)≡1 MO 98
8.940 The Chebyshev polynomials 993
4. Cλ
n(1)≡/parenleftbigg2λ+n−1
n/parenrightbigg
MO 98
8.938 A differential equation leading to the polynomials Cλ
n(t):
y/prime/prime+(2λ+1 )t
t2−1y/prime−n(2λ+n)
t2−1y=0 ( c f . 9.174 ) WH
For series of products of Bessel functions and the polynomials Cλ
n(x), see8.532, 8.534 .
8.93910Differentiation and Rodrigues’ formulas and orthogonality relation
1.d
dtCλ
n(t)=2λCλ+1
n−1(t) MS 5.3.2
2.dm
dtmCλ
n(t)=2mλ(λ+1 ) (λ+2 )...(λ+m−1)Cλ+m
n−m(t) MS 5.3.2
3.d
dtCλ
n−1(t)=td
dtCλ
n(t)−nCλ
n(t) MS 5.3.2
4.d
dtCλ
n+1(t)=td
dtCλ
n(t)+( 2 λ+n)Cλ
n(t) MS 5.3.2
5./parenleftbig
1−t2/parenrightbigd
dtCλ
n(t)=(n+2λ−1)Cλ
n−1(t)−ntCλ
n(t)=(n+2λ)tCλ
n(t)−(n+1 )Cλ
n+1(t)
=2λ/parenleftbig
1−t2/parenrightbig
Cλ+1
n−1(t)
MS 5.3.2
6.d
dt/bracketleftBig
Cλ
n+1(t)−Cλ
n−1(t)/bracketrightBig
=2 (n+λ)Cλ
n(t) MS 5.3.2
7. Cλ
n(t)=(−1)n2λ(2λ+ 1)(2 λ+2 )...(2λ+n−1)/parenleftbig
1−t2/parenrightbig1
2−λ
2nn!/parenleftbig
λ+1
2/parenrightbig/parenleftbig
λ+3
2/parenrightbig
.../parenleftbig
λ+n−1
2/parenrightbigdn
dtn/bracketleftBig/parenleftbig
1−t2/parenrightbign+λ−1
2/bracketrightBig
=(−1)nΓ/parenleftbig
λ+1
2/parenrightbig
Γ(n+2λ)/parenleftbig
1−t2/parenrightbig1
2−λ
2nn!Γ ( 2λ)Γ/parenleftbig
n+λ+1
2/parenrightbigdn
dtn/bracketleftBig/parenleftbig
1−t2/parenrightbign+λ−1
2/bracketrightBig
[Rodrigues’ formula] MS 5.3.2
8./integraldisplay1
−1Cλ
n(t)Cλ
m(t)/parenleftbig
1−t2/parenrightbigλ−1
2dt=0 n/negationslash=m
=π21−2λΓ(n+2λ)
n!(λ+n)[Γ(λ)]2n=m
[λ/negationslash= 0] [Orthogonality relation] MS 5.3.2
8.94 The Chebyshev polynomials Tn(x)andUn(x)
8.940 Definition
1. Chebyshev’s polynomials of the first kind
Tn(x)=c os( narccos x)=1
2/bracketleftBig/parenleftBig
x+i/radicalbig
1−x2/parenrightBign
+/parenleftBig
x−i/radicalbig
1−x2/parenrightBign/bracketrightBig
=xn−/parenleftBign
2/parenrightBig
xn−2/parenleftbig
1−x2/parenrightbig
+/parenleftBign
4/parenrightBig
xn−4/parenleftbig
1−x2/parenrightbig2−/parenleftBign
6/parenrightBig
xn−6/parenleftbig
1−x2/parenrightbig3+...
NA 66, 71
994 Orthogonal Polynomials 8.941
2. Chebyshev’s polynomials of the second kind:
Un(x)=sin [(n+ 1)arccos x]
sin [arccos x]=1
2i√
1−x2/bracketleftbigg/parenleftBig
x+i/radicalbig
1−x2/parenrightBign+1
−/parenleftBig
x−i/radicalbig
1−x2/parenrightBign+1/bracketrightbigg
=/parenleftbiggn+1
1/parenrightbigg
xn−/parenleftbiggn+1
3/parenrightbigg
xn−2/parenleftbig
1−x2/parenrightbig
+/parenleftbiggn+1
5/parenrightbigg
xn−4/parenleftbig
1−x2/parenrightbig2−...
Functional relations
8.941 Recursion formulas:
1. Tn+1(x)−2xTn(x)+Tn−1(x)=0 NA 358
2. Un+1(x)−2xUn(x)+Un−1(x)=0
3. Tn(x)=Un(x)−xUn−1(x) EH II 184(3)
4./parenleftbig
1−x2/parenrightbig
Un−1(x)=xTn(x)−Tn+1(x) EH II 184(4)
For the orthogonality, see 7.343 and8.904 .
8.942 Relations with other functions:
1. Tn(x)=F/parenleftbigg
n,−n;1
2;1−x
2/parenrightbigg
MO 104
2. Tn(x)=(−1)n√
1−x2
(2n−1)!!dn
dxn/parenleftbig
1−x2/parenrightbign−1
2MO 104
3. Un(x)=(−1)n(n+1 )√
1−x2(2n+1 ) ! !dn
dxn/parenleftbig
1−x2/parenrightbign+1
2EH II 185(15)
See also 8.962 3.
8.94310Special cases
1. T0(x)=1
2. T1(x)=x
3. T2(x)=2x2−1
4. T3(x)=4x3−3x
5. T4(x)=8x4−8x2+1
6. T5(x)=1 6 x5−20x3+5x
7. T6(x)=3 2 x6−48x4+1 8x2−1
8. T7(x)=6 4 x7−112x5+5 6x3−7x
9. T8(x) = 128 x8−256x6+ 160x4−32x2+110. U0(x)=1
11. U1(x)=2x
12. U2(x)=4x2−1
13. U3(x)=8x3−4x
14. U4(x)=1 6 x4−12x2+1
15. U5(x)=3 2 x5−32x3+6x
16. U6(x)=6 4 x6−80x4+2 4x2−1
17. U7(x) = 128 x7−192x5+8 0x3−8x
18. U8(x) = 256 x8−448x6+ 240x4−40x2+1
8.949 The Chebyshev polynomials 995
8.944 Particular values:
1. Tn(1) = 1
2. Tn(−1) = (−1)n
3. T2n(0) = ( −1)n
4. T2n+1(0) = 05. U2n+1(0) = 0
6. U2n(0) = ( −1)n
8.945 The generating function:
1.111−t2
1−2tx+t2=T0(x)+2∞/summationdisplay
k=1Tk(x)tk[|t|<1] MO 104
2.111
1−2tx+t2=∞/summationdisplay
k=0Uk(x)tk[|t|<1] MO 104a, EH II 186(31)
8.946 Zeros. The polynomials Tn(x)a n d Un(x) only have real simple zeros. All these zeros lie in
the interval ( −1,+1).
8.947 The functions Tn(x)a n d√
1−x2Un−1(x) are two linearly independent solutions of the differ-
ential equation
/parenleftbig
1−x2/parenrightbigd2y
dx2−xdy
dx+n2y=0. NA 69(58)
8.948 Of all polynomials of degree nwith leading coefficient equal to 1, the one that deviates the least
from zero on the interval [ −1,+1] is the polynomial 2−n+1Tn(x).
8.94910Differentiation and Rodrigues’ formulas and orthogonality relations
1.d
dxTn(x)=nUn−1(x) MS 5.7.2
2.dm
dxmTn(x)=2m−1Γ(m)nCm
n−m(x) MS 5.7.2
3./parenleftbig
1−x2/parenrightbigd
dxTn(x)=n[Tn−1(x)−xTn(x)] =n[xTn(x)−Tn+1(x)] MS 5.7.2
4.d
dxUn(x)=2C2
n−1(x) MS 5.7.2
5.dm
dxmUn(x)=2mm!Cm+1
n−m(x) MS 5.7.2
6./parenleftbig
1−x2/parenrightbigd
dxUn(x)=(n+1 )Un−1(x)−nxUn(x)=(n+2 )xUn(x)−(n+1 )Un+1(x)
MS 5.7.2
7. Tn(x)=(−1)nπ1/2/parenleftbig
1−x2/parenrightbigc1
2
2n+1Γ/parenleftbig
n+1
2/parenrightbigdn
dxn/bracketleftBig/parenleftbig
1−x2/parenrightbign−1
2/bracketrightBig
[Rodrigues’ formula] MS 5.7.2
8. Un(x)=(−1)nπ1/2(n+1 )/parenleftbig
1−x2/parenrightbig−1/2
2n+1Γ/parenleftbig
n+3
2/parenrightbigdn
dxn/bracketleftBig/parenleftbig
1−x2/parenrightbign+1
2/bracketrightBig
[Rodrigues’ formula] MS 5.7.2
996 Orthogonal Polynomials 8.950
9./integraldisplay1
−1Tm(x)Tn(x)/parenleftbig
1−x2/parenrightbig−1/2dx=⎧
⎪⎨
⎪⎩0,m /negationslash=n
π/2,m=n/negationslash=0
π, m =n=0
[Orthogonality relation] MS 5.7.2
10./integraldisplay1
−1Um(x)Un(x)/parenleftbig
1−x2/parenrightbig−1/2dx=/braceleftBigg
0,m /negationslash=n
π/8,m=n
[Orthogonality relation] MS 5.7.2
8.95 The Hermite polynomials Hn(x)
8.950 Definition
1. Hn(x)=(−1)nex2dn
dxn/parenleftBig
e−x2/parenrightBig
SM 567(14)
or
2. Hn(x)=2nxn−2n−1/parenleftBign
2/parenrightBig
xn−2+2n−2·1·3·/parenleftBign
4/parenrightBig
xn−4−2n−3·1·3·5·/parenleftBign
6/parenrightBig
xn−6+... MO 105a
3.10H0(x)=1
4.10H1(x)=2x
5.10H2(x)=4x2−2
6.10H3(x)=8x3−12x
7.10H4(x)=1 6 x4−48x2+1 2
8.10H5(x)=3 2 x5−160x3+ 120 x
9.10H6(x)=6 4 x6−480x4+ 720 x2−120
10.10H7(x) = 128 x7−1344x5+ 3360 x3−1680x
11.10H8(x) = 256 x8−3584x6+ 13440 x4−13440 x2+ 1680
8.951 The integral representation:
Hn(x)=2n
√π/integraldisplay∞
−∞(x+it)ne−t2dt MO 106a
Functional relations
8.952 Recursion formulas:
1.dHn(x)
dx=2nHn−1(x) SM 569(22)
2. Hn+1(x)=2xHn(x)−2nHn−1(x) SM 570(23)
For the orthogonality, see 7.374 1a n d8.904 .
3.10nHn(x)=−nH/prime
n−1(x)+xH/prime
n(x) MS 5.6.2
8.957 Hermite polynomials 997
4.10Hn(x)=2xHn−1(x)−H/prime
n−1(x) MS 5.6.2
8.953 The connection with other functions:
1. H2n(x)=(−1)n(2n)!
n!Φ/parenleftbig
−n,1
2;x2/parenrightbig
MO 106a
2. H2n+1(x)=(−1)n2(2n+1 ) !
n!xΦ/parenleftbig
−n,3
2;x2/parenrightbig
MO 106a
•For a connection with the polynomials Cλ
n(x), see8.936 5.
•For a connection with the Laguerre polynomials, see 8.972 2a n d8.972 3.
•For a connection with functions of a parabolic cylinder, see 9.253 .
8.954 Inequalities:
1.10|Hn(x)|≤2n
2−⌊n
2⌋n!
⌊n/2⌋!e2x√
⌊n/2⌋MO 106a
2.10|Hn(x)|<k√
n!2n/2ex2/2,k≈1.086435 SA 324
8.955 Asymptotic representation:
1. H2n(x)=(−1)n2n(2n−1)!!ex2/2/bracketleftbigg
cos/parenleftbig√
4n+1x/parenrightbig
+O/parenleftbigg1
4√n/parenrightbigg/bracketrightbigg
SM 579
2. H2n+1(x)=(−1)n2n+1
2(2n−1)!!√
2n+1ex2/2/bracketleftbigg
sin/parenleftbig√
4n+3x/parenrightbig
+O/parenleftbigg1
4√n/parenrightbigg/bracketrightbigg
SM 579
8.956 Special cases and particular values:
1. H0(x)=1
2. H1(x)=2x
3. H2(x)=4x2−2
4. H3(x)=8x3−12x
5. H4(x)=1 6 x4−48x2+1 2
6. H2n(0) = ( −1)n2n(2n−1)!! SM 570(24)
7. H2n+1(0) = 0
Series of Hermite polynomials
8.957 The generating function:
1. exp/parenleftbig
−t2+2tx/parenrightbig
=∞/summationdisplay
k=0tk
k!Hk(x) SM 569(21)
2.1
esinh 2x=∞/summationdisplay
k=01
(2k+1 ) !H2k+1(x) MO 106a
998 Orthogonal Polynomials 8.958
3.1
ecosh 2 x=∞/summationdisplay
k=01
(2k)!H2k(x) MO 106a
4. esin 2x=∞/summationdisplay
k=0(−1)k1
(2k+1 ) !H2k+1(x) MO 106a
5. ecos2x=∞/summationdisplay
k=0(−1)k1
(2k)!H2k(x) MO 106a
8.958 “The summation theorem”:
1.11/parenleftBiggr/summationdisplay
k=1a2
k/parenrightBiggn
2
n!Hn⎛
⎜⎜⎜⎜⎝r/summationdisplay
k=1akxk
/radicalbig/summationtexta2
k⎞
⎟⎟⎟⎟⎠=/summationdisplay
m1+m2+···+mr=nr/productdisplay
k=1/braceleftbiggamk
k
mk!Hmk(xk)/bracerightbigg
MO 106a
2. A special case:
2n
2Hn(x+y)=n/summationdisplay
k=0/parenleftBign
k/parenrightBig
Hn−k/parenleftBig
x√
2/parenrightBig
Hk/parenleftBig
y√
2/parenrightBig
MO 107a
8.959 Hermite polynomials satisfy the differential equation
1.d2un
dx2−2xdun
dx+2nun=0 ; SM 566(9)
A second solution of this differential equation is provided by the functions ( AandBare arbitrary
constants):
2. u2n=AxΦ/parenleftbig1
2−n;3
2;x2/parenrightbig
,
3. u2n+1=BΦ/parenleftbig
−1
2−n;1
2;x2/parenrightbig
MO 107
8.959(1)10Rodrigues’ formula and orthogonality relation
1. Hn(x)=(−1)nex2dn
dxn/bracketleftBig
e−x2/bracketrightBig
[Rodrigues’ formula] MS 5.6.2
2./integraldisplay∞
−∞e−x2Hm(x)Hn(x)dx=/braceleftBigg
0f o r m/negationslash=n
π1/22nn!f o r m=nMS 5.6.2
8.96 Jacobi’s polynomials
8.960 Definition
1. P(α,β)
n(x)=(−1)n
2nn!(1−x)−α(1 +x)−βdn
dxn/bracketleftbig
(1−x)α+n(1 +x)β+n/bracketrightbig
EH II 169(10), CO
=1
2nn/summationdisplay
m=0/parenleftbiggn+α
m/parenrightbigg/parenleftbiggn+β
n−m/parenrightbigg
(x−1)n−m(x+1 )mEH II 169(2)
8.962 Jacobi’s polynomials 999
8.961 Functional relations:
1.11P(α,α)
n(−x)=(−1)nP(α,α)
n(x) EH II 169(13)
2. 2( n+1 ) (n+α+β+ 1)(2 n+α+β)P(α,β)
n+1(x)
=( 2n+α+β+1 )/bracketleftbig
(2n+α+β)(2n+α+β+2 )x+α2−β2/bracketrightbig
P(α,β)
n(x)
−2(n+α)(n+β)(2n+α+β+2 )P(α,β)
n−1(x)
EH II 169(11)
3. (2 n+α+β)/parenleftbig
1−x2/parenrightbigd
dxP(α,β)
n(x)=n[(α−β)−(2n+α+β)x]P(α,β)
n(x)
+2(n+α)(n+β)P(α,β)
n−1(x)
EH II 170(15)
4.11dm
dxm/bracketleftBig
P(α,β)
n(x)/bracketrightBig
=1
2mΓ(n+m+α+β+1 )
Γ(n+α+β+1 )P(α+m,β+m)
n−m (x)
[m=1,2,...,n ] EH II 170(17)
5./parenleftbig
n+1
2α+1
2β+1/parenrightbig
(1−x)P(α+1,β)
n (x)=(n+α+1)P(α,β)
n(x)−(n+1)P(α,β)
n+1(x) EH II 173(32)
6./parenleftbig
n+1
2α+1
2β+1/parenrightbig
(1+x)P(α,β+1)
n (x)=(n+β+1)P(α,β)
n(x)+(n+1)P(α,β)
n+1(x) EH II 173(33)
7. (1 −x)P(α+1,β)
n (x)+( 1+ x)P(α,β+1)
n (x)=2P(α,β)
n(x) EH II 173(34)
8. (2 n+α+β)P(α−1,β)
n (x)=(n+α+β)P(α,β)
n(x)−(n+β)P(α,β)
n−1(x) EH II 173(35)
9. (2 n+α+β)P(α,β−1)
n (x)=(n+α+β)P(α,β)
n(x)+(n+α)P(α,β)
n−1(x) EH II 173(36)
10. P(α,β−1)
n (x)−P(α−1,β)
n (x)=P(α,β)
n−1(x) EH II 173(37)
8.962 Connections with other functions:
1. P(α,β)
n(x)=(−1)nΓ(n+1+ β)
n!Γ( 1+ β)F/parenleftbigg
n+α+β+1,−n;1+β;1+x
2/parenrightbigg
CO, EH II 170(16)
=Γ(n+1+ α)
n!Γ ( 1+ α)F/parenleftbigg
n+α+β+1,−n;1+α;1−x
2/parenrightbigg
EH II 170(16)
=Γ(n+1+ α)
n!Γ ( 1+ α)/parenleftbigg1+x
2/parenrightbiggn
F/parenleftbigg
−n,−n−β;α+1 ;x−1
x+1/parenrightbigg
EH II 170(16)
=Γ(n+1+ β)
n!Γ ( 1+ β)/parenleftbiggx−1
2/parenrightbiggn
F/parenleftbigg
−n,−n−α;β+1 ;x+1
x−1/parenrightbigg
EH II 170(16)
2. Pn(x)=P(0,0)
n(x) CO, EH II 179(3)
3. Tn(x)=22n(n!)2
(2n)!P(−1
2,−1
2)
n (x) CO, EH II 184(5)a
4. Cν
n(x)=Γ(n+2ν)Γ/parenleftbig
ν+1
2/parenrightbig
Γ(2ν)Γ/parenleftbig
n+ν+1
2/parenrightbigP(ν−1/2,ν−1/2)
n (x) MO 108a, EH II 174(4)
1000 Orthogonal Polynomials 8.963
8.963 The generating function:
∞/summationdisplay
n=0P(α,β)
n(x)zn=2α+βR−1(1−z+R)−α(1 +z+R)−β,
R=/radicalbig
1−2xz+z2 [|z|<1]
EH II 172(29)
8.964 The Jacobi polynomials constitute the unique rational solution of the differential (hypergeomet-
ric) equation/parenleftbig
1−x2/parenrightbig
y/prime/prime+[β−α−(α+β+2 )x]y/prime+n(n+α+β+1 )y=0. EH II 169(14)
8.965 Asymptotic representation
P(α,β)
n(cosθ)=
cos/braceleftbig/bracketleftbig
n+1
2(α+β+1 )/bracketrightbig
θ−/parenleftbig1
2α+1
4/parenrightbig
π/bracerightbig
√πn/parenleftbig
sin1
2θ/parenrightbigα+1
2/parenleftbig
cos1
2θ/parenrightbigβ+1
2+O/parenleftBig
n−3/2/parenrightBig
[Imα=I mβ=0,0<θ<π ]EH II 198(10)
8.966 A limit relationship:
lim
n→∞/bracketleftBig
n−αP(α,β)
n/parenleftBig
cosz
n/parenrightBig/bracketrightBig
=/parenleftBigz
2/parenrightBig−α
Jα(z) EH II 173(41)
8.967 Ifα>−1a n d β>−1, all the zeros of the polynomial P(α,β)
n(x) are simple, and they lie in the
interval ( −1,1).
8.97 The Laguerre polynomials
8.970 Definition.
1. Lα
n(x)=1
n!exx−αdn
dxn/parenleftbig
e−xxn+α/parenrightbig[Rodrigues’ formula] EH II 188(5), MO 108
=n/summationdisplay
m=0(−1)m/parenleftbiggn+α
n−m/parenrightbiggxm
m!MO 109, EH II 188(7)
2. L0
n(x)=Ln(x) ET I 369
3.10Lα
0(x)=1
4.10Lα
1(x)=−x+α+1
5.10Lα
2(x)=1
2/bracketleftbig
x2−2(α+2 )x+(α+1 ) (α+2 )/bracketrightbig
6.10Lα
3(x)=−1
6/bracketleftbig
x3−3(α+3 )x2+3 (α+2 ) (α+3 )x−(α+1 ) (α+2 ) (α+3 )/bracketrightbig
7.10Lα
4(x)=1
24/bracketleftbigg
x4−4(α+4 )x3+6 (α+3 )(α+4 )x2−4(α+2 ) (α+3 ) (α+4 )x
+(α+1 ) (α+2 ) (α+3 ) (α+4 )/bracketrightbigg
8.10Lα
5(x)=−1
120/bracketleftbigg
x5−5(α+5 )x4+ 10(α+4 ) (α+5 )x3−10(α+3 ) (α+4 ) (α+5 )x2
+5 (α+2 ) (α+3 ) (α+4 ) (α+5 )x−(α+1 ) (α+2 ) (α+3 ) (α+4 ) (α+5 )/bracketrightbigg
8.973 The Laguerre polynomials 1001
8.971 Functional relations:
1.d
dx/bracketleftbig
Lα
n(x)−Lα
n+1(x)/bracketrightbig
=Lα
n(x) EH II 189(16)
2.11d
dxLα
n(x)=−Lα+1
n−1(x)=nLα
n(x)−(n+α)Lα
n−1(x)
xEH II 189(15), SM 575(42)a
3. xd
dxLα
n(x)=nLα
n(x)−(n+α)Lα
n−1(x)
=(n+1 )Lα
n+1(x)−(n+α+1−x)Lα
n(x)
E HI I1 8 9 ( 1 2 ) ,M O1 0 9
4. xLα+1
n(x)=(n+α+1 )Lα
n(x)−(n+1 )Lα
n+1(x)
=(n+α)Lα
n−1(x)−(n−x)Lα
n(x)
SM 575(43)a, EH II 190(23)
5. Lα−1
n(x)=Lα
n(x)−Lα
n−1(x) SM 575(44)a, EH II 190(24)
6. ( n+1 )Lα
n+1(x)−(2n+α+1−x)Lα
n(x)+(n+α)Lα
n−1(x)=0
[n=1,2,...]MO 109, EH II 190(25, 24)
7.10(n+α)Lα−1
n(x)=(n+1 )Lα
n+1(x)−(n+1−x)Lα
n(x) MS 5.5.2
8.10nLα
n(x)=( 2 n+α−1−x)Lα
n−1(x)−(n+α−1)Lα
n−2(x)
[n=2,3,...] MS 5.5.2
8.972 Connections with other functions:
1. Lα
n(x)=/parenleftbiggn+α
n/parenrightbigg
Φ(−n, α+1 ;x) MO 109, FI II 189(14)
2. H2n(x)=(−1)n22nn!L−1/2
n/parenleftbig
x2/parenrightbig
EH II 193(2), SM 576(47)
3. H2n+1(x)=(−1)n22n+1n!xL1/2
n/parenleftbig
x2/parenrightbig
EH II 193(3), SM 577(48)
8.973 Special cases:
1. Lα
0(x)=1 EH II 188(6)
2. Lα
1(x)=α+1−x EH II 188(6)
3. Lα
n(0) =/parenleftbiggn+α
n/parenrightbigg
EH II 189(13)
4. L−n
n(x)=(−1)nxn
n!MO 109
5. L1(x)=1−x
6. L2(x)=1−2x+x2
2MO 109
1002 Orthogonal Polynomials 8.974
8.974 Finite sums:
1.n/summationdisplay
m=0m!
Γ(m+α+1 )Lα
m(x)Lα
m(y)=(n+1 ) !
Γ(n+α+1 ) (x−y)/bracketleftbig
Lα
n(x)Lα
n+1(y)−Lα
n+1(x)Lα
n(y)/bracketrightbig
EH II 188(9)
2.11n/summationdisplay
m=0Γ(α−β+m)
Γ(α−β)m!Lβ
n−m(x)=Lβ
n(x) MO 110, EH II 192(39)
3.n/summationdisplay
m=0Lα
m(x)=Lα+1
n(x) EH II 192(38)
4.11n/summationdisplay
m=0Lα
m(x)Lβ
n−m(y)=Lα+β+1
n (x+y) EH II 192(41)
8.975 Arbitrary functions:
1. (1 −z)−α−1expxz
z−1=∞/summationdisplay
n=0Lα
n(x)zn[|z|<1] EH II 189(17), MO 109
2. e−xz(1 +z)α=∞/summationdisplay
n=0Lα−n
n(x)zn[|z|<1] MO 110, EH II 189(19)
3. Jα/parenleftbig
2√xz/parenrightbig
ez(xz)−1
2α=∞/summationdisplay
n=0zn
Γ(n+α+1 )Lα
n(x)[ α>−1] EH II 189(18), MO 109
8.976 Other series of Laguerre polynomials:
1.∞/summationdisplay
n=0n!Lα
n(x)Lα
n(y)zn
Γ(n+α+1 )=(xyz)−1
2α
1−zexp/parenleftbigg
−zx+y
1−z/parenrightbigg
Iα/parenleftbigg
2√xyz
1−z/parenrightbigg
[|z|<1] EH II 189(20)
2.∞/summationdisplay
n=0Lα
n(x)
n+1=exx−αΓ(α,x)[ α>−1,x > 0] EH II 215(19)
3.6Lα
n(x)2=Γ(n+α+1 )
22nn!n/summationdisplay
k=0/parenleftbigg2n−2k
n−k/parenrightbigg(2k)!
k!1
Γ(α+k+1 )L2α
2k(2x) MO 110
4.6Lα
n(x)Lα
n(y)=Γ(1 + α+n)
n!n/summationdisplay
k=0Lα+2k
n−k(x+y)
Γ(1 + α+k)(xy)k
k!MO 110, EH II 192(42)
8.977 Summation theorems:
1. Lα1+α2+···+αk+k−1
n (x1+x2+···+xk)=/summationdisplay
i1+i2+···+i2=nLα1
i1(x1)Lα2
i2(x2)···Lαk
ik(xk) MO 110
2. Lα
n(x+y)=ey∞/summationdisplay
k=0(−1)k
k!ykLα+k
n(x) MO 110
8.982 The Laguerre polynomials 1003
8.978 Limit relations and asymptotic behavior:
1. Lα
n(x) = lim
β→∞P(α,β)
n/parenleftbigg
1−2x
β/parenrightbigg
EH II 191(35)
2. lim
n→∞/bracketleftBig
n−αLα
n/parenleftBigx
n/parenrightBig/bracketrightBig
=x−1
2αJα/parenleftbig
2√x/parenrightbig
EH II 191(36)
3. Lα
n(x)=1√πe1
2xx−1
2α−1
4n1
2α−1
4cos/bracketleftBig
2√nx−απ
2−π
4/bracketrightBig
+O/parenleftBig
n1
2α−3
4/parenrightBig
[Imα=0,x > 0] EH II 199(1)
8.979 Laguerre polynomials satisfy the following differential equation:
xd2u
dx2+(α−x+1 )du
dx+nu=0 EH II 188(10), SM 574(34)
8.98011Orthogonality relation/integraldisplay∞
0e−xxαLα
n(x)Lα
m(x)dx=/braceleftBigg
0,m /negationslash=n
Γ(1 + α)/parenleftbign+α
n/parenrightbig
,m=nMS 5.5.2
8.98110Behavior of relative maxima of |Lα
n(x)|
1. Let αbe arbitrary and real. The sequence formed by the relative maxima of |Lα
n(x)|and by the
value of this function at x= 0, is decreasing for x<α +1
2, and increasing for x>α +1
2.T h e
successive relative maxima of |Lα
n(x)|form a decreasing sequence for x≤0, and an increasing
sequence for x≥0. SZ 174(7.6.1)
2. Let αbe an arbitrary real number. The successive relative maxima of
e−x/2x(α+1)/2|Lα
n(x)|ande−x/2xα/2+1
4|Lα
n(x)|
form an increasing sequence, provided x>x 0. In the first case
x0=⎧
⎨
⎩0i f α2≤1,
α2−1
2n+α+1ifα2>1
In the second case,
x0=/braceleftBigg
0i f α2≤q1
4,
/parenleftbig
α2−1
4/parenrightbig1
2ifα2>1
4SZ 174(7.6.2)
In the first case, we take nso large that 2 n+α+1>0.
8.98210Asymptotic and limiting behavior of Lα
n(x)
1. Let αbe arbitrary and real, candwfixed positive constants, and let n→∞.T h e n
Lα
n(x)=/braceleftBigg
x−α/2−1
4O/parenleftBig
nα/2−1
4/parenrightBig
ifcn−1≤qx≤qω
O(nα)i f 0 ≤qx≤qcn−1
These bounds are precise as regards their orders in n.F o r α≥q−1
2, both bounds hold in both
intervals, that is,
1004 Orthogonal Polynomials 8.982
Lα
n(x)=/braceleftBigg
x−α/2−1
4O/parenleftBig
nα/2−1
4/parenrightBig
,
O(nα),0<x≤qω, α ≥q−1
2SZ 175(7.6.4)
2. Let αbe arbitrary and real. Then for an arbitrary complex z
lim
n→∞n−αLα
n(x)=z−α/2Jα/parenleftBig
2z1/2/parenrightBig
, SZ 191(8.1.3)
uniformly if zis bounded.
9.114 Integral representations 1005
9.1 Hypergeometric Functions
9.10 Definition
9.100 Ahypergeometric series is a series of the form
F(α,β;γ;z)=1+α·β
γ·1z+α(α+1 )β(β+1 )
γ(γ+1 )·1·2z2+α(α+1 ) (α+2 )β(β+1 ) (β+2 )
γ(γ+1 ) (γ+2 )·1·2·3z3+...
9.101 A hypergeometric series terminates if αorβis equal to a negative integer or to zero. For
γ=−n(n=0,1,2,...), the hypergeometric series is indeterminate if neither αnorβis equal to −m
(where m<n andmis a natural number). However,
1. lim
γ→−nF(α,β;γ;z)
Γ(γ)=α(α+1 )...(α+n)β(β+1 )...(β+n)
(n+1 ) !
×zn+1F(α+n+1,β+n+1 ;n+2 ;z)
EH I 62(16)
9.102 If we exclude these values of the parameters α,β,γ , a hypergeometric series converges in the unit
circle|z|<1.Fthen has a branch point at z= 1. Then we have the following conditions for convergence
on the unit circle:
1. 1 >Re(α+β−γ)≥0. The series converges throughout the entire unit circle, except at the
point z=1 .
2. Re( α+β−γ)<0. The series converges (absolutely) throughout the entire unit circle.
3. Re( α+β−γ)≥1. The series diverges on the entire unit circle. FI II 410, WH
9.11 Integral representations
9.111 F(α,β;γ;z)=1
B(β,γ−β)/integraldisplay1
0tβ−1(1−t)γ−β−1(1−tz)−αdt [Reγ>Reβ>0] WH
9.1128F/parenleftbig
p, n+p;n+1 ;z2/parenrightbig
=z−n
2πΓ(p)n!
Γ(p+n)/integraldisplay2π
0cosnt dt
(1−2zcost+z2)p
[n=0,1,2,...;p/negationslash=0,−1,−2,...;|z|<1]W H ,M O1 6
9.113 F(α,β;γ;z)=Γ(γ)
Γ(α)Γ(β)1
2πi/integraldisplay∞i
−∞iΓ(α+t)Γ(β+t)Γ(−t)
Γ(γ+t)(−z)tdt
Here,|arg(−z)|<πand the path of integration are chosen in such a way that the poles of the functions
Γ(α+t)a n dΓ ( β+t) lie to the left of the path of integration and the poles of the function Γ( −t) lie to
the right of it.
9.114 F/parenleftbigg
−m,−p+m
2;1−p+m
2;−1/parenrightbigg
=(−2)m(p+m)
sinpπ/integraldisplayπ
0cosmϕcospϕ dϕ
[m+ 1 is a natural number; p/negationslash=0 , ±1,...]E HI8 0 ( 8 ) ,M O1 6
See also 3.194 1, 2, 5, 3.196 1,3.197 6, 9,3.259 3,3.312 3,3.518 4–6,3.665 2,3.671 1, 2,3.681 1,
3.984 7.
1006 Hypergeometric Functions 9.121
9.12 Representation of elementary functions in terms of a hypergeometric functions
9.121
1.8F(−n, β;β;−z)=( 1+ z)nEH I 101(4), GA 127 Ia
2. F/parenleftbigg
−n
2,−n−1
2;1
2;z2
t2/parenrightbigg
=(t+z)n+(t−z)n
2tnGA 127 II
3. lim
ω→∞F/parenleftBig
−n, ω;2ω;−z
t/parenrightBig
=/parenleftBig
1+z
2t/parenrightBign
GA 127 IIIa
4. F/parenleftbigg
−n−1
2,−n−2
2;3
2;z2
t2/parenrightbigg
=(t+z)n−(t−z)n
2nztn−1GA 127 IV
5. F/parenleftBig
1−n,1;2;−z
t/parenrightBig
=(t+z)n−tn
nztn−1GA 127 V
6. F(1,1;2;−z)=ln(1 + z)
zGA 127 VI
7. F/parenleftbigg1
2,1;3
2;z2/parenrightbigg
=ln1+z
1−z
2zGA 127 VII
8. lim
k→∞F/parenleftBig
1,k;1;z
k/parenrightBig
=1+ zlim
k→∞F/parenleftBig
1,k;2;z
k/parenrightBig
=1+ z+z2
2lim
k→∞F/parenleftBig
1,k;3;z
k/parenrightBig
=···=ez
GA 127 VIII
9. lim
k→∞
k/prime→∞F/parenleftbigg
k,k/prime;1
2;z2
4kk/prime/parenrightbigg
=ez+e−z
2=c o s h z GA 127 IX
10. lim
k→∞
k/prime→∞F/parenleftbigg
k,k/prime;3
2;z2
4kk/prime/parenrightbigg
=ez−e−z
2z=sinhz
zGA 127 X
11. lim
k→∞
k/prime→∞F/parenleftbigg
k,k/prime;3
2;−z2
4kk/prime/parenrightbigg
=sinz
zGA 127 XI
12. lim
k→∞
k/prime→∞F/parenleftbigg
k,k/prime;1
2;−z2
4kk/prime/parenrightbigg
=c o s z GA 127 XII
13. F/parenleftbigg1
2,1
2;3
2;s in2z/parenrightbigg
=z
sinzGA 127 XIII
14. F/parenleftbigg
1,1;3
2;s in2z/parenrightbigg
=z
sinzcoszGA 127 XIV
15. F/parenleftbigg1
2,1;3
2;−tan2z/parenrightbigg
=z
tanzGA 127 XV
16. F/parenleftbiggn+1
2,−n−1
2;3
2;s in2z/parenrightbigg
=sinnz
nsinzGA 127 XVI
17. F/parenleftbiggn+2
2,−n−2
2;3
2;s in2z/parenrightbigg
=sinnz
nsinzcoszGA 127 XVII
9.121 Elementary functions as hypergeometric function 1007
18. F/parenleftbigg
−n−2
2,−n−1
2;3
2;−tan2z/parenrightbigg
=sinnz
nsinzcosn−1zGA 127 XVIII
19. F/parenleftbiggn+2
2,n+1
2;3
2;−tan2z/parenrightbigg
=sinnzcosn+1z
nsinzGA 127 XIX
20. F/parenleftbiggn
2,−n
2;1
2;s in2z/parenrightbigg
=c o s nz EH I 101(11), GA 127 XX
21. F/parenleftbiggn+1
2,−n−1
2;1
2;s in2z/parenrightbigg
=cosnz
coszEH I 101(11), GA 127 XXI
22. F/parenleftbigg
−n
2,−n−1
2;1
2;−tan2z/parenrightbigg
=cosnz
cosnzEH I 101(11), GA 127 XXII
23. F/parenleftbiggn+1
2,n
2;1
2;−tan2z/parenrightbigg
=c o s nzcosnz GA 127 XXIII
24. F/parenleftbigg1
2,1;2;4z(1−z)/parenrightbigg
=1
1−z/bracketleftbig
|z|≤1
2;|z(1−z)|≤1
4/bracketrightbig
25. F/parenleftbigg1
2,1;1;sin2z/parenrightbigg
=s e c z
26. F/parenleftbigg1
2,1
2;3
2;z2/parenrightbigg
=arcsin z
z(cf.9.121 13)
27. F/parenleftbigg1
2,1;3
2;−z2/parenrightbigg
=arctan z
z(cf.9.121 15)
28. F/parenleftbigg1
2,1
2;3
2;−z2/parenrightbigg
=arcsinh z
z(cf.9.121 26)
29. F/parenleftbigg1+n
2,1−n
2;3
2;z2/parenrightbigg
=sin (narcsin z)
nz(cf.9.121 16)
30. F/parenleftbigg
1+n
2,1−n
2;3
2;z2/parenrightbigg
=sin (narcsin z)
nz√
1−z2(cf.9.121 17)
31. F/parenleftbiggn
2,−n
2;1
2;z2/parenrightbigg
=c o s( narcsin z) (cf. 9.121 20)
32. F/parenleftbigg1+n
2,1−n
2;1
2;z2/parenrightbigg
=cos (narcsin z)√
1−z2(cf.9.121 21)
The representation of special functions in terms of a hypergeometric function:
•for complete elliptic integrals, see 8.113 1a n d8.114 1;
•for integrals of Bessel functions, see 6.574 1, 3,6.576 2–5,6.621 1–3;
•for Legendre polynomials, see 8.911 and8.916 . (All these hypergeometric series terminate; that
is, these series are finite sums);
•for Legendre functions, see 8.820 and8.837 ;
•for associated Legendre functions, see 8.702, 8.703, 8.751, 8.77, 8.852 ,a n d8.853 ;
•for Chebyshev polynomials, see 8.942 1;
•for Jacobi’s polynomials, see 8.962 ;
1008 Hypergeometric Functions 9.122
•for Gegenbauer polynomials, see 8.932 ;
•for integrals of parabolic cylinder functions, see 7.725 6.
9.122 Particular values:
1. F(α,β;γ;1)=Γ(γ)Γ(γ−α−β)
Γ(γ−α)Γ(γ−β)[Reγ>Re(α+β)]
GA 147(48), FI II 793
2. F(α,β;γ;1)= F(−α,−β;γ−α−β;1) [R e γ>Re(α+β)] GA 148(49)
=1
F(−α,β;γ−α;1)[Reγ>Re(α+β)] GA 148(50)
=1
F(α,−β;γ−β;1)[Reγ>Re(α+β)] GA 148(51)
3. F/parenleftbigg
1,1;3
2;1
2/parenrightbigg
=π
2(cf.9.121 14)
9.13 Transformation formulas and the analytic continuation of functions defined by
hypergeometric series
9.130 The series F(α,β;γ;z) defines an analytic function that, speaking generally, has singularities at
the points z=0 ,1 ,a n d ∞. (In the general case, there are branch points.) We make a cut in the z-plane
along the real axis from z=1t o z=∞; that is, we require that |arg(−z)|<πfor|z|≥1. Then, the
series f(α,β;γ;z) will, in the cut plane, yield a single-valued analytic continuation, which we can obtain
by means of the formulas below (provided γ+ 1 is not a natural number and α−βandγ−α−βare
not integers). These formulas make it possible to calculate the values of Fin the given region, even in
the case in which |z|>1. There are other closely related transformation formulas that can also be used
to get the analytic continuation when the corresponding relationships hold between α,β,γ .
Transformation formulas
9.131
1.11F(α,β;γ;z)=(1 −z)−αF/parenleftbigg
α,γ−β;γ;z
z−1/parenrightbigg
GA 218(91)
=( 1−z)−βF/parenleftbigg
β,γ−α;γ;z
z−1/parenrightbigg
GA 218(92)
=( 1−z)γ−α−βF(γ−α,γ−β;γ;z)
2. F(α,β;γ;z)=Γ(γ)Γ(γ−α−β)
Γ(γ−α)Γ(γ−β)F(α,β;α+β−γ+1 ;1 −z)
+(1−z)γ−α−βΓ(γ)Γ(α+β−γ)
Γ(α)Γ(β)F(γ−α,γ−β;γ−α−β+1 ;1 −z)
EH I 94, MO 13
9.136 Transformation formulas for hypergeometric series 1009
9.132
1. F(α,β;γ;z)=(1−z)−αΓ(γ)Γ (β−α)
Γ(β)Γ(γ−α)F/parenleftbigg
α,γ−β;α−β+1 ;1
1−z/parenrightbigg
+(1−z)−βΓ(γ)Γ(α−β)
Γ(α)Γ(γ−β)F/parenleftbigg
β,γ−α;β−α+1 ;1
1−z/parenrightbigg
MO 13
2.11F(α,β;γ;z)=Γ(γ)Γ(β−α)
Γ(β)Γ(γ−α)(−z)−αF/parenleftbigg
α,α+1−γ;α+1−β;1
z/parenrightbigg
+Γ(γ)Γ(α−β)
Γ(α)Γ(γ−β)(−z)−βF/parenleftbigg
β,β+1−γ;β+1−α;1
z/parenrightbigg
[|argz|<π , α −β/negationslash=±m, m =0,1,2,...]GA 220(93)
9.133 F/parenleftbig
2α,2β;α+β+1
2;z/parenrightbig
=F/parenleftbig
α,β;α+β+1
2;4z(1−z)/parenrightbig
/bracketleftbig
|z|≤1
2,|z(1−z)|≤1
4/bracketrightbig
WH
9.134
1. F(α,β;2β;z)=/parenleftBig
1−z
2/parenrightBig−α
F/parenleftBigg
α
2,α+1
2;β+1
2;/parenleftbiggz
2−z/parenrightbigg2/parenrightBigg
MO 13, EH I 111(4)
2. F(2α,2α+1−γ;γ;z)=( 1+ z)−2αF/parenleftbigg
α,α+1
2;γ;4z
(1 +z)2/parenrightbigg
GA 225(100)
3. F/parenleftbigg
α,α+1
2−β;β+1
2;z2/parenrightbigg
=( 1+ z)−2αF/parenleftbigg
α,β;2β;4z
(1 +z)2/parenrightbigg
GA 225(101)
9.135 F/parenleftbigg
α,β;α+β+1
2;s in2ϕ/parenrightbigg
=F/parenleftbigg
2α,2β;α+β+1
2;s in2ϕ
2/parenrightbigg
/bracketleftBigg
x=s i n2ϕ
2real;1−√
2
2<x<1
2/bracketrightBigg
MO 13
9.1368We set
A=Γ/parenleftbig
α+β+1
2/parenrightbig√π
Γ/parenleftbig
α+1
2/parenrightbig
Γ/parenleftbig
β+1
2/parenrightbig,B =−Γ/parenleftbig
α+β+1
2/parenrightbig
2√π
Γ(α)Γ(β);
then
1. F/parenleftbigg
2α,2β;α+β+1
2;1−√z
2/parenrightbigg
=AF/parenleftbigg
α,β;1
2;z/parenrightbigg
+B√zF/parenleftbigg
α+1
2,β+1
2;3
2;z/parenrightbigg
GA 227(106)
2. F/parenleftbigg
2α,2β;α+β+1
2;1+√z
2/parenrightbigg
=AF/parenleftbigg
α,β;1
2;z/parenrightbigg
−B√zF/parenleftbigg
α+1
2,β+1
2;3
2;z/parenrightbigg
GA 227(107)
3./parenleftbig
α−1
2/parenrightbig/parenleftbig
β−1
2/parenrightbig
α+β−1
2A√zF/parenleftbigg
α,β;3
2;z/parenrightbigg
=F/parenleftbigg
2α−1,2β−1;α+β−1
2;1+√z
2/parenrightbigg
−F/parenleftbigg
2α−1,2β−1;α+β−1
2;1−√z
2/parenrightbigg
GA 229(110)
1010 Hypergeometric Functions 9.137
9.1377Gauss’ recursion functions:
1. γ[γ−1−(2γ−α−β−1)z]F(α,β;γ;z)+(γ−α)(γ−β)zF(α,β;γ+1 ;z)+γ(γ−1)(z−
1)F(α,β;γ−1;z)=0
2. (2 α−γ−αz+βz)F(α,β;γ;z)+(γ−α)F(α−1,β;γ;z)+α(z−1)F(α+1,β;γ;z)=0
3. (2 β−γ−βz+αz)F(α,β;γ;z)+(γ−β)F(α,β−1;γ;z)+β(z−1)F(α,β+1 ;γ;z)=0
4. γF(α,β−1;γ;z)−γF(α−1,β;γ;z)+(α−β)zF(α,β;γ+1 ;z)=0
5.8γ(α−β)F(α,β;γ;z)−α(γ−β)F(α+1,β;γ+1 ;z)+β(γ−α)F(α,β+1 ;γ+1 ;z)=0
6. γ(γ+1 )F(α,β;γ;z)−γ(γ+1 )F(α,β;γ+1 ;z)−αβzF(α+1,β+1 ;γ+2 ;z)=0
7. γF(α,β;γ;z)−(γ−α)F(α,β+1 ;γ+1 ;z)−α(1−z)F(α+1,β+1 ;γ+1 ;z)=0
8. γF(α,β;γ;z)+(β−γ)F(α+1,β;γ+1 ;z)−β(1−z)F(α+1,β+1 ;γ+1 ;z)=0
9. γ(γ−βz−α)F(α,β;γ;z)−γ(γ−α)F(α−1,β;γ;z)+αβz(1−z)F(α+1,β+1 ;γ+1 ;z)=0
10. γ(γ−αz−β)F(α,β;γ;z)−γ(γ−β)F(α,β−1;γ;z)+αβz(1−z)F(α+1,β+1 ;γ+1 ;z)=0
11. γF(α,β;γ;z)−γF(α,β+1 ;γ;z)+αzF(α+1,β+1 ;γ+1 ;z)=0
12.8γF(α,β;γ;z)−γF(α+1,β;γ;z)+βzF(α+1,β+1 ;γ+1 ;z)=0
13. γ[α−(γ−β)z]F(α,β;γ;z)−αγ(1−z)F(α+1,β;γ;z)+(γ−α)(γ−β)zF(α,β;γ+1 ;z)=0
14. γ[β−(γ−α)z]F(α,β;γ;z)−βγ(1−z)F(α,β+1 ;γ;z)+(γ−α)(γ−β)zF(α,β;γ+1 ;z)=0
15.8γ(γ+1 )F(α,β;γ;z)−γ(γ+1 )F(α,β+1 ;γ+1 ;z)+α(γ−β)zF(α+1,β+1 ;γ+2 ;z)=0
16. γ(γ+1 )F(α,β;γ;z)−γ(γ+1 )F(α+1,β;γ+1 ;z)+β(γ−α)zF(α+1,β+1 ;γ+2 ;z)=0
17. γF(α,β;γ;z)−(γ−β)F(α,β;γ+1 ;z)−βF(α,β+1 ;γ+1 ;z)=0
18.8γF(α,β;γ;z)−(γ−α)F(α,β;γ+1 ;z)−αF(α+1,β;γ+1 ;z)=0 MO 13–14
9.14 A generalized hypergeometric series
The series
1. pFq(α1,α2,...,α p;β1,β2,...,β q;z)=∞/summationdisplay
k=0(α1)k(α2)k...(αp)k
(β1)k(β2)k...(βq)kzk
k!MO 14
is called a generalized hypergeometric series (see also 9.210).
2. 2F1(α,β;γ;z)≡F(α,β;γ;z) MO 15
For integral representations, see 3.254 2,3.259 2, and 3.478 3.
9.15 The hypergeometric differential equation
9.151 A hypergeometric series is one of the solutions of the differential equation
z(1−z)d2u
dz2+[γ−(α+β+1 )z]du
dz−αβu=0, WH
which is called the hypergeometric equation .
9.153 The hypergeometric differential equation 1011
The solution of the hypergeometric differential equation
9.152 The hypergeometric differential equation 9.151 possesses two linearly independent solutions .
These solutions have analytic continuations to the entire z-plane, except possibly for the three points
0, 1, and ∞. Generally speaking, the points z=0,1,∞are branch points of at least one of the branches
of each solution of the hypergeometric differential equation. The ratio w(z) of two linearly independent
solutions satisfies the differential equation
2w/prime/prime/prime
w/prime−3/parenleftbiggw/prime/prime
w/prime/parenrightbigg2
=1−a2
1
z2+1−a2
2
(z−1)2+a2
1+a2
2−a2
3−1
z(z−1),
where
a2
1=( 1−γ)2,a2
2=(γ−α−β)2,a2
3=(α−β)2.
Ifα,β,γ are real, the function w(z) maps the upper (Im z>0) or the lower (Im z<0) half-plane onto
a curvilinear triangle whose angles are πa1,πa2,πa3. The vertices of this triangle are the images of the
points z=0,z=1 ,a n d z=∞.
9.153 Within the unit circle |z|<1, the linearly independent solutions u1(z)a n d u2(z)o ft h eh y p e r g e -
ometric differential equation are given by the following formulas:
1. If γis not an integer,
u1=F(α,β;γ;z),
u2=z1−γeF(α−γ+1,β−γ+1 ;2 −γ;z)
2. If γ=1,then
u1=F(α,β;1;z),
u2=F(α,β;1;z)lnz+∞/summationdisplay
k=1zk(α)k(β)k
(k!)2
×{ψ(α+k)−ψ(α)+ψ(β+k)−ψ(β)−2ψ(k+1 )+2 ψ(1)}
(see9.142)
3. If γ=m+1( w h e r e mis a natural number), and if neither αnorβis a positive number not
exceeding m,t h e n
u1=F(α,β;m+1 ;z),
u2=F(α,β;m+1 ;z)lnz+∞/summationdisplay
k=1zk(α)k(β)k
(1 +m)k{h(k)−h(0)}−m/summationdisplay
k=1(k−1)!(−m)k
(1−α)k(1−β)kz−k
(see9.142)
where
h(n)=ψ(α+n)+ψ(β+n)−ψ(m+1+ n)−ψ(n+1 ) [ n+ 1 is a natural number]
4.11Suppose that γ=m+1(where mis a natural number) and that αorβis equal to m/prime+1,where
0≤m/prime<m. Then, for example, for α=m/prime+ 1, we obtain
u1=F(1 +m/prime,β;1+m;z),
u2=z−mF(1 +m/prime−m, β−m;1−m;z)
In this case, u2is a polynomial in z−1.
1012 Hypergeometric Functions 9.154
5. If γ=1−m(where mis a natural number) and if αandβare both different from the numbers
0,−1,−2,...,1−m,t h e n
u1=zmF(α+m, β+m;1+m;z),
u2=zmF(α+m, β+m;1+m;z)lnz+∞/summationdisplay
k=1zk(α+m)k(β+m)k
(1 +m)kk!{h∗(k)−h∗(0)}
−∞/summationdisplay
k=1(k−1)!(−m)k
(1−α−m)k(1−β−m)kzm−n
(see9.142)
where
h∗(n)=ψ(α+m+n)+ψ(β+m+n)−ψ(1 +m+n)−ψ(1 +n)
We note that
ψ(α+n)−ψ(α)=1
α+1
α+1+···+1
α+n−1(cf.8.365 3)
and that, for α=−λ,w h e r e λis a natural number or zero and n=λ+1,λ+2,...the expression
(α)k[ψ(α+n)−ψ(α)]
in formulas 9.153 2–5 should be replaced with the expression
(−1)λλ!(n−λ−1)!
6. Suppose that γ=1−m(where mis a natural number) and that αorβis an integer ( −m/prime),
where m/primeis one of the following numbers: 0 ,1,...,m −1. Suppose, for example, that α=−m/prime.
Then,
u1=F(−m/prime,β;1−m;z),
u2=F(−m/prime+m, β+m;1+m;z)
MO 18
7. For γ=1
2(α+β+1 )
u1=F/parenleftbig
α,β;1
2(α+β+1 ) ;z/parenrightbig
,
u2=F/parenleftbig
α,β;1
2(α+β+1 ) ;1 −z/parenrightbig
are two linearly independent solutions of the hypergeometric differential equation, provided α,β,
andγare not zero or negative integers. MO 17–19
The analytic continuation of a solution that is regular at the point z=0
9.154 Formulas 9.153 make possible the analytic continuation, by means of the hypergeometric series,
of the function F(α,β;γ;z) defined inside the circle |z|<1 to the region |z|>1, and |arg(−z)|<π. Here,
it is assumed that α−βis not an integer. In the event that α−βis an integer (for example, if β=α+m,
where mis a natural number), then, for |z|>1, and |arg(−z)|<πwe have:
9.155 The hypergeometric differential equation 1013
1.Γ(α)Γ(α+m)
Γ(γ)F(α,α+m;γ;z)
=sinπ(γ−α)
π/braceleftBiggm−1/summationdisplay
k=0Γ(α+k)Γ(1−γ+α+k)Γ(m−k)
k!(−z)−α−k
+(−z)−α−m∞/summationdisplay
k=0Γ(α+m+k)Γ(1−γ+α+m+k)
k!(k+m)!g(k)z−k/bracerightBigg
where
2. g(n)=ln ( −z)+πcotπ(γ−α)+ψ(n+1 )+ ψ(n+m+1 )
−ψ(α+m+n)−ψ(1−γ+α+m+n)
Form= 0, we should setm−1/summationdisplay
k=0=0 .
9.155 This formula loses its meaning when α,γ,o rα−γ+1 is equal to one of the numbers 0 ,−1,−2,....
In this last case, we have
1. If αis a non-positive integer and γis not an integer, F(α,α+m;γ;z) is a polynomial in z.
2. Suppose that γis a non-positive integer and that αis not an integer. We then set γ=−λ,w h e r e
λ=0,1,2,....Then,
Γ(α+λ+1 )Γ ( α+λ+m+1 )
Γ(λ+2 )zλ+1F(α+λ+1,α+λ+m+1 ;λ+2 ;z)
is a solution of the hypergeometric equation that is regular at the point z= 0. This solution is
equal to the right-hand member of formula 9.154 1 if we replace γwithλin this equation and
in formula 9.154 2.
3. If α−γ+ 1 is a non-positive integer and if αandγare not themselves integers, we may use the
formula
F(α,α+m;γ;z)=( 1 −z)γ−2α−mF(γ−α−m, γ−α;γ;z)
and apply formula 9.154 1 to its right-hand member, provided γ−α−m> 0. However, if
α−γ−m≤0, the right member of this expression is a polynomial taken to the (1 −z)thpower.
4. If α,β,a n d γare integers, the hypergeometric differential equation always has a solution that is
regular for z= 0 and that is of the form
R1(z)+l n ( 1 −z)R2(z),
where R1(z)a n d R2(z) are rational functions of z. To get a solution of this form, we need to
apply formulas 9.137 1–9.137 3 to the function F(α,β;γ;z). However, if γ=−λ,w h e r e λ+1
is a natural number, formulas 9.137 1a n d9.137 2 should be applied not to F(α,β;γ;z) but to
the function zλ+1F(α+λ+1,β+λ+1 ;λ+2,z).
By successive applications of these formulas, we can reduce the positive values of the parameters
to the pair, unity and zero. Furthermore, we can obtain the desired form of the solution from
the formulas
F(1,1;2;z)=−z−1ln(1−z),
F(0,β;γ;z)=F(α,0;γ;z)=1
MO 19–20
1014 Hypergeometric Functions 9.160
9.16 Riemann’s differential equation
9.160 The hypergeometric differential equation is a particular case of Riemann’s differential equation
1.11d2u
dz2+/bracketleftbigg1−α−α/prime
z−a+1−β−β/prime
z−b+1−γ−γ/prime
z−c/bracketrightbiggdu
dz
+⎡
⎣αα/prime(a−b)(a−c)
z−a+ββ/prime(b−c)(b−a)
z−bγγ/prime(c−a)(c−b)
z−c⎤
⎦u
(z−a)(z−b)(z−c)=0
WH
The coefficients of this equation have poles at the points a,b,a n dc, and the numbers α,α/prime;β,β/prime;
γ,γ/primeare called the indices corresponding to these poles. The indices α,α/prime;β,β/prime;γ,γ/primeare related
by the following equation:
α+α/prime+β+β/prime+γ+γ/prime−1=0 WH
2. The differential equations 9.160 1 are written diagramatically as follows:
3. u=P⎧
⎨
⎩abc
αβγz
α/primeβ/primeγ/prime⎫
⎬
⎭
The singular points of the equation appear in the first row in this scheme, the indices corresponding to
them appear beneath them, and the independent variable appears in the fourth column. WH
9.161 The two following transformation formulas are valid for Riemann’s P-equation:
1./parenleftbiggz−a
z−b/parenrightbiggk/parenleftbiggz−c
z−b/parenrightbiggl
P⎧
⎨
⎩abc
αβγz
α/primeβ/primeγ/prime⎫
⎬
⎭=P⎧
⎨
⎩abc
α+kβ−k−1γ+lz
α/prime+kβ/prime−k−lγ/prime+l⎫
⎬
⎭WH
2. P⎧
⎨
⎩abc
αβγz
α/primeβ/primeγ/prime⎫
⎬
⎭=P⎧
⎨
⎩a1b1c1
αβγz 1
α/primeβ/primeγ/prime⎫
⎬
⎭WH
The first of these formulas means that if
u=P⎧
⎨
⎩abc
αβγz
α/primeβ/primeγ/prime⎫
⎬
⎭,
then the function
u1=/parenleftbiggz−a
z−b/parenrightbiggk/parenleftbiggz−c
z−b/parenrightbiggl
u
satisfies a second-order differential equation having the same singular points as equation 9.161 2a n d
indices equal to α+k,α/prime+k;β−k−l,β/prime−k−l;γ+l,γ/prime+l. The second transformation formula
converts a differential equation with singularities at the points a,b, and c, indices α,α/prime;β,β/prime;γ,γ/prime,a n da n
independent variable zinto a differential equation with the same indices, singular points a1,b1,andc1,a n d
independent variable z1. The variable z1is connected with the variable zby the fractional transformation
9.164 Riemann’s differential equation 1015
z=Az1+B
Cz1+D[AD−BC/negationslash=0 ]
The same transformation connects the points a1,b1,a n d c1with the points a,b,a n d c.
W H ,M O2 0
9.162 By the successive application of the two transformation formulas 9.161 1a n d9.161 2, we can
convert Riemann’s differential equation into the hypergeometric differential equation. Thus, the solution
of Riemann’s differential equation can be expressed in terms of a hypergeometric function.
Fork=−α,l=−γ,a n d z1=(z−a)(c−b)
(z−b)(c−a),w eh a v e
1. u=P⎧
⎨
⎩abc
αβγz
α/primeβ/primeγ/prime⎫
⎬
⎭=/parenleftbiggz−a
z−b/parenrightbiggα/parenleftbiggz−c
z−b/parenrightbiggγ
P⎧
⎨
⎩abc
0 β+α+γ 0 z
α/prime−αβ/prime+α+γγ/prime−γ⎫
⎬
⎭
=/parenleftbiggz−a
z−b/parenrightbiggα/parenleftbiggz−c
z−b/parenrightbiggγ
P⎧
⎨
⎩0 ∞ 1
0 β+α+γ 0(z−a)(c−b)
(z−b)(c−a)
α/prime−αβ/prime+α+γγ/prime−γ⎫
⎬
⎭
MO 23
Thus, this solution can be expressed as a hypergeometric series as follows:
2. u=/parenleftbiggz−a
z−b/parenrightbiggα/parenleftbiggz−c
z−b/parenrightbiggγ
F/parenleftbigg
α+β+γ,α+β/prime+γ;1+α−α/prime;(z−a)(c−b)
(z−b)(c−a)/parenrightbigg
If the constants a,b,c;α,α/prime;β,β/prime;γ,γ/primeare permuted in a suitable manner, Riemann’s equation remains
unchanged. Thus, we obtain a set of 24 solutions of differential equations having the following form(provided none of the differences α−α
/prime,β−β/prime,γ−γ/primeis an integer): W H ,M O2 3
9.163
1. u1=/parenleftbiggz−a
z−b/parenrightbiggα/parenleftbiggz−c
z−b/parenrightbiggγ
F/braceleftbigg
α+β+γ,α+β/prime+γ;1+α−α/prime;(c−b)(z−a)
(c−a)(z−b)/bracerightbigg
2. u2=/parenleftbiggz−a
z−b/parenrightbiggα/prime/parenleftbiggz−c
z−b/parenrightbiggγ
F/braceleftbigg
α/prime+β+γ,α/prime+β/prime+γ;1+α/prime−α;(c−b)(z−a)
(c−a)(z−b)/bracerightbigg
3. u3=/parenleftbiggz−a
z−b/parenrightbiggα/parenleftbiggz−c
z−b/parenrightbiggγ/prime
F/braceleftbigg
α+β+γ/prime,α+β/prime+γ/prime;1+α−α/prime;(c−b)(z−a)
(c−a)(z−b)/bracerightbigg
4. u4=/parenleftbiggz−a
z−b/parenrightbiggα/prime/parenleftbiggz−c
z−b/parenrightbiggγ/prime
F/braceleftbigg
α/prime+β+γ/prime,α/prime+β/prime+γ;1+α/prime−α;(c−b)(z−a)
(c−a)(z−b)/bracerightbigg
9.164
1.10u5=/parenleftbiggz−b
z−c/parenrightbiggβ/parenleftbiggz−a
z−c/parenrightbiggα
F/braceleftbigg
β+γ+α,β+γ/prime+α;1+β−β/prime;(a−c)(z−b)
(a−b)(z−c)/bracerightbigg
2. u6=/parenleftbiggz−b
z−c/parenrightbiggβ/prime/parenleftbiggz−a
z−c/parenrightbiggα
F/braceleftbigg
β/prime+γ+α,β/prime+γ/prime+α;1+β/prime−β;(a−c)(z−b)
(a−b)(z−c)/bracerightbigg
3. u7=/parenleftbiggz−b
z−c/parenrightbiggβ/parenleftbiggz−a
z−c/parenrightbiggα/prime
F/braceleftbigg
β+γ+α/prime,β+γ/prime+α/prime;1+β−β/prime;(a−c)(z−b)
(a−b)(z−c)/bracerightbigg
1016 Hypergeometric Functions 9.165
4. u8=/parenleftbiggz−b
z−c/parenrightbiggβ/prime/parenleftbiggz−a
z−c/parenrightbiggα/prime
F/braceleftbigg
β/prime+γ+α/prime,β/prime+α/prime+γ/prime;1+β/prime−β;(a−c)(z−b)
(a−b)(z−c)/bracerightbigg
9.165
1. u9=/parenleftbiggz−c
z−a/parenrightbiggγ/parenleftbiggz−b
z−a/parenrightbiggβ
F/braceleftbigg
γ+α+β,γ+α/prime+β;1+γ−γ/prime;(b−a)(z−c)
(b−c)(z−a)/bracerightbigg
2. u10=/parenleftbiggz−c
z−a/parenrightbiggγ/prime/parenleftbiggz−b
z−a/parenrightbiggβ
F/braceleftbigg
γ/prime+α+β,γ/prime+α/prime+β;1+γ/prime−γ;(b−a)(z−c)
(b−c)(z−a)/bracerightbigg
3. u11=/parenleftbiggz−c
z−a/parenrightbiggγ/parenleftbiggz−b
z−a/parenrightbiggβ/prime
F/braceleftbigg
γ+α+β/prime,γ+α/prime+β/prime;1+γ−γ/prime;(b−a)(z−c)
(b−c)(z−a)/bracerightbigg
4. u12=/parenleftbiggz−c
z−a/parenrightbiggγ/prime/parenleftbiggz−b
z−a/parenrightbiggβ/prime
F/braceleftbigg
γ/prime+α+β/prime,γ/prime+α/prime+β/prime;1+γ/prime−γ;(b−a)(z−c)
(b−c)(z−a)/bracerightbigg
9.166
1. u13=/parenleftbiggz−a
z−c/parenrightbiggα/parenleftbiggz−b
z−c/parenrightbiggβ
F/braceleftbigg
α+γ+β,α+γ/prime+β;1+α−α/prime;(b−c)(z−a)
(b−a)(z−c)/bracerightbigg
2. u14=/parenleftbiggz−a
z−c/parenrightbiggα/prime/parenleftbiggz−b
z−c/parenrightbiggβ
F/braceleftbigg
α/prime+γ+β,α/prime+γ/prime+β;1+α/prime−α;(b−c)(z−a)
(b−a)(z−c)/bracerightbigg
3. u15=/parenleftbiggz−a
z−c/parenrightbiggα/parenleftbiggz−b
z−c/parenrightbiggβ/prime
F/braceleftbigg
α+γ+β/prime,α+γ/prime+β/prime;1+α−α/prime;(b−c)(z−a)
(b−a)(z−c)/bracerightbigg
4. u16=/parenleftbiggz−a
z−c/parenrightbiggα/prime/parenleftbiggz−b
z−c/parenrightbiggβ/prime
F/braceleftbigg
α/prime+γ+β/prime,α/prime+γ/prime+β/prime;1+α/prime−α;(b−c)(z−a)
(b−a)(z−c)/bracerightbigg
9.167
1. u17=/parenleftbiggz−c
z−b/parenrightbiggγ/parenleftbiggz−a
z−b/parenrightbiggα
F/braceleftbigg
γ+β+α,γ+β/prime+α;1+γ−γ/prime;(a−b)(z−c)
(a−c)(z−b)/bracerightbigg
2. u18=/parenleftbiggz−c
z−b/parenrightbiggγ/prime/parenleftbiggz−a
z−b/parenrightbiggα
F/braceleftbigg
γ/prime+β+α,γ/prime+β/prime+α;1+γ/prime−γ;(a−b)(z−c)
(a−c)(z−b)/bracerightbigg
3. u19=/parenleftbiggz−c
z−b/parenrightbiggγ/parenleftbiggz−a
z−b/parenrightbiggα/prime
F/braceleftbigg
γ+β+α/prime,γ+β/prime+α/prime;1+γ−γ/prime;(a−b)(z−c)
(a−c)(z−b)/bracerightbigg
4. u20=/parenleftbiggz−c
z−b/parenrightbiggγ/prime/parenleftbiggz−a
z−b/parenrightbiggα/prime
F/braceleftbigg
γ/prime+β+α/prime,γ/prime+β/prime+α/prime;1+γ/prime−γ;(a−b)(z−c)
(a−c)(z−b)/bracerightbigg
9.168
1. u21=/parenleftbiggz−b
z−a/parenrightbiggβ/parenleftbiggz−c
z−a/parenrightbiggγ
F/braceleftbigg
β+α+γ,β+α/prime+γ;1+β−β/prime;(c−a)(z−b)
(c−b)(z−a)/bracerightbigg
2. u22=/parenleftbiggz−b
z−a/parenrightbiggβ/prime/parenleftbiggz−c
z−a/parenrightbiggγ
F/braceleftbigg
β/prime+α+γ,β/prime+α/prime+γ;1+β/prime−β;(c−a)(z−b)
(c−b)(z−a)/bracerightbigg
9.175 Second-order differential equations 1017
3. u23=/parenleftbiggz−b
z−a/parenrightbiggβ/parenleftbiggz−c
z−a/parenrightbiggγ/prime
F/braceleftbigg
β+α+γ/prime,β+α/prime+γ/prime;1+β−β/prime;(c−a)(z−b)
(c−b)(z−a)/bracerightbigg
4. u24=/parenleftbiggz−b
z−a/parenrightbiggβ/prime/parenleftbiggz−c
z−a/parenrightbiggγ/prime
F/braceleftbigg
β/prime+α+γ/prime,β/prime+α/prime+γ/prime;1+β/prime−β;(c−a)(z−b)
(c−b)(z−a)/bracerightbigg
WH
9.17 Representing the solutions to certain second-order differential equations using
a Riemann scheme
9.171 The hypergeometric equation (see 9.151 ):
u=P⎧
⎨
⎩0∞ 1
0 α 0 z
1−γβγ −α−β⎫
⎬
⎭WH
9.172 The associated Legendre’s equation defining the functions Pm
n(z)f o rnandmintegers (see 8.700 1):
1. u=P⎧
⎪⎨
⎪⎩0 ∞ 1
1
2mn +11
2m1−z
2
−1
2m−n−1
2m⎫
⎪⎬
⎪⎭WH
2. u=P⎧
⎪⎪⎨
⎪⎪⎩0 ∞ 1
−1
2n1
2m01
1−z2
n+1
2−1
2m1
2⎫
⎪⎪⎬
⎪⎪⎭WH
9.173 The function Pm
n/parenleftbigg
1−z2
2n2/parenrightbigg
satisfies the equation
u=P⎧
⎨
⎩4n2∞ 0
1
2mn +11
2mz2
−1
2m−n−1
2m⎫
⎬
⎭WH
The function Jm(z) satisfies the limiting form of this equation obtained as n→∞.
9.174 The equation defining the Gegenbauer polynomials Cλ
n(z)( s e e8.938 ):
u=P⎧
⎨
⎩−1 ∞ 1
1
2−λn+2λ1
2−λz
0 −n 0⎫
⎬
⎭WH
9.175 Bessel’s equation (see 8.401 ) is the limiting form of the equations:
1. u=P⎧
⎨
⎩0∞ c
ni c1
2+ic z
−n−ic1
2−ic⎫
⎬
⎭WH
2. u=eizP⎧
⎨
⎩0 ∞ c
n1
20 z
−n3
2−2ic2ic−1⎫
⎬
⎭WH
3. u=P⎧
⎨
⎩0 ∞ c2
1
2n1
2(c−n)0 z2
−1
2n−1
2(c+n)n+1⎫
⎬
⎭WH
asc→∞.
1018 Hypergeometric Functions 9.180
9.18 Hypergeometric functions of two variables
9.180
1. F1(α,β,β/prime,γ;x, y)=∞/summationdisplay
m=0∞/summationdisplay
n=0(α)m+n(β)m(β/prime)n
(γ)m+nm!n!xmyn
[|x|<1,|y|<1]
EH I 224(6), AK 14(11)
2. F2(α,β,β/prime,γ,γ/prime;x, y)=∞/summationdisplay
m=0∞/summationdisplay
n=0(α)m+n(β)m(β/prime)n
(γ)m(γ/prime)nm!n!xmyn
[|x|+|y|<1] EH I 224(7), AK 14(12)
3. F3(α,α/prime,β,β/prime,γ;x, y)=∞/summationdisplay
m=0∞/summationdisplay
n=0(α)m(α/prime)n(β)m(β/prime)n
(γ)m+nm!n!xmyn
[|x|<1,|y|<1]
EH I 224(8), AK 14(13)
4. F4(α,β,γ,γ/prime;x, y)=∞/summationdisplay
m=0∞/summationdisplay
n=0(α)m+n(β)m+n
(γ)m(γ/prime)nm!n!xmyn/bracketleftbig/vextendsingle/vextendsingle√x/vextendsingle/vextendsingle+|√y|<1/bracketrightbig
EH I 224(9), AK 14(14)
9.181 The functions F1,F2,F3,a n d F4satisfy the following systems of partial differential equations
forz:
1. System of equations for z=F1:
x(1−x)∂2z
∂x2+y(1−x)∂2z
∂x∂y+[γ−(α+β+1 )x]∂z
∂x−βy∂z
∂y−αβz=0, EH I 233(9)
y(1−y)∂2z
∂y2+x(1−y)∂2z
∂x∂y+[γ−(α+β/prime+1 )y]∂z
∂x−β/primex∂z
∂x−αβ/primez=0
2. System of equations for z=F2:
x(1−x)∂2z
∂x2−xy∂2z
∂x∂y+[γ−(α+β+1 )x]∂z
∂x−βy∂z
∂y−αβz=0, EH I 234(10)
y(1−y)∂2z
∂y2−xy∂2z
∂x∂y+[γ/prime−(α+β/prime+1 )y]∂z
∂y−β/primex∂z
∂x−αβ/primez=0
3. System of equations for z=F3:
x(1−x)∂2z
∂x2+y∂2z
∂x∂y+[γ−(α+β+1 )x]∂z
∂x−αβz=0,
y(1−y)∂2z
∂y2+x∂2z
∂x∂y+[γ−(α/prime+β/prime+1 )y]∂z
∂y−α/primeβ/primez=0
EH I 234(11)
9.182 Hypergeometric functions of two variables 1019
4. System of equations for z=F4:
x(1−x)∂2z
∂x2−y2∂2z
∂y2−2xy∂2z
∂x∂y+[γ−(α+β+1 )x]∂z
∂x−(α+β+1 )y∂z
∂y−αβz=0,
EH I 234(12)
y(1−y)∂2z
∂y2−x2∂2z
∂x2−2xy∂2z
∂x∂y+[γ/prime−(α+β+1 )y]∂z
∂y−(α+β+1 )x∂z
∂x−αβz=0
AK 44
9.182 For certain relationships between the parameters and the argument, hypergeometric functions of
two variables can be expressed in terms of hypergeometric functions of a single variable or in terms ofelementary functions:
1. F
1(α,β,β/prime,β+β/prime;x, y)=( 1 −y)−αF/parenleftbigg
α,β;β+β/prime;x−y
1−y/parenrightbigg
EH I 238(1), AK 24(28)
2. F2(α,β,β/prime,β,γ/prime;x, y)=( 1 −x)−αF/parenleftbigg
α,β/prime;γ/prime;y
1−x/parenrightbigg
EH I 238(2), AK 23
3. F2(α,β,β/prime,α,α;x, y)=( 1 −x)−β(1−y)−β/prime
F/parenleftbigg
β,β/prime;α;xy
(1−x)(1−y)/parenrightbigg
EH I 238(3)
4. F3(α,γ−α,β,γ −β,γ;x, y)=( 1 −y)α+β−γF(α,β;γ;x+y−xy) EH I 238(4), AK 25(35)
5. F4(α,γ+γ/prime−α−1,γ,γ/prime;x(1−y),y(1−x))
=F(α,γ+γ/prime−α−1;γ;x)F(α,γ+γ/prime−α−1;γ/prime;y)
EH I 238(5)
6. F4/parenleftbigg
α,β,α,β ;−x
(1−x)(1−y),−y
(1−x)(1−y)/parenrightbigg
=(1−x)β(1−y)α
(1−xy)EH I 238(6)
7. F4/parenleftbigg
α,β,β,β ;−x
(1−x)(1−y),−y
(1−x)(1−y)/parenrightbigg
=( 1−x)α(1−y)αF(α,1+α−β;β;xy)
EH I 238(7)
8. F4/parenleftbigg
α,β,1+α−β,β;−x
(1−x)(1−y),−y
(1−x)(1−y)/parenrightbigg
=( 1−y)αF/bracketleftbigg
α,β;1+α−β;−x(1−y)
1−x/bracketrightbigg
EH I 238(8)
9. F4/parenleftbigg
α,α+1
2,γ,1
2;x, y/parenrightbigg
=1
2(1 +√y)−2αF/parenleftBigg
α,α+1
2;γ;x
/parenleftbig
1+√y/parenrightbig2/parenrightBigg
+1
2(1−√y)−2αF/parenleftBigg
α,α+1
2;γ;x
/parenleftbig
1−√y/parenrightbig2/parenrightBigg
AK 23
10. F1(α,β,β/prime,γ;x,1) =Γ(γ)Γ(γ−α−β/prime)
Γ(γ−α)Γ(γ−β/prime)F(α,β:γ−β/prime;x) EH I 239(10), AK 22(23)
1020 Hypergeometric Functions 9.183
11. F1(α,β,β/prime,γ;x, x)=F(α,β+β/prime;γ;x) EH I 239(11), AK 23(25)
9.183 Functional relations between hypergeometric functions of two variables:
1. F1(α,β,β/prime,γ;x, y)=( 1 −x)−β(1−y)−βF1/parenleftbigg
γ−α,β,β/prime,γ;x
x−1,y
y−1/parenrightbigg
EH I 239(1)
=( 1−x)−αF1/parenleftbigg
α,γ−β−β/prime,β/prime,γ;x
x−1,y−x
1−x/parenrightbigg
EH I 239(2)
=( 1−y)−αF1/parenleftbigg
α,β,γ −β−β/prime,γ;y−x
y−1,y
y−1/parenrightbigg
EH I 239(3)
=( 1−x)γ−α−β(1−y)−β/prime
F1/parenleftbigg
γ−α,γ−β−β/prime,β/prime,γ;x,x−y
1−y/parenrightbigg
EH I 240(4)
=( 1−x)−β(1−y)γ−α−β/prime
F1/parenleftbigg
γ−α,β,γ −β−β/prime,γ;x−y
x−1,y/parenrightbigg
EH I 240(5), AK 30(5)
2.8F2(α,β,β/prime,γ,γ/prime;x, y)=(1 −x)−αF2/parenleftbigg
α,γ−β,β/prime,γ,γ/prime;x
x−1,y
1−x/parenrightbigg
EH I 240(6)
=( 1−y)−αF2/parenleftbigg
α,β,γ/prime−β/prime,γ,γ/prime;x
1−y,y
y−1/parenrightbigg
EH I 240(7)
=( 1−x−y)−αF2/parenleftbigg
α,γ−β,γ/prime−β/prime,γ,γ/prime;x
x+y−1,y
x+y−1/parenrightbigg
EH I 240(8), AK 32(6)
3.7F4(α,β,γ,γ/prime;x, y)=Γ(γ/prime)Γ(β−α)
Γ(γ/prime−α)Γ (β)(−y)−αF4/parenleftbigg
α,α+1−γ/prime,γ,α+1−β;x
y,1
y/parenrightbigg
+Γ(γ/prime(Γ(α−β)
Γ(γ/prime−β)Γ(α)(−y)βF4/parenleftbigg
β+1−γ/prime,β,γ,β +1−α;x
y,1
y/parenrightbigg
EH I 240(9), AK 26(37)
9.185 Hypergeometric functions of two variables 1021
9.184 Integral representations: Double integrals of the Euler type
1. F1(α,β,β/prime,γ;x, y)=Γ(γ)
Γ(β)Γ(β/prime)Γ(γ−β−β/prime)
×/integraldisplay/integraldisplay
u≥0,v≥0
u+v≤1uβ−1vβ/prime−1(1−u−v)γ−β−β/prime−1(1−ux−vy)−αdu dv
[Reβ>0,Reβ/prime>0,Re (γ−β−β/prime)>0]EH I 230(1), AK 28(1)
2. F2(α,β,β/prime,γ,γ/prime;x, y)=Γ(γ)Γ(γ/prime)
Γ(β)Γ(β/prime)Γ(γ−β)Γ(γ/prime−β/prime)
×/integraldisplay1
0/integraldisplay1
0uβ−1vβ/prime−1(1−u)γ−β−1(1−v)γ/prime−β/prime−1(1−ux−vy)−αdu dv
[Reβ>0,Reβ/prime>0,Re (γ−β)>0,Re (γ/prime−β/prime)>0]EH I 230(2), AK 28(2)
3. F3(α,α/prime,β,β/prime,γ;x, y)
=Γ(γ)
Γ(β)Γ(β/prime)Γ(γ−β−β/prime)
×/integraldisplay/integraldisplay
u≥0,v≥0
u+v≤1uβ−1vβ/prime−1(1−u−v)−γ−β−β/prime−1(1−ux)−α(1−vy)−α/primedu dv
[Reβ>0,Reβ/prime>0,Re (γ−β−β/prime)>0]EH I 230(3), AK 28(3)
4. F4(α,β,γ,γ/prime;x(1−y),y(1−x))
=Γ(γ)Γ(γ/prime)
Γ(α)Γ(β)Γ(γ−α)Γ(γ/prime−β)/integraldisplay1
0/integraldisplay1
0uα−1vβ−1(1−u)γ−α−1(1−v)γ/prime−β−1
×(1−ux)α−γ−γ/prime+1(1−vy)β−γ−γ/prime+1(1−ux−vy)γ+γ/prime−α−β−1du dv
[Reα>0,Reβ>0,Re (γ−α)>0,Re (γ/prime−β)>0]EH I 230(4)
9.185 Integral representations: Integrals of the Mellin–Barnes type
The functions F1,F2,F3,a n d F4can be represented by means of double integrals of the following
form:
F(x, y)=Γ(γ)
Γ(α)Γ(β)(2πi)2/integraldisplayi∞
−i∞/integraldisplayi∞
−i∞Ψ(s, t)Γ(−s)Γ(−t)(−x)s(−y)tdsdt
1022 Confluent Hypergeometric Functions 9.201
Ψ(s, t) F(x, y)
Γ(α+s+t)Γ(β+s)Γ(β/prime+t)
Γ(β/prime)Γ(γ+s+t)F1(α,β,β/prime,γ;x, y)
Γ(α+s+t)Γ(β+s)Γ(β/prime+t)Γ(γ/prime)
Γ(β/prime)Γ(γ+s)Γ(γ/prime+t)F2(α,β,β/prime,γ,γ/prime;x, y)
Γ(α+s)Γ(α/prime+t)Γ(β+s)Γ(β/prime+t)
Γ(α/prime)Γ(β/prime)Γ(γ+s+t)F3(α,α/prime,β,β/prime,γ;x, y)
Γ(α+s+t)Γ(β+s+t)Γ(γ/prime)
Γ(γ+s)Γ(γ/prime+t)F4(α,β,γ,γ/prime;x, y)
[α,α/prime,β,β/primemay not be negative integers] EH I 232(9–13), AK 41(33)
9.19 A hypergeometric function of several variables
FA(α;β1,...,β n;γ1,...,γ n;z1,...,z n)
=∞/summationdisplay
m1=0∞/summationdisplay
m2=0...∞/summationdisplay
mn=0(α)m1+···+mn(β1)m1···(βn)mn
(γ1)m1···(γn)mnm1!···mn!zm1
1zm2
2···zmn
n
ET I 385
9.2 Confluent Hypergeometric Functions
9.20 Introduction
9.20110Aconfluent hypergeometric function is obtained by taking the limit as c→∞ in the solution
of Riemann’s differential equation
u=P⎧
⎨
⎩0∞ c
1
2+μ−cc−λz
1
2−μ0 λ⎫
⎬
⎭WH
9.202 The equation obtained by means of this limiting process is of the form
1.d2u
dz2+du
dz+/parenleftbiggλ
z+1
4−μ2
z2/parenrightbigg
u=0 WH
Equation 9.202 1 has the following two linearly independent solutions:
2. z1
2+μe−zΦ/parenleftbig1
2+μ−λ,2μ+1 ;z/parenrightbig
3. z1
2−μe−zΦ/parenleftbig1
2−μ−λ,−2μ+1 ;z/parenrightbig
which are defined for all values of μ/negationslash=±1
2,±2
2,±3
2,... MO 111
9.214 The functions Φ(α, γ;z)andΨ(α, γ;z) 1023
9.21 The functions Φ(α, γ;z)andΨ(α, γ;z)
9.21010The series
1. Φ( α,γ;z)=1+α
γz
1!+α(α+1 )
γ(γ+1 )z2
2!+α(α+1 ) (α+2 )
γ(γ+1 ) (γ+2 )z3
3!+...
is also called a confluent hypergeometric function .
A second notation: Φ( α,γ;z)= 1F1(α;γ;z).
2. Ψ( α,γ;z)=Γ(1−γ)
Γ(α−γ+1 )Φ(α,γ;z)+Γ(γ−1)
Γ(α)z1−γΦ(α−γ+1,2−γ;z) EH I 257(7)
3. Bateman’s function k ν(x) is defined by
kν(x)=2
π/integraldisplayπ/2
0cos(xtanθ−νθ)dθ [x,νreal] EH I 267
9.211 Integral representation:
1. Φ( α,γ;z)=21−γe1
2z
B(α,γ−α)/integraldisplay1
−1(1−t)γ−α−1(1 +t)α−1e1
2ztdt
[0<Reα<Reγ] MO 114
2. Φ( α,γ;z)=1
B(α,γ−α)z1−γ/integraldisplayz
0ettα−1(z−t)γ−α−1dt
[0<Reα<Reγ] MO 114
3. Φ( −ν,α+1 ;z)=Γ(α+1 )
Γ(α+ν+1 )ezz−α
2/integraldisplay∞
0e−ttν+α
2Jα/parenleftBig
2√
zt/parenrightBig
dt
/bracketleftBig
Re(α+ν+1 )>0,|argz|<π
2/bracketrightBig
MO 115
4.8Ψ(α,γ;z)=1
Γ(α)/integraldisplay∞
0e−zttα−1(1 +t)γ−α−1dt [Reα>0,Rez>0] EH I 255(2)
Functional relations
9.212
1. Φ( α,γ;z)=ezΦ(γ−α,γ;−z) MO 112
2.z
γΦ(α+1,γ+1 ;z)=Φ ( α+1,γ;z)−Φ(α,γ;z) MO 112
3. αΦ(α+1,γ+1 ;z)=(α−γ)Φ (α,γ+1 ;z)+γΦ(α,γ;z) MO 112
4. αΦ(α+1,γ;z)=(z+2a−γ)Φ(α,γ;z)+(γ−α)Φ(α−1,γ;z) MO 112
9.213dΦ
dz=α
γΦ(α+1,γ+1 ;z) MO 112
9.214 lim
γ→−n1
Γ(γ)Φ(α,γ;z)=zn+1/parenleftbiggα+n
n+1/parenrightbigg
Φ(α+n+1,n+2 ;z)[ n=0,1,2,...] MO 112
1024 Confluent Hypergeometric Functions 9.215
9.21510
1. Φ( α,α;z)=ezMO 15
2. Φ( α,2α;2z)=2α−1
2exp/bracketleftbig1
4(1−2α)πi/bracketrightbig
Γ/parenleftbig
α+1
2/parenrightbig
ezz1
2−αJα−1
2/parenleftbig
zeπ
2i/parenrightbig
MO 112
3. Φ/parenleftbig
p+1
2,2p+1 ;2iz/parenrightbig
=Γ (p+1 )/parenleftBigz
2/parenrightBig−p
eizJp(z) MO 15
For a representation of special functions in terms of a confluent hypergeometric function Φ( α,γ;z), see:
•for the probability integral, 9.236 ;
•for integrals of Bessel functions, 6.631 1;
•for Hermite polynomials, 8.953 and8.959 ;
•for Laguerre polynomials, 8.972 1;
•for parabolic cylinder functions, 9.240 ;
•for the Whittaker functions Mλ,μ(z),9.220 2a n d9.220 3.
9.216 The function Φ( α,γ;z) is a solution of the differential equation
1. zd2F
dz2+(γ−z)dF
dz−αF=0 MO 111
This equation has two linearly independent solutions:
2. Φ( α,γ;z)
3. z1−γΦ(α−γ+1,2−γ;z) MO 112
9.22–9.23 The Whittaker functions Mλ,μ(z)andWλ,μ(z)
9.220 If we make the change of variable u=e−z
2Win equation 9.202 1, we obtain the equation
1.d2W
dz2+/parenleftbigg
−1
4+λ
z+1
4−μ2
z2/parenrightbigg
W=0 MO 115
Equation 9.220 1 has the following two linearly independent solutions:
2. Mλ,μ(z)=zμ+1
2e−z/2Φ/parenleftbig
μ−λ+1
2,2μ+1 ;z/parenrightbig
3.11Mλ,−μ(z)=z−μ+1
2e−z/2Φ/parenleftbig
−μ−λ+1
2,−2μ+1 ;z/parenrightbig
MO 115
To obtain solutions that are also suitable for 2 μ=±1,±2,...,we introduce Whittaker’s function
4. Wλ,μ(z)=Γ(−2μ)
Γ/parenleftbig1
2−μ−λ/parenrightbigMλ,μ(z)+Γ(2μ)
Γ/parenleftbig1
2+μ−λ/parenrightbigMλ,−μ(z) WH
which, for 2 μapproaching an integer, is also a solution of equation 9.220 1.
For the functions Mλ,μ(z)a n d Wλ,μ(z),z= 0 is a branch point and z=∞is an essential singular
point. Therefore, we shall examine these functions only for |argz|<π.
These functions Wλ,μ(z)a n dW−λ,μ(−z) are linearly independent solutions of equation 9.220 1.
9.226 The Whittaker functions Mλ,μ(z)andWλ,μ(z) 1025
Integral representations
9.221 Mλ,μ(z)=zμ+1
2
22μB/parenleftbig
μ+λ+1
2,μ−λ+1
2/parenrightbig/integraldisplay1
−1(1 +t)μ−λ−1
2(1−t)μ+λ−1
2e1
2ztdt, WH
if the integral converges. See also 6.631 1a n d7.623 3.
9.222
1.11Wλ,μ(z)=zμ+1
2e−z/2
Γ/parenleftbig
μ−λ+1
2/parenrightbig/integraldisplay∞
0e−zttμ−λ−1
2(1 +t)μ+λ−1
2dt
/bracketleftBig
Re(μ−λ)>−1
2,|argz|<π
2/bracketrightBig
MO 118
2. Wλ,μ(z)=zλe−z/2
Γ/parenleftbig
μ−λ+1
2/parenrightbig/integraldisplay∞
0tμ−λ−1
2e−t/parenleftbigg
1+t
z/parenrightbiggμ+λ−1
2
dt
/bracketleftbig
Re(μ−λ)>−1
2,|argz|<π/bracketrightbig
WH
9.223 Wλ,μ(z)=e−z
2
2πi/integraldisplayi∞
−i∞Γ(u−λ)Γ/parenleftbig
−u−μ+1
2/parenrightbig
Γ/parenleftbig
−u+μ+1
2/parenrightbig
Γ/parenleftbig
−λ+μ+1
2/parenrightbig
Γ/parenleftbig
−λ−μ+1
2/parenrightbig zudu
[the path of integration is chosen in such a way that the poles of the function Γ( u−λ) are separated from
the poles of the functions Γ/parenleftbig
−u−μ+1
2/parenrightbig
and Γ/parenleftbig
−u+μ+1
2/parenrightbig
.] See also 7.142 . MO 118
9.224 Wμ,1
2+μ(z)=zμ+1e−1
2z/integraldisplay∞
0(1 +t)2μe−ztdt=z−μe1
2z/integraldisplay∞
zt2μe−tdt [Rez>0] WH
9.225
1. Wλ,μ(x)W−λ,μ(x)=−x/integraldisplay∞
0tanh2λt
2{J2μ(xsinht)sin(μ−λ)π
+Y2μ(xsinht)cos(μ−λ)π}dt
/bracketleftbig
|Reμ|−Reλ<1
2;x>0/bracketrightbig
MO 119
2. Wκ,μ(z1)Wλ,μ(z2)=(z1z2)μ+1
2exp/bracketleftbig
−1
2(z1+z2)/bracketrightbig
Γ(1−κ−λ)
×/integraldisplay∞
0e−tt−κ−λ(z1+t)−1
2+κ−μ(z2+t)−1
2+λ−μ
×F/parenleftbig1
2−κ+μ,1
2−λ+μ;1−κ−λ;Θ/parenrightbig
dt
Θ=t(z1+z2+t)
(z1+t)(z2+t),[z1/negationslash=0,z2/negationslash=0,|argz1|<π , |argz2|<π , Re(κ+λ)<1]
MO 119
See also 3.334 ,3.381 6,3.382 3,3.383 4, 8,3.384 3,3.471 2.
9.226 Series representations
M0,μ(z)=z1
2+μ/braceleftBigg
1+∞/summationdisplay
k=1z2k
24kk!(μ+1 ) (μ+2 )...(μ+k)/bracerightBigg
WH
1026 Confluent Hypergeometric Functions 9.227
Asymptotic representations
9.2277For large values of |z|
Wλ,μ(z)∼e−z/2zλ⎛
⎝1+∞/summationdisplay
k=1/bracketleftBig
μ2−/parenleftbig
λ−1
2/parenrightbig2/bracketrightBig/bracketleftBig
μ2−/parenleftbig
λ−3
2/parenrightbig2/bracketrightBig
.../bracketleftBig
μ2−/parenleftbig
λ−k+1
2/parenrightbig2/bracketrightBig
k!zk⎞
⎠
[|argz|≤π−α<π ] WH
9.228 For large values of |λ|
Mλ,μ(z)∼1√πΓ(2μ+1 )λ−μ−1
4z1/4cos/parenleftbigg
2√
λz−μπ−1
4π/parenrightbigg
MO 118
9.229
1. Wλ,μ∼−/parenleftbigg4z
λ/parenrightbigg1
4
e−λ+λlnλsin/parenleftBig
2√
λz−λπ−π
4/parenrightBig
MO 118
2. W−λ,μ∼/parenleftBigz
4λ/parenrightBig1
4eλ−λlnλ−2√
λzMO 118
Formulas 9.228 and9.229 are applicable for
|λ|/greatermuch1,|λ|/greatermuch|z|,|λ|/greatermuch|μ|,z/negationslash=0 , |arg√z|<3π
4and|argλ|<π
2. MO 118
Functional relations
9.231
1. Mn+μ+1
2,μ(z)=z1
2−μe1
2z
(2μ+ 1)(2 μ+2 )...(2μ+n)dn
dzn/parenleftbig
zn+2μe−z/parenrightbig
[n=0,1,2,...;2μ/negationslash=−1,−2,−3,...]
MO 117
2. z−1
2−μMλ,μ(z)=(−z)−1
2−μM−λ,μ(−z)[ 2 μ/negationslash=−1,−2,−3,...] WH
9.232
1. Wλ,μ(z)=Wλ,−μ(z) MO 116
2. W−λ,μ(−z)=Γ(−2μ)
Γ/parenleftbig1
2−μ+λ/parenrightbigM−λ,μ(−z)+Γ(2μ)
Γ/parenleftbig1
2+μ+λ/parenrightbigM−λ,−μ(−z)
/bracketleftbig
|arg(−z)|<3
2π/bracketrightbig
WH
9.233
1. Mλ,μ(z)=Γ(2μ+1 )
Γ/parenleftbig
μ−λ+1
2/parenrightbigeiπλW−λ,μ/parenleftbig
eiπz/parenrightbig
+Γ(2μ+1 )
Γ/parenleftbig
μ+λ+1
2/parenrightbigexp/bracketleftbig
iπ/parenleftbig
λ−μ−1
2/parenrightbig/bracketrightbig
Wλ,μ(z)
/bracketleftbig
−3
2π<argz<1
2π;2μ/negationslash=−1,−2,.../bracketrightbig
MO 117
2. Mλ,μ(z)=Γ(2μ+1 )
Γ/parenleftbig
μ−λ+1
2/parenrightbige−iπλW−λ,μ/parenleftbig
e−iπz/parenrightbig
+Γ(2μ+1 )
Γ/parenleftbig
μ+λ+1
2/parenrightbigexp/bracketleftbig
−iπ/parenleftbig
λ−μ−1
2/parenrightbig/bracketrightbig
Wλ,μ(z)
/bracketleftbig
−1
2π<argz<3
2π;2μ/negationslash=−1,−2,.../bracketrightbig
MO 117
9.237 The Whittaker functions Mλ,μ(z)andWλ,μ(z) 1027
9.234 Recursion formulas
1. Wλ,μ(z)=√zWλ−1
2,μ−1
2(z)+/parenleftbig1
2+μ−λ/parenrightbig
Wλ−1,μ(z) WH
2.11Wλ,μ(z)=√zWλ−1
2,μ+1
2(z)+/parenleftbig1
2−μ−λ/parenrightbig
Wλ−1,μ(z) WH
3. zd
dzWλ,μ(z)=/parenleftbig
λ−1
2z/parenrightbig
Wλ,μ(z)−/bracketleftBig
μ2−/parenleftbig
λ−1
2/parenrightbig2/bracketrightBig
Wλ−1,μ(z) WH
4./bracketleftbigg/parenleftbigg
μ+1−z
2/parenrightbigg
Wλ,μ(z)−zd
dzWλ,μ(z)/bracketrightbigg/parenleftbig
μ+1
2+λ/parenrightbig
=/bracketleftbigg/parenleftbigg
μ+1+z
2/parenrightbigg
Wλ,μ+1(z)+zd
dzWλ,μ+1(z)/bracketrightbigg/parenleftbig
μ+1
2−λ/parenrightbig
MO 117
5./parenleftbig3
2+λ+μ/parenrightbig/parenleftbig1
2+λ+μ/parenrightbig
zWλ,μ(z)=z(z+2μ+1 )d
dzWλ+1,μ+1(z)
+/bracketleftbig1
2z2+/parenleftbig
μ−λ−1
2/parenrightbig
z+2μ2+2μ+1
2/bracketrightbig
Wλ+1,μ+1(z)
MO 117
Connections with other functions
9.235
1. M0,μ(z)=22μΓ(μ+1 )√zIμ/parenleftBigz
2/parenrightBig
MO 125a
2. W0,μ(z)=/radicalbiggz
πKμ/parenleftBigz
2/parenrightBig
MO 125
9.236
1. Φ( x)=1−ex2
2√πxW−1
4,1
4/parenleftbig
x2/parenrightbig
=2x√πΦ/parenleftbig1
2,3
2;−x2/parenrightbig
W H ,M O1 2 6
2. li( z)=−√z/radicalBig
ln1
2W−1
2,0(−lnz) WH
3. Γ( α,x)=e−xΨ(1−α,1−α;x) EH I 266(21)
4. γ(α,x)=xα
αΦ(α,α+1 ;−x) EH I 266(22)
9.237
1. Wλ,μ(z)=(−1)2μzμ+1
2e−1
2z
Γ/parenleftbig1
2−μ−λ/parenrightbig
Γ/parenleftbig1
2+μ−λ/parenrightbig
×/braceleftBigg∞/summationdisplay
k=0Γ/parenleftbig
μ+k−λ+1
2/parenrightbig
k!(2μ+k)!zk/bracketleftbig
Ψ(k+1 )+Ψ ( 2 μ+k+1 )−Ψ/parenleftbig
μ+k−λ+1
2/parenrightbig
−lnz/bracketrightbig
+(−z)−2μ2μ−1/summationdisplay
k=0Γ(2μ−k)Γ/parenleftbig
k−μ−λ+1
2/parenrightbig
k!(−z)k/bracerightBigg
/bracketleftbig
|argz|<3
2π;2μ+ 1 is a natural number/bracketrightbig
MO 116
1028 Confluent Hypergeometric Functions 9.238
2. Set λ−μ−1
2=l,w h e r e l+ 1 is a natural number. Then
3. Wl+μ+1
2,μ(z)=(−1)lzμ+1
2e−1
2z(2μ+ 1)(2 μ+2 )···(2μ+l)Φ(−l,2μ+1 ;z)
=(−1)lzμ+1
2e−1
2zL2μ
l(z)
MO 116
9.238
1. Jν(x)=2−ν
Γ(ν+1 )xνe−ixΦ/parenleftbig1
2+ν,1+2ν;2ix/parenrightbig
EH I 265(9)
2. Iν(x)=2−ν
Γ(ν+1 )xνe−xΦ/parenleftbig1
2+ν,1+2ν;2x/parenrightbig
EH I 265(10)
3. Kν(x)=√πe−x(2x)νΨ/parenleftbig1
2+ν,1+2ν;2x/parenrightbig
EH I 265(13)
9.24–9.25 Parabolic cylinder functions Dp(z)
9.240 Dp(z)=21
4+p
2W1
4+p
2,−1
4/parenleftbiggz2
2/parenrightbigg
z−1/2
=2p
2e−z2
4⎧
⎪⎪⎨
⎪⎪⎩√
π
Γ/parenleftbigg1−p
2/parenrightbiggΦ/parenleftbigg
−p
2,1
2;z2
2/parenrightbigg
−√
2πz
Γ/parenleftBig
−p
2/parenrightBigΦ/parenleftbigg1−p
2,3
2;z2
2/parenrightbigg⎫
⎪⎪⎬
⎪⎪⎭
MO 120a
are called parabolic cylinder functions .
Integral representations
9.241
1. Dp(z)=1√π2p+1
2e−π
2piez2
4/integraldisplay∞
−∞xpe−2x2+2ixzdx [Rep>−1; for x<0,argxp=pπi]
MO 122
2. Dp(z)=e−z2
4
Γ(−p)/integraldisplay∞
0e−xz−x2
2x−p−1dx [Rep<0] (cf. 3.462 1) MO 122
9.242
1.10Dp(z)=−Γ(p+1 )
2πie−1
4z2/integraldisplay(0+)
∞e−zt−1
2t2(−t)−p−1dt [|arg(−t)|≤π] WH
2. Dp(z)=21
2(p−1)Γ/parenleftbigp
2+1/parenrightbig
iπ/integraldisplay(−1+)
−∞e1
4z2t(1 +t)−1
2p−1(1−t)1
2(p−1)dt
/bracketleftBig
|argz|<π
4;|arg(1 + t)|≤π/bracketrightBig
WH
3. Dp(z)=1
2πie−1
4z2/integraldisplay∞i
−∞iΓ/parenleftbig1
2t−1
2p/parenrightbig
Γ(−t)
Γ(−p)/parenleftBig√
2/parenrightBigt−p−2
ztdt
/bracketleftbig
|argz|<3
4π;pis not a positive integer/bracketrightbig
WH
9.246 Parabolic cylinder functions Dp(z) 1029
4. Dp(z)=1
2πie−1
4z2/integraldisplay(0−)
∞Γ/parenleftbig1
2t−1
2p/parenrightbig
Γ(−t)
Γ(−p)/parenleftBig√
2/parenrightBigt−p−2
ztdt
[for all values of arg z; also, the contours encircle the poles of the function Γ( −t), but they do not encircle
the poles of the function Γ/parenleftbig1
2t−1
2p/parenrightbig
]. WH
9.243
1. Dn(z)=(−1)μ/parenleftBigπ
2/parenrightBig−1/2/parenleftbig√n/parenrightbign+1e1
4z2−1
2n⎧
⎨
⎩/integraldisplay∞
−∞e−n(t−1)2cos
sin/parenleftbig
zt√n/parenrightbig
dt
+/integraldisplay∞
0/bracketleftBig
e1
2n(1−t2)tn−e−n(t−1)2/bracketrightBigcos
sin/parenleftbig
zt√n/parenrightbig
dt−/integraldisplay0
−∞e−n(t−1)2cos
sin/parenleftbig
zt√n/parenrightbig
dt⎫
⎬
⎭
[nis a natural number] WH
2. Dn(z)=(−1)μ2n+2(2π)−1/2e1
4z2/integraldisplay∞
0tne−2t2cos
sin(2zt)dt
[nis a natural number, μ=/floorleftBign
2/floorrightBig
, and the cosine or sine is chosen accordingly as nis even or odd]
WH
9.244
1. D−p−1[(1 +i)z]=e−iz2
2
2p−1
2Γ/parenleftbigp+1
2/parenrightbig/integraldisplay∞
0e−ix2z2xp
(1 +x2)1+p
2dx/bracketleftbig
Rep>−1,Re/parenleftbig
iz2/parenrightbig
≥0/bracketrightbig
MO 122
2. Dp[(1 +i)z]=2p+1
2
Γ/parenleftbig
−p
2/parenrightbig/integraldisplay∞
1e−i
2z2x(x+1 )p−1
2
(x−1)1+p
2dx/bracketleftbig
Rep<0; Re/parenleftbig
iz2/parenrightbig
≥0/bracketrightbig
MO 122
See also 3.383 6, 7,3.384 2, 6,3.966 5, 6.
9.245
1.10Dp(x)D−p−1(x)=−1√π/integraldisplay∞
0cothp+1
2/parenleftbiggt
2/parenrightbigg1√
sinhtsin/parenleftbiggx2sinht+pπ
2/parenrightbigg
dt
[xis real ,Rep<0] MO 122
2. Dp/parenleftbig
zeπ
4i/parenrightbig
Dp/parenleftbig
ze−π
4i/parenrightbig
=1
Γ(−p)/integraldisplay∞
0cothptexp/parenleftbigg
−z2
2sinh 2t/parenrightbiggdt
sinht/bracketleftBig
|argz|<π
4;R e p<0/bracketrightBig
MO 122
See also 6.613 .
9.246 Asymptotic expansions. If |z|/greatermuch1a n d |z|/greatermuch|p|,t h e n
1. Dp(z)∼e−z2
4zp/parenleftbigg
1−p(p−1)
2z2+p(p−1)(p−2)(p−3)
2·4z4−.../parenrightbigg
/bracketleftbig
|argz|<3
4π/bracketrightbig
MO 121
2.11Dp(z)∼e−z2/4zp/parenleftbigg
1−p(p−1)
2z2+p(p−1)(p−2)(p−3)
2·4z4−.../parenrightbigg
−√
2π
Γ(−p)epπiez2/4z−p−1⎛
⎝1+(p+1 ) (p+2 )
2z2+(p+1 ) (p+2 ) (p+3 ) (p+4 )
2·4z4+...⎞
⎠
/bracketleftbig1
4π<argz<5
4π/bracketrightbig
MO 121
1030 Confluent Hypergeometric Functions 9.247
3.11Dp(z)∼e−z2/4zp/parenleftbigg
1−p(p−1)
2z2+p(p−1)(p−2)(p−3)
2·4z4−.../parenrightbigg
−√
2π
Γ(−p)e−pπiez2/4z−p−1⎛
⎝1+(p+1 ) (p+2 )
2z2+(p+1 ) (p+2 ) (p+3 ) (p+4 )
2·4z4+...⎞
⎠
/bracketleftbig
−1
4π>argz>−5
4π/bracketrightbig
MO 121
Functional relations
9.247 Recursion formulas:
1. Dp+1(z)−zDp(z)+pDp−1(z)=0 WH
2.d
dzDp(z)+1
2zDp(z)−pDp−1(z)=0 WH
3.d
dzDp(z)−1
2zDp(z)+Dp+1(z)=0 MO 121
9.248 Linear relations:
1. Dp(z)=Γ(p+1 )√
2π/bracketleftBig
eπ/2D−p−1(iz)+e−πpi/2D−p−1(−iz)/bracketrightBig
=e−pπiDp(−z)+√
2π
Γ(−p)e−π(p+1)i/2D−p−1(iz)
=epπiDp(−z)+√
2π
Γ(−p)eπ(p+1)i/2D−p−1(−iz)
MO 121
9.24910Dp[(1 +i)x]+Dp[−(1 +i)x]=21+p/2
Γ(−p)exp/bracketleftbigg
−i
2/parenleftBig
x2+pπ
2/parenrightBig/bracketrightbigg/integraldisplay∞
0cosxt
tp+1e−it2/4dt
[xreal; −1<Rep<0] MO 122
9.25110Dn(z)=(−1)nez2/4dn
dzn/parenleftBig
e−z2/2/parenrightBig
[n=0,1,2,...] WH
9.252 Dp(ax+by) = exp(bx−ay)2
4/parenleftbigga√
a2+b2/parenrightbiggp∞/summationdisplay
k=0/parenleftBigp
k/parenrightBig
Dp−k/parenleftBig/radicalbig
a2+b2x/parenrightBig
Dk/parenleftBig/radicalbig
a2+b2y/parenrightBig/parenleftbiggb
a/parenrightbiggk
[a>b> 0,x > 0,y > 0,Rep≥0] “summation theorem” MO 124
Connections with other functions
9.25311Dn(z)=2−n
2e−z2
4Hn/parenleftbiggz√
2/parenrightbigg
MO 123a
9.254
1. D−1(z)=ez2
4/radicalbiggπ
2/bracketleftbigg
1−Φ/parenleftbiggz√
2/parenrightbigg/bracketrightbigg
MO 123
2.11D−2(z)=ez2
4/radicalbiggπ
2/braceleftBigg/radicalbigg
2
πe−z2
2−z/bracketleftbigg
1−Φ/parenleftbiggz√
2/parenrightbigg/bracketrightbigg/bracerightBigg
MO 123
9.262 Confluent hypergeometric series of two variables 1031
9.255 Differential equations leading to parabolic cylinder functions:
1.d2u
dz2+/parenleftbigg
p+1
2−z2
4/parenrightbigg
u=0
The solutions are u=Dp(z),Dp(−z),D−p−1(iz), and D−p−1(−iz).
(These four solutions are linearly dependent. See 9.248 .)
2.d2u
dz2+/parenleftbig
z2+λ/parenrightbig
u=0,u =D−1+iλ
2[±(1 +i)z]
EH II 118(12,13)a, MO 123
3.7d2u
dz2+zdu
dz+(p+1 )u=0,u =e−z2
4Dp(z) MO 123
9.26 Confluent hypergeometric series of two variables
9.261
1.6Φ1(α,β,γ,x,y )=∞/summationdisplay
m,n=0(α)m+n(β)m
(γ)m+nm!n!xmyn[|x|<1] EH I 225(20)
2. Φ 2(β,β/prime,γ,x ,y )=∞/summationdisplay
m,n=0(β)m(β/prime)m
(γ)m+nm!n!xmynEH I 225(21)a, ET I 385
3. Φ 3(β,γ,x,y )=∞/summationdisplay
m,n=0(β)m
(γ)m+nm!n!xmynEH I 225(22)
The functions Φ 1,Φ2,Φ3satisfy the following systems of partial differential equations:
9.262
1. z=Φ1(α,β,γ,x,y ) EH I 235(23)
x(1−x)∂2z
∂x2+y(1−x)∂2z
∂x∂y+[γ−(α+β+1 )x]∂z
∂x−βy∂z
∂y−αβz=0,
y∂2z
∂y2+x∂2z
∂x∂y+(γ−y)∂z
∂y−x∂z
∂x−αz=0
2. z=Φ2(β,β/prime,γ,x ,y ) EH I 235(24)
x∂2z
∂x2+y∂2z
∂x∂y+(γ−x)∂z
∂x−βz=0,
y∂2z
∂y2+x∂2z
∂x∂y+(γ−y)∂z
∂y−β/primez=0
1032 Meijer’s G-Function 9.301
3. z=Φ3(β,γ,x,y ) EH I 235(25)
x∂2z
∂x2+y∂2z
∂x∂y+(γ−x)∂z
∂x−βz=0,
y∂2z
∂y2+x∂2z
∂x∂y+γ∂z
∂y−z=0
9.3 Meijer’s G-Function
9.30 Definition
9.301 Gm,n
p,q/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
=1
2πi/integraldisplaym/productdisplay
j=1Γ(bj−s)n/productdisplay
j=1Γ( 1−aj+s)
q/productdisplay
j=m+1Γ( 1−bj+s)p/productdisplay
j=n+1Γ(aj−s)xsds
[0≤m≤q,0≤n≤p, and the poles of Γ ( bj−s) must not coincide with the poles of Γ(1 −ak+s)
for any jandk(where j=1,...,m ;k=1,...,n ]). Besides 9.301 , the following notations are also
used:
Gmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
r
bs/parenrightbigg
,Gmn
pq(x),G (x) EH I 207(1)
9.302 Three types of integration paths Lin the right member of 9.301 can be exhibited:
1. The path Lruns from −∞to +∞in such a way that the poles of the functions Γ (1 −ak+s)
lie to the left, and the poles of the functions Γ ( bj−s) lie to the right of L(forj=1,2,...,m
andk=1,2,...,n ). In this case, the conditions under which the integral 9.301 converges are of
the form
p+q<2(m+n),|argx|</parenleftbig
m+n−1
2p−1
2q/parenrightbig
π. EH I 207(2)
2. Lis a loop, beginning and ending at + ∞, that encircles the poles of the functions Γ ( bj−s)( f o r
j=1,2,...,m ) once in the negative direction. All the poles of the functions Γ (1 −ak+s)m u s t
remain outside this loop. Then, the conditions under which the integral 9.301 converges are:
q≥1 and either p<q orp=qand|x|<1. EH I 207(3)
3. Lis a loop, beginning and ending at −∞, that encircles the poles of the functions Γ (1 −ak+s)
(fork=1,2,...,n ) once in the positive direction. All the poles of the functions Γ ( bj−s)( f o r
j=1,2,...,m ) must remain outside this loop.
The conditions under which the integral in 9.301 converges are
p≥1 and either p>q orp=qand|x|>1. EH I 207(4)
The function Gmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
r
bs/parenrightbigg
is analytic with respect to x; it is symmetric with respect to the
parameters a1,...,a nand also with respect to an+1,...,a p;b1,...,b m;bm+1,...,b q.
EH I 208
9.304 Functional relations 1033
9.30311If no two bj(forj=1,2,...,n ) differ by an integer, then, under the conditions that either
p<q orp=qand|x|<1,
Gmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglear
bs/parenrightbigg
=m/summationdisplay
h=1m/productdisplay
j=1Γ(bj−bh)n/productdisplay
j=1Γ(1+ bh−aj)
q/productdisplay
j=m+1Γ(1+ bh−bj)p/productdisplay
j=n+1Γ(aj−bh)xbh
×pFq−1/bracketleftbigg
1+bh−a1,...,1+bh−ap;1 + bh−b1,...
...,∗,...,1+bh−bq;(−1)p−m−nx/bracketrightbigg
EH I 208(5)
The prime by the product symbol denotes the omission of the product when j=h. The asterisk in
the function pFq−1denotes the omission of the hthparameter.
9.3047If no two ak(fork=1,2,...,n ) differ by an integer then, under the conditions that q<p or
q=pand|x|>1,
Gmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
r
bs/parenrightbigg
=n/summationdisplay
h=1n/productdisplay/prime
j=1Γ(ah−aj)m/productdisplay
j=1Γ(bj−ah+1 )
p/productdisplay
j=n+1Γ(aj−ah+1 )q/productdisplay
j=m+1Γ(ah−bj)xah−1
×qFp−1/bracketleftbigg
1+b1−ah,...,1+bq−ah;1 + a1−ah,...
...,∗,...,1+ap−ah;(−1)q−m−nx−1/bracketrightbigg
EH I 208(6)
9.31 Functional relations
If one of the parameters aj(forj=1,2,...,n ) coincides with one of the parameters bj(forj=m+
1,m +2,...,q ), the order of the G-function decreases. For example,
1. Gmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q−1,a1/parenrightbigg
=Gm,n−1
p−1,q−1/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
2,...,a p
b1,...,b q−1/parenrightbigg
[n, p, q ≥1]
An analogous relationship occurs when one of the parameters bj(forj=1,2,...,m ) coincides
with one of the aj(forj=n+1,...,p ). In this case, it is mand not nthat decreases by one
unit.
TheG-function with p>q can be transformed into the G-function with p<q by means of the
relationships:
2. Gmn
pq/parenleftbigg
x−1/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
r
bs/parenrightbigg
=Gnm
qp/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−b
s
1−ar/parenrightbigg
EH I 209(9)
1034 Meijer’s G-Function 9.304
3. xd
dxGmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
r
bs/parenrightbigg
=Gmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1−1,a2,...,a p
b1,...,b q/parenrightbigg
+(a1−1)Gmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
r
bs/parenrightbigg
[n≥1] EH I 210(13)
4. Gm+1,n
p+1,q+1/parenleftbigg
z/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
p,1−r
0,bq/parenrightbigg
=(−1)rGm,n+1
p+1,q+1/parenleftbigg
z/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−r,a
p
bq,1/parenrightbigg
[r=0,1,2,...] MS2 6 (1.2.2)
5. zkGmn
pq/parenleftbigg
z/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
p
bq/parenrightbigg
=Gmn
pq/parenleftbigg
z/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
p+k
bq+k/parenrightbigg
MS2 7 (1.2.7)
9.32 A differential equation for the G-function
Gmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
r
bs/parenrightbigg
satisfies the following linear qth-order differential equation:
⎡
⎣(−1)p−m−nxp/productdisplay
j=1/parenleftbigg
xd
dx−aj+1/parenrightbigg
−q/productdisplay
j=1/parenleftbigg
xd
dx−bj/parenrightbigg⎤
⎦y=0 [ p≤q] EH I 210(1)
9.33 Series of G-functions
Gmn
pq/parenleftbigg
λx/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q/parenrightbigg
=λb1∞/summationdisplay
r=01
r!(1−λ)rGmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1+r, b2,...,b q/parenrightbigg
[|λ−1|<1,m≥1,ifm=1a n d p<q,λmay be arbitrary]
EH I 213(1)
=λbq∞/summationdisplay
r=01
r!(λ−1)rGmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p
b1,...,b q−1,bq+r/parenrightbigg
[m<q , |λ−1|<1]
EH I 213(2)
=λa1−1∞/summationdisplay
r=01
r!/parenleftbigg
λ−1
λ/parenrightbiggr
Gmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1−r, a2,...,a p
b1,...,b q/parenrightbigg
/bracketleftbig
n≥1,Reλ>1
2,(ifn=1a n d p>q,t h e n λmay be arbitrary)/bracketrightbig
EH I 213(3)
=λap−1∞/summationdisplay
r=01
r!/parenleftbigg1
λ−1/parenrightbiggr
Gmn
pq/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1,...,a p−1,ap−r
b1,...,b q/parenrightbigg
/bracketleftbig
n<p , Reγ>1
2/bracketrightbig
EH I 213(4)
For integrals of the G-function, see 7.8.
9.34 Connections with other special functions
1. Jν(x)xμ=2μG10
02/parenleftbigg1
4x2/vextendsingle/vextendsingle/vextendsingle/vextendsingle
1
2ν+1
2μ,1
2μ−1
2ν/parenrightbigg
EH I 219(44)
2. Yν(x)xμ=2μG20
13/parenleftBigg
1
4x2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2μ−1
2ν−1
2
1
2μ−1
2ν,1
2μ+1
2ν,1
2μ−1
2ν−1
2/parenrightBigg
EH I 219(46)
9.304 Functional relations 1035
3. Kν(x)xμ=2μ−1G20
02/parenleftbigg1
4x2/vextendsingle/vextendsingle/vextendsingle/vextendsingle
1
2μ+1
2ν,1
2μ−1
2ν/parenrightbigg
EH I 219(47)
4. Kν(x)=ex√πG20
12/parenleftbigg
2x/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2
ν,−ν/parenrightbigg
EH I 219(49)
5. Hν(x)xμ=2μG11
13/parenleftBigg
1
4x2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2+1
2ν+1
2μ
1
2+1
2ν+1
2μ,1
2μ−1
2ν,1
2μ+1
2ν/parenrightBigg
EH I 220(51)
6. Sμ,ν(x)=2μ−1 1
Γ/parenleftbig1−μ−ν
2/parenrightbig
Γ/parenleftbig1−μ+ν
2/parenrightbigG31
13/parenleftBigg
1
4x2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2+1
2μ
1
2+1
2μ,1
2ν,−1
2ν/parenrightBigg
EH I 220(55)
7.72F1(a,b;c;−x)=Γ(c)x
Γ(a)Γ(b)G12
22/parenleftbigg
x/vextendsingle/vextendsingle/vextendsingle/vextendsingle−a,−b
−1,−c/parenrightbigg
EH I 222(74)a
8. pFq(a1,...,a p;b1,...,b q;x)=/producttextq
j=1Γ(bj)/producttextp
j=1Γ(aj)G1,p
p,q+1/parenleftbigg
−x/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−a
1,...,1−ap
0,1−b1,...,1−bq/parenrightbigg
=/producttextq
j=1Γ(bj)/producttextp
j=1Γ(aj)Gp,1
q+1,p/parenleftbigg
−1
x/vextendsingle/vextendsingle/vextendsingle/vextendsingle1,b
1,...,b q
a1,...,a p/parenrightbigg
EH I 215(1)
9. Wk,m(x)=2k√xe1
2x
√
2πG40
24/parenleftBigg
x2
4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
4−1
2k,3
4−1
2k
1
2+1
2m,1
2−1
2m,1
2m,−1
2m/parenrightBigg
EH I 221(70)
9.4 MacRobert’s E-Function
9.41 Representation by means of multiple integrals
E(p;αr:q;/rho1s:x)=Γ(αq+1)
Γ(/rho11−α1)Γ(/rho12−α2)···Γ(/rho1q−αq)
×q/productdisplay
μ=1/integraldisplay∞
0λ/rho1μ−αμ−1
μ (1−λμ)−/rho1μdλμp−q−1/productdisplay
ν=2/integraldisplay∞
0e−λq+νλαq+ν−1
q+νdλq+ν
×/integraldisplay∞
0e−λpλαp−1
p/bracketleftbigg
1+λq+2λq+3···λp
(1 +λ1)···(1 +λq)x/bracketrightbigg−αq+1
dλp
[|argx|<π,p≥q+1 ,αrand/rho1sare bounded by the condition that the integrals on the right be
convergent.] EH I 204(3)
9.42 Functional relations
1. α1xE(α1,...,α p:/rho11,...,/rho1 q:x)=xE(α1+1,α2,...,α p:/rho11,...,/rho1 q:x)
+E(α1+1,α2+1,...,α p+1:/rho11+1,...,/rho1 q+1:x)
EH I 205(7)
1036 Riemann’s Zeta Functions ζ(z,q)andζ(z), and the Functions Φ(z,s,v)andξ(s) 9.511
2. ( /rho11−1)xE(α1,...,α p:/rho11,...,/rho1 q:x)=xE(α1,...,α p:/rho11−1,/rho12,...,/rho1 q:x)
+E(α1+1,...,α p+1:/rho11+1,...,/rho1 q+1:x)
EH I 205(9)
3.d
dxE(α1,...,α p:/rho11,...,/rho1 q:x)=x−2E(α1+1,...,α p+1:/rho11+1,...,/rho1 q+1:x)
EH I 205(8)
9.5 Riemann’s Zeta Functions ζ(z,q)andζ(z), and the Functions
Φ(z,s,v)andξ(s)
9.51 Definition and integral representations
9.511 ζ(z,q)=1
Γ(z)/integraldisplay∞
0tz−1e−qt
1−e−tdt;
WH
=1
2q−z+q1−z
z−1+2/integraldisplay∞
0/parenleftbig
q2+t2/parenrightbig−z
2/bracketleftbigg
sin/parenleftbigg
zarctant
q/parenrightbigg/bracketrightbiggdt
e2πt−1
[0<q< 1,Rez>1]
WH
9.512 ζ(z,q)=−Γ(1−z)
2πi/integraldisplay(0+)
∞(−θ)z−1e−qθ
1−e−θdθ
This equation is valid for all values of z, except for z=1,2,3,.... It is assumed that the path of
integration (see drawing below) does not pass through the points 2 nπi(where nis a natural number).
See also 4.251 4,4.271 1, 4, 8, 4.272 9, 12,4.294 11.
9.513
1. ζ(z)=1
(1−21−z)Γ (z)/integraldisplay∞
0tz−1
et+1dt [Rez>0] WH
2. ζ(z)=2z
(2z−1)Γ(z)/integraldisplay∞
0tz−1et
e2t−1dt [Rez>1] WH
3.11ζ(z)=πz
2
Γ/parenleftbigz
2/parenrightbig/bracketleftBigg
1
z(z−1)+/integraldisplay∞
1/parenleftBig
t1−z
2+tz
2/parenrightBig
t−1∞/summationdisplay
k=1e−k2πtdt/bracketrightBigg
WH
4. ζ(z)=2z−1
z−1−2z/integraldisplay∞
0/parenleftbig
1+t2/parenrightbig−z
2sin (zarctan t)dt
eπt+1WH
5. ζ(z)=2z−1
2z−1z
z−1+2
2z−1/integraldisplay∞
0/parenleftbigg1
4+t2/parenrightbigg−z/2
sin (zarctan 2 t)dt
e2πt−1WH
See also 3.411 1,3.523 1,3.527 1, 3,4.271 8.
9.532 Functional relations 1037
9.52 Representation as a series or as an infinite product
9.521
1. ζ(z,q)=∞/summationdisplay
n=01
(q+n)z[Rez>1,q/negationslash=0,−1,−2,...] WH
2. ζ(z,q)=2Γ ( 1−z)
(2π)1−z/bracketleftBigg
sinzπ
2∞/summationdisplay
n=1cos2πqn
n1−z+c o szπ
2∞/summationdisplay
n=1sin2πqn
n1−z/bracketrightBigg
[Rez<0,0<q≤1] WH
3.8ζ(z,q)=N/summationdisplay
n=01
(q+n)z−1
(1−z)(N+q)z−1−∞/summationdisplay
n=NFn(z),
where
Fn(z)=1
1−z/parenleftbigg1
(n+1+ q)z−1−1
(n+q)z−1/parenrightbigg
−1
(n+1+ q)z
=z/integraldisplayn+1
n(t−n)dt
(t+q)z+1WH
9.522
1. ζ(z)=∞/summationdisplay
n=11
nz[Rez>1] WH
2. ζ(z)=1
1−21−z∞/summationdisplay
n=1(−1)n+11
nz[Rez>0] WH
9.523 The following product and summation are taken over all primes p:
1.7ζ(z)=/productdisplay
p1
1−p−z[Rez>1] WH
2. ln ζ(z)=/summationdisplay
p∞/summationdisplay
k=11
kpkz[Rez>1] WH
9.52411ζ/prime(z)
ζ(z)=−∞/summationdisplay
k=1Λ(k)
kz, [Rez>1]
where Λ( k)=0w h e n kis not a power of a prime and Λ( k)=l n pwhen kis a power of a prime p. WH
9.53 Functional relations
9.531 ζ(−n, q)=−B/prime
n+2(q)
(n+1 ) (n+2 )=−Bn+1(q)
n+1
[nis a nonnegative integer] see EH I 27 (11) WH
9.532∞/summationdisplay
k=2(−1)k−1
kzkζ(k,q)=l ne−CzΓ(q)
Γ(z+q)−z
q+∞/summationdisplay
k=1qz
k(q+k)[|z|<q] WH
1038 Riemann’s Zeta Functions ζ(z,q)andζ(z), and the Functions Φ(z,s,v)andξ(s) 9.533
9.533
1. lim
z→1ζ(z,q)
Γ(1−z)=−1 WH
2. lim
z→1/braceleftbigg
ζ(z,q)−1
z−1/bracerightbigg
=−Ψ(q) WH
3./braceleftbiggd
dzζ(z,q)/bracerightbigg
z=0=l nΓ ( q)−1
2ln2π WH
9.534 ζ(z,1) =ζ(z)
9.535
1. ζ(z)=1
2z−1ζ/parenleftbig
z,1
2/parenrightbig
[Rez>1] WH
2.112zΓ(1−z)ζ(1−z)sin/parenleftBigzπ
2/parenrightBig
=π1−zζ(z) WH
3. 21−zΓ(z)ζ(z)coszπ
2=πzζ(1−z) WH
4. Γ/parenleftBigz
2/parenrightBig
π−z
2ζ(z)=Γ/parenleftbigg1−z
2/parenrightbigg
πz−1
2ζ(1−z) WH
9.536 lim
z→1/braceleftbigg
ζ(z)−1
z−1/bracerightbigg
=C
9.537 Setz=1
2+it. Then, Ξ( t)=(z−1)Γ/parenleftbigz
2+1/parenrightbig
√
πzζ(z)=Ξ ( −t) is an even function of twith real
coefficients in its expansion in powers of t2. JA
9.54 Singular points and zeros
9.5417
1. z= 1 is the only singular point of the function ζ(z) WH
2. The function ζ(z) has simple zeros at the points −2n,w h e r e nis a natural number. All other
zeros of the function ζ(z) lie in the strip 0 ≤Rez<1.
3.8Riemann’s hypothesis: All zeros of the function ζ(z) lie on the straight line Re z=1
2.I t h a s
been shown that a countably infinite set of zeros of the zeta function lie on this line. The first1,500,000,001 zeros lying in 0 <Imz<545,439,823.215 are known to have Re z=
1
2. WH
9.542 Particular values:
1. ζ(2m)=22m−1π2m|B2m|
(2m)![mis a natural number] WH
2. ζ(1−2m)=−B2m
2m[mis a natural number] WH
3. ζ(−2m)=0 [ mis a natural number] WH
4. ζ/prime(0) =−1
2ln2π WH
9.559 The Lerch function Φ(z,s,v) 1039
9.55 The Lerch function Φ(z,s,v)
9.550 Definition:
Φ(z,s,v)=∞/summationdisplay
n=0(v+n)−szn[|z|<1,v/negationslash=0,−1,...] EH I 27(1)
Functional relations
9.551 Φ(z,s,v)=zmΦ(z,s,m +v)+m−1/summationdisplay
n=0(v+n)−szn[m=1,2,3,..., v /negationslash=0,−1,−2,...]
EH I 27(1)
9.552 Φ(z,s,v)
=iz−v(2π)s−1Γ(1−s)/bracketleftbigg
e−iπs
2Φ/parenleftbigg
e−2πiv,1−s,lnz
2πi/parenrightbigg
−eiπ(s
2−2v)Φ/parenleftbigg
e2πiv,1−s,1−lnz
2πi/parenrightbigg/bracketrightbigg
EH I 29(7)
Series representation
9.553 Φ(z,s,v)=z−vΓ(1−s)∞/summationdisplay
n=−∞(−lnz+2πni)s−1e2πnvi
[0<v≤1,Res<0,|arg (−lnz+2πni)|≤π]EH I 28(6)
9.554 Φ(z,m,v )=z−v/braceleftBigg∞/summationdisplay/prime
n=0ζ(m−n, v)(lnz)n
n!+(lnz)m−1
(m−1)!/bracketleftbigg
Ψ(m)−Ψ(v)−ln/parenleftbigg
ln1
z/parenrightbigg/bracketrightbigg/bracerightBigg∗
[m=2,3,4,..., |lnz|<2π, v /negationslash=0,−1,−2,...]EH I 30(9)
9.555 Φ(z,−m, v)=m!
zv/parenleftbigg
ln1
z/parenrightbigg−m−1
−1
zv∞/summationdisplay
r=0Bm+r+1(v)(lnz)r
r!(m+r+1 )[|lnz|<2π] EH I 30(11)
Integral representation
9.556 Φ(z,s,v)=1
Γ(s)/integraldisplay∞
0ts−1e−vt
1−ze−tdt=1
Γ(s)/integraldisplay∞
0ts−1e−(v−1)tdt
et−z
[Rev>0,or|z|≤1,z/negationslash=1,Res>0,orz=1,Res>1]EH I 27(3)
Limit relationships
9.557 lim
z→1(1−z)1−sΦ(z,s,v)=Γ ( 1 −s)[ R e s<1] EH I 30(12)
9.558 lim
z→1Φ(z,1,v)
−ln(1−z)=1 EH I 30(13)
A connection with a hypergeometric function
9.559 Φ(z,1,v)=v−1
2F1(1,v;1+v;z)[ |z|<1] EH I 30(10)
∗In 9.554 the prime on the symbol/summationtextmeans that the term corresponding to n=m−1 is omitted.
1040 Bernoulli Numbers and Polynomials, Euler Numbers 9.561
9.56 The function ξ(s)
9.561 ξ(s)=1
2s(s−1)Γ/parenleftbig1
2s/parenrightbig
π1
2sζ(s) EH III 190(10)
9.562 ξ(1−s)=ξ(s) EH III 190(11)
9.6 Bernoulli Numbers and Polynomials, Euler Numbers, the Functions
ν(x),ν(x, α),μ(x, β),μ(x, β, α ),λ(x, y)and Euler Polynomials
9.61 Bernoulli numbers
9.610 The numbers Bn, representing the coefficients oftn
n!in the expansion of the function
t
et−1=∞/summationdisplay
n=0Bntn
n![0<|t|<2π],
are called Bernoulli numbers. Thus, the functiont
et−1is a generating function for the Bernoulli num-
bers. GE 48(57), FI II 520
9.611 Integral representations
1. B2n=(−1)n−14n/integraldisplay∞
0x2n−1
e2πx−1dx [n=1,2,...] (cf. 3.411 2, 4)
FI II 721a
2. B2n=(−1)n−1π−2n/integraldisplay∞
0x2n
sinh2xdx [n=1,2,...]
3. B2n=(−1)n−12n(1−2n)
π/integraldisplay∞
0x2n−2ln/parenleftbig
1−e−2πx/parenrightbig
dx
[n=1,2,...]
4.∗Bn= lim
x→0dn
dxn/parenleftbiggx
ex−1/parenrightbigg
See also 3.523 2,4.271 3.
Properties and functional relations
9.6128A symbolic notation:
(B+α)[n]=n/summationdisplay
k=0/parenleftBign
k/parenrightBig
Bkαn−k[n≥2]
in particular
Bn=(B+1 )[n]=n/summationdisplay
k=0/parenleftBign
k/parenrightBig
Bk [n≥2]
hence by recursion
9.622 Bernoulli polynomials 1041
Bn=−n!n−1/summationdisplay
k=0Bk
k!(n+1−k)![n≥2]
9.613 All the Bernoulli numbers are rational numbers.
9.614 Every number Bncan be represented in the form
Bn=Cn−/summationdisplay1
k+1,
where Cnis an integer and the sum is taken over all k>0 such that k+ 1 is a prime and kis a divisor
ofn. GE 64
9.61511All the Bernoulli numbers with odd index are equal to zero, except that B1=−1
2;t h a ti s ,
B2n+1=0f o r na natural number. G E5 2 ,F II I5 2 1
B2n=−1
2n+1+1
2−n−1/summationdisplay
k=1;keven2n(2n−1)...(2n−2k+2 )
(2k)!Bk/2 [n≥1]
9.616 B2n=(−1)n−1(2n)!
22n−1π2nζ(2n)[ n≥0] (cf.9.542) GE 56(79), FI II 721a
9.6177B2n=(−1)n−12(2n)!
(2π)2n1
∞/productdisplay
p=2/parenleftbigg
1−1
p2n/parenrightbigg [n≥1] (cf. 9.523 )
(where the product is taken over all primes p).
•For a connection with Riemann’s zeta function, see 9.542 .
•For a connection with the Euler numbers, see 9.635 .
•For a table of values of the Bernoulli numbers, see 9.71
9.619 An inequality/vextendsingle/vextendsingle/vextendsingle(B−θ)[n]/vextendsingle/vextendsingle/vextendsingle≤|B
n|[0<θ< 1]
9.62 Bernoulli polynomials
9.620 The Bernoulli polynomials Bn(x) are defined by
Bn(x)=n/summationdisplay
k=0/parenleftBign
k/parenrightBig
Bkxn−kGE 51(62)
or symbolically, Bn(x)=(B+x)[n]. GE 52(68)
9.621 The generating function
ext
et−1=∞/summationdisplay
n=0Bn(x)tn−1
n![0<|t|<2π] (cf. 1.213) GE 65(89)a
9.622 Series representation
1.7Bn(x)=−2n!
(2π)n∞/summationdisplay
k=1cos/parenleftbig
2πkx−1
2πn/parenrightbig
kn
[n>1,1≥x≥0;n=1,1>x> 0]AS 805(23.1.16)
1042 Bernoulli Numbers and Polynomials, Euler Numbers 9.623
2.7B2n−1(x)=2(−1)n2(2n−1)!
(2π)2n−1∞/summationdisplay
k=1sin 2kπx
k2n−1
[n>1,1≥x≥0;n=1,1>x> 0]AS 805(23.1.17)
3.10B2n(x)=(−1)n−12(2n)!
(2π)2n∞/summationdisplay
k=1cos 2kπx
k2n[0≤x≤1,n=1,2,...] GE 71
9.623 Functional relations and properties:
1. Bm+1(n)=Bm+1+(m+1 )n−1/summationdisplay
k=1km
[n and m are natural numbers] (see also 0.121 )GE 51(65)
2. Bn(x+1 )−Bn(x)=nxn−1GE 65(90)
3. B/prime
n(x)=nBn−1(x)[ n=1,2,...] GE 66
4. Bn(1−x)=(−1)nBn(x) GE 66
5.10(−1)nBn(−x)=Bn(x)+nxn−1[n=0,1,...] AS 804(23.1.9)
9.6247Bn(mx)=mn−1m−1/summationdisplay
k=0Bn/parenleftbigg
x+k
m/parenrightbigg
[m=1,2,...n =0,1,...] ; “summation theorem” GE 67
9.625 Fornodd, the differences
Bn(x)−Bn
vanish on the interval [0 ,1] only at the points 0 ,1
2, and 1. They change sign at the point x=1
2.F o r n
even, these differences vanish at the end points of the interval [0 ,1]. Within this interval, they do not
change sign, and their greatest absolute value occurs at the point x=1
2.
9.626 The polynomials
B2n(x)−B2nandB2n+2(x)−B2n+2
have opposite signs in the interval (0 ,1). GE 87
9.627 Special cases:
1. B1(x)=x−1
2GE 70
2. B2(x)=x2−x+1
6GE 70
3. B3(x)=x3−3
2x2+1
2x GE 70
4. B4(x)=x4−2x3+x2−1
30GE 70
5. B5(x)=x5−5
2x4+5
3x3−1
6x GE 70
9.628 Particular values:
1. Bn(0) = Bn
2. B1(1) =−B1=1
2,Bn(1) = Bn [n/negationslash=1 ] GE 76
9.640 The functions ν(x),ν(x, α),μ(x, β),μ(x, β, α ),a n d λ(x, y) 1043
9.63 Euler numbers
9.630 The numbers En, representing the coefficients oftn
n!in the expansion of the function
1
cosht=∞/summationdisplay
n=0Entn
n!/bracketleftBig
|t|<π
2/bracketrightBig
,
are known as the Euler numbers . Thus, the function1
coshtis a generating function for the Euler numbers.
CE 330
9.631 A recursion formula
(E+1 )[n]+(E−1)[n]=0 [ n≥1],E 0=1 CE 329
Properties of the Euler numbers
9.632 The Euler numbers are integers.
9.633 The Euler numbers of odd index are equal to zero; the signs of two adjacent numbers of even
indices are opposite; that is,
E2n+1=0,E 4n>0,E 4n+2<0. CE 329
9.634 Ifα,βγ,... are the divisors of the number n−m, the difference E2n−E2mis divisible by those
of the numbers 2 α+1,2β+1,2γ+1,...that are primes.
9.635 A connection with the Bernoulli numbers (symbolic notation):
1.11En−1+4 (−1)n/parenleftbig
3n−1−1/parenrightbig
B1=(4B−1)[n]−(4B−3)[n]
2n+4 (−1)n+1/parenleftbig
3n−1−1/parenrightbig
B1 CE 330
2. Bn=n(E+1 )[n−1]
2n(2n−1)[n≥2] CE 330
3.6/parenleftbig
B+1
4/parenrightbig[2n+1]=−4−2n−1(2n+1 )E2n [n≥0] CE 341
4. En−1=(4B+3 )[n]−(4B+1 )[n]
2n[n≥1]
For a table of values of the Euler numbers, see 9.72.
9.64 The functions ν(x),ν(x, α),μ(x, β),μ(x, β, α ), and λ(x, y)
9.640
1. ν(x)=/integraldisplay∞
0xtdt
Γ(t+1 )EH III 217(1)
2. ν(x, α)=/integraldisplay∞
0xα+tdt
Γ(α+t+1 )EH III 217(1)
3. μ(x, β)=/integraldisplay∞
0xttβdt
Γ(β+1 )Γ ( t+1 )EH III 217(2)
4. μ(x, β, α )=/integraldisplay∞
0xα+ttβdt
Γ(β+1 )Γ ( α+t+1 )EH III 217(2)
5. λ(x, y)=/integraldisplayy
0Γ(u+1 )du
xuMI 9
1044 Bernoulli Numbers and Polynomials, Euler Numbers 9.650
9.6510Euler polynomials
9.650 The Euler polynomials are defined by
En(x)=n/summationdisplay
k=0/parenleftBign
k/parenrightBigEk
2k/parenleftbigg
x−1
2/parenrightbiggn−k
AS 804 (23.1.7)
9.651 The generating function:
2ext
et+1=∞/summationdisplay
n=0En(x)tn
n!AS 804 (23.1.1)
9.652 Series representation:
1. En(x)=4n!
πn+1∞/summationdisplay
k=0sin/parenleftbig
(2k+1 )πx−1
2πn/parenrightbig
(2k+1 )n+1
[n>0,1≥x≥0,n=1,1>x> 0]AS 804 (23.1.16)
2.10E2n−1(x)=(−1)n4(2n−1)!
π2n∞/summationdisplay
k=0cos(2k+1 )πx
(2k+1 )2n[n=1,2,..., 1≥x≥0]
AS 804 (23.1.17)
3. E2n(x)=(−1)n4(2n)!
π2n+1∞/summationdisplay
k=0sin(2k+1 )πx
(2k+1 )2n+1
[n>0,1≥x≥0,n=0,1>x> 0]AS 804 (23.1.18)
9.653 Functional relations and properties:
1. Em(n+1 )=2n/summationdisplay
k=1(−1)n−kkm+(−1)n+1Em(0), [mandnare natural numbers]
AS 804 (23.1.4)
2. E/prime
n(x)=nEn−1(x). [n=1,2,...] AS 804 (23.1.5)
3. En(x+1 )+ En(x)=2xn[n=0,1,...] AS 804 (23.1.6)
4.8En(mx)=mnm−1/summationdisplay
k=0(−1)kEn/parenleftbigg
x−k
m/parenrightbigg
[n=0,1,...,m =1,3,...]
AS 804 (23.1.10)
5. En(mx)=−2
n+1mnm−1/summationdisplay
k=0(−1)kBn+1/parenleftbigg
x+k
m/parenrightbigg
[n=0,1,...,m =2,4,...]
AS 804 (23.1.10)
9.654 Special cases:
1. E1(x)=x−1
2
2. E2(x)=x2−x
9.655 Euler numbers 1045
3. E3(x)=x3−3
2x2+1
4
4. E4(x)=x4−2x3+x
5. E5(x)=x5−5
2x4+5
2x2−1
2
9.655 Particular values:
1. E2n+1=0. [n=0,1,...] AS 805 (23.1.19)
2. En(0) =−En(1) =−2(n+1 )−1/parenleftbig
2n+1−1/parenrightbig
Bn+1 [n=1,2,...] AS 805 (23.1.20)
3. En/parenleftbig1
2/parenrightbig
=2−nEn [n=0,1,...] AS 805 (23.1.21)
4. E2n−1/parenleftbig1
3/parenrightbig
=−E2n−1/parenleftbig2
3/parenrightbig
=−(2n)−1/parenleftbig
1−31−2n/parenrightbig/parenleftbig
22n−1/parenrightbig
B2n
[n=1,2,...] AS 806 (23.1.22)
9.7 Constants
9.71 Bernoulli numbers
•B0=1
•B1=−1/2
•B2=1/6
•B4=−1/30
•B6=1/42
•B8=−1/30
•B10=5/66
•B12=−691/2730
•B14=7/6
•B16=−3617/510•B18=43867/798
•B20=−174611/330
•B22=854513/138
•B24=−236364091 /2730
•B26=8553103 /6
•B28=−23749461029 /870
•B30=8615841276005 /14322
•B32=−7709321041217 /510
•B34=2577687858367 /6
9.72 Euler numbers
•E0=1
•E2=−1
•E4=5
•E6=−61
•E8= 1385
•E10=−50521•E12= 2702765
•E14=−199360981
•E16= 19391512145
•E18=−2404879675441
•E20= 370371188237525
The Bernoulli and Euler numbers of odd index (with the exception of B1) are equal to zero.
1046 Constants 9.740
9.73 Euler’s and Catalan’s constants
Euler’s constant
C=0.577215664901532860606512 ... (cf.8.367 )
Catalan’s constant
G=∞/summationdisplay
k=0(−1)k
(2k+1 )2=0.915965594 ...
9.7410Stirling numbers
9.740 TheStirling number of the first kind S(m)
nis defined by the requirement that ( −1)n−mS(m)
n
is the number of permutations of nsymbols which have exactly mcycles. AS 824 (23.1.3)
9.741 Generating functions:
1. x(x−1)···(x−n+1 )=n/summationdisplay
m=0S(m)
nxmAS 824 (24.1.3)
2. {ln(1 + x)}m=m!∞/summationdisplay
n=mS(m)
nxn
n![|x|<1] AS 824 (24.1.3)
9.742 Recurrence relations:
1.8S(m)
n+1=S(m−1)
n −nS(m)
n;S(0)
n=δ0n;S(1)
n=(−1)n−1(n−1)!; S(n)
n=1
[n≥m≥1] AS 824 (24.1.3)
2./parenleftBigm
r/parenrightBig
S(m)
n=n−r/summationdisplay
k=m−r/parenleftBign
k/parenrightBig
S(r)
n−kS(m+r)
k[n≥m≥r] AS 824 (24.1.3)
9.743 Functional relations and properties
1. x(x−h)(x−2h)···(x−mh+h)=hmΓ/parenleftbigx
h+1/parenrightbig
Γ/parenleftbigx
h−m+1/parenrightbig=hmm/summationdisplay
k=1/parenleftBigx
h/parenrightBigk
S(m)
k
2. [( x+1 ) (x+2 )···(x+m)]−1=/bracketleftbigg/parenleftbiggx+m
m/parenrightbigg
m!/bracketrightbigg−1
=/bracketleftBiggp/summationdisplay
k=1(x+m)kS(m)
k/bracketrightBigg−1
3. [( x+h)(x+2h)···(x+mh)]−1=Γ/parenleftbigx
h+1/parenrightbig
hmΓ/parenleftbigx
h+m+1/parenrightbig=/bracketleftBigg
hmm/summationdisplay
k=1/parenleftBigx
h+m/parenrightBigk
S(m)
k/bracketrightBigg−1
9.744 The Stirling number of the second kind S(m)
nis the number of ways of partitioning a set of n
elements into mnon-empty subsets.
9.748 Stirling numbers 1047
9.745 Generating functions:
1. xn=n/summationdisplay
m=0S(m)
nx(x−1)···(x−m+1 ) AS 824 (24.1.4)
2. ( ex−1)m=m!∞/summationdisplay
n=mS(m)
nxn
n!AS 824 (24.1.4)
3. [(1 −x)(1−2x)···(1−mx)]−1=∞/summationdisplay
n=mS(m)
nxn−m/bracketleftbig
|x|<m−1/bracketrightbig
AS 824 (24.1.4)
9.746 Closed form expression:
1. S(m)
n=1
m!m/summationdisplay
k=0(−1)m−k/parenleftBigm
k/parenrightBig
knAS 824 (24.1.4)
9.747 Recurrence relations:
1.8S(m)
n+1=mS(m)
n+S(m−1)
n,S(0)
n=δ0n,S(1)
n=S(n)
n=1
[n≥m≥1] AS 825(24.1.4)
2./parenleftBigm
r/parenrightBig
S(m)
n=n−r/summationdisplay
k=m−r/parenleftBign
k/parenrightBig
S(r)
n−kS(m−r)
k[n≥m≥r] AS 825 (24.1.4)
3. S(m)
n=n−m/summationdisplay
k=0(−1)k/parenleftbiggn−1+k
n−m+k/parenrightbigg/parenleftbigg2n−m
n−m−k/parenrightbigg
S(k)
n−m+kAS 824 (24.1.3)
9.7487Particular values:
Stirling numbers of the first kind S(m)
n
mS(m)
1 S(m)
2 S(m)
3 S(m)
4 S(m)
5 S(m)
6 S(m)
7 S(m)
8 S(m)
9
1 1 -1 2 -6 24 -120 720 -5040 40320
2 1 -3 11 -50 274 -1764 13068 -109584
3 1 -6 35 -225 1624 -13132 118121
4 1 -10 85 -735 6769 -67284
5 1 -15 175 -1960 22449
6 1 -21 332 -4536
7 1 -28 546
8 1- 3 6
9 1
1048 Constants 9.749
Stirling numbers of the second kind S(m)
n
mS(m)
1 S(m)
2 S(m)
3 S(m)
4 S(m)
5 S(m)
6 S(m)
7 S(m)
8 S(m)
9
1 111111111
2 1 3 71 53 16 3 1 2 7 2 5 5
3 1 6 25 90 301 966 3025
4 1 10 65 350 1701 7770
5 1 15 140 1050 6951
6 1 21 266 2646
7 1 28 462
8 13 6
9 1
9.7498Relationship between Stirling numbers of the first kind and derivatives of (ln x)−m:
1.dn
dxn/parenleftbigg1
lnmx/parenrightbigg
=1
lnmxn/summationdisplay
k=1(−1)k(m)kS(k)
n
xnlnkx
where ( m)k=Γ (m+k)/Γ(m),[m, nare positive integers]
10 Vector Field Theory
10.1–10.8 Vectors, Vector Operators, and Integral Theorems
10.11 Products of vectors
Leta=(a1,a2,a3),b=(b1,b2,b3), and c=(c1,c2,c2) be arbitrary vectors, and i,j,kbe the set of
orthogonal unit vectors in terms of which the components of a,b,a n dcare expressed. Two different
products involving pairs of vectors are defined, namely, the scalar product, written a·b, and the vector
product, written either a×bora∧b. Their properties are as follows:
1. a·b=a1b1+a2b2+a3b3 (scalar product)
2. a×b=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleijk
a
1a2a3
b1b2b3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(vector product)
3. a×b·c=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
1a2a3
b1b2b3
c1c2c3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(triple scalar product)
4. a×(b×c)=(a·c)b−(a·b)c (triple vector product)
10.12 Properties of scalar product
1. a·b=b·a (commutative)
2. a×b·c=b×c·a=c×a·b=−a×c·b=−b×a·c=−c×b·a.
Note:a×b·cis also written [ a,b,c]; thus (2) may also be written
3. [ a,b,c]=[b,c,a]=[c,a,b]=−[a,c,b]=−[b,a,c]=−[c,b,a]
10.13 Properties of vector product
1. a×b=−b×a (anticommutative)
2. a×(b×c)=−a×(c×b)=−(b×c)×a
3. a×(b×c)+b×(c×a)+c×(a×b)=0
1049
1050 Vectors, Vector Operators, and Integral Theorems
10.14 Differentiation of vectors
Ifa(t)=(a1(t),a2(t),a3(t)),b(t)=(b1(t),b2(t),b3(t)),c(t)=(c1(t),c2(t),c3(t)),φ(t) is a scalar and all
functions of tare differentiable, then
1.da
dt=da1
dti+da2
dtj+da3
dtk
2.d
dt(a+b)=da
dt+db
dt
3.d
dt(φa)=dφ
dta+φda
dt
4.d
dt(a·b)=da
dt·b+a·db
dt
5.d
dt(a×b)=da
dt×b+a×db
dt
6.d
dt(a×b·c)=da
dt×b·c+a×db
dt·c+a×b·dc
dt
7.d
dt{a×(b×c)}=da
dt×(b×c)+a×/parenleftbiggdb
dt×c/parenrightbigg
+a×/parenleftbigg
b×dc
dt/parenrightbigg
10.21 Operators grad, div, and curl
In cartesian coordinates O{x1,x2,x3}, in which system it is convenient to denote the triad of unit vectors
bye1,e2,e3, the vector operator ∇, called either “del” or “nabla,” has the form
1. ∇≡e1∂
∂x1+e2∂
∂x2+e3∂
∂x3
If Φ(x, y, z) is any differentiable scalar function, the gradient of Φ, written grad Φ, is
2. gradΦ ≡∇Φ=∂Φ
∂x1e1+∂Φ
∂x2e2+∂Φ
∂x3e3
The divergence of the differentiable vector function f=(f1,f2,f3), written div f,i s
3. div f≡∇·f=∂f1
∂x1+∂f2
∂x2+∂f3
∂x3
The curl, or rotation, of the differentiable vector function f=(f1,f2,f3), written either curl for
rotf,i s
4. curl f≡rotf≡∇×f=/parenleftbigg∂f3
∂x2−∂f2
∂x3/parenrightbigg
e1+/parenleftbigg∂f1
∂x3−∂f3
∂x1/parenrightbigg
e2+/parenleftbigg∂f2
∂x1−∂f1
∂x2/parenrightbigg
e3,
or equivalently,
curlf=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglee
1e2e2
∂
∂x1∂
∂x2∂
∂x3
f1f2f3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
Properties of the operator ∇ 1051
10.31 Properties of the operator ∇
Let Φ ( x1,x2,x3), Ψ(x1,x2,x3) be any two differentiable scalar functions, f(x1,x2,x3),g(x1,x2,x3)a n y
two differentiable vector functions, and aan arbitrary vector. Define the scalar operator ∇2, called the
Laplacian, by
∇2≡∂2
∂x2
1+∂2
∂x2
2+∂2
∂x2
3Then, in terms of the operator ∇, we have the following: MF I 114
1. ∇(Φ + Ψ) = ∇Φ+∇Ψ
2. ∇(ΦΨ) = Φ ∇Ψ+Ψ ∇Φ
3. ∇(f·g)=(f·∇)g+(g·∇)f+f×(∇×g)+g×(∇×f)
4. ∇·(Φf)=Φ( ∇·f)+f·∇Φ
5. ∇·(f×g)=g·(∇×f)−f·(∇×g)
6. ∇×(Φf)=Φ( ∇×f)+(∇Φ)×f
7. ∇×(f×g)=f(∇·g)−g(∇·f)+(g·∇)f−(f·∇)g
8. ∇×(∇×f)=∇(∇·f)−∇2f
9. ∇×(∇Φ)≡0
10. ∇·(∇×f)≡0
11.10∇2(ΦΨ) = Φ ∇2Ψ+2( ∇Φ)·(∇Ψ) + Ψ ∇2Φ
The equivalent results in terms of grad, div, and curl are as follows:
1. grad(Φ + Ψ) = gradΦ + gradΨ
2. grad(ΦΨ) = Φ gradΨ + Ψgrad Φ
3. grad( f·g)=(f·grad)g+(g·grad)f+f×curlg+g×curlf
4. div (Φ f)=Φd i v f+f·gradΦ
5. div ( f×g)=g·curlf−f·curlg
6. curl (Φ f)=Φc u r l f+g r a dΦ ×f
7. curl ( f×g)=fdivg−gdivf+(g·grad)f−(f·grad)g
8. curl (curl f) = grad(div f)−∇2f
9. curl (gradΦ) ≡0
10. div (curl f)≡0
11. ∇2(ΦΨ) = Φ ∇2Ψ+2g r a dΦ ·gradΨ + Ψ ∇2Φ
The expression ( a·∇) or, equivalently ( a·grad), defined by
(a·∇)≡a1∂
∂x1+a2∂
∂x2+a3∂
∂x3,
is the directional derivative operator in the direction of vector a.
1052 Vectors, Vector Operators, and Integral Theorems 10.411
10.41 Solenoidal fields
A vector field fis said to be solenoidal if div f≡0. We have the following representation:
10.411 Representation theorem for vector Helmholtz equation. Ifuis a solution of the scalar Helmholtz
equation
∇2u+λ2u=0,
andmis a constant unit vector, then the vectors
X=c u r l( mu),Y=1
λcurlX
are independent solutions of the vector Helmholtz equation
∇2H+λ2H=0
involving a solenoidal vector H. The general solution of the equation is
H=c u r l( mu)+1
λcurl curl ( mu).
10.51–10.61 Orthogonal curvilinear coordinates
Consider a transformation from the cartesian coordinates O{x1,x2,x3}to the general orthogonal curvi-
linear coordinates O{u1,u2,u3}:
x1=x1(u1,u2,u3),x 2=x2(u1,u2,u3),x 3=x3(u1,u2,u3)
Then,
1. dxi=∂xi
∂u1du1+∂xi
∂u2du2+∂xi
∂u3du3 (i=1,2,3),
and the length element dlmay be determined from
2. dl2=g11du2
1+g22du2
2+g33du2
3+2g23du2du3+2g31du3du1+2g12du1du2,
where
3.3gij=∂x1
∂ui∂x1
∂uj+∂x2
∂ui∂x2
∂uj+∂x3
∂ui∂x3
∂uj=gji,g ij=0,i/negationslash=j,
provided the Jacobian of the transformation
4. J=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂x1
∂u1∂x2
∂u1∂x3
∂u1∂x1
∂u2∂x2
∂u2∂x3
∂u2∂x1
∂u3∂x2
∂u3∂x3
∂u3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
does not vanish (see 14.313 ).
Define the metrical coefficients
5. h
1=√g11,h 2=√g22,h 3=√g33;
then the volume element dVin orthogonal curvilinear coordinates is
6. dV=h1h2h3du1du2du3,
and the surface elements of area dsion the surfaces ui= constant, for i=1 ,2 ,3 ,a r e
7. ds1=h2h3du2du3,d s 2=h1h3du1du3,d s 3=h1h2du1du2
Denote by e1,e2,a n de3the triad of orthogonal unit vectors that are tangent to the u1,u2,
andu3coordinate lines through any given point P, and choose their sense so that they form a
right-handed set in this order. Then in terms of this triad of vectors and the components fu1,fu2,
andfu3offalong the coordinate line,
10.613 Orthogonal curvilinear coordinates 1053
8. f=fu1e1+fu2e2+fu3e3 MF I 115
10.611 ∇Φ, div f,c u r lf,and∇2in general orthogonal curvilinear coordinates.
1. gradΦ =e1
h1∂Φ
∂u1+e2
h2∂Φ
∂u2+e3
h3∂Φ
∂u3
2.3divf=1
h1h2h3/parenleftbigg∂
∂u1(h2h3fu1)+∂
∂u2(h3h1fu2)+∂
∂u3(h1h2fu3)/parenrightbigg
3. curl f=1
h1h2h3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleh
1e1h2e2h3e3
∂
∂u1∂
∂u2∂
∂u3
h1fu1h2fu2h3fu3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
4. ∇
2≡1
h1h2h3/parenleftbigg∂
∂u1/parenleftbiggh2h3
h1∂
∂u1/parenrightbigg
+∂
∂u2/parenleftbiggh3h1
h2∂
∂u2/parenrightbigg
+∂
∂u3/parenleftbiggh1h2
h3∂
∂u3/parenrightbigg/parenrightbigg
MF I 21-31
10.612 Cylindrical polar coordinates. In terms of the coordinates O{r, φ, z},t h a ti s , u1=r,u2=φ,
u3=z,w h e r e x1=rcosφ,x2=rsinφ,x3=zfor−π<φ ≤π, it follows that
1. h1=1,h2=r, h 3=1,
and
2. gradΦ =∂Φ
∂rer+1
r∂Φ
∂φeφ+∂Φ
∂zez,
3. div f=1
r∂
∂r(rfr)+1
r∂fφ
∂φ+∂fz
∂z,
4. curl f=1
r/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglee
rreφez
∂
∂r∂
∂φ∂
∂z
frrfφfz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,
5. ∇
2≡1
r∂
∂r/parenleftbigg
r∂
∂r/parenrightbigg
+1
r2∂2
∂φ2+∂2
∂z2MF I 116
10.613 Spherical polar coordinates. In terms of the coordinates O{r, θ, φ},t h a ti s , u1=r,u2=θ,
u3=φ,w h e r e x1=rsinθcosφ,x2=rsinθsinφ,x3=rcosθ,f o r0 ≤θ≤π,−π<φ ≤π,w eh a v e
1. h1=1,h2=r, h 3=rsinθ,
and
2.10gradΦ =∂Φ
∂rer+1
r∂Φ
∂θeθ+1
rsinθ∂Φ
∂φeφ,
3. div f=1
r2∂
∂r/parenleftbig
r2fr/parenrightbig
+1
rsinθ∂
∂θ(fθsinθ)+1
rsinθ∂fφ
∂φ,
4. curl f=1
r2sinθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglee
rreθrsinθeφ
∂
∂r∂
∂θ∂
∂φ
frrfθrsinθfφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,
5. ∇
2≡1
r2∂
∂r/parenleftbigg
r2∂
∂r/parenrightbigg
+1
r2sinθ∂
∂θ/parenleftbigg
sinθ∂
∂θ/parenrightbigg
+1
r2sin2θ∂2
∂φ2MF I 116
1054 Vectors, Vector Operators, and Integral Theorems 10.614
Special Orthogonal Curvilinear Coordinates and their Metrical Coefficients h1,h2,h3
10.614 Elliptic cylinder coordinates O{u1,u2,u3}.
1. x1=u1u2,x 2=/radicalBig
(u2
1−c2)(1−u2
2),x 3=u3
2. h1=/radicalBigg
u2
1−c2u2
2
u21−c2,h 2=/radicalBigg
u2
1−c2u2
2
1−u2
2,h 3=1 MF I 657
10.615 Parabolic cylinder coordinates O{u1,u2,u3}.
1. x1=1
2/parenleftbig
u2
1−u2
2/parenrightbig
,x 2=u1u2,x 3=u3
2. h1=/radicalBig
u2
1+u2
2,h 2=/radicalBig
u2
1+u2
2,h 3=1 MF I 658
10.616 Conical coordinates O{u1,u2,u3}.
1. x1=u1
a/radicalBig
(a2−u2
2)(a2+u2
3),x 2=u1
b/radicalBig
(b2+u2
2)(b2−u2
3),x 3=u1u2u3
ab
witha2+b2=1
2. h1=1,h 2=u1/radicalBigg
u2
2+u2
3
(a2−u2
2)(b2+u2
2),h 3=u1/radicalBigg
u2
2+u2
3
(a2+u2
3)(b2−u2
3)MF I 659
10.617 Rotational parabolic coordinates O{u1,u2,u3}.
1. x1=u1u2u3,x 2=u1u2/radicalBig
1−u2
3,x 3=1
2/parenleftbig
u2
1−u2
2/parenrightbig
2. h1=/radicalBig
u2
1+u2
2,h 2=/radicalBig
u2
1+u2
2,h 3=u1u2/radicalbig
1−u2
3MF I 660
10.618 Rotational prolate spheroidal coordinates O{u1,u2,u3}.
1. x1=/radicalBig
(u2
1−a2)(1−u2
2),x 2=/radicalBig
(u2
1−a2)(1−u2
2)(1−u2
3),x 3=u1u2
2. h1=/radicalBigg
u2
1−a2u2
2
u21−a2,h 2=/radicalBigg
u2
1−a2u2
2
1−u2
2,h 3=/radicalBigg
(u2
1−a2)(1−u2
2)
1−u2
3MF I 661
10.619 Rotational oblate spheroidal coordinates O{u1,u2,u3}.
1. x1=u3/radicalBig
(u2
1+a2)(1−u2
2),x 2=/radicalBig
(u2
1+a2)(1−u2
2)( 1−u2
3),x 3=u1u2
2. h1=/radicalBigg
u2
1+a2u2
2
u21+a2,h 2=/radicalBigg
u2
1+a2u2
2
1−u2
2,h 3=/radicalBigg
(u2
1+a2)(1−u2
2)
1−u2
3MF I 662
10.713 Vector integral theorems 1055
10.620 Ellipsoidal coordinates O{u1,u2,u3}.
1. x1=/radicalBigg
(u2
1−a2)(u2
2−a2)(u2
3−a2)
a2(a2−b2),x2=/radicalBigg
(u2
1−b2)(u2
2−b2)(u2
3−b2)
b2(b2−a2),x3=u1u2u3
ab
2. h1=/radicalBigg
(u2
1−u2
2)(u2
1−u2
3)
(u2
1−a2)(u2
1−b2),h 2=/radicalBigg
(u2
2−u2
1)(u2
2−u2
3)
(u2
2−a2)(u2
2−b2),h 3=/radicalBigg
(u2
3−u2
1)(u2
3−u2
2)
(u2
3−a2)(u2
3−b2)
MF I 663
10.621 Paraboloidal coordinates O{u1,u2,u3}.
1. x1=/radicalBigg
(u2
1−a2)(u2
2−a2)(u2
3−a2)
a2−b2,x 2=/radicalBigg
(u2
1−b2)(u2
2−b2)(u2
3−b2)
b2−a2,
x3=1
2/parenleftbig
u2
1+u2
2+u2
3−a2−b2/parenrightbig
2. h1=/radicalBigg
(u2
1−u2
2)(u2
1−u2
3)
(u2
1−a2)(u2
1−b2),h 2=u2/radicalBigg
(u2
3−u2
1)(u2
3−u2
2)
(u2
2−a2)(u2
2−b2),h 3=u3/radicalBigg
(u2
3−u2
1)(u2
3−u2
2)
(u2
3−a2)(u2
3−b2)
MF I 664
10.622 Bispherical coordinates O{u1,u2,u3}.
1. x1=au3/radicalbig
1−u2
2
u1−u2,x 2=a/radicalbig
(1−u2
2)( 1−u2
3)
u1−u2,x 3=/radicalbig
u2
1−1
u1−u2
2. h1=a
(u1−u2)/radicalbig
u2
1−1,
h2=a
(u1−u2)/radicalbig
1−u2
2,h 3=/parenleftbigga
u1−u2/parenrightbigg/radicalBigg
1−u2
2
1−u2
3MF I 665
10.71–10.72 Vector integral theorems
10.711 Gauss’s divergence theorem. LetVbe a volume bounded by a simple closed surface Sand let
fbe a continuously differentiable vector field defined in Vand on S. Then, if dSis the outward drawn
vector element of area,/integraldisplay
Sf·dS=/integraldisplay
VdivfdV KE 39
10.712 Green’s theorems. Let Φ and Ψ be scalar fields which, together with ∇2Φa n d ∇2Ψ, are defined
both in a volume Va n do ni t ss u r f a c e S, which we assume to be simple and closed. Then, if ∂/∂n denotes
differentiation along the outward drawn normal to S,w eh a v e
10.713 Green’s first theorem/integraldisplay
SΦ∂Ψ
∂ndS=/integraldisplay
V/parenleftbig
Φ∇2Ψ + gradΦ ·gradΨ/parenrightbig
dV KE 212
1056 Vectors, Vector Operators, and Integral Theorems 10.714
10.714 Green’s second theorem/integraldisplay
S/parenleftbigg
Φ∂Ψ
∂n−Ψ∂Φ
∂n/parenrightbigg
dS=/integraldisplay
V/parenleftbig
Φ∇2Ψ−Ψ∇2Φ/parenrightbig
dV KE 215
10.715 Special cases
1./integraldisplay
S(ΦgradΦ) ·dS=/integraldisplay
V/parenleftBig
Φ∇2Φ+( g r a dΦ )2/parenrightBig
dV
2./integraldisplay
S∂Φ
∂ndS=/integraldisplay
V∇2ΦdV MV 81
10.716 Green’s reciprocal theorem. If Φ and Ψ are harmonic, so that ∇2Φ=∇2Ψ=0 ,t h e n
3./integraldisplay
SΦ∂Ψ
∂ndS=/integraldisplay
SΨ∂Φ
∂ndS MM 105
10.717 Green’s representation theorem. If Φ and ∇2Φ are defined within a volume Vbounded by a
simple closed surface S,a n d Pis an interior point of V, then in three dimensions
4. Φ( P)=−1
4π/integraldisplay
V1
r∇2ΦdV+1
4π/integraldisplay
S1
r∂Φ
∂ndS−1
4π/integraldisplay
SΦ∂
∂n/parenleftbigg1
r/parenrightbigg
dS KE 219
If Φ is harmonic within V,s ot h a t ∇2Φ = 0, then the previous result becomes
5. Φ( P)=1
4π/integraldisplay
S1
r∂Φ
∂ndS−1
4π/integraldisplay
SΦ∂
∂n/parenleftbigg1
r/parenrightbigg
dS
In the case of two dimensions, result (4) takes the form
6. Φ( p)=1
2π/integraldisplay
S∇2Φ(q)ln|p−q|dS
+1
2π/integraldisplay
CΦ(q)∂
∂nqln|p−q|dq−1
2π/integraldisplay
ln|p−q|∂
∂nqΦ(q)dq
MM 116
where Cis the boundary of the planar region S, and result (5) takes the form
7. Φ( p)=1
2π/integraldisplay
CΦ(q)∂
∂nqln|p−q|dq−1
2π/integraldisplay
Cln|p−q|∂
∂nqΦ(q)dq VL 280
10.718 Green’s representation theorem in Rn.If Φ is twice differentiable within a region Ω in Rn
bounded by the surface Σ with outward drawn unit normal n,t h e nf o r p/negationslash∈Σa n d n>3
Φ(p)=−1
(n−2)σn/integraldisplay
Ω∇2Φ(q)
|p−q|n−2dΩq+1
(n−2)σn/integraldisplay
Σ/parenleftBigg
1
|p−q|n−2∂Φ(q)
∂nq−Φ(q)∂
∂nq1
|p−q|n−2/parenrightBigg
dΣq,
where
σn=2πn/2
Γ(n/2)VL 279
is the area of the unit sphere in Rn.
10.719 Green’s theorem of the arithmetic mean. If Φ is harmonic in a sphere, then the value of Φ at
the center of the sphere is the arithmetic mean of its value on the surface. KE 223
10.720 Poisson’s integral in three dimensions. If Φ is harmonic in the interior of a spherical volume
Vof radius Rand is continuous on the surface of the sphere on which, in terms of the spherical polar
coordinates ( r, θ, φ), it satisfies the boundary condition Φ ( R,θ,φ )=f(θ,φ), then
10.811 Integral rate of change theorems 1057
Φ(r, θ, φ)=R/parenleftbig
R2−r2/parenrightbig
4π/integraldisplayπ
0/integraldisplayπ
−πf(θ/prime,φ/prime)s i nθ/primedθ/primedφ/prime
(r2+R2−2rRcosγ)3/2,
where
cosγ=c o s θcosθ/prime+s i nθsinθ/primecos(φ−φ/prime). KE 241
10.721 Poisson’s integral in two dimensions. If Φ is harmonic in the interior of a circular disk Sof
radius Rand is continuous on the boundary of the disk on which, in terms of the polar coordinates ( r, θ),
it satisfies the boundary condition Φ( R,θ)=f(θ), then
Φ(r, θ)=/parenleftbig
R2−r2/parenrightbig
2π/integraldisplayπ
−πf(φ)dφ
r2+R2−2rRcos(θ−φ).
10.722 Stokes’ theorem. Let a simple closed curve Cbe spanned by a surface S. Define the positive
normal ntoS, and the positive sense of description of the curve Cwith line element dr, such that the
positive sense of the contour Cis clockwise when we look through the surface Sin the direction of the
normal. Then, if fis continuously differentiable vector field defined on SandCwith vector element
S=ndS,/contintegraldisplay
Cf·dr=/integraldisplay
Scurlf·dS, MM 143
where the line integral around Ci st a k e ni nt h ep o s i t i v es e n s e .
10.723 Planar case of Stokes’ theorem. If a region Rin the ( x, y)-plane is bounded by a simple closed
curve C,a n d f1(x, y),f2(x, y) are any two functions having continuous first derivatives in Rand on C,
then/contintegraldisplay
C(f1dx+f2dy)=/integraldisplay/integraldisplay
R/parenleftbigg∂f2
∂x−∂f1
∂y/parenrightbigg
dxdy, MM 143
where the line integral is taken in the counterclockwise sense.
10.81 Integral rate of change theorems
10.811 Rate of change of volume integral bounded by a moving closed surface. Letfbe a continuous
scalar function of position and time tdefined throughout the volume V(t), which is itself bounded by a
simple closed surface S(t) moving with velocity v. Then the rate of change of the volume integral of fis
given by
D
Dt/integraldisplay
V(t)fd V=/integraldisplay
V(t)∂f
∂tdV+/integraldisplay
S(t)fv·dS,
where dSis the outward drawn vector element of area, and
D
Dt≡∂
∂t+v·∇.
By virtue of Gauss’s theorem, this also takes the form
D
Dt/integraldisplay
V(t)fd V=/integraldisplay
V(t)/parenleftbiggDf
Dt+fdivv/parenrightbigg
dV. MV 88
1058 Vectors, Vector Operators, and Integral Theorems 10.812
10.812 Rate of change of flux through a surface. Letqbe a vector function that may also depend on
the time t,a n dnbe the unit outward drawn normal to the surface Sthat moves with velocity v. Defining
the flux of qthrough Sas
m=/integraldisplay
Sq·ndS,
then
Dm
Dt=/integraldisplay
S/parenleftbigg∂q
∂t+vdivq+c u r l( q×v)/parenrightbigg
·ndS. MV 90
10.813 Rate of change of the circulation around a given moving curve. LetCbe a closed curve, moving
with velocity v, on which is defined a vector field q. Defining the circulation ζofqaround Cby
ζ=/integraldisplay
Cq·dr,
then
Dζ
Dt=/integraldisplay
C/parenleftbigg∂q
∂t+( c u r l q)×v/parenrightbigg
·dr. MV 94
11 Algebraic Inequalities
11.1–11.3 General Algebraic Inequalities
11.11 Algebraic inequalities involving real numbers
11.111 Lagrange’s identity. Leta1,a2,...,a nandb1,b2,...,b nbe any two sets of real numbers; then
/parenleftBiggn/summationdisplay
k=1akbk/parenrightBigg2
=/parenleftBiggn/summationdisplay
k=1a2
k/parenrightBigg/parenleftBiggn/summationdisplay
k=1b2
k/parenrightBigg
−/summationdisplay
(akbj−ajbk)2BB 3
11.112 Cauchy–Schwarz–Buniakowsky inequality. Leta1,a2,...,a nandb1,b2,...,b nbe any two arbi-
trary sets of real numbers; then
/parenleftBiggn/summationdisplay
k=1akbk/parenrightBigg2
≤/parenleftBiggn/summationdisplay
k=1a2
k/parenrightBigg/parenleftBiggn/summationdisplay
k=1b2
k/parenrightBigg
.
The equality holds if, and only if, the sequences a1,a2,...,a nandb1,b2,...,b nare proportional.
MT 30
11.113 Minkowski’s inequality. Leta1,a2,...,a nandb1,b2,...,b nbe any two sets of nonnegative real
numbers, and let p>1; then
/parenleftBiggn/summationdisplay
k=1(ak+bk)p/parenrightBigg1/p
≤/parenleftBiggn/summationdisplay
k=1ap
k/parenrightBigg1/p
+/parenleftBiggn/summationdisplay
k=1bp
k/parenrightBigg1/p
.
The equality holds if, and only if, the sequences a1,a2,...,a nandb1,b2,...,b nare proportional.
MT 55
11.114 H¨older’s inequality. Leta1,a2,...,a nandb1,b2,...,b nbe any two sets of nonnegative real
numbers, and let1
p+1
q= 1, with p>1; then
/parenleftBiggn/summationdisplay
k=1ap
k/parenrightBigg1/p/parenleftBiggn/summationdisplay
k=1bq
k/parenrightBigg1/q
≥n/summationdisplay
k=1akbk.
The equality holds if, and only if, the sequences ap
1,ap2,...,ap
nandbq
1,bq2,...,bq
nare proportional.
MT 50
11.115 Chebyshev’s inequality. Leta1,a2,...,a nandb1,b2,...,b nbe two arbitrary sets of real numbers
such that either a1≥a2≥···≥ anandb1≥b2≥···≥ bn,o ra1≤a2≤···≤ anandb1≤b2≤···≤ bn;
then/parenleftbigga1+a2+···+an
n/parenrightbigg/parenleftbiggb1+b2+···+bn
n/parenrightbigg
≤1
nn/summationdisplay
k=1akbk.
The equality holds if, and only if, either a1=a2=···=anorb1=b2=···=bn.
1059
1060 General Algebraic Inequalities 11.116
11.116 Arithmetic-geometric inequality. Leta1,a2,...,a nbe any set of positive numbers, with arith-
metic mean
An=/parenleftbigga1+a2+···+an
n/parenrightbigg
and geometric mean
Gn=(a1a2...a n)1/n;
thenAn≥Gnor, equivalently,/parenleftbigga1+a2+···+an
n/parenrightbigg
≥(a1a2...a n)1/n.
The equality holds only in the event that all of the numbers aiare equal. BB 4
11.117 Carleman’s inequality. Ifa1,a2,...,a nis any finite set of non-negative numbers, then
n/summationdisplay
r=1(a1a2...a r)1/r≤e(a1+a2+···+an),
where eis the best possible constant in this inequality. The inequality is strict except for the trivial case
when ar=0f o r r=1,2,...,n . MT 131
11.118 An inequality involving absolute values. Leta1,a2,...,a nandb1,b2,...,b nbe two arbitrary
sets of real numbers; then
n/summationdisplay
i,j=1{|ai−bj|p+|bi−aj|p−|ai−aj|p−|bi−bj|p}≥0,0<p≤2.
11.21 Algebraic inequalities involving complex numbers
Ifα,βare any two real numbers, the complex number z=α+iβwith real part αand imaginary part β
has for its modulus |z|the nonnegative number
|z|=/radicalbig
α2+β2,
and for its argument (amplitude)arg zthe angle arg z=θsuch that
cosθ=α
|z|and sin θ=β
|z|,
where −π<θ ≤π. The complex number z=α−iβis said to be the complex conjugate ofz=α+iβ.
Ifz=reiθ=r(cosθ+isinθ),
then
zn=rneinθ=rn(cosnθ+isinnθ),
and, setting r=1 ,w eh a v e de Moivre’s theorem
(cosθ+isinθ)n=c o s nθ+isinnθ.
It follows directly that, if z=eiθ,t h e n
cosθ=1
2/parenleftbigg
z+1
z/parenrightbigg
,sinα=−i
2/parenleftbigg
z−1
z/parenrightbigg
,
and
cosrθ=1
2/parenleftbigg
zr+1
zr/parenrightbigg
,sinrθ=−i
2/parenleftbigg
zr−1
zr/parenrightbigg
.
Ifw=zp/qwithp,qintegral, and z=reiθ, then the qroots of w0,w1,...,w q−1ofzare
11.313 Inequalities for sets of complex numbers 1061
wk=rp/q/bracketleftbigg
cos/parenleftbiggpθ+2kπ
q/parenrightbigg
+isin/parenleftbiggpθ+2kπ
q/parenrightbigg/bracketrightbigg
,
withk=0,1,2,...,q −1.
11.2117Simple properties and inequalities involving the modulus and the complex conjugate. If the real
part of zis denoted by Re zand the imaginary part by Im z,t h e n
z+z=2R e z=2α,
z−z=2I m z=2iβ,
z=(z),
1
z=/parenleftbigg1
z/parenrightbigg
,
(zn)=(z)n,/vextendsingle/vextendsingle/vextendsingle/vextendsinglez1
z2/vextendsingle/vextendsingle/vextendsingle/vextendsingle=|
z1|
|z2|,
(z1+z2+···+zn)=z1+z2+···+zn,
z1z2···zn=z1z2···zn.
11.212 Inequalities for pairs of complex numbers .I f a,b are any two complex numbers, then
(i) |a+b|≤|a|+|b| (triangle inequality),
(ii) |a−b|≥| |a|−|b||.
11.31 Inequalities for sets of complex numbers
11.311 Complex Cauchy–Schwarz–Buniakowsky inequality. Leta1,a2,...,a nandb1,b2,...,b nbe any
two arbitrary sets of complex numbers; then/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay
k=1akbk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
≤/parenleftBiggn/summationdisplay
k=1|ak|2/parenrightBigg/parenleftBiggn/summationdisplay
k=1|bk|2/parenrightBigg
.
The equality holds if, and only if, the sequences a1,a2,...,anandb1,b2,...,b nare proportional.
MT 42
11.312 Complex Minkowski inequality. Leta1,a2,...,a nandb1,b2,...,b nbe any two arbitrary sets
of complex numbers, and let the real number pbe such that p>1; then
/parenleftBiggn/summationdisplay
k=1|ak+bk|p/parenrightBigg1/p
≤/parenleftBiggn/summationdisplay
k=1|ak|p/parenrightBigg1/p
+/parenleftBiggn/summationdisplay
k=1|bk|p/parenrightBigg1/p
. MT 56
11.313 Complex H¨ older inequality. Leta1,a2,...,a nandb1,b2,...,b nbe any two arbitrary sets of
complex numbers, and let the real numbers p,q be such that p>1a n d1
p+1
q=1 ;t h e n
/parenleftBiggn/summationdisplay
k=1|ak|p/parenrightBigg1/p/parenleftBiggn/summationdisplay
k=1|bk|q/parenrightBigg1/p
≥/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay
k=1akbk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
The equality holds if, and only if, the sequences
|a
1|p,|a2|p,...,|an|pand|b1|p,|b2|p,...|bn|p,
are proportional and arg akbkis independent of kfork=1,2,...,n . MT 53
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12 Integral Inequalities
12.11 Mean Value Theorems
12.111 First mean value theorem
Letf(x)a n d g(x) be two bounded functions integrable in [ a,b], and let g(x) be of one sign in this interval.
Then/integraldisplayb
af(x)g(x)dx=f(ξ)/integraldisplayb
ag(x)dx, CA 105
witha≤ξ≤b.
12.112 Second mean value theorem
(i) Let f(x) be a bounded, monotonic decreasing, and nonnegative function in [ a,b], and let g(x)b e
a bounded integrable function. Then,
/integraldisplayb
af(x)g(x)dx=f(a)/integraldisplayξ
ag(x)dx,
witha≤ξ≤b.
(ii) Let f(x) be a bounded, monotonic increasing, and nonnegative function in [ a,b], and let g(x)b e
a bounded integrable function. Then,
/integraldisplayb
af(x)g(x)dx=f(b)/integraldisplayb
ηg(x)dx,
witha≤η≤b.
(iii) Let f(x) be bounded and monotonic in [ a,b], and let g(x) be a bounded integrable function which
experiences only a finite number of sign changes in [ a,b]. Then,
/integraldisplayb
af(x)g(x)dx=f(a+0 )/integraldisplayξ
ag(x)dx+f(b−0)/integraldisplayb
ξg(x)dx, CA 107
witha≤ξ≤b.
12.113 First mean value theorem for infinite integrals
Letf(x) be bounded for x≥a, and integrable in the arbitrary interval [ a,b], and let g(x) be of one sign
inx≥aand such that/integraltext∞
ag(x)dxis finite. Then,
1063
1064 Differentiation of Definite Integral Containing a Parameter
/integraldisplay∞
af(x)g(x)dx=μ/integraldisplay∞
ag(x)dx, CA 123
where m≤μ≤Mandm, M are, respectively, the lower and upper bounds of f(x)f o rx≥a.
12.114 Second mean value theorem for infinite integrals
Letf(x) be bounded and monotonic when x≥a,a n d g(x) be bounded and integrable in the arbitrary
interval [ a,b] in which it experiences only a finite number of changes of sign. Then, provided/integraltext∞
ag(x)dx
is finite,
/integraldisplay∞
af(x)g(x)dx=f(a+0 )/integraldisplayξ
ag(x)dx+f(∞)/integraldisplay∞
ξg(x)dx, CA 123
witha≤ξ≤∞.
12.21 Differentiation of Definite Integral Containing a Parameter
12.211 Differentiation when limits are finite
Letφ(α)a n d ψ(α) be twice differentiable functions in some interval c≤α≤d,a n dl e t f(x, α)b eb o t h
integrable with respect to xover the interval φ(α)≤x≤ψ(α) and differentiable with respect to α. Then,
d
dα/integraldisplayψ(α)
φ(α)f(x, α)dx=/parenleftbiggdψ
dα/parenrightbigg
f(ψ(α),α)−/parenleftbiggdφ
dα/parenrightbigg
f(φ(α),α)+/integraldisplayψ(α)
φ(α)∂f
∂αdx. FI II 680
12.212 Differentiation when a limit is infinite
Letf(x, α)a n d ∂f/∂α both be integrable with respect to xover the semi-infinite region x≥a,b≤α<c.
Then, if the integral
f(α)=/integraldisplay∞
af(x, α)dx
exists for all b≤α≤c,a n di f/integraltext∞
a∂f
∂αdxis uniformly convergent for αin [b,c], it follows that
d
dα/integraldisplay∞
af(x, α)dx=/integraldisplay∞
a∂f
∂αdx
12.31 Integral Inequalities
12.311 Cauchy-Schwarz-Buniakowsky inequality for integrals
Letf(x)a n d g(x) be any two real integrable functions on [ a,b]. Then,
/parenleftBigg/integraldisplayb
af(x)g(x)dx/parenrightBigg2
≤/parenleftBigg/integraldisplayb
af2(x)dx/parenrightBigg/parenleftBigg/integraldisplayb
ag2(x)dx/parenrightBigg
,
and the equality will hold if, and only if, f(x)=kg(x), with kreal. BB 21
12.312 H¨ older’s inequality for integrals
Letf(x)a n dg(x) be any two real functions for which |f(x)|pand|g(x)|qa r ei n t e g r a b l eo n[ a,b] with p>1
and1
p+1
q=1 ;t h e n
Gram’s inequality for integrals 1065
/integraldisplayb
af(x)g(x)dx≤/parenleftBigg/integraldisplayb
a|f(x)|pdx/parenrightBigg1/p/parenleftBigg/integraldisplayb
a|g(x)|qdx/parenrightBigg1/q
.
The equality holds if, and only if, α|f(x)|p=β|g(x)|q,w h e r e αandβare positive constants. BB 21
12.313 Minkowski’s inequality for integrals
Letf(x)a n d g(x)be any two real functions for which |f(x)|pand|g(x)|pare integrable on [ a,b]f o rp>0;
then/parenleftBigg/integraldisplayb
a|f(x)+g(x)|pdx/parenrightBigg1/p
≤/parenleftBigg/integraldisplayb
a|f(x)|pdx/parenrightBigg1/p
+/parenleftBigg/integraldisplayb
a|g(x)|pdx/parenrightBigg1/p
.
The equality holds if, and only if, f(x)=kg(x) for some real k≥0. BB 21
12.314 Chebyshev’s inequality for integrals
Letf1,f2,... ,f nbe nonnegative integrable functions on [ a,b] which are all either monotonic increasing
or monotonic decreasing; then
/integraldisplayb
af1(x)dx/integraldisplayb
af2(x)d x.../integraldisplayb
afn(x)dx≤(b−a)n−1/integraldisplayb
af1(x)f2(x)...f n(x)dx MT 39
12.315 Young’s inequality for integrals
Letf(x) be a real-valued continuous strictly monotonic increasing function on the interval [0 ,a], with
f(0) = 0 and b≤f(a). Then
ab≤/integraldisplaya
0f(x)dx+/integraldisplayb
0f−1(y)dy,
where f−1(y) denotes the function inverse to f(x). The equality holds if, and only if, b=f(a). BB 15
12.316 Steffensen’s inequality for integrals
Letf(x) be nonnegative and monotonic decreasing in [ a,b], and g(x) be such that 0 ≤g(x)≤1i n[a,b].
Then/integraldisplayb
b−kf(x)dx≤/integraldisplayb
af(x)g(x)dx≤/integraldisplaya+k
af(x)dx,
where k=/integraltextb
ag(x)dx. MT 107
12.317 Gram’s inequality for integrals
Letf1(x),f2(x),... ,f n(x) be real square integrable functions on [ a,b]; then/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraltext
b
af2
1(x)dx/integraltextb
af1(x)f2(x)dx···/integraltextb
af1(x)fn(x)dx/integraltextb
af2(x)f1(x)dx/integraltextb
af2
2(x)dx ···/integraltextb
af2(x)fn(x)dx
............/integraltextb
afn(x)f1(x)dx/integraltextb
afn(x)f2(x)dx···/integraltextb
af2
n(x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥0.
MT 47
1066 Convexity and Jensen’s Inequality
12.318 Ostrowski’s inequality for integrals
Letf(x) be a monotonic function integrable on [ a,b], and let f(a)f(b)≥0,|f(a)|≥|f(b)|. Then, if gis
a real function integrable on [ a,b],/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
b
af(x)g(x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤|f(a)|max
a≤ξ≤b/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
ξ
ag(x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
12.41 Convexity and Jensen’s Inequality
A function f(x) is said to be convex on an interval [ a,b] if for any two points x1,x2in [a,b]
f/parenleftbiggx1+x2
2/parenrightbigg
≤f(x1)+f(x2)
2.
A function f(x) is said to be concave on an interval [ a,b] if for any two points x1,x2in [a,b] the function
−f(x) is convex in that interval.
If the function f(x) possesses a second derivative in the interval [ a,b], then a necessary and sufficient
condition for it to be convex on that interval is that f/prime/prime(x)≥0 for all xin [a,b].
A function f(x) is said to be logarithmically convex on the interval [ a,b]i ff>0 and log f(x)i s
concave on [ a,b].
Iff(x)a n d g(x) are logarithmically convex on the interval [ a,b], then the functions f(x)+g(x)a n d
f(x)g(x) are also logarithmically convex on [ a,b]. MT 17
12.411 Jensen’s inequality
Letf(x),p(x) be two functions defined for a≤x≤bsuch that α≤f(x)≤βandp(x)≥0, with p(x)/negationslash≡0.
Letφ(u) be a convex function defined on the interval α≤u≤β;t h e n
φ/parenleftBigg/integraltextb
af(x)p(x)dx
/integraltextb
ap(x)dx/parenrightBigg
≤/integraltextb
aφ(f)p(x)dx
/integraltextb
ap(x)dx. HL 151
12.412 Carleman’s inequality for integrals
Iff(x)≥0 and the integrals exist, then/integraldisplay∞
0exp/parenleftbigg1
x/integraldisplayx
0f(t)dt/parenrightbigg
dx≤e/integraldisplay∞
0f(x)dx.
12.51 Fourier Series and Related Inequalities
The trigonometric Fourier series representation of the function f(x)i n t e g r a b l eo n[ −π,π]i s
f(x)∼a0
2+∞/summationdisplay
n=1(ancosnx+bnsinnx),
where the Fourier coefficients anandbnoff(x) are given by
an=1
2π/integraldisplayπ
−πf(x)cosnxdx, b n=1
2π/integraldisplayπ
−πf(x)sinnxdx.
(See0.320–0.328 for convergence of Fourier series on ( −l,l).) TF 1
Generalized Fourier series 1067
12.511 Riemann-Lebesgue lemma
Iff(x)i si n t e g r a b l eo n[ −π,π], then
lim
t→∞/integraldisplayπ
−πf(x)sintxdx→0
and
lim
t→∞/integraldisplayπ
−πf(x)costxdx→0. TF 11
12.512 Dirichlet lemma
/integraldisplayπ
0sin/parenleftbig
n+1
2/parenrightbig
x
2s in1
2xdx=π
2,
in which sin/parenleftbig
n+1
2/parenrightbig
x/slashbig
2s in1
2xis called the Dirichlet kernel . ZY 21
12.513 Parseval’s theorem for trigonometric Fourier series
Iff(x)i ss q u a r ei n t e g r a b l eo n[ −π,π], then
a2
0
2+∞/summationdisplay
r=1/parenleftbig
a2
r+b2
r/parenrightbig
=1
π/integraldisplayπ
−πf2(x)dx. Y1 0
12.514 Integral representation of the nthpartial sum
Iff(x)i si n t e g r a b l eo n[ −π,π], then the nthpartial sum
sn(x)=a0
2+n/summationdisplay
r=1(arcosrx+brsinrx)
has the following integral representation in terms of the Dirichlet kernel:
sn(x)=1
π/integraldisplayπ
−πf(x−t)sin/parenleftbig
n+1
2/parenrightbig
t
2s in1
2tdt. Y2 0
12.515 Generalized Fourier series
Let the set of functions {φn}∞
n=0form an orthonormal set over [a,b], so that
/integraldisplayb
aφm(x)φn(x)dx=/braceleftBigg
1f o r m=n,
0f o r m/negationslash=n.
Then the generalized Fourier series representation of an integrable function f(x)o n[a,b]i s
f(x)∼∞/summationdisplay
n=0cnφn(x),
where the generalized Fourier coefficients of f(x) are given by
cn=/integraldisplayb
af(x)φn(x)dx.
1068 Fourier Series and Related Inequalities
12.516 Bessel’s inequality for generalized Fourier series
For any square integrable function defined on [ a,b],
∞/summationdisplay
n=0c2
n≤/integraldisplayb
af2(x)dx,
where the cnare the generalized Fourier coefficients of f(x).
12.517 Parseval’s theorem for generalized Fourier series
Iff(x) is a square integrable function defined on [ a,b]a n d{φn(x)}∞
n=0is acomplete orthonormal set
of continuous functions defined on [ a,b], then
∞/summationdisplay
n=0c2
n=/integraldisplayb
af2(x)dx,
where the cnare generalized Fourier coefficients of f(x).
13 Matrices and Related Results
13.11–13.12 Special Matrices
13.111 Diagonal matrix
A square matrix Aof the form
A=⎡
⎢⎢⎢⎢⎢⎣λ
100 ... 0
0λ20... 0
00 λ3 0
.........
000 λn⎤
⎥⎥⎥⎥⎥⎦
in which all entries away from the leading diagonal are zero.
13.112 Identity matrix and null matrix
Theidentity matrix is a diagonal matrix Iin which all entries in the leading diagonal are unity. The
null matrix is all zeros.
13.113 Reducible and irreducible matrices
Then×nmatrix A=[aij] is said to be reducible , if the indices 1 ,2,...,n can be divided into two
disjoint non-empty sets i1,i2,...,i μ;j1,j2,...,j νwith ( μ+ν=n), such that
aiαjβ=0 ( α=1,2,...,μ ;β=1,2,...,ν ).
Otherwise, Awill be said to be irreducible. GA 61
13.114 Equivalent matrices
Anm×nmatrix Aisequivalent to anm×nmatrix Bif, and only if, B=PAQ for suitable non-singular
m×mandn×nmatrices PandQ, respectively.
13.115 Transpose of a matrix
IfA=[aij]i sa n m×nmatrix with element aijin the ithrow and the jthcolumn, then the transpose
ATofAis the n×mmatrix
AT=[bij] with bij=aji,
that is, the matrix derived from Aby interchanging rows and columns.
1069
1070 Special Matrices
13.116 Adjoint matrix
IfAis ann×nmatrix, then its adjoint , denoted by adj A, is the transpose of the matrix of cofactors
AijofA,s ot h a t
adjA=[Aij]T(see14.13 ).
13.117 Inverse matrix
IfA=[aij]i sa n n×nmatrix with a nonsingular determinant |A|,t h e ni t s inverse A−1is given by
A−1=adjA
|A|.
13.118 Trace of a matrix
The trace of an n×nmatrix A=[aij], written tr A, is defined to be the sum of the terms on the leading
diagonal, so that
trA=a11+a22+...+ann.
13.119 Symmetric matrix
Then×nmatrix A=[aij]i ssymmetric ifaij=ajifori,j=1,2,...,n .
13.120 Skew-symmetric matrix
Then×nmatrix A=[aij]i sskew-symmetric ifaij=−ajifori,j=1,2,...,n .
13.121 Triangular matrices
Ann×nmatrix A=[aij]i so f upper triangular type ifaij=0f o r i>j and of lower triangular
type ifaij=0f o r j>i.
13.122 Orthogonal matrices
A real n×nmatrix Aisorthogonal if, and only if, AAT=I.
13.123 Hermitian transpose of a matrix
IfA=[aij]i sa n n×nmatrix with complex elements, then its hermitian transpose AHis defined to
be
AH=[aji],
with the bar denoting the complex conjugate operation.
13.124 Hermitian matrix
Ann×nmatrix Aishermitian ifA=AH, or equivalently, if A=AT, with the bar denoting the
complex conjugate operation.
Diagonally dominant 1071
13.125 Unitary matrix
Ann×nmatrix Aisunitary ifAAH=AHA=I.
13.126 Eigenvalues and eigenvectors
IfAis ann×nmatrix, each eigenvector xcorresponding to λsatisfies the equation
AX=λx,
while the eigenvalues λsatisfy the characteristic equation
|A−λI|=0 ( s e e 15.61 ).
13.127 Nilpotent matrix
Ann×nmatrix Aisnilpotent ifAk=0for some k.
13.128 Idempotent matrix
Ann×nmatrix Aisidempotent ifA2=A.
13.129 Positive definite
Ann×nmatrix Aispositive definite ifxTAx>0, forx/negationslash=0annelement column vector.
13.130 Non-negative definite
Ann×nmatrix Aisnon-negative definite ifxTAx≥0, forx/negationslash=0annelement column vector.
13.131 Diagonally dominant
Ann×nmatrix Aisdiagonally dominant if|aii|>/summationtext
j/negationslash=i|aij|for all i.
13.21 Quadratic Forms
Aquadratic form involving the nreal variables x1,x2,...,x nthat are associated with the real n×n
matrix A=[aij] is the scalar expression
Q(x1,x2,...,x n)=n/summationdisplay
i=1n/summationdisplay
j=1aijxixj.
In terms of matrix notation, if xis the n×1 column vector with real elements x1,x2,... ,x n,a n dxTis
the transpose of x,t h e n
Q(x)=xTAx.
Employing the inner product notation, this same quadratic form may also be written
Q(x)≡(x,Ax).
If the n×nmatrix Ais hermitian, so that AT=A, where the bar denotes the complex conjugate
operation, then the quadratic form associated with the hermitian matrix Aand the vector x,w h i c hm a y
have complex elements, is the real quadratic form
1072 Quadratic Forms
Q(x)=(x,Ax).
It is always possible to express an arbitrary quadratic form
Q(x)=n/summationdisplay
i=1n/summationdisplay
j=1αijxixj
in the form
Q(x)=(x,Ax),
where A=[aij] is a symmetric matrix, by defining
aii=αii fori=1,2,...,n
and
aij=1
2(αij+αji)f o r i,j=1,2,...,n and i/negationslash=j.
13.211 Sylvester’s law of inertia
When a quadratic form Qinnvariables is reduced by a nonsingular linear transformation to the form
Q=y2
1+y2
2+...+y2
p−y2
p+1−y2
p+2−...−y2
r,
the number pof positive squares appearing in the reduction is an invariant of the quadratic form Q,a n d
it does not depend on the method of reduction itself. ML 377
13.212 Rank
Therank of the quadratic form Qin the above canonical form is the total number rof squared terms
(both positive and negative) appearing in its reduced form. ML 360
13.213 Signature
Thesignature of the quadratic form Qabove is the number sof positive squared terms appearing in its
reduced form. It is sometimes also defined to be 2 s−r. ML 378
13.214 Positive definite and semidefinite quadratic form
The quadratic form Q(x)=(x,Ax) is said to be positive definite when Q(x)>0f o rx/negationslash=0. It is said
to bepositive semidefinite ifQ(x)≥0f o rx/negationslash=0 . ML 394
13.215 Basic theorems on quadratic forms
1. Two real quadratic forms are equivalent under the group of linear transformations if, and only
if, they have the same rank and the same signature.
2. A real quadratic form in nvariables is positive definite if, and only if, its canonical form is
Q=z2
1+z2
2.+...+z2
n.
3. A real symmetric matrix Ais positive definite if, and only if, there exists a real nonsingular
matrix Msuch that A=MMT.
4. Any real quadratic form in nvariables may be reduced to the diagonal form
Basic theorems on quadratic forms 1073
Q=λ1z2
1+λ2z2
2+...+λnz2
n,λ1≥λ2≥...≥λn
by a suitable orthogonal point-transformation.
5. The quadratic form Q=(x,Ax) is positive definite if, and only if, every eigenvalue of Ais
positive; it is positive semidefinite if, and only if, all the eigenvalues of Aare nonnegative, and
it is indefinite if the eigenvalues of Aare of both signs.
6. The necessary conditions for an hermitian matrix Ato be positive definite are
(i) aii>0 for all i,
(ii) aiiaij>|aij|2fori/negationslash=j,
(iii) the element of largest modulus must lie on the leading diagonal,(iv) |A|>0.
7. The quadratic form Q=(x,Ax) with Ahermitian will be positive definite if all the principal
minors in the top left-hand corner of Aare positive, so that
a
11>0,/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
11a12
a21a22/vextendsingle/vextendsingle/vextendsingle/vextendsingle>0,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
11a12a13
a21a22a23
a31a32a33/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>0,....
ML 353-379
13.31 Differentiation of Matrices
If the n×mmatrices A(t)a n dB(t) have elements that are differentiable functions of t,s ot h a t
A(t)=[aij(t)],B(t)=[bij(t)]
then
1.d
dtA(t)=/bracketleftbiggd
dtaij(t)/bracketrightbigg
2.d
dt[A(t)±B(t)] =/bracketleftbiggd
dtaij(t)±d
dtbij(t)/bracketrightbigg
=d
dtA(t)±d
dtB(t).
3. If the matrix product A(t)B(t) is defined, then
d
dt[A(t)B(t)] =/parenleftbiggd
dtA(t)/parenrightbigg
B(t)+A(t)/parenleftbiggd
dtB(t)/parenrightbigg
.
4. If the matrix product A(t)B(t) is defined, then
d
dt[A(t)B(t)]T=/parenleftbiggd
dtB(t)/parenrightbiggT
AT(t)+BT(t)/parenleftbiggd
dtA(t)/parenrightbiggT
.
5. If the square matrix Ais nonsingular, so that |A|/negationslash=0,then
d
dt/bracketleftbig
A−1/bracketrightbig
=−A−1(t)/parenleftbiggd
dtA(t)/parenrightbigg
A−1(t)
6./integraldisplayT
t0A(τ)dτ=/bracketleftBigg/integraldisplayT
t0aij(τ)dτ/bracketrightBigg
1074 The Matrix Exponential
13.41 The Matrix Exponential
IfAis a square matrix, and zis any complex number, then the matrix exponential eAzis defined to be
eAz=I+Az+...+Anzn
n!+...=∞/summationdisplay
r=01
r!Arzr.
3.411 Basic properties
1. e0=I,eIz=Iez,eA(z1+z2)=eAz1·eAz2,
e−Az=/parenleftbig
eAz/parenrightbig−1,eAz·eBz=e(A+B)z[when A+Bis defined and AB=BA]
2.dr
dzr/parenleftbig
eAz/parenrightbig
=AreAz=eAzAr.
ML 340
3. If the square matrix Ac a nb ee x p r e s s e di nt h ef o r m A=/bracketleftbigg
B0
0C/bracketrightbigg
,withBandCsquare matrices,
then
eAz=/bracketleftbigg
eBz0
0eCz/bracketrightbigg
.
14 Determinants
14.11 Expansion of Second- and Third-Order Determinants
1./vextendsingle/vextendsingle/vextendsingle/vextendsinglea
11a12
a21a22/vextendsingle/vextendsingle/vextendsingle/vextendsingle=a
11a22−a12a21.
2./vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
11a12a13
a21a22a23
a31a32a33/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=a
11a22a33−a11a23a32+a12a23a31−a12a21a33+a13a21a32−a13a22a31.
14.12 Basic Properties
LetA=[aij]a n dB=[bij]b en×nmatrices. Then the following results are true:
1. If any two adjacent rows (or columns) of a square matrix are interchanged, then the sign of the
associated determinant is changed.
2. If any two rows (or columns) of a determinant are identical, the determinant is zero.3. A determinant is not changed in value if any multiple of a row (or column) is added to any other
row (or column).
4. |kA|=k
n|A| for any scalar k.
5./vextendsingle/vextendsingleAT/vextendsingle/vextendsingle=|A| where ATis the transpose of A.
6. |AB|=|A||B|.
7./vextendsingle/vextendsingleA−1/vextendsingle/vextendsingle=1
|A|when the inverse exists.
8. If the elements aijofAare functions of x,t h e n
d|A|
dx=n/summationdisplay
i,j=1daij
dxAij (see14.13 ).
14.13 Minors and Cofactors of a Determinant
Theminor Mijof the element aijin the nth-order determinant |A|associated with the square n×n
matrix Ais the ( n−1)th-order determinant derived from Aby deletion of the ithrow and jthcolumn.
The cofactor Aijof the element aijis defined to be
Aij=(−1)i+jMij. ML 20
1075
1076 Principal Minors
14.14 Principal Minors
Aprincipal minor is one whose elements are situated symmetrically with respect to the leading diagonal
ofA. ML 197
14.15*Laplace Expansion of a Determinant
Thenth-order determinant denoted by |A|,o rd e t A, associated with the n×nmatrix A=[aij]m a yb e
expanded either by elements of the ithrow as
|A|=n/summationdisplay
j=1aijAij,
or by elements of the jthcolumn as
|A|=n/summationdisplay
i=1aijAij,
where Aijis the cofactor of element aij. The cofactors Aijsatisfy the following nlinear equations:
n/summationdisplay
j=1aijAkj=δik|A|,n/summationdisplay
i=1aijAik=δjk|A|,
fori,j,k=1,2,...,n andδij=/braceleftBigg
1f o r i=j
0f o r i/negationslash=j.ML 21
14.16 Jacobi’s Theorem
LetMrbe an r-rowed minor of the nth-order determinant |A|, associated with the n×nmatrix A=[aij],
in which the rows i1,i2,...,i rare represented together with the columns k1,k2,...,k r.
Define the complementary minor toMrto be the ( n−k)-rowed minor obtained from |A|by deleting
all the rows and columns associated with Mr,a n dt h e signed complementary minor M(r)toMrto
be
M(r)=(−1)i1+i2+···+ir+k1+k2+···+kr×(complementary minor to Mr).
Then, if Δ is the matrix of cofactors given by
Δ=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA
11A12···A1n
A21A22···A2n
............
An1An2···Ann/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,
andM
randM/prime
rare corresponding r-rowed minors of |A|and Δ, it follows that
M/prime
r=|A|r−1M(r). ML 25
Corollary. If|A|=0 ,t h e n
ApkAnq=AnkApq.
Cramer’s Rule 1077
14.17 Hadamard’s Theorem
If|A|is ann×ndeterminant with elements aijthat may be complex, then |A|/negationslash=0i f
|aii|>n/summationdisplay
j=1,j/negationslash=i|aij|.
14.18 Hadamard’s Inequality
LetA=[aij] be an arbitrary n×nnonsingular matrix with real elements and determinant |A|.T h e n
|A|2≤n/productdisplay
i=1/parenleftBiggn/summationdisplay
k=1a2
ik/parenrightBigg
.
This result is also true when Ais hermitian. ML 418
Deductions.
1. If M=m a x |aij|,then
|A|≤Mnnn/2. ML 419
2. If the n×nmatrix A=[aij] is positive definite, then
|A|≤a11a22...a nn. BL 126
3. If the real n×nmatrix Ais diagonally dominant, so that/summationtextn
j/negationslash=1|aij|<|aii|fori=1,2,...,n ,
then|A|/negationslash=0 .
14.21 Cramer’s Rule
If the nlinear equations
a11x1+a12x2+···+a1nxn=b1,
a21x1+a22x2+···+a2nxn=b2,
...............
an1x1+an2x2+···+annxn=bn,
have a nonsingular coefficient matrix A=[aij], so that |A|/negationslash= 0, then there is a unique solution
xj=A1jb1+A2jb2+···+Anjbj
|A|
forj=1,2,...,n ,w h e r e Aijis the cofactor of element aijin the coefficient matrix A. ML 134
1078 Some Special Determinants
14.31 Some Special Determinants
14.311 Vandermonde’s determinant (alternant)
Third order./vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle111
x
1x2x3
x2
1x22x23/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=(x
3−x2)(x3−x1)(x2−x1),
and, in general, the nth-order Vandermonde’s determinant is/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle11 ··· 1
x
1 x2··· xn
x2
1 x22··· x2
n
............
xn−1
1 xn−1
2···xn−1
n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/productdisplay
1≤i<j≤n(xj−xi),
where the right-hand side is the continued product of all the differences that can be formed from the
1
2n(n−1) pairs of numbers taken from x1,x2,...,x n, with the order of the differences taken in the
reverse order of the suffixes that are involved. ML 17
14.312 Circulants
Second order./vextendsingle/vextendsingle/vextendsingle/vextendsinglex
1x2
x2x1/vextendsingle/vextendsingle/vextendsingle/vextendsingle=(x
1+x2)(x1−x2).
Third order./vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex
1x2x3
x3x1x2
x2x3x1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=(x
1+x2+x3)/parenleftbig
x1+ωx2+ω2x3/parenrightbig/parenleftbig
x1+ω2x2+ωx3/parenrightbig
,
where ωandω2are the complex cube roots of 1. In general, the nth-order circulant determinant is/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex
1x2x3··· xn
xnx1x2···xn−1
xn−1xnx1···xn−2
...............
x2x3x4··· x1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=
n/productdisplay
j=1/parenleftbig
x1+x2ωj+x3ω2
j+···+xnωn−1
j/parenrightbig
,
where ωjis annthroot of 1. The eigenvalues λ(see15.61 )o fa n n×ncirculant matrix are
λj=x1+x2ωj+x3ω2
j+···+xnωn−1
j,
where ωjis again an nthroot of 1. ML 36
14.313 Jacobian determinant
Iff1,f2,...,f narenreal-valued functions which are differentiable with respect to x1,x2,...,x n,t h e n
the Jacobian Jf(x)o ft h e fiwith respect to the xjis the determinant
Jf(x)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂f1
∂x1∂f1
∂x2···∂f1
∂xn∂f2
∂x1∂f2
∂x2···∂f2
∂xn............
∂fn
∂x1∂fn
∂x2···∂fn
∂xn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
Properties 1079
The notation
∂(f1,f2,...,f n)
∂(x1,x2,...,x n)
is also used to denote the Jacobian Jf(x).
14.314 Hessian determinants
The Jacobian of the derivatives∂φ
∂x1,∂φ
∂x2,...,∂φ
∂xnof a function φ(x1,x2,...,x n) with respect to x1,x2,
...,xnis called the Hessian Hofφ,s ot h a t
H=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2φ
∂x2
1∂2φ
∂x1∂x2∂2φ
∂x1∂x3···∂2φ
∂x1∂xn
∂2φ
∂x2∂x1∂2φ
∂x22∂2φ
∂x2∂x3···∂2d2φ
∂x2∂xn
...............
∂2φ
∂xn∂x1∂2φ
∂xn∂x2∂2φ
∂xn∂x3···∂2φ
∂x2n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
14.315 Wronskian determinants
Letf1,f2,...,fnbenfunctions each ntimes differentiable with respect to xin some open interval
(a,b). Then the Wronskian W(x)o ff1,f2,...,f nis defined by
W(x)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglef
1 f2··· fn
f(1)
1 f(1)
2 ··· f(1)
n
f(2)
1 f(2)
2 ··· f(2)
n
............
f(n−1)
1 f(n−1)
2 ···f(n−1)
n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,
where f
(r)
i=drfi
dxr.
14.316 Properties
1.dW
dxfollows from W(x) by replacing the last row of the determinant defining W(x)b yt h e nth
derivatives f(n)
1,f(n)
2,...,f(n)
n.
2. If constants k1,k2,...,k nexist, not all zero, such that
k1f1+k2f2+···+knfn=0
for all xin (a,b), then W(x) = 0 for all xin (a,b).
3. The vanishing of the Wronskian throughout ( a,b) is necessary, but not sufficient, for the linear
dependence of f1,f2,...,f n.
1080 Some Special Determinants 14.318
14.317 Gram-Kowalewski theorem on linear dependence
A necessary and sufficient condition for nfunctions f1,f2,...,f nsquare integrable over a≤n≤bto be
linearly dependent in this interval is the vanishing of the Gram determinant
G(f1,f2,...,f n)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraltext
b
af2
1(x)dx/integraltextb
af1(x)f2(x)dx···/integraltextb
af1(x)fn(x)dx/integraltextb
af2(x)f1(x)dx/integraltextb
af2
2(x)dx ···/integraltextb
af2(x)fn(x)dx
............/integraltextb
afn(x)f1(x)dx/integraltextb
afn(x)f2(x)dx···/integraltextb
af2
n(x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
SA 2 (Theorem 3)
14.318 If the nfunctions f1,f2,...,f nare square integrable over a≤n≤b, then the Gram determinant
G(f1,f2,...,f n)≥0,
and the equality sign holds only when the functions are linearly dependent in a≤n≤b.
SA 4 (Corollary 1)
14.319 The rank of the matrix corresponding to the Gram determinant G(f1,f2,...,f n) gives the
maximum number of linearly independent functions f1,f2,...,fnina≤x≤b. If the rank is r,t h e n r
of the functions are linearly independent, and the other n−rfunctions are linearly dependent on these.
SA 3 (Theorem 4)
15 Norms
15.1–15.9 Vector Norms
15.11 General Properties
Thevector norm ||x||of an n×1 column vector xis a nonnegative number having the property that
1. ||x||>0w h e n x/negationslash=0and||x||= 0 if, and only if, x=0;
2. ||kx||=|k|||x||for any scalar k;
3. ||x+y|| ≤ ||x||+||y||.
15.21 Principal Vector Norms
15.211 The norm ||x||1
Ifxis a vector with complex components x1,x2,...,x n,t h e n
||x||1=n/summationdisplay
r=1|xr|. VA 15
15.212 The norm ||x||2(Euclidean or L2norm)
Ifxis a vector with complex components x1,x2,...,x n,t h e n
||x||2=/parenleftBiggn/summationdisplay
r=1|xr|2/parenrightBigg1/2
. VA 8
15.213 The norm ||x||∞
Ifxis a vector with complex components x1,x2,...,x n,t h e n
||x||∞=m a x
i|xi|. VA 15
1081
1082 Matrix Norms
15.31 Matrix Norms
15.311 General properties
Thematrix norm ||A||of a square matrix Ais a nonnegative number associated with Ahaving the
properties that
1. ||A||>0w h e n A/negationslash=0and||A||= 0 if, and only if, A=0;
2. ||kA||=|k|||A||for any scalar k;
3. ||A+B|| ≤ ||A||+||B||;
4. ||AB|| ≤ ||A||||B||.V A 9
The matrix norm ||A||associated with A=[aij], and the vector norm ||x||associated with the column
vector xfor which the matrix product Axis defined, are said to be compatible if
||Ax|| ≤ ||A||||x||.
15.312 Induced norms
When a vector zwith norm ||z||exists such that the maximum is attained in the expression
||A||=m a x
||z||=1||Az||,
then||A||is a matrix norm and is said to be the natural norm induced by, or subordinate to, the
vector norm ||z||. NO 428
15.313 Natural norm of unit matrix
IfIis the unit matrix, then for any natural norm
||I||=1. NO 429
15.41 Principal Natural Norms
The natural matrix norms induced on matrix A=[aij] by the 1, 2, and ∞vector norms are as follows:
15.411 Maximum absolute column sum norm
||A||1=m a x
jn/summationdisplay
i=1|aij| NO 429
15.412 Spectral norm
IfAHdenotes the Hermitian transpose of the square matrix A=[aij], so that AH=[aji] with a bar
denoting the complex conjugate operation, then
||A||2=/radicalbig
maximum eigenvalue of AHA,
or, equivalently,
||A||2=m a x
||x||2/negationslash=0||Ax||2
||x||2. NO 429
Deductions from Gerschgorin’s theorem (see 15.814 ) 1083
15.413 Maximum absolute row sum norm
||A||∞=m a x
in/summationdisplay
j=1|aij| NO 429
15.51 Spectral Radius of a Square Matrix
LetA=[aij]b ea n n×nmatrix with elements that may be complex, and with eigenvalues λ1,λ2,...,λ n.
Then the spectral radius ρ(A)o fAis the number
ρ(A)= m a x
1≤i≤n|λi|. VA 9
15.511 Inequalities concerning matrix norms and the spectral radius
1. ||A||22≤| |A||1||A||∞. NO 431
2. If Ais any arbitrary n×nmatrix with elements that may be complex, and the n×nmatrix Uis
unitary, so that UH=U−1, withHdenoting the Hermitian transpose of A(see13.123 ), then
||AU||=||UA||=||A||. VA 15
3. If Ais any nonsingular n×nmatrix with elements that may be complex with eigenvalues λ1,
λ2,λn,t h e n
1
||A−1||≤|λ|≤| |A||. VA 16
4. For any square matrix Awith spectral radius ρ(A) and any natural norm ||A||,
ρ(A)≤| |A||. NO 430
5. If the square matrix Ais Hermitian, then
ρ(A)=||A||.
6. If the square matrix Ais Hermitian and Pm(x) is any polynomial of degree mwith real coefficients,
then
||Pm(A)||=ρ(Pm(A)).
7. If Ais any arbitrary n×nmatrix with elements that may be complex, then the sequence of
matrices A,A2,A3,...converges to the null matrix as n→∞ if, and only if, ρ(A)<1.
NO 303
15.512 Deductions from Gerschgorin’s theorem (see 15.814)
1. Let Abe any arbitrary n×nmatrix with elements that may be complex; then ρ(A)≤
min⎛
⎝max
1≤i≤nn/summationdisplay
j=1|aij|,max
1≤j≤nn/summationdisplay
i=1|aij|⎞
⎠. VA 17
1084 Inequalities Involving Eigenvalues of Matrices
2. Let Abe any arbitrary n×nmatrix with elements that may be complex, and x1,x2,...,x nbe any
set of npositive numbers; then ρ(A)≤min/parenleftBigg
max
1≤i≤n/parenleftBigg/summationtextn
j=1|aij|xj
xi/parenrightBigg
,max
1≤j≤n/parenleftBigg
xjn/summationdisplay
i=1|aij|
xi/parenrightBigg/parenrightBigg
.
VA 18
15.61 Inequalities Involving Eigenvalues of Matrices
Theeigenvalues (characteristic values orlatent roots )λof an n×nmatrix A=[aij]a r et h e
solutions to the characteristic equation
|A−λI|=0.
When expanded, the determinant |A−λI|is called the characteristic polynomial , and it has the
form
|A−λI|=(−1)nλn+cn−1λn−1+cn−2λn−2+···+c1λ+c0.
The zeros of this polynomial satisfy the characteristic equation and so are the eigenvalues of A.I n t h e
characteristic polynomial the coefficients have the form
cn−r=(−1)n−r(sum of all principal minors of |A|of order r).
It then follows that
bn−1=(−1)n(a11+a22+···+ann),
bn−2=(−1)n/summationdisplay
i<j(aiiajj−aijaji),
b0=|A|.
Since the sum of the elements of the leading diagonal of Ais called the trace ofA, written tr A,i t
follows that bn−1=(−1)ntrA. ML 198
15.611 Cayley-Hamilton theorem
Every square matrix Asatisfies its characteristic equation, so that
(−1)nAn+cn−1An−1+cn−2An−2+···+c1A+c0I=0. ML 206
15.612 Corollaries
1. If Ais nonsingular, then its adjoint, denoted by adj A,i s
adjA=−/bracketleftbig
(−1)nAn−1+cn−1An−2+cn−2An−3+···+c2A+c1I/bracketrightbig
.
2. If Ais nonsingular, then the characteristic polynomial of A−1is
(−1)n/parenleftbigg
λn+c1
|A|λn−1+c2
|A|λn−2+···+(−1)n
|A|/parenrightbigg
.
15.71 Inequalities for the Characteristic Polynomial
The first group of inequalities that follow, which relate to the characteristic polynomial of an n×nmatrix
Awhose elements may be complex, refer directly to the coefficients of the polynomial when written in
the form
Named and unnamed inequalities 1085
P(λ)≡|λI−A|=λn+b1λn−1+b2λn−2+···+bn−1λ+bn,
and only implicitly to the coefficients aijofAthat give rise to the bi.
15.711 Named and unnamed inequalities
The first group of inequalities relating to the eigenvalues λsatisfying P(λ) = 0 are unnamed and are as
follows:
1. All the eigenvalues λlie within or on the circle ||z|| ≤r,w h e r e ris the positive root of .
|bn|+|bn−1|z+|bn−2|z2+···+|b1|zn−1−zn=0 MG 122
2. All the eigenvalues λlie within the circle
|z|<1+m a x
i|bi|. MG 123
3. When bn/negationslash= 0 the eigenvalue λof smallest modulus lies in the annulus R≤|z|≤R
21/n−1,w h e r e
Ris the positive root of
|bn|−|bn−1|z−|bn−2|z2−···− zn=0. MG 126
4. All the eigenvalues λlie on or outside the circle
|z|=m i n
k/bracketleftbigg|bn|
(|bn|+|bk|)/bracketrightbigg
. MG 126
5. If the eigenvalues λare ordered so that
|λ1|≥|λ2|≥···≥| λp|>1≥|λp+1|≥···≥| λn|,
then
|z1z2...z p|≤N, |zp|≤N1
p,
where
N2=1+ |b1|2+|b2|2+···+|bn|2. MG 129
6. All the eigenvalues λlie in or on the circle
|z|≤n/summationdisplay
j=1|bj|1/j. MG 126
7. All the eigenvalues λlie on the disk
/vextendsingle/vextendsingle/vextendsingle/vextendsinglez+b
1
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
1
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle+|b
2|1/2+|b3|1/3+···+|bn|1/n. MG 145
8. All the eigenvalues λlie in the annulus m≤| |z|| ≤M,w h e r e
m2=m a x/braceleftbigg
0,min
1≤j≤n−1/bracketleftBig
1−|bj|,|bn|2/bracketrightBig/bracerightbigg
and
1086 Inequalities for the Characteristic Polynomial
M2=m a x⎧
⎨
⎩1+|bj|,|bn|2+2n−1/summationdisplay
j=1|bj|2⎫
⎬
⎭.
The next group of inequalities are named theorems that apply to the explicit form of the characteristic
polynomial P(λ). MG 145
15.712 Parodi’s theorem
The eigenvalues λsatisfying P(λ) = 0 lie in the union of the disks
|z|≤1,|z+b1|≤n/summationdisplay
j=1|bj|. MG 143
15.713 Corollary of Brauer’s theorem
If
|b1|>1+n/summationdisplay
j=2|bj|,
then one and only one eigenvalue satisfying P(λ) = 0 lies on the disk
|z+b1|≤n/summationdisplay
j=2|bj|. MG 141
15.714 Ballieu’s theorem
For any set μ=(μ1,μ2,...,μ n) of positive numbers, let μ0=0a n d
Mμ=m a x
0≤k≤n−1/bracketleftbiggμk+μn|bn−k|
μk+1/bracketrightbigg
.
Then all the eigenvalues satisfying P(λ) = 0 lie on the disk ||z|| ≤Mμ. MG 144
15.715 Routh-Hurwitz theorem
Consider the characteristic equation
|λI−A|=λn+b1λn−1+···+bn−1λ+bn=0
determining the neigenvalues λof the real n×nmatrix A. Then the eigenvalues λall have negative real
parts if
Δ1>0,Δ2>0, ..., Δn>0,
where
Δk=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleb
1 10000 ... 0
b3 b2 b1 100 ... 0
b5 b4 b3 b2 b1 0... 0
...............
b2k−1b2k−2b2k−3b2k−4b2k−5b2k−6... b k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
GM 230
Poincare’s separation theorem 1087
15.81–15.82 Named Theorems on Eigenvalues
In the following theorems involving eigenvalue inequalities the elements aijof matrix Aenter directly,
and not in the form of the coefficients of the characteristic polynomial.
15.811 Schur’s inequalities
IfA=[aij]i sa n n×nmatrix with elements that may be complex, and eigenvalues λ1,λ2,...,λ n,t h e n
1.n/summationdisplay
i=1|λi|2≤n/summationdisplay
i,j=1|aij|2
2.n/summationdisplay
i=1|Reλi|2≤n/summationdisplay
i,j=1/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
ij+aji
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
3.n/summationdisplay
i=1|Imλi|2≤n/summationdisplay
i,j=1/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
ij−aji
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
ML 309
15.812 Sturmian separation theorem
LetAr=[aij] with i,j=1,2,...,r andr=1,2,...,N be a sequence of Nsymmetric matrices of
increasing order. Then if λk(Ar)f o rk=1,2,...,r denotes the ktheigenvalue of Ar, where the ordering
is such that
λ1(Ar)≥λ2(Ar)≥···≥ λr(Ar),
it follows that
λk+1(Ai+1)≤λk(Ai)≤λk(Ai+1). BL 115
15.813 Poincare’s separation theorem
Let/braceleftbig
yk/bracerightbig
, with k=1,2,...,K , be a set of orthonormal vectors so that the inner product ( yk,yk)=1 .
Set
x=K/summationdisplay
k=1ukyk,
so that for any square matrix Afor which the product Axis defined, the quadratic form
(x,Ax)=K/summationdisplay
k,l=1ukul/parenleftbig
yk,Ayl/parenrightbig
.
Then if
bK=/parenleftbig
yk,Ayl/parenrightbig
fork,l=1,2,...,K,
it follows that
λi(bK)≤λi(A)f o r i=1,2,...,K,
λK−j(bK)≥λN−j(A)f o r j=0,1,2,...,K −1.BL 117
1088 Named Theorems on Eigenvalues
15.814 Gerschgorin’s theorem
LetA=[aij] be any arbitrary n×nmatrix with elements that may be complex, and let
Λi≡n/summationdisplay
j=1,i/negationslash=j|aij|fori=1,2,...,n .
Then all of the eigenvalues λiofAlie in the union of the ndisks Γ i,w h e r e
Γi:|z−aii|≤Λifori=1,2,...,n . VA 16
15.815 Brauer’s theorem
If in Gerschgorin’s theorem for a given m
|ajj−amm|≥Λj+Λm
for all j/negationslash=m, then one and only one eigenvalue of Alies in the disk Γ m. MG 141
15.816 Perron’s theorem
Ifμ=(μ1,μ2,...,μ n) is an arbitrary set of positive numbers, then all the eigenvalues λof the n×n
matrix A=[aij] lie on the disk |z|≤Mμ,w h e r e
Mμ=m a x
1≤i≤nn/summationdisplay
j=1μj
μi|aij|. MG 141
15.817 Frobenius theorem
IfA=[aij] is a matrix with positive coefficients, so that aij>0 for all i,j=1,2,...,n ,t h e n Ahas a
positive eigenvalue λ0, and all its eigenvalues lie on the disk
|z|≤λ0. MG 142
15.818 Perron–Frobenius theorem
If all elements aijof an irreducible matrix Aare nonnegative, then R=m i n Mλis a simple eigenvalue
ofA, and all the eigenvalues of Alie on the disk |z|≤R,w h e r e ,i f λ=(λ1,λ2,...,λ n)i sas e to f
nonnegative numbers, not all zero,
Mλ=i n f⎧
⎨
⎩μ:μλi>n/summationdisplay
j=1|aij|λj,1≤i≤n⎫
⎬
⎭
andR=m i n Mλ.
Furthermore, if Ahas exactly peigenvalues ( p≤n)o nt h ec i r c l e |z|=R, then the set of all its
eigenvalues is invariant under rotations 2 π/pabout the origin. GM 69
15.819 Wielandt’s theorem
If the n×nmatrix Asatisfies the conditions of the Perron–Frobenius theorem and if in the n×nmatrix
C=[cij]
|cij|≤aij,i , j =1,2,...,n ,
then any eigenvalue λ0ofCsatisfies the inequality |λ0|≤R. The equality sign holds only when there
exists an n×nmatrix D=[±δij] such that δii= 1 for all i,δij= 0 for all i/negationslash=j,a n d
Hermitian matrices and diophantine relations 1089
C=(λ0/R)DAD−1. GM 69
15.820 Ostrowski’s theorem
IfA=[aij] is a matrix with positive coefficients and λ0is the positive eigenvalue in Frobenius’ theorem,
then the n−1 eigenvalues λj/negationslash=λ0satisfy the inequality
|λj|≤λ0M2−m2
M2+m2,
where
M=m a x aij,m =m i n aij fori,j=1,2,...,n . MG 145
15.821 First theorem due to Lyapunov
In order that all the eigenvalues of the real n×nmatrix Ahave negative real parts, it is necessary and
sufficient that if V is an n×nmatrix, the equation
ATV+VA=−I
has as a solution the matrix of coefficients Vof some positive-definite quadratic form ( x,Vx)( s e e13.21 ).
GM 224
15.822 Second theorem due to Lyapunov
If all the eigenvalues of the real matrix Ahave negative real parts, then to an arbitrary negative-definite
quadratic form ( x,Wx) with x=x(t) there corresponds a positive-definite quadratic form ( x,Vx)s u c h
that if one takes
dx
dt=Ax
then (x,Vx)a n d( x,Wx) satisfy
d
dt(x,Vx)=(x,Wx).
Conversely, if for some negative-definite form ( x,Wx) there exists a positive-definite form ( x,Vx)
connected to ( x,Wx) by the preceding two equations, then all the eigenvalues of Ahave negative real
parts (see 13.21, 13.31 ). GM 222
15.823 Hermitian matrices and diophantine relations involving circular functions of
rational angles due to Calogero and Perelomov
1. The off-diagonal Hermitian matrix Aof rank nwhose elements are given by
ajk=( 1−δjk)/braceleftbigg
1+icot/bracketleftbigg(j−k)π
n/bracketrightbigg/bracerightbigg
,
has the integer eigenvalues
λ(a)
s=2s−n−1f o rs=1,2,...,n ,
and the corresponding eigenvectors v(s)have the components
1090 Named Theorems on Eigenvalues
v(s)
j=e x p/parenleftbigg
−2πisj
n/parenrightbigg
forj=1,2,...,n .
2. The two off-diagonal Hermitian matrices BandCwhose elements are defined by the formulas
bjk=( 1−δjk)sin−2/bracketleftbigg(j−k)π
n/bracketrightbigg
,
cjk=( 1−δjk)sin−4/bracketleftbigg(j−k)π
n/bracketrightbigg
,
are related to the matrix Ain (1) by the equations
B=1
2/parenleftBig
A2+2A−σ(1)
nI/parenrightBig
,
C=−1
6/parenleftBig
B2−2/parenleftBig
2+σ(1)
n/parenrightBig
B−σ(2)
nI/parenrightBig
,
where Iis the unit matrix and
σ(1)
n=1
3/parenleftbig
n2−1/parenrightbig
,σ(2)
n=1
45/parenleftbig
n2−1/parenrightbig/parenleftbig
n2+1 1/parenrightbig
.
The eigenvalues of BandCcorresponding to the eigenvector v(s)
jin (1) have the form
λ(b)
s=σ(1)
n−2s(n−s)f o r s=1,2,...,n ,
λ(c)
s=σ(2)
n−2s(n−s)s(n−s)+2
3fors=1,2,...,n .
3. Together, the above two results imply the following diophantine summation rules:
(a)n−1/summationdisplay
k=1cot/parenleftbiggkπ
n/parenrightbigg
sin/parenleftbigg2skπ
n/parenrightbigg
=n−2s fors=1,2,...,n −1
(b)n−1/summationdisplay
k=1sin−2/parenleftbiggkπ
n/parenrightbigg
cos/parenleftbigg2skπ
n/parenrightbigg
=bs fors=1,2,...,n −1,
(c)n−1/summationdisplay
k=1sin−4/parenleftbiggkπ
n/parenrightbigg
cos/parenleftbigg2skπ
n/parenrightbigg
=cs fors=1,2,...,n −1,
(d)n−1/summationdisplay
k=1sin−2p/parenleftbiggkπ
n/parenrightbigg
=σ(p)
n,
withσ(1)
nandσ(2)
nas defined in (2), and
σ(3)
n=σ(1)
n2n4+2 3n2+ 191
315,σ(4)
n=σ(2)
n3n4+1 0n2+ 227
315
bs=σ(1)
n−2s(n−s),c s=σ(2)
n−2
3s(n−s)[s(n−s)+2 ].
Basic theorems 1091
15.91 Variational Principles
15.911 Rayleigh quotient
IfAis an Hermitian matrix, the Rayleigh quotient ρ(x) is the expression
ρ(x)=(x,Ax)
(x,x). NO 407
15.912 Basic theorems
1. If the n×nmatrix Ais Hermitian and has eigenvalues λ1≤λ2≤···≤ λn,t h e n
λ1≤ρ≤λn,
where ρis the Rayleigh quotient for any x/negationslash=0,a n d
λ1=m i n
x/negationslash=0(x,Ax)
(x,x)and λn=m a x
x/negationslash=0(x,Ax)
(x,x). NO 407
2. If the n×nmatrix Ais Hermitian and has eigenvalues λ1≤λ2≤···≤ λncorresponding to the
eigenvectors x1,x2,... ,xn, respectively, and x/negationslash=0is such that
(x,x1)=(x,x2)=···=(x,xn)=0,
then
λj=m i n
x(x,Ax)
(x,x),
and
λj≤(x,Ax)
(x,x)≤λn. NO 410
3. If the n×nmatrix Ais Hermitian, then the eigenvalue
λr=m a x/parenleftbigg
min(x,Ax)
(x,x)/parenrightbigg
,
where first the minimum over xis taken subject to ( bi,x)=0,i=1,2,...,r −1, with the bi
regarded as fixed vectors, and then the maximum over all possible bi. Also, the eigenvalue
λr=m i n/parenleftbigg
max(x,Ax)
(x,x)/parenrightbigg
,
where now the maximum over xis taken first subject to ( bi,x)=0,i=r+1,r+2,...,n for
fixedbi, and then the minimum over all possible bi. NO 414
4. The ( n−1) eigenvalues λ/prime
1,λ/prime2,...,λ/primen−1obtained from the ( n−1)×(n−1) matrix derived from
an Hermitian matrix Afrom which the last row and column have been omitted separate the n
eigenvalues of A,s ot h a t
λ1<λ/prime
1<λ2<λ/prime2<···<λ/prime
n−1<λn (see15.812 ).
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16 Ordinary Differential Equations
16.1–16.9 Results Relating to the Solution of Ordinary Differential
Equations
16.11 First-Order Equations
16.111 Solution of a first-order equation
Consider the real function f(t, x) that is defined and continuous in an open set D⊂R2.T h e na solution
to the first-order differential equation
dx
dt=f(t, x)
in the open interval I⊂Ris a real function u(t) that is defined and is both continuous and differentiable
inI, with the property that
(i) ( t, u(t))∈Dfort∈I,
(ii)du
dt=f(t, u(t)) for t∈I.
16.112 Cauchy problem
TheCauchy problem for the differential equation
dx
dt=f(t, x)
is the problem of existence and uniqueness of the solution to this equation satisfying the initial condition
u(t0)=x0,
where ( t0,u(t0))∈D, the open set defined above. The solution to the initial value problem may be
expressed in the form of the integral equation
u(t)=x0+/integraldisplayt
t0f(τ,u(τ))dτ (see16.316 ).
16.113 Approximate solution to an equation
The real function φ(t) is said to be an approximate solution , to within the error /epsilon1, of the differential
equation
1093
1094 Fundamental Inequalities and Related Results
dx
dt=f(t, x)
ifφ/primeis piecewise continuous, and for a given /epsilon1>0 and an open interval I⊂R,
|φ/prime(t)−f(t, φ(t))|≤/epsilon1,
except at points of discontinuity of the derivative. HU 3
16.114 Lipschitz continuity of a function
The real function f(t, x) defined and continuous in some open set D⊂R2is said to be Lipschitz
continuous with respect to xfor some constant k>0 if, for all points ( t, x1)a n d( t, x2) belonging
toD
|f(t, x1)−f(t, x2)|≤k|x1−x2|. HU 5
16.21 Fundamental Inequalities and Related Results
16.211 Gronwall’s lemma
Let the three piecewise continuous, non-negative functions u,v,a n dwbe defined in the interval [0 ,a]a n d
satisfy the inequality
w(t)≤u(t)+/integraldisplayt
0v(τ)w(τ)dτ,
except at points of discontinuity of the functions. Then, except at these same points,
w(t)≤u(t)+/integraldisplayt
0u(τ)v(τ)exp/parenleftbigg/integraldisplayt
τv(σ)dσ/parenrightbigg
dτ. BB 135
16.212 Comparison of approximate solutions of a differential equation
Letfbe a real function that is defined in an open set D⊂R2, in which it is both continuous and
Lipschitz continuous. In addition, let u1andu2be two approximate solutions of
dx
dt=f(t, x)
in an open set I⊂Rin the sense already defined, with
|u/prime
1(t)−f(t, u1(t))|≤/epsilon11,|u/prime
2(t)−f(t, u2(t))|≤/epsilon12,
except where the derivatives are discontinuous. Then, if for all t0∈I
|u1(t0)−u2(t0)|≤δ,
it follows that
|u1(t)−u2(t)|≤δexp{|t−t0|}+/parenleftbigg/epsilon11+/epsilon12
k/parenrightbigg
[exp{k|t−t0|} −1]. HU 6
16.31 First-Order Systems
16.311 Solution of a system of equations
Thesystem ofnfirst-order differential equations
Lipschitz continuity of a vector 1095
dx1
dt=f1(t, x1,x2,...,x n),
dx2
dt=f2(t, x1,x2,...,x n),
...
dxn
dt=fn(t, x1,x2,...,x n),
in which the functions f1,f2,...,f nare real and continuous in an open set D⊂Rn+1, may be written
in the concise matrix form
dx
dt=f(t,x),
where xandfaren×1 column vectors. Its solution in the open interval I⊂Ris the vector u(t) with
elements u1(t),u2(t),...,u n(t) with the property that
(i) ( t,u(t))∈Dfort∈I,
(ii)du
dt=f(t,u(t)) for t∈I. HU 24
16.312 Cauchy problem for a system
TheCauchy problem for the system
dx
dt=f(t,x)
is the problem of existence and uniqueness of the solution to this system satisfying the initial vector
condition
u(t0)=x0,
where ( t0,u(t0)∈D, the open set defined above in connection with the system. The solution to the
initial value problem may be expressed in the form of the vector integral equation
u(t)=x0+/integraldisplayt
t0f(τ,u(τ))dτ.
16.313 Approximate solution to a system
The real vector φ(t) is said to be an approximate vector solution , to within the order /epsilon1, of the system
dx
dt=f(t,x),
if the elements of φ/primeare piecewise continuous, and for a given /epsilon1>0 and open interval I⊂R,
||φ/prime(t)−f(t, φ(t))|| ≤/epsilon1,
except at points of discontinuity of the derivative, where ||w||denotes the supremum norm
||w||=s u p( |w1|,|w2|,...,|wn|). HU 25
16.314 Lipschitz continuity of a vector
The real vector f(t, x) defined and continuous in some open set D⊂Rnis said to be Lipschitz contin-
uous with respect to xfor some constant k>0 if, for all points ( t,x1), (t,x2) belonging to D,
||f(t,x1)−f(t,x2)|| ≤k||x1−x2||. HU 26
1096 First-Order Systems
16.315 Comparison of approximate solutions of a system
Letfbe a real vector defined in an open set D⊂R×Rnin which it is both continuous and Lipschitz
continuous. In addition, let u1andu2be two approximate solutions of the system
dx
dt=f(t,x)
in an open set I⊂Rin the sense already defined, with
|u/prime
1(t)−f(t,u1(t))|≤/epsilon11,|u/prime
2(t)−f(t,u2(t))|≤/epsilon12,
except where the derivatives are discontinuous. Then, if for all t0∈I
||u1(t0)−u2(t0)|| ≤δ,
it follows that
||u1(t)−u2(t)|| ≤δexp{k|t−t0|}+/parenleftbigg/epsilon11+/epsilon12
k/parenrightbigg
[exp{k|t−t0|} −1]. HU 27
16.316 First-order linear differential equation
Thefirst-order linear differential equation when expressed in the canonical form
dy
dt+P(t)y=Q(t)
has an integrating factor
μ(t) = exp/parenleftbigg/integraldisplay
P(t)dt/parenrightbigg
,
and a general solution
y(t)=1
μ(t)/parenleftbigg
μ(t0)y0+/integraldisplayt
t0μ(ξ)Q(ξ)dξ/parenrightbigg
,
where y0=y(t0).
16.317 Linear systems of differential equations
Consider the homogeneous system of linear differential equations
dx
dt=A(t)x,
wherexis ann×1 column vector and A(t)a nn×nmatrix. Then a fundamental system of solutions of
this system is a set of nlinearly independent solution vectors φ1(t),φ2(t),...,φn(t), The square matrix
K(t) whose columns comprise the vectors φ1(t),φ2(t),... ,φn(t) is called the fundamental matrix of
the differential equation, and we have the representation
|K(t)|=|K(t0)|exp/parenleftbigg/integraldisplayt
t0trA(τ)dτ/parenrightbigg
.
Using the fundamental matrix K(t) defined in terms of the homogeneous system, the unique solution to
the inhomogeneous system
dx
dt=A(t)x+b(t),
assuming the initial value x(t0)=x0,i s
φ(t)=K(t)[K(t0)]−1x0+K(t)/integraldisplayt
t0[K(τ)]−1b(τ)dτ, HU 43
where b(t)i sa n n×1 column vector. CL 69
Homogeneous differential equations 1097
16.41 Some Special Types of Elementary Differential Equations
16.411 Variables separable
A first-order differential equation is said to be variables separable if it is of the form
dy
dx=M(x)N(y),
or
P(x)Q(y)dx+R(x)S(y)dy=0.
It may then be written in the form
M(x)dx−1
N(y)dy=0,
or
P(x)
R(x)dx+S(y)
Q(y)dy=0,
provided R(x)Q(y)/negationslash=0 .
16.412 Exact differential equations
A differential equation
M(x, y)dx+N(x, y)dy=0
is said to be exact if there exists a function h(x, y) such that
d[h(x, y)] =M(x, y)dx+N(x, y)dy. IN 16
16.413 Conditions for an exact equation
A necessary and sufficient condition that an equation of this form is exact is that the functions M(x, y)
andN(x, y) together with their partial derivatives ∂M/∂y and∂N/∂x exist and are continuous in a
region in which
∂M
∂y=∂N
∂x. IN 16
16.414 Homogeneous differential equations
A differential equation
M(x, y)dx+N(x, y)dy=0
is said to be algebraically homogeneous if, for arbitrary k,
M(kx,ky )
N(kx,ky )=M(x, y)
N(x, y).
Setting y=sx, it may then be expressed in the form
[M(1,s)+sN(1,s)]dx+xN(1)dx=0,
in which the variables sandxare separable. IN 18
1098 Second-Order Equations
16.51 Second-Order Equations
16.511 Adjoint and self-adjoint equations
The linear second-order differential equation
L(u)≡a(x)d2u
dx2+b(x)du
dx+c(x)u=0
has associated with it the adjoint equation
M(v)≡d2
dx2[a(x)v]−d
dx[b(x)v]+c(x)v=0.
The equation L(u) = 0 is said to be self-adjoint ifL(u)≡M(u).
A linear self-adjoint second-order differential equation defined on [ α,β] can always be expressed in the
form
d
dx/parenleftbigg
p(x)du
dx/parenrightbigg
+q(x)u=0,
where p(x)a n d q(x) are continuous on [ α,β]a n d p(x)>0. The general equation L(u) = 0 can always be
made self-adjoint and written in this form by multiplication by the factor
1
a(x)/bracketleftbigg
exp/integraldisplayb(x)
a(x)dx/bracketrightbigg
,
when
p(x) = exp/integraldisplayb(x)
a(x)dxand q(x)=c(x)
a(x)/bracketleftbigg
exp/integraldisplayb(x)
a(x)dx/bracketrightbigg
.
In general, if
L(u)=p0dnu
dxn+p1dn−1u
dxn−1...+pn−1du
dx+pnu,
then its adjoint is
M(v)=(−1)ndn
dxn[p0v]+(−1)n−1dn−1
dxn−1[p1v]+...−d
dx[pn−1v]+pnv. HI 391
16.512 Abel’s identity
Ifp(x)a n d q(x) are continuous in [ α,β]i nw h i c h p(x)>0, and u(x)a n d v(x) are suitably differentiable
with
d
dx/parenleftbigg
p(x)du
dx/parenrightbigg
+q(x)u=0,
then the result
p(x)/parenleftbigg
udv
dx−vdu
dx/parenrightbigg
≡const.
is known as Abel’s identity .
More generally, if we consider the linear nth-order equation
p0dnu
dxn+p1dn−1u
dxn−1+...+pn−1du
dx+pn=0,
and Δ is the Wronskian of a (fundamental) set of linearly independent solutions u1,u2,...,u n,t h eA b e l
identity takes the form
Δ=Δ 0exp/parenleftbigg
−/integraldisplayx
x0p1(x)
p0(x)dx/parenrightbigg
,
where Δ 0is the value of Δ at x=x0. IN 119
Solutions of the Riccati equation 1099
16.513 Lagrange identity
If the linear nth-order equation L(u) = 0 is defined by
L(u)≡p0dnu
dxn+p1dn−1u
dxn−1+...+pn−1du
dx+pnu,
then the expression
vL(u)−uM(v)=d
dx{P(u,v)},
where M(v) is the adjoint of L(u), is called the Lagrange identity . The expression P(u,v), which is
linear and homogeneous in
u,du
dx,... ,dn−1u
dxn−1and v,dv
dx,...,dn−1v
dxn−1,
is then known as the bilinear concomitant . In the case of the second-order equation
L(u)=a(x)d2u
dx2+b(x)du
dx+c(x)u=0,
with adjoint M(v), the Lagrange identity becomes
vL(u)−uM(v)=d
dx/parenleftbigg
a(x)vdu
dx−d
dx(a(x)v)u+b(x)uv/parenrightbigg
. IN 124
16.514 The Riccati equation
The general Riccati equation has the form
dz
dx+a(x)z+b(x)z2+c(x)=0,
and an equation of this form results from the substitution
z=/parenleftbig
p(x)du
dx/parenrightbig
uin the general self-adjoint equation
d
dx/parenleftbigg
p(x)du
dx/parenrightbigg
+q(x)u=0.
The further substitution v=u/parenleftbig
exp/integraltextx
αa(x)dx/parenrightbig
in the Riccati equation then gives the more convenient
form
dv
dx+r(x)v2+s(x)=0,
with
r(x)=b(x)exp/parenleftbigg
−/integraldisplayx
αa(x)dx/parenrightbigg
and s(x)=c(x)exp/parenleftbigg/integraldisplayx
αa(x)dx/parenrightbigg
. HI 273
16.515 Solutions of the Riccati equation
If in the Riccati equation
dv
dx+r(x)v2+s(x)=0,
r(x)/negationslash= 0, while r(x)a n d s(x) are continuous on the interval [ α,β], then every solution v(x)m a yb e
expressed in the form
1
r(x)Au/prime(x)+Bv/prime(x)
Au(x)+Bv(x),
withA,Barbitrary constants, not both zero, and the prime denoting differentiation, while uandvare
linearly independent solutions of
1100 Oscillation and Non-Oscillation Theorems for Second-Order Equations
d
dx/parenleftbigg1
r(x)dz
dx/parenrightbigg
+s(x)z=0.
Conversely, if u(x)a n d v(x) are linearly independent solutions of this last equation and AandBare
arbitrary constants, not both zero, the function
1
r(x)Au/prime(x)+Bv/prime(x)
Au(x)+Bv(x)
is a solution of the Riccati equation wherever Au(x)=Bv(x)/negationslash=0 . IN 24
16.516 Solution of a second-order linear differential equation
Afundamental system of solutions of a homogeneous second-order linear differential equation in the
canonical form
d2x
dt2+a(t)dx
dt+b(t)x=0
is a system of two linearly independent solutions φ1(t)a n d φ2(t). The Wronskian of these solutions is
W(t)=/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ
1(t)φ2(t)
φ/prime
1(t)φ/prime
2(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=φ
1(t)φ/prime
2(t)−φ2(t)φ/prime
1(t),
and the solution to the inhomogeneous equation
d2x
dt2+a(t)dx
dt+b(t)x=f(t),
subject to the initial conditions x(t0)=x0andx/prime(t0)=x1may be written
x(t)=c1φ1(t)+c2φ2(t)+/integraldisplayt
t0φ1(ξ)φ2(t)−φ2(ξ)φ1(t)
W(ξ)f(ξ)dξ,
where the constants c1andc2are chosen such that x(t) satisfies the initial conditions.
The linear combination c1φ1(t)+c2φ2(t) is known as the complementary function where c1and
c2are arbitrary constants.
16.61–16.62 Oscillation and Non-Oscillation Theorems for Second-
Order Equations
Equations whose solutions possess an infinite number of zeros in the interval (0 ,∞) are said to have
oscillatory solutions. The following theorems relate to such properties:
16.611 First basic comparison theorem
If all solutions of the equation
d2u
dx2+φ(x)u=0
are oscillatory, and if
ψ(x)≥φ(x),
then all the solutions of
d2v
dx2+ψ(x)v=0
are oscillatory, and conversely. That is, if ψ(x)≥φ(x) and some solutions vare non-oscillatory, then so
also must some solutions ube non-oscillatory. BS 119
Szeg¨o’s comparison theorem 1101
16.622 Second basic comparison theorem
If all the solutions of the self-adjoint equation
d
dx/parenleftbigg
p1(x)du
dx/parenrightbigg
+q1(x)u=0
are oscillatory as x→∞,a n di f
q2(x)≥q1(x),
p2(x)≥p1(x)>0,
then all the solutions of the self-adjoint equation
d
dx/parenleftbigg
p2(x)dv
dx/parenrightbigg
+q2(x)v=0
are oscillatory. BS 120
16.623 Interlacing of zeros
Lety1(x)a n d y2(x) be two linearly independent solutions of
d2y
dx2+F(x)y=0,
and suppose that y1(x) has at least two zeros in the interval ( a,b). Then if x1andx2are two consecutive
zeros of y1(x), the function y2(x) has one, and only one, zero in the interval ( x1,x2). HI 374
16.624 Sturm separation theorem
Letu(x)a n d v(x) be two linearly independent solutions of the self-adjoint equation
d
dx/parenleftbigg
p(x)dy
dx/parenrightbigg
+q(x)=0,
in which p(x)>0a n d p(x),q(x) are continuous on [ a,b]. Then, between any two consecutive zeros of
u(x) there will be one, and only one, zero of v(x). IN 224
16.625 Sturm comparison theorem
Letp1(x)≥p2(x)>0a n d q1(x)≥q2(x) be continuous functions in the differential equations
d
dx/parenleftbigg
p1(x)du
dx/parenrightbigg
+q1(x)u=0,
d
dx/parenleftbigg
p2(x)dv
dx/parenrightbigg
+q2(x)v=0.
Then between any two zeros of a non-trivial solution u(x) of the first equation there will be at least one
zero of every non-trivial solution v(x) of the second equation. IN 228
16.626 Szeg¨ o’s comparison theorem
Suppose, under the conditions of the Sturm comparison theorem, that p1(x)≡p2(x),q1(x)/negationslash≡q2(x), and
u(x)>0,v(x)>0f o ra<x<b , together with
lim
x→ap1(x)/parenleftbiggdu
dxv−dv
dxu/parenrightbigg
=0.
Then, if u(b) = 0, there is a point ξin (a,b) such that v(ξ)=0 . HI 379
1102 Oscillation and Non-Oscillation Theorems for Second-Order Equations
16.627 Picone’s identity
Consider the equations
d
dx/parenleftbigg
p1(x)du
dx/parenrightbigg
+q1(x)u=0,
d
dx/parenleftbigg
p2(x)dv
dx/parenrightbigg
+q2(x)v=0,
withp1,p2,q1,a n d q2positive and continuous for a<x<b ,w h e r e q2(x)>q1(x)a n d p1(x)>p2(x).
Then with a<α<β<b , Picone’s identity is/parenleftbiggu
v/parenleftbigg
p1du
dxv−p2dv
dxu/parenrightbigg/parenrightbiggβ
α=/integraldisplayβ
α(q2−q1)u2ds+/integraldisplayβ
α(p1−p2)/parenleftbiggdu
ds/parenrightbigg2
ds+/integraldisplayβ
αp2
v2/parenleftbigg
vdu
ds−udv
ds/parenrightbigg2
ds.
IN 226
16.628 Sturm-Picone theorem
Consider the self-adjoint equations
d
dx/parenleftbigg
p1(x)du
dx/parenrightbigg
+q1(x)u=0
and
d
dx/parenleftbigg
p2(x)dv
dx/parenrightbigg
+q2(x)v=0.
Letp1,p2,q1,a n d q2be positive and continuous for a<x<b ,w h e r e q2(x)>q1(x)a n d p1(x)>p2(x).
Then, if x1andx2is a pair of consecutive zeros of u(x)i n(a,b),v(x) has at least one zero in the open
interval ( a,b). IN 225
16.629 Oscillation on the half line
Consider the self-adjoint equation
d
dx/parenleftbigg
p(x)du
dx/parenrightbigg
+q(x)u=0.
We then have the following results:
(i) Let p(x)>0a n d p, qbe continuous on [0 ,∞). If the two improper integrals
/integraldisplay∞
1dx
p(x)and/integraldisplay∞
1q(x)dx
diverge, then every solution u(x) has infinitely many zeros on the interval [1 ,∞). Also, if the two
integrals
/integraldisplay1
0dx
p(x)=+∞and/integraldisplay1
0q(x)dx=+∞,
then every solution u(x) has infinitely many zeros on the interval (0, 1).
(ii) (Moore’s theorem). Every non-trivial solution u(x) has at most a finite number of zeros on the
interval [ a,∞) if the improper integral
/integraldisplay∞
adx
p(x)
converges, and if
Kneser’s non-oscillation theorem 1103
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
x
aq(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle<M for a≤x<∞
withM>0 a finite constant.
16.71 Two Related Comparison Theorems
16.711 Theorem 1
Consider the equations in the Sturm comparison theorem with the same assumptions on p(x)a n d q(x),
and let u(x),v(x) be solutions such that
u(x1)=v(x1)=0,u/prime(x)=v/prime(x1)>0.
Then if u(x) is increasing in [ x1,x2] and reaches a maximum at x2, the function v(x) reaches a maximum
at some point x3such that x1<x3<x2. HI 376
16.712 Theorem 2
Consider the equation
d2y
dx2+F(x)y=0,
in which F(x) is continuous in ( a,b)a n ds u c ht h a t
0<m≤F(x)≤M.
Then, if the solution y(x) has two successive zeros x1,x2, it follows that
πM−1/2≤x2−x1≤πm−1/2.
16.81–16.82 Non-Oscillatory Solutions
The real solution y(x)o f
d2y
dx2+F(x)y=0
is said to be non-oscillatory in the wide sense in (0 ,∞) if there exists a finite number csuch that the
solution has no zeros in [ c,∞). HI 376
16.811 Kneser’s non-oscillation theorem
Consider the equation
d2y
dx2+F(x)y=0,
and let
limsup/bracketleftbig
x2F(x)/bracketrightbig
=γ∗,
liminf/bracketleftbig
x2F(x)/bracketrightbig
=γ∗.
Then the solution y(x) is non-oscillatory if γ∗<1
4, oscillatory if1
4<γ∗and no conclusion can be drawn
if either γ∗orγ∗equals1
4. HI 461
1104 Some Growth Estimates for Solutions of Second-Order Equations
16.822 Comparison theorem for non-oscillation
Consider the differential equations
d2y
dx2+F(x)y=0,f(x)=x/integraldisplay∞
xF(s)ds,
d2y
dx2+G(x)y=0,g(x)=x/integraldisplay∞
xG(s)ds,
where 0 <g(x)<f(x). Then if the first equation is non-oscillatory in the wide sense, so also is the
second. HI 460
16.823 Necessary and sufficient conditions for non-oscillation
Consider the equation
d2y
dx2+F(x)y=0.
Then, if
lim
x→∞sup/parenleftbigg
x/integraldisplay∞
xF(s)ds/parenrightbigg
=F∗,
lim
x→∞inf/parenleftbigg
x/integraldisplay∞
xF(s)ds/parenrightbigg
=F∗,
it follows that:
(i) a necessary condition that the solution y(x) be non-oscillatory is that F∗≤1
4andF∗≤1;
(ii) a sufficient condition that the solution y(x) be non-oscillatory is that F∗<1
4.
16.91 Some Growth Estimates for Solutions of Second-Order Equa-
tions
16.911 Strictly increasing and decreasing solutions
Suppose that G(x)>0 be continuous in ( −∞,∞)a n ds u c ht h a t xG(x)/negationslash∈L(0,∞). Then the equation
d2y
dx2−G(x)y= 0 has one, and only one, solution y+(x) passing through the point (0 ,1), which is positive
and strictly monotonic decreasing for all x, and one and only one solution y−(x) through the point (0 ,1),
which is positive and strictly increasing for all x. The solution y+(x) has the property that
[G(x)]1/2y+(x)∈L2(0,∞)a n ddy+(x)
dx∈L2(0,∞).
If, in addition, 0 <α2≤G(x)≤β2<∞,t h e n
e−βx≤y+(x)≤e−αxfor x>0. HI 359
16.912 General result on dominant and subdominant solutions
Consider the equations
d2y
dx2−g(x)y=0,d2Y
dx2−G(x)Y=0,
where gandGare continuous on (0 ,∞) with 0 <g(x)<G(x), and xg(x)/negationslash∈L(0,∞). In addition, let yα
andYαbe the solutions of these respective equations corresponding to
A theorem due to Lyapunov 1105
yα(0) = Yα(0) = 1 ,y/prime
α(0) = Y/prime
α(0) = αfor−∞<α< ∞.
LetyωandYωbe determined, respectively, by
yω(0) = Yω(0) = 0 ,y/prime
ω(0) = Y/prime
ω(0) = 1 ,
and let y+andY+be the subdominant solutions for which
y+(0) = Y+(0) = 1
while/bracketleftbig
y/prime
+(x)/bracketrightbig2,g(x)[y+(x)]2,/bracketleftbig
Y/prime
+(x)/bracketrightbig2,a n d G(x)/bracketleftbig
Y/prime
+(x)/bracketrightbig2belong to L(0,∞). Then, if βandγare such
thaty−β=y+andY−γ=Y+, it follows that β<γ and
yα(x)<Yα(x),0<x< ∞,−γ≤α,
yω(x)<Yω(x),
y+(x)>Y+(x).HI 440
16.913 Estimate of dominant solution
LetG(x) be positive and continuous with continuous first- and second-order derivatives satisfying
G(x)G/prime(x)<5
4[G/prime(x)]2.
Then there exists a dominant solution y(x) of the fundamental solutions Y0(x)a n d Y1(x)o f
d2y
dx2−G(x)y=0,
determined by the initial conditions
2Y0(0) = 0 ,Y 1(0)= 1 ,
Y/prime
0(0) = 1 ,Y/prime
1(0) = 0 ,
such that
y(x)<[G(x)]−1/4exp/parenleftbigg/integraldisplayx
0[G(ξ)]1/2dξ/parenrightbigg
,
and a positive constant Csuch that the normalized subdominant solution y+(x), for which y+(0) = 1
and/bracketleftbig
y/prime
+(x)/bracketrightbig2∈L(0,∞),G(x)[y+(x)]2∈L(0,∞), satisfies
y+(x)>C G (x)−1/4exp/parenleftbigg
−/integraldisplayx
0[G(ξ)]1/2dξ/parenrightbigg
. HI 443
16.914 A theorem due to Lyapunov
Lety(x) be any solution of
d2y
dx2−G(x)y=0
withG(x) positive and continuous in (0 ,∞) with xG(x)∈L(0,∞). Then
exp/parenleftbigg
−/integraldisplayx
0[G(ξ)+1 ] dξ/parenrightbigg
<[y(x)]2+[y/prime(x)]2
<Cexp/parenleftbigg/integraldisplayx
0[G(ξ)+1 ] dξ/parenrightbigg
,HI 446
where C=[y(0)]2+[y/prime(0)]2.
1106 Boundedness Theorems
16.92 Boundedness Theorems
16.9216All solutions of the equation
d2u
dx2+( 1+ φ(x)+ψ(x))u=0
are bounded, provided that
(i)/integraltext∞|φ(x)|dx <∞,
(ii)/integraldisplay∞
|ψ(x)|dx <∞and ψ(x)→0a sx→∞. BS 112
16.922 If all solutions of the equation
d2u
dx2+a(x)u=0
are bounded, then all solutions of
d2u
dx2+(a(x)+b(x))u=0
are also bounded if/integraldisplay∞
|b(x)|dx <∞. BS 112
16.923 If a(x)→∞ monotonically as x→∞, then all solutions of
d2u
dx2+a(x)u=0
are bounded as x→∞. BS 113
16.924 Consider the equation
d2u
dx2+a(x)u=0
in which/integraldisplay∞
x|a(x)|dx <∞.
Then lim
x→∞/parenleftbiggdu
dx/parenrightbigg
exists, and the general solution is asymptotic to d0+d1xasx→∞,w h e r e d0andd1
may be zero, but not simultaneously. BS 114
16.9310Growth of maxima of |y|
Sonin’s theorem generalized by P´ olya may be stated as follows: Let y(x) satisfy the differential equation
{k(x)y/prime}/prime+φ(x)y=0,
where k(x)>0,φ(x)>0, and both functions k(x),φ(x)have a continuous derivative. Then the relative
maxima of |y|form an increasing or decreasing sequence according as k(x)φ(x)is decreasing or increasing.
SZ 164
17 Fourier, Laplace, and Mellin
Transforms
17.1–17.4 Integral Transforms
17.11 Laplace transform
TheLaplace transform of the function f(x), denoted by F(s), is defined by the integral
F(s)=/integraldisplay∞
0f(x)e−sxdx, Res>0.
The functions f(x)a n d F(s) are called a Laplace transform pair , and knowledge of either one enables
the other to be recovered.
Iffis summable over all finite intervals, and there is a constant cfor which/integraldisplay∞
0|f(x)|e−c|x|dx
is finite, then the Laplace transform exists when s=σ+iτis such that σ≥c.
Setting
F(s)=L[f(x);s]
to emphasize the nature of the transform, we have the symbolic inverse result
f(x)=L−1[F(s);x].
The inversion of the Laplace transform is accomplished for analytic functions F(s)o fo r d e r O/parenleftbig
s−k/parenrightbig
with
k>1 by means of the inversion integral
f(x)=1
2πi/integraldisplayγ+i∞
γ−i∞F(s)esxds,
where γis a real constant that exceeds the real part of all the singularities of F(s). SN 30
17.12 Basic properties of the Laplace transform
1.8Foraandbarbitrary constants,
L[af(x)+bg(x)] =aF(s)+bG(s) (linearity)
2. If n>0 is an integer and lim
x→∞f(x)e−sx=0 ,t h e nf o r x>0,
L/bracketleftBig
f(n)(x);s/bracketrightBig
=snF(s)−sn−1f(0)−sn−2f(1)(0)−···− f(n−1)(0) (transform of a derivative)
SN 32
1107
1108 Integral Transforms
3.11If lim
x→∞/parenleftbig
e−sx/integraltextx
0f(ζ)dζ/parenrightbig
=0 ,t h e n
L/bracketleftbigg/integraldisplayx
0f(ξ)dξ;s/bracketrightbigg
=1
sF(s) (transform of an integral) SN 37
4. L/bracketleftbig
e−axf(x);s/bracketrightbig
=F(s+a) (shift theorem) SU 143
5. The Laplace convolution f∗gof two functions f(x)a n d g(x) is defined by the integral
f∗g(x)=/integraldisplayx
0f(x−ξ)g(ξ)dξ,
and it has the property that f∗g=g∗fandf∗(g∗h)=(f∗g)∗h. In terms of the convolution
operation
L[f∗g(x);s]=F(s)G(s) (convolution (Faltung) theorem). SN 30
17.13 Table of Laplace transform pairs
f(x) F(s)
1 1 1/s
2 xn,n =0,1,2,...n!
sn+1, Res>0 ET I 133(3)
3 xν,ν > −1Γ(ν+1 )
sν+1, Res>0 ET I 137(1)
4 xn−1
2Γ/parenleftbig
n+1
2/parenrightbig
sn+1
2, Res>0 ET I 135(17)
5 x−1/2(x+a)−1, |arga|<π πa−1/2easerfc/parenleftBig
a1/2s1/2/parenrightBig
,
Res≥0 ET I 136(25)
6/braceleftBigg
xfor 0<x< 1
1f o r x>11−e−s
s2, Res>0 ET I 142(14)
7 e−ax 1
s+a, Res>−Rea ET I 143(1)
8 xe−ax 1
(s+a)2, Res>−Rea ET I 144(2)
9ae−ax−e−bx
b−a(s+a)−1(s+b)−1,
Res>{−Rea,−Reb} AS 1022(29.3.12)
c o n t i n u e do nn e x tp a g e
Table of Laplace transform pairs 1109
continued from previous page
f(x) F(s)
9b11 αe−ax+βe−bx+γe−cx
(a−b)(b−c)(c−a)
a,b,c distinct , α=c−b,
β=a−c,γ=b−a(s+a)−1(s+b)−1(s+c)−1,
Res>{−Rea,−Reb,−Rec}
1011ae−ax−be−bx
b−as(s+a)−1(s+b)−1,
Res>{−Rea,−Reb} AS 1022(29.3.13)
11eax−1
as−1(s−a)−1, Res>Rea
12eax−ax−1
a2s−2(s−a)−1, Res>Rea
13/parenleftbig
eax−1
2a2x2−ax−1/parenrightbig
a3s−3(s−a)−1, Res>Rea
14 (1 +ax)eax s
(s−a)2, Res>Rea
151+(ax−1)eax
a2s−1(s−a)−2, Res>Rea
162+ax+(ax−2)eax
a3s−2(s−a)−2, Res>Rea
17 xneax,n =0,1,2,... n!(s−a)−(n+1), Res>Rea
18/parenleftbig
x+1
2ax2/parenrightbig
eax s
(s−a)3, Res>Rea
19/parenleftbig
1+2ax+1
2a2x2/parenrightbig
eax s2
(s−a)3, Res>Rea
201
6x3eax(s−a)−4, Res>Rea
21/parenleftbig1
2x2+1
6ax3/parenrightbig
eax s
(s−a)4, Res>Rea
22/parenleftbig
x+ax2+1
6a2x3/parenrightbig
eaxs2(s−a)−4, Res>Rea
c o n t i n u e do nn e x tp a g e
1110 Integral Transforms
continued from previous page
f(x) F(s)
23/parenleftbig
1+3ax+3
2a2x2+1
6a3x3/parenrightbig
eaxs3(s−a)−4, Res>Rea
24aeax−bebx
a−bs(s−a)−1(s−b)−1, Res>{Rea,Reb}
25/parenleftbig1
aeax−1
bebx+1
b−1
a/parenrightbig
a−bs−1(s−a)−1(s−b)−1, Res>{Rea,Reb}
26 xν−1e−ax, Reν>0Γ(ν)(s+a)−ν,Res>−Rea ET I 144(3)
27 xe−x2/(4a), Rea>02a−2π1/2a3/2seas2erfc/parenleftBig
sa1/2/parenrightBig
ET I 146(22)
28 exp (−aex), Rea>0asΓ(−s, a) ET I 147(37)
298x1/2e−a/(4x), Rea≥01
2π1/2s−3/2/parenleftBig
1+a1/2s1/2/parenrightBig
exp/bracketleftBig
(−as)1/2/bracketrightBig
,
Res>0 ET I 146(26)
308x−1/2e−a/(4x), Rea≥0π1/2s−1/2exp/bracketleftBig
(−as)1/2/bracketrightBig
,
Res>0 ET I 146(27)
318x−3/2e−a/(4x), Rea>02π1/2a−1/2exp/bracketleftBig
(−as)1/2/bracketrightBig
,
Res≥0 ET I 146(28)
32 sin(ax) a/parenleftbig
s2+a2/parenrightbig−1, Res>|Ima| ET I 150(1)
33 cos(ax) s/parenleftbig
s2+a2/parenrightbig−1, Res>|Ima| ET I 154(3)
34 |sin(ax)|,a > 0a/parenleftbig
s2+a2/parenrightbig−1coth/parenleftBigπs
2a/parenrightBig
,
Res>0 ET I 150(2)
c o n t i n u e do nn e x tp a g e
Table of Laplace transform pairs 1111
continued from previous page
f(x) F(s)
3511|cos(ax)|,a > 0/parenleftbig
s2+a2/parenrightbig−1/bracketleftBig
s+acosech/parenleftBigπs
2a/parenrightBig/bracketrightBig
,
Res>0 ET I 155(44)
361−cos(ax)
a2s−1/parenleftbig
s2+a2/parenrightbig−1,
Res>|Ima| AS 1022(29.3.19)
37ax−sin(ax)
a3s−2/parenleftbig
s2+a2/parenrightbig−1,
Res>|Ima| AS 1022(29.3.20)
38sin(ax)−axcos(ax)
2a3/parenleftbig
s2+a2/parenrightbig−2,
Res>|Ima| AS 1022(29.3.21)
39xsin(ax)
2as/parenleftbig
s2+a2/parenrightbig−2,Res>|Ima| ET I 152(14)
40sin(ax)+axcos(ax)
2as2/parenleftbig
s2+a2/parenrightbig−2,
Res>|Ima| AS 1023(29.3.23)
41 xcos(ax)/parenleftbig
s2−a2/parenrightbig/parenleftbig
s2+a2/parenrightbig−2,
Res>|Ima| ET I 157(57)
42cos(ax)−cos(bx)
b2−a2s/parenleftbig
s2+a2/parenrightbig−1/parenleftbig
s2+b2/parenrightbig−1,
Res>{|Ima|,|Imb|} AS 1023(29.3.25)
43/bracketleftbig1
2a2x2−1+c o s ( ax)/bracketrightbig
a4s−3/parenleftbig
s2+a2/parenrightbig−1, Res>|Ima|
44/bracketleftbig
1−cos(ax)−1
2axsin(ax)/bracketrightbig
a4s−1/parenleftbig
s2+a2/parenrightbig−2, Res>|Ima|
45/bracketleftbig1
bsin(bx)−1
asin(ax)/bracketrightbig
a2−b2/parenleftbig
s2+a2/parenrightbig−1/parenleftbig
s2+b2/parenrightbig−1,
Res>{|Ima|,|Imb|}
4611/bracketleftbig
1−cos(ax)+1
2axsin(ax)/bracketrightbig
a2s−1/parenleftbig
s2+a2/parenrightbig−2/parenleftbig
2s2+a2/parenrightbig
, Res>|Ima|
c o n t i n u e do nn e x tp a g e
1112 Integral Transforms
continued from previous page
f(x) F(s)
47asin(ax)−bsin(bx)
a2−b2s2/parenleftbig
s2+a2/parenrightbig−1/parenleftbig
s2+b2/parenrightbig−1,
Res>{|Ima|,|Imb|}
48 sin(a+bx) (ssina+bcosa)/parenleftbig
s2+b2/parenrightbig−1, Res>|Imb|
49 cos(a+bx) (scosa−bsina)/parenleftbig
s2+b2/parenrightbig−1, Res>|Imb|
50/bracketleftbig1
asinh(ax)−1
bsin(bx)/bracketrightbig
a2+b2/parenleftbig
s2−a2/parenrightbig−1/parenleftbig
s2+b2/parenrightbig−1,
Res>{|Rea|,|Imb|}
51cosh(ax)−cos(bx)
a2+b2s/parenleftbig
s2−a2/parenrightbig−1/parenleftbig
s2+b2/parenrightbig−1,
Res>{|Rea|,|Imb|}
52asinh(ax)+bsin(bx)
a2+b2s2/parenleftbig
s2−a2/parenrightbig−1/parenleftbig
s2+b2/parenrightbig−1,
Res>{|Rea|,|Imb|}
53 sin(ax)sin(bx) 2abs/bracketleftbig
s2+(a−b)2/bracketrightbig−1/bracketleftbig
s2+(a+b)2/bracketrightbig−1,
Res>{|Ima|,|Imb|}
54 cos(ax)cos(bx) s/parenleftbig
s2+a2+b2/parenrightbig/bracketleftbig
s2+(a−b)2/bracketrightbig−1/bracketleftbig
s2+(a+b)2/bracketrightbig−1,
Res>{|Ima|,|Imb|}
55 sin(ax)cos(bx) a/parenleftbig
s2+a2−b2/parenrightbig/bracketleftbig
s2+(a−b)2/bracketrightbig−1/bracketleftbig
s2+(a+b)2/bracketrightbig−1,
Res>{|Ima|,|Imb|}
56 sin2(ax) 2a2s−1/parenleftbig
s2+4a2/parenrightbig−1, Res>|Ima|
57 cos2(ax)/parenleftbig
s2+2a2/parenrightbig
s−1/parenleftbig
s2+4a2/parenrightbig−1, Res>|Ima|
58 sin(ax)cos(ax) a/parenleftbig
s2+4a2/parenrightbig−1, Res>|Ima|
59 e−axsin(bx) b/bracketleftbig
(s+a)2+b2/bracketrightbig−1,Res>{−Rea,|Imb|}
c o n t i n u e do nn e x tp a g e
Table of Laplace transform pairs 1113
continued from previous page
f(x) F(s)
60 e−axcos(bx) (s+a)/bracketleftbig
(s+a)2+b2/bracketrightbig−1,
Res>{−Rea,|Imb|}
61 x−1sin(ax) arctan( a/s), Res>|Ima| ET I 152(16)
62 x−1[1−cos(ax)]1
2ln/parenleftbig
1+a2/s2/parenrightbig
,
Res>|Ima| ET I 157(59)
63 sinh(ax) a/parenleftbig
s2−a2/parenrightbig−1, Res>|Rea| ET I 162(1)
64 cosh(ax) s/parenleftbig
s2−a2/parenrightbig−1, Res>|Rea| ET I 162(2)
65 xν−1sinh(ax), Reν>−11
2Γ(ν)/bracketleftbig
(s−a)−ν−(s+a)−ν/bracketrightbig
,
Res>|Rea| ET I 164(18)
66 xν−1cosh(ax), Reν>01
2Γ(ν)/bracketleftbig
(s−a)−ν+(s+a)−ν/bracketrightbig
,
Res>|Rea| ET I 164(19)
67 xsinh(ax) 2as/parenleftbig
s2−a2/parenrightbig−2, Res>|Rea|
68 xcosh(ax)/parenleftbig
s2+a2/parenrightbig/parenleftbig
s2−a2/parenrightbig−2, Res>|Rea|
69 sinh(ax)−sin(ax) 2a3/parenleftbig
s4−a4/parenrightbig−1,
Res>{|Rea|,|Ima|} AS 1023(29.3.31)
70 cosh(ax)−cos(ax) 2a2s/parenleftbig
s4−a4/parenrightbig−1,
Res>{|Rea|,|Ima|} AS 1023(29.3.32)
71 sinh(ax)+axcosh(ax) 2as2/parenleftbig
a2−s2/parenrightbig−2, Res>|Rea|
72 axcosh(ax)−sinh(ax) 2a3/parenleftbig
a2−s2/parenrightbig−2, Res>|Rea|
c o n t i n u e do nn e x tp a g e
1114 Integral Transforms
continued from previous page
f(x) F(s)
73 xsinh(ax)−cosh(ax) s/parenleftbig
a2+2a−s2/parenrightbig/parenleftbig
a2−s2/parenrightbig−2, Res>|Rea|
74/bracketleftbig1
asinh(ax)−1
bsinh(bx)/bracketrightbig
a2−b2/parenleftbig
a2−s2/parenrightbig−1/parenleftbig
b2−s2/parenrightbig−1,
Res>{|Rea|,|Reb|}
75cosh(ax)−cosh(bx)
a2−b2s/parenleftbig
s2−a2/parenrightbig−1/parenleftbig
s2−b2/parenrightbig−1,
Res>{|Rea|,|Reb|}
76asinh(ax)−bsinh(bx)
a2−b2s2/parenleftbig
s2−a2/parenrightbig−1/parenleftbig
s2−b2/parenrightbig−1,
Res>{|Rea|,|Reb|}
77 sinh(a+bx) (bcosha+ssinha)/parenleftbig
s2−b2/parenrightbig−1,Res>|Reb|
78 cosh(a+bx) (scosha+bsinha)/parenleftbig
s2−b2/parenrightbig−1,Res>|Reb|
79 sinh(ax)sin h( bx) 2abs/bracketleftbig
s2−(a+b)2/bracketrightbig−1/bracketleftbig
s2−(a−b)2/bracketrightbig−1,
Res>{|Rea|,|Reb|}
808cosh(ax)cosh( bx) s/parenleftbig
s2−a2−b2/parenrightbig/bracketleftbig
s2−(a+b)2/bracketrightbig−1/bracketleftbig
s2−(a−b)2/bracketrightbig−1,
Res>{|Rea|,|Reb|}
81 sinh(ax)cosh( bx) a/parenleftbig
s2−a2+b2/parenrightbig/bracketleftbig
s2−(a+b)2/bracketrightbig−1/bracketleftbig
s2−(a−b)2/bracketrightbig−1,
Res>{|Rea|,|Reb|}
82 sinh2(ax) 2a2s−1/parenleftbig
s2−4a2/parenrightbig−1, Res>|Rea|
83 cosh2(ax)/parenleftbig
s2−2a2/parenrightbig
s−1/parenleftbig
s2−4a2/parenrightbig−1, Res>|Rea|
84 sinh(ax)cosh( ax) a/parenleftbig
s2−4a2/parenrightbig−1, Res>|Rea|
85cosh(ax)−1
a2s−1/parenleftbig
s2−a2/parenrightbig−1, Res>|Rea|
c o n t i n u e do nn e x tp a g e
Table of Laplace transform pairs 1115
continued from previous page
f(x) F(s)
86sinh(ax)−ax
a3s−2/parenleftbig
s2−a2/parenrightbig−1, Res>|Rea|
87/bracketleftbig
cosh(ax)−1
2a2x2−1/bracketrightbig
a4s−3/parenleftbig
s2−a2/parenrightbig−1, Res>|Rea|
88/bracketleftbig
1−cosh(ax)+1
2axsinh(ax)/bracketrightbig
a4s−1/parenleftbig
s2−a2/parenrightbig−2, Res>|Rea|
89 x1/2sinh(ax)/parenleftBig
π1/2/4/parenrightBig/bracketleftBig
(s−a)3/2−(s+a)3/2/bracketrightBig
,
Res>|Rea|
90 lnx −s−1ln (Cs), Res>0 ET I 148(1)
91 ln(1 + ax), |arga|<π s−1es/aEi(−s/a), Res>0 ET I 148(4)
92 x−1/2lnx −(π/s)1/2ln (4Cs),Res>0 ET I 148(9)
93 H(x−a)=/braceleftBigg
0f o r x<a
1f o r x>a
(Heaviside step function)s−1e−as,a ≥0
94 δ(x) (Dirac delta function) 1
95 δ(x−a) e−as,a ≥0
96 δ/prime(x−a) se−as,a ≥0
97 Si(x)≡/integraldisplayx
0sinξ
ξdξ≡1
2π+s i (x)s−1arccot s, Res>0 ET I 177(17)
98 Ci(x)≡ci(x)≡−/integraldisplay∞
xcosξ
ξdξ −1
2s−1ln/parenleftbig
1+s2/parenrightbig
,Res>0 ET I 178(19)
998erf/parenleftBigx
2a/parenrightBig
s−1ea2s2erfc(as),
Res>0,|arga|<π /4 ET I 176(2)
c o n t i n u e do nn e x tp a g e
1116 Integral Transforms
continued from previous page
f(x) F(s)
100 erf/parenleftbig
a√x/parenrightbig
as−1/parenleftbig
s+a2/parenrightbig−1/2,
Res>/braceleftbig
0,−Rea2/bracerightbig
ET I 176(4)
101 erfc/parenleftbig
a√x/parenrightbig
s−1/parenleftbig
s+a2/parenrightbig−1
2/bracketleftBig/parenleftbig
s+a2/parenrightbig1/2−a/bracketrightBig
,
Res>0 ET I 177(9)
1028erfc/parenleftbigga√x/parenrightbigg
s−1e−2a√s,
Res>0,Rea>0 ET I 177(11)
1038Jν(ax), Reν>−1a−ν/parenleftBig/radicalbig
s2+a2−s/parenrightBigν/parenleftbig
s2+a2/parenrightbig−1/2,
Res>|Ima|, ET I 182(1)
104 xJν(ax), Reν>−2aν/bracketleftBig
s+ν/parenleftbig
s2+a2/parenrightbig1/2/bracketrightBig/bracketleftBig
s+/parenleftbig
s2+a2/parenrightbig1/2/bracketrightBig−ν
×/parenleftbig
s2+a2/parenrightbig−3/2,
Res>|Ima|, ET I 182(2)
105Jν(ax)
xaνν−1/bracketleftBig
s+/parenleftbig
s2+a2/parenrightbig1/2/bracketrightBig−ν
,
Res≥|Ima| ET I 182(5)
106 xnJn(ax) 1·3·5···(2n−1)an/parenleftbig
s2+a2/parenrightbig−(n+1
2),
Res>|Ima| ET I 182(4)
107 xνJν(ax), Reν>−1
22νπ−1/2Γ/parenleftbig
ν+1
2/parenrightbig
aν/parenleftbig
s2+a2/parenrightbig−(ν+1
2),
Res>|Ima|, ET I 182(7)
108 xν+1Jν(ax), Reν>−12ν+1π−1/2Γ/parenleftbig
ν+3
2/parenrightbig
aνs/parenleftbig
s2+a2/parenrightbig−(ν+3
2),
Res>|Ima| ET I 182(8)
1098Iν(ax), Reν>−1a−ν/bracketleftBig
s−/radicalbig
s2−a2/bracketrightBigν/parenleftbig
s2−a2/parenrightbig−1/2,
Res>|Rea| ET I 195(1)
c o n t i n u e do nn e x tp a g e
Fourier transform 1117
continued from previous page
f(x) F(s)
110 xνIν(ax), Reν>−1
22νπ−1/2Γ/parenleftbig
ν+1
2/parenrightbig
aν/parenleftbig
s2−a2/parenrightbig−(ν+1
2),
Res>|Rea| ET I 195(6)
111 xν+1Iν(ax), Reν>−12ν+1π−1/2Γ/parenleftbig
ν+3
2/parenrightbig
aνs/parenleftbig
s2−a2/parenrightbig−(ν+3
2),
Res>|Rea| ET I 196(7)
112 x−1Iν(ax), Reν>0ν−1aν/bracketleftBig
s+/parenleftbig
s2−a2/parenrightbig1/2/bracketrightBig−ν
,
Res>|Rea| ET I 195(4)
113 sin/parenleftBig
2a1/2x1/2/parenrightBig
(πa)1/2s−3/2e−a/s,Res>0 ET I 153(32)
114 x−1/2cos/parenleftBig
2a1/2x1/2/parenrightBig
π1/2s−1/2e−a/s, Res>0 ET I 158(67)
115 x−1e−axI1(ax)/bracketleftBig
(s+2a)1/2−s1/2/bracketrightBig/bracketleftBig
(s+2a)1/2+s1/2/bracketrightBig−1
,
Res>|Rea| AS 1024(29.3.52)
116Jk(ax)
xk−1a−k/bracketleftBig/parenleftbig
s2+a2/parenrightbig1/2−s/bracketrightBigk
,
Res>|Ima|,k>−1 AS 1025(29.3.58)
117/parenleftBigx
2a/parenrightBigk−1
2Jk−1
2(ax) Γ(k)π−1/2/parenleftbig
s2+a2/parenrightbigk,
Res>|Ima|,k > 0 AS 1024(29.3.57)
118 J0(ax)−axJ1(ax) s2/parenleftbig
s2+a2/parenrightbig−3/2, Res>|Ima|
119 I0(ax)+axI1(ax) s2/parenleftbig
s2−a2/parenrightbig−3/2, Res>|Ima|
17.21 Fourier transform
TheFourier transform , also called the exponential orcomplex Fourier transform , of the function
f(x), denoted by F(ξ), is defined by the integral
F(ξ)=1√
2π/integraldisplay∞
−∞f(x)eiξxdx.
The functions f(x)a n d F(ξ) are called a Fourier transform pair , and knowledge of either one enables
the other to be recovered. Setting F(ξ)=F[f(x);ξ],to emphasize the nature of the transform, we have
1118 Integral Transforms
the symbolic inverse result f(x)=F−1[F(ξ);x].The inversion of the Fourier transform is accomplished
by means of the inversion integral
f(x)=1√
2π/integraldisplay∞
−∞F(ξ)e−iξxdξ.
17.22 Basic properties of the Fourier transform
1. For aandbarbitrary constants,
F[af(x)+bg(x)] =aF(ξ)+bG(ξ) (linearity)
2. If n>0 is an integer, and lim
|x|→∞f(r)(x)=0f o r r=0,1,...,n −1 with f(0)(x)≡f(x), then
F/bracketleftBig
f(n)(x);ξ/bracketrightBig
=(−iξ)nF(ξ) (transform of a derivative) SN 27
3. The Fourier convolution f∗gof two functions f(x)a n d g(x) is defined by the integral
f∗g(x)=1√
2π/integraldisplay∞
−∞f(x−ξ)g(ξ)dξ,
and it has the property f∗g=g∗f,a n d f∗(g∗h)=(f∗g)∗h. In terms of the convolution
operation,
F[f∗g(x);ξ]=F(ξ)G(ξ) (convolution [Faltung] theorem). SN 24
17.23 Table of Fourier transform pairs
f(x) F(ξ)
1 1 (2π)1/2δ(ξ) SU 496
271
x(π/2)1/2isignξ SU 50
3 δ(x) (2π)−1/2SU 496
48δ(ax+b),a , b ∈R,a/negationslash=0 (2π)−1/2eibξ/aSU 517
5/braceleftBigg
1|x|<a
0|x|>a,a > 0(2/π)1/2ξ−1sin(aξ)
68H(x)=/braceleftBigg
0x<0
1x>0−1
iξ√
2π+/radicalbiggπ
2δ(ξ) SN 523
c o n t i n u e do nn e x tp a g e
Table of Fourier transform pairs 1119
continued from previous page
f(x) F(ξ)
71
|x|a, 0<Rea<1(2/π)1/2Γ(1−a)sin/parenleftbig1
2aπ/parenrightbig
|ξ|1−aSN 523
8 eiax,a ∈R(2π)1/2δ(ξ+a) SU 50
9 e−a|x|,a > 0a(2/π)1/2
a2+ξ2SU 50
107xe−a|x|,a > 02aiξ(2/π)1/2
(a2+ξ2)2,ξ > 0 SU 50
11 |x|e−a|x|,a > 0(2/π)1/2/parenleftbig
a2−ξ2/parenrightbig
(a2+ξ2)2SU 50
12e−a|x|
|x|1/2,a > 0/bracketleftBig
a+/parenleftbig
a2+ξ2/parenrightbig1/2/bracketrightBig1/2
x(a2+ξ2)1/2SN 523
13 e−a2x2,a > 0/parenleftBig
a√
2/parenrightBig−1
e−ξ2/4a2SU 51
141
a2+x2, Rea>0(π/2)1/2e−a|ξ|
aSU 51
157x
a2+x2, Rea>0isignξ(π/2)1/2e−a|ξ|
169sin/parenleftbig
ax2/parenrightbig 1
(2a)1/2cos/parenleftbiggξ2
4a+π
4/parenrightbigg
SN 523
17 cos/parenleftbig
ax2/parenrightbig 1
(2a)1/2cos/parenleftbiggξ2
4a−π
4/parenrightbigg
SN 523
18 e−a|x|cos(bx),a > 0,b > 0a(2π)−1/2/bracketleftbigg1
a2+(b+ξ)2+1
a2+(b−ξ)2/bracketrightbigg
19 e−1
2ax2sin(bx),a > 0,b > 01
2ia−1/2/braceleftbigg
exp/bracketleftbigg
−1
2(ξ−b)2
a/bracketrightbigg
−exp/bracketleftbigg
−1
2(ξ+b)2
a/bracketrightbigg/bracerightbigg
209sinh(ax)
sinh(bx), |a|<|b|(π/2)1/2sin(πa/b)
b[cosh ( πξ/b)+c o s ( πa/b)]SU 123
219cosh(ax)
sinh(bx), |a|<|b|i(π/2)1/2sinh (πξ/b)
b[cosh ( πξ/b)+c o s ( πa/b)]SU 123
c o n t i n u e do nn e x tp a g e
1120 Integral Transforms
continued from previous page
f(x) F(ξ)
22sin(ax)
x/braceleftBigg
(π/2)1/2|ξ|<a ,
0 |ξ|>aSN 523
2311x
sinhx/parenleftbig
2π3/parenrightbig1/2eπξ
(1 +eπξ)2SU 123
247xnsignx, n =1,2,... (2/π)1/2(−iξ)−(1+n)n! SU 506
257|x|ν,
−1<ν< 0,but not integral(2/π)1/2Γ(ν+1 )|ξ|−ν−1cos [π(ν+1 )/2]
SU506
267|x|νsignx,
−1<ν< 0,but not integralisignξ(2/π)1/2sin [(π/2)(ν+1 ) ]Γ ( ν+1 )
|ξ|ν+1
SU 506
27 e−axln/vextendsingle/vextendsingle1−e−x/vextendsingle/vextendsingle,
−1<Rea<0/parenleftBigπ
2/parenrightBig1/2cot(πa−iξπ)
a−iξET I 121(26)
28 e−axln/parenleftbig
1+e−x/parenrightbig
,
−1<Rea<0/parenleftBigπ
2/parenrightBig1/2csc (πa−iξπ)
a−iξET I 121 (27)
In deriving results for the preceding table from ET I, account has been taken of the fact that the normal-
ization factor 1 /(2π)1/2employed in our definition of Fhas not been used in those tables, and that there
is a difference of sign between the exponents used in the definitions of the exponential Fourier transform.
17.24 Table of Fourier transform pairs for spherically symmetric functions
f(||r||)=1
(2π)3/2/integraldisplay/integraldisplay/integraldisplay
E(||k||)eik·rdkE(||k||)=1
(2π)3/2/integraldisplay/integraldisplay/integraldisplay
f(||r||)e−ik·rdr
1 f(r)=/radicalbigg
2
π1
r/integraldisplay∞
0E(k)sin(kr)kd k E(k)=/radicalbigg
2
π1
k/integraldisplay∞
0f(r)sin(kr)rd r
2 e−ar/radicalbigg
2
π2a
(a2+k2)2
311e−ar
r/radicalbigg
2
π1
(a2+k2)2
4111 (2π)3/2δ(k)
Basic properties of the Fourier sine and cosine transforms 1121
17.31 Fourier sine and cosine transforms
TheFourier sine andcosine transforms of the function f(x), denoted by Fs(ξ)a n d Fc(ξ), respectively,
are defined by the integrals
Fs(ξ)=/radicalbigg
2
π/integraldisplay∞
0f(x)sin(ξx)dxand Fc(ξ)=/radicalbigg
2
π/integraldisplay∞
0f(x)cos(ξx)dx.
The functions f(x)a n d Fs(ξ) are called a Fourier sine transform pair , and the functions f(x)a n d
Fc(ξ)aFourier cosine transform pair , and knowledge of either Fs(ξ)o rFc(ξ) enables f(x)t ob e
recovered.
Setting
Fs(ξ)=Fs[f(x);ξ]a n d Fc(ξ)=Fc[f(x);ξ],
to emphasize the nature of the transforms, we have the symbolic inverses
f(x)=Fs−1[Fs(ξ);x]a n d f(x)=Fc−1[Fc(ξ);x].
The inversion of the Fourier sine transform is accomplished by means of the inversion integral
f(x)=/radicalbigg
2
π/integraldisplay∞
0Fs(ξ)sin(ξx)dξ [x≥0]
and the inversion of the Fourier cosine transform is accomplished by means of the inversion integral
f(x)=/radicalbigg
2
π/integraldisplay∞
0Fc(ξ)cos(ξx)dξ [x≥0]. SN 17
17.32 Basic properties of the Fourier sine and cosine transforms
1. For aandbarbitrary constants,
Fs[af(x)+bg(x)] =aFs(ξ)+bGs(ξ)
and
Fc[af(x)+bg(x)] =aFc(ξ)+bGc(ξ) (linearity)
2. If lim
x→∞f(r−1)(x) = 0 and lim
x→∞/radicalBig
2
πf(r−1)(x)=ar−1, then denoting the Fourier sine and cosine
transforms of f(r)(x)b yFs(r)andFc(r), respectively,
(i) Fc(r)(ξ)=−ar−1+ξFs(r−1).
(ii) Fs(r)(ξ)=−ξFc(r−1)(ξ),
(iii) Fc(2r)(ξ)=−r−1/summationdisplay
n=0(−1)na2r−2n−1ξ2n+(−1)rξ2nFc(ξ),
(iv) Fc(2r+1)(ξ)=−r−1/summationdisplay/prime
n=0(−1)na2r−2nξ2n+(−1)rξ2r+1Fs(ξ),
(v) Fs(r)(ξ)=ξar−2−ξ2Fs(r−2)(ξ),
(vi)6Fs(2r)(ξ)=−r/summationdisplay
n=1(−1)nξ2n−1a2r−2n+(−1)rξ2rFs(ξ),
(vii) Fs(2r+1)(ξ)=−r/summationdisplay/prime
n=1(−1)nξ2n−1a2r−2n+1+(−1)r+1ξ2r+1Fc(ξ). SN 28
1122 Integral Transforms
3. (i)/integraldisplay∞
0Fs(ξ)Gs(ξ)cos(ξx)dξ=1
2/integraldisplay∞
0g(s)[f(s+x)+f(s−x)]ds,
(ii)/integraldisplay∞
0Fc(ξ)Gc(ξ)cos(ξx)dξ=1
2/integraldisplay∞
0g(s)[f(s+x)+f(|x−s|)]ds
(convolution (Faltung) theorem) SN 24
4. (i) If Fs(ξ) is the Fourier sine transform of f(x), then the Fourier sine transform of Fs(x)i s
f(ξ).
(ii) If Fc(ξ) is the Fourier cosine transform of f(x), then the Fourier cosine transform of Fc(x)
isf(ξ).
(iii) If f(x) is an odd function in ( −∞,∞), then the Fourier sine transform of f(x)i n( 0 ,∞)i s
−iF(ξ).
(iv) If f(x) is an even function in ( −∞,∞), then the Fourier cosine transform of f(x)i n( 0 ,∞)
isF(ξ).
(v) The Fourier sine transform of f(x/a)i saFs(aξ).
(vi) The Fourier cosine transform of f(x/a)i saFc(aξ).
(vii) Fs[f(x);ξ]=Fs(|ξ|)signξ SU 45
17.33 Table of Fourier sine transforms
f(x) Fs(ξ)( ξ>0)
1 x−1(π/2)1/2,ξ > 0 ET I 64(3)
2 x−ν, 0<Reν<2(2/π)1/2ξν−1Γ(1−ν)cos(νπ/2),
ξ>0 ET I 68(1)
3 x−1/2ξ−1/2,ξ > 0 ET I 64(6)
4 x−3/22ξ1/2,ξ > 0 ET I 64(9)
5/braceleftBigg
10<x<a
0x>a(2/π)1/2ξ−1[1−cos(aξ)],ξ > 0 ET I 63(1)
6/braceleftBigg
x−10<x<a
0 x>a(2/π)1/2Si(aξ),ξ > 0 ET I 64(4)
71
a−x,a > 0(2/π)1/2/braceleftbig
sin(aξ)Ci(aξ)−cos(aξ)/bracketleftbig1
2π+S i (aξ)/bracketrightbig/bracerightbig
,
ξ>0 ET I 64(11)
c o n t i n u e do nn e x tp a g e
Table of Fourier sine transforms 1123
continued from previous page
f(x) Fs(ξ)( ξ>0)
871
x2+a2,a > 0(2π)−1/2a−1/bracketleftbig
e−aξEi(aξ)−eaξEi(−aξ)/bracketrightbig
,
ξ>0 ET I 65(14)
9 x/parenleftbig
x2+a2/parenrightbig−3/2, Rea>0(2/π)1/2ξK0(aξ),ξ > 0 ET I 66(27)
10 x−1/2/parenleftbig
x2+a2/parenrightbig−1/2,Rea>0ξ1/2I1
4/parenleftbig1
2aξ/parenrightbig
K1
4/parenleftbig1
2aξ/parenrightbig
,ξ > 0 ET I 66(28)
117x/parenleftbig
x2+a2/parenrightbig−ν−3
2,
Reν>−1,Rea>0ξν+1
√
2(2a)νΓ/parenleftbig
ν+3
2/parenrightbigKν(aξ),
12x
a2+x2, Rea>0/parenleftBigπ
2/parenrightBig1/2
e−aξ,ξ > 0 ET I 65(15)
13x
(a2+x2)2/radicalbig
π/8a−1ξe−aξ,ξ > 0 ET I 67(35)
14 x−1/parenleftbig
x2+a2/parenrightbig−1, Rea>0/radicalbig
π/2
a2/parenleftbig
1−e−aξ/parenrightbig
,ξ > 0 ET I 65(20)
15 x−1e−ax, Rea>0(2/π)1/2tan−1/parenleftbiggξ
a/parenrightbigg
,ξ > 0 ET I 72(2)
16 xν−1e−ax,
Reν>−1,Rea>0(2/π)1/2Γ(ν)/parenleftbig
a2+ξ2/parenrightbig−ν/2sin/bracketleftbigg
νtan−1/parenleftbiggξ
a/parenrightbigg/bracketrightbigg
,
ξ>0 ET I 72(7)
17 e−ax, Rea>0/radicalbig
2/πξ
a2+ξ2,ξ > 0 ET I 72(1)
18 xe−ax, Rea>0(2/π)1/22aξ
(a2+ξ2)2,ξ > 0 ET I 72(3)
19 xe−ax2, |arga|<π /2(2a)−3/2ξexp/parenleftbigg−ξ2
4a/parenrightbigg
,ξ > 0 ET I 73(19)
20sinax
x,a > 01
(2π)1/2ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ+a
ξ−a/vextendsingle/vextendsingle/vextendsingle/vextendsingle,ξ > 0
ET I 78(1)
21sinax
x2,a > 0/braceleftBigg
ξ/parenleftbigπ
2/parenrightbig1/20<ξ<a
a/parenleftbigπ
2/parenrightbig1/2a<ξ< ∞,ξ > 0 ET I 78(2)
c o n t i n u e do nn e x tp a g e
1124 Integral Transforms
continued from previous page
f(x) Fs(ξ)( ξ>0)
22 sin/parenleftbigga2
x/parenrightbigg
,a > 0a/parenleftBigπ
2/parenrightBig1/2
ξ−1/2J1/parenleftBig
2aξ1
2/parenrightBig
,
ξ>0 ET I 83(6)
23 x−1sin/parenleftbigga2
x/parenrightbigg
,a > 0/parenleftBigπ
2/parenrightBig1/2
Y0/parenleftBig
2aξ1/2/parenrightBig
+/parenleftbigg2
π/parenrightbigg1/2
K0/parenleftBig
2aξ1/2/parenrightBig
ET I 83(7)
24 x−2sin/parenleftbigga2
x/parenrightbigg
,a > 0/parenleftBigπ
2/parenrightBig1/2
a−1ξ1/2J1/parenleftBig
2aξ1/2/parenrightBig
,
ξ>0 ET I 83(8)
2510cosech( ax), Rea>0(π/2)1/2a−1tanh/parenleftbig1
2πa−1ξ/parenrightbig
,
ξ>0 ET I 88(2)
26 coth/parenleftbigg1
2ax/parenrightbigg
−1, Rea>0(2π)1/2a−1coth/parenleftbig
πa−1ξ/parenrightbig
−ξ,
ξ>0 ET I 88(3)
27/parenleftbig
1−x2/parenrightbig−1sin(πx)/braceleftBigg
(2/π)1/2sinξ0≤ξ≤π
0 π<ξET I 78(4)
28 e−ax2sin(bx), Rea>0(2a)−1/2exp/bracketleftbig
−/parenleftbig
ξ2+b2/parenrightbig
/(4a)/bracketrightbig
sinh (bξ/2a),
ξ>0 ET I 78(7)
29sin2(ax)
x,a > 0⎧
⎪⎨
⎪⎩π1/22−3/20<ξ< 2a
π1/22−5/2ξ=2a
02 a<ξET I 78(8)
30 sin/parenleftbig
ax2/parenrightbig
,a > 0a−1/2/braceleftBig
cos/parenleftbig
ξ2/4a/parenrightbig
C/bracketleftBig
(2πa)−1/2ξ/bracketrightBig/bracerightBig
+sin/parenleftbig
ξ2/4a/parenrightbig
S/bracketleftBig
(2πa)−1/2ξ/bracketrightBig
,
ξ>0 ET I 82(1)
31 cos/parenleftbig
ax2/parenrightbig
,a > 0a−1/2/braceleftBig
sin/parenleftbig
ξ2/4a/parenrightbig
C/bracketleftBig
(2πa)−1/2ξ/bracketrightBig/bracerightBig
−cos/parenleftbig
ξ2/4a/parenrightbig
S/bracketleftBig
(2πa)−1/2ξ/bracketrightBig
,
ξ>0
c o n t i n u e do nn e x tp a g e
Table of Fourier sine transforms 1125
continued from previous page
f(x) Fs(ξ)( ξ>0)
32 arctan/parenleftBigx
a/parenrightBig
,a > 0(π/2)1/2ξ−1e−aξ,ξ > 0 ET I 87(3)
337arctan/parenleftbigg2a
x/parenrightbigg
, Rea>0(2π)−1/2e−aξsinh(aξ),ξ > 0 ET I 87(8)
34lnx
x−(π/2)1/2(C+l nξ),ξ > 0 ET I 76(2)
35 ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglex+a
x−a/vextendsingle/vextendsingle/vextendsingle/vextendsingle,a > 0
(2π)1/2ξ−1sin(aξ),ξ > 0 ET I 77(11)
367ln/parenleftbig
1+a2x2/parenrightbig
x,a > 0−(2π)1/2Ei(−ξ/a),ξ > 0 ET I 77(14)
37 J0(ax),a > 0/braceleftBigg
00 <ξ<a
(2/π)1/2/parenleftbig
ξ2−a2/parenrightbig−1/2a<ξ< ∞
ET I 99(1)
38 Jν(ax), Reν>−2,a > 0(2/π)1/2/parenleftbig
a2−ξ2/parenrightbig−1/2sin/bracketleftbigg
νsin−1/parenleftbiggξ
a/parenrightbigg/bracketrightbigg
for 0 <ξ<a
aνcos/parenleftbig1
2νπ/parenrightbig
(ξ2−a2)1/2/bracketleftBig
ξ+(ξ2−a2)1/2/bracketrightBigνfora<ξ< ∞
ET I 99(3)
39J0(ax)
x,a > 0/braceleftBigg
(2/π)1/2sin−1/parenleftBig
ξ
a/parenrightBig
0<ξ<a
(π/2)1/2a<ξ< ∞
ET I 99(4)
407/parenleftbig
x2+b2/parenrightbig−1J0(ax),
a>0,Reb>0(2/π)1/2sinh(bξ)K0(ab)/b,
0<ξ<a ET I 100(12)
41 x/parenleftbig
x2+b2/parenrightbig−1J0(ax),
a>0,Reb>0(π/2)1/2e−bξI0(ab),
a<ξ< ∞ ET I 100(13)
In deriving results for the preceding table from ET I, account has been taken of the fact that the normal-
ization factor/radicalbig
2/πemployed in our definition of Fshas not been used in those tables.
1126 Integral Transforms
17.34 Table of Fourier cosine transforms
f(x) Fc(ξ)
1 x−ν, 0<Reν<1(π/2)1/2[Γ(ν)]−1sec/parenleftbig1
2νπ/parenrightbig
ξν−1,
ξ>0 ET I 10(1)
2/braceleftBigg
10<x<a
0x>a(2/π)1/2sin(aξ)
ξ,ξ > 0 ET I 7(1)
3/braceleftBigg
00 <x<a
1/x x > a−(2/π)1/2Ci(aξ),ξ > 0 ET I 8(3)
4/braceleftBigg
x−1/20<x<a
0 x>a2ξ−1/2C(aξ),ξ > 0 ET I 8(5)
5/braceleftBigg
00 <x<a
x−1/2x>a2ξ−1/2/bracketleftbig1
2−C(aξ)/bracketrightbig
,ξ > 0 ET I 8(6)
69xν−1, 0<ν< 1(2/π)1/2Γ(ν)ξ−νcos/parenleftbig1
2νπ/parenrightbig
,
0<ν< 1 ET I 10(1)
71
x2+a2, Rea>0(π/2)1/2e−aξ
a,ξ > 0 ET I 11(7)
8111
(x2+a2)2, Rea>0(π/2)1
2(1 +aξ)e−aξ
2a3,ξ > 0 ET I 11(7)
9/parenleftbig
x2+a2/parenrightbig−ν−1
2,
Rea>0,Reν>−1
2√
2/parenleftbiggξ
2a/parenrightbiggνKν(aξ)
Γ/parenleftbig
ν+1
2/parenrightbig,ξ > 0 ET I 11(7)
10/braceleftBigg/parenleftbig
a2−x2/parenrightbigν0<x<a
0 x>a,
Reν>−12νΓ(ν+1 ) (a/ξ)ν+1
2Jν+1
2(aξ),
ξ>0 ET I 11(8)
11/braceleftBigg
00 <x<a
/parenleftbig
x2−a2/parenrightbig−ν−1
2x>a,
−1
2<Reν<1
2−2−(ν+1
2)Γ/parenleftbig1
2−ν/parenrightbig
(ξ/a)νYν(aξ),
ξ>0 ET I 11(9)
12 e−ax, Rea>0(2/π)1/2a/parenleftbig
a2+ξ2/parenrightbig−1,ξ > 0 ET I 14(1)
continued on next page
Table of Fourier cosine transforms 1127
continued from previous page
f(x) Fc(ξ)
13 xe−ax, Rea>0(2/π)1/2/parenleftbig
a2−ξ2/parenrightbig/parenleftbig
a2+ξ2/parenrightbig−2,
ξ>0 ET I 15(7)
147xν−1e−ax,
Rea>0,Reν>a(2/π)1/2Γ(ν)/parenleftbig
a2+ξ2/parenrightbig−ν/2cos/bracketleftbigg
νtan−1/parenleftbiggξ
a/parenrightbigg/bracketrightbigg
,
ξ>0 ET I 15(7)
15 x−1/2e−ax, Rea>0/parenleftbig
a2+ξ2/parenrightbig−1/2/bracketleftBig/parenleftbig
a2+ξ2/parenrightbig1/2+a/bracketrightBig1/2
,
ξ>0 ET I 14(4)
167e−a2x2, Rea>02−1/2|a|−1e−ξ2/4a2,ξ > 0 ET I 15(11)
17 x−1e−xsinx (2π)−1/2tan−1/parenleftbigg2
ξ2/parenrightbigg
,ξ > 0 ET I 19(7)
18 sin/parenleftbig
ax2/parenrightbig
,a > 01
2√a/bracketleftbigg
cos/parenleftbiggξ2
4a/parenrightbigg
−sin/parenleftbiggξ2
4a/parenrightbigg/bracketrightbigg
,
ξ>0 ET I 23(1)
19 cos/parenleftbig
ax2/parenrightbig
,a > 01
2√a/bracketleftbigg
cos/parenleftbiggξ2
4a/parenrightbigg
+s i n/parenleftbiggξ2
4a/parenrightbigg/bracketrightbigg
,
ξ>0 ET I 24(7)
20sin(ax)
x,a > 0⎧
⎪⎨
⎪⎩(π/2)1/2ξ<a
1
2(π/2)1/2ξ=a
0 ξ>aET I 18(1)
217sin2(ax)
x2,a > 0/braceleftBigg
(π/2)1/2/parenleftbig
a−1
2ξ/parenrightbig
ξ<2a
02 a<ξET I 19(8)
227e−bxsin(ax),a > 0,Reb>0(2π)−1/2/bracketleftbigga+ξ
b2+(a+ξ)2+a−ξ
b2+(a−ξ)2/bracketrightbigg
,
ξ>0 ET I 19(6)
23sin/bracketleftBig
b/parenleftbig
x2+a2/parenrightbig1/2/bracketrightBig
(x2+a2)2a>0(b/a)(π/2)1/2e−aξ,ξ > 0 ET I 26(29)
24/parenleftbig
x2+a2/parenrightbig−1/2sin/bracketleftBig
b/parenleftbig
x2+a2/parenrightbig1/2/bracketrightBig
,
a>0/braceleftBigg
(π/2)1/2J0/bracketleftBig
a/parenleftbig
b2−ξ2/parenrightbig1/2/bracketrightBig
0<ξ<b
0 b<ξ
ET I 26(30)
continued on next page
1128 Integral Transforms
continued from previous page
f(x) Fc(ξ)
251−cos(ax)
x2,a > 0/braceleftBigg
(π/2)1/2(a−ξ)ξ<a
0 a<ξET I 20(16)
26 e−ax2sin/parenleftbig
bx2/parenrightbig
,Rea>|Imb|2−1/2/parenleftbig
a2+b2/parenrightbig−1/4exp/braceleftbig
−aξ2//bracketleftbig
4/parenleftbig
a2+b2/parenrightbig/bracketrightbig/bracerightbig
×sin/bracketleftBig
1
2arctan( b/a)−1
4bξ2/parenleftbig
a2+b2/parenrightbig−1/bracketrightBig
,
ξ>0 ET I 23(5)
27 e−ax2cos/parenleftbig
bx2/parenrightbig
,Rea>|Imb|2−1/2/parenleftbig
a2+b2/parenrightbig−1/4exp/braceleftbig
−aξ2//bracketleftbig
4/parenleftbig
a2+b2/parenrightbig/bracketrightbig/bracerightbig
×cos/bracketleftBig
1
4bξ2/parenleftbig
a2+b2/parenrightbig−1−1
2arctan( b/a)/bracketrightBig
,
ξ>0 ET I 24(6)
28sinh(ax)
sinh(bx)|Rea|<Reb/parenleftBigπ
2/parenrightBig1/2 sin(πa/b)
b[cosh ( πξ/b)+c o s ( πa/b)],
ξ>0 ET I 31(14)
29cosh(ax)
cosh(bx), |Rea|<Reb(2π)1/2cos(πa/2b)c os h( πξ/2b)
b[cosh ( πξ/b)+c o s ( πa/b)],
ξ>0 ET I 31(12)
30 sech(ax), Rea>0a−1(π/2)1/2sech (πξ/2a),
ξ>0 ET I 30(1)
31/parenleftbig
x2+a2/parenrightbig
sech/parenleftBigπx
2a/parenrightBig
,Rea>02(2/π)1/2a3sech3(aξ),ξ > 0 ET I 32(19)
32 ln/parenleftbigg
1+a2
x2/parenrightbigg
, Rea>0(2π)1/2ξ−1/parenleftbig
1−e−aξ/parenrightbig
,ξ > 0 ET I 18(10)
337ln/parenleftbigga2+x2
b2+x2/parenrightbigg
,
Rea>0,Reb>0(2π)1/2/parenleftbig
e−bξ−e−aξ/parenrightbig
,ξ > 0 ET I 18(12)
34/parenleftbig
x2+b2/parenrightbig−1J0(ax),
a>0,Reb>0(π/2)1/2b−1e−bξI0(ab),
a<ξ< ∞ ET I 45(14)
continued on next page
Mellin transform 1129
continued from previous page
f(x) Fc(ξ)
35 x/parenleftbig
x2+b2/parenrightbig−1J0(ax),
a>0,Reb>0(2/π)1/2cosh(bξ)K0(ab),
0<ξ<a ET I 45(15)
In deriving results for the preceding table from ET I, account has been taken of the fact that the normal-
ization factor/radicalbig
2/πemployed in our definition of Fchas not been used in those tables.
17.35 Relationships between transforms
The following relationships exist between transforms, and they may be used to derive further transform
pairs from among the results given in Sections 17.13–17.34. The appropriate sections of the main bodyof the tables may also be used to extend the list of transform pairs.
17.351
Fourier cosine transform and Laplace transform relationship
F
c[f(x);ξ]=1√
2πL[f(x);iξ]+1√
2πL[f(x);−iξ].
17.352
Fourier sine transform and Laplace transform relationship
Fs[f(x);ξ]=i√
2πL[f(x);iξ]−i√
2πL[f(x);−iξ].
17.353
Exponential Fourier transform and Laplace transform relationship
F[f(x);ξ]=√
2πL[f(x);−iξ]+√
2πL[f(−x);iξ].
17.4110Mellin transform
TheMellin transform of the function f(x), denoted by f∗(s), is defined by the integral
f∗(s)=/integraldisplay∞
0f(x)xs−1dx.
The functions f(x)a n d f∗(s) are called a Mellin transform pair , and knowledge of either one enables
the other to be recovered.
The transform exists, provided the integral/integraldisplay∞
0|f(x)|xk−1dx
is bounded for some k>0, and then the inversion of the Mellin transform is accomplished by means of
theinversion integral
1130 Integral Transforms
f(x)=1
2πi/integraldisplayc+i∞
c−i∞f∗(s)x−sds,
where c>k.
Setting
f∗(s)=M[f(x);s]
to denote the Mellin transform, we have the symbolic expression for the inverse result
f(x)=M−1[f∗(s);x]. MS 397(6)
17.42 Basic properties of the Mellin transform
1. For aandbarbitrary constants,
M[af(x)+bg(x)] =af∗(s)+bg∗(s) (linearity)
2. If lim
x→0xs−r−1f(r)(x)=0,r=0,1,...,n −1,
(i) M/bracketleftBig
f(n)(x);s/bracketrightBig
=(−1)nΓ(s)
Γ(s−n)f∗(s−n)
(transform of a derivative) SU 267 (4.2.3)
(ii) M/bracketleftBig
xnf(n)(x);s/bracketrightBig
=(−1)nΓ(s+n)
Γ(s)f∗(s)
(transform of a derivative) SU 267 (4.2.5)
3. Denoting the nthrepeated integral of f(x)b yIn[f(x)], where
In[f(x)] =/integraldisplayx
0In−1[f(u)]du,
(i) M[In[f(x)] ;s]=(−1)nΓ(s)
Γ(n+s)f∗(s+n)
(transform of an integral) SU 269 (4.2.15)
(ii) M[I∞
n[f(x)] ;s]=Γ(s)
Γ(s+n)f∗(s+n),
where
I∞
n[f(x)] =/integraldisplay∞
xI∞
n−1[f(u)]du (transform of an integral) SU 269 (4.2.18)
4. M[f(x)g(x);s]=1
2πi/integraldisplayc+i∞
c−i∞f∗(u)g∗(s−u)du
(Mellin convolution theorem) SU 275(4.4.1)
Table of Mellin transforms 1131
17.43 Table of Mellin transforms
f(x) f∗(s)
1 e−xΓ(s), Res>0 SU 521(M13)
2 e−x21
2Γ/parenleftbig1
2s/parenrightbig
, Res>0 SU 521(M14)
3 cosx Γ(s)cos/parenleftbig1
2πs/parenrightbig
,0<Res<1 SU 521(M15)
4 sinx Γ(s)sin/parenleftbig1
2πs/parenrightbig
,0<Res<1 SU 521(M16)
51
1−xπcot(πs), 0<Res<1 SU 521(M1)
61
1+xπcosec( πs), 0<Res<1 SU 521(M2)
7 (1 +xa)−b Γ(s/a)Γ(b−s/a)
aΓ(b),0<Res<a b
SU 521(M3)
8Tn(x)H(1−x)/radicalbig
(1−x2)2−sπΓ(s)
Γ/parenleftbig1
2+1
2s+1
2n/parenrightbig
Γ/parenleftbig1
2+1
2s−1
2n/parenrightbig,
Res>0 SU 521(M4)
9Tn/parenleftbig
x−1/parenrightbig
H(1−x)/radicalbig
(1−x2)2s−2Γ/parenleftbig1
2n+1
2s/parenrightbig
Γ/parenleftbig1
2s−1
2n/parenrightbig
Γ(s),
Res>n SU 521(M5)
10 Pn(x)H(1−x)Γ/parenleftbig1
2s/parenrightbig
Γ/parenleftbig1
2s+1
2/parenrightbig
2Γ/parenleftbig1
2s−1
2n+1
2/parenrightbig
Γ/parenleftbig1
2s+1
2n+1/parenrightbig,
Res>0 SU 521(M6)
11 Pn/parenleftbig
x−1/parenrightbig
H(1−x)2s−1Γ/parenleftbig1
2s+1
2n+1
2/parenrightbig
Γ/parenleftbig1
2s−1
2n/parenrightbig
√πΓ(s+1 ),
Res>n SU 521(M7)
121+xcosφ
1−2xcosφ+x2πcos(sφ)
sin(sπ), 0<Res<1 SU 521(M11)
13xsinφ
1−2xcosφ+x2,−π<φ<ππsin(sφ)
sin(sπ), 0<Res<1 SU 521(M12)
c o n t i n u e do nn e x tp a g e
1132 Integral Transforms
continued from previous page
f(x) f∗(s)
14 e−xcosφcos (xsinφ),
1
2π<φ<1
2πΓ(s)cos(sφ), Res>0 SU 522(M17)
15 e−xsinφsin (xsinpφ),
−1
2π<φ<1
2πΓ(s)sin(sφ), Res>−1 SU 522(M18)
16 x−νJν(x),ν > −1
22s−ν−1Γ/parenleftbig1
2s/parenrightbig
Γ/parenleftbig
ν−1
2s+1/parenrightbig,0<Res<1 SU 522(M19)
17 Yν(x),ν ∈R−2s−1π−1Γ/parenleftbig1
2s+1
2ν/parenrightbig
Γ/parenleftbig1
2s−1
2ν/parenrightbig
×cos/parenleftbig1
2s−1
2ν/parenrightbig
π,
|ν|<Res<3
2SU 522(M20)
18 Kν(x),ν ∈R2s−2Γ/parenleftbig1
2s+1
2ν/parenrightbig
Γ/parenleftbig1
2s−1
2ν/parenrightbig
,
Res>ν> 0 SU 522(M21)
19 H ν(x),ν ∈R2s−1tan/parenleftbig1
2πs+1
2πν/parenrightbig
Γ/parenleftbig1
2s+1
2ν/parenrightbig
Γ/parenleftbig1
2ν−1
2s+1/parenrightbig ,
−1−ν<Res<min/parenleftbig3
2,1−ν/parenrightbig
SU 522(M22)
201
a+xn,
|arga|<π , n =1,2,3,...,πn−1cosec/parenleftBigπs
n/parenrightBig
a(s/n)−1,
0<Res<n MS 453
21/parenleftbig
1+axh/parenrightbig−ν,
h>0,|arga|<πh−1a−s/hB(s/h, ν −(s/h))
0<Res<h Reν MS 454
22/braceleftBigg/parenleftbig
1−xh/parenrightbigν−1for 0<x< 1
0f o r x>1,
h>0,Reν>0h−1B(ν,s/h) MS 454
23 ln(1 + ax), |arga|<π πs−1a−scosec( πs),−1<Res<0 MS 454
24 arctan x −1
2πs−1sec(πs/2),−1<Res<0 MS 454
c o n t i n u e do nn e x tp a g e
Table of Mellin transforms 1133
continued from previous page
f(x) f∗(s)
25 arccot x1
2πs−1sec(πs/2), 0<Res<1 MS 454
26 cosech( ax)R e a>0a−s2/parenleftbig
1−2−s/parenrightbig
Γ(s)ζ(s),Res>1 MS 454
27 sech2(ax), Rea>04a−s(1−22−s)Γ(s)2−sζ(s−1),
Res>2 MS 454
28 cosech2(ax), Rea>04a−sΓ(s)2−sζ(s−1), Res>2 MS 454
2911/parenleftbig
x2+b2/parenrightbig−1
2νJν/bracketleftBig
a/parenleftbig
x2+b2/parenrightbig1/2/bracketrightBig
21
2s−1a−1
2sb1
2s−νΓ/parenleftbig1
2s/parenrightbig
Jν−s/2(ab),
0<Res<3
2+R eν ET I 328
30⎧
⎪⎪⎨
⎪⎪⎩/parenleftbig
a2−x2/parenrightbig1
2νJν/bracketleftBig
a/parenleftbig
b2−x2/parenrightbig1/2/bracketrightBig
for 0<x<a
0f o r x>a,
Reν>−121
2s−1Γ/parenleftbig1
2s/parenrightbig
b−1
2saν+1
2sJν+1
2s(ab),
Res>0 MS 455
31⎧
⎪⎪⎨
⎪⎪⎩/parenleftbig
a2−x2/parenrightbig−1
2νJν/bracketleftBig
b/parenleftbig
a2−x2/parenrightbig1/2/bracketrightBig
for 0<x<a
0f o r x>a, 21−ν[Γ(ν)]−1a1
2s−νb−1
2νsν−1+1
2s,1
2s−ν(ab),
Res>0 MS 455
32 Kν(αx) α−s2s−2Γ/parenleftbig1
2s−1
2ν/parenrightbig
Γ/parenleftbig1
2s+1
2ν/parenrightbig
,
Res>|Reν| MS 455
33/parenleftbig
βa2+x2/parenrightbig−1
2ν
×Kν/bracketleftBig
α/parenleftbig
βa2+x2/parenrightbig1/2/bracketrightBig ,
Re (α,β)>0α−1
2s21
2s−1β1
2s−νΓ/parenleftbig1
2s/parenrightbig
Kν−1
2s(αβ),
Res>0 MS 455
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18 The z-Transform
18.1–18.3 Definition, Bilateral, and Unilateral z-Transforms
18.1 Definitions
Thez-transform converts a numerical sequence x[n] into a function of the complex variable z,a n di t
takes two different forms. The bilateral ortwo-sided z-transform , denoted here by Zb{x[n]},i su s e d
mainly in signal and image processing, while the unilateral orone-sided z-transform , denoted here
byZu{x[n]}, is used mainly in the analysis of discrete time systems and the solution of linear difference
equations.
Thebilateral z-transform ,Xb(z) of the sequence x[n]={xn}∞
n=−∞is defined as
Zb{x[n]}=∞/summationdisplay
n=−∞xnz−n=Xb(z),
and the unilateral z-transform Xu(z) of the sequence x[n]={xn}∞
n=0is defined as
Zb{x[n]}=∞/summationdisplay
n=0xnz−n=Xu(z),
where each has its own domain of convergence (DOC). The series Xb(z) is a Laurent series, and Xu(z)
is the principal part of the Laurent series for Xb(z). When xn=0f o r n<0, the two z-transforms
Xb(z)a n d Xu(z) are identical. In each case the sequence x[n] and its associated z-transform is a called
az-transform pair .
The inverse z-transformation x[n]=Z−1{X(z)}is given by
x[n]=1
2πi/integraldisplay
ΓX(z)zn−1dz,
where X(z)i se i t h e r Xb(z)o rXu(z), and Γ is a simple closed contour containing the origin and lying
entirely within the domain of convergence of X(z). In many practical situations, the z-transform is either
found by using a series expansion of X(z) in the inversion integral or, if X(z)=N(z)/D(z)w h e r e N(z)
andD(z) are polynomials in z, by means of partial fractions and the use of an appropriate table of
z-transform pairs. In order for the inverse z-transform to be unique, it is necessary to specify the domain
of convergence, as can be seen by comparison of entries 3 and 4 of Table 18.2. Table 18.1 lists generalproperties of the bilateral z-transform, and Table 18.2 lists some bilateral z-transform pairs. In what
follows, use is made of the unit integer function h(n)=/braceleftBigg
0f o r n<0
1f o r n≥0, that is, a generalization of
the Heaviside step function, and the unit integer pulse function Δ(n−k)=/braceleftBigg
1f o r n=k
0f o r n/negationslash=k,t h a ti s ,
a generalization of the delta function.
1135
1136 Definition, Bilateral, and Unilateral z-Transforms
18.2 Bilateral z-transform
Table 18.1 General properties of the bilateral z-transform Xb(n)=∞/summationdisplay
n=−∞xnz−n.
Term in sequence z-Transform Xb(z) Domain of Convergence
1αxn+βyn αXb(z)+βYb(z)Intersection of DOC’s of Xb(z)
andYb(z) with α,βconstants
2xn−N z−nXb(z)DOC of Xb(z), to which it may
be necessary to add or delete the
origin or the point at infinity
3nxn −zdXb(z)
dzDOC of Xb(z), to which it may
be necessary to add or delete theorigin and the point at infinity
4zn
0xn Xb/parenleftbiggz
z0/parenrightbigg
DOC of Xb(z) scaled by |z0|
5nzn
0xn −zdXb(z/z0)
dzDOC of Xb(z) scaled by |z0|to
which it may be necessary toadd or delete the origin and thepoint at infinity
6x−n Xb(1/z)DOC of radius 1 /R,w h e r e Ris
the radius of convergence ofDOC of X
b(z)
7nx−n −zdXb(1/z)
dzDOC of radius 1 /R,w h e r e Ris
the radius of convergence of
DOC of Xb(z)
8¯xn Xb(z) The same DOC as xn
9R e xn1
2/bracketleftbig
Xb(z)+Xb(z)/bracketrightbig
DOC contains the DOC of xn
10 Im xn1
2i/bracketleftbig
Xb(z)−Xb(z)/bracketrightbig
DOC contains the DOC of xn
11∞/summationdisplay
k=−∞xkyn−k Xb(z)Yb(z)DOC contains the intersection of
the DOCs of Xb(z)a n d Yb(z)
(convolution theorem)
12xnyn1
2πi/integraldisplay
ΓXb(ξ)Yb/parenleftbiggz
ξ/parenrightbigg
ξ−1dξDOC contains the DOCs of
Xb(z)a n d Yb(z), with Γ inside
the DOC and containing theorigin (convolution theorem)
13 Parseval formula∞/summationdisplay
n=−∞xn¯yn=1
2πi/integraldisplay
ΓXb(ξ)Yb/parenleftbiggz
ξ/parenrightbigg
ξ−1dξDOC contains the intersection of
DOCs of Xb(z)a n d Yb(z), with
Γ inside the DOC andcontaining the origin
14Initial value
theorem forx
nh(n)x0= lim
z→∞Xb(z)
Bilateral z-transform 1137
Table 18.2 Basic bilateral z-transforms
Term in sequence z-Transform Xb(z) Domain of Convergence
1Δ ( n)1 Converges for all z
2Δ ( n−N) z−nWhen N>0 convergence is for
allzexcept at the origin. When
N<0 convergence is for all z
except at ∞
3anh(n)z
z−a|z|>|a|
4anh(−n−1)z
z−a|z|<|a|
5nanh(n)az
(z−a)2|z|>a> 0
6nanh(−n−1)az
(z−a)2|z|<a , a> 0
7n2anh(n)az(z+a)
(z−a)3|z|>a> 0
8/parenleftbigg1
an+1
bn/parenrightbigg
h(n)az
az−1+bz
bz−1|z|>max/parenleftBig
1
|a|,1
|b|/parenrightBig
9anh(n−N)z/parenleftbig
1−(a/z)N/parenrightbig
z−a|z|>0
10anh(n)sinΩ nazsin Ω
z2−2azcos Ω + a2|z|>a> 0
11anh(n)cosΩ nz(z−acosΩ)
z2−2azcos Ω + a2|z|>a> 0
12eanh(n)z
z−ea|z|>e−a
13e−anh(n)sinΩ nzeasin Ω
z2e2a−2zeacos Ω + 1|z|>e−a
14e−anh(n)cosΩ nzea(zea−cosΩ)
z2e2a−2zeacos Ω + 1|z|>e−a
1138 Definition, Bilateral, and Unilateral z-Transforms
18.3 Unilateral z-transform
The relationship between the Laplace transform of a continuous function x(t) sampled at t=0 ,T,2T,
...and the unilateral z-transform of the function ˆ x(t)=∞/summationdisplay
n=0x(nT)δ(t−nT) follows from the result
L{ˆx(t)}=/integraldisplay∞
0/bracketleftBigg∞/summationdisplay
k=0x(kT)δ(t−kT)/bracketrightBigg
e−stdt
=∞/summationdisplay
k=0x(kT)e−ksT.
Setting z=esT, this becomes:
L{ˆx(t)}=∞/summationdisplay
k=0x(kT)z−k=X(z),
showing that the unilateral z-transform Xu(z) can be considered to be the Laplace transform of a con-
tinuous function x(t)f o rt≥0 sampled at t=0 ,T,2T,....
Table 18.3 lists some general properties of the unilateral z-transform, and Table 18.4 lists some uni-
lateral z-transform pairs.
Unilateral z-transform 1139
Table 18.3 General properties of the unilateral z-transform
Term in sequence z-Transform Xu(z) Domain of Convergence
1αxn+βyn αXu(z)+βYu(z)Intersection of DOC’s of Xu(z)
andYu(z) with α,βconstants
2xn+kzkXu(z)−zkx0−zk−1x1
−zk−2x2−···− zxk−1
3nxn −zdXu(z)
dzDOC of Xu(z), to which it may
be necessary to add or delete theorigin and the point at infinity
4zn
0xn Xu/parenleftbiggz
z0/parenrightbiggDOC of Xb(z) scaled by |z0|,t o
which it may be necessary toadd or delete the origin and thepoint at infinity
5nzn
0xn −zdXu(z/z0)
dzDOC of Xu(z) scaled by |z0|,t o
which it may be necessary toadd or delete the origin and thepoint at infinity
6¯xn Xu(z) The same DOC as xn
7R e xn1
2/bracketleftbig
Xu(z)+Xu(z)/bracketrightbig
DOC contains the DOC of xn
8∂
∂αxn(α)∂
∂αXu(z,α) Same DOC as xn(α)
9Initial value
theoremx0= lim
z→∞Xu(z)
10Final value
theoremlim
n→∞xn= lim
z→1/bracketleftbigg/parenleftbiggz−1
z/parenrightbigg
Xu(z)/bracketrightbiggWhen Xu(z)=N(z)/D(z) with
N(z),D(z) polynomials in zand
the zeros of D(z) inside the unit
circle|z|= 1 or at z=1
1140 Definition, Bilateral, and Unilateral z-Transforms
Table 18.4 Basic unilateral z-transforms
Term in sequence z-Transform Xu(z) Domain of Convergence
1Δ ( n)1 Converges for all z
2Δ ( n−k) z−kConvergence for all z/negationslash=0
3anh(n)z
z−a|z|>|a|
4nanh(n)az
(z−az)2|z|>a> 0
5n2anh(n)az(z+a)
(z−a)3|z|>a> 0
6nan−1h(n)z
(z−a)2|z|>a> 0
7(n−1)anh(n)z(2a−z)
(z−a)2|z|>a> 0
8e−anh(n)zea
zea−1|z|>e−a
9ne−anh(n)zea
(zea−1)2|z|>e−a
10n2e−anh(n)zea(1 +zea)
(zea−1)3|z|>e−a
11e−anh(n)sinΩ nzeasin Ω
z2e2a−2zeacosΩ + 1|z|>e−a
12e−anh(n)cosΩ nzea(zea−cosΩ)
z2e2a−2zeacosΩ + 1|z|>e−a
13h(n)sin h anzsinha
z2−2zcosha+1|z|>e−a
14h(n)cosh anz(z−cosha)
z2−2zcosha+1|z|>e−a
15h(n)an−1e−ansin Ωnzeasin Ω
z2e2a−2zaeacosΩ + a2|z|>e−a
16h(n)ane−ancos Ωnzea(zea−acosΩ)
z2−2zaeacosΩ + a2|z|>e−a
Bibliographic References Used in
Preparation of Text
(See the introduction for an explanation of the letters preceding each bibliographic reference.)
AS Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions , Dover Publications,
New York, 1972.
AD Adams, E. P. and Hippisley, R. L., Smithsonian Mathematical Formulae and Tables of Elliptic
Functions , Smithsonian Institute, Washington, D.C., 1922.
AK Appell, P. and Kamp´ ed eF ´ eriet,Fonctions hyperg´ eometriques et hypersph´ eriques, polynomes
d’Hermite , Gauthier Villars, Paris, 1926.
BB Beckenbach, E. F. and Bellman, R., Inequalities , 3rd printing. Springer–Verlag, Berlin, 1971.
BE Bertrand, J., Traite de calcul diff´ erentiel et de calcul int´ egral, vol. 2, Calcul int´ egral, int´ egrales
d´efinies et ind´ efinies , Gauthier-Villars, Paris, 1870.
BI Bierens de Haan, D., Nouvelles tables d’int´ egrales d´ efinies , Amsterdam, 1867. (Reprint) G. E.
Stechert & Co., New York, 1939.
BL Bellman, R., Introduction to Matrix Analysis , McGraw Hill, New York, 1960.
BR Bromwich, T. I’A., An Introduction to the Theory of Infinite Series , Macmillan, London, 1908,
2nd edition, 1926.∗
BS Bellman, R., Stability Theory of Differential Equations , McGraw-Hill, New York, 1953.
BU Buchholz, H., Die konfluente hypergeometrische Funktion mit besonderer Ber¨ ucksichtigung ihrer
Anwendungen , Springer–Verlag, Berlin, 1953. Also an English edition: The confluent
Hypergeometric Function , Springer–Verlag, Berlin, 1969.
BY B y r d ,P .F .a n dF r i e d m a n ,M .D . , Handbook of Elliptic Integrals for Engineers and Physicists ,
Springer–Verlag, Berlin, 1954.
CA Carslaw, H. S., Introduction to the Theory of Fourier’s Series and Integrals , Macmillan, London,
1930.
CE Ces`aro, Z., Elementary Class Book of Algebraic Analysis and the Calculation of Infinite Limits ,
1st ed. ONTI, Moscow and Leningrad, 1936.
CL Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations , McGraw Hill,
New York, 1955.
CO Courant, R. and Hilbert, D., Methods of Mathematical Physics , vol. I, Wiley (Interscience), New
York, 1953.
DW Dwight, H. B., Tables of Integrals and Other Mathematical Data , Macmillan, New York, 1934.
∗The Bibliographic Reference BR* refers to the 1908 edition of Bromwich T. I.’A., An Introduction to the Theory of
Infinite Series ; BR refers to the 1926 edition.
1141
1142 References
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EH Erd´elyi, A., et al., Higher Transcendental Functions , vols. I, II, and III. McGraw Hill, New York,
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EU Euler, L., Introductio in Analysin Infinitorum , Bousquet, Lausanne, 1748.
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GA Gauss, K. F., Werke , Bd. III. G¨ ottingen, 1876.
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GH2 Gr¨obner, W. and Hofreiter, N., Integraltafel , vol. 2, Bestimmte Integrale , Springer, Wien, 1961.
GI Giunter, N. M. and Kuz’min, R. O. (eds.), Sbornik zadach po vysshey matematike (Collection of
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GM Gantmacher, F. R., Applications of the Theory of Matrices , translation by J. L. Brenner. Wiley
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GW Gr¨obner, W. and Hofreiter, N., Integraltafel ,T e i lI I , Bestimmte Integrale , Springer–Verlag, Wien
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HI Hille, E., Lectures on Ordinary Differential Equations , Addison- Wesley, Reading,
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JAC Jackson, J. D., Classical Electrodynamics , Wiley, New York, 1975.
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KE Kellogg, O. D., Foundations of Potential Theory , Dover, New York, 1958.
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KU Kuzmin, R. O., Besselevy funktsii (Bessel functions). ONTI, Moscow and Leningrad, 1935.
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ME M c L a c h l a n ,N .W .a n dH u m b e r t ,P . , Formulaire pour le calcul symbolique , L’Acad. des Sciences
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ML Mirsky L., An Introduction to Linear Algebra , Oxford University Press, London, 1963.
MM MacMillan, W. D., The Theory of the Potential , Dover, New York, 1958.
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MS2 Mathai, A. M. and Saxens, R. K. , Generalized Hypergeometrics Functions With Applications in
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MZ Meyer Zur Capellen, W., Integraltafeln, Sammlung unbestimmer Integrale elementarer
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NA Natanson, I. P., Konstruktivnaya teoriya funktsiy (Constructive theory of functions).
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1144 References
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Classified Supplementary References
(Prepared by Alan Jeffrey for the English language edition.)
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18. National Bureau of Standards, Handbook of Mathematical Functions , U.S. Government Printing Office,
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19. Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series , Vols. 1–5, Gordon and
Breach, New York, 1986–1992.
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1145
1146 Supplemental References
21. Vein, R. and Dale, P., Determinants and Their Applications in Mathematical Physics , Springer–Verlag,
New York, 1999.
22. Whittaker, E. T. and Watson, G. N., AC o u r s eo fM o d e r nA n a l y s i s , 4th ed., Cambridge University Press,
London, 1940.
Asymptotic expansions
1. De Bruijn, N. G., Asymptotic Methods in Analysis , North-Holland Publishing Co., Amsterdam, 1958.
2. Cesari, L., Asymptotic Behavior and Stability Problems in Ordinary Differential Equations ,3 r de d . ,
Springer, New York, 1971.
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6. Hardy, G. H., Divergent Series , Clarendon Press, Oxford, 1949.
7. Watson, G. N., A Treatise on the Theory of Bessel Functions , 2nd ed., Cambridge University Press,
London, 1958.
Bessel functions
1. Bickley, W. G., Bessel Functions and Formulae , Cambridge University Press, London, 1953.
2. Erd´ elyi, A. et al., Higher Transcendental Functions , vols. I and II. McGraw Hill, New York, 1954.
3. Erd´ elyi, A. et al., Tables of Integral Transforms , vols. I and II. McGraw Hill, New York, 1954.
4. Gray, A., Mathews, G. B. and MacRobert, T. M., A Treatise on Bessel Functions and Their Applications
to Physics , 2nd ed., Macmillan, 1922.
5. McLachlan, N. W., Bessel Functions for Engineers , 2nd ed., Oxford University Press, London, 1955.
6. Luke, Y. L., Integrals of Bessel Functions , McGraw Hill, New York, 1962.
7. Petiau, G., La th´eorie des fonctions de Bessel , Centre National de la Recherche Scientifique, Paris, 1955.
8. Relton, F. E., Applied Bessel Functions , Blackie, London, 1946.
9. Watson, G. N., A Treatise on the Theory of Bessel Functions , 2nd ed., Cambridge University Press,
London, 1958.
10. Wheelon, A. D., Tables of Summable Series and Integrals Involving Bessel Functions , Holden-Day, San
Francisco, 1968.
Complex analysis
1. Ahlfors, L. V., Complex Analysis , 3rd ed., McGraw Hill, New York, 1979.
2. Ahlfors, L. V. and Sario, L., Riemann Surfaces , Princeton University Press, Princeton, New Jersey, 1971.
3. Bieberbach, L., Conformal Mapping , Chelsea, New York, 1964.
4. Henrici, P., Applied and Computational Complex Analysis , 3 vols, Wiley, New York, 1988, 1991, 1977.
5. Hille, E., Analytic Function Theory , 2 vols. 2nd ed., Chelsea, New York, 1990, 1987.
6. Kober, H., Dictionary of Conformal Representations , Dover Publications, New York, 1952.
7. Titchmarsh, E. C., The Theory of Functions , 2nd ed., Oxford University Press, London, 1939. (Reprinted
1975).
Supplemental References 1147
Error function and Fresnel integrals
1. Erd´ elyi, A. et al., Higher Transcendental Functions , vol. II, McGraw Hill, New York, 1953.
2. Erd´ elyi, A. et al., Tables of Integral Transforms , vol. I, McGraw Hill, New York, 1954.
3. Slater, L. J., Confluent Hypergeometric Functions , Cambridge University Press, London, 1960.
4. Tricomi, F. G., Funzioni ipergeometriche confluenti , Edizioni Cremonese, Turan, Italy, 1954.
5. Watson, G. N., A Treatise on the Theory of Bessel Functions , 2nd ed., Cambridge University Press,
London, 1958.
Exponential integrals, gamma function and related functions
1. Artin, E., The Gamma Function , Holt, Rinehart, and Winston, New York, 1964.
2. Busbridge, I. W., The Mathematics of Radiative Transfer , Cambridge University Press, London, 1960.
3. Erd´ elyi, A. et al., Higher Transcendental Functions , vol. II, McGraw Hill, New York, 1953.
4. Erd´ elyi, A. et al., Tables of Integral Transforms , vols. I and II, McGraw Hill, New York, 1954.
5. Hastings, Jr., C., Approximations for Digital Computers , Princeton University Press, Princeton, New
Jersey, 1955.
6. Kourganoff, V., Basic Methods in Transfer Problems , Oxford University Press, London, 1952.
7. L¨ osch, F. and Schoblik, F., Die Fakult¨ at (Gammafunktion) und verwandte Funktionen , Teubner, Leipzig,
1951.
8. Nielsen, N., Handbuch der Theorie der Gammafunktion , Teubner, Leipzig, 1906.
9. Oberhettinger, F., Tabellen zur Fourier Transformation , Springer–Verlag, Berlin, 1957.
Hypergeometric and confluent hypergeometric functions
1. Appell, P., Sur les Fonctions Hyperg´ eometriques de Plusieures Variables , Gauthier-Villars, Paris, 1926.
2. Bailey, W. N., Generalized Hypergeometric Functions , Cambridge University Press, London, 1935.
3. Buchholz, H., Die konfluente hypergeometrische Funktion , Springer–Verlag, Berlin, 1953.
4. Erd´ elyi, A. et al., Higher Transcendental Functions , vol. I, McGraw Hill, New York, 1953.
5. Jeffreys, H. and Jeffreys, B. S., Methods of Mathematical Physics , Cambridge University Press, London,
1956.
6. Klein, F., Vorlesungen ¨ uber die hypergeometrische Funktion , Springer–Verlag, Berlin, 1933.
7. N¨ orlund, N. E., Sur les Fonctions Hyperg´ eometriques d’Ordre Superior , North–Holland, Copenhagen,
1956.
8. Slater, L. J., Confluent Hypergeometric Functions , Cambridge University Press, London, 1960.
9. Slater, L. J. Generalized Hypergeometric Functions , Cambridge University Press, London, 1966.
10. Snow, C., The Hypergeometric and Legendre Functions with Applications to Integral Equations of Poten-
tial Theory , 2nd ed., National Bureau of Standards, Washington, D.C., 1952.
11. Swanson, C. A. and Erd´ elyi, A., Asymptotic Forms of Confluent Hypergeometric Functions ,M e m o i r2 5 ,
American Mathematical Society, Providence, Rhode Island, 1957.
12. Tricomi, F. G., Lezioni sulla funzioni ipergeometriche confluenti , Gheroni, Torino, 1952.
Integral transforms
1. Bochner, S., Vorlesungen ¨ uber Fouriersche Integrale , Akad. Verlag, Leipzig, 1932. Reprint Chelsea, New
York, 1948.
1148 Supplemental References
2. Bochner, S. and Chandrasekharan, K., Fourier Transforms , Princeton University Press, Princeton, New
Jersey, 1949.
3. Campbell, G. and Foster, R., Fourier Integrals for Practical Applications , Van Nostrand, New York, 1948.
4. Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids , Oxford University Press, London, 1948.
5. Doetsch, G., Theorie und Anwendung der Laplace-Transformation , Springer–Verlag, Berlin, 1937. (Reprinted
by Dover Publications, New York, 1943)
6. Doetsch, G., Theory and Application of the Laplace-Transform ,C h e l s e a ,N e wY o r k ,1 9 6 5 .
7. Doetsch, G., Handbuch der Physik, Mathematische Methoden II , 1st ed., Springer–Verlag, Berlin, 1955.
8. Doetsch, G., Guide to the Applications of the Laplace and Z-Transforms , 2nd ed., Van Nostrand-Reinhold,
London, 1971.
9. Doetsch, G., Handbuch der Laplace-Transformation , Vols. I–IV, Birkh¨ auser Verlag, Basel, 1950–56.
10. Doetsch, G., Kniess, H., and Voelker, D., Tabellen zur Laplace- Transformation, Springer–Verlag, Berlin,
1947.
11. Erd´ elyi, A., Operational Calculus and Generalized Functions , Holt, Rinehart and Winston, New York,
1962.
12. Exton, H., Multiple Hypergeometric Functions and Applications, Horwood, Chichester, 1976.
13. Exton, H., Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs ,
Horwood, Chichester, 1978.
14. Hirschmann, J. J. and Widder, D. V., The Convolution Transformation , Princeton University Press,
Princeton, New Jersey, 1955.
15. Marichev, O. I., Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Al-
gorithmic Tables , Ellis Horwood Ltd., Chichester (1982).
16. Oberhettinger, F., Tabellen zur Fourier Transformation, Springer–Verlag, Berlin (1957).
17. Oberhettinger, F., Tables of Bessel Transforms , Springer–Verlag, New York (1972).
18. Oberhettinger, F., Fourier Expansions: A Collection of Formulas, Academic Press, New York, 1973.
19. Oberhettinger, F., Fourier Transforms of Distributions and Their Inverses , Academic Press, New York,
1973.
20. Oberhettinger, F., Tables of Mellin Transforms , Springer–Verlag, Berlin, 1974.
21. Oberhettinger, F. and Badii, L., Tables of Laplace Transforms, Springer–Verlag, Berlin, 1973.
22. Oberhettinger, F. and Higgins, T. P., Tables of Lebedev, Mehler and Generalized Mehler Transforms ,
Math. Note No. 246, Boeing Scientific Research Laboratories, Seattle, Wash., 1961.
23. Roberts, G. E. and Kaufman, H., Table of Laplace Transforms, McAinsh, Toronto, 1966.
24. Sneddon, I. N., Fourier Transforms , McGraw Hill, New York, 1951.
25. Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals , Oxford University Press, London,
1937.
26. Van der Pol, B. and Bremmer, H., Operational Calculus Based on the Two Sided Laplace Transformation ,
Cambridge University Press, London, 1950.
27. Widder, D. V., The Laplace Transform , Princeton University Press, Princeton, New Jersey, 1941.
28. Wiener, N., The Fourier Integral and Certain of its Applications , Dover Publications, New York, 1951.
Jacobian and Weierstrass elliptic functions and related functions
1. Erd´ elyi, A. et al., Higher Transcendental Functions , vol. II, McGraw Hill, New York, 1953.
2. Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Physicists , Springer–
Verlag, Berlin, 1954.
3. Graeser, E., Einf¨uhrung in die Theorie der Elliptischen Funktionen und deren Anwendungen , Oldenbourg,
Munich, 1950.
4. Hancock, H., Lectures on the Theory of Elliptic Functions , vol. I, Dover Publications, New York, 1958.
Supplemental References 1149
5. Neville, E. H., Jacobian Elliptic Functions , Oxford University Press, London, 1944 (2nd ed. 1951).
6. Oberhettinger, F. and Magnus, W., Anwendungen der Elliptischen Funktionen in Physik und Technik ,
Springer–Verlag, Berlin, 1949.
7. Roberts, W. R. W., Elliptic and Hyperelliptic Integrals and Allied Theory , Cambridge University Press,
London, 1938.
8. Tannery, J. and Molk, J., El´ements de la Th´ eorie des Fonctions Elliptiques , 4 volumes. Gauthier-Villars,
Paris, 1893–1902.
9. Tricomi, F. G., Elliptische Funktionen , Akad. Verlag, Leipzig, 1948.
Legendre and related functions
1. Erd´ elyi, A. et al., Higher Transcendental Functions , vol. I, McGraw Hill, New York, 1953.
2. Helfenstein, H., Ueber eine Spezielle Lam´ esche Differentialgleichung , Brunner and Bodmer, Zurich, 1950
(Bibliography).
3. Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics , Cambridge University Press, London,
1931. Reprinted by Chelsea, New York, 1955.
4. Lense, J., Kugelfunktionen , Geest and Portig, Leipzig, 1950.
5. MacRobert, T. M., Spherical Harmonics: An Elementary Treatise on Harmonic Functions with Applica-
tions, Methuen, England, 1927. (Revised ed. 1947; reprinted Dover Publications, New York, 1948).
6. Snow, C., The Hypergeometric and Legendre Functions with Applications to Integral Equations of Poten-
tial Theory , 2nd ed., National Bureau of Standards, Washington, D.C., 1952.
7. Stratton, J. A., Morse, P. M., Chu, L. J. and Hunter, R. A., Elliptic Cylinder and Spheroidal Wave
Functions Including Tables of Separation Constants and Coefficients , Wiley, New York, 1941.
Mathieu functions
1. Erd´ elyi, A., Higher Transcendental Functions , vol. III, McGraw Hill, New York, 1955.
2. McLachlan, N. W., Theory and Application of Mathieu Functions , Oxford University Press, London,
1947.
3. Meixner, J. and Sch¨ afke, F. W., Mathieusche Funktionen und Sph¨ aroidfunktionen mit Anwendungen auf
Physikalische und Technische Probleme , Springer–Verlag, Heidelberg, 1954.
4. Strutt, M. J. O., Lam´esche, Mathieusche und verwandte Funktionen in Physik und Technik ,E r g e b .M a t h .
Grenzgeb. 1, 199–323 (1932). Reprint Edwards Bros., Ann Arbor, Michigan, 1944.
Orthogonal polynomials and functions
1. Bibliography on Orthogonal Polynomials , Bulletin of National Research Council No. 103, Washington,
D.C., 1940.
2. Courant, R. and Hilbert, D., Methods of Mathematical Physics , vol. I, Interscience, New York, 1953.
3. Erd´ elyi, A. et al., Higher Transcendental Functions , vol. II, McGraw Hill, New York, 1954.
4. Kaczmarz, St. and Steinhaus, H., Theorie der Orthogonalreihen ,C h e l s e a ,N e wY o r k ,1 9 5 1 .
5. Lorentz, G. G., Bernstein Polynomials , University of Toronto Press, Toronto, 1953.
6. Sansone, G., Orthogonal Functions , Interscience, New York, 1959.
7. Shohat, J. A. and Tamarkin, J. D., The Problem of Moments , American Mathematical Society, Provi-
dence, Rhode Island, 1943.
1150 Supplemental References
8. Szeg¨ o, G., Orthogonal Polynomials , American Mathematical Society Colloquim Pub. No. 23, Providence,
Rhode Island, 1959.
9. Titchmarsh, E. C., Eigenfunction Expansions Associated with Second Order Differential Equations , Ox-
ford University Press, London, part I (1946), part II (1958).
10. Tricomi, F. G., Vorlesungen ¨ uber Orthogonalreihen , Springer–Verlag, Berlin, 1955.
Parabolic cylinder functions
1. Buchholz, H., Die konfluente hypergeometrische Funktion , Springer–Verlag, Berlin, 1953.
2. Erd´ elyi, A. et al., Higher Transcendental Functions , vol. II, McGraw Hill, New York, 1954.
Probability function
1. Cramer, H., Mathematical Methods of Statistics , Princeton University Press, Princeton, New Jersey, 1951.
2. Erd´ elyi, A. et al., Higher Transcendental Functions , vols. I, II, and III. McGraw Hill, New York, 1953–
1955.
3. Kendall, M. G. and Stuart, A., The Advanced Theory of Statistics, vol. I: Distribution Theory ,G r i ffi n ,
London, 1958.
Riemann zeta function
1. Titchmarsh, E. C., The Zeta Function of Riemann , Cambridge University Press, London, 1930.
2. Titchmarsh, E. C., The Theory of the Riemann Zeta Function , Oxford University Press, London, 1951.
Struve functions
1. Erd´ elyi, A. et al., Higher Transcendental Functions , vol. II, McGraw Hill, New York, 1954.
2. Gray, A., Mathews, G. B. and MacRobert, T. M., A Treatise on Bessel Functions and Their Applications
to Physics , 2nd ed., Macmillan, London, 1922.
3. Watson, G. N., A Treatise on the Theory of Bessel Functions , 2nd ed., Cambridge University Press,
London, 1958.
Index of Functions and Constants
This index shows the occurrence of functions and constants used in the expressions within the text. The
numbers refer to pages on which the function or constant appears.
Symbols
!a n d! ! ............................... seefactorials
S(m)
n............ seeStirling numbers, second kind
Z(m)
n........................ seeBessel functions, Z
∇................. xliv, 767, 1050–1053, 1055–1057
β ................................ seebeta function
δ(x)............................. seedelta function
δij............................. seeKronecker delta
γand Γ ...................... seegamma functions
λfunction ............................. xxxix, 1043
μfunction ............................. xxxix, 1043
νfunction ............................. xxxix, 1043
Φ........... seeLerch function and hypergeometric
functions, confluent
Ψ........... seeEuler function and hypergeometric
functions, confluent
Θ function ............... seeJacobi theta function
℘(x)...................... seeWeierstrass function
ξfunction .............................. xxxix, 1040
| |·| |....... 1081–1083, 1085, 1086, 1095, 1096, 1120
| |·| |1................................... 1081–1083
| |·| |2................................... 1081–1083
| |·| | ∞.................................. 1081, 1083
A
Airy function (Ai) .......................... xxxviii
am function ................... xxxix, 625, 866, 867
Anger function ( J).....xli, 339, 352, 371, 384, 421,
423, 444–446, 670, 671, 946, 948, 949, 992
arccos function ..xxxi, xxxii, 23, 56–60, 64, 99, 135,
139, 173, 179–183, 210, 211, 241–244, 263, 272,279, 293–296, 307, 312, 313, 318, 393, 454, 511,558, 562, 589, 600, 601, 607, 610, 624, 695, 730,
742, 767, 768, 861, 890, 936, 993, 994
arccosec function ..................... 242, 244, 728
arccosh function ...... xxxi, xxxii, 56, 60–62, 64, 97,
126, 133, 135, 137, 138, 241, 382, 386, 511, 532,621, 624, 729, 768arccot function ...... xxxi, xxxii, 51, 56–58, 64, 242,
244, 245, 263, 274, 279, 325, 326, 499, 556, 561,
599, 601–607, 625, 767, 892, 1115, 1133
arccoth function ......... xxxi, xxxii, 56, 60, 62, 75,
131–134, 172, 177, 178, 241, 644, 647
arcosech function ............................... 62
arcsec function ................. 61, 66, 99, 242, 244
arcsech function ................................. 62
arcsin function ............... xxxi, xxxii, 27, 56–61,
64, 66, 94, 97, 99, 116, 124–126, 133–138, 173,179–182, 186–193, 195, 196, 198, 202–213, 225,241–245, 254, 263, 265, 272, 275, 279, 297, 307,382, 558, 563, 566–568, 583, 588, 589, 591, 600,
601, 604, 605, 607, 621, 622, 624, 625, 631, 632,
637, 638, 662, 668, 700–702, 713, 717, 718, 727,728, 743, 744, 748, 755, 767, 768, 793, 815, 860,914, 989, 1007
arcsinh function ..xxxi, xxxii, 54, 56, 60–62, 64, 94,
97, 126, 133, 135, 139, 240, 241, 371, 382, 386,
448, 588, 624, 637, 638, 1007
arctan function .......... xxxi, xxxii, 27, 30, 49, 51,
52, 55–61, 63–67, 71–77, 79, 83–85, 87, 90, 97,
103, 104, 106, 114, 116, 117, 119, 126, 128–133,147, 148, 171–175, 177, 178, 190, 205, 239, 240,
242–245, 254, 263, 272, 274, 279, 294, 295, 307,
317, 324, 329, 346, 363, 372, 373, 381, 393, 409,453, 493–501, 507, 509, 516–521, 524, 556, 557,560, 563–565, 593, 599–607, 612, 622–624, 631,632, 637, 639, 640, 643, 644, 646–649, 748, 763,860, 884, 885, 890–893, 898, 923, 1007, 1036,
1113, 1125, 1128, 1132
arctanh function ............. xxxi, xxxii, 56, 60–62,
64, 75, 79, 97, 125–128, 131, 132, 134, 172, 177,178, 241, 621–623, 977
1151
1152 INDEX OF FUNCTIONS AND CONSTANTS
associated Legendre functions
first kind ( P)....xli, 326, 327, 333, 336, 374, 406,
407, 486, 660, 661, 665, 666, 686, 699, 703, 705,706, 727, 755, 760, 761, 767–789, 792, 793, 797,806–808, 810, 823, 831, 839, 840, 848, 958–972,974, 975, 980–983, 992
second kind ( Q)...... xli, 333, 336, 374, 383, 407,
511, 661, 662, 666, 685, 686, 700, 702, 703, 705,727, 769–781, 783–785, 791, 795, 823, 831, 839,840, 958–973, 981
B
Bn(x)................... seeBernoulli polynomials
B(x)............................. seeBeta function
Bateman function (k) ................ xli, 349, 1023
bei(z)...................... seeThomson functions
ber(z)...................... seeThomson functions
Bernouli number ( Bn)..xxxii, xxxiii, xxxix, 1–3, 8,
26, 42, 43, 46, 55, 145, 146, 148, 221–224, 353,356, 376, 379–382, 387, 472, 550–552, 554, 560,567, 574, 580, 581, 587, 589, 591, 764–766, 899,906, 936, 1038–1045
Bernouli number ( B
∗
n).................... xxxiii, 62
Bernoulli polynomial ( Bn(x))...xxxii, xxxiii, xxxix,
46, 1037, 1041, 1042, 1045
Bessel functions
In(x).................................... xxxviii,
xli, 13, 320, 339, 340, 345, 347, 350, 351, 368,382, 385, 419, 435, 441, 444, 445, 470, 477–480,
491, 494, 496, 507, 513–515, 524, 595, 605, 616,
617, 660, 661, 663–681, 684–687, 689, 691, 692,695–699, 702–716, 719, 720, 722–725, 727, 729,730, 735, 736, 738, 741, 743, 745–747, 751–760,762, 779–781, 783–787, 789, 794, 797, 800, 820,832, 833, 838, 846, 847, 901, 911, 916, 917, 919,
920, 925–933, 943, 954, 1002, 1027, 1028, 1116,
1117, 1123, 1125, 1128
J
n(x)........... xxxvii, xxxviii, xli, 13, 339, 350,
352, 371, 384, 385, 417–421, 423, 435, 440–443,445, 446, 477–480, 482, 483, 491, 492, 507, 514,
515, 522, 524, 525, 578, 629, 642, 653, 659–694,
696–753, 756–759, 761–763, 767, 768, 777, 779,780, 782–787, 792–794, 797–799, 802, 803, 808,811, 812, 818–820, 830–838, 841, 845–848, 854,855, 900, 910–914, 916, 918–931, 933–950, 954–957, 963, 964, 972, 992, 1000, 1002–1004, 1017,
1023–1025, 1028, 1034, 1116, 1117, 1124–1129,
1132, 1133Bessel functions ( continued )
K
n(x)............................. xxxviii, xli, 2,
337, 339, 345–348, 350–353, 364–368, 370, 371,384, 385, 417, 419, 435, 442, 444, 445, 477–482,490, 491, 504, 505, 507, 514, 515, 529, 573, 575,576, 578, 595, 638, 645, 648, 653, 654, 657, 660–
682, 684–696, 698–700, 702–716, 718–724, 726,
727, 729–732, 735, 736, 738, 740, 742, 745–753,756–759, 761, 768, 776–787, 789, 794, 800, 803,811, 814, 817–820, 828, 832–834, 837, 838, 841,845, 846, 848, 854, 855, 900, 911, 917–920, 923,925–933, 939, 942, 945, 955, 957, 1027, 1028,
1035, 1123–1126, 1129, 1132, 1133
N
n(x)............................... xxxvii, 910
Yn(x)............. xxxvii, xlii, 338, 339, 345, 346,
351–353, 371, 384, 385, 419, 435, 440, 442, 443,445, 446, 477–480, 482, 483, 492, 507, 514, 515,573, 578, 647, 654, 659–664, 666–682, 684–693,
695–700, 705–709, 711, 714–720, 722–724, 726,
728, 729, 732–736, 738, 740–742, 745, 747–752,756–759, 761, 767, 768, 777, 779, 782, 783, 785,787, 793, 794, 799, 817–819, 832, 833, 835–837,847, 848, 854, 855, 910, 911, 914, 918–920, 922,923, 925–931, 933, 937–939, 941–943, 945, 946,
949, 954–957, 975, 1025, 1034, 1124, 1126, 1132
Z
n(x)...... xlii, 483, 629, 630, 767, 910, 911, 926,
931–933, 937, 940, 941, 975
Zn(x).............................. xlii, 629, 630
Hankel ...................... seeHankel function
beta function ( β)........... 319, 322, 324, 325, 334,
371, 375, 383, 395, 396, 403, 432, 471, 553, 558,562–564, 573, 586, 602, 904, 906, 907
Beta function (B) ......................... xxxix, 6,
129, 175, 315–318, 320, 322–330, 332, 333, 335,338, 347–349, 351, 359, 360, 364, 368, 370, 372,
374, 375, 382, 383, 395–397, 399–402, 407, 408,
411–413, 440, 442, 444, 460, 469, 472, 485, 486,490, 512, 539–543, 548, 553, 559, 585, 705, 749,754, 760, 801, 810, 813, 814, 816, 821, 894, 895,908–910, 991, 1005, 1023, 1025
Bi function ................................. xxxviii
bilateral ztransform ........ 1135–1137, 1139, 1159
binomial coefficients .................... xxxiii, xliii,
1–6, 11, 12, 15, 22, 23, 25, 31, 33, 46, 84, 86,88, 89, 100–103, 106, 110, 111, 114, 115, 119,120, 140, 143, 148, 153, 157, 161, 173, 215, 220,221, 223, 228, 232–238, 241, 316, 326, 329, 354,
357, 361, 362, 386, 393, 394, 397–402, 416, 431,
437, 459–461, 466, 469, 470, 478, 488, 498, 499,504, 505, 545, 546, 548, 549, 552, 612, 808, 910,934, 993, 994, 996, 998, 1003, 1023, 1030, 1040,1041, 1044, 1046, 1047
INDEX OF FUNCTIONS AND CONSTANTS 1153
C
Cn(x)................. seeGegenbauer polynomials
C(x)..seeFresnel sine integral and Young function
Catalan constant ( G)......... xxxii, xl, 9, 375, 380,
433, 434, 448, 449, 452, 453, 470–472, 530–534,536–538, 556, 558, 560, 563, 564, 580, 600–603,
632, 633, 1046
cd function .................................. xxxiii
Ce function ........... xxxviii, xl, 763–766, 953, 954
ce function ............ xxxviii, xl, 763–767, 951–957
Chebyshev polynomials
first kind ( T
n(x))......... xli, 448, 667, 718, 790,
800–803, 983, 988, 993–996, 999, 1131
second kind ( Un(x))......... xxxvii, xli, 800–802,
994–996
chi function ....... xxxvi, xl, 142–144, 644, 645, 886
ci function ....xxxv, xl, 219–221, 340–344, 423–426,
436, 437, 447, 495, 505, 506, 528, 529, 571, 572,
578, 581, 594, 595, 599, 605, 628, 629, 638–645,
647, 656, 658, 748, 762, 886, 887, 1115
Cin function ................................. xxxvi
cn function .......... xxxiii, xxxiv, xl, 623–626, 714,
866–873, 875, 879, 880
complex conjugate .............. xliii, 293, 341, 342,
421, 422, 424–426, 511, 528, 529, 927, 931, 933,
1060, 1061, 1070, 1071, 1082, 1087, 1136, 1139
confluent hypergeometric functions ............. see
hypergeometric functions, confluent
constants
Catalan .................... seeCatalan constant
Euler ......................... seeEuler constant
cos function .......... xxix, xxxvi, xxxviii, 4, 13, 19,
20, 26–39, 41–52, 54–56, 64, 74–76, 78, 79, 126,147, 151–237, 249, 250, 253, 254, 317, 318, 320,322, 323, 326–329, 331–333, 338–345, 353, 358,
372, 373, 377–383, 385, 388–534, 537, 540–545,
550, 551, 561, 563, 565, 567–573, 576, 578, 579,581–601, 604–608, 610–612, 616, 621–623, 628,629, 631, 632, 634–644, 647–650, 652, 655, 656,658, 659, 662–664, 667, 669, 671, 673–675, 677–681, 686, 688, 689, 691, 692, 695, 703–707, 709,
711, 713, 715, 717–748, 750–752, 754, 756–770,
779–782, 784, 785, 787–789, 792–794, 797–800,802, 806, 808, 811, 812, 815, 817, 818, 823, 825,829–831, 833, 836, 837, 844, 845, 847–849, 854,862–869, 877–880, 882–886, 888–894, 896, 898–900, 904, 906–910, 912–918, 920, 922, 924, 925,
928, 930, 933–943, 945–951, 953, 954, 958–964,
966–981, 984–993, 997, 998, 1000, 1003, 1005–1007, 1023, 1025, 1026, 1029, 1030, 1037, 1038,1041, 1042, 1044, 1053, 1057, 1060, 1061, 1066,1067, 1090, 1110–1113, 1115, 1117, 1119–1122,
1124–1128, 1131, 1132, 1137, 1140
cosec function ..xxvii, xxix, 36–39, 43, 44, 49, 50, 64,
113–115, 126, 155, 156, 160, 225, 254, 315–323,325, 327–332, 334, 335, 339, 349, 352, 354, 355,358, 373, 381–383, 387, 388, 400, 401, 403–408,
410–414, 421, 422, 434, 437–439, 446, 453, 454,
471, 479, 480, 484, 496, 508, 509, 529, 540, 541,547, 551, 565, 586, 588, 593, 600, 602, 604, 658,659, 663, 664, 669, 670, 677–682, 689–692, 695,697, 700, 713, 715, 717, 718, 723, 724, 755, 760,761, 780, 787, 788, 844, 853, 869, 900, 921, 927,
929, 930, 932, 964, 967, 1131, 1132
cosech function ...... 27, 43, 113–115, 126, 387, 501,
513, 634, 715, 751, 1111, 1124, 1133
cosh function .............. xxviii, xxxvi, 27–36, 38,
42, 43, 45, 47, 48, 50–52, 64, 110–151, 231–237,251, 323, 338, 339, 371–390, 407, 419, 425, 432,
433, 439, 448, 451, 452, 454, 455, 468, 482, 484,
485, 502, 504, 509–527, 568, 570, 573, 578–581,595, 596, 605, 610, 611, 621, 622, 634, 643–645,648, 686, 702, 705, 710, 713–717, 722, 723, 729,735, 747, 750–752, 755, 760, 763–767, 778, 781,787–789, 791, 792, 802, 806, 843, 844, 886, 888,
896, 906, 908, 909, 912–917, 921, 922, 952–958,
960–963, 967–969, 973, 977, 980–982, 998, 1006,1043, 1112–1115, 1119, 1128, 1129, 1140
cosine integral (Ci) ....xxxv, xxxvi, 886, 930, 1115,
1122, 1126
cot function ..xxvii, xxviii, 28, 36, 37, 39, 42, 44, 46,
49, 52, 56, 64, 147, 157–161, 168, 174, 176–178,185, 188, 189, 194, 195, 204–207, 213, 222–225,229, 230, 274, 318–323, 330, 331, 334, 354, 355,358, 372, 379, 381–384, 388, 395, 396, 400, 401,403–406, 411–414, 422, 434, 448, 454, 455, 472,
484, 485, 492, 493, 496, 506, 509, 511, 532–534,
540, 542, 543, 546, 558, 559, 565, 567, 568, 587,588, 590, 594, 595, 604, 606, 611, 623, 631, 632,637, 663, 664, 669, 670, 676, 678–681, 690–692,697, 700, 717, 723, 753, 754, 849, 867, 868, 876,882, 903–905, 912, 914, 915, 922, 927, 929, 930,
932, 954, 967–969, 971, 979, 989, 1013, 1089,
1090, 1120
coth function ....... 28, 39, 40, 42, 44, 64, 110, 116,
118–120, 124, 130–134, 138, 145–148, 381, 384,386–388, 485, 489, 502, 509–513, 520, 523, 580,582, 595, 622, 634, 715, 716, 876, 921, 922, 956,
957, 967, 981, 1029, 1110, 1124
cs function ................................... xxxiii
curl......................... 1050–1053, 1057, 1058
cylinder function ....seeparabolic cylinder function
1154 INDEX OF FUNCTIONS AND CONSTANTS
D
dc function .................................. xxxiii
degrees .................................... 263–265
delta function ( δ(x)).......... 661, 1115, 1118–1120
determinant .....1070, 1075–1077, 1084–1086, 1096,
1100
dilogarithm ( L2)............................... 642
div.............. 1050, 1051, 1053, 1055, 1057, 1058
dn function ......... xxxiii, xxxiv, xl, 623–626, 714,
866–873, 875, 879, 880
double factorials ............... seefactorial, double
ds function .................................. xxxiii
E
En(x)....................... seeEuler polynomials
E(x)...................... seeMacRobert function
elliptic functions ............................... 859
D..................... ................... xl, 860
complete ........................... xl, 860, 861
E ................ xxxi, xl, 60, 135–139, 179–183,
185–195, 197, 198, 200, 202–214, 225, 255–262,264–274, 280–283, 285–296, 300–315, 410, 604,
606, 621–623, 625, 632, 672, 777, 814, 852, 853,
855, 857, 860, 862–864, 880
complete .......... xxxiv, xl, 313, 394, 408–410,
472–475, 562, 592, 593, 596–600, 619–622, 632,633, 671, 696, 704, 714, 729, 860–865, 869, 880,
990
F....... xxxi, 60, 61, 134–139, 179–183, 186–195,
197–200, 202–214, 225, 250, 251, 254–315, 407,408, 410, 411, 470, 486, 563, 568, 602, 604, 606,621–623, 631, 632, 654, 660, 661, 683, 684, 687,699, 700, 704, 707, 730, 731, 733, 734, 758, 804,
809–820, 822, 823, 843, 849, 856, 860–864, 889,
918, 959, 963, 964, 968, 970, 971, 974–976, 979,
981, 982, 984, 986, 991, 994
K...................... ..................... 672
complete .............. xxxiv, xli, 274, 275, 313,
394, 408–410, 472–475, 538, 539, 562, 567, 585,
588, 592, 593, 596–600, 611, 612, 619, 620, 622,
631–633, 696, 704, 713, 714, 719, 720, 729, 756,788, 860–870, 875, 879–881, 990
Π....... xxxi, xxxiv, xxxix, 51, 52, 135, 137, 138,
180, 181, 183, 197, 198, 200–202, 204, 205, 210,212–214, 255, 262, 263, 265, 276–279, 284, 285,
293, 295–300, 605, 620, 623, 632, 860, 880
erf........... xxxvi, xl, 107–109, 336, 365, 635, 646,
887–889, 1115, 1116
erfc....... xxxvi, xl, 887, 888, 890, 891, 1108, 1110,
1115, 1116
error functions ..................... seeerf and erfcEuler constant ( C)....xxxii, xxxv, xxxvi, xl, 3, 15,
321, 323, 330–332, 334, 335, 359, 361, 362, 364,367, 369–371, 411, 412, 422, 447, 476, 478, 483,484, 501, 534, 535, 538, 541, 553–555, 558, 559,570–574, 578–581, 585, 586, 588, 593, 594, 599,605, 628, 639, 644, 656, 658, 747, 748, 883, 884,
886, 894–896, 898, 903–906, 911, 919, 937–939,
944, 1037, 1038, 1046, 1125
Euler function ( ψ)............ xxxix, 318, 321, 323,
330–332, 334, 335, 356, 359, 360, 364, 369, 382,387, 388, 400, 403, 405, 411–414, 466, 486, 490,501, 509, 510, 523, 535–538, 540–543, 553–555,
558, 559, 562, 570–574, 576–579, 585, 586, 588,
594, 595, 607, 617, 658, 659, 747, 769, 770, 820,842, 902–907, 911, 919, 929, 969, 1011–1013
Euler number ( E
n).................... xxxii, xxxiii,
xl, 8, 43, 145, 146, 221, 223, 376, 379, 380, 533,550, 580, 1043–1045
Euler polynomial ( E
n(x))..xxxii, xxxiii, 1044, 1045
exponential function (exp) ...................... 27,
108, 109, 143, 144, 215, 335–340, 345–349, 352,364–371, 383–385, 390, 426, 428–430, 439, 440,480, 482, 484–486, 488, 489, 492–498, 501–509,513, 516, 521, 523, 524, 526, 574–576, 581, 617,
639, 645–649, 651, 652, 655–658, 693, 697–699,
706–710, 712, 713, 722, 723, 748–753, 759, 760,768, 778, 781, 785, 791, 805, 810, 811, 815, 826,828, 829, 831, 834, 837, 841–844, 848–850, 855,876, 877, 879, 880, 886–888, 890–893, 895, 896,
913, 915–917, 920–923, 928, 933, 935, 967, 997,
1002, 1024–1026, 1029, 1030, 1066, 1090, 1094,1096, 1098, 1099, 1105, 1110, 1119, 1123, 1124,1128
exponential integral ( E
n(x)).................. xxxv
exponential integral (Ei( x)).................. xxxv,
xl, 107, 109, 143, 144, 150, 151, 338, 340–344,
361, 370, 375, 386, 421, 422, 424–426, 432, 461,468, 483, 484, 492, 495, 527–530, 535, 553, 555,571–573, 577, 578, 594, 595, 605–607, 627, 628,638–647, 649, 656, 658, 748, 883–887, 900, 902,931, 1115, 1123, 1125
F
F............................ seeFourier transform
F(x).................. seehypergeometric function
INDEX OF FUNCTIONS AND CONSTANTS 1155
factorial
!............................ xxxii–xxxiv, xxxvii,
xliii, 2–5, 8, 12, 13, 18, 19, 22, 23, 25–27, 33,34, 42–44, 46, 49–51, 54, 55, 60–62, 66, 68, 77,79, 85, 90, 92, 94, 106–109, 114, 115, 127, 128,140–143, 145, 146, 148, 152, 155, 156, 163–167,
174, 215–219, 221–224, 228, 230, 241, 315, 316,
321, 323, 325, 326, 328–330, 332, 336, 340–342,344, 346, 353, 354, 359, 361, 364, 365, 367, 379,381, 382, 386, 389, 393, 396–398, 400–402, 405,408, 417, 419, 430, 431, 436, 437, 441, 444, 466,469, 470, 472, 486–488, 495–499, 502–505, 508,
510, 512, 517, 522, 528, 530, 531, 533, 535, 550–
552, 555, 559, 560, 567, 572–578, 580, 585–587,591, 593, 601, 607, 612, 613, 616, 619, 627, 635,636, 660, 672, 677, 687, 688, 698, 704, 705, 707,709, 725, 769–771, 789–793, 795–801, 803–812,840–842, 844, 860–862, 866, 867, 869, 884–886,
889, 892, 893, 895–897, 899–901, 904, 907, 909–
911, 913, 918–921, 923–925, 929, 930, 934–936,940, 941, 944, 949, 950, 961, 962, 968, 973–975,977, 979, 982–984, 986, 988–993, 995, 997–1002,1005, 1010–1013, 1018, 1022, 1023, 1025–1027,1031, 1034, 1038–1044, 1046, 1047, 1074, 1108,
1109, 1120
double (!!) ....................... xliii, 23, 77, 79,
94, 110, 111, 113, 114, 127, 128, 146, 152, 155,156, 174, 222, 226, 245, 250, 316, 319, 324–326,336, 345, 346, 363, 364, 367, 369, 395–398, 401,
402, 405, 408–410, 420, 430, 435, 459, 460, 466,
467, 469, 470, 472, 478, 488, 531, 538, 539, 543,551, 573–576, 585, 586, 601, 607, 616, 793, 805,860–862, 889, 897, 909, 923, 934, 940, 974, 977,982, 984, 986, 988–990, 992, 994, 997
Fe function ............................. xl, 953–955
fe function ............................. xl, 953–955
Fek function ........................... xl, 955, 957
Fey function .................. xl, 765, 767, 955, 956
Fourier transform ...... xliv, 1117, 1118, 1121, 1122,
1129
cosine ................ xliv, 1121, 1122, 1126–1129
sine........................ xliv, 1121–1125, 1129
Fresnel integral
cosine (C) .....xxxvi, xl, 171, 225, 226, 415, 434,
475, 476, 492, 629, 641, 649, 650, 659, 887–890,935, 1126
sine (S) ..xxxvi, xli, 170, 171, 225, 226, 415, 434,
475, 476, 492, 629, 641, 649, 650, 659, 887–890,
935, 1057, 1124G
Gpq
nm(x|a1,...
b1,...)............... seeMeijer Gfunction
gamma function
Γ(x).....xxxiv, xxxvii–xxxix, xliii, 6, 9, 68, 107–
109, 121–123, 163–167, 264, 296, 317, 318, 321,322, 324, 326–333, 336–338, 346–355, 358–361,
365–368, 370, 374, 376, 377, 379–384, 386–390,
395, 396, 398–401, 406, 407, 411, 414, 419, 421,423, 436–445, 459, 460, 462, 466, 472, 479, 486,491, 492, 497–499, 503, 506, 509, 511, 512, 515,521–523, 529, 535, 538, 539, 545–548, 550–553,555, 560, 566–568, 570–574, 576–580, 585, 588,
594, 595, 602, 604, 605, 613–617, 632–640, 645,
646, 648–663, 665–668, 670, 672–688, 690–694,696–700, 702–712, 715–717, 724–727, 730–734,736–738, 741, 744–749, 752–761, 769–789, 791–801, 803–853, 856, 864, 889, 892–902, 904, 909,910, 912–921, 923, 929, 940–949, 959–964, 966–
975, 978, 979, 981–983, 991–993, 995, 999, 1002,
1003, 1005, 1008, 1009, 1013, 1019–1030, 1032,1033, 1035–1040, 1043, 1046, 1048, 1056, 1108,1110, 1113, 1116, 1117, 1119, 1120, 1122, 1123,1126, 1127, 1130–1133
γ(x)..2 1 5 ,3 3 5 ,3 3 8 ,3 4 0 ,3 4 6 ,3 4 7 ,3 7 0 ,4 4 0 ,4 9 2 ,
496, 639, 657, 677, 706, 899, 902, 1027
incomplete (Γ( x, y))........ xxxix, 215, 338, 340,
346–348, 352, 366, 368, 436, 438, 498, 576, 657,658, 710, 749, 787, 899–902, 1002, 1027, 1110
incomplete ( γ(x, y))......... xxxix, 439, 899–902
gd(x).................. seeGudermannian function
Ge function ............................ xl, 953–955
ge function ............................ xl, 953, 955
Gegenbauer polynomial ( C
n(x))....... xl, 327, 406,
795–800, 927, 940, 941, 969, 983, 990–993, 995,997, 999, 1017
Gek function ........................... xl, 955, 957
Gey function ...................... xl, 765, 955–957
grad.................. 1050, 1051, 1053, 1055, 1056
Gudermannian (gd) ................. xl, 52, 53, 116
H
H function ............................ xli, 879, 880
Hn(x).................... seeHermite polynomials
H(x)........................... seeStruve function
H(x).......................... seeHankel function
H(x)........................ seeHeaviside function
Hankel function ( Hn(x)).................... xxxvii,
xli, 339, 350, 351, 368, 370, 385, 492, 653, 663,688, 691, 693–695, 698, 702, 709, 723, 750, 752,753, 768, 778, 789, 850, 910, 911, 914–916, 920,
922, 923, 925–928, 931, 940, 944
He
n(x)................... seeHermite polynomials
1156 INDEX OF FUNCTIONS AND CONSTANTS
Heaviside Function (H( x)).....xliv, 642, 750, 1115,
1118, 1131
heiν(z)..................... seeThomson functions
herν(z)..................... seeThomson functions
Hermite polynomials
Hn(x)...... xxxvi, xxxvii, xli, 365, 503, 803–806,
810–812, 983, 992, 996–998, 1001, 1030
Hen(x)............................ xxxvi, xxxvii
Hermitian .............. xliv, 1070, 1071, 1082, 1083
hyperbolic
cosine integral ................... seechi function
sine integral ..................... seeshi function
hypergeometric functions
F ............................ xxxix, xl, 315–318,
320, 327, 329, 330, 335, 347–349, 351, 368, 370,374, 375, 394, 398, 436, 438–440, 442, 444, 488,490, 503, 512, 517, 639, 646, 648, 654, 657, 663,670, 671, 673, 677–681, 683, 685, 688, 690, 699,
703, 704, 706, 707, 711, 712, 736, 737, 745, 749,
754, 755, 759, 760, 771–776, 779, 780, 784, 791,792, 794–797, 801, 803, 805, 807–810, 813–818,821–824, 826–835, 838, 841, 844, 846, 848, 849,889, 910, 946, 982, 999, 1005–1013, 1015–1023,1025, 1033, 1035, 1037, 1039
confluent (Φ) ...... xxxix, 1022–1024, 1027, 1028,
1030–1032
confluent (Ψ) ..816, 1023, 1027, 1028, 1038, 1039
I
incomplete beta function
I........................................ xli, 910
B............................... xxxix, 910, 1132
incomplete Gamma function ..seegamma function,
incomplete
inverse functions ........................ 1118, 1121
J
Jacobi elliptic functions ...seecd, cn, cs, dc, dn, ds,
nc, nd, ns, sc, sd, sn
Jacobi polynomial ( pn(x))...... xli, 998–1000, 1003
Jacobi theta function (Θ) ....xxxiv, xxxix, 879, 880
Jacobi zeta function (zn) ..................... xxxiv
K
kei(z)...................... seeThomson functions
ker(z)...................... seeThomson functions
K/prime............................................. 867
k/prime.......................... xliv, 134, 135, 184–200,
204, 206, 225, 263, 410, 472–475, 562, 567, 568,585, 588, 592, 593, 596–602, 604–606, 619–626,631–633, 859–868, 870–873, 875, 879, 881, 1006
Kronecker delta ........ xliv, 1046, 1047, 1076, 1088L
L............................ seeLaplace transform
L2(x)..................... seedilogarithm function
Ln(x)o rLα
n(x)........... seeLaguerre polynomials
L(x).................... seeLobachevskiy function
L(x)........................... seeStruve function
Laguerre
function ( Lα
n(x))....348, 441, 707, 709, 803–806,
808–812, 840, 901, 983, 1000–1004, 1028
polynomial ( Ln(x))..xli, 344, 806, 808, 809, 811,
812, 844
Laplace transform ...... xliv, 1107, 1108, 1129, 1138
Legendre functions
first kind ( Pn(x)).....xli, 93, 106, 327, 390, 405,
406, 409, 513, 612, 698, 707, 719, 769–772, 774,776–782, 785, 786, 788–794, 801, 809, 815, 829,933, 936, 940, 941, 959–961, 963–969, 972–990,992, 999, 1017, 1131
second kind ( Q
n(x))...... xli, 324, 373, 383, 696,
719, 769–771, 773, 777, 780, 785, 788, 790, 791,959, 960, 965, 966, 968, 972, 973, 975–981, 986
Lerch function (Φ) ................ xxxix, 642, 1039
li function ...xxxv, xli, 238, 340, 527, 553, 636, 637,
883, 884, 887, 902, 1027
limit....... xxxii, 6–8, 14, 21, 26, 53, 250–252, 511,
610, 611, 617, 635, 883, 887, 890, 894, 895, 904,905, 931, 951, 963, 992, 1000, 1003–1006, 1023,1038–1040, 1067, 1101, 1104, 1106–1108, 1118,1121, 1130, 1136, 1139
ln function ....xxvii–xxix, xxxi, xxxii, xxxiv–xxxvii,
3, 9–11, 23, 26, 27, 43, 44, 46, 47, 49, 51–56,61–67, 69–85, 87, 90, 94, 97, 99, 103, 104, 106,113–120, 123–130, 133, 143, 145–148, 150, 155–161, 167–172, 174–176, 178, 186–197, 199, 200,204–207, 220–225, 237–245, 250, 316, 321, 324,
326, 330–332, 334, 338–340, 353–364, 369–373,
375, 376, 378–381, 383, 386–390, 395, 402, 410,431, 433, 434, 438, 447–449, 451–457, 462–466,470–473, 483–485, 495, 497, 499–502, 517–521,527–607, 622–628, 631–633, 636–645, 647–649,656, 658, 659, 661, 668, 671, 672, 695, 702, 718,
719, 728, 747, 748, 755, 763, 861, 862, 868, 880,
882–887, 891–893, 895, 898–900, 902–907, 909,911, 914, 919, 929, 937–939, 944, 963, 969, 972,977–979, 981, 982, 990, 1006, 1011–1013, 1026,1027, 1037–1040, 1046, 1048, 1056, 1113, 1115,1120, 1123, 1125, 1128, 1132, seelog function
Lobachevskiy function ( L)...xli, 147, 225, 375, 380,
381, 530–534, 588, 589, 593, 891
log function ................ 27, 642, seeln function
INDEX OF FUNCTIONS AND CONSTANTS 1157
Lommel function (S) ...... xxxvi, xli, 339, 346, 352,
371, 384, 386, 417, 670, 674, 676–678, 680, 681,756, 758, 760, 761, 779, 782, 783, 785, 787, 788,794, 815, 816, 819, 828, 945–947, 950, 1035
Lommel function (s) ....xli, 419–421, 439, 443, 670,
692, 725, 760, 761, 945, 946, 1133
Lommel function (U) ...................... xlii, 947
Lommel function (V) ................. xlii, 947, 948
M
M............................ seeMellin transform
Mλ,μ(z)................... seeWhittaker functions
MacRobert function ( E)................ 1035, 1036
Mathieu functions
Se................. xxxviii, xli, 764–766, 953, 954
se................. xxxviii, xli, 763–766, 951–957
max.....851, 854, 856, 987, 1066, 1081–1086, 1088,
1091
Meijer function (G) .....xl, 351, 444, 654, 690, 691,
704, 711, 758, 776, 778, 817–819, 825–832, 835,838, 844, 845, 847, 850–856, 1032–1035
Mellin transform ......................... xliv, 1130
min...................... 851, 854, 856, 1085, 1091
N
Nν(z)....................... seeBessel function, Y
nc function .................................. xxxiii
nd function .................................. xxxiii
Neumann function ........... seeBessel function, Y
Neumann polynomial ( On(x))...... xxxvii, xli, 346,
384, 386, 946, 949, 950
norm ........................... see| |·| |and| |·| | p
ns function .................................. xxxiii
O
On(x)................... seeNeumann polynomials
orthogonal function ............................ 798
P
Pn(x)........ seeJacobi polynomials and Legendre
polynomials
Pn(x)........... seeLegendre functions (first kind)
Pm
n(x).....seeLegendre functions (associated, first
kind)
parabolic cylinder function (D) .......... xxxviii, xl,
348, 349, 352, 365, 384, 390, 503, 504, 506, 653,
657, 658, 697, 708, 712, 740, 746, 802, 805, 811,
841–850, 1028–1031Phi function (Φ) ...................... xxxvi, xxxix,
xl, 239, 336–338, 344, 345, 353, 354, 358, 364,367, 371, 376, 379, 381, 384, 390, 489, 503, 504,526, 574, 604, 629, 640, 645–649, 748, 749, 755,781, 802, 835, 838, 887–891, 899, 902, 997, 998,1001, 1050, 1051, 1056, 1057
Pochhammer symbol ............ xliii, 321, 330, 635,
672, 705, 900, 918, 947, 1010–1012, 1018, 1022,1031, 1048
polynomials ...................... seespecific name
principal value (PV) ...xliii, 322, 329, 335, 337, 433,
454, 528, 534, 563, 572, 883
Q
Qn(x)........ seeLegendre functions (second kind)
Qm
n(x)...seeLegendre function (associated, second
kind)
R
root............ 15, 84, 104, 331, 539, 542, 553, 576
3....7 2 ,8 6 – 8 8 ,2 6 4 ,3 3 0 ,3 3 1 ,3 6 3 ,3 6 4 ,5 3 9 ,5 7 0 ,
918
4......... 73, 78, 83, 105, 135, 136, 139, 210, 211,
263–265, 272, 295, 296, 312–315, 483, 493–495,507, 524, 525, 868, 875, 878–881, 997
8............................................ 105
2
k...................... ..................... 894
rot...................... ..................... 1050
S
Sn(x)...................... seeSchlafli polynomials
S(m)
n................ seeStirling numbers, first kind
s(x).......................... seeLommel function
S(x)....... seeLommel function and Fresnel cosine
integral
sc function ................................... xxxiii
Schlafli polynomial ( Sn(x))............ xli, 949, 950
sd function .................................. xxxiii
Se(x)........................ seeMathieu functions
se(x)........................ seeMathieu functions
sec function .................. 36, 39, 43, 44, 50, 52,
64, 113, 114, 155, 156, 315, 323, 328, 329, 371,372, 377–379, 389, 395, 396, 400, 401, 403–405,410–414, 421, 422, 436, 438, 439, 446, 471, 472,479, 541, 551, 586, 646, 653, 661, 663, 664, 669,670, 674, 682, 689, 691, 699, 706–708, 710, 713,
716, 718, 719, 726, 728, 734, 740, 749, 757, 761,
783, 806, 820, 825, 845, 846, 864, 921, 922, 932,1007, 1126, 1132, 1133
sech function ............................ 27, 43, 62,
113–115, 323, 387, 509, 634, 715, 750, 751, 787,
800, 802, 841, 883, 1128, 1133
1158 INDEX OF FUNCTIONS AND CONSTANTS
shi function ..xxxvi, xli, 142–144, 495, 644, 645, 886
Si function ......... xxxv, 643, 886, 930, 1115, 1122
si function ........ xxxv, xli, 219–221, 340–344, 421,
423–426, 447, 495, 505, 506, 528, 529, 571, 572,578, 581, 594, 595, 599, 605, 628, 629, 638–644,647, 649, 650, 656, 658, 748, 762, 886, 887, 992,
1115
sigma function ..................... xxxix, 876, 877
sign function ..xlv, 46, 177, 241, 243, 251, 322, 350,
351, 365, 370, 423, 437, 438, 447, 465, 485, 594,596, 603, 604, 610, 611, 640, 642, 652, 750, 768,885, 1118–1120, 1122
sin function ................ xxvii, xxix, xxxi, xxxii,
xxxiv–xxxviii, 4, 13, 19, 20, 23, 26–52, 55, 56,64, 74–76, 79, 147, 151–237, 249, 250, 253, 254,263–265, 317, 318, 321–323, 325–329, 331–333,339–345, 348, 354, 355, 358, 359, 371–373, 375,377–383, 385, 387–534, 537, 539–547, 550, 551,
554, 558, 559, 561–563, 565, 567–573, 576, 578,
579, 581–601, 604–606, 608, 610–612, 616, 621–623, 628, 629, 631–633, 635–644, 647–651, 655,656, 658, 659, 662–665, 667, 669, 671–675, 677,678, 680, 684, 686, 688, 689, 691, 692, 695, 698,703, 705, 706, 708, 709, 711, 713–715, 717–748,
751, 752, 754–756, 758–760, 762–770, 773, 774,
776, 777, 779–782, 784, 789, 793, 794, 797–800,802, 806, 808, 810–812, 817, 820, 825, 829, 830,833, 836, 837, 839, 842, 844–846, 848–850, 853,859–869, 876–880, 882–893, 896, 898, 900, 904,
906–910, 912–918, 920, 922, 924, 925, 927, 928,
930, 933–940, 942, 943, 945–951, 954, 958, 959,961–964, 966–981, 984–992, 994, 997, 998, 1000,1005–1007, 1009, 1013, 1025, 1026, 1029, 1036–1038, 1042, 1044, 1053, 1057, 1060, 1061, 1066,1067, 1090, 1110–1113, 1115, 1117–1128, 1131,
1132, 1137, 1140
sinh function ......................... xxxvi, 27–36,
38, 40, 42, 43, 45, 47, 48, 50–52, 64, 110–151,231–237, 251, 338, 339, 358, 371–390, 407, 419,425, 432, 433, 438, 439, 448, 451, 452, 454, 455,461, 466–468, 477, 484, 485, 489, 491, 496, 502,
504, 508–527, 570, 573, 578–582, 595, 596, 606,
610, 611, 621, 622, 634, 643–645, 664, 686, 698,702, 704, 705, 710, 711, 713–716, 723, 729, 735,747, 751–753, 755, 760, 763–766, 778, 787–789,806, 843, 844, 886, 888, 896, 898, 913–915, 917,943, 953, 955–957, 960–963, 967–969, 980, 981,
997, 1006, 1025, 1029, 1040, 1112–1115, 1119,
1120, 1124, 1125, 1128, 1140
sn function ......... xxxiii, xxxiv, xli, 623–626, 714,
866–873, 875, 879, 880
special functions ............................... 859square root .................................. xxxii,
xxxiv, xxxvi, xxxviii, xliv, 2, 9–11, 14, 15, 23,25, 26, 30, 37, 43, 44, 54–61, 63–67, 71–79, 83–99, 103–109, 125–139, 158, 170–175, 177–184,197, 199, 200, 202–214, 225, 226, 230, 239–245,249, 251, 254–315, 317–319, 321, 324–328, 330,
333, 336, 337, 339, 344–353, 355, 359, 363–376,
380, 382–385, 390, 391, 393–396, 400–402, 404–411, 414–419, 421, 425, 426, 428–430, 434–436,440–446, 448, 451–454, 456, 457, 460, 472–479,481–483, 485, 486, 488–497, 499, 501–507, 511,513–515, 517, 518, 522–527, 529, 531, 532, 534,
535, 537–539, 542, 543, 545, 549–551, 553, 554,
556–558, 560, 562, 563, 565–568, 570–576, 578–581, 584, 585, 588, 590–593, 595–606, 608–613,615–617, 619, 621–623, 629, 631–635, 637–642,644–651, 653, 657, 659, 661–668, 670, 672–675,677, 678, 680–683, 685–763, 766–768, 771, 773,
777, 778, 780–782, 785–789, 791–794, 800–808,
810–812, 814, 815, 819, 828, 829, 837, 841, 843–845, 848, 853, 854, 856, 859–866, 868, 870–876,879–881, 887–891, 893–902, 905, 908, 909, 913–915, 917, 918, 920–926, 928, 931–946, 950, 951,958, 960–974, 976–983, 985, 987, 988, 990–998,
1000, 1002, 1003, 1007, 1009, 1018, 1019, 1023,
1026–1030, 1035, 1038, 1052, 1054, 1055, 1060,1082, 1116–1121, 1123, 1125–1127, 1129, 1131
step function ................................... 798
Stirling number
first kind ( S
m
n)................... xlv, 1046–1048
second kind ( Sm
n)................ xlv, 1046–1048
Struve function
H(x)..xli, 345, 351, 421, 435, 442, 443, 573, 647,
659, 660, 663, 664, 669, 675, 677, 679, 680, 692,694, 708, 722, 725, 735, 753–759, 787, 838, 848,
856, 942, 943, 946, 1035, 1132
modified ( L(x))...... xli, 345, 350, 351, 435, 441,
515, 595, 605, 663, 664, 669, 671, 675, 676, 678,679, 692, 722, 736, 753–759, 787, 794, 942, 943
INDEX OF FUNCTIONS AND CONSTANTS 1159
T
Tn(x).................. seeChebyshev polynomials
tan function .............. xxvii–xxix, 27–30, 35, 36,
39, 40, 42, 44, 46, 50, 52, 53, 55, 56, 60, 64, 126,151, 155–161, 167–178, 180, 181, 183–185, 188,189, 194, 195, 199, 200, 202–207, 209, 212, 214,
222–225, 229, 230, 274, 322, 332, 339, 340, 355,
371, 375–379, 381, 382, 384, 388, 389, 393, 395,396, 400, 403–405, 409–414, 421–423, 433, 434,451–455, 457–459, 471–475, 483–486, 492, 493,496, 498, 505, 506, 509, 515, 518, 532–535, 537,541, 545, 546, 567–570, 579, 582, 586–589, 591–
594, 597–599, 604, 606, 622, 631, 632, 659, 664,
669–671, 682, 686, 699, 716, 717, 725, 728, 744,745, 754, 766, 849, 862–865, 867–869, 883, 886,890, 894, 905, 909, 922, 932, 954, 969, 971, 979,986, 990, 1006, 1007, 1023, 1123, 1127, 1132
tanh function ......... 12, 27–29, 31, 39, 40, 42, 44,
51, 52, 62, 64, 110, 113–120, 123–126, 128–132,
134–137, 139, 145–148, 338, 380–383, 387, 390,472, 484, 485, 489, 502, 509, 512, 513, 516, 518,520, 569, 606, 621, 622, 716, 717, 751, 753, 766,788, 789, 841, 921, 922, 956, 957, 1025, 1124
theta function ( θ)....xxxiv, 521, 633, 634, 877–883
Thomson functions
bei(x).................. xxxix, 761–763, 944, 945
ber(x).................. xxxix, 761–763, 944, 945
hei(x)................................... xli, 944
her(x)................................... xli, 944
kei(x)...... xli, 641, 663, 672, 674, 748, 762, 763,
944, 945
ker(x)...... xli, 641, 663, 671, 674, 747, 762, 763,
944, 945
toroidal function ............................... 981
tr........................................ seetrace
trace ......................................... 1084
transpose ............... xlv, 1069–1073, 1075, 1089U
Un(x).................. seeChebyshev polynomials
unilateral ztransform ....... 1135, 1138–1140, 1159
W
Wλ,μ(z)................... seeWhittaker functions
Weber function ( E)....338, 339, 346, 353, 371, 384,
421, 423, 670, 671, 751, 943, 946, 948, 949
Weierstrass function ( ℘)....xxxix, xl, 626, 873–877,
880
Whittaker functions
M..... xli, 338, 348, 445, 654, 682, 697, 703, 705,
706, 709, 710, 715–717, 736, 748, 749, 784, 785,787, 819–841, 1024–1027
W ......... xlii, 338, 346–349, 367, 368, 384, 423,
445, 635, 652, 654, 682, 697, 698, 704, 706, 707,709, 710, 712, 715, 716, 726, 727, 736, 745, 749,
756, 759, 761, 776–778, 781, 782, 784–788, 803,
814–817, 819–841, 843, 844, 846, 847, 857, 979,1024–1028, 1035
X
Xb........................ seebilateral ztransform
Xu...................... seeunilateral ztransform
Y
Y ........................... seeBessel function, Y
Young function ( C)......... xxxvi, xl, 417, 439, 440
Z
zeta function ( ζ)..xxxix, 8, 338, 353, 354, 358, 359,
376, 377, 379–381, 386–389, 433, 434, 449, 471,509, 540, 542, 543, 550, 552, 560, 567, 569, 576,577, 580, 587, 591, 593, 607, 626, 633, 634, 658,659, 802, 876, 877, 880, 894, 898, 903–905, 907,
909, 1036–1041, 1133
zn(x)...................... seeJacobi zeta function
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Index of Concepts
This index refers to concepts appearing in the text.
A
Abel’s identity ................................ 1098
absolute convergence ............................. 6
absolute values .................................. 63
addition theorems ......................... 973, 975
adjoint ....................................... 1099
equations .................................. 1098
algebraic
inequalities ........................... 1059, 1060
algebraic functions ......................... 82, 253
and arccosine ................................ 242
and arccotangent ............................ 244
and arcsine .................................. 242
and arctangent .............................. 244
and associated Legendre functions ........... 789
and Bessel functions .................... 674, 715
and exponentials ........................ 344, 363
and hyperbolic functions ........... 132, 375, 715
and logarithmic functions .................... 538
and logarithms .............................. 238
and powers .................................. 363
and rational functions ....................... 789
and trigonometric functions .................. 434
alternating series ................................. 7
amplitudes ..................................... 866
analytic continuation ..................... 970, 1012
Anger functions ................................ 948
angle of parallelism ............................. 51
anticommutative .............................. 1049
approximate solution .................... 1093–1096
approximation by tangents ..................... 921
arccosecant .................................... 242
and powers .................................. 244
arccosine ...................................... 241
and algebraic functions ...................... 242
arccotangent ................................... 242
and algebraic functions ...................... 244
arcsecant ...................................... 242
and powers .................................. 244arcsine ......................................... 241
and algebraic functions ...................... 242
arctangent ..................................... 242
and algebraic functions ...................... 244
and Bessel functions ......................... 747
argument ...................................... 866
of a complex number ........................ xliv
arithmetic mean theorem ..................... 1056
arithmetic progression ............................ 1
arithmetic-geometric inequality ............... 1060
arithmetic-geometric progression ................. 1
associated Legendre functions ...769, 788, 958, 972,
974
and algebraic functions ...................... 789
and Bessel functions .................... 782, 787
and exponentials ............................ 776
and hyperbolic functions ..................... 778
and powers ........................ 770, 776, 779
and probability integral ...................... 781
and rational functions ....................... 789
and trigonometric functions .................. 779
associated Mathieu functions ................... 952
asymptotic expansions ........................ 1146
asymptotic result .......... 21, 356, 895, 1026, 1029
asymptotic series ................................ 21
B
Ballieu theorem ............................... 1086
basic theorems ................................ 1091
Bateman’s function ........................... 1023
Bernoulli
numbers .............................. 1040, 1045
polynomials .......................... 1040, 1041
Bessel functions ...629, 659, 748, 749, 753, 910, 912,
914, 916–920, 924, 925, 928, 931, 933–937, 940,
941, 954, 1146, seeConstant/Function index
and algebraic functions ................. 674, 715
and arctangent .............................. 747
and associated Legendre functions ...... 782, 787
1161
1162 INDEX OF CONCEPTS
Bessel functions ( continued )
and Chebyshev polynomials ................. 803
and exponentials ....694, 699, 708, 711, 713, 715,
742, 834
and Gegenbauer functions ................... 798
and hyperbolic functions ........... 713, 715, 747
and hypergeometric functions ................ 817
confluent ......................... 830, 831, 834
and Legendre polynomials ................... 794
and logarithms .............................. 747
and MacRobert functions .................... 854
and Mathieu functions ....................... 767
and Meijer functions ......................... 854
and parabolic cylinder functions ............. 845
and powers .....664, 675, 689, 699, 708, 711, 727,
742, 831, 834
and rational functions ....................... 670
and Struve functions ........................ 756
and trigonometric functions ...717, 727, 742, 747
generating functions ......................... 933
imaginary arguments ........................ 911
Bessel inequality .............................. 1068
bilateral z-transform .................... 1135, 1136
bilinear concomitant .......................... 1099
binomial coefficients .............................. 3
symbol ...................................... xliii
binomials ................... .................... 25
and powers ............................. 315, 322
Bonnet–Heine formula ......................... 988
bounded variation ............................... 20
boundedness theorems ........................ 1106
branch points ............................ 866, 1024
Brauer theorem ......................... 1086, 1088
Buniakowsky inequality ........... 1059, 1061, 1064
C
Calogero ...................................... 1089
Carleman inequality .................... 1060, 1066
Catalan constant ................... xxxii, 1046, see
Constant/Function index
Cauchy principal value ......................... 528
Cauchy problem ........................ 1093, 1095
Cauchy–Schwarz–Buniakowsky inequality .....1059,
1061, 1064
Cayley–Hamilton theorem .................... 1084
change of variables ................... 248, 607, 608
characteristic equation ........................ 1071
characteristic polynomial ...................... 1084
characteristic values .......................... 1084
Chebyshev inequality ................... 1059, 1065Chebyshev polynomials .................... 988, 993
and Bessel functions ......................... 803
and elementary functions .................... 802
and powers .................................. 800
Christoffel formula ............................. 983
Christoffel summation formula ................. 986
circle of convergence ............................ 16
circulants ..................................... 1078
classification system ........................... xxxi
classified references ........................... 1145
column norm ................................. 1082
comparison of approximate solutions ....1094, 1096
comparison theorem ......... 1100, 1101, 1103, 1104
complementary error function ...seeerror functions,
complementary
complementary modulus ....................... 859
complete elliptic integrals ............. 619, 632, 859
complex analysis .............................. 1146
complex conjugate ............................. xliii
conditional convergence .......................... 6
conditions, Dirichlet ............................. 19
confluent hypergeometric function .............. see
hypergeometric function, confluent
conical functions ............................... 980
constant of integration .......................... 63
constants ............. seeConstant/Function index
Catalan .................... seeCatalan constant
Euler ......................... seeEuler constant
continued fraction .............................. 902
continuity, Lipschitz .................... 1094, 1095
converge
absolutely ..................................... 6
conditionally .................................. 6
uniformly ..................................... 15
convergence
circle ......................................... 16
radius ........................................ 16
tests....................................... 6, 19
convexity ..................................... 1066
convolution ................................... 1118
theorem ............ 1108, 1118, 1122, 1130, 1136
coordinates, curvilinear ....................... 1052
cosine
and rational functions .................. 171, 390
and square roots ............................. 472
integral ............................ 628, 639, 886
hyperbolic ............................. 644, 886
multiple angles .............................. 161
cosine-amplitude ............................... 866
Cramer’s rule ................................. 1077
cube roots ...................................... 86
curl.......................................... 1050
INDEX OF CONCEPTS 1163
curvilinear coordinates ........................ 1052
cycles ......................................... 1046
cylinder function ....seeparabolic cylinder function
D
Darboux–Christoffel formula ................... 983
de Moivre’s theorem .......................... 1060
decreasing solutions ........................... 1104
definite integrals ......... 247,seeintegrals, definite
delta amplitude ................................ 866
derivative of a composite function ............... 22
determinants ...................... 1075, 1076, 1078
Gram ...................................... 1080
Hessian .................................... 1079
Jacobian ................................... 1078
Vandermonde .............................. 1078
Wronskian ................................. 1079
differential equations ..................... 873, 874,
910, 931, 944, 947–950, 952, 958, 974, 975, 980,981, 983, 993, 995, 998, 1000, 1003, 1011, 1013,1015, 1024, 1031, 1034, 1093
adjoint ..................................... 1098
exact....................................... 1097
homogeneous ............................... 1097
hypergeometric ............................. 1010
partial ............................... 1018, 1031
Riccati ..................................... 1099
Riemann ............................. 1014, 1022
second-order .............. 1017, 1098, 1100, 1104
self-adjoint ................................. 1098
special types ............................... 1097
variables separable ......................... 1097
differentiation
of integrals ............................. 21, 1064
of matrices ................................. 1073
of vectors .................................. 1050
dilogarithm .................................... 642
diophantine relations .......................... 1089
directional derivative .......................... 1051
Dirichlet conditions ............................. 19
Dirichlet lemma .............................. 1067
div...................... ..................... 1050
divergence theorem ........................... 1055
DOC (domain of convergence) ................ 1135
domain of convergence (DOC) ................ 1135
dominant solutions ...................... 1104, 1105
double factorial symbol ........................ xliii
double integrals .......................... 610, 1021
doubling formula ............................... 896
doubly-periodic function .................. 866, 874E
eigenvalues ................... 951, 1071, 1084, 1087
eigenvectors .................................. 1071
elementary functions ................. 25, 247, 1006
and Chebyshev polynomials ................. 802
and Gegenbauer polynomials ................ 797
and Legendre polynomials ................... 792
and MacRobert functions .................... 850
and Meijer functions ......................... 850
indefinite integrals ............................ 63
elliptic functions ................ 619, 631, 865, 1148
Jacobian ................................ 866, 870
order........................................ 865
Weierstrass .......... 626,seeWeierstrass elliptic
functions
elliptic integrals ...104, 184, 619, 621, 631, 632, 859
complete ................. 394, 472–474, 632, 859
derivatives ......................... 394, 863, 865
functional relations .......................... 863
generalized .................................. 635
Jacobian .................................... 623
kinds ........................................ 859
equations
differential ............. seedifferential equations
first-order ............................ 1093, 1096
linear ...................................... 1096
special types ............................... 1097
system ............................... 1094, 1095
error functions ........................... 887, 1147
complementary .............................. 887
essential singularity ........................... 1024
Euclidean norm ............................... 1081
Euler
constant .....xxxii, seeConstant/Function index
dilogarithm .................................. 642
integrals ................................ 892, 908
numbers ........................ 1040, 1043, 1045
polynomials ................................ 1044
substitutions ................................. 92
exact differential equations .................... 1097
expansion of determinants ............... 1075, 1076
expansions, asymptotic ....................... 1146
expansions, Weierstrass ........................ 869
exponential integrals ..627, 636, 638, 883, 885, 1147
and exponentials ............................ 628
and powers .................................. 627
exponentials ........................... 2 6 ,1 0 6 ,3 3 4
and algebraic functions ................. 344, 363
and associated Legendre functions ........... 776
and Bessel functions .....694, 699, 708, 711, 713,
715, 742, 834
and complicated arguments .................. 336
1164 INDEX OF CONCEPTS
exponentials ( continued )
and exponential integrals .................... 628
and gamma functions ........................ 652
and hyperbolic functions ...... 148, 338, 382, 386,
522, 525, 713, 715
and hypergeometric functions ................ 814
confluent .............................. 822, 834
and inverse trigonometric functions .......... 605
and logarithmic functions .....339, 571, 573, 599
and parabolic cylinder functions ............. 842
and powers .....148, 346, 353, 363, 364, 386, 497,
525, 573, 699, 708, 711, 742, 754, 776, 834, 842
and rational functions .............. 106, 340, 353
and Struve functions ........................ 754
and trigonometric functions ...227, 339, 485, 493,
495, 497, 522, 525, 599, 742
matrix ..................................... 1074
of exponentials .............................. 338
series ......................................... 27
F
factorial symbol ............................... xliii
field theory ................................... 1049
figures ............ 608–610, 892, 913, 915, 916, 1036
final value theorem ........................... 1139
finite sums ....................................... 1
first mean value theorem ...................... 1063
first-order equations .................... 1093, 1096
first-order systems ............................ 1094
footnotes ..................................... xxix,
xxxi, 82, 132, 247, 248, 274, 397, 410, 547, 656,859, 867, 908, 920, 931, 981, 991, 1039, 1141
Fourier series .................... 19, 46, 1066, 1067
generalized ........................... 1067, 1068
Fourier transform ....................... 1107, 1117
basic properties ............................ 1118
cosine ................................ 1121, 1129
properties ................................. 1121
table...................................... 1126
exponential ................................ 1129
sine.................................. 1121, 1129
properties ................................. 1121
table...................................... 1122
tables ................................ 1118, 1120
fourth roots .................................... 313
fractional transformation ...................... 1014
Fresnel integrals ................ 629, 649, 887, 1147
Frobenius theorem ............................ 1088
functional series ................................. 15functions ............. seeConstant/Function index
inner...................................... xxviii
ordering ................................... xxviii
orthogonal .................................. 798
outer ...................................... xxvii
fundamental inequalities ...................... 1094
fundamental system ........................... 1100
G
gamma functions .......... 650, 892, 894, 895, 1147
and exponentials ............................ 652
and logarithms .............................. 656
and powers .................................. 652
and trigonometric functions .................. 655
incomplete .............................. 657, 899
Gauss divergence theorem ..................... 1055
Gegenbauer functions and Bessel functions .....798
Gegenbauer polynomials ....................... 990
and elementary functions .................... 797
and powers .................................. 795
general formulas ............................ 65, 249
generalized elliptic integrals .................... 635
generalized Fourier series ................ 1067, 1068
generalized Legendre polynomials .............. 990
generating functions
Bernoulli numbers .......................... 1040
Bernoulli polynomials ................ xxxii, 1041
Bessel functions ............................. 933
Chebyshev polynomials ...................... 995
Euler numbers ............................. 1043
Euler polynomials .................... xxxii, 1044
Hermite polynomials ........................ 997
Jacobi polynomials ......................... 1000
Legendre polynomials ....................... 988
Neumann polynomials ................ xxxvii, 950
Stirling numbers ...................... 1046, 1047
geometric progression ............................ 1
Gerschgorin theorem .................... 1083, 1088
grad.......................................... 1050
gradient ...................................... 1050
Gram determinant ............................ 1080
Gram inequality .............................. 1065
Gram–Kowalewski theorem ................... 1080
Green theorem .......................... 1055, 1056
Gronwall’s lemma ............................. 1094
growth estimates .............................. 1104
growth of maxima ............................ 1106
Gudermannian (gd) ............................. 52
INDEX OF CONCEPTS 1165
H
Hadamard’s inequality ........................ 1077
Hadamard’s theorem .......................... 1077
Hankel functions .......................... 910, 925
Heaviside step function ........................ xliv
Heine formula .................................. 988
Helmholtz equation ...................... 767, 1052
Hermite method ................................ 67
Hermite polynomials ................. 803, 996, 997
Hermitian matrix ....................... 1077, 1089
Hessian determinant .......................... 1079
H¨older inequality ................. 1059, 1061, 1064
homogeneity ................................... 875
homogeneous differential equations ............ 1097
hyperbolic
amplitude .................................... 52
cosine integral .......................... 644, 886
sine integral ............................ 644, 886
hyperbolic functions ................... 28, 110, 371
and algebraic functions ............. 132, 375, 715
and associated Legendre functions ........... 778
and Bessel functions ............... 713, 715, 747
and exponentials ....148, 338, 382, 386, 522, 525,
713, 715
and inverse trigonometric functions .......... 605
and linear functions ......................... 120
and logarithmic functions .................... 578
and Mathieu functions ....................... 763
and parabolic cylinder functions ............. 843
and powers ............... 139, 148, 386, 516, 525
and rational functions ....................... 125
and trigonometric functions ...231, 509, 516, 522,
525, 747, 763
inverse ................................... 56, 240
and logarithms ............................. 237
powers ................................. 110, 120
hypergeometric
differential equation ........................ 1010
series ................................. 1005, 1008
confluent .................................. 1031
generalized ................................ 1010
hypergeometric functions ....... 812, 841, 946, 1005,
1006, 1039, 1147
and Bessel functions ......................... 817
and exponentials ............................ 814
and powers .................................. 812
and trigonometric functions .................. 817hypergeometric functions ( continued )
confluent .............. 820, 841, 1022, 1023, 1147
and Bessel functions .............. 830, 831, 834
and exponentials ...................... 822, 834
and Legendre functions ..................... 839
and parabolic cylinder functions ............ 849
and polynomials ............................ 840
and powers ....................... 820, 831, 834
and special functions ....................... 839
and Struve functions ....................... 838
and trigonometric functions ................ 829
several variables ............................ 1022
two variables ............................... 1018
I
identities
Abel....................................... 1098
Lagrange ................................... 1099
Picone ..................................... 1102
improper integrals ......................... 251, 252
incomplete beta functions ...................... 910
incomplete gamma function .................... 657
increasing solutions ........................... 1104
indefinite integrals
elementary functions .......................... 63
special functions ............................. 619
induced norm ................................. 1082
inequalities .....950, 963, 979, 987, 997, 1041, 1061,
1083–1085, 1094
algebraic ............................. 1059, 1060
Carleman ............................ 1060, 1066
for sets ..................................... 1061
Hadamard ................................. 1077
integral ............................... 1063–1066
Schur ...................................... 1087
triangle .................................... 1061
inertia ........................................ 1072
infinite products ......................... 6, 14, 862
initial value theorem .................... 1136, 1139
inner function ................................. xxxi
integer function ............................... 1135
integer pulse function ......................... 1135
integral
differentiation .......................... 21, 1064
formula ..................................... 985
inequalities ........................... 1063–1066
inversion ................. 1107, 1118, 1121, 1129
part (symbol) ............................... xliii
representations ...... 887, 888, 892, 898, 900, 902,
906, 908, 912, 914, 916, 942, 946, 950, 960, 974,976, 980, 981, 985, 991, 996, 1005, 1021, 1023,
1025, 1028, 1035, 1036, 1039, 1040, 1067
1166 INDEX OF CONCEPTS
integral ( continued )
theorems ................................... 1049
transforms ........................... 1107, 1147
relationships .............................. 1129
integrals
definite ...................................... 247
special functions ........................... 631
double ................................. 610, 1021
elliptic .................................. 104, 859
Fresnel ..................................... 1147
improper ............................... 251, 252
indefinite ................. seeindefinite integrals
Mellin–Barnes .............................. 1021
multiple ................................ 607, 612
pseudo-elliptic ............................... 105
triple ........................................ 610
integration
constant ...................................... 63
techniques .................................... 92
termwise ..................................... 16
interlacing of zeros ............................ 1101
invariants ...................................... 874
inverse z-transformation ...................... 1135
inverse hyperbolic functions ......... seehyperbolic
functions
inverse trigonometric functions ............. 599,see
trigonometric functions
and exponentials ............................ 605
and hyperbolic functions ..................... 605
and logarithms .............................. 607
and powers ........................ 600, 601, 607
and trigonometric functions ............. 605, 607
inversion integral ............ 1107, 1118, 1121, 1129
J
Jacobi polynomials ........................ 806, 998
Jacobi theorem ............................... 1076
Jacobian determinant ......................... 1078
Jacobian elliptic functions ...... 866, 870, 879, 1148
Jacobian elliptic integrals ...................... 623
Jensen inequality ............................. 1066
K
Kneser’s non-oscillation theorem .............. 1103
Kowalewski theorem .......................... 1080
L
L2norm ...................................... 1081
Lagrange identity ....................... 1059, 1099
Laguerre polynomials .................... 808, 1000
Laplace formula ................................ 987
Laplace integral formula ....................... 985Laplace transform ...................... 1107, 1129
basic properties ............................ 1107
table....................................... 1108
Laplacian ................................ 767, 1051
latent roots values ............................ 1084
Laurent series ................................ 1135
least common factor ........................... 798
least common multiple ......................... 798
Lebesgue lemma .............................. 1067
Legendre functions ....................... 975, 1149
and hypergeometric functions
confluent ................................... 839
associated .....seeassociated Legendre functions
special values ................................ 969
Legendre normal form ......................... 859
Legendre polynomials ..................... 983, 988
and Bessel functions ......................... 794
and elementary functions .................... 792
and powers .................................. 791
lemmas
Dirichlet ................................... 1067
Gronwall ................................... 1094
Riemann–Lebesgue ......................... 1067
letters, conventions .............................. 63
linear dependence ............................. 1080
linear equations ............................... 1096
L∞norm ..................................... 1081
Lipschitz continuity ..................... 1094, 1095
Lobachevskiy’s “angle of parallelism” ............ 51
Lobachevskiy’s function ........................ 891
logarithm integrals ........................ 636, 887
logarithms ........................ 53, 237, 527, 529
and algebraic functions ................. 238, 538
and Bessel functions ......................... 747
and exponentials .............. 339, 571, 573, 599
and gamma functions ........................ 656
and hyperbolic functions ..................... 578
inverse ..................................... 237
and inverse trigonometric functions .......... 607
and powers .......... 540, 542, 553, 555, 573, 594
and rational functions .................. 535, 553
and trigonometric functions ...339, 581, 594, 599
gamma functions ............................ 898
Lommel functions ......................... 760, 945
two variables ................................ 947
Lyapunov theorem ...................... 1089, 1105
M
MacRobert functions ..................... 850, 1035
and Bessel functions ......................... 854
and elementary functions .................... 850
and special functions ........................ 856
INDEX OF CONCEPTS 1167
Mathieu functions .....763, 950, 951, 953, 954, 1149
and Bessel functions ......................... 767
and hyperbolic functions ..................... 763
and trigonometric functions .................. 763
imaginary argument ......................... 952
matrix
adjoint ..................................... 1070
cofactors ................................... 1075
determinants ................... seedeterminants
diagonal .................................... 1069
diagonally dominant ........................ 1071
differentiation .............................. 1073
equivalent .................................. 1069
exponential ................................ 1074
Hermitian ...................... 1070, 1077, 1089
idempotent ................................. 1071
identity .................................... 1069
inverse ..................................... 1070
irreducible ................................. 1069
minors ..................................... 1075
principal .................................. 1076
nilpotent ................................... 1071
non-negative definite ....................... 1071
norm................................. 1082, 1083
null........................................ 1069
orthogonal ................................. 1070
positive definite ............................ 1071
reducible ................................... 1069
skew-symmetric ............................ 1070
special ..................................... 1069
symmetric .................................. 1070
trace....................................... 1070
transpose ............................. 1069, 1070
triangular .................................. 1070
unitary ..................................... 1071
maxima ...................................... 1106
mean value theorems ............... 247, 1063, 1064
Meijer functions .......................... 850, 1032
and Bessel functions ......................... 854
and elementary functions .................... 850
and special functions ........................ 856
Mellin transform ........................ 1107, 1129
basic properties ............................ 1130
table....................................... 1131
Mellin–Barnes integrals ....................... 1021
metric coefficients ............................. 1052
metrical coefficients ........................... 1054
Minkowski inequality .............. 1059, 1061, 1065
modulus .............................. 632, 859, 860
multiple angle expansion ........................ 31
multiple integrals ......................... 607, 612N
named theorems .............................. 1087
natural norm ................................. 1082
natural numbers ............................... xliv
necessary conditions .......................... 1104
Neumann functions ............................ 910
Neumann polynomials ......................... 949
nome .......................................... 877
non-oscillation .................... 1100, 1103, 1104
normal form ................................... 859
norms ........................................ 1081
column ..................................... 1082
compatible ................................. 1082
Euclidean .................................. 1081
induced .................................... 1082
matrix ............................... 1082, 1083
natural ..................................... 1082
row........................................ 1083
spectral .................................... 1082
vector ...................................... 1081
notation ....................................... xliii
O
one-sided z-transform ......................... 1135
order of presentation ......................... xxvii
ordinary differential equations ................. 1093
orthogonal curvilinear coordinates ............. 1052
orthogonal polynomials .............. 795, 982, 1149
oscillation .............................. 1100, 1102
Ostrogradskiy–Hermite method ................. 67
Ostrowski inequality .......................... 1066
Ostrowski theorem ............................ 1089
outer function ................................. xxxi
P
parabolic cylinder functions ....841, 849, 1028, 1150
and Bessel functions ......................... 845
and exponentials ............................ 842
and hyperbolic functions ..................... 843
and hypergeometric functions ................ 849
and powers .................................. 842
and Struve functions ........................ 848
and trigonometric functions .................. 844
parameter ..................................... 877
parameter of the integral ....................... 859
Parodi theorem ............................... 1086
Parseval formula .............................. 1136
Parseval theorem ....................... 1067, 1068
partial fractions ................................. 66
partial sums .................................. 1067
Perelomov .................................... 1089
1168 INDEX OF CONCEPTS
periodic ...................... ................... 19
Mathieu functions ........................... 951
periods ................................... 865, 870
permutations ................................. 1046
Perron theorem ............................... 1088
Perron–Frobenius theorem .................... 1088
Picone identity ............................... 1102
Picone theorem ............................... 1102
Pochhammer symbol .......................... xliii
Poincare’s separation theorem ................. 1087
points, singular ................................ 958
Poisson integral ......................... 1056, 1057
poles............................ 865, 870, 874, 892
polynomials .......................... 254, 313, 322
and hypergeometric functions confluent ...... 840
characteristic ............................... 1084
Chebyshev ........... seeChebyshev polynomials
degree 3 or 4 ................................ 859
Gegenbauer ................................. 990
Hermite ................ seeHermite polynomials
Jacobi ................... seeJacobi polynomials
Laguerre ............... seeLaguerre polynomials
Legendre .............. seeLegendre polynomials
orthogonal ........................ 795, 982, 1149
positive definite ......................... 1071, 1072
positive semidefinite .......................... 1072
power series .............................. 16–18, 25
expansion .................................... 42
powers ......................................... 253
and algebraic functions ...................... 363
and arccosecant ............................. 244
and arcsecant ............................... 244
and associated Legendre functions ..770, 776, 779
and Bessel functions .....664, 675, 689, 699, 708,
711, 727, 742, 831, 834
and binomials .......................... 315, 322
and Chebyshev polynomials ................. 800
and exponential integrals .................... 627
and exponentials ....148, 346, 353, 363, 364, 386,
497, 525, 573, 699, 708, 711, 742, 754, 776, 834,842
and gamma functions ........................ 652
and Gegenbauer polynomials ................ 795
and hyperbolic functions ..139, 148, 386, 516, 525
and hypergeometric functions ................ 812
confluent ......................... 820, 831, 834
and inverse trigonometric functions .....600, 601,
607
and Legendre polynomials ................... 791
and logarithmic functions .....540, 542, 553, 555,
573, 594
and parabolic cylinder functions ............. 842powers ( continued )
and rational functions .............. 353, 401, 553
and square roots ............................. 472
and Struve functions ........................ 754
and trigonometric functions ...214, 397, 401, 405,
411, 436, 459, 475, 497, 516, 525, 594, 607, 727,
742, 779
binomials ..................................... 25
hyperbolic functions .................... 110, 120
trigonometric functions ................. 151, 395
principal
function .................................... xxxi
natural norms .............................. 1082
values ............................... 56, 252, 528
vector norms ............................... 1081
probability function ........................... 1150
probability integrals .................. 629, 645, 887
and associated Legendre functions ........... 781
problem, Cauchy ........................ 1093, 1095
product
finite..................... .................... 41
infinite ................................. 6, 14, 45
of vectors .................................. 1049
theorem ..................................... 896
progressions ................................... 1, 8
pseudo-elliptic integrals ................... 105, 184
pulse function ................................ 1135
Q
q-series ........................................ 880
quadratic forms ............................... 1071
quasiperiodicity ................................ 878
R
radius of convergence ........................... 16
rank.......................................... 1072
rate of change theorems ....................... 1057
rational functions ...................... 66, 253, 254
and algebraic functions ...................... 789
and associated Legendre functions ........... 789
and Bessel functions ......................... 670
and cosine .............................. 171, 390
and exponentials ................... 106, 340, 353
and hyperbolic functions ..................... 125
and logarithmic functions ............... 535, 553
and powers ........................ 353, 401, 553
and sine ................................ 171, 390
and trigonometric functions ........ 401, 423, 447
Rayleigh quotient ............................. 1091
real numbers ................................... xlv
reciprocal theorem ............................ 1056
reciprocals .................................... 3, 12
INDEX OF CONCEPTS 1169
references ..................................... 1141
supplementary ............................. 1145
remainder ................... .................... 18
representation theorem ........................ 1056
residues ..................... ................... 870
Riccati equation .............................. 1099
Riemann differential equation ................. 1014
Riemann hypothesis .......................... 1038
Riemann zeta functions ................. 1036, 1150
Riemann–Lebesgue lemma .................... 1067
Rodrigues’ formula ............. 993, 995, 998, 1000
roots...... seesquare roots and Constant/Function
index
fourth .................... ................... 313
Routh–Hurwitz theorem ...................... 1086
row norm ..................................... 1083
S
saltus ....................... .................... 19
scalar product ................................ 1049
Schl¨afli integral formula ........................ 985
Schl¨afli polynomials ............................ 949
Schur’s inequalities ........................... 1087
Schwarz inequality ................ 1059, 1061, 1064
second mean value theorem ............. 1063, 1064
second-order equations ...... 1017, 1098, 1100, 1104
self-adjoint equations ......................... 1098
semiconvergent series ............................ 21
separation theorem ..................... 1087, 1101
series ......................... 860,seespecific type
alternating .................................... 7
asymptotic ................................... 21
convergence ................................... 6
diverge ................... .................... 21
Fourier ........................ 19, 46, 1066–1068
generalized .......................... 1067, 1068
functional .................................... 15
hyperbolic functions .......................... 51
hypergeometric ....................... 1005, 1008
generalized ................................ 1010
of exponentials ............................... 27
of logarithms ................................. 55
power ..................................... 16–18
rational fractions ............................. 26
remainder .................................... 18
semiconvergent ............................... 21
Taylor ..................... ................... 18
trigonometric ............................ 46, 862
sign function ................................... xlv
signature ..................................... 1072
signum function ................................ xlvsine
and rational functions .................. 171, 390
and square roots ............................. 472
integral ............................ 628, 639, 886
hyperbolic ............................. 644, 886
multiple angles .............................. 161
sine-amplitude ................................. 866
singular points ........................... 958, 1038
solenoidal fields ............................... 1052
Sonin theorem ................................ 1106
special functions ............................. xxxix
and hypergeometric functions
confluent ................................... 839
and MacRobert functions .................... 856
and Meijer functions ......................... 856
indefinite integrals ........................... 619
spectral norm ................................. 1082
spectral radius ................................ 1083
spherical functions ............................. 974
square roots ...84, 88, 92, 94, 99, 103, 179, 184, 254
and cosine ................................... 472
and powers .................................. 472
and sin ...................................... 472
trigonometric functions ...................... 408
Steffensen inequality .......................... 1065
step function .................................. xliv
Stieltjes’ theorems ............................. 987
Stirling numbers ........................ 1046, 1048
table................................. 1047, 1048
Stokes phenomenon ............................ 920
Stokes theorem ............................... 1057
Struve functions ..................... 753, 942, 1150
and Bessel functions ......................... 756
and exponentials ............................ 754
and hypergeometric functions confluent ...... 838
and parabolic cylinder functions ............. 848
and powers .................................. 754
and trigonometric functions .................. 755
Sturm comparison theorem ................... 1101
Sturm separation theorem .................... 1101
Sturm–Picone theorem ........................ 1102
Sturmian separation theorem ................. 1087
subdominant solutions ........................ 1104
subordinate norm ............................. 1082
substitutions, Euler ............................. 92
sufficient conditions ........................... 1104
summation formula ............................ 986
summation theorems ...... 940, 986, 992, 998, 1002,
1030, 1042
1170 INDEX OF CONCEPTS
sums
binomial coefficients ........................... 3
partial ......................................... 7
powers ........................................ 1
powers of trigonometric functions ............. 37
products ................... .................... 3
products of trigonometric functions ........... 38
reciprocals ................................. 3, 12
tangents of multiple angles .................... 39
trigonometric and hyperbolic functions ........ 36
supplementary references ..................... 1145
Sylvester’s law of inertia ...................... 1072
symbol
binomial coefficient .......................... xliii
factorial ..................................... xliii
double ..................................... xliii
integral part ................................ xliii
Pochhammer ................................ xliii
synonyms .................................... xxvii
system of equations ..................... 1094, 1095
linear ...................................... 1096
Szeg¨o comparison theorem .................... 1101
T
table usage .................................... xxxi
tangent approximation ......................... 921
Taylor series .................................... 18
termwise integration ............................ 16
tests, convergence ............................ 6, 19
theorems
addition ................................ 973, 975
arithmetic mean ............................ 1056
Ballieu ..................................... 1086
basic................... .................... 1091
boundedness ............................... 1106
Brauer ............................... 1086, 1088
Cayley–Hamilton ........................... 1084
comparison ............... 1100, 1101, 1103, 1104
convolution ......... 1108, 1118, 1122, 1130, 1136
de Moivre .................................. 1060
divergence .................................. 1055
final value .................................. 1139
Frobenius .................................. 1088
Gauss ...................................... 1055
general nature ............................... 247
Gerschgorin .......................... 1083, 1088
Gram–Kowalewski .......................... 1080
Green ................................ 1055, 1056
Hadamard ................................. 1077
initial value .......................... 1136, 1139
integral .................................... 1049
Jacobi ...................................... 1076theorems ( continued )
Kneser ..................................... 1103
Lyapunov ............................ 1089, 1105
mean value ...................... 247, 1063, 1064
named ..................................... 1087
non-oscillation .................. 1100, 1103, 1104
oscillation .................................. 1100
Ostrowski .................................. 1089
Parodi ..................................... 1086
Parseval .............................. 1067, 1068
Perron ..................................... 1088
Perron–Frobenius .......................... 1088
Poincare’s .................................. 1087
product ..................................... 896
quadratic forms ............................ 1072
rate of change .............................. 1057
reciprocal .................................. 1056
representation .............................. 1056
Routh–Hurwitz ............................. 1086
second-order equations ..................... 1100
separation .................................. 1087
Sonin ...................................... 1106
Stieltjes’ .................................... 987
Stokes ...................................... 1057
Sturm comparison .......................... 1101
Sturm separation ........................... 1101
Sturm–Picone .............................. 1102
Sturmian ................................... 1087
summation ..940, 986, 992, 998, 1002, 1030, 1042
Szeg¨o comparison .......................... 1101
vector integral .............................. 1055
Wielandt ................................... 1088
theta functions ............................ 633, 877
Thomson functions ........................ 761, 944
total variation .................................. 20
trace ......................................... 1084
transformation formulas ....................... 1008
transforms
Fourier .................... seeFourier transform
fractional .................................. 1014
Hankel ..................... seeHankel transform
integral .................................... 1147
Laplace ................... seeLaplace transform
Mellin ...................... seeMellin transform
of a derivative .............................. 1118
triangle inequality ............................ 1061
trigonometric functions ........... 28, 151, 390, 415
and algebraic functions ...................... 434
and associated Legendre functions ........... 779
and Bessel functions ........... 717, 727, 742, 747
INDEX OF CONCEPTS 1171
trigonometric functions ( continued )
and exponentials ....227, 339, 485, 493, 495, 497,
522, 525, 599, 742
and gamma functions ........................ 655
and hyperbolic functions ...... 231, 509, 516, 522,
525, 747, 763
and hypergeometric functions ................ 817
confluent ................................... 829
and inverse trigonometric functions .....605, 607
and logarithmic functions .....339, 581, 594, 599
and Mathieu functions ....................... 763
and parabolic cylinder functions ............. 844
and powers .....214, 395, 397, 401, 405, 411, 436,
459, 475, 497, 516, 525, 594, 607, 727, 742, 779
and rational functions .............. 401, 423, 447
and square roots ............................. 408
and Struve functions ........................ 755
inverse ................................... 56, 241
powers ................................. 151, 459
trigonometric series ........................ 46, 862
triple integrals ................................. 610
triple vector product .......................... 1049
two-sided z-transform ......................... 1135
U
uniform convergence ............................ 15
unilateral z-transform ................... 1135, 1138
unit integer function .......................... 1135
unit integer pulse function .................... 1135
use of the tables .............................. xxxiV
Vandermonde determinant .................... 1078
variables separable ............................ 1097
variational principles .......................... 1091
vector
differentiation .............................. 1050
field theory ................................. 1049
integral theorems ........................... 1055
norms ...................................... 1081
operators ................................... 1049
product .................................... 1049
W
Weber functions ............................... 948
Weierstrass elliptic functions ....626, 873, 880, 1148
Weierstrass expansions ......................... 869
weight function ................................ 982
Whittaker functions .......................... 1024
Wielandt theorem ............................ 1088
Wronskian determinant ....................... 1079
Y
Young inequality .............................. 1065
Z
zeros ...................... 865, 870, 879, 972, 1038
interlacing ................................. 1101
simple ...................................... 1000
zeta function ................................. 1150
z-transforms .................................. 1135
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