Schaum Mathematical Handbook of Formulas and Tables 3rd ed 2008

SCHAUM’S outlines ’|Solved Mathematical Handbook ofFormulas and Tables ———— Third Edition More than 2,400 formulas andtables *Covers elementary toadvanced math topics *Arranged bytopicsforeasyreference College Mathematics *Numerical Analysis *Calculus *Calculus II Calculus III*Differential Equations *Probability andStatistics Murray R.Spiegel, Ph.D. *Seymour Lipschutz, Ph.D. *John Liu, Ph.D. Mathematical Handbook of Formulas and Tables SCHAUM'S outlines This page intentionally left blank Mathematical Handbook of Formulas and Tables Third Edition Murray R. Spiegel, PhD Former Professor and Chairman Mathematics Department Rensselaer Polytechnic Institute Hartford Graduate Center Seymour Lipschutz, PhD Mathematics Department Temple University John Liu, PhD Mathematics Department University of Maryland Schaum’s Outline Series New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto SCHAUM'S outlines Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of Ameri ca. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0-07-154856-4The material in this eBook also appears in the print version of this title: 0-07-154855-6.All trademarks are trademarks of their respective owners. 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If you’d like more information about this book,its author, or related books and websites,please click here.Professional Want to learn more? vPreface This handbook supplies a collection of mathematical formulas and tables which will be valuable to students and research workers in the fields of mathematics, physics, engineering, and other sciences. Care has been taken to include only those formulas and tables which are most likely to be needed in practice, rather than highly specialized results which are rarely used. It is a “user-friendly” handbook with material mostly rooted in university mathematics and scientific courses. In fact, the first edition can already be found in many libraries and offices, and it most likely has moved with the owners from office to office since their college times. Thus, this handbook has survived the test of time (while most other college texts have been thrown away). This new edition maintains the same spirit as the second edition, with the following changes. First of all, we have deleted some out-of-date tables which can now be easily obtained from a simple calculator, and we have deleted some rarely used formulas. The main change is that sections on Probability and Random Variables have been expanded with new material. These sections appear in both the physical and social sciences, including education. Topics covered range from elementary to advanced. Elementary topics include those from algebra, geometry, trigonometry, analytic geometry, probability and statistics, and calculus. Advanced topics include those from differential equations, numerical analysis, and vector analysis, such as Fourier series, gamma and beta functions, Bessel and Legendre functions, Fourier and Laplace transforms, and elliptic and other special functions of importance. This wide coverage of topics has been adopted to provide, within a single volume, most of the important mathematical results needed by student and research workers, regardless of their particular field of interest or level of attainment. The book is divided into two main parts. Part A presents mathematical formulas together with other mate- rial, such as definitions, theorems, graphs, diagrams, etc., essential for proper understanding and application of the formulas. Part B presents the numerical tables. These tables include basic statistical distributions (normal, Student’s t , chi-square, etc.), advanced functions (Bessel, Legendre, elliptic, etc.), and financial functions (compound and present value of an amount, and annuity). McGraw-Hill wishes to thank the various authors and publishers—for example, the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., Dr. Frank Yates, F.R.S., and Oliver and Boyd Ltd., Edinburgh, for Table III of their book Statistical Tables for Biological, Agricultural and Medical Research —who gave their permission to adapt data from their books for use in several tables in this handbook. Appropriate references to such sources are given below the corresponding tables. Finally, I wish to thank the staff of the McGraw-Hill Schaum’s Outline Series, especially Charles Wall, for their unfailing cooperation. S EYMOUR LIPSCHUTZ Temple University Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. This page intentionally left blank viiContents Part A FORMULAS 1 Section I Elementary Constants, Products, Formulas 3 1. Greek Alphabet and Special Constants 3 2. Special Products and Factors 5 3. The Binomial Formula and Binomial Coefficients 7 4. Complex Numbers 10 5. Solutions of Algebraic Equations 13 6. Conversion Factors 15 Section II Geometry 16 7. Geometric Formulas 16 8. Formulas from Plane Analytic Geometry 22 9. Special Plane Curves 28 10. Formulas from Solid Analytic Geometry 34 11. Special Moments of Inertia 41 Section III Elementary Transcendental Functions 43 12. Trigonometric Functions 43 13. Exponential and Logarithmic Functions 53 14. Hyperbolic Functions 56 Section IV Calculus 62 15. Derivatives 62 16. Indefinite Integrals 67 17. Tables of Special Indefinite Integrals 71 18. Definite Integrals 108 Section V Differential Equations and Vector Analysis 116 19. Basic Differential Equations and Solutions 116 20. Formulas from Vector Analysis 119 Section VI Series 134 21. Series of Constants 134 22. Taylor Series 138 23. Bernoulli and Euler Numbers 142 24. Fourier Series 144For more information about this title, click here viii Section VII Special Functions and Polynomials 149 25. The Gamma Function 149 26. The Beta Function 152 27. Bessel Functions 153 28. Legendre and Associated Legendre Functions 164 29. Hermite Polynomials 169 30. Laguerre and Associated Laguerre Polynomials 171 31. Chebyshev Polynomials 175 32. Hypergeometric Functions 178 Section VIII Laplace and Fourier Transforms 180 33. Laplace Transforms 180 34. Fourier Transforms 193 Section IX Elliptic and Miscellaneous Special Functions 198 35. Elliptic Functions 198 36. Miscellaneous and Riemann Zeta Functions 203 Section X Inequalities and Infinite Products 205 37. Inequalities 205 38. Infinite Products 207 Section XI Probability and Statistics 208 39. Descriptive Statistics 208 40. Probability 217 41. Random Variables 223 Section XII Numerical Methods 227 42. Interpolation 227 43. Quadrature 231 44. Solution of Nonlinear Equations 233 45. Numerical Methods for Ordinary Differential Equations 235 46. Numerical Methods for Partial Differential Equations 237 47. Iteration Methods for Linear Systems 240 Part B TABLES 243 Section I Logarithmic, Trigonometric, Exponential Functions 245 1. Four Place Common Logarithms log10Nor log N 245 2. Sin x(x in degrees and minutes) 247 3. Cos x (x in degrees and minutes) 248 4. Tan x (x in degrees and minutes) 249CONTENTS ix 5. Conversion of Radians to Degrees, Minutes, and Seconds or Fractions of Degrees 250 6. Conversion of Degrees, Minutes, and Seconds to Radians 251 7. Natural or Napierian Logarithms log e x or ln x 252 8. Exponential Functions ex 254 9. Exponential Functions e/H11546x 255 10. Exponential, Sine, and Cosine Integrals 256 Section II Factorial and Gamma Function, Binomial Coefficients 257 11. Factorial n 257 12. Gamma Function 258 13. Binomial coefficients 259 Section III Bessel Functions 261 14. Bessel Functions J0(x) 261 15. Bessel Functions J1(x) 261 16. Bessel Functions Y0(x) 262 17. Bessel Functions Y1(x) 262 18. Bessel Functions I0(x) 263 19. Bessel Functions I1(x) 263 20. Bessel Functions K0(x) 264 21. Bessel Functions K1(x) 264 22. Bessel Functions Ber(x) 26523. Bessel Functions Bei( x) 265 24. Bessel Functions Ker(x) 26625. Bessel Functions Kei(x) 266 26. Values for Approximate Zeros of Bessel Functions 267 Section IV Legendre Polynomials 268 27. Legendre Polynomials Pn(x) 268 28. Legendre Polynomials Pn(cos /H9258) 269 Section V Elliptic Integrals 270 29. Complete Elliptic Integrals of First and Second Kinds 270 30. Incomplete Elliptic Integral of the First Kind 271 31. Incomplete Elliptic Integral of the Second Kind 271 Section VI Financial Tables 272 32. Compound amount: (1 + r)n 272 33. Present Value of an Amount: (1+r)/H11002n 273 34. Amount of an Annuity: (1+)–1r rn 274 35. Present Value of an Annuity: 1–( 1+ )–r rn 275CONTENTS x Section VII Probability and Statistics 276 36. Areas Under the Standard Normal Curve 276 37. Ordinates of the Standard Normal curve 277 38. Percentile Values (tp) for Student's t Distribution 278 39. Percentile Values (/H92732 p) for /H92732 (Chi-Square) Distribution 279 40. 95th Percentile Values for the F distribution 280 41. 99th Percentile Values for the F distribution 281 42. Random Numbers 282 Index of Special Symbols and Notations 283 Index 285CONTENTS FORMULAS PART A Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. This page intentionally left blank Section I: Elementary Constants, Products, Formulas 1 GREEK ALPHABET and SPECIAL CONSTANTS Greek Alphabet Special Constants 1.1. p = 3.14159 26535 89793 … 1.2. e = 2.71828 18284 59045 … = lim nn n →∞+⎛ ⎝⎞ ⎠11 = natural base of logarithms 1.3. γ = 0.57721 56649 01532 86060 6512 … = Euler’s constant =+ + + + −⎛ ⎝⎜⎞ ⎠⎟→∞lim n nn 11 21 31/midhorizellipsis ln 1.4. eγ=1.78107 24179 90197 9852 … [see 1.3]Greek Greek letter name Lower case Capital Alpha a A Beta b B Gamma g /H9003 Delta d /H9004 Epsilon /H9280 E Zeta z Z Eta h H Theta u /H9008 Iota i I Kappa k K Lambda l /H9011 Mu m MGreek Greek letter name Lower case Capital Nu n N Xi j /H9014 Omicron o O Pi p /H9016 Rho r P Sigma s /H9018 Tau t T Upsilon y /H9020 Phi f /H9021 Chi x X Psi c /H9023 Omega v /H9024 3 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 1.5. e=1.64872 12707 00128 1468 … 1.6. π= =Γ()1 21.77245 38509 05516 02729 8167 … where Γ is the gamma function [see 25.1]. 1.7. Γ()1 3=2.67893 85347 07748 … 1.8. Γ()1 4=3.62560 99082 21908 … 1.9. 1 radian = 180°/p = 57.29577 95130 8232 …° 1.10. 1° = p/180 radians = 0.01745 32925 19943 29576 92 … radiansGREEK ALPHABET AND SPECIAL CONSTANTS 4 2 SPECIAL PRODUCTS and FACTORS 2.1. ()xyx x y y+= ++22 22 2.2. ()xy x x yy−= −+22 22 2.3. ()xy x x y x y y+= + + +33 2 2 333 2.4. ()xy x x y x y y−= − + −33 2 2 333 2.5. ()xy x x y x y x y y+= + + + +44 3 2 2 3 446 4 2.6. ()xy x x y x y x y y−= − + − +44 3 2 2 3 446 4 2.7. ()xy x x y x y x y x y y+ = + +++ +55 4 3 2 2 3 4 551 0 1 0 5 2.8. ()xy x x y x y x y x y y−= − + − + −55 4 3 2 2 3 4 551 0 1 0 5 2.9. ()xy x x y x y x y x y x y y+= + + + + + +66 5 4 2 3 3 2 4 5 661 5 2 0 1 5 6 2.10. ()xy x x y x y x y x y x y y−= − + − + − +66 5 4 2 3 3 2 4 5 661 5 2 0 1 5 6 The results 2.1 to 2.10 above are special cases of the binomial formula [see 3.3]. 2.11. xy x y x y22−=− + () () 2.12. xy x y xx y y33 2 2−=− ++ () ( ) 2.13. xy x y xx y y33 2 2+=+ −+ () ( ) 2.14. xy x y x y xy44 22−=− + + () () ( ) 2.15. xy x y xx y x yx yy55 43 2 2 34−=− + + + + () ( ) 2.16. xy x y xx y x yx yy55 43 2 2 34+=+ − + − + () ( ) 2.17. x y xy xy x x yyx x yy66 2 2 2 2−=− + ++ −+ () () ( ) ( ) 55 2.18. x x y y x x yyx x yy42 24 2 2 2 2++ = + + − + () () 2.19. x y x x yy x x yy44 2 2 2 242 2 2 2+ =++ −+ () () Some generalizations of the above are given by the following results where n is a positive integer. 2.20. xy x y x x y x y ynn n n n n 21 21 2 21 22 2 2 ++ − −−= − + + + + () ( ) /midhorizellipsis ==− −++⎛ ⎝⎜⎞ ⎠⎟− () c o s c o s xy x x ynyx x y22 222 2124 2ππ n ny xx yn ny++⎛ ⎝⎜⎞ ⎠⎟ −++⎛ ⎝⎜⎞ ⎠⎟1 22 212 22/midhorizellipsis cosπ 2.21. xy x y x x y x y ynn n n n n 21 21 2 21 22 2 2 ++ − −+= + − + − + () ( ) /midhorizellipsis ==+ +++⎛ ⎝⎜⎞ ⎠⎟+ () c o s c o s xyx x ynyx x y22 222 2124 2ππ n ny xx yn ny++⎛ ⎝⎜⎞ ⎠⎟ +++⎛ ⎝⎜⎞ ⎠⎟1 22 212 22/midhorizellipsis cosπ 2.22. xy x y x y x x y x y xnn n n n n 22 1 2 3 2−= − + + + +−− − −() () ( ) (/midhorizellipsis112 3 2 222−+ − =− + − +−−xy xy xy xy x x ynynn/midhorizellipsis) () () c o sπ ⎛ ⎛ ⎝⎜⎞ ⎠⎟−+⎛ ⎝⎜⎞ ⎠⎟ −−xx yny xx yn22 222 21cos cos()π /midhorizellipsisπ π ny+⎛ ⎝⎜⎞ ⎠⎟2 2.23. xy x x ynyx x ynnn22 2 2 22223 2+=+ +⎛ ⎝⎜⎞ ⎠⎟++ cos cosππy y xx yn ny2 22221 2⎛ ⎝⎜⎞ ⎠⎟ +−+⎛ ⎝⎜⎞ ⎠⎟/midhorizellipsis cos() πSPECIAL PRODUCTS AND FACTORS 6 3 THE BINOMIAL FORMULA and BINOMIAL COEFFICIENTS Factorial n For n = 1, 2, 3, …, factorial n or n factorial is denoted and defined by 3.1. nn n!( )=− ⋅ ⋅ ⋅ ⋅ 13 2 1/midhorizellipsis Zero factorial is defined by 3.2. 0! = 1 Alternately, n factorial can be defined recursively by 0! = 1 and n! = n ⋅ (n – 1)! EXAMPLE: 4! = 4 ⋅ 3 ⋅ 2 ⋅ 1 = 24, 5! = 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 5 ⋅ 4! = 5(24) = 120, 6! = 6 ⋅ 5! = 6(120) = 720 Binomial Formula for Positive Integral n For n = 1, 2, 3, …, 3.3. ()() !() ( )xy x n xynnxynn nnn n n+= + +−+−−−−12 2 1 212 3 333 !xy ynn−++/midhorizellipsis This is called the binomial formula. It can be extended to other values of n, and also to an infinite series [see 22.4]. EXAMPLE: (a) ( ) () () () (ab a a b a b ab b−= + − + − + − + −24 2 6 2 4 2 244 3 2 2 3) )44 3 2 2 3 482 4 3 21 6 2=− + − + == −aa ba b a b b xa y Here and b b. (b) See Fig. 3-1a. Binomial Coefficients Formula 3.3 can be rewritten in the form 3.4. ()xy xnxynxynnn n n+= +⎛ ⎝⎜⎞ ⎠⎟ +⎛ ⎝⎜⎞ ⎠⎟ +⎛ ⎝−− 12 312 2⎜ ⎜⎞ ⎠⎟ ++⎛ ⎝⎜⎞ ⎠⎟−xyn nynn33/midhorizellipsis 7 where the coefficients, called binomial coefficients, are given by 3.5. n knn n n k kn kn kn ⎛ ⎝⎜⎞ ⎠⎟=−− − +=−=() ( ) ( ) !! !( )!12 1 /midhorizellipsis nnk−⎛ ⎝⎜⎞ ⎠⎟ EXAMPLE: 9 49876 123412612 512 11 ⎛ ⎝⎜⎞ ⎠⎟=⋅⋅⋅ ⋅⋅⋅=⎛ ⎝⎜⎞ ⎠⎟=⋅,⋅⋅ ⋅ ⋅ ⋅⋅⋅⋅=⎛ ⎝⎜⎞ ⎠⎟=⎛ ⎝⎜⎞ ⎠⎟=10 9 8 1234579210 710 31,0098 123120⋅⋅ ⋅⋅= Note that n r⎛ ⎝⎜⎞ ⎠⎟ has exactly r factors in both the numerator and the denominator. The binomial coefficients may be arranged in a triangular array of numbers, called Pascal’s triangle, as shown in Fig. 3.1b. The triangle has the following two properties: (1) The first and last number in each row is 1. (2) Every other number in the array can be obtained by adding the two numbers appearing directly above it. For example 10 = 4 + 6, 15 = 5 + 10, 20 = 10 + 10 Property (2) may be stated as follows: 3.6. n kn kn k⎛ ⎝⎜⎞ ⎠⎟++⎛ ⎝⎜⎞ ⎠⎟=+ +⎛ ⎝⎜⎞ ⎠⎟11 1 Fig. 3-1 Properties of Binomial Coefficients The following lists additional properties of the binomial coefficients: 3.7. nnn n nn 0122⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟++⎛ ⎝⎜⎞ ⎠⎟= /midhorizellipsis 3.8. nnn n nn 01210⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟−−⎛ ⎝⎜⎞ ⎠⎟= /midhorizellipsis() 3.9. n nn nn nnm n⎛ ⎝⎜⎞ ⎠⎟++⎛ ⎝⎜⎞ ⎠⎟++⎛ ⎝⎜⎞ ⎠⎟+++⎛ ⎝⎜⎞ ⎠⎟12/midhorizellipsis = =++ +⎛ ⎝⎜⎞ ⎠⎟nm n1 1THE BINOMIAL FORMULA AND BINOMIAL COEFFICIENTS 8 3.10.nnnn 02421 ⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟+=−/midhorizellipsis 3.11.nnnn 13521 ⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟+=−/midhorizellipsis 3.12.nnn n n 0122222 2⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟++⎛ ⎝⎜⎞ ⎠⎟= /midhorizellipsisn n n⎛ ⎝⎜⎞ ⎠⎟ 3.13.mn pmn pm p 01 1⎛ ⎝⎜⎞ ⎠⎟⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝⎜⎞ ⎠⎟++⎛ ⎝⎜/midhorizellipsis⎞ ⎞ ⎠⎟⎛ ⎝⎜⎞ ⎠⎟=+⎛ ⎝⎜⎞ ⎠⎟nm n p 0 3.14. () ( ) () ( )112233nnnnn n⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟++/midhorizellipsis⎛ ⎛ ⎝⎜⎞ ⎠⎟=−nn21 3.15. () ( ) () ( )1122331nnnn ⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟−−+/midhorizellipsis1 10 ()nn n⎛ ⎝⎜⎞ ⎠⎟= Multinomial Formula Let n1, n2, …, nr be nonnegative integers such that nn n nr 12+++= /midhorizellipsis . Then the following expression, called a multinomial coefficient, is defined as follows: 3.16.n nn nn nn nr r 12 12,,,! !! ! … /midhorizellipsis⎛ ⎝⎜⎞ ⎠⎟= EXAMPLE: 7 2327 2322108 42208 ,,! !!!,,,,! ⎛ ⎝⎜⎞ ⎠⎟==⎛ ⎝⎜⎞ ⎠⎟=44220420!!!!= The name multinomial coefficient comes from the following formula: 3.17. (),, ,xx xn nn nxx xpn rn n r 12 12121 2 +++ =⎛ ⎝⎜⎞ ⎠⎟ ∑ /midhorizellipsis…/midhorizellipsisn nr where the sum, denoted by Σ, is taken over all possible multinomial coefficients.THE BINOMIAL FORMULA AND BINOMIAL COEFFICIENTS 9 4 COMPLEX NUMBERS Definitions Involving Complex Numbers A complex number z is generally written in the form z = a + bi where a and b are real numbers and i, called the imaginary unit, has the property that i2 = −1. The real num- bers a and b are called the real and imaginary parts of z = a + bi, respectively. The complex conjugate of z is denoted by z; it is defined by ab i ab i +=− Thus, a + bi and a – bi are conjugates of each other. Equality of Complex Numbers 4.1. ab icd i+= + if and only if ac bd==and Arithmetic of Complex Numbers Formulas for the addition, subtraction, multiplication, and division of complex numbers follow: 4.2. () () ( ) ( )ab i cd i ac bd i++ += + + + 4.3. () () ( ) ( )ab i cd i ac bd i+− += − + − 4.4. () () ( ) ( )a bi c di ac bd ad bc i+ + =−++ 4.5. ab i cd iab i cd icd i cd iac bd cdbc ad + +=+ +− −=+ ++−i22ccdi22+⎛ ⎝⎞ ⎠ Note that the above operations are obtained by using the ordinary rules of algebra and replacing i2 by −1 wherever it occurs. EXAMPLE: Suppose z = 2 + 3i and w = 5 − 2i. Then zw i i i i i zw i+=+ +− = + +−= + =+ −() () () (23 52 253 2 7 235 221 0 1 5 4 61 6 1 1 23 23 52ii i ii zi i w)=+ − − =+ =+ =− =− and 225 2 52 235223 2323ii w zi iii ii=+ =− +=−− +−() () () ( ) )=−=−41 9 134 1319 13ii 10 Complex Plane Real numbers can be represented by the points on a line, called the real line, and, similarly, complex num- bers can be represented by points in the plane, called the Argand diagram or Gaussian plane or, simply, the complex plane. Specifically, we let the point ( a, b) in the plane represent the complex number z = a + bi. For example, the point P in Fig. 4-1 represents the complex number z = −3 + 4i. The complex number can also be interpreted as a vector from the origin O to the point P. The absolute value of a complex number z = a + bi, written || ,z is defined as follows: 4.6. ||za b z z=+ =22 We note ||z is the distance from the origin O to the point z in the complex plane. Fig. 4-1 Fig. 4-2 Polar Form of Complex Numbers The point P in Fig. 4-2 with coordinates (x, y) represents the complex number zx i y=+ . The point P can also be represented by polar coordinates (r, q). Since x = r cos q and y = r sin q , we have 4.7. zx i yr i=+ = + (cos sin )θθ called the polar form of the complex number. We often call rz x y== +||22 the modulus and q the amplitude of z = x + iy. Multiplication and Division of Complex Numbers in Polar Form 4.8. [ (cos sin )][ (cos sin )] [cri r i r r11 1 2 2 2 1 2 θθ θθ++ = oos( ) sin( )]θθ θθ12 12++ + i 4.9. ri rir r11 1 22 21 2(cos sin ) (cos sin )[cos (θθ θθθ+ +=112 12−+ −θθ θ)s i n ( ) ]i De Moivre’s Theorem For any real number p, De Moivre’s theorem states that 4.10. [ (cos sin )] (cos sin )ri r p i pppθθ θ θ+= +COMPLEX NUMBERS 11 Roots of Complex Numbers Let p = 1/n where n is any positive integer. Then 4.10 can be written 4.11. [ (cos sin )] cos sin//ri rk nik nnnθθθπ θπ+=+++11 22 ⎛ ⎛ ⎝⎜⎞ ⎠⎟ where k is any integer. From this formula, all the nth roots of a complex number can be obtained by putting k = 0, 1, 2, …, n – 1.COMPLEX NUMBERS 12 5 SOLUTIONS of ALGEBRAIC EQUATIONS Quadratic Equation: ax + bx + c =20 5.1. Solutions: xbb a c a=−± −24 2 If a, b, c are real and if D = b2 − 4ac is the discriminant, then the roots are (i) real and unequal if D > 0 (ii) real and equal if D = 0(iii) complex conjugate if D < 0 5.2. If x 1, x2 are the roots, then x1 + x2 = −b/a and x1x2 = c/a. Cubic Equation: x+ a x+ a x + a=3 12 230 Let QaaRaa a a SR Q R T=−=−− =+ +3 992 7 2 54212 12 3 13 32 3,, , ==− +RQ R32 3 where ST = – Q. 5.3. Solutions: xS T a xS T a i S T x11 31 21 21 311 2 31 23=+− =− + − + − =−() () (() ()ST a i ST+− − −⎧ ⎨⎪ ⎩⎪ 1 311 23 If a1, a2, a3, are real and if D = Q3 + R2 is the discriminant, then (i) one root is real and two are complex conjugate if D > 0 (ii) all roots are real and at least two are equal if D = 0(iii) all roots are real and unequal if D < 0. If D < 0, computation is simplified by use of trigonometry. 5.4. Solutions: if DxQ a xQ <=− − =− + ° − 02 2 12011 31 31 21 3 :cos( ) cos( )θ θ1 1 31 31 31 3 2 240a xQ a=− + ° −⎧ ⎨⎪ ⎩⎪cos( ) θ where cos / θ=−RQ3 13 5.5. x x x a xx xx xx a xxx a1 2 3 1 12 23 31 2 123 3++= − + + = = − ,, where x1, x2, x3 are the three roots. Quartic Equation: x +ax +a x +a x+a =4 13 22 340 Let y1 be a real root of the following cubic equation: 5.6. ya y a a a y a aaa a3 22 13 4 24 32 12 444 0 −+ − + − − = () ( ) The four roots of the quartic equation are the four roots of the following equation: 5.7. za a a y z y y a2 1 2 112 211 2 112 444 4 0 +±− +( )+−( )= ∓ Suppose that all roots of 5.6 are real; then computation is simplified by using the particular real root that produces all real coefficients in the quadratic equation 5.7. 5.8. xxxx a xx xx xx xx xx xx1234 1 12 23 34 41 13 2+++= − +++++442 123 234 124 134 3123= +++= −a xxx xxx xxx xxx axxxx 444=⎧ ⎨⎪⎪ ⎩⎪ ⎪ x where x1, x2, x3, x4 are the four roots.SOLUTIONS OF ALGEBRAIC EQUATIONS 14 6 CONVERSION FACTORS Length 1 kilometer (km) = 1000 meters (m) 1 inch (in) = 2.540 cm 1 meter (m) = 100 centimeters (cm) 1 foot (ft) = 30.48 cm 1 centimeter (cm) = 10−2 m 1 mile (mi) = 1.609 km 1 millimeter (mm) = 10−3 m 1 millimeter = 10−3 in 1 micron (m) = 10−6 m 1 centimeter = 0.3937 in 1 millimicron (mm) = 10−9 m 1 meter = 39.37 in 1 angstrom (Å) = 10−10 m 1 kilometer = 0.6214 mi Area 1 square meter (m2) = 10.76 ft2 1 square mile (mi2) = 640 acres 1 square foot (ft2) = 929 cm2 1 acre = 43,560 ft2 Volume 1 liter (l) = 1000 cm3 = 1.057 quart (qt) = 61.02 in3 = 0.03532 ft3 1 cubic meter (m3) = 1000 l = 35.32 ft3 1 cubic foot (ft3) = 7.481 U.S. gal = 0.02832 m3 = 28.32 l 1 U.S. gallon (gal) = 231 in3 = 3.785 l; 1 British gallon = 1.201 U.S. gallon = 277.4 in3 Mass 1 kilogram (kg) = 2.2046 pounds (lb) = 0.06852 slug; 1 lb = 453.6 gm = 0.03108 slug 1 slug = 32.174 lb = 14.59 kg Speed 1 km/hr = 0.2778 m/sec = 0.6214 mi/hr = 0.9113 ft/sec 1 mi/hr = 1.467 ft/sec = 1.609 km/hr = 0.4470 m/sec Density 1 gm/cm3 = 103 kg/m3 = 62.43 lb/ft3 = 1.940 slug/ft3 1 lb/ft3 = 0.01602 gm/cm3; 1 slug/ft3 = 0.5154 gm/cm3 Force 1 newton (nt) = 105 dynes = 0.1020 kgwt = 0.2248 lbwt 1 pound weight (lbwt) = 4.448 nt = 0.4536 kgwt = 32.17 poundals 1 kilogram weight (kgwt) = 2.205 lbwt = 9.807 nt 1 U.S. short ton = 2000 lbwt; 1 long ton = 2240 lbwt; 1 metric ton = 2205 lbwt Energy 1 joule = 1 nt m = 10 7 ergs = 0.7376 ft lbwt = 0.2389 cal = 9.481 × 10−4 Btu 1 ft lbwt = 1.356 joules = 0.3239 cal = 1.285 × 10–3 Btu 1 calorie (cal) = 4.186 joules = 3.087 ft lbwt = 3.968 × 10–3 Btu 1 Btu (British thermal unit) = 778 ft lbwt = 1055 joules = 0.293 watt hr 1 kilowatt hour (kw hr) = 3.60 × 10 6 joules = 860.0 kcal = 3413 Btu 1 electron volt (ev) = 1.602 × 10−19 joule Power 1 watt = 1 joule/sec = 107 ergs/sec = 0.2389 cal/sec 1 horsepower (hp) = 550 ft lbwt/sec = 33,000 ft lbwt/min = 745.7 watts 1 kilowatt (kw) = 1.341 hp = 737.6 ft lbwt/sec = 0.9483 Btu/sec Pressure 1 nt/m 2 = 10 dynes/cm2 = 9.869 × 10−6 atmosphere = 2.089 × 10−2 lbwt/ft2 1 lbwt/in2 = 6895 nt/m2 = 5.171 cm mercury = 27.68 in water 1 atm = 1.013 × 105 nt/m2 = 1.013 × 106 dynes/cm2 = 14.70 lbwt/in2 = 76 cm mercury = 406.8 in water 15 16Section II: Geometry 7GEOMETRIC FORMULAS Rectangle of Length b and Width a 7.1. Area = ab 7.2. Perimeter = 2a + 2b Parallelogram of Altitude h and Base b 7.3. Area = bh = ab sin u 7.4. Perimeter = 2a + 2b Triangle of Altitude h and Base b 7.5. Area ==1 21 2bh ab sinθ = −−−s s as bs c() () () where sa b c=+ + =1 2() semiperimeter 7.6. Perimeter = a + b + c Trapezoid of Altitude h and Parallel Sides a and b 7.7. Area =+1 2ha b() 7.8. Perimeter =++ +⎛ ⎝⎜⎞ ⎠⎟abh11 sin sinθφ =++ +abh (csc csc )θφ Fig. 7-1 Fig. 7-1 Fig. 7-2 Fig. 7-2 Fig. 7-3 Fig. 7-3 Fig. 7-4 Fig. 7-4 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. GEOMETRIC FORMULAS 17 Regular Polygon of n Sides Each of Length b 7.9. Area ==1 42 1 42nbnnbn ncotcos( / ) sin( / )ππ π 7.10. Perimeter = nb Circle of Radius r 7.11. Area = pr2 7.12. Perimeter = 2pr Sector of Circle of Radius r 7.13. Area =1 22rθ [q in radians] 7.14. Arc length s = rq Radius of Circle Inscribed in a Triangle of Sides a, b, c 7.15. rs s as bs c s=−−−() () () where sa b c=+ + =1 2() semiperimeter. Radius of Circle Circumscribing a Triangle of Sides a, b, c 7.16. Rabc s s as bs c= −−− 4 ( )( )( ) where sa b c=+ + =1 2() semiperimeter.Fig. 7-5 Fig. 7-5 Fig. 7-6 Fig. 7-6 Fig. 7-7 Fig. 7-7 Fig. 7-8 Fig. 7-8 Fig. 7-9 Fig. 7-9 GEOMETRIC FORMULAS Regular Polygon of n Sides Inscribed in Circle of Radius r 7.17. Area ==°1 22 1 22 2 360nrnnrnsin sinπ 7.18. Perimeter ==°22180nrnnrnsin sinπ Regular Polygon of n Sides Circumscribing a Circle of Radius r 7.19. Area ==°nrnnrn22 180tan tanπ 7.20. Perimeter ==°22180nrnnrntan tanπ Segment of Circle of Radius r 7.21. Area of shaded part =−1 22r(s i n )θθ Ellipse of Semi-major Axis a and Semi-minor Axis b 7.22. Area = pab 7.23. Perimeter =−∫4122 02ak d sin/θθπ =+21 222π ()ab [approximately] where ka b a=−22/. See Table 29 for numerical values. Segment of a Parabola 7.24. Area =2 3ab 7.25. Arc length ABC b ab a=++1 2222 168 ln 41 622ab a b++⎛ ⎝⎜⎞ ⎠⎟Fig. 7-10 Fig. 7-10 Fig. 7-11 Fig. 7-11 Fig. 7-12 Fig. 7-12 Fig. 7-13 Fig. 7-13 Fig. 7-14 Fig. 7-14 18 GEOMETRIC FORMULAS 19 Rectangular Parallelepiped of Length a, Height b, Width c 7.26. V olume = abc 7.27. Surface area = 2(ab + ac + bc) Parallelepiped of Cross-sectional Area A and Height h 7.28. V olume = Ah = abc sinq Sphere of Radius r 7.29. V olume =4 33πr 7.30. Surface area = 4πr2 Right Circular Cylinder of Radius r and Height h 7.31. V olume = pr2h 7.32. Lateral surface area = 2prh Circular Cylinder of Radius r and Slant Height l 7.33. V olume = pr2h = pr2l sin u 7.34. Lateral surface area == =222 ππ θπθ rlrhrhsincscFig. 7-15 Fig. 7-15 Fig. 7-16 Fig. 7-16 Fig. 7-17 Fig. 7-17 Fig. 7-18 Fig. 7-18 Fig. 7-19 Fig. 7-19 GEOMETRIC FORMULAS 20 Cylinder of Cross-sectional Area A and Slant Height l 7.35. V olume = Ah = Al sinq 7.36. Lateral surface area = ph = pl sinq Note that formulas 7.31 to 7.34 are special cases of formulas 7.35 and 7.36. Right Circular Cone of Radius r and Height h 7.37. V olume =1 32πrh 7.38. Lateral surface area =+ =ππrr h r l22 Pyramid of Base Area A and Height h 7.39. V olume =1 3Ah Spherical Cap of Radius r and Height h 7.40. V olume (shaded in figure) =−1 323 πhr h() 7.41. Surface area = 2p rh Frustum of Right Circular Cone of Radii a, b and Height h 7.42. V olume =+ +1 322πha a b b() 7.43. Lateral surface area =+ + −π() ()abh ba22 = p(a + b)lFig. 7-20 Fig. 7-20 Fig. 7-21 Fig. 7-21 Fig. 7-22 Fig. 7-22 Fig. 7-23 Fig. 7-23 Fig. 7-24 Fig. 7-24 GEOMETRIC FORMULAS 21 Spherical Triangle of Angles A, B, C on Sphere of Radius r 7.44. Area of triangle ABC = (A + B + C − p)r2 Torus of Inner Radius a and Outer Radius b 7.45. V olume =+ −1 422π() ( )ab b a 7.46. Surface area = p 2(b2 − a2) Ellipsoid of Semi-axes a, b, c 7.47. V olume =4 3πabc Paraboloid of Revolution 7.48. V olume =1 22πbaFig. 7-25 Fig. 7-25 Fig. 7-26 Fig. 7-26 Fig. 7-27 Fig. 7-27 Fig. 7-28 Fig. 7-28 8FORMULAS from PLANE ANALYTIC GEOMETRY Distance d Between Two Points P1(x1,y1) and P2(x2,y2) 8.1. dx x y y=− + −() ()212 212 Slope m of Line Joining Two Points P1(x1,y1) and P2(x2,y2) 8.2. myy xx=− −=21 21tanθ Equation of Line Joining Two Points P1(x1,y1) and P2(x2,y2) 8.3. yy xxyy xxmy y m x x− −=− −=− = −1 121 2111or ( ) 8.4. y = mx + b where by m xxy xy xx=− =− −1121 12 21 is the intercept on the y axis, i.e., the y intercept. Equation of Line in Terms of x Intercept a ≠0 and y Intercept b ≠0 8.5. x ay b+= 1 Fig. 8-1 Fig. 8-1 Fig. 8-2 Fig. 8-2 22 23 Normal Form for Equation of Line 8.6. xcos a + y sin a = p where p = perpendicular distance from origin O to line and a = angle of inclination of perpendicular with positive x axis. General Equation of Line 8.7. Ax + By + C = 0 Distance from Point (x1, y1) to Line Ax +By+C=0 8.8. Ax By C AB11 22++ ±+ where the sign is chosen so that the distance is nonnegative. Anglex Between Two Lines Having Slopes m1 and m2 8.9. tanψ=− +mm mm21 121 Lines are parallel or coincident if and only if m1 = m2. Lines are perpendicular if and only if m2 = −1/m1. Area of Triangle with Vertices at (x1,y1), (x2,y2), (x3,y3) 8.10. Area =±1 21 1111 2233xy xy xy = ± ++−−−1 212 13 32 23 12 13()xy yx yx yx yx xy where the sign is chosen so that the area is nonnegative. If the area is zero, the points all lie on a line. Fig. 8-3 Fig. 8-3 Fig. 8-4 Fig. 8-4 Fig. 8-5 Fig. 8-5FORMULAS FROM PLANE ANALYTIC GEOMETRY Transformation of Coordinates Involving Pure Translation 8.11.xx x yy yxx x yy y=′+ =′+⎧⎨⎩′=− ′=−⎧⎨⎩0 00 0or where (x , y) are old coordinates (i.e., coordinates relative to xy system), (x ′, y′) are new coordinates (relative to x ′, y′ system), and (x0, y0) are the coordinates of the new origin O ′ relative to the old xy coordinate system. Transformation of Coordinates Involving Pure Rotation 8.12. xx y yx yxx y =′ −′ =′ +′ {′=+ cos sin sin coscos αα αααorssin cos sinα αα ′=−{yy x where the origins of the old [xy] and new [x′y′] coordinate systems are the same but the x′ axis makes an angle a withthe positive x axis. Transformation of Coordinates Involving Translation and Rotation 8.13. xx y x yx y y x=′ −′ + =′ +′ +⎧⎨⎩ ′cos sin sin cosαα αα0 0 or==− +− ′=− −−() c o s () s i n () c o s ()xx yy yy y x x00 00αα α ssinα⎧⎨⎩ where the new origin O′ of x′y′ coordinate system has coordinates (x0, y0) relative to the old xy coordinate system and the x′ axis makes an angle a with the positive x axis. Polar Coordinates (r, p) A point P can be located by rectangular coordinates (x, y) or polar coordinates (r, u). The transformation between these coordinates isas follows: 8.14. xr yrrx y yx= ={=+ =⎧ ⎨ ⎩−cos sin tan ( / )θ θ θor22 1 Fig. 8-6 Fig. 8-6 Fig. 8-7 Fig. 8-7 Fig. 8-8 Fig. 8-8 Fig. 8-9 Fig. 8-9 24 FORMULAS FROM PLANE ANALYTIC GEOMETRY 25 Equation of Circle of Radius R, Center at (x0,y0) 8.15. ( x − x0)2 + (y − y0)2 = R2 Equation of Circle of Radius R Passing Through Origin 8.16. r = 2R cos(u − a) where (r, u) are polar coordinates of any point on the circle and (R, a) are polar coordinates of the center ofthe circle. Conics (Ellipse, Parabola, or Hyperbola) If a point P moves so that its distance from a fixed point (called the focus) divided by its distance from a fixed line(called the directrix) is a constant /H9280 (called the eccentricity), then the curve described by P is called a conic (so-called because such curves can be obtained by intersecting a plane and a cone at different angles). If the focus is chosen at origin O, the equation of a conic in polar coordinates (r, u) is, if OQ = p and LM = D (see Fig. 8-12), 8.17. rpD=−=− 11/H9280/H9280 /H9280 cos cosθθ The conic is (i) an ellipse if /H9280 < 1(ii) a parabola if /H9280 = 1(iii) a hyperbola if /H9280 > 1Fig. 8-10 Fig. 8-10 Fig. 8-11 Fig. 8-11 Fig. 8-12 Fig. 8-12FORMULAS FROM PLANE ANALYTIC GEOMETRY 26 Ellipse with Center C(x0,y0) and Major Axis Parallel to x Axis 8.18. Length of major axis A′A = 2a 8.19. Length of minor axis B′B = 2b 8.20. Distance from center C to focus F or F′ is ca b=−22 8.21. Eccentricity == =−/H9280c aab a22 8.22. Equation in rectangular coordinates: () ()xx ayy b−+−=02 202 21 8.23. Equation in polar coordinates if C is at O: rab ab222 22 2 2=+ sin cosθθ 8-24. Equation in polar coordinates if C is on x axis and F′ is at O: ra=− −() cos1 12/H9280 /H9280 θ 8.25. If P is any point on the ellipse, PF +PF′= 2a If the major axis is parallel to the y axis, interchange x and y in the above or replace u by 1 2πθ− (or 90°−u ). Parabola with Axis Parallel to x Axis If vertex is at A (x0,y0) and the distance from A to focus F is a > 0, the equation of the parabola is 8.26. ( y−y0)2= 4a(x −x0) if parabola opens to right (Fig. 8-14) 8.27. ( y−y0)2=−4a(x −x0) if parabola opens to left (Fig. 8-15) If focus is at the origin (Fig. 8-16), the equation in polar coordinates is 8.28. ra=−2 1c o s θ Fig. 8-13 Fig. 8-13 Fig. 8-14 Fig. 8-15 Fig. 8-16 FORMULAS FROM PLANE ANALYTIC GEOMETRY In case the axis is parallel to the y axis, interchange x and y or replace u by 1 2πθ−(or 90°−u). 27 Hyperbola with Center C(x0,y0) and Major Axis Parallel to x Axis Fig. 8-17 8.29. Length of major axis A′A = 2a 8.30. Length of minor axis B′B = 2b 8.31. Distance from center C to focus F or ′== +Fc ab22 8.32. Eccentricity /H9280==+ c aab a22 8.33. Equation in rectangular coordinates: () ()xx ayy b−−−=02 202 21 8.34. Slopes of asymptotes G′H and GHb a′=± 8.35. Equation in polar coordinates if C is at O: rab ba222 22 2 2=− cos sinθθ 8.36. Equation in polar coordinates if C is on x axis and F′ is at O: ra=− −() cos/H9280 /H928021 1 θ 8.37. If P is any point on the hyperbola, PF − PF′ = ±2a (depending on branch) If the major axis is parallel to the y axis, interchange x and y in the above or replace u by 1 2πθ− (or 90° − u).FORMULAS FROM PLANE ANALYTIC GEOMETRY Lemniscate 9.1. Equation in polar coordinates: r2 = a2 cos 2u 9.2. Equation in rectangular coordinates: ( x2 + y2)2 = a2(x2 − y2) 9.3. Angle between AB′ or A′B and x axis = 45° 9.4. Area of one loop = a2 Cycloid 9.5. Equations in parametric form: xa ya=− =−{(s i n ) (c o s )φφ φ 1 9.6. Area of one arch = 3πa2 9.7. Arc length of one arch = 8a This is a curve described by a point P on a circle of radius a rolling along x axis. Hypocycloid with Four Cusps 9.8. Equation in rectangular coordinates: x2/3 + y2/3 = a2/3 9.9. Equations in parametric form: xa ya= =⎧⎨⎩cos sin3 3θ θ 9.10. Area bounded by curve =3 82πa 9.11. Arc length of entire curve = 6a This is a curve described by a point P on a circle of radius a/4 as it rolls on the inside of a circle of radius a.Fig. 9-1 Fig. 9-1 Fig. 9-2 Fig. 9-2 Fig. 9-3 Fig. 9-3 9SPECIAL PLANE CURVES 28 29 Cardioid 9.12. Equation: r = 2a(1 + cos u) 9.13. Area bounded by curve = 6pa2 9.14. Arc length of curve = 16a This is the curve described by a point P of a circle of radius a as it rolls on the outside of a fixed circle of radius a. The curveis also a special case of the limacon of Pascal (see 9.32). Catenary 9.15. Equation: yaee ax axa xa=+ =− 2() c o s h// This is the curve in which a heavy uniform chain would hang if suspended vertically from fixed points A and B. Three-Leaved Rose 9.16. Equation: r = a cos 3u The equation r = a sin 3u is a similar curve obtained by rotating the curve of Fig. 9-6 counterclockwise through 30°or p/6 radians. In general, r = a cos nu or r = a sin nu has n leaves if n is odd. Four-Leaved Rose 9.17. Equation: r = a cos 2u The equation r = a sin 2u is a similar curve obtained by rotating the curve of Fig. 9-7 counterclockwise through 45°or p/4 radians. In general, r = a cos nu or r = a sin nu has 2n leaves if n is even. Fig. 9-4 Fig. 9-4 Fig. 9-5 Fig. 9-5 Fig. 9-6 Fig. 9-6 Fig. 9-7 Fig. 9-7 SPECIAL PLANE CURVES Epicycloid 9.18. Parametric equations: xa b bab b ya b b=+ −+⎛ ⎝⎜⎞ ⎠⎟ =+ −() c o s c o s () s i n s i nθθ θaab b+⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨⎪⎪ ⎩⎪ ⎪θ This is the curve described by a point P on a circle of radius b as it rolls on the outside of a circle of radius a. The cardioid (Fig. 9-4) is a special case of an epicycloid. General Hypocycloid 9.19. Parametric equations: xa b bab b ya b b=− +−⎛ ⎝⎜⎞ ⎠⎟ =− −() c o s c o s () s i n s i nφφ φaab b−⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨⎪ ⎪ ⎩⎪ ⎪φ This is the curve described by a point P on a circle of radius b as it rolls on the inside of a circle of radius a. If b = a/4, the curve is that of Fig. 9-3. Trochoid 9.20. Parametric equations: xa b yab=− =−{φφ φsin cos This is the curve described by a point P at distance b from the center of a circle of radius a as the circle rolls on the x axis. If b < a, the curve is as shown in Fig. 9-10 and is called a curtate cycloid. If b > a, the curve is as shown in Fig. 9-11 and is called a prolate cycloid. If b = a, the curve is the cycloid of Fig. 9-2. Fig. 9-8 Fig. 9-8 Fig. 9-9 Fig. 9-9 Fig. 9-11 Fig. 9-1030 SPECIAL PLANE CURVES 31 Tractrix 9.21. Parametric equations: xa ya=− =⎧⎨⎩(c o t c o s ) sinln1 2φφ φ This is the curve described by endpoint P of a taut string PQ of length a as the other end Q is moved along the x axis. Witch of Agnesi 9.22. Equation in rectangular coordinates: ya xa=+8 43 22 9.23. Parametric equations: xa ya= =−⎧⎨⎩2 12cot (c o s )θ θ In Fig. 9-13 the variable line QA intersects y = 2a and the circle of radius a with center (0, a) at A and B, respectively. Any point P on the “witch” is located by constructing lines parallel to the x and y axes through B and A, respectively, anddetermining the point P of intersection. Folium of Descartes 9.24. Equation in rectangular coordinates: x3 + y3 = 3axy 9.25. Parametric equations: xat t yat t=+ =+⎧ ⎨⎪⎪ ⎩⎪ ⎪3 1 3 13 2 3 9.26. Area of loop =3 22a 9.27. Equation of asymptote: x + y + a = 0 Involute of a Circle 9.28. Parametric equations: xa ya=+ =−⎧⎨⎩(cos sin ) (sin cos )φφ φ φφ φ This is the curve described by the endpoint P of a string as it unwinds from a circle of radius a while held taut. Fig. 9-12 Fig. 9-12 Fig. 9-13 Fig. 9-13 Fig. 9-14 Fig. 9-14 Fig. 9-15 Fig. 9-15SPECIAL PLANE CURVES 32 Evolute of an Ellipse 9.29. Equation in rectangular coordinates: ( ax)2/3 + (by)2/3 = (a2 − b2)2/3 9.30. Parametric equations: ax a b by a b=− =−⎧⎨⎩() c o s () s i n22 3 22 3θ θ This curve is the envelope of the normals to the ellipse x2/a2 + y2/b2 = 1 shown dashed in Fig. 9-16. Ovals of Cassini 9.31. Polar equation: r4 + a4 − 2a2r2 cos 2u = b4 This is the curve described by a point P such that the product of its distance from two fixed points (distance 2a apart) is a constant b2. The curve is as in Fig. 9-17 or Fig. 9-18 according as b < a or b > a, respectively. If b = a, the curve is a lemniscate (Fig. 9-1). Fig. 9-17 Fig. 9-18 Limacon of Pascal 9.32. Polar equation: r = b + a cos u Let OQ be a line joining origin O to any point Q on a circle of diameter a passing through O. Then the curve is the locus of all points P such that PQ = b. The curve is as in Fig. 9-19 or Fig. 9-20 according as 2a > b > a or b < a, respectively. If b = a, the curve is a cardioid (Fig. 9-4). If ba/H110842, the curve is convex. Fig. 9-19 Fig. 9-20Fig. 9-16 Fig. 9-16 SPECIAL PLANE CURVES 33 Cissoid of Diocles 9.33. Equation in rectangular coordinates: yx ax22 2=− 9.34. Parametric equations: xa ya= =⎧ ⎨⎪ ⎩⎪2 22 3sin sin cosθ θ θ This is the curve described by a point P such that the distance OP = distance RS. It is used in the problem ofduplication of a cube, i.e., finding the side of a cube which has twice the volume of a given cube. Spiral of Archimedes 9.35. Polar equation: r = au Fig. 9-21 Fig. 9-21 Fig. 9-22 Fig. 9-22SPECIAL PLANE CURVES Distance d Between Two Points P 1(x1,y1,z1) and P 2(x2,y2,z2) 10.1. dx x y y z z=− + − + −() () ( )212 212 212 Direction Cosines of Line Joining Points P 1(x1,y1,z1) and P 2(x2,y2,z2) 10.2. lxx dmyy dnzz d==−==−==−cos , cos , cos αβ γ21 21 2 1 where a, b, g are the angles that line P1P2 makes with the positive x, y, z axes, respectively, and d is given by 10.1 (see Fig. 10-1). Relationship Between Direction Cosines 10.3. cos cos cos22 2 2 2 211 αβ γ++= + + = orlmn Direction Numbers Numbers L, M, N, which are proportional to the direction cosines l , m, n, are called direction numbers. The relationship between them is given by 10.4. lL LM NmM LM NnN LM N= ++= ++= ++22 2 22 2 22 2,, Equations of Line Joining P 1(x1,y1,z1) and P 2(x2,y2,z2) in Standard Form 10.5. xx xxyy yyzz zzxx lyy m− −=− −=− −−=−=1 211 211 2111orzzz n−1 These are also valid if l, m, n are replaced by L, M, N, respectively.Fig. 10-1αγz y xOP2(x2, y2, z2) P1(x1, y1, z1)d β Fig. 10-1αγz y xOP2(x2, y2, z2) P1(x1, y1, z1)d β10FORMULAS from SOLID ANALYTIC GEOMETRY 34 35 Equations of Line Joining P 1(x1,y1,z1) and P 2(x2,y2,z2) in Parametric Form 10.6. xx l t yy m t zz n t=+ =+ = +11 1,, These are also valid if l, m, n are replaced by L, M, N, respectively. Anglee Between Two Lines with Direction Cosines l 1,m1,n1 and l 2,m2,n2 10.7. cosφ=+ +ll mm nn12 1 2 1 2 General Equation of a Plane 10.8. Ax By Cz D+++ = 0 (A, B, C, D are constants) Equation of Plane Passing Through Points (x 1,y1,z1), (x 2,y2,z2), (x 3,y3,z3) 10.9. xx yy zz xxyyzzxxyyzz−−− −−− −−−= 11 1 21212 1 31313 10 0 or 10.10. yy zz yyzzxxzzxx zzx2121 3131121 2 1 31−− −−−+−− −() 33112121 313110−−+−− −−−=xyyxxyy xxyyzz () () Equation of Plane in Intercept Form 10.11.x ay bz c++ = 1 where a, b, c are the intercepts on the x, y, z axes, respectively. Equations of Line Through (x 0,y0,z0) and Perpendicular to Plane Ax +By+Cz+D=0 10.12. xx Ayy Bzz Cxx A t yy B t zz C t−=−=−=+ =+ =+00 0 00 0or , , Note that the direction numbers for a line perpendicular to the plane Ax + By + Cz + D = 0 are A, B, C.Fig. 10-2by xc aOz Fig. 10-2by xc aOzFORMULAS FROM SOLID ANALYTIC GEOMETRY Distance from Point (x 0,y0,z0) to Plane Ax +By+Cz+D=0 10.13. Ax By Cz D ABC000 22 2+++ ±+ + where the sign is chosen so that the distance is nonnegative. Normal Form for Equation of Plane 10.14. xyz pcos cos cosαβ γ++ = where p = perpendicular distance from O to plane at P and a, b, g are angles between OP and positive x, y, z axes. Transformation of Coordinates Involving Pure Translation 10.15.xx x yy y zz zxx x yy y=′+ =′+ =′+⎧ ⎨⎪ ⎩⎪′=− ′=− ′0 0 00 0or zzz z=−⎧ ⎨⎪ ⎩⎪ 0 where (x, y, z) are old coordinates (i.e., coordinates relative to xyz system), (x′, y′, z′) are new coordinates (relative to x′y′z′ system) and (x 0, y0, z0) are the coordinates of the new origin O′ relative to the old xyz coordinate system. Transformation of Coordinates Involving Pure Rotation 10.16. xl x l y l z ym x m y m z zn x n y=′+ ′+ ′ = ′+ ′+ ′ = ′+ ′123 123 12 + + ′⎧ ⎨⎪ ⎩⎪ ′=+ + ′=+ + ′nz xl x m y n z yl x m y n z z3 11 1 22 2 or ==+ +⎧ ⎨⎪ ⎩⎪ lx my nz33 3 where the origins of the xyz and x′y′z′ systems are the same and l1, m1, n1; l2, m2, n2; l3, m3, n3 are the direction cosines of the x′,y′,z′ axes relative to the x, y, z axes, respectively.Fig. 10-3xz P Oβp αγ y Fig. 10-3xz P Oβp αγ y Fig. 10-4x'O'y'z' (x0, y0, z0) Oz y x Fig. 10-4x'O'y'z' (x0, y0, z0) Oz y x Fig. 10-5yy'z z' x' xO Fig. 10-5yy'z z' x' xO36 FORMULAS FROM SOLID ANALYTIC GEOMETRY 37 Transformation of Coordinates Involving Translation and Rotation 10.17. xl x l y l z x ym x m y m z y zn=′+ ′+ ′+ = ′+ ′+ ′+ = ′1230 123 0 1xxn yn zz xl x x m y y+ ′+ ′+⎧ ⎨⎪ ⎩⎪ ′=− + − +23 0 10 10 or() () nnz z yl x x m y y n z z zl10 20 20 2 0 3() () () ( )− ′=− + − +− ′=(() () ( )xx m yy n zz−+ −+ −⎧ ⎨⎪ ⎩⎪03 0 3 0 where the origin O′ of the x′y′z′ system has coordinates (x0, y0, z0) relative to the xyz system and lmnlmnlmn11 1 2 2 2 33 3,, ; ,, ; ,, are the direction cosines of the x′, y′, z′ axes relative to the x, y, z axes, respectively. Cylindrical Coordinates (r, p,z) A point P can be located by cylindrical coordinates (r, u, z) (see Fig. 10-7) as well as rectangular coordinates (x, y, z). The transformation between these coordinates is 10.18.xr yr zzrx y yx= ==⎧ ⎨⎪ ⎩⎪=+ = −cos sin tan ( / )θ θθor22 1 zzz=⎧ ⎨⎪ ⎩⎪ Spherical Coordinates (r, p,e ) A point P can be located by spherical coordinates (r, u,f) (see Fig. 10-8) as well as rectangular coordinates (x, y, z). The transformation between those coordinates is 10.19. xr yr zr rx y= ==⎧ ⎨⎪ ⎩⎪ =+sin cos sin sin cosθφ θφ θ or22+ + = =+ +⎧ ⎨⎪ ⎩⎪− −z yx zx y z2 1 12 2 2φ θtan ( / ) cos ( / )O' Ozz'y' (x0, y0, z0) y x' x Fig. 10-6O' Ozz'y' (x0, y0, z0) y x' x Fig. 10-6 Fig. 10-7 Fig. 10-7 Fig. 10-8 Fig. 10-8FORMULAS FROM SOLID ANALYTIC GEOMETRY 38 Equation of Sphere in Rectangular Coordinates 10.20. () () ( )xx yy zz R−+ −+ −=02 02 022 where the sphere has center (x0, y0, z0) and radius R. Equation of Sphere in Cylindrical Coordinates 10.21. rr r r z z R2 00 02 0222−− + + − = cos( ) ( )θθ where the sphere has center (r0, u0, z0) in cylindrical coordinates and radius R. If the center is at the origin the equation is 10.22. rzR22 2+= Equation of Sphere in Spherical Coordinates 10.23. rr r r R2 02 00 022 +− − = sin sin cos( )θθ φ φ where the sphere has center (r0, u0, f0) in spherical coordinates and radius R. If the center is at the origin the equation is 10.24. r = R Equation of Ellipsoid with Center (x 0,y0,z0) and Semi-axes a, b,c 10.25.() () ( )xx ayy bzz c−+−+−=02 202 202 21Fig. 10-9 Fig. 10-9 Fig. 10-10 Fig. 10-10 FORMULAS FROM SOLID ANALYTIC GEOMETRY Elliptic Cylinder with Axis as z Axis 10.26. x ay b2 22 21 += where a, b are semi-axes of elliptic cross-section. If b = a it becomes a circular cylinder of radius a. Elliptic Cone with Axis as z Axis 10.27. x ay bz c2 22 22 2 += Hyperboloid of One Sheet 10.28.x ay bz c2 22 22 21 +−= Hyperboloid of Two Sheets 10.29.x ay bz c2 22 22 21 −−= Note orientation of axes in Fig. 10-14.Fig. 10-11 Fig. 10-11 Fig. 10-12 Fig. 10-12 Fig. 10-13 Fig. 10-13 Fig. 10-14 Fig. 10-14 FORMULAS FROM SOLID ANALYTIC GEOMETRY 39 40 Elliptic Paraboloid 10.30. x ay bz c2 22 2+= Hyperbolic Paraboloid 10.31. x ay bz c2 22 2−= Note orientation of axes in Fig. 10-16.Fig. 10-15 Fig. 10-15 Fig. 10-16 Fig. 10-16 FORMULAS FROM SOLID ANALYTIC GEOMETRY 11SPECIAL MOMENTS of INERTIA The table below shows the moments of inertia of various rigid bodies of mass M. In all cases it is assumed the body has uniform (i.e., constant) density. TYPE OF RIGID BODY MOMENT OF INERTIA 11.1. Thin rod of length a (a) about axis perpendicular to the rod through the center of mass (b) about axis perpendicular to the rod through one end1 122Ma 1 32Ma 11.2. Rectangular parallelepiped with sides a, b,c 1 1222Ma b()+ 1 12224Ma b() +(a) about axis parallel to c and through center of face ab (b) about axis through center of face bc and parallel to c 11.3. Thin rectangular plate with sides a, b (a) about axis perpendicular to the plate through center (b) about axis parallel to side b through center1 1222Ma b()+ 1 122Ma 11.4. Circular cylinder of radius a and height h (a) about axis of cylinder (b) about axis through center of mass and perpendicular to cylindrical axis (c) about axis coinciding with diameter at one end1 22Ma 1 12223 Mh a()+ 1 122243Mh a() + 11.5. Hollow circular cylinder of outer radius a, inner radius b and height h (a) about axis of cylinder (b) about axis through center of mass and perpendicular to cylindrical axis (c) about axis coinciding with diameter at one end1 222Ma b()+ 1 1222 233Ma b h() ++ 1 12222334Ma b h() ++ 11.6. Circular plate of radius a (a) about axis perpendicular to plate through center (b) about axis coinciding with a diameter1 22Ma 1 42Ma 41 42 11.7. Hollow circular plate or ring with outer radius a and inner radius b (a) about axis perpendicular to plane of plate through center (b) about axis coinciding with a diameter1 222Ma b()+ 1 422Ma b()+ 11.8. Thin circular ring of radius a (a) about axis perpendicular to plane of ring through center (b) about axis coinciding with diameterMa2 1 22Ma 11.9. Sphere of radius a (a) about axis coinciding with a diameter (b) about axis tangent to the surface2 52Ma 7 52Ma 11.10. Hollow sphere of outer radius a and inner radius b (a) about axis coinciding with a diameter (b) about axis tangent to the surface2 555 33Ma b a b() / ()−− 2 555 33 2Ma b a b M a() / ()−− + 11.11. Hollow spherical shell of radius a (a) about axis coinciding with a diameter (b) about axis tangent to the surface2 32Ma 5 32Ma 11.12. Ellipsoid with semi-axes a, b, c (a) about axis coinciding with semi-axis c (b) about axis tangent to surface, parallel to semi-axis c and at distance a from center1 522Ma b()+ 1 5226Ma b() + 11.13. Circular cone of radius a and height h (a) about axis of cone (b) about axis through vertex and perpendicular to axis(c) about axis through center of mass and perpendicular to axis3 102Ma 3 20224 Ma h()+ 3 80224Ma h() + 11.14. Torus with outer radius a and inner radius b (a) about axis through center of mass and perpendicular to the plane of torus (b) about axis through center of mass and in the plane of torus1 422763Ma a b b() −+ 1 42291 0 5Ma a b b() −+SPECIAL MOMENTS OF INERTIA 4343Section III: Elementary Transcendental Functions 12 TRIGONOMETRIC FUNCTIONS Definition of Trigonometric Functions for a Right Triangle Triangle ABC has a right angle (90°) at C and sides of length a, b, c. The trigonometric functions of angle A are defined as follows: 12.1. sine of A = sin A ==a copposite hypotenuse 12.2. cosine of A = cos A ==b cadjacent hypotenuse 12.3. tangent of A = tan A ==a bopposite adjacent 12.4. cotangent of A = cot A ==b aadjacent opposite 12.5. secant of A = sec A ==c bhypotenuse adjacent 12.6. cosecant of A = csc A ==c ahypotenuse opposite Extensions to Angles Which May be Greater Than 90° Consider an xy coordinate system (see Figs. 12-2 and 12-3). A point P in the xy plane has coordinates (x , y) where x is considered as positive along OX and negative along OX′ while y is positive along OY and nega tive along OY′. The distance from origin O to point P is positive and denoted by rx y=+22. The angle A described counter- clockwise from OX is considered positive. If it is described clockwise from OX it is considered negative. We call X′OX and Y ′OY the x and y axis, respectively. The various quadrants are denoted by I, II, III, and IV called the first, second, third, and fourth quadrants, respectively. In Fig. 12-2, for example, angle A is in the second quadrant while in Fig. 12-3 angle A is in the third quadrant. Fig. 12-2 Fig. 12-3Fig. 12-1 Fig. 12-1 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 44 TRIGONOMETRIC FUNCTIONS For an angle A in any quadrant, the trigonometric functions of A are defined as follows. 12.7. sin A = y/r 12.8. cos A = x/r 12.9. tan A = y/x 12.10. cot A = x/y 12.11. sec A = r/x 12.12. csc A = r/y Relationship Between Degrees and Radians A radian is that angle q subtended at center O of a circle by an arc MN equal to the radius r. Since 2p radians = 360° we have 12.13. 1 radian = 180°/p = 57.29577 95130 8232 …° 12.14. 1° = p/180 radians = 0.01745 32925 19943 29576 92 … radians Relationships Among Trigonometric Functions 12.15. tansin cosAA A= 12.19. sin cos221 AA+= 12.16. cottancos sinAAA A==1 12.20. sec tan221 AA−= 12.17. seccosAA=1 12.21. csc cot221 AA−= 12.18. cscsinAA=1 Signs and Variations of Trigonometric Functions Quadrant sin A cos A tan A cot A sec A csc A I + 0 to 1+ 1 to 0+ 0 to ∞+ ∞ to 0+ 1 to ∞+ ∞ to 1 II + 1 to 0– 0 to –1– –∞ to 0– 0 to –∞– –∞ to –1+ 1 to ∞ III – 0 to –1– –1 to 0+ 0 to ∞+ ∞ to 0– –1 to –∞– –∞ to –1 IV – –1 to 0+ 0 to 1– –∞ to 0– 0 to –∞+ ∞ to 1– –1 to –∞Fig. 12-4 Fig. 12-4 44 Exact Values for Trigonometric Functions of Various Angles Angle A in degreesAngle A in radians sin A cos A tan A cot A sec A csc A 0° 0 0 1 0 ∞ 1 ∞ 15° p/121 462()−1 462()+ 23− 23+ 62− 62+ 30° p/61 21231 33 32 33 2 45° p/41 22122 11 2 2 60° p/31 231231 33 22 33 75° 5p/121462()+1 462()− 23+ 23− 62+ 62− 90° p/2 1 0 ±∞ 0 ± ∞ 1 105° 7p/121 462()+ −−1 462() −+()23 −−()23 −+()62 62− 120° 2p/31 23 −12−3 −1 33 –22 33 135° 3p/4122 −122 –1 –1 −2 2 150° 5p/61 2 −123 −1 33 −3 −2 33 2 165° 11p/121 462()− −+1 462() −−()23 −+()23 −−()62 62+ 180° p 0– 1 0 +−∞ –1 ±∞ 195° 13p/12 −−1 462() −+1 462() 23− 23+ −−()62 −+()62 210° 7p/6 −1 2 −1231 33 3 −2 33 –2 225° 5p/4 −1 22 −122 11 −2 −2 240° 4p/3 −1 23 −1231 33 –2 −2 33 255° 17p/12 −+1 462() −−1 462() 23+ 23− −+()62 −−()62 270° 3p/2 –1 0 ±∞ 0 +−∞ –1 285° 19p/12 −+1 462()1 462()− −+()23 −−()23 62+ −−()62 300° 5p/3 −1 2312−3 −1 33 2 −2 33 315° 7p/4 −122122 –1 –1 2 −2 330° 11p/6 −1 2123 −1 33 −32 33 –2 345° 23p/12 −−1 462()1 462()+ −−()23 −+()23 62− −+()62 360° 2p 01 0 +−∞ 1 +− ∞ For other angles see Tables 2, 3, and 4.TRIGONOMETRIC FUNCTIONS 45 Graphs of Trigonometric Functions In each graph x is in radians. 12.22. y = sin x 12.23. y = cos x Fig. 12-5 Fig. 12-6 12.24. y = tan x 12.25. y = cot x Fig. 12-7 Fig. 12-8 12.26. y = sec x 12.27. y = csc x Fig. 12-9 Fig. 12-10 Functions of Negative Angles 12.28. sin(–A) = – sin A 12.29. cos(–A) = cos A 12.30. tan(–A) = – tan A 12.31. csc(–A) = – csc A 12.32. sec(–A) = sec A 12.33. cot(–A) = – cot ATRIGONOMETRIC FUNCTIONS 46 Addition Formulas 12.34. sin (A ± B) = sin A cos B ± cos A sin B 12.35. cos (A ± B) = cos A cos B +− sin A sin B 12.36. tan ( )tan tan tan tanABAB AB±=± +−1 12.37. cot ( )cot cot cot cotABAB BA±=+− ±1 Functions of Angles in All Quadrants in Terms of Those in Quadrant I –A90° ± A π 2±A180° ± A p ± A270° ± A 3 2π±Ak(360°) ± A 2kp ± A k = integer sin – sin A cos A sin A – cos A ± sin A cos cos A +− sin A – cos A +− sin A cos A tan – tan A +− cot A ± tan A +− cot A ± tan A csc – csc A sec A +− csc A – sec A ± csc A sec sec A +− csc A – sec A ± csc A sec A cot – cot A +− tan A ± cot A +− tan A ± cot A Relationships Among Functions of Angles in Quadrant I sin A = u cos A = u tan A = u cot A = u sec A = u csc A = u sin Au 12−u uu/12+ 112/+u uu21−/ 1/u cos A 12−u u 112/+u uu/12+ 1/u uu21−/ tan A uu/12− 12−uu/ u 1/u u21− 112/u− cot A 12−uu/ uu/12− 1/uu 112/u− u21− sec A 112/−u 1/u 12+u 12+uu/ u uu/21− csc A 1/u 112/−u 12+uu/ 12+u uu/21− u For extensions to other quadrants use appropriate signs as given in the preceding table.TRIGONOMETRIC FUNCTIONS 47 Double Angle Formulas 12.38. sin 2A = 2 sin A cos A 12.39. cos 2A = cos2 A – sin2 A = 1 – 2 sin2 A = 2 cos2 A – 1 12.40. tantan tan22 12AA A=− Half Angle Formulas 12.41. sincos/AAA 21 22 =±−+ −if in quadrant I or II is iif is in quadrant III or IVA/2⎡ ⎣⎢ ⎢⎤ ⎦⎥ ⎥ 12.42. coscos/AAA 21 22 =±++ −if is in quadrant I or IV iif is in quadrant II or IIIA/2⎡ ⎣⎢ ⎢⎤ ⎦⎥ ⎥ 12.43. tancos cos/AA AA 21 12 =±− ++if is in quadrant I or r III if is in quadrant II or IV−⎡ ⎣⎢ ⎢⎤ ⎦⎥ ⎥ A/2 = =+=−=−sin coscos sincsc cotA AA AAA11 Multiple Angle Formulas 12.44. sin 3A = 3 sin A – 4 sin3 A 12.45. cos 3A = 4 cos3 A –3 cos A 12.46. tantan tan tan33 133 2AAA A=− − 12.47. sin 4A = 4 sin A cos A – 8 sin3 A cos A 12.48. cos 4A = 8 cos4 A – 8 cos2 A + 1 12.49. tantan tan tan tan444 163 24AAA AA=− −+ 12.50. sin 5A = 5 sin A – 20 sin3 A + 16 sin5 A 12.51. cos 5A = 16 cos5 A – 20 cos3 A + 5 cos A 12.52. tantan tan tan tan tan510 5 11 0 553 24AAA A AA=−+ −+ See also formulas 12.68 and 12.69. Powers of Trignometric Functions 12.53. sin cos2 1 21 2 2 AA=− 12.57. sin cos cos4 3 81 21 8 24 AA A=− + 12.54. cos cos2 1 21 2 2 AA=+ 12.58. cos cos cos4 3 81 21 8 24 AA A=+ + 12.55. sin sin sin3 3 41 4 3 AA A=− 12.59. sin sin sin sin5 5 85 161 16 35 AA A A=− + 12.56. cos cos cos3 3 41 4 3 AA A=+ 12.60. cos cos cos cos5 5 85 161 16 35 AA A A=+ + See also formulas 12.70 through 12.73.TRIGONOMETRIC FUNCTIONS 48 Sum, Difference, and Product of Trignometric Functions 12.61. sin sin sin ( ) cos ( )AB A B A B+= + − 21 21 2 12.62. sin sin cos ( )sin ( )AB A B A B−= + − 21 21 2 12.63. cos cos cos ( )cos ( )AB A B A B+= + − 21 21 2 12.64. cos cos sin ( )sin ( )AB A B B A−= + − 21 21 2 12.65. sin sin {cos( ) cos ( )}A B AB AB =− − −1 2 12.66. cos cos {cos( ) cos( )}A B AB AB =− + +1 2 12.67. sin cos {sin( ) sin( )}AB A B A B =− + +1 2 General Formulas 12.68. sin sin ( cos ) ( cos )nA A AnAnnn=−−⎛ ⎝⎜⎞ ⎠⎟ +−−22 1213− −⎛ ⎝⎜⎞ ⎠⎟ −⋅⋅⋅⎧⎨⎩⎫⎬⎭−3 225(c o s ) An 12.69. cos ( cos ) ( cos )nA AnAnnnn=− +−⎛ ⎝⎜⎞ ⎠⎟⎧− 1 221223 12⎨ ⎨⎩ −−⎛ ⎝⎜⎞ ⎠⎟ +⋅⋅⋅⎫⎬⎭− −(c o s ) (c o s )2 34 224 6A nnAn n 12.70. sin()sin ( )211 221 22121 1nn n An An−− − =−−−−⎛ ⎝⎜⎞ ⎠⎟⎧ ⎧⎨⎩− +⋅⋅⋅ −− −⎛ ⎝⎜⎞ ⎠⎟⎫−sin ( ) ( ) sin23 121 11nAn nAn⎬ ⎬ ⎭ 12.71. cos cos ( ) cos (21 221 22121 12n n An Ann− − =− +−⎛ ⎝⎜⎞ ⎠⎟ −− +⋅⋅⋅+− −⎛ ⎝⎜⎞ ⎠⎟⎧⎨⎩⎫⎬⎭321 1)c os An nA 12.72. sin()cos2 22 11 22 1 222 1n nn nAn nnAn=⎛ ⎝⎜⎞ ⎠⎟+−−⎛ ⎝− ⎜ ⎜⎞ ⎠⎟ − +⋅⋅⋅ −−⎛ ⎝⎜⎞ ⎠⎟−cos ( ) ( ) cos22 12 121nAn nAn⎧ ⎧ ⎨ ⎩⎫ ⎬ ⎭ 12.73. cos cos2 22 11 22 1 222 1n nnAn nnAn=⎛ ⎝⎜⎞ ⎠⎟++⎛ ⎝⎜⎞ ⎠⎟ −ccos ( ) cos 222 12 nAn nA − +⋅⋅⋅+−⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨ ⎩⎫ ⎬ ⎭ Inverse Trigonometric Functions If x = sin y, then y = sin–1x, i.e. the angle whose sine is x or inverse sine of x is a many-valued function of x which is a collection of single-valued functions called branches. Similarly, the other inverse trigonometric functions are multiple-valued. For many purposes a particular branch is required. This is called the principal branch and the values for this branch are called principal values.TRIGONOMETRIC FUNCTIONS 49 Principal Values for Inverse Trigonometric Functions Principal values for x /H11084 0 Principal values for x < 0 0 /H11017 sin–1 x /H11017 p/2 –p/2 /H11017 sin–1 x < 0 0 /H11017 cos–1 x /H11017 p/2 p/2 < cos–1 x /H11017 p 0 /H11017 tan–1 x < p/2 –p/2 < tan–1 x < 0 0 < cot–1 x /H11017 p/2 p/2 < cot–1 x < p 0 /H11017 sec–1 x < p/2 p/2 < sec–1 x /H11017 p 0 < csc–1 x /H11017 p/2 –p/2 /H11017 csc–1 x < 0 Relations Between Inverse Trigonometric Functions In all cases it is assumed that principal values are used. 12.74. sin cos /−−+=112 xx π 12.80. sin ( ) sin−−−= −11xx 12.75. tan cot /−−+=112 xx π 12.81. cos ( ) cos−−−=−11xxπ 12.76. sec csc /−−+=112 xx π 12.82. tan ( ) tan−−−= −11xx 12.77. csc sin ( / )−−=111 xx 12.83. cot ( ) cot−−−=−11xxπ 12.78. sec cos ( / )−−=111 xx 12.84. sec ( ) sec−−−=−11xxπ 12.79. cot tan ( / )−−=111 xx 12.85. csc ( ) csc−−−= −11xx Graphs of Inverse Trigonometric Functions In each graph y is in radians. Solid portions of curves correspond to principal values. 12.86. yx=−sin1 12.87. yx=−cos1 12.88. yx=−tan1 Fig. 12-11 Fig. 12-12 Fig. 12-13TRIGONOMETRIC FUNCTIONS 50 12.89. yx=−cot1 12.90. yx=−sec1 12.91. yx=−csc1 Fig. 12-14 Fig. 12-15 Fig. 12-16 Relationships Between Sides and Angles of a Plane Triangle The following results hold for any plane triangle ABC with sides a, b, c and angles A, B, C. 12.92. Law of Sines: a Ab Bc C sin sin sin== 12.93. Law of Cosines: ca b a b C22 22 =+− cos with similar relations involving the other sides and angles. 12.94. Law of Tangents: ab abAB AB+ −=+ −tan ( ) tan ( )1 2 1 2 with similar relations involving the other sides and angles. 12.95. sin ( )( )( )Abcs s as bs c =− − −2 where sa b c=+ +1 2() is the semiperimeter of the triangle. Similar relations involving angles B and C can be obtained. See also formula 7.5. Relationships Between Sides and Angles of a Spherical Triangle Spherical triangle ABC is on the surface of a sphere as shown in Fig. 12-18. Sides a , b, c (which are arcs of great circles) are measured by their angles subtended at center O of the sphere. A, B, C are the angles opposite sides a, b, c, respectively. Then the following results hold. 12.96. Law of Sines: sin sinsin sinsin sina Ab Bc C== 12.97. Law of Cosines: cos a = cos b cos c + sin b sin c cos A cos A = –cos B cos C + sin B sin C cos a with similar results involving other sides and angles.Fig. 12-17 Fig. 12-17 Fig. 12-18 Fig. 12-18TRIGONOMETRIC FUNCTIONS 51 12.98. Law of Tangents: tan ( ) tan ( )tan ( ) tan ( )1 2 1 21 2 1 2AB ABab ab+ −=+ − with similar results involving other sides and angles. 12.99. cossin sin ( ) sin sinAs s c bc 2=− where sa b c=+ +1 2() . Similar results hold for other sides and angles. 12.100. coscos( )cos( ) sin sinaS B S C BC 2=−− where SA B C=+ +1 2() . Similar results hold for other sides and angles. See also formula 7.44. Napier’s Rules for Right Angled Spherical Triangles Except for right angle C , there are five parts of spherical triangle ABC which, if arranged in the order as given in Fig. 12-19, would be a, b, A, c, B. Fig. 12-19 Fig. 12-20 Suppose these quantities are arranged in a circle as in Fig. 12-20 where we attach the prefix “co” (indicat- ing complement) to hypotenuse c and angles A and B. Any one of the parts of this circle is called a middle part, the two neighboring parts are called adjacent parts, and the two remaining parts are called opposite parts. Then Napier’s rules are 12.101. The sine of any middle part equals the product of the tangents of the adjacent parts. 12.102. The sine of any middle part equals the product of the cosines of the opposite parts. EXAMPLE :Since co-A = 90° – A, co-B = 90° – B, we have sin a = tan b (co-B) or sin a = tan b cot B sin (co-A) = cos a cos (co-B) or cos A = cos a sin B These can of course be obtained also from the results of 12.97.TRIGONOMETRIC FUNCTIONS 52 13 EXPONENTIAL and LOGARITHMIC FUNCTIONS Laws of Exponents In the following p, q are real numbers, a, b are positive numbers, and m, n are positive integers. 13.1. aa apq p q⋅=+ 13.2. aa apq p q/=− 13.3. ()aapq p q= 13.4. aa0=≠10, 13.5. aapp−=1/ 13.6. ()ab a bpp p= 13.7. aan n=1/ 13.8. aam n mn=/ 13.9. ab a bn nn//= Inap,p is called the exponent, a is the base, and ap is called the pth power of a. The function y=ax is called an exponential function. Logarithms and Antilogarithms Ifap=N where a ≠ 0 or 1, then p = logaN is called the logarithm ofNto the base a. The number N=ap is called the antilogarithm of p to the base a, written antilogap. Example: Since 32= 9 we have log3 9 = 2. antilog3 2 = 9. The function y = logax is called a logarithmic function. Laws of Logarithms 13.10. logaMN= logaM+ logaN 13.11. log log logaa aM NMN =− 13.12. logaMp=p logaM Common Logarithms and Antilogarithms Common logarithms and antilogarithms (also called Briggsian) are those in which the base a= 10. The common logarithm of N is denoted by log10N or briefly log N. For numerical values of common logarithms, see Table 1. Natural Logarithms and Antilogarithms Natural logarithms and antilogarithms (also called Napierian ) are those in which the base a=e= 2.71828 18 … [see page 3]. The natural logarithm of N is denoted by loge N or In N . For numerical values of natural logarithms see Table 7. For values of natural antilogarithms (i.e., a table giving ex for values of x ) see Table 8. 53 Change of Base of Logarithms The relationship between logarithms of a number N to different bases a and b is given by 13.13. loglog logab bNN a= In particular, 13.14. loge N = ln N = 2.30258 50929 94 … log10 N 13.15. log10 N = log N = 0.43429 44819 03 … loge N Relationship Between Exponential and Trigonometric Functions 13.16. eiθ = cos θ + i sin θ, e–iθ = cos θ – i sin θ These are called Euler’s identities. Here i is the imaginary unit [see page 10].13.17. sinθθθ =−−ee iii 2 13.18. cosθθθ =+−eeii 2 13.19. tan()θθθ θθθθ θθ=− +=−− +− −− −ee ie eiee eeii iiii ii⎛ ⎛ ⎝⎜⎞ ⎠⎟ 13.20. cotθθθ θθ=+ −⎛ ⎝⎜⎞ ⎠⎟− −iee eeii ii 13.21. secθθθ=+−2 eeii 13.22. cscθθθ=−−2i eeii Periodicity of Exponential Functions 13.23. ei(θ + 2kp) = eiθ k = integer From this it is seen that ex has period 2pi. Polar Form of Complex Numbers Expressed as an Exponential The polar form (see 4.7) of a complex number z = x + iy can be written in terms of exponentials as follows: 13.24. zx i yr i r ei=+ = + = (cos sin )θθθEXPONENTIAL AND LOGARITHMIC FUNCTIONS 54 Operations with Complex Numbers in Polar Form Formulas 4.8 to 4.11 are equivalent to the following: 13.25. () ( )()re re rreii i 12 1 212 1 2 θθ θ θ=+ 13.26. re rer rei ii 1 21 21 212θ θθθ=−() 13.27. ()re r eip p i pθθ= (De Moivre’s theorem) 13.28. () [ ]/( ) / / ( ) /re re r ei n ik n n ik nθθ π θ π12 1 1 2==++ Logarithm of a Complex Number 13.29. ln ( ) lnre r i k i kiθθπ =+ + = 2 integerEXPONENTIAL AND LOGARITHMIC FUNCTIONS 55 14 HYPERBOLIC FUNCTIONS Definition of Hyperbolic Functions 14.1. Hyperbolic sine of x ==−− sinhxeexx 2 14.2. Hyperbolic cosine of x ==+− cosh xeexx 2 14.3. Hyperbolic tangent of x ==− +− −tanh xee eexx xx 14.4. Hyperbolic cotangent of x ==+ −− −coth xee eexx xx 14.5. Hyperbolic secant of x ==+−sech xeexx2 14.6. Hyperbolic cosecant of x ==−−csch xeexx2 Relationships Among Hyperbolic Functions 14.7. tanhsinh coshxx x= 14.8. cothtanhcosh sinhxxx x==1 14.9. sech xx=1 cosh 14.10. csch xx=1 sinh 14.11. cosh sinh221 xx−= 14.12. sech221 xx+=tanh 14.13. coth221 xx−=csc h Functions of Negative Arguments 14.14. sinh (–x) = – sinh x 14.15. cosh (–x) = cosh x 14.16. tanh (–x) = – tanh x 14.17. csch (–x) = – csch x 14.18. sech (–x) = sech x 14.19. coth (–x) = – coth x 56 Addition Formulas 14.20. sinh( ) sinh cosh cosh sinh xy x y x y±= ± 14.21. cosh( ) cosh cosh sinh sinh xy x y x y±= ± 14.22. tanh ( )tanh tanh tanh tanhxyxy xy±=± ±1 14.23. coth ( )coth coth coth cothxyxy yx±=± ±1 Double Angle Formulas 14.24. sinh sinh cosh22xx x= 14.25. cosh cos h sin h cos h sin h 22 11222 2 2xx x x x=+ = − = + 14.26. tanhtan h tanh22 12xx x=+ Half Angle Formulas 14.27. sinhcosh[, ]xxxx21 200 =±−+> −<if if 14.28. coshcosh xx 21 2=+ 14.29. tanhcosh cosh[, ] sinhxx xxx x21 100 =±− ++> −< =if if ccoshcosh sinh xx x +=− 11 Multiple Angle Formulas 14.30. sinh sinh sinh33 43xx x=+ 14.31. cosh cosh cosh 34 33xx x=− 14.32. tanhtanh tanh tanh33 133 2xxx x=+ + 14.33. sinh sinh cosh sinh cosh48 43xx x x x=+ 14.34. cosh cosh cosh 48 8 142xxx=−+ 14.35. tanhtanh tanh tanh tanh444 163 24xxx xx=+ ++HYPERBOLIC FUNCTIONS 57 Powers of Hyperbolic Functions 14.36. sinh cosh2 1 21 2 2 xx=− 14.37. cosh cosh2 1 21 2 2 xx=+ 14.38. sinh sinh sinh3 1 43 4 3 xx x=− 14.39. cosh cosh cosh3 1 43 4 3 xx x=+ 14.40. sinh cosh cosh4 3 81 21 8 24 xx x=− + 14.41. cosh cosh cosh4 3 81 21 8 24 xx x=+ + Sum, Difference, and Product of Hyperbolic Functions 14.42. sinh sinh sinh ( ) cosh ( )xy x y x y+= + − 21 21 2 14.43. sinh sinh cosh ( )sinh ( )xy x y x y−= + − 21 21 2 14.44. cosh cosh cosh ( )cosh ( ) xy x y x y+= + − 21 21 2 14.45. cosh cosh sinh ( )sinh ( ) xy x y x y−= + − 21 21 2 14.46. sinh sinh {cosh ( ) cosh ( )}x y xy xy =+ − −1 2 14.47. cosh cosh {cosh ( ) cosh ( )} xy x y x y =+ + −1 2 14.48. sinh cosh {sinh ( ) sinh ( )}xy x y x y =+ + −1 2 Expression of Hyperbolic Functions in Terms of Others In the following we assume x > 0. If x < 0, use the appropriate sign as indicated by formulas 14.14 to 14.19. sinh x = u cosh x = u tanh x = u coth x = u sech x = u csch x = u sinh xu u21− uu/12− 112/u− 12−uu/ 1/u cosh x 12+u u 112/−u uu/21− 1/u 12+uu/ tanh x uu/12+ uu21−/ u 1/u 12−u 112/+u coth x uu21+/ uu/21− 1/uu 112/−u 12+u sech x 112/+u 1/u 12−u uu21−/ u uu/12+ csch x 1/u 112/u− 12−uu/ u21− uu/12− uHYPERBOLIC FUNCTIONS 58 Graphs of Hyperbolic Functions 14.49. y = sinh x 14.50. y = cosh x 14.51. y = tanh x Fig. 14-1 Fig. 14-2 Fig. 14-3 14.52. y = coth x 14.53. y = sech x 14.54. y = csch x Fig. 14-4 Fig. 14-5 Fig. 14-6 Inverse Hyperbolic Functions If x = sinh y, then y = sinh–1 x is called the inverse hyperbolic sine of x. Similarly we define the other inverse hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as in the case of inverse trigo-nometric functions [see page 49] we restrict ourselves to principal values for which they can be considered as single-valued. The following list shows the principal values (unless otherwise indicated) of the inverse hyperbolic func- tions expressed in terms of logarithmic functions which are taken as real valued. 14.55. sinh ln ( )−=+ + − < <121 xx x x/H11009/H11009 14.56. cosh ln ( )−−=+ − >12 110 xx x x x /H110841 (cosh is prin ncipal value) 14.57. tanh ln−=+ −⎛ ⎝⎜⎞ ⎠⎟ −< <1 1 21 111 xx xx 14.58. coth ln−=+ −⎛ ⎝⎞ ⎠>< −1 1 21 111 xx xxx or 14.59. sech sech is pr− −=+ −⎛ ⎝⎜⎞ ⎠⎟<>1 21 1110 1 0 xx xxx ln ( /H11017 iincipal value) 14.60. csch−=+ +⎛ ⎝⎜⎞ ⎠⎟≠1 21110 xx xx lnHYPERBOLIC FUNCTIONS 59 Relations Between Inverse Hyperbolic Functions 14.61. csch−−=111 xxsinh ( / ) 14.62. sech−−=111 xxcosh ( / ) 14.63. coth tanh ( / )−−=111 xx 14.64. sinh ( ) sinh−−−= −11xx 14.65. tanh ( ) tanh−−−= −11xx 14.66. coth ( ) coth−−−= −11xx 14.67. csch csch−−−= −11()xx Graphs of Inverse Hyperbolic Functions 14.68. yx=−sinh1 14.69. yx=−cosh1 14.70. yx=−tanh1 Fig. 14-7 Fig. 14-8 Fig. 14-9 14.71. yx=−coth1 14.72. yx=−sech1 14.73. yx=−csch1 Fig. 14-10 Fig. 14-11 Fig. 14-12HYPERBOLIC FUNCTIONS 60 Relationship Between Hyperbolic and Trigonometric Functions 14.74. sin ( ) sinhix i x= 14.75. cos ( ) coshix x= 14.76. tan ( ) tanhix i x= 14.77. csc ( )ix i x=− csch 14.78. sec ( )ix x=sech 14.79. cot ( ) cothix i x=− 14.80. sinh ( ) sin ix i x= 14.81. cosh ( ) cos ix x= 14.82. tanh ( ) tan ix i x= 14.83. csch ( ) csc ix i x=− 14.84. sech ( ) sec ix x= 14.85. coth ( ) cot ix i x=− Periodicity of Hyperbolic Functions In the following k is any integer. 14.86. sinh ( ) sinh xk i x+=2π 14.87. cosh ( ) cosh xk i x+=2π 14.88. tanh ( ) tanh xk i x+=π 14.89. csch csch ()xk i x+=2π 14.90. sech sech ()xk i x+=2π 14.91. coth ( ) coth xk i x+=π Relationship Between Inverse Hyperbolic and Inverse Trigonometric Functions 14.92. sin ( ) sin−−=11ix i x 14.93. sinh ( ) sin−−=11ix i x 14.94. cos cosh−−=±11xi x 14.95. cosh cos−−=±11xi x 14.96. tan ( ) tanh−−=11ix i x 14.97. tanh ( ) tan−−=11ix i x 14.98. cot ( ) coth−−=11ix i x 14.99. coth ( ) cot−−=−11ix i x 14.100. sec−−=±11xi x sech 14.101. sech−−=±11xi x sec 14.102. csc ( )−−=−11ix i x csch 14.103. csch−−=−11() c s cix i xHYPERBOLIC FUNCTIONS 61 Section IV: Calculus 15 DERIVATIVES Definition of a Derivative Suppose y = f (x). The derivative of y or f (x) is defined as 15.1. dy dxfx h fx hfx x fx hx=+−=+− →→lim() ( )lim() ( ) 00 ΔΔ Δ Δx where h = Δx. The derivative is also denoted by y′ , df/dx or f′(x). The process of taking a derivative is called differentiation. General Rules of Differentiation In the following, u, /H9271, w are functions of x; a, b, c, n are constants (restricted if indicated); e = 2.71828 … is the natural base of logarithms; ln u is the natural logarithm of u (i.e., the logarithm to the base e ) where it is assumed that u > 0 and all angles are in radians. 15.2. d dxc()=0 15.3. d dxcx c()= 15.4. d dxcx ncxnn() =−1 15.5. d dxuwdu dxd dxdw dx()±± ± = ± ± ±/H9271/H9271/midhorizellipsis/midhorizellipsis 15.6. d dxcu cdu dx()= 15.7. d dxuud dxdu dx()/H9271/H9271/H9271 =+ 15.8. d dxuw udw dxuwd dxwdu dx()/H9271/H9271/H9271/H9271 =++ 15.9. d dxud u d x u d d x /H9271/H9271/H9271 /H9271⎛ ⎝⎜⎞ ⎠⎟=− (/) (/) 2 15.10. d dxun udu dxnn()=−1 15.11. dy dxdy dudu dx= (Chain rule) 62 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 15.12. du dx dx du=1 / 15.13. dy dxdy du dx du=/ / Derivatives of Trigonometric and Inverse Trigonometric Functions 15.14. d dxuudu dxsin cos = 15.15. d dxuudu dxcos sin =− 15.16. d dxuudu dxtan sec =2 15.17. d dxuudu dxcot csc =−2 15.18. d dxuu udu dxsec sec tan = 15.19. d dxuu udu dxcsc csc cot =− 15.20. d dxu udu dxu sin sin−−= −−< <⎡ ⎣⎢⎤ ⎦⎥1 21 1 1 22ππ 15.21. d dxu udu dxu cos [ cos ]−−=− −<<1 21 1 10 π 15.22. d dxuudu dxu tan tan−−=+−< <⎡ ⎣⎢⎤ ⎦⎥1 21 1 1 22ππ 15.23. d dxuudu dxu cot [ cot ]−−=− +<<1 21 1 10 π 15.24. d dxu uudu dx uudu dxsec ||se−= −=± −+<1 221 11 10if c c/ /s e c− −< −< <⎡ ⎣⎢⎤ ⎦⎥1 12 2u uπ ππif 15.25. d dxu uudu dx uudu dxcsc ||c−=− −= −−<1 221 11 10 ∓ if s sc / /c s c− −< +− < <⎡ ⎣⎢⎤ ⎦⎥1 12 20u uπ πif Derivatives of Exponential and Logarithmic Functions 15.26. d dxue udu dxaaaloglog, =≠ 01 15.27. d dxud dxuudu dxe ln log==1 15.28. d dxaaadu dxuu= lnDERIVATIVES 63 15.29. d dxeedu dxuu= 15.30. d dxud dxeed dxuudu dxuuu /H9271/H9271 /H9271 /H9271 /H9271/H9271/H9271 == =+− ln ln[l n] l1n nud dx/H9271 Derivatives of Hyperbolic and Inverse Hyperbolic Functions 15.31. d dxuudu dxsinh cosh = 15.32. d dxuudu dxcosh sinh = 15.33. d dxuudu dxtanh =sech2 15.34. d dxuudu dxcoth =−csch2 15.35. d dxuu udu dxsech sech =− tanh 15.36. d dxuu udu dxcsch csch =− coth 15.37. d dxu udu dxsinh−= +1 21 1 15.38. d dxu udu dxcosh−=± −1 21 1 +> > −< >⎡ ⎣⎢⎤ ⎦⎥− −if ifcosh , cosh ,1 101 01uu uu 15.39. d dxuudu dxtanh−=−1 21 1 [–1 < u < 1] 15.40. d dxuudu dxcoth−=−1 21 1 [ u > 1 or u < –1] 15.41. d dxu uudu dxsech−= −1 21 1∓ −> < < +< < <⎡ ⎣⎢− −if sech if sech1 100 1 00 1uu uu, ,⎤ ⎤ ⎦⎥ 15.42. d dxu uudu dx uudu dxcsch−=− += +1 221 11 1 ||∓ [– if u > 0, + if u < 0] Higher Derivatives The second, third, and higher derivatives are defined as follows. 15.43. Second derivative =⎛ ⎝⎜⎞ ⎠⎟== ′′ =′′d dxdy dxdy dxfx y2 2 () 15.44. Third derivative =⎛ ⎝⎜⎞ ⎠⎟== ′′′ =′′′d dxdy dxdy dxfx y2 23 3 () 15.45. nth derivative =⎛ ⎝⎜⎞ ⎠⎟== =− −d dxdy dxdy dxfxyn nn nnn1 1() ()()DERIVATIVES 64 Leibniz’s Rule for Higher Derivatives of Products Let Dp stand for the operator d dxp p so that Dudu dxpp p == the pth derivative of u. Then 15.46. Du u DnDu DnDnn n() ( ) ( ) (/H9271/H9271 /H9271=+⎛ ⎝⎜⎞ ⎠⎟ +⎛ ⎝⎜⎞ ⎠⎟− 1212uuD D unn)( )−++2/H9271/H9271/midhorizellipsis where nn 12⎛ ⎝⎜⎞ ⎠⎟⎛ ⎝⎜⎞ ⎠⎟,,… are the binomial coefficients (see 3.5). As special cases we have 15.47. d dxuud dxdu dxd dxdu dx2 22 22 22 ()/H9271/H9271/H9271/H9271 =+ + 15.48. d dxuud dxdu dxd dxdu dxd dx3 33 32 22 233 ()/H9271/H9271/H9271/H9271/H9271 = +++ddu dx3 3 Differentials Let y = f(x) and ΔΔyf x x f x=+−() ( ) . Then 15.49. Δ ΔΔ Δy xfx x fx xfxdy dx=+−=′+= +() ( )() /H9280/H9280 where /H9280 → 0 as Δx → 0. Thus, 15.50. ΔΔ Δyf x x x=′ + () /H9280 If we call Δx = dx the differential of x, then we define the differential of y to be 15.51. dy f x dx=′() Rules for Differentials The rules for differentials are exactly analogous to those for derivatives. As examples we observe that 15.52. du w d u d d w()±± ± = ± ± ±/H9271/H9271 /midhorizellipsis/midhorizellipsis 15.53. du u d d u()/H9271/H9271 /H9271=+ 15.54. dud u u d /H9271/H9271/H9271 /H9271⎛ ⎝⎜⎞ ⎠⎟=− 2 15.55. d u nu dunn()=−1 15.56. du u d u(sin ) cos = 15.57. du u d u(cos ) sin =−DERIVATIVES 65 Partial Derivatives Let z = f(x, y) be a function of the two variables x and y. Then we define the partial derivative of z or f(x, y) with respect to x, keeping y constant, to be 15.58. ∂ ∂=+− →f xfx xy fxy x xlim(, ) ( , ) ΔΔ Δ 0 This partial derivative is also denoted by ∂∂zxf zxx/, , . or Similarly the partial derivative of z = f (x, y) with respect to y, keeping x constant, is defined to be 15.59. ∂ ∂=+− →f yfxy y fxy y ylim(, ) (,) ΔΔ Δ 0 This partial derivative is also denoted by ∂∂zyf zyy/, , . or Partial derivatives of higher order can be defined as follows: 15.60. ∂ ∂=∂ ∂∂ ∂⎛ ⎝⎜⎞ ⎠⎟∂ ∂=∂ ∂∂ ∂⎛ ⎝⎜⎞ ⎠⎟2 22 2f x xf xf y yf y, 15.61. ∂ ∂∂=∂ ∂∂ ∂⎛ ⎝⎜⎞ ⎠⎟∂ ∂∂=∂ ∂∂ ∂⎛ ⎝⎜⎞ ⎠⎟22f xy xf yf yx yf x, The results in 15.61 will be equal if the function and its partial derivatives are continuous; that is, in such cases, the order of differentiation makes no difference. Extensions to functions of more than two variables are exactly analogous. Multivariable Differentials The differential of z = f(x, y) is defined as 15.62. dz dff xdxf ydy ==∂ ∂+∂ ∂ where dx = Δx and dy = Δy. Note that dz is a function of four variables, namely x, y, dx, dy, and is linear in the variables dx and dy. Extensions to functions of more than two variables are exactly analogous. EXAMPLE: Let z = x2 + 5xy + 2y3. Then zx = 2x + 5y and zy = 5x + 6y2 and hence dz = (2x + 5y) dx + (5x + 6y2) dy Suppose we want to find dz for dx = 2, dy = 3 and at the point P (4, 1), i.e., when x = 4 and y = 1. Substitution yields dz = (8 + 5)2 + (20 + 6)3 = 26 + 78 = 104DERIVATIVES DERIVATIVES 66 16 INDEFINITE INTEGRALS Definition of an Indefinite Integral If dy dxfx=() , then y is the function whose derivative is f (x) and is called the anti-derivative of f (x) or the indefi- nite integral of f (x), denoted by fxd x() .∫ Similarly if yf u d u=∫() , then dy dufu=() . Since the derivative of a constant is zero, all indefinite integrals differ by an arbitrary constant. For the definition of a definite integral, see 18.1. The process of finding an integral is called integration. General Rules of Integration In the following, u, /H9271, w are functions of x; a, b, p, q, n any constants, restricted if indicated; e = 2.71828 … is the natural base of logarithms; ln u denotes the natural logarithm of u where it is assumed that u > 0 (in general, to extend formulas to cases where u < 0 as well, replace ln u by ln |u|); all angles are in radians; all constants of integration are omitted but implied. 16.1. ad x a x =∫ 16.2. af x dx a f x dx() () =∫ ∫ 16.3. ()u w dx u dx dx w dx±± ± = ± ± ± ∫ ∫∫ ∫/H9271/H9271 /midhorizellipsis/midhorizellipsis 16.4. ud u d u/H9271/H9271 /H9271=−∫∫(Integration by parts) For generalized integration by parts, see 16.48. 16.5. fa xd xafud u () ( ) = ∫∫1 16.6. Ffx d x F udx duduFu fxdu u {() } ()() ()==′= ∫∫ ∫where ffx() 16.7. ud uu nnnnn =+≠− =−+ ∫1 111 1 6 8 ,( , . ) For see 16.8. du uuu u u u=> − < =∫ln ln( ) ln | |if or if 00 16.9. ed u euu=∫ 16.10. ad u e d ue aa aaauu aua u == => ≠∫∫lnln ln ln,, 01 67 16.11. sin cosud u u∫=− 16.12. cos sinud u u∫= 16.13. tan ln sec ln cosud u u u∫== − 16.14. cot ln sinud u u∫= 16.15. sec ln (sec tan ) ln tanud u u uu∫=+ =+⎛ ⎝⎜⎞ ⎠⎟24π 16.16. csc ln(csc cot ) ln tanud u u uu∫=− =2 16.17. sec tan2ud u u∫= 16.18. csc cot2ud u u∫=− 16.19. tan tan2ud u u u∫=− 16.20. cot cot2ud u u u∫=− − 16.21. sinsin(s i n c o s )2 22 41 2ud uuuuu u ∫=− = − 16.22. cossin(s i n c o s )2 22 41 2ud uuuuu u ∫=+ = + 16.23. sec tan secuu d u u∫= 16.24. csc cot cscuu d u u∫=− 16.25. sinh cosh ud u u = ∫ 16.26. cosh sinh ud u u = ∫ 16.27. tanh ln cosh ud u u = ∫ 16.28. coth ln sinh ud u u = ∫ 16.29. sech or ud u u eu=−−∫sin (tanh ) tan112 16.30. csch or ud uueu∫=−−ln tanh coth21 16.31. sech2ud u u∫=tanhINDEFINITE INTEGRALS 68 16.32. csch2∫=− ud u u coth 16.33. tanh tanh2∫=− ud u u u 16.34. coth coth2∫=− ud u u u 16.35. sinhsinh(sinh cosh )2 2 421 2 ∫=− = −ud uuuuu u 16.36. coshsinh(sinh cosh )2 2 421 2 ∫=+ = +ud uuuuu u 16.37. sech sech uu d u utanh =− ∫ 16.38. csch csch uu d u ucoth =− ∫ 16.39. du ua au a221 1 +=−∫tan 16.40. du ua aua ua au aua2212 2 1 21 −=− +⎛ ⎝⎜⎞ ⎠⎟=− > ∫−ln coth 16.41. du au aau au au aua2212 2 1 21 −=+ −⎛ ⎝⎜⎞ ⎠⎟=< ∫−ln tanh 16.42. du auu a 221 −= ∫−sin 16.43. du uauu au a 2222 1 +=++−∫ln( ) sinhor 16.44. du uauu a 2222 −=+ − ∫ln ( ) 16.45. du uu a au a 221 1 −=−∫sec 16.46. du uu a aau a u 22221 +=−++⎛ ⎝⎜⎞ ⎠⎟ ∫ln 16.47. du ua u aaa u u 22221 −=−+−⎛ ⎝⎜⎞ ⎠⎟ ∫ln 16.48. fg d x f g f g f g fnn n n n() ( ) ( ) ( )() =− ′+ ′′−−−− −∫12 31/midhorizellipsis ggd xn()∫ This is called generalized integration by parts.INDEFINITE INTEGRALS 69 Important Transformations Often in practice an integral can be simplified by using an appropriate transformation or substitution together with Formula 16.6. The following list gives some transformations and their effects. 16.49. Fa x bd xaFu d u () ( )+= ∫ ∫1 where u = ax + b 16.50. Fa xb d xauFud u () ( ) += ∫∫2 where ua x b=+ 16.51. Fa xb d xn auF u d un n() ( ) += ∫∫−1 where ua x bn=+ 16.52. Fa xd xa F a u u d u( ) ( cos ) cos22−= ∫ ∫ where x = a sin u 16.53. Fx ad xa F a u u d u( ) ( sec )sec22 2+= ∫ ∫ where x = a tan u 16.54. Fx ad xa F a u u u d u() ( t a n ) s e c t a n22−= ∫∫ where x = a sec u 16.55. Fe d xaFu uduax()()=∫ ∫1 where u = eax 16.56. Fx d x F u e d uu(ln ) ( ) =∫ ∫ where u = ln x 16.57. Fx adx a F u u du sin ( ) cos−⎛ ⎝⎜⎞ ⎠⎟= ∫∫1 where ux a=−sin1 Similar results apply for other inverse trigonometric functions. 16.58. Fx x d x Fu uu udu(sin , cos ) , =+− +⎛ ⎝⎜⎞ ⎠⎟ ∫∫22 11 1122 2+ +u2 where ux=tan2INDEFINITE INTEGRALS 70 17TABLES of SPECIAL INDEFINITE INTEGRALS Here we provide tables of special indefinite integrals. As stated in the remarks on page 67, here a, b, p, q, n are constants, restricted if indicated; e = 2.71828 . . . is the natural base of logarithms; ln u denotes the natural logarithm of u , where it is assumed that u > 0 (in general, to extend formulas to cases where u < 0 as well, replace ln u by ln |u|); all angles are in radians; and all constants of integration are omitted but implied. It is assumed in all cases that division by zero is excluded. Our integrals are divided into types which involve the following algebraic expressions and functions: (1) ax + b (13) ax bx c2++ (25) eax (2) ax b+ (14) x3 + a3 (26) ln x (3) ax + b and px + q (15) xa44± (27) sinh ax (4) ax b+ and px + q (16) xann± (28) cosh ax (5) ax b px q++and (17) sin ax (29) sinh ax and cosh ax (6) x2 + a2 (18) cos ax (30) tanh ax (7) x2 – a2, with x2 > a2 (19) sin ax and cos ax (31) coth ax (8) a2 – x2, with x2 < a2 (20) tan ax (32) sech ax (9) xa22+ (21) cot ax (33) csch ax (10) xa22− (22) sec ax (34) inverse hyperbolic functions (11) ax22− (23) csc ax (12) ax2 + bx + c (24) inverse trigonometric functions Some integrals contain the Bernouilli numbers Bn and the Euler numbers En defined in Chapter 23. (1) Integrals Involving ax/H11545b 17.1.1. dx ax b aax b+=+ ∫1ln ( ) 17.1.2. xd x ax bx ab aax b+=− + ∫ 2ln ( ) 17.1.3. xd x ax bax b aba x b ab aax b22 332 322 +=+−+++()()ln ( ) ∫ ∫ 17.1.4. dx xa x b bx ax b ()ln+=+⎛ ⎝⎜⎞ ⎠⎟ ∫1 17.1.5. dx xa x b bxa bax b x221 ()ln+=− ++⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.1.6. dx ax b aa x b () () +=− + ∫ 21 17.1.7. xd x ax bb aa x b aax b() ()ln ( )+=+++ ∫ 22 21 17.1.8. xd x ax bax b ab aa x bb aax b2 232 332 () ()ln ( )+=+−+−+ ∫ 17.1.9. dx xa x b ba x b bx ax b () ()ln+=+++⎛ ⎝⎜⎞ ⎠⎟ ∫ 2211 71 72 17.1.10. dx xa x ba ba x b b xa bax b x22 2 2 312 () ()ln+=− +−++⎛ ⎝⎜⎞ ⎠ ⎠⎟ ∫ 17.1.11. dx ax b ax b() ()+=− + ∫ 321 2 17.1.12. xd x ax b a ax bb aa x b ()() ()+=− +++ ∫ 32 2 21 2 17.1.13. xd x ax bb aa x bb aa x b aa2 332 32 32 21 ()()()ln (+=+−++ xxb+ ∫) 17.1.14. ()() ()., . . ax b dxax b nannn +=+ +=−+1 111 7 1 If see 1 1. ∫ 17.1.15. xa x b d xax b naba x b nnnn ()() ()() (+=+ +−+ + ∫++2 21 21 ) ),,an2 12≠− − If n = –1, –2, see 17.1.2 and 17.1.7. 17.1.16. xa x bd xax b naba x b nnnn 23 32 32()() ()() (+=+ +−+∫++ + +++ ++ 21321 3)() () aba x b nan If n = –1, –2, –3, see 17.1.3, 17.1.8, and 17.1.13. 17.1.17. xa xb d xxa x b mnnb mnxa x mnmn m ∫+=+ +++++++ ()()(1 11bbd x xa xb mn amb mn axn mn m) () () ()− + −∫ + ++−++1 1 1 11(() () () ()ax b dx xa x b nbmn nbn mn+ −+ ++++ +∫ ++11 12 1xxa xb d xmn∫+⎧ ⎨⎪ ⎪⎪ ⎩⎪ ⎪ ⎪+()1 (2) Integrals Involving ax b/H11545 17.2.1. dx ax bax b a +=+∫2 17.2.2. xd x ax bax b aax b+=−+ ∫22 32() 17.2.3. xd x ax ba x abx b aax b22 2 2 323 4 8 15 +=−++ ∫() 17.2.4.dx xa x bbax b b ax b b bax b +=+− ++⎛ ⎝⎜⎞ ⎠⎟ −+ −−1 21ln tanb b⎧ ⎨⎪ ⎪ ⎩⎪ ⎪∫ 17.2.5.dx xa x bax b bxa bdx xa x b2 2 +=−+− +∫ ∫(see 17.2.12.) 17.2.6. ax b dxax b a+=+∫2 33() 17.2.7. xa x b d xax b aax b +=−+ ∫23 2 1523 ()()TABLES OF SPECIAL INDEFINITE INTEGRALS 17.2.8. xa xb d xa x abx b aax b222 2 33 21 5 1 2 8 105+=−++ ∫()() 17.2.9.ax b xdx ax b bdx xa x b+=+ + +∫ ∫2 (See 17.2.12.) 17.2.10.ax b xdxax b xad x xa x b+=−++ +∫∫ 22(See 17.2.12.) 17.2.11.x ax bdxxa x b mamb max ax bmm m +=+ +−+ + ∫−2 212 211 () ()ddx ∫ 17.2.12.dx xa x bax b mb xma mbdx xm m m+=−+ −−− −−()() () 123 221 − −+ ∫ ∫ 1ax b 17.2.13. xa x b d xx maax bmb maxmm m+=++−+− 2 232 2332 1 ()()()/aax b dx+ ∫ ∫ 17.2.14.ax b xdxax b mxa mdx xa x bmm m+=−+ −+− +− − ∫∫ () () 1 211 1 17.2.15.ax b xdxax b mb xma mmm+=−+ −−− −−() ()() ()/32 1125 22 b bax b xdxm+ −∫ ∫ 1 17.2.16. ()() ()/() / ax b dxax b ammm +=+ + ∫+ 222 22 2 17.2.17. xa x b d xax b amba x bmm ()() ()(/() / +=+ +−+∫+ 242 22 42 ) ) ()() /m am+ +22 22 17.2.18. xa x b d xax b amba xmm 2262 32 64()() ()(/() / +=+ +−+∫+b b amba x b ammm) ()() ()() / () / ++ +++ +42 322 2 342 2 17.2.19.() () ()// ( )/ax b xdxax b mbax b xdmmm+=+++∫∫− 22 2 22x x 17.2.20.() () ()/( )/ax b xdxax b bxma bax bmm m+=−++++ ∫2 222 2//2 xdx ∫ 17.2.21.dx xa x b m ba x b bdx xa xmm() ( ) () (/( )/+=−+++− 22 22 21 b bm)() /− ∫ ∫ 22 (3) Integrals Involving ax/H11545b and px /H11545q 17.3.1. dx ax b px q bp aqpx q ax b () ()ln++=−+ +⎛ ⎝⎜⎞ ⎠⎟ ∫1 17.3.2.xd x ax b px q bp aqb aax bq ppx q() ()ln ( ) ln ( )++=−+− +1 ⎧ ⎧ ⎨ ⎩⎫ ⎬ ⎭∫ TABLES OF SPECIAL INDEFINITE INTEGRALS 73 74 17.3.3. dx ax b px q bp aq ax bp bp aqpx q ax () ()ln++=−++−+ +211 b b⎛ ⎝⎜⎞ ⎠⎟⎧⎨⎩⎫⎬⎭∫ 17.3.4. xd x ax b px q bp aqq bp aqax b px q () ()ln++=−−+ +⎛ ⎝⎜⎞ 21 ⎠ ⎠⎟−+⎧⎨⎩⎫⎬⎭∫b aa x b() 17.3.5.xd x ax b px qb bp aq a ax b bp aq2 22 21 () () ( ) () (++=−++−) )ln ( )()ln ( )22 22 q ppx qbb p a q aax b ++−+⎧ ⎨ ⎩⎫ ⎬ ⎭∫ 17.3.6.dx ax b px q nb p a q ax b pmn m() () () ( ) () ( ++=− −− +−1 11 1xxq am ndx ax b px qn mn+ { ++ −++ }− −∫ ∫) ()() ()1 1 2 17.3.7.ax b px qdxax pbp aq ppx q+ +=+−+ ∫ 2ln( ) 17.3.8.() ()() ( )() ( ax b px qdxnb p a qax b px m nm + +=− −−++1 11 + ++−−+ +⎧ ⎨ ⎩⎫ ⎬ ⎭ −−− ∫qnm aax b px qdxnm n)()() ()112 1 (()() ()()() nm pax b px qmb p a qax bm nm −−+ ++−+ −− 111 (() ()() ()px qdx npax b px qmn m n+⎧ ⎨ ⎩⎫ ⎬ ⎭ − −+ +−∫ −1 11a aax b px qdxm n() ()+ +⎧ ⎨ ⎩⎫ ⎬ ⎭⎧ ⎨⎪ ⎪ ⎪ ⎩⎪ ⎪⎪− − ∫∫ 1 1 (4) Integrals Involving ax b/H11545and px/H11545q 17.4.1. px q ax bdxapx aq bp aax b+ +=+−+ ∫23 2 32() 17.4.2. dx px q ax bbp aq pp ax b bp aq pa x b ()ln() () ++=−+− − +∫1 ++−⎛ ⎝⎜⎞ ⎠⎟ −+ −⎧ ⎨⎪ ⎪ ⎩⎪−bp aq aq bp ppa x b aq bp21tan() ⎪ ⎪ 17.4.3. ax b px qdxax b pbp aq ppp ax b bp aq pa + +=++−+ − − 2ln() (xx b bp aq ax b paq bp pppa x b++ −⎛ ⎝⎜⎞ ⎠⎟ +−− +−) tan( 2 21 ) ) aq bp−⎧ ⎨⎪ ⎪ ⎩⎪ ⎪∫ 17.4.4. ()() () (px q ax b dxpx q ax b npbp aqnn ++ =++ ++−+2 23 21 nnppx q ax bn ++ +∫ ∫ 3)() 17.4.5. dx px q ax bax b na q b p p x qn n n() () ( ) ( )( ++=+ −− ++−12 1− − −− ++ ∫∫ −3 21 1) () ( ) ()a na q b pdx px q ax bn 17.4.6. () () ()() px q ax bdxpx q ax b nana q b pnn+ +=++ ++− 2 212 (()() 211 napx q dx ax bn ++ +− ∫ ∫ 17.4.7. ax b px qdxax b np p x qa npnn+ +=−+ −++−−() ( ) () () 1 211 ∫∫∫++−dx px q ax bn()1TABLES OF SPECIAL INDEFINITE INTEGRALS 75 (5) Integrals Involving ax b/H11545 and px q+ 17.5.1.dx ax b px qapa p xq p a xb ap() ()ln ( ) ( ) t++=++ + ( ) −∫2 2aan() ()−−+ +⎧ ⎨⎪⎪ ⎩⎪ ⎪1 pa x b ap x q 17.5.2.xd x ax b px qax b px q apbp aq apdx () ()() () ( ++=++−+∫ 2 aax b px q++ ∫)( ) 17.5.3. () () () ()(ax b px q dxapx bp aq apax b px qb++ =++++ −2 4ppa q apdx ax b px q− ++ ∫ ∫) () ()2 8 17.5.4.px q ax bdxax b px q aaq bp adx ax b p+ +=+++− + ∫() () () ( 2 xxq+ ∫) 17.5.5.dx px q ax b px qax b aq bp px q () () () ( )++ +=+ −+ ∫2 (6) Integrals Involving x2/H11545a2 17.6.1.dx xa ax a221 1 +=−∫tan 17.6.2.xd x xaxa2222 1 2 +=+ ∫ln ( ) 17.6.3.xd x xaxax a2 221 +=−−∫tan 17.6.4.xd x xaxaxa3 2222 22 22 +=− + ∫ln ( ) 17.6.5.dx xx a ax xa ()ln22 22 221 2 +=+⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.6.6.dx xx a a x ax a22 2 2 31 11 ()tan+=− −−∫ 17.6.7.dx xx a a x ax xa32 2 2 2 42 221 21 2 ()ln+=− −+⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.6.8.dx xax ax a ax a () ()tan22 2 2 22 31 21 2 +=++−∫ 17.6.9.xd x xa xa() ()22 2 221 2 +=− + ∫ 17.6.10.xd x xax xa ax a2 22 2 221 21 2 () ()tan+=− ++−∫TABLES OF SPECIAL INDEFINITE INTEGRALS 76 17.6.11.xd x xaa xaxa3 22 22 2222 21 2 () ()ln ( )+=+++ ∫ 17.6.12.dx xx a a x a ax xa () ()ln22 2 2 22 42 221 21 2 +=+++⎛ ⎝⎜⎞ ⎠⎟ ∫ ∫ 17.6.13.dx xx a a xx ax a ax a22 2 2 4 42 2 51 1 23 2 () ()tan+=− −+−−∫ 17.6.14.dx xx a a x ax a ax x32 2 2 4 2 4 2 2 62 21 21 21 () ()ln+=− −+−+a a2⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.6.15.dx xax na x an nann() ( ) () ( )22 2 22 1 22123 22 +=−++− −− ∫∫∫ +−dx xan()22 1 17.6.16.xd x xa n xann() ( ) ()22 22 11 21 +=− −+− ∫ 17.6.17.dx xx a n a x a adx xxnn() ( ) () (22 2 22 1 2 21 211 +=−+++ ∫ −a an21)− ∫ 17.6.18.xd x xaxd x xaaxd x xm nm nm () () (222 22 122 2+=+−− −− ∫ ∫ + +∫ an2) 17.6.19. dx xx a adx xx a adx xxmn mn m() () (22 2 22 1 2 211 +=+− ∫ −− 222+ ∫ ∫ an) (7) Integrals Involving x2/H11546a2,x2>a2 17.7.1.dx xa axa xa ax a221 1 21 −=− +⎛ ⎝⎜⎞ ⎠⎟ − ∫−ln coth or 17.7.2.xd x xaxa2222 1 2 −=− ∫ln ( ) 17.7.3.xd x xaxax a xa2 22 2 −=+− +⎛ ⎝⎜⎞ ⎠⎟ ∫ln 17.7.4.xd x xaxaxa3 2222 22 22 −=+ − ∫ln ( ) 17.7.5.dx xx a axa x ()ln22 222 21 2 −=−⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.7.6.dx xx a a x axa xa22 2 2 311 2 ()ln−=+− +⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.7.7.dx xx a a x ax xa32 2 2 2 42 221 21 2 ()ln−=−−⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.7.8.dx xax ax a axa xa () ()ln22 2 2 22 321 4 −=− −−− +⎛ ⎝⎜⎞ ⎠⎟ ∫TABLES OF SPECIAL INDEFINITE INTEGRALS 77 17.7.9. xd x xa xa() ()22 2 221 2 −=− − ∫ 17.7.10.xd x xax xa axa xa2 22 2 2221 4 () ()ln−=− −+− +⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.7.11.xd x xaa xaxa3 22 22 2222 21 2 () ()ln ( )−=− −+− ∫ 17.7.12.dx xx a a x a ax xa () ()ln22 2 2 22 42 221 21 2 −=− −+−⎛ ⎝⎜⎞ ⎠⎟ ⎟ ∫ 17.7.13.dx xx a a xx ax a axa xa22 2 2 4 4 2 2 51 23 4 () ()ln−=− −−−− +⎛ ⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.7.14.dx xx a a x ax a ax x32 2 2 4 2 4 2 2 62 21 21 21 () ()ln−=− −−+−a a2⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.7.15.dx xax na x an nann() ( ) () ( )22 2 22 12123 22 −=− −−−− −− 222 2 1 ∫∫ −−dx xan() 17.7.16.xd x xa n xann() ( ) ()22 22 11 21 −=− −−− ∫ 17.7.17.dx xx a n a x a adx xxnn() ( ) () (22 2 22 1 2 21 211 −=− −−−− ∫ − −− ∫ an21) 17.7.18.xd x xaxd x xaaxd x xam nm nm () () (222 22 122 2−=−+−− −− 2 2)n ∫ ∫ ∫ 17.7.19.dx xx a adx xx a adx xx amn m n m() () (22 2 2 22 2 211 −=−−−− 221)n− ∫ ∫ ∫ (8) Integrals Involving x2/H11546a2,x2<a2 17.8.1.dx ax aax ax ax a221 1 21 −=+ −⎛ ⎝⎜⎞ ⎠⎟ ∫−ln tanh or 17.8.2.xd x axax2222 1 2 −=− − ∫ln ( ) 17.8.3.xd x axxaa x ax2 22 2 −=− ++ −⎛ ⎝⎜⎞ ⎠⎟ ∫ln 17.8.4.xd x axxaax3 2222 22 22 −=− − − ∫ln ( ) 17.8.5.dx xa x ax ax ()ln22 22 221 2 −=−⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.8.6.dx xa x a x aax ax22 2 2 311 2 ()ln−=− ++ −⎛ ⎝⎜⎞ ⎠⎟ ∫TABLES OF SPECIAL INDEFINITE INTEGRALS 78 17.8.7. dx xa x a x ax ax32 2 2 2 42 221 21 2 ()ln−=− +−⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.8.8. dx axx aa x aax ax () ()ln22 2 2 22 321 4 −=−++ −⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.8.9. xd x ax ax() ()22 2 221 2 −=− ∫ 17.8.10.xd x axx ax aax ax2 22 2 2221 4 () ()ln−=−−+ −⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.8.11.xd x axa axax3 22 22 2222 21 2 () ()ln ( )−=−+− ∫ 17.8.12.dx xa x a a x ax ax () ()ln22 2 2 22 42 221 21 2 −=−+−⎛ ⎝⎜⎞ ⎠⎟ ∫ ∫ 17.8.13.dx xa x a xx aa x aax ax22 2 2 4 4 2 2 51 23 4 () ()ln−=−+−++ −⎛ ⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.8.14.dx xa x a x aa x ax a32 2 2 4 2 4 2 2 62 21 21 21 () ()ln−=−+−+−x x2⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.8.15.dx axx na a xn nann() ( ) () ( )22 2 22 1 22123 22 −=−−+− −− ∫∫∫ −−dx axn()22 1 17.8.16.xd x ax n axnn() ( ) ()22 22 11 21 −=−−− ∫ 17.8.17.dx xa x n a a x adx xa xnn() ( ) () (22 2 22 1 2 21 211 −=−−+−− 221)n− ∫ ∫ 17.8.18.xd x axaxd x axxd x axm nm nm () () (2222 222 2−=−−− ∫∫−− 221)n− ∫ 17.8.19.dx xa x adx xa x adx xamn m n m() () (22 2 22 1 2 211 −=−+−− ∫ 222− ∫ ∫ xn) (9) Integrals Involving xa22+ 17.9.1. dx xaxx ax a 2222 1 +=+ + ∫−ln ( ) sinhor 17.9.2.xd x xaxa 2222 +=+ ∫ 17.9.3.xd x xaxx a axx a2 2222 2 22 22 +=+−+ + ∫ln( ) 17.9.4.xd x xaxaax a3 2222 3 2 22 2 3 +=+−+ ∫()/TABLES OF SPECIAL INDEFINITE INTEGRALS 79 17.9.5. dx xx a aax a x 22221 +=−++⎛ ⎝⎜⎞ ⎠⎟ ∫ln 17.9.6. dx xx axa ax 22 222 2+=−+∫ 17.9.7. dx xx axa ax aax a x 32 222 22 322 21 2 +=−++++⎛ ⎝⎜⎞ ⎠⎟ ∫ln 17.9.8. xa d xxx a axx a2222 2 22 22+=+++ + ∫ln ( ) 17.9.9. xx ad xxa2222 3 2 3+=+∫()/ 17.9.10. xx a d xxx a ax x a axx22 222 3 2 2 22 4 2 48 8+=+−+−+()ln (/ + + ∫a2) 17.9.11. xx a d xxa a xa32 222 5 2 2 22 3 2 53+=+−+∫() ()// 17.9.12.xa xdx x a aax a x22 2222+=+ −++⎛ ⎝⎜⎞ ⎠⎟ ∫ln 17.9.13.xa xdxxa xxx a22 222 22 +=−++++ ∫ln ( ) 17.9.14.xa xdxxa x aax a x22 322 222 21 2+=−+−++⎛ ⎝⎜⎞ ⎠⎟ ∫ln 17.9.15.dx xax ax a ()/ 22 3 222 2 += +∫ 17.9.16.xd x xa xa ()/ 22 3 2221 +=− +∫ 17.9.17.xd x xax xaxx a2 22 3 22222 ()ln ( )/+=− +++ + ∫ 17.9.18.xd x xaxaa xa3 22 3 2222 22 ()/+=+ + +∫ 17.9.19.dx xx a ax a aax a x ()ln/ 22 3 222 232211 += +−++⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.9.20.dx xx axa axx ax a22 2 3 222 442 2 ()/+=−+− +∫ 17.9.21.dx xx a ax x a a x a a32 2 3 222 2 2 4 2 251 23 23 2 ()l/+=− +− ++ n nax a x++⎛ ⎝⎜⎞ ⎠⎟ ∫22 17.9.22. ()()l// xa d xxx a ax x aa22 3 222 3 2 2 22 4 43 83 8+=++++ ∫nn( )xx a++22 17.9.23. xx a d xxa()()// 22 3 222 5 2 5+=+∫TABLES OF SPECIAL INDEFINITE INTEGRALS 80 17.9.24. xx a d xxx a axx a22 2 3 222 5 2 2 22 3 2 62()() ()/// +=+−+∫441 6 1 642 26 22−+−+ +ax x a axx aln ( ) 17.9.25. xx a d xxa a xa32 2 3 222 7 2 2 22 5 2 75()() ()/// +=+−+∫ 17.9.26.() ()ln//xa xdxxaax a aax2 2 32 2 2 32 22 2 32 3+=+++ −++ a a x2 ⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.9.27.() ()ln//xa xdxxa xxx aa22 3 2 222 3 2 22 2 3 23 2+=−++++ (()xx a++ ∫22 17.9.28.() ()ln//xa xdxxa xxa a22 3 2 322 3 2 222 23 23 2+=−+++ −aax a x++⎛ ⎝⎜⎞ ⎠⎟ ∫22 (10) Integrals Involving xa22− 17.10.1.dx xaxx axd x xaxa 2222 2222 −=+ − −=− ∫∫ln ( ), 17.10.2.xd x xaxx a axx a2 2222 2 22 22 −=−++ − ∫ln ( ) 17.10.3.xd x xaxaax a3 2222 3 2 22 2 3 −=−+− ∫()/ 17.10.4.dx xx a ax a 221 1 −=−∫sec 17.10.5.dx xx axa ax 22 222 2−=−∫ 17.10.6.dx xx axa ax ax a 32 222 22 31 21 2 −=−+−∫sec 17.10.7. xa d xxx a axx a2222 2 22 22−=−−+ − ∫ln ( ) 17.10.8. xx ad xxa2222 3 2 3−=−∫()/ 17.10.9. xx a d xxx a ax x a axx22 222 3 2 2 22 4 2 48 8−=−+−−+()ln (/ − − ∫a2) 17.10.10. xx a d xxa a xa32 222 5 2 2 22 3 2 53−=−+−∫() ()// 17.10.11.xa xdx x a ax a22 22 1 −=− −−∫sec 17.10.12.xa xdxxa xxx a22 222 22 −=−−++ − ∫ln ( )TABLES OF SPECIAL INDEFINITE INTEGRALS 81 17.10.13.xa xdxxa xax a22 322 21 21 2−=−−+−∫sec 17.10.14.dx xax ax a ()/ 22 3 222 2 −=− −∫ 17.10.15.xd x xa xa ()/ 22 3 2221 −=− −∫ 17.10.16.xd x xax xaxx a2 22 3 22222 ()ln ( )/−=− −++− ∫ 17.10.17.xd x xaxaa xa3 22 3 2222 22 ()/−=− − −∫ 17.10.18.dx xx a ax a ax a ()sec/ 22 3 222 231 11 −=− −−−∫ 17.10.19.dx xx axa axx ax a22 2 3 222 442 2 ()/−=−−− −∫ 17.10.20.dx xx a ax x a a x a a32 2 3 222 2 2 4 2 251 23 23 2 ()se/−= −− −− c c−∫1x a 17.10.21. ()()l// xa d xxx a ax x aa22 3 222 3 2 2 22 4 43 83 8−=−−−+ ∫nn( )xx a+−22 17.10.22. xx a d xxa()()// 22 3 222 5 2 5−=−∫ 17.10.23. xx a d xxx a axx a22 2 3 222 5 2 2 22 3 2 62()() ()/// −=−+−∫ 441 6 1 642 26 22−−++ −ax x a axx aln ( ) 17.10.24. xx a d xxa a xa32 2 3 222 7 2 2 22 5 2 75()() ()/// −=−+−∫ 17.10.25.() ()sec//xa xdxxaax a ax22 3 2 22 3 2 22 2 3 1 3−=−−− + ∫− a a 17.10.26.() ()l//xa xdxxa xxx aa22 3 2 222 3 2 22 2 3 23 2−=−−+−− ∫nn( )xx a+−22 17.10.27.() ()se//xa xdxxa xxaa22 3 2 322 3 2 222 23 23 2−=−−+−− c c−∫1x a (11) Integrals Involving ax22− 17.11.1.dx axx a 221 −=−∫sin 17.11.2.xd x axax 2222 −=− − ∫TABLES OF SPECIAL INDEFINITE INTEGRALS 82 17.11.3. xd x axxa x a x a2 2222 2 1 22 −=−−+ ∫−sin 17.11.4. xd x axaxaa x3 2222 3 2 22 2 3 −=−−− ∫()/ 17.11.5. dx xa x aaa x x 22221 −=−+−⎛ ⎝⎜⎞ ⎠⎟ ∫ln 17.11.6. dx xa xax ax 22 222 2−=−−∫ 17.11.7. dx xa xax ax aaa x x 32 222 22 322 21 2 −=−−−+−⎛ ⎝⎜⎞ ⎠⎟ ∫ln 17.11.8. ax d xxa x a x a2222 2 1 22−=−+ ∫−sin 17.11.9. xa x d xax2222 3 2 3−= −−∫()/ 17.11.10. xa x d xxa x ax a x a22 222 3 2 2 22 4 1 48 8−= −−+−+ ∫− ()sin/x x a 17.11.11. xa x d xax a ax32 222 5 2 2 22 3 2 53−=−−−∫() ()// 17.11.12.ax xdx a x aaa x x22 2222−=− −+−⎛ ⎝⎜⎞ ⎠⎟ ∫ln 17.11.13.ax xdxax xx a22 222 1 −=−−−−∫sin 17.11.14.ax xdxax x aaa x x22 322 222 21 2−=−−++−⎛ ⎝⎜⎞ ⎠⎟ ∫ln 17.11.15.dx axx aa x ()/ 22 3 222 2 −= −∫ 17.11.16.xd x ax ax ()/ 22 3 2221 −= −∫ 17.11.17.xd x axx axx a2 22 3 2221 ()sin/−= −− ∫− 17.11.18.xd x axaxa ax3 22 3 2222 22 ()/−=− + −∫ 17.11.19.dx xa x aa x aaa x x ()ln/ 22 3 222 232211 −= −−+−⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.11.20.dx xa xax axx aa x22 2 3 222 442 2 ()/−=−−+ −∫ 17.11.21.dx xa x ax a x a a x a32 2 3 222 2 2 4 2 251 23 23 2 ()/−=− −+ −− ∫llnaa x x+−⎛ ⎝⎜⎞ ⎠⎟22TABLES OF SPECIAL INDEFINITE INTEGRALS 83 17.11.22. ()()s// ax d xxa x ax a xa22 3 222 3 2 2 22 4 43 83 8−=−+−+ ∫iin−1x a 17.11.23. xa x d xax()()// 22 3 222 5 2 5−= −−∫ 17.11.24. xa x d xxa x axa x22 2 3 222 5 2 2 22 3 2 6()() ()/// −= −−+−∫ 224 16 1642 26 1+−+− ax a x a x asin 17.11.25. xa x d xax a ax32 2 3 222 7 2 2 22 5 2 75()() ()/// −=−−−∫ 17.11.26.() ()ln//ax xdxaxaa x aaa22 3 2 22 3 2 22 2 32 3−=−+− −+∫− − ⎛ ⎝⎜⎞ ⎠⎟x x2 17.11.27.() ()s//ax xdxax xxa xa22 3 2 222 3 2 22 2 3 23 2−=−−−−− ∫iin−1x a 17.11.28.() ()l//ax xdxax xaxa22 3 2 322 3 2 222 23 23 2−=−−−−+ ∫n naa x x+−⎛ ⎝⎜⎞ ⎠⎟22 (12) Integrals Involving ax2/H11545bx/H11545c 17.12.1.dx ax bx cac bax b ac b ba c221 2 22 42 4 1 4++=−+ − −−tan ln224 242 2ax b b ac ax b b ac+− − ++ −⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨⎪⎪ ⎩⎪ ⎪∫ If ba c a x b x c a x b a22 242=+ + = + ,( / ) and the results 17.1.6 to 17.1.10 and 17.1.14 to 17.1.17 can be used. If b = 0 use results on page 75. If a or c = 0 use results on pages 71–72. 17.12.2.xd x ax bx c aax bx cb adx ax bx c22 21 22 ++=+ + −++∫ ∫ln ( ) 17.12.3.xd x ax bx cx ab aax bx cba c a2 2222 222 2 ++=− + ++−∫ln ( )ddx ax bx c2++∫ 17.12.4.xd x ax bx cx mac axd x ax bx cb amm m 212 2 1 ++=−−++−−− ()xxd x ax bx cm− ++∫ ∫ ∫1 2 17.12.5.dx xa x b x c cx ax bx cb cdx a ()ln22 21 22 ++=++⎛ ⎝⎜⎞ ⎠⎟− ∫ xxb x c2++∫ 17.12.6.dx xa x b x cb cax bx c x cx22 22 221 ()ln++=++ ⎛ ⎝⎜⎞ ⎠⎟−+ ∫bba c cdx ax bx c2 222 2− ++∫ 17.12.7.dx xa x b x c n c xb cdx xa xb xnn n() ( ) (21 1 21 1 ++=−−−+−−+ +−++−∫ ∫ ∫ ca cdx xa xb x cn)()22 17.12.8.dx ax bx cax b ac b ax bx ca ac () ( ) ()22 2 22 42 4 ++=+ −+ ++−−+ +∫ ∫ bdx ax bx c22 17.12.9.xd x ax bx cbx c ac b ax bx cb a () ( ) ()22 2 22 44 ++=−+ −+ +−ccbdx ax bx c −+ + ∫∫ 22TABLES OF SPECIAL INDEFINITE INTEGRALS 84 17.12.10.xd x ax bx cba c x b c aa cb a x b2 222 222 4 ()() () ( ++=−+ −+ xxcc ac bdx ax bx c ++−+ + ∫∫ )2 422 17.12.11.xd x ax bx cx nm a a x b xcm nm n() ( ) ()21 2121 ++=−−− + +− −+ +− −− ++ −−− ∫ ∫() () () ()mc nm axd x ax bx c nm bm n1 212 2 (() () 211 2 nm axd x ax bx cm n −− ++− ∫ 17.12.12.xd x ax bx c axd x ax bx ccn nn n21 223 211−− −++=++−() () a axd x ax bx cb axd x ax bx cn nn n ∫∫−− ++−++23 222 2() ()∫ ∫ ∫ 17.12.13.dx x a xb x c c a xb x cb cdx ax bx () () (22 2 21 2 2 ++=++−++ ∫ c c cdx xa x b x c )( )221+++ ∫ ∫ 17.12.14.dx x ax bx c cx ax bx ca cdx ax b22 2 2 213 () () ( ++=−++−+ ∫ xxcb cdx xa x b x c +−++ ∫ ∫ )( )22 22 17.12.15.dx x a xb x c m c xa xb x cmn m n() ( ) ()( 21 2 11 1 ++=−−+ +−−−mmn a mcdx xa x b x c mn bmn+− − ++ −+−∫∫ −23 1 222) () () () (() () mcdx xa xb x cmn − ++−∫ 112 (13) Integrals Involving ax bx c2/H11545/H11545 In the following results if ba c a x b x c a x b a2242=+ + = + ,( / ) and the results 17.1 can be used. If b = 0 use the results 17.9. If a = 0 or c = 0 use the results 17.2 and 17.5. 17.13.1.dx ax bx caaa x b x c a x b a22 1122 12 ++=++ + + −−−ln ( ) sinaax b ba c aax b ac b+ −⎛ ⎝⎜⎞ ⎠⎟+ −⎛ ⎝⎜⎞ ⎠− 21 2412 4or sinh⎟ ⎟⎧ ⎨⎪⎪ ⎩⎪ ⎪∫ 17.13.2.xd x ax bx cax bx c ab adx ax bx c22 2 2 ++=++− ++∫∫ 17.13.3.xd x ax bx cax b aax bx cba c ad2 2222 223 434 8 ++=−++ +−∫x x ax bx c2++∫ 17.13.4.dx xa x b x ccca x b x c b x c x 2212 2 1 ++=−++ ++ ⎛ ⎝⎜⎞ ⎠⎟ −∫ln c cbx c xb a c cbx csin ||sinh−− + −⎛ ⎝⎜⎞ ⎠⎟−+1 21 2 412or || |xa c b42−⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨⎪⎪ ⎩⎪ ⎪ 17.13.5.dx xa x b x cax bx c cxb cdx xa x b x c222 2 2 ++=−++− ++∫∫ 17.13.6. ax bx c dxax b ax bx c aac b adx ax222 22 44 8++ =++ ++−∫() +++∫bx cTABLES OF SPECIAL INDEFINITE INTEGRALS 85 17.13.7. xa x b x c d xax bx c aba x b aax223 2 22 32 8++ =++−++() ( )/ bbx c ba cb adx ax bx c+ −− ++∫ ∫()4 162 22 17.13.8. xa x b x c d xax b aax bx cba22 223 2265 2454++ =−++ +−()/ c c aax bx c dx1622∫∫++ 17.13.9.ax bx c xdx ax bx cbd x ax bx ccdx xa x2 2 22 2++=+ + + +++ ∫∫+++∫bx c 17.13.10.ax bx c xdxax bx c xadx ax bx cbd x xa2 22 2 2++=−+++ +++ ∫xxb x c2++∫ ∫ 17.13.11.dx ax bx cax b ac b ax bx c ()() ()/ 23 22222 4 ++=+ −+ +∫ 17.13.12.xd x ax bx cbx c ba c a x b x c ()() ()/ 23 22222 4 ++=+ −+ +∫ 17.13.13.xd x ax bx cba c x b c aa cb a x2 23 22 224 2 4 ()() ()/++=−+ −2221 +++ ++∫ ∫bx c adx ax bx c 17.13.14.dx xa x b x c ca x b x c cdx xa x b x cb ()/ 23 22211 ++= +++ ++−2 223 2 cdx ax bx c()/++∫ ∫ ∫ 17.13.15.dx xa x b x cax bx c c x ax bx cb 22 3 22 2222 ()/++=−++ +++ ∫− − ++ − ++∫ ∫2 2 3 2223 2 22ac cdx ax bx c b cdx xa x b x c()/ 17.13.16. ()() ( )// ax bx c dxax b ax bx c ann 21 221 22 4++ =++ +++ ∫(()() ( ) ()()/ nna c b anax bx cn +++− +++− 121 4 812 21 2ddx ∫ 17.13.17. xa x b x c d xax bx c annn ()() ()// 21 223 2 23++ =++ +−++ ∫b b aax bx c dxn 221 2()/+++∫ 17.13.18.dx ax bx cax b na c b a xn()() () ( ) (/ 21 2 222 21 4 ++=+ −−+ 221 2 2281 21 4++ +− −− +− ∫ bx c an na c bdx axn) () () ( ) (/ bbx cn+− ∫ )/12 17.13.19.dx x ax bx c n c ax bx cnn() ( ) ()// 21 2 21 21 21 ++=−+ ++− ∫ + +++−++−+ ∫1 221 2 21 cdx xa x b x cb cdx ax bx cnn() ()// 2 2 ∫TABLES OF SPECIAL INDEFINITE INTEGRALS 86 (14) Integrals Involving x3/H11545a3 Note that for formulas involving x3 – a3 replace a with –a. 17.14.1. dx xa axa xa x a a33 22 2221 61 3 +=+ −+⎛ ⎝⎜⎞ ⎠⎟+−ln()tan1 12 3xa a−∫ 17.14.2.xd x xa axa x a xa a3322 21 1 61 3 +=−+ +⎛ ⎝⎜⎞ ⎠⎟+−ln()tan2 2 3xa a−∫ 17.14.3.xd x xaxa2 3333 1 3 +=+ ∫ln ( ) 17.14.4. dx xx a ax xa ()ln33 33 331 3 +=+⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.14.5. dx xx a a x axa x a xa23 3 3 422 211 6 ()ln() +=− −−+ +⎛ ⎝⎜⎞ ⎠⎟ ⎟−−∫− 1 32 341 axa atan 17.14.6. dx xax ax a axa xa x a () ()ln() 33 2 3 33 52 2231 9 +=+++ −+⎛ ⎛ ⎝⎜⎞ ⎠⎟+−−∫2 332 351 axa atan 17.14.7. xd x xax ax a axa x a x () ()ln(33 22 33 3 422 31 18 +=++−+ +a a axa a )tan241 1 332 3⎛ ⎝⎜⎞ ⎠⎟+−−∫ 17.14.8. xd x xa xa2 33 2 331 3 ()()+=−+ ∫ 17.14.9. dx xx a a x a ax xa () ()ln33 2 3 33 63 331 31 3 +=+++⎛ ⎝⎜⎞ ⎠⎟ ∫ ∫ 17.14.10.dx xx a a xx ax a axd x x23 3 2 62 63 3 6 31 34 3 () ()+=− −+−+ ∫a a3(See 17.14.2.) ∫ 17.14.11.xd x xax maxd x xamm m 332 33 332 +=−−+−− ∫∫ 17.14.12.dx xx a an x adx xxann n() ( ) ()33 3 1 3 3 331 11 +=− −−+−− ∫∫ ∫ (15) Integrals Involving x4/H11550a4 17.15.1.dx xa axa x a xa x a44 322 221 422 21 2 +=++ −+⎛ ⎝⎜⎞ ⎠⎟− ∫lna ax ax a 311 21212tan tan−−−⎛ ⎝⎜⎞ ⎠⎟−+⎛ ⎝⎜⎞ ⎠⎟⎡ ⎣⎢ ⎢⎤ ⎦⎥ ⎥ ⎥ 17.15.2.xd x xa ax a44 212 21 2 +=−∫tan 17.15.3.xd x xa axa x a xa x a2 4422 221 422 21 +=−+ ++⎛ ⎝⎜⎞ ⎠⎟− ∫ln222121211 ax ax atan tan−−−⎛ ⎝⎜⎞ ⎠⎟−+⎛ ⎝⎜⎞ ⎠⎟⎡ ⎣⎢ ⎢⎤ ⎦⎥ ⎥ ⎥TABLES OF SPECIAL INDEFINITE INTEGRALS 87 17.15.4. xd x xaxa3 4444 1 4 +=+ ∫ln ( ) 17.15.5. dx xx a ax xa ()ln44 44 441 4 +=+⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.15.6.dx xx a a x axa x a xa x a24 4 4 522 2211 422 2 ()ln+=− −−+ ++⎛ ⎛ ⎝⎜⎞ ⎠⎟ +−⎛ ⎝⎜⎞ ⎠⎟−+⎛ ⎝∫ −− 1 221212 511 ax ax atan tan⎜ ⎜⎞ ⎠⎟⎡ ⎣⎢ ⎢⎤ ⎦⎥ ⎥ 17.15.7. dx xx a a x ax a34 4 4 2 612 21 21 2 ()tan+=− −−∫ 17.15.8. dx xa axa xa ax a44 3 31 1 41 2 −=− +⎛ ⎝⎜⎞ ⎠⎟− ∫−ln tan 17.15.9. xd x xa axa xa44 222 221 4 −=− +⎛ ⎝⎜⎞ ⎠⎟ ∫ln 17.15.10.xd x xa axa xa ax a2 441 1 41 2 −=− +⎛ ⎝⎜⎞ ⎠⎟+ ∫−ln tan 17.15.11.xd x xaxa3 4444 1 4 −=− ∫ln ( ) 17.15.12.dx xx a axa x ()ln44 444 41 4 −=−⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.15.13.dx xx a a x axa xa a34 4 4 5 511 41 2 ()ln tan−=+− +⎛ ⎝⎜⎞ ⎠⎟+− −∫1x a 17.15.14.dx xx a a x axa xa34 4 4 2 622 221 21 4 ()ln−=+− +⎛ ⎝⎜⎞ ⎠⎟ ∫ (16) Integrals Involving xn/H11550an 17.16.1.dx xx a n ax xann nn nn()ln+=+⎛ ⎝⎜⎞ ⎠⎟ ∫1 17.16.2.xd x xa nxan nnnn− +=+ ∫11ln ( ) 17.16.3.xd x xaxd x xaaxd x xm nn rmn nn rnmn n() () (+=+−+− −− ∫ 1a anr) ∫ ∫ 17.16.4.dx xx a adx xx a adx xxmn n r n mn n r n m n() () (+=+−−− ∫11 1 nnn ra+ ∫ ∫ ) 17.16.5.dx xx a naxa a xa ann nnn n nn n+=+− ++⎛ ⎝⎜⎞ ⎠⎟ ∫1ln 17.16.6.dx xx a n axa xnn nnn n()ln−=−⎛ ⎝⎜⎞ ⎠⎟ ∫1TABLES OF SPECIAL INDEFINITE INTEGRALS 88 17.16.7. xd x xa nxan nnnn− −=− ∫11ln ( ) 17.16.8.xd x xaaxd x xaxd x xam nn rnmn nn rmn nn() () ()−=−+−−− r r− ∫ ∫ ∫ 1 17.16.9. dx xx a adx xx a adx xx am n nr n mn n nr n m n() () (−=−−−−11 nnr)− ∫ ∫ ∫ 17.16.10.dx xx a naa x nn nn n−=−∫21cos 17.16.11.xd x xa m akp mxap mm m p− −− +=−+∫1 22 21 12 1 2sin()tanπ ccos[( ) / ] sin[( ) / ]21 2 21 21km ak mkm− −⎛ ⎝⎜⎞ ⎠⎟ =∑π π − −−+− − =∑1 221 222 2 12 makp mxa xk mp km cos()cos( πln1 1 22 )π ma+⎛ ⎝⎜⎞ ⎠⎟ where 0 < p /H11017 2m. 17.16.12.xd x xa m akp mxa xk mp mm m p− −−=−1 22 22 1 22 cos ln cosππ+ +⎛ ⎝⎜⎞ ⎠⎟ −−=− −−∑ ∫a makp mxakm mp2 11 21 1sin tanco π ss( / ) sin( / ) {lnkm ak m makm mpπ π⎛ ⎝⎜⎞ ⎠⎟ +=− −∑ 11 21 2(() ( ) l n () }xa xap−+ − + 1 where 0 < p /H11017 2m. 17.16.13.xd x xa m ap mmp mp− ++− −++=− +1 21 211 2121 212 () ()sinkkp mxa k m ak mππ π 21221 221 +++−tancos[ /( )] sin[ /( + +⎛ ⎝⎜⎞ ⎠⎟ −− += − −+∑ ∫1 1 2121 1 21)] () ()coskm p mpmakp p mxa xk ma kmππ 2122 21 122 1 ++++⎛ ⎝⎜⎞ ⎠⎟ +−=∑ ln cos ())l n ( ) ()p mpxa ma− −++ +1 2121 where 0 < p /H11017 2m +1 . 17.16.14.xd x xa m akp mp mm m p− ++ − +−=− ++1 21 21 2 12 212 2 ()sinπ 1 1221 2211tancos[ /( )] sin[ /( )]− −+ +⎛ ⎝xa k m ak mπ π⎜ ⎜⎞ ⎠⎟ +++−= −+∑ ∫ km mpmakp mx1 212 1 212 212()cos lnπaaxk ma xa makm cos ln ( ) ()2 21 212 1 2π ++⎛ ⎝⎜⎞ ⎠⎟ +− +=∑ mmp−+1 where 0 < p /H11017 2m +1 .TABLES OF SPECIAL INDEFINITE INTEGRALS 89 (17) Integrals Involving sinax 17.17.1. sincosax dxax a=− ∫ 17.17.2. xa x d xax axa x asinsin cos=− ∫ 2 17.17.3. xa x d xx aaxax aax2 23222sin sin cos =+ −⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.17.4. xa x d xx aaaxx ax a32 24 3336 6sin sin =−⎛ ⎝⎜⎞ ⎠⎟ +−⎛ ⎝⎜⎞ ⎠ ⎠⎟ ∫cosax 17.17.5. sin ( ) !() !ax xdx axax ax=−⋅+⋅−⋅⋅⋅ ∫35 33 55 17.17.6. sin sin cos(ax xdxax xaax xdx2=− + ∫ ∫See 17.18.5.) 17.17.7.dx axax axax sinln(csc cot ) ln tan=− = ∫11 2 αα 17.17.8. xd x ax aaxax axn sin() () (=+ ++ +−1 187 180022 235 21 /midhorizellipsis− − ++⎧ ⎨⎪ ⎩⎪⎫ ⎬⎪ ⎭⎪+ ∫1 2121)() () !Ba x nnn /midhorizellipsis 17.17.9. sinsin2 22 4ax dxxa x a=− ∫ 17.17.10. xa x d xxx a x aax asinsin cos22 2 42 42 8=− − ∫ 17.17.11. sincos cos33 3ax dxax aax a ∫=− + 17.17.12. sinsin sin4 3 82 44 32 ∫=− + ax dxxa x aax a 17.17.13.dx ax aaxsincot21∫=− 17.17.14.dx axax aa x aax sincos sinln tan3221 22=− + ∫ 17.17.15. sin sinsin ( ) ()sin ( ) (px qx dxpq x pqpq x pq=− −−+ + 22 ) )(, ∫=±If see 17.17.9. ) pq 17.17.16.dx ax aax 11 42 −=+⎛ ⎝⎜⎞ ⎠⎟ ∫ sintanπ 17.17.17.xdx axx aax aax 14 22 422−=+⎛ ⎝⎜⎞ ⎠⎟+−sintan ln sinππ ⎛ ⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.17.18.dx axax 11 42 +=− −⎛ ⎝⎜⎞ ⎠⎟ ∫ sintanαπ 17.17.19.xd x axx aax aax 14 22 42 +=− −⎛ ⎝⎜⎞ ⎠⎟++sintan ln sinππ 2 2⎛ ⎝⎜⎞ ⎠⎟ ∫TABLES OF SPECIAL INDEFINITE INTEGRALS 90 17.17.20.dx ax aax a (s i n )tan tan11 24 21 6423 −=+⎛ ⎝⎜⎞ ⎠⎟++ ∫ππ aax 2⎛ ⎝⎜⎞ ⎠⎟ 17.17.21.dx ax aax a (s i n )tan tan11 24 21 6423 +=− −⎛ ⎝⎜⎞ ⎠⎟− ∫ππ− −⎛ ⎝⎜⎞ ⎠⎟ax 2 17.17.22.dx pq a xap qpa x q pq aq+=−+ − ∫− sintantan 2 12211 2 22 2− −+− − ++ −⎛ ⎝⎜⎞ ⎠ ppa x q q p pa x q q p21 222 1 222lntan tan⎟ ⎟⎧ ⎨⎪ ⎪ ⎩⎪ ⎪ (If p = ± q, see 17.17.16 and 17.17.18.) 17.17.23.dx pq a xqa x ap q p q a xp p (s i n )cos () ( s i n ) +=−++−22 2 2q qdx pq a x2 +∫ ∫ sin (If p = ± q, see 17.17.20 and 17.17.21.) 17.17.24.dx pq a x ap p qpq a x p22 2221221 += ++−∫ sintantan 17.17.25.dx pq a xap p qpq a x p ap q22 2221221 1 2−=−−− sintantan 22222 22−−+ −−⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨⎪ ⎪ ⎩⎪ pqp a x p qp a x plntan tan⎪ ⎪∫ 17.17.26. xa x d xxa x amx ax amm axmmm sincos sin ( )=− + −−∫−1 221m max dx−∫2sin 17.17.27.sin sin ()cos ax xdxax nxa nax xdxnn n∫∫=−−+−−−1111((See 17.18.30.) 17.17.28. sinsin cossinnn nax dxax ax ann nax dx =− +−− −∫∫1 2 1 17.17.29.dx axax an a xn ndx nn nsincos () s i n s i n=− −+− −−−12 11 2 2ax ∫ ∫ 17.17.30.xd x axxa x an a x a n nnnsincos () s i n () (=− −−−−−11 1122 22 122)s in s innnaxn nxd x ax−− +− − ∫∫ (18) Integrals Involving cosax 17.18.1. cossinax dxax a= ∫ 17.18.2. xa x d xax axa x acoscos sin=+ ∫ 2 17.18.3. xa x d xx aaxx a aax2 22 322cos cos sin =+ −⎛ ⎝⎜⎞ ⎠⎟ ∫TABLES OF SPECIAL INDEFINITE INTEGRALS 91 17.18.4. xa x d xx aaaxx ax a32 243 336 6cos cos =−⎛ ⎝⎜⎞ ⎠⎟ +−⎛ ⎝⎜⎞ ⎠ ⎠⎟ ∫sinax 17.18.5. cosln() !() !() !ax xdx xax ax ax=−⋅+⋅−⋅+246 22 44 66/midhorizellipsis /midhorizellipsis ∫ 17.18.6. cos cos sin(ax xdxax xaax xdx2=− − ∫ ∫See 17.17.5.) 17.18.7. dx ax aax axaax cosln (sec tan ) ln tan=+ =+⎛ ⎝⎜⎞ 11 42π ⎠ ⎠⎟ ∫ 17.18.8.xd x ax aax ax ax Ea xn cos() () () (=+ + + +1 285 144224 6 /midhorizellipsis) ) () ( ) !22 22 2n nn+ ++⎧⎨⎩⎫⎬⎭∫/midhorizellipsis 17.18.9. cossin2 22 4ax dxxa x a=+ ∫ 17.18.10. xa x d xxx a x aax acossin cos22 2 42 42 8=+ + ∫ 17.18.11. cossin sin33 3ax dxax aax a=− ∫ 17.18.12. cossin sin4 3 82 44 32ax dxxa x aax a=+ + ∫ 17.18.13.dx axax a costan 2= ∫ 17.18.14.dx axax aa x aax cossin cosln tan3221 24 2=+ +⎛ ⎝⎜⎞ ⎠⎟π∫ ∫ 17.18.15. cos cossin( ) ()sin( ) (ax px dxap x apap x a ∫=− −++ + 22 p pap)(,If see 17.18.9.)=± 17.18.16.dx ax aax 11 2 −=− ∫ coscot 17.18.17.xd x axx aax aax 122 22 −=− + ∫ coscot ln sin 17.18.18.dx ax aax 11 2 += ∫ costan 17.18.19.xd x axx aax aax 122 22 +=+ ∫ costan ln cos 17.18.20.dx ax aax aax (c o s )cot cot11 221 6223 −=− − ∫ 17.18.21.dx ax aax aax (c o s )tan tan11 221 6223 +=+ ∫ 17.18.22.dx pq a xap qpq pq a x a+=−−+− costan ( ) / ( ) tan21 2 1221 qqpax q p q p ax q p q221 2 1 2−++ − −+lntan ( ) / ( ) tan ( ) / ( − −⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨⎪ ⎪ ⎩⎪ ⎪∫ p)(,If see 17.18.16 and 17.18.18.)pq=±TABLES OF SPECIAL INDEFINITE INTEGRALS 92 17.18.23.dx pq a xqa x aq p p q a xp q (c o s )sin () ( c o s ) +=−+−−22 2 2p pdx pq a x2 +∫ ∫ cos(If see 17.18.19 and 17.18.20.)pq=± 17.18.24.dx pq a x ap p qpa x pq222221 221 += ++−∫ costantan 17.18.25.dx pq a xap p qpa x pq ap q222221 221 1 2−=−−− costantan 22222 22−−− +−⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨⎪⎪ ⎩⎪ ppa x qp pa x qplntan tan ⎪ ⎪∫ 17.18.26. xa x d xxa x amx aaxmm axmmm m∫=+ −−− cossincos()1 221− −∫2cosax dx 17.18.27.cos cos ()sin(ax xdxax nxa nax xdxnn n=−−−−−−1111Seee 17.17.27.) ∫ ∫ 17.18.28. cossin coscosnn nax dxax ax ann nax dx =+−− −∫∫1 2 1 17.18.29.dx axax an a xn bdx nn ncossin () c o s c o s=−+− −−−12 112aax ∫ ∫ 17.18.30.xd x axxa x an a x a n nnncossin () c o s () (=−−−−−11 1212))c o s c o snnaxn nxd x ax−− +− − ∫∫ 222 1 (19) Integrals Involving sinax and cos ax 17.19.1. sin cossinax ax dxax a= ∫2 2 17.19.2. sin coscos( ) ()cos( ) (px qx dxpq x pqpq x p=−− −−+ + 22 q q) 17.19.3. sin cossin ()(,nn ax ax dxax nan ∫=+=−+1 11 If see 1 17.21.1.) 17.19.4. cos sincos (),nn ax ax dxax nan ∫=−+=−+1 11 (If see 17.20.1.) 17.19.5. sin cossin22 84 32ax ax dxxa x a=− ∫ 17.19.6.dx ax ax aaxsin cosln tan= ∫1 17.19.7.dx ax ax aax aa x sin cosln tansin21 421=+⎛ ⎝⎜⎞ ⎠⎟− ∫π 17.19.8.dx ax ax aax aa x sin cosln tancos21 21=+ ∫ 17.19.9.dx ax axax a sin coscot 2222=− ∫TABLES OF SPECIAL INDEFINITE INTEGRALS 93 17.19.10.sin cossinln tan21 24ax axdxax aaax=− + +⎛ ⎝⎜⎞ ⎠⎟ ∫π 17.19.11.cos sincosln tan21 2ax axdxax aaax=+ ∫ 17.19.12.dx ax ax a ax aax cos ( sin ) ( sin )ln tan11 211 22 ±=±+ ∫∓ + +⎛ ⎝⎜⎞ ⎠⎟π 4 17.19.13.dx ax ax a ax aax sin ( cos ) ( cos )ln tan11 211 22 ±=±±+ ∫ 17.19.14.dx ax ax aax sin cosln tan±=±⎛ ⎝⎜⎞ ⎠⎟ ∫1 2 28π 17.19.15.sin sin cosln (sin cos )ax dx ax axx aax ax±=± ∫ 21 2∓ 17.19.16.cos sin cosln (sin cos )ax dx ax axx aax ax±=± + ± ∫ 21 2 17.19.17.sin cosln ( cos )ax dx p q ax aqpq a x+=− + ∫1 17.19.18.cos sinln ( sin )ax dx p q ax aqpq a x+=+ ∫1 17.19.19.sin (c o s ) ( ) (c o s )ax dx p q ax aq n p q axnn+=−+− ∫1 11 17.19.20.cos (s i n ) ( ) (s i n )ax dx p q ax aq n p q axnn+=− −+− ∫1 11 17.19.21.dx pa x q a x ap qax q p sin cosln tantan ( / ) += ++⎛−1 2 221 ⎝ ⎝⎜⎞ ⎠⎟ ∫ 17.19.22.dx pa x q a x rar p qpr q sin costan() t a n ++=−−+−− 2 22 21 ((/ ) lnax rp q ap q rpp q r2 122 2 222222−−⎛ ⎝⎜⎞ ⎠⎟ +−−+ − ++− ++ − + −⎛ ⎝() t a n ( / ) () t a n ( / )rq a x pp q rr q a x2 2222⎜ ⎜⎞ ⎠⎟⎧ ⎨⎪ ⎪ ⎩⎪ ⎪∫ (If r = q see 17.19.23. If r2 = p2 + q2 see 17.19.24.) 17.19.23.dx p ax q ax apqpax sin ( cos )ln tan++=+⎛ ⎝⎜⎞ ⎠⎟ ∫ 11 2 17.19.24.dx pa x q a x pq a pqax sin costantan +± +=− ++− 22 221 4π∓1 1 2(/)qp ⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.19.25.dx pa x q a x apqpa x q22 2 21 1 sin costantan +=⎛ ⎝⎜⎞ ⎠⎟−∫ 17.19.26.dx pa x q a x apqpa x q pa x q22 2 21 2 sin coslntan tan −=− +⎛ ⎛ ⎝⎜⎞ ⎠⎟ ∫TABLES OF SPECIAL INDEFINITE INTEGRALS 94 17.19.27. sin cossin cos ()mnmn ax ax dxax ax am nm m=−++−−+111 + + ++− +−∫nax ax dx ax ax am nmn mnsin cos sin cos ()2 11n n mnax ax dxmn − +⎧ ⎨⎪ ⎩⎪−∫∫12sin cos 17.19.28.sin cossin () c o s m nm n ax axdxax an a xm n =−−− −− −1 111 1 1 12 2 1 1sin cos sin () c o sm n m nax axdx ax an a x− − + −∫ −− −−+ − − −− −∫mn nax axdx ax am nm n m2 12 1sin cos sin () c oossin cosnm naxm mnax axdx−− +− −⎧ ⎨⎪ ⎪⎪ ⎩⎪ ⎪⎪∫∫ 121 17.19.29.cos sincos () s i n m nm n ax axdxax an a xm n =− −−−− −1 111 − − − −− − + −∫1 12 2 1 1cos sin cos () s i nm n m nax axdx ax an aaxmn nax axdx ax am nm n m−−+ − −− −∫2 12 1cos sin cos () ssincos sinnm naxm mnax axdx−− +− −⎧ ⎨⎪ ⎪⎪ ⎩⎪ ⎪ ⎪ ∫∫ 121 17.19.30.dx ax axa n ax axmn mnmn sin cos() s i n c o s=−++− −−1 1112 2 1 1 12 1ndx ax ax am a xmn mn− − −− −−∫sin cos () s i n c o s1122 1 axmn mdx ax axmn ++− −⎧ ⎨⎪ ⎩⎪ ⎪−∫∫ sin cos (20) Integrals Involving tanax 17.20.1. tan ln cos ln secax dxaaxaax =− = ∫11 17.20.2. tantan2ax dxax ax =− ∫ 17.20.3. tantanln cos32 21ax dxax aaax =+ ∫ 17.20.4. tan sectan ()nn ax ax dxax na21 1=++ ∫ 17.20.5.sec tanln tan21 ax axdxaax = ∫ 17.20.6.dx ax aaxtanln sin= ∫1 17.20.7. xa x d xaax ax axn tan() () () (=+ + + +1 31 52 1052 235 7 2 /midhorizellipsis221 2122 1 n nnBa x n− ++⎧⎨⎩⎫⎬⎭+ ∫)() () !/midhorizellipsis 17.20.8.t a n () () () ( ax xdx axax ax Bnn n=+ + + +−35 22 92 7522 1/midhorizellipsisaax nnn) () ( ) !21 21 2− −+ ∫/midhorizellipsis 17.20.9. xa x d xxa x a aaxxtantanln cos2 221 2=+ − ∫TABLES OF SPECIAL INDEFINITE INTEGRALS 95 17.20.10.dx pq a xpx pqq ap qqa x p a+=++++tan ()ln ( sin cos22 22 x x) ∫ 17.20.11. tantan ()tannn nax dxax naax dx =−−− −∫ ∫1 2 1 (21) Integrals Involving cotax 17.21.1. cot ln sinax dxaax ∫=1 17.21.2. cotcot2ax dxax ax =− − ∫ 17.21.3. cotcotln sin32 21∫=− − ax dxax aaax 17.21.4. cot csccot ()nn ax ax dxax na ∫=−++ 21 1 17.21.5. csc cotln cot21 ax axdxaax =− ∫ 17.21.6. dx ax aaxcotln cos =− ∫1 17.21.7. xa x d xaaxax ax Ba xn ncot() () ()∫= −−− −1 9 2252 235 2 /midhorizellipsis221 21n n+ +−⎧ ⎨⎪ ⎩⎪⎫ ⎬⎪ ⎭⎪ () !/midhorizellipsis 17.21.8.cot ( ) () (ax xdxaxax ax Ba xn nn =− − − − −−1 3 13523 22 1 /midhorizellipsis221 2nn−− ∫)( )!/midhorizellipsis 17.21.9. xa x d xxa x aaaxxcotcotln sin2 221 2 ∫=− + − 17.21.10.dx pq a xpx pqq ap qqa x q a+=+−++cot ( )ln ( sin cos22 22x x) ∫ 17.21.11. cotcot ()cotnn nax dxax naax dx ∫∫=−−−− −1 2 1 (22) Integrals Involving secax 17.22.1. sec ln (sec tan ) ln tanax dxaax axaax=+ = +⎛ ⎝⎜⎞ 11 24π ⎠ ⎠⎟ ∫ 17.22.2. sectan2∫= ax dxax a 17.22.3. secsec tanln (sec tan )3 21 2 ∫=+ + ax dxax ax aaax axTABLES OF SPECIAL INDEFINITE INTEGRALS 96 17.22.4. sec tansecnn ax ax dxax na ∫= 17.22.5. dx axax a secsin= ∫ 17.22.6. xa x d xaax ax ax Ea xnsec() () () (=+ + + +1 285 144224 6 /midhorizellipsis) ) () ( ) !22 22 2n nn+ ++⎧ ⎨⎪ ⎩⎪⎫ ⎬⎪ ⎭⎪∫/midhorizellipsis 17.22.7.secln() () () ax xdx xax ax ax=+ + + + +24 6 45 9661 4320/midhorizellipsisEEa x nnnn() () !2 22+ ∫/midhorizellipsis 17.22.8. xa x d xx aaxaax sec tan ln cos2 21∫=+ 17.22.9.dx qp a xx qp qdx pq a x +=−+ ∫∫sec cos 17.22.10. secsec tan ()secnn nax dxax ax ann n ∫∫=−+− −− −2 2 12 1aax dx (23) Integrals Involving cscax 17.23.1. csc ln (csc cot ) ln tanax dxaax axaax=− = ∫11 2 17.23.2. csccot2∫=− ax dxax a 17.23.3. csccsc cotln tan3 21 22 ∫=− + ax dxax ax aaax 17.23.4. csc cotcscnn ax ax dxax na ∫=− 17.23.5. dx axax a csccos=− ∫ 17.23.6. xa x d xaaxax axn csc() () (=+ ++ + ∫−1 187 180022 235 2 /midhorizellipsis112 11 21− ++⎧ ⎨⎪ ⎩⎪⎫ ⎬⎪ ⎭⎪+)() () !Ba x nnn /midhorizellipsis 17.23.7. csc ( ) () ax xdxaxax ax Bn n=− + + + +−−1 67 108022 13 21 /midhorizellipsis(() () ( ) !ax nnn21 21 2− −+ ∫/midhorizellipsis 17.23.8. xa x d xxa x aaax csccotln sin2 21∫=− + 17.23.9. dx qp a xx qp qdx pq a x +=−+ ∫∫csc sin(See 17.17.22.) 17.23.10. csccsc cot ()cscnn nax dxax ax ann n=−−+− −− −∫2 2 12 1∫ ∫ax dxTABLES OF SPECIAL INDEFINITE INTEGRALS 97 (24) Integrals Involving Inverse Trigonometric Functions 17.24.1. sin sin−−=+ − ∫11 2 2 x adx xx aax 17.24.2. xx adxxa x axa xsin sin−−=−⎛ ⎝⎜⎞ ⎠⎟ +−∫122 122 24 4 17.24.3. xx adxxx axa a x213 122 2 2 32 9 ∫−−=++−sin sin() 17.24.4.s i n (/) (/) (/)− =+ +13 5 23313 245xa xdxx axa xa iii iii iii iiii/midhorizellipsis5135 246777 ++ ∫(/)xa 17.24.5.sin ( / ) sin ( / )ln−− =− −+−⎛ ⎝∫1 212 21 xa xdxxa xaaa x x ⎜ ⎜⎞ ⎠⎟ 17.24.6. sin sin−−⎛ ⎝⎜⎞ ⎠⎟ =⎛ ⎝⎜⎞ ⎠⎟−+ − ∫12 12 2222x adx xx axa x ssin−1x a 17.24.7. cos cos−−=− − ∫11 2 2 x adx xx aax 17.24.8. xx adxxa x axa xcos cos−−∫=−⎛ ⎝⎜⎞ ⎠⎟ −−122 122 24 4 17.24.9. xx adxxx axa a x213 122 2 2 32 9 ∫−−=−+−cos cos() 17.24.10.cos ( / )lnsin ( / )(−− =− ∫ ∫11 2xa xdx xxa xdxπSee 17.2 24.4.) 17.24.11.cos ( / ) cos ( / )ln−− =− ++−⎛ ⎝∫1 212 21 xa xdxxa xaaa x x ⎜ ⎜⎞ ⎠⎟ 17.24.12. cos cos−−⎛ ⎝⎜⎞ ⎠⎟ =⎛ ⎝⎜⎞ ⎠⎟−− − ∫12 12 2222x adx xx axa x ccos−1x a 17.24.13. tan tan ln ( )−−=− + ∫11 2 2 2x adx xx aaxa 17.24.14. xx adx x ax aaxtan ( ) tan−−=+ − ∫12 2 1 1 22 17.24.15. xx adxxx aax axa213 123 22 36 6tan tan ln ( )−−=− + + ∫ 17.24.16.t a n (/) (/) (/) (/)− = −+−13 25 27 2357xa xdxx axa xa xa+ + ∫/midhorizellipsis 17.24.17.tan ( / )tan ln− −=− −+⎛ ⎝⎜⎞ ⎠1 2122 211 2xa xdxxx aaxa x⎟ ⎟ ∫TABLES OF SPECIAL INDEFINITE INTEGRALS 98 17.24.18. cot cot )−−∫=+ +11 2 2 2x adx xx aaxaln ( 17.24.19. xx adx x ax aaxcot ( ) cot−−∫=+ +12 2 1 1 22 17.24.20. xx adxxx aax axa213 123 22 36 6cot cot ( )−−∫=+ − + ln 17.24.21.c o t (/) t a n (/)−− ∫∫=−11 2xa xdx xxa xdxπln (See 17.24.16.) 17.24.22.cot ( / ) cot ( / )−− ∫=++⎛ ⎝⎜1 212 2 21 2xa xdxxa xaxa xln⎞ ⎞ ⎠⎟ 17.24.23. secsec ( ) sec −−− =−+ − < < 112 2 102 x adxxx aax xax alnπ x xx aax x ax asec sec−−++ − < <⎧ ⎨⎪ ⎩⎪∫ 12 2 1 2ln ( )ππ 17.24.24. xx adxxx aax a x a xsecsec sec −−− =−−<< 12 122 1 22202π 222 2122 1sec sec−−+−<<⎧ ⎨⎪⎪ ⎩⎪ ⎪∫x aax a x aππ 17.24.25. xx adxxx aax x a axx 213 122 3 2 36 6secsec −− =−−−+ − ln( a ax a xx aax x a ax21 3 122 302 36 6)s e c sec<< +−+− −π ln(++− < <⎧ ⎨⎪⎪ ⎩⎪ ⎪−∫ xax a22 1 2)s e cππ 17.24.26.sec ( / ) ( / ) ( /− =+ + + ∫13 22 3 313 xa xdx xa xax ax πlniii ))( / )57 245 5135 24677 ii iii iiii+ +⋅⋅⋅ax 17.24.27.sec ( / )sec ( / )sec −− − =−+−< 1 212 2 10 xa xdxxa xxa axx a a xa xxa axx a< −−−<<⎧ ⎨⎪ ⎪⎪ ⎩− −π ππ2 212 2 1 sec ( / )sec ⎪ ⎪ ⎪⎪∫ 17.24.28. csccsc ( ) csc −−− =++ − < < 112 2 102 x adxxx aaxx ax alnπ x xx aaxx ax acsc ( ) csc−−−+ −− < <⎧ ⎨⎪ ⎩⎪∫ 12 2 1 20 lnπ 17.24.29. xx adxxx aax a x a xcsccsc csc −−− =+−<< 12 122 1 22202π 222 20122 1csc csc−−−−−< <⎧ ⎨⎪⎪ ⎩⎪ ⎪∫x aax a x aπ 17.24.30. xx adxxx aax x a axx 213 122 3 2 36 6csccsc ( −− =+−++ ∫ln −−< < −−−− −ax a xx aax x a a21 3 122 302 36 6)c s c csc (π lnxxx ax a+− − < <⎧ ⎨⎪⎪ ⎩⎪ ⎪− 22 1 20 )c s cπTABLES OF SPECIAL INDEFINITE INTEGRALS 99 17.24.31.csc ( ) ( ) ( )− ∫=− + +13 513 4xa xdxa xax ax // 233/ 2 iii iiiiii iiii 55/ 26 7 7+ +⋅⋅⋅⎛ ⎝⎜⎞ ⎠⎟135 47()ax 17.24.32.csc ( )csc ( )csc−− − ∫=−−−<1 212 2 10 xa xdxxa xxa ax // x x a xa xxa axx a< −+−−< <⎧ ⎨⎪⎪ − −π π2 2012 2 1 csc ( )csc/ ⎩ ⎩⎪ ⎪ 17.24.33. xx adxx mx amx axdxmmm sin sin−+ −+ =+−+ −∫11 11 22 11 1 ∫ ∫ 17.24.34. xx adxx mx amx axdxmmm cos cos−+ −+ =+++ −∫11 11 22 11 1∫ ∫ 17.24.35. xx adxx mx aa mx xadxmmm tan tan−+ −+ =+−++ ∫11 11 2211 ∫ ∫ 17.24.36. xx adxx mx aa mx xadxmmm cot cot−+ −+ =++++ ∫11 11 2211 ∫ ∫ 17.24.37. xx adxxx a ma mxd x xa mmm secsec ( / ) −+− =+−+ −∫111 2 11 2 21 11 202 11∫<< +++− +−sec sec ( / )x a xx a ma mxd x xmmπ − −<<⎧ ⎨⎪ ⎪ ⎩⎪ ⎪ ∫− ax a 21 2ππ sec 17.24.38. xx adxxx a ma mxd x xa mmm cscsec ( ) −+− =+++ −∫111 2 11/ 2 21 11 202 11∫<< +−+− +−csc csc ( )x a xx a ma mxd x xmmπ / − −−< <⎧ ⎨⎪ ⎪ ⎩⎪ ⎪ ∫− ax a 21 20πcsc (25) Integrals Involving eax 17.25.1. ed xe aaxax ∫= 17.25.2. xe dxe axaaxax ∫=−⎛ ⎝⎜⎞ ⎠⎟1 17.25.3. xe d xe axx aaaxax 22 222∫=− +⎛ ⎝⎜⎞ ⎠⎟ 17.25.4. xe d xxe an axe d x e axnx ana xna x na x ax nn∫∫=− =− +− −1 1nnn x an annn n() ( ) !−−⋅⋅⋅− ⎛ ⎝⎜⎞ ⎠⎟ =−112 2 if positive e integer 17.25.5.e xdx xax ax axax = + + + +⋅⋅⋅ ∫ln11 2 2 3323 ii i!() !() ! 17.25.6.e xdxe nxa ne xdxax nax nax n ∫∫=− −+−−−()1 111TABLES OF SPECIAL INDEFINITE INTEGRALS 100 17.25.7. dx pq ex pa ppq eaxax +=− + ∫1ln ( ) 17.25.8. dx pq ex p ap p qe appq eax axax () ())+=++−+ ∫ 22 211ln ( 17.25.9.dx pe qeap qp qe ap qeax axax +=⎛ ⎝⎜⎞ ⎠⎟ −−− ∫1 1 21tan lnaax axqp eq p−− +−⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨⎪ ⎪⎪ ⎩⎪ ⎪ ⎪/ / 17.25.10. eb x d xea b xb b x abaxax sin(s i n c o s )=− + ∫ 22 17.25.11. eb x d xea b xb b x abaxax cos(c o s s i n )=+ + ∫ 22 17.25.12. xe bx dxxe a bx b bx abeaxax ax sin(s i n c o s ) { (=− +− ∫ 22aa b bx ab bx ab22 22 22 −− +)s i n c os } () 17.25.13. xe bx dxxe a bx b bx abeaxax ax cos(c o s s i n ) { (=+ +− ∫ 22aa b bx ab bx ab22 22 22 −+ +)c o s s i n } () 17.25.14. ex d xex aae xdxaxax ax lnln∫∫=−1 17.25.15. eb x d xeb x an bab x n bax nax n ∫=+−− sinsin(s i n c o1 22 2 ss)()sin bxnn b an beb x d xax n+− +−∫12 22 22 17.25.16. eb x d xeb x an bab x n bax nax n ∫=++− coscos(c o s s i1 22 2 nn)()cos bxnn b an beb x d xax n+− +−∫12 22 22 (26) Integrals Involving ln x 17.26.1. ln lnxd x x x x=− ∫ 17.26.2. xx d xxx ln ln =−⎛ ⎝⎜⎞ ⎠⎟ ∫2 21 2 17.26.3. xx d xx mxmmmm ln ln If see =+−+⎛ ⎝⎜⎞ ⎠⎟=−+ ∫1 11 111 7 (, . . . . ) 26 4 17.26.4.lnlnx xdx x= ∫1 22 17.26.5.ln lnx xdxx xx21=− − ∫ 17.26.6. ln ln ln2222 xd x x x x x x∫=−+ 17.26.7.ln lnIf seennxd x xx nn =+=−+ ∫1 11 1 72 68 (, . . . ) 17.26.8.dx xxxlnln ln= ∫()TABLES OF SPECIAL INDEFINITE INTEGRALS 101 17.26.9. dx xxxxx lnln ln= + + + +⋅⋅⋅ ∫() l nln !ln !23 22 33ii 17.26.10.xd x xxm xmx mm lnln ln=+ + ++++() ( ) l n() l n !(11 2222 i1 1 3333)l n !x i ∫+⋅⋅⋅ 17.26.11. ln lnnn nxd x x x n xd x =−−∫ ∫ln1 17.26.12. xx d xxx mn mxx d xmnmn mnlnlnln =+−++ −∫∫1 1 11 If m = –1, see 17.26.7. 17.26.13. ln ( ln ( xa d x xxa xax a22 22 122 += + − +−∫)) t an 17.26.14. ln ( ) ln ( ) lnxa d x xxa x axa xa22 222 −= − − ++ −⎛ ⎝⎜⎞ ⎠⎟ ∫ 17.26.15. xx a d xxx a mmx xammm ln )ln ( )(2212 2 2 2 12 1±=± +−+ ±++ 2 2dx ∫ ∫ (27) Integrals Involving sinh ax 17.27.1. sinhcoshax dxax a ∫= 17.27.2. xa x d xxa x aax asinhcosh sinh∫=−2 17.27.3. xa x d xx aaaxx aax22 3222sinh cosh sinh ∫=+⎛ ⎝⎜⎞ ⎠⎟− 17.27.4.sinh ( ) !() !ax xdx axax ax= + + +⋅⋅⋅ ∫35 33 55ii 17.27.5.sinh sinh coshax xdxax xaax xdx2 =− + ∫∫ (See 17.28.4.) 17.27.6.dx ax aax sinhln tanh= ∫1 2 17.27.7.xd x ax aaxax ax sinh() () (= − + −⋅⋅⋅+−∫1 187 18002 235112 1 2122 1)( ) ( ) () !nn nnBa x n− ++⋅⋅⋅⎧⎨⎩⎫⎬⎭+ 17.27.8. sinhsinh cos2 22ax dxax ax ax=− ∫h 17.27.9. xa x d xxa x aax axsinhsinh cosh2 222 42 8 4=− − ∫TABLES OF SPECIAL INDEFINITE INTEGRALS 102 17.27.10.dx axax a sinhcoth 2=− ∫ 17.27.11. sinh sinhsinh ( ) ()sinh ( )ax px dxap x apap x=+ +−−∫2 22( )ap− For a = ± p see 17.27.8. 17.27.12. xa x d xxa x am axa x d xmm msinhcoshcosh =− ∫∫−1 (See 17.28.12.) 17.27.13. sinhsinh coshsinhnn nax dxax ax ann nax d =−−∫− −1 2 1x x ∫ 17.27.14.sinh sinh ()cosh ax xdxax nxa nax xnnn∫=− −+−−−1111ddx ∫(See 17.28.14.) 17.27.15.dx axax an a xn ndx nnsinhcosh () s i n h s ∫=− −−− −−12 11iinhnax−∫ 2 17.27.16.xd x axxa x an a x a nnnsinhcosh () s i n h ( ∫=− −−−−11 112))( )sinh sinh na xn nxd x axnn−−− −−− ∫22 122 (28) Integrals Involving cosh ax 17.28.1. coshsinhax dxax a ∫= 17.28.2. xa x d xxa x aax acoshsinh cosh∫=−2 17.28.3. xa x d xxa x ax aaax2 22 322coshcoshsinh ∫=− + +⎛ ⎝⎜⎞ ⎠⎟ 17.28.4.coshln() !() !() ax xdx xax ax ax=+ + + ∫246 22 44 66iii ! !+⋅⋅⋅ 17.28.5.cosh cosh sinh ax xdxax xaax xdx2 =− + ∫∫ (See 17.27.4.) 17.28.6.dx ax aeax coshtan=−∫21 17.28.7.xd x ax aax ax ax cosh() () () (= − + +⋅⋅⋅+ ∫1 285 144224 6− − ++⋅⋅⋅⎧⎨⎩⎫⎬⎭+1 22 222)( ) () ( ) !n nnEa x nn 17.28.8. coshsinh cosh2 22ax dxxa x a x a=+ ∫ 17.28.9. xa x d xxx a x aax acoshsinh cosh22 2 42 42 8=+ − ∫TABLES OF SPECIAL INDEFINITE INTEGRALS 103 17.28.10.dx axax a coshtanh 2= ∫ 17.28.11. cosh coshsinh( ) ()sinh( )ax px dxap x apap x=− −++∫2 2 2()ap+ 17.28.12. xa x d xxa x am axa x d xmm mcoshsinhsinh =− ∫∫−1 (See 17.27.12.) 17.28.13. coshcosh sinhcoshnn nax dxax ax ann nax d =+−∫− −1 2 1x x ∫ 17.28.14.cosh cosh ()sinh ax xdxax nxa nax xnn n∫=− −+−−−1111ddx ∫ (See 17.27.14.) 17.28.15.dx axax an a xn ndx nncoshsinh ( ) cosh co ∫=−+− −−12 11sshnax− ∫ 2 17.28.16.xd x axxa x an a x n nnncoshsinh () c o s h () ( ∫=−+−−11 11− −+− −−− ∫22 122 2) cosh cosh aa xn nxd x axnn (29) Integrals Involving sinh ax and cosh ax 17.29.1. sinh coshsinhax ax dxax a= ∫2 2 17.29.2. sinh coshcosh ( ) ()cosh ( )px qx dxpq x pqpq x=+ ++−∫2 22( )pq− 17.29.3. sinh coshsinh22 4 32 8ax ax dxax ax=− ∫ 17.29.4.dx ax ax aaxsinh coshln tanh ∫=1 17.29.5.dx ax axax a sinh coshcoth 2222∫=− 17.29.6.sinh coshsinhtan sinh2 1 1 ax axdxax aaax ∫=− 17.29.7.cosh sinhcoshln tanh21 2ax axdxax aaax∫=+TABLES OF SPECIAL INDEFINITE INTEGRALS 104 (30) Integrals Involving tanh ax 17.30.1. tanh ln coshax dxaax = ∫1 17.30.2. tanhtanh2ax dx xax a=− ∫ 17.30.3. tanh ln coshtanh321 2ax dxaaxax a=− ∫ 17.30.4. xa x d xaax ax axtanh() () () (= − + −⋅⋅⋅− 1 31 52 105235 7 112 2 1 2112 2 21)( ) ( ) () !nn n nnBa x n−+− ++⋅⋅⋅⎧ ⎨⎪ ⎩⎪⎫ ⎬ ⎬⎪ ⎭⎪∫ 17.30.5. xa x d xxx a x aaax tanhtanhln cosh22 221=− + ∫ 17.30.6.t a n h () () () ( ax xdx axax axnn = − + −⋅⋅⋅−− 35 12 92 7512 221 21 222 1 n nnBa x nn− −+⋅⋅⋅− ∫)( ) () ( ) ! 17.30.7.dx pq a xpx pqq ap qqa x p+=−−−+ ∫ tanh ()ln ( sinh c22 22 oosh )ax 17.30.8. tanhtanh ()tanhnn nax dxax aaax dx =− −+− −∫ ∫1 2 1 (31) Integrals Involving coth ax 17.31.1. coth ln sinhax dxaax ∫=1 17.31.2. cothcoth2ax dx xax a ∫=− 17.31.3. coth ln sinhcoth321 2ax dxaaxax a ∫=− 17.31.4. xa x d xaaxax axn coth() () ()∫= + − +⋅⋅⋅−−1 9 2251 235 12 2 2122 1n nnBa x n() () !+ ++⋅⋅⋅⎧⎨⎩⎫⎬⎭ 17.31.5. xa x d xxx a x a aax cothcothln sinh22 2 21∫=− + 17.31.6.coth ( ) () ( ax xdxaxax ax Bann n= − + − +⋅⋅⋅− 1 3 135123 2x x nnn) () ( ) !21 21 2− −+⋅⋅⋅ ∫ 17.31.7.dx pq a xpx pqq ap qpa x q+=−−−+ ∫ coth ()ln ( sinh c22 22 oosh )ax 17.31.8. cothcoth ()cothnn nax dxax anax dx =−−+− −∫ ∫1 2 1TABLES OF SPECIAL INDEFINITE INTEGRALS 105 (32) Integrals Involving sech ax 17.32.1. sech ax dxaeax= ∫− 21tan 17.32.2. sech2ax dxax a= ∫tanh 17.32.3. sech3ax dxax ax aaax =+ ∫− sech tanhtan sinh21 21 17.32.4. xa x d xaax ax axsech∫= − + +⋅⋅⋅− 1 285 144224 6() () () (1 1 22 222)( ) () ( ) !n nnEa x nn+ ++⋅⋅⋅⎧⎨⎩⎫⎬⎭ 17.32.5. xa x d xxa x aaax sech2=− ∫tanhln cosh1 2 17.32.6.sechln() () () ax xdx xax ax ax=− + − + ⋅24 6 45 9661 4320⋅⋅ ⋅−+⋅⋅⋅ ∫() () () !1 222 n nnEa x nn 17.32.7. sech sechnn ax dxax ax ann n=−+− − ∫−sech tanh ()2 12 1n nax dx−∫2 (33) Integrals Involving csch ax 17.33.1. csch ax dxaax= ∫1 2ln tanh 17.33.2. csch2ax dxax a=− ∫coth 17.33.3. csch3ax dxax ax aaax=− − ∫csch cothln tanh21 22 17.33.4. xa x d xaaxax axcsch = − + +⋅⋅⋅+−∫1 187 18002 235() () (112 1 2121 21)( ) ( ) () !nn nnBa x n−+− ++⋅⋅⋅⎧⎨⎩⎫⎬⎭ 17.33.5. xa x d xxa x a aax csch2=− + ∫cothlnsinh1 2 17.33.6.csch ( ) ()( ax xdxaxax axnn = −−+ + ⋅ ⋅ ⋅− 1 67 108012 23 2−−−− −+⋅⋅⋅ ∫12 11 21 2)() () ( ) !Ba x nnnn 17.33.7. csch cscnn ax dxax ax ann n=− −−− − ∫−csch coth ()2 12 1h h2nax dx−∫TABLES OF SPECIAL INDEFINITE INTEGRALS 106 (34) Integrals Involving Inverse Hyperbolic Functions 17.34.1. sinh sinh−−∫=− +11 2 2 x adx xx axa 17.34.2. xx adxxa x axxasinh sinh−−∫=+⎛ ⎝⎜⎞ ⎠⎟ −+122 122 24 4 17.34.3.sinh ( )() () − ∫=−+ 135 23313 24 xa xdxx axa xa /// iii iiiiii iiii 55135 2467 7 27 2− +⋅⋅⋅() ln (xaxa xa/|| < /))() () () 2 22213 244413524 −+ −ax ax ax// / iii iiiii6 6 2224666 2 2 222iiii ii+⋅⋅⋅ −−+−xa xa ax> // ln ( ) ( ) 113 2444135 2466646i iiiii iiii() ()ax axx//+ −⋅⋅⋅ < <−⎧ ⎨⎪ ⎪⎪⎪ ⎩⎪ ⎪⎪⎪a 17.34.4. coshcosh ( ) , cosh ( ) −−− ∫=−− > 112 2 1x adxxx a x a x a // 0 0 012 2 1xx a x a x acosh ( ) , cosh ( )−−+− <⎧ ⎨⎪ ⎩⎪ // 17.34.5. xx adxxa x a x xa cosh() c o s h ( ) −− ∫=−− − 122 1 22 1 421 4/ ,, c o s h ( ) () c o s h ( )− −> −+ −1 22 1 20 1 421 4xa xa x a x x/ / aax a210 ,c o s h ( )−<⎧ ⎨⎪⎪ ⎩⎪ ⎪/ 17.34.6.cosh ( )ln ( )()− ∫=± + +1 221 221 xa xdx x aax /// 222iii331 3 546() ()ax ax/ 2444/ 24666 iiiii iiii+ +⋅⋅⋅⎡ ⎣⎢⎤ ⎤ ⎦⎥ +> −<−−if / if /cosh ( ) , cosh ( )1100 xa xa 17.34.7. tanh tanh ln( )−−∫=+ −11 2 2 2x adx xx aaax 17.34.8. xx adxaxxax atanh ( ) tanh−−∫=+ −12 2 1 21 2 17.34.9.t a n h () () ()− ∫= + + +⋅⋅⋅13 25 235xa xdxx axa xa // / 17.34.10. coth coth ln ( )−−∫=+ −11 2 2 2x adx x xaxa 17.34.11. xx adxaxxax acoth ( )coth−−∫=+ −12 2 1 21 2 17.34.12.c o t h () () ()− ∫= − + + +⋅⋅⋅⎛ ⎝13 25 235xa xdxa xax ax // / ⎜ ⎜⎞ ⎠⎟ 17.34.13. sechsech ( ) sin ( ), sech −−− − ∫=+ 111 1x adxxx a a x a // ( () sech ( ) sin ( ), sech (xa x xa a xa xa/ // /> −−− −0 11 1) )<⎧ ⎨⎪ ⎩⎪ 0 17.34.14. csch sinh ( ,−− −∫=± + > −11 10x adx xx aax axx csch if if < <0)TABLES OF SPECIAL INDEFINITE INTEGRALS 107 17.34.15. xx adxx mx amx xammm sinh sinh−+ −+ =+−+ +∫11 11 22 11 1ddx ∫ 17.34.16. xx adxx mx amx xa mmm coshcosh −+ −+ ∫=+−+ − 11 11 22 11 1ddx x a x mx amx xmm∫− + −+> +++cosh ( ) cosh1 1 11 20 11 1/ − −<⎧ ⎨⎪ ⎪ ⎩⎪ ⎪ ∫− adx x a 210 cosh ( ) / 17.34.17. xx adxx mx aa mx axmmm tanh tanh−+ −+ =+−+ − ∫11 11 22 11ddx ∫ 17.34.18. xx adxx mx amx axmmm coth coth−+ −+ =+−+ − ∫11 11 22 11 1ddx ∫ 17.34.19. xx adxx mx aa mxd x ax mmm sechsech −+ − ∫=+++ − 11 1 22 11 ∫ ∫− + −> +−+ −sech ( ) sech1 1 1 20 11 1xa x mx amxd x axmm/ 2 210 ∫−<⎧ ⎨⎪ ⎪ ⎩⎪ ⎪sech ( ) xa/ 17.34.20. xx adxx mx aa mxd x xammm csch csch−+ −=+±+ +∫11 1 22 11 ∫ ∫+> −<(, )if ifxx00TABLES OF SPECIAL INDEFINITE INTEGRALS 18DEFINITE INTEGRALS Definition of a Definite Integral Let f(x) be defined in an interval a /H11017 x /H11017 b. Divide the interval into n equal parts of length /H9004x = (b − a)/n. Then the definite integral of f (x) between x = a and x = b is defined as 18.1. f x d x f axf a xxf a x n ab( ) l i m { ( )( )( ) =+ + + + →∞ ∫ΔΔ Δ Δ Δ 2 xxf a n x x++ +−/midhorizellipsis (( ) ) } 1ΔΔ The limit will certainly exist if f (x) is piecewise continuous. If fxd dxgx () () ,= then by the fundamental theorem of the integral calculus the above definite integral can be evaluated by using the result 18.2. fxd xd dxgxd x gx gb ga ab ab ab() () () () () == = − ∫∫ If the interval is infinite or if f (x) has a singularity at some point in the interval, the definite integral is called an improper integral and can be defined by using appropriate limiting procedures. For example, 18.3. fxd x fxd x b aab() l i m () = →∞∞∫∫ 18.4. fxd x fxd x a ab b() l i m () = →−∞ −∞∞ →∞∫∫ 18.5. fxd x fxd x b() l i m () = →∈0if is a singular point. . α∈b ab −∫ ∫ 18.6. f x dx dx a ab ab( ) lim = → +∫ ∫∈ ∈ 0fx( ) if is a singula ar point. General Formulas Involving Definite Integrals 18.7. {() () () } () ()f x gx hx d x f xd x gxd x ab ab ab±±± = ± ± ∫∫/midhorizellipsis ∫∫∫± hxd x ab() /midhorizellipsis 18.8. cf x dx c f x dx c ab a() () = ∫where is any constant.b b∫ 18.9. fxd x aa() = ∫0 18.10. fxd x fxd x ab ba() () =− ∫∫ 18.11. fxd x fxd x fxd x ab ac cb() () () =+ ∫∫∫ 108 109 18.12. fxd x b a fc c a b ab() ( ) () . =− ∫where is between and This is called the mean value theorem for definite integrals and is valid if f (x) is continuous in a /H11017 x /H11017 b. 18.13. fx gxd x fc gxd x c a ab() () () () = ∫where is between and b b ab∫ This is a generalization of 18.12 and is valid if f (x) and g (x) are continuous in a /H11017 x /H11017 b and g (x) /H11084 0. Leibnitz’s Rules for Differentiation of Integrals 18.14.d dFx d xF ddx Fd d αααφαφ α φαφα φ(, ) ( , ) ()()=∂+ ∫ 12 22 1 12 11 ()()(,) αφαφαφ α ∫−Fd d Approximate Formulas for Definite Integrals In the following the interval from x = a to x = b is subdivided into n equal parts by the points a = x0, x1, x2, …, xn–1, xn = b and we let y0 = f(x0), y1 = f(x1), y2 = f(x2), …, yn = f(xn), h = (b – a)/n. Rectangular formula: 18.15. fxd x h y y y ynab() ( ) ≈+ + + +− ∫ 012 1 /midhorizellipsis Trapezoidal formula: 18.16. fxd xhyyy yynnab() ( ) ≈+ + + ++− ∫ 222 2012 1 /midhorizellipsis Simpson’s formula (or parabolic formula) for n even: 18.17. fxd xhyyyy y yynn na() ( ) ≈+ + + + +++−− 3424 2 40123 2 1 /midhorizellipsisb b∫ Definite Integrals Involving Rational or Irrational Expressions 18.18.dx xa a220 2 +=∞∫π 18.19.xd x xppp−∞ +=< < ∫1 0101π π sin, 18.20.xd x xaa nm nmnm nnmn +=+<+ <∞+− ∫01 101π π sin[( ) / ], 18.21.xd x xx mmm 1220++=∞∫ cos sinsin sin βπ πβ β 18.22.dx axa 22 0 2 −= ∫π 18.23. ax d xa a22 02 4−= ∫πDEFINITE INTEGRALS 110 18.24. xa x d xam n p nmmn n pmn p ()[( )/ ] ( ) [(−=++ +++111 1ΓΓ Γ ))/ ]npa ++ ∫ 1 0 18.25.xd x xaam n nm nn rrm n r ()() [ ( ) / ] sin[ +=−+−+ −1111π Γ (() / ] ( ) ! [ () / ],mn r m n rmn r+−+ − +<+ <∞∫ 111101 0 π Γ Definite Integrals Involving Trigonometric Functions All letters are considered positive unless otherwise indicated. 18.26. sin sin, /,mx nx dxmn m n mn=≠ 0 2integers and intege π rrs and mn=⎧ ⎨⎪ ⎩⎪∫0π 18.27. cos cos, /,mx nx dxmn m n mn00 2π π∫=≠ integers and inntegers and mn=⎧ ⎨⎪ ⎩⎪ 18.28. sin cos, /(mx nx dxmn m n mm=+ 0 22integers and even −−+⎧ ⎨⎪ ⎩⎪∫nm n m n2 0) , integers and oddπ 18.29. sin cos/ /22 02 02 4xd x xd x ==∫ ∫π π π 18.30. sin cos/22 02 135 2 1 246 2 2mmxd x xd xm m==−∫π π ii/midhorizellipsis ii/midhorizellipsis,, , ,/m= ∫12 02…π 18.31. sin cos/21 0221 246 2 135 2mmxd x xd xm m++∫==π ii/midhorizellipsis ii/midhorizellipsis + += ∫ 112 02π/,, ,m … 18.32. sin cos() () ()/21 21 02 2pqxx d xpq pq−−=+ ∫ΓΓ Γπ 18.33.sin/ /px xdxp p p=> = −<⎧ ⎨⎪⎪ ⎩⎪ ⎪∞∫π π20 00 200 18.34.sin cos/ /px qx xdxpq pq pq000 2040∞∫=>> << =>⎧ ⎨⎪⎪π π⎩ ⎩⎪ ⎪ 18.35.sin sin/ /px qx xdxpp q qp q2020 20=< >⎧ ⎨⎪ ⎩⎪∞∫π π/H11017 /H11084 18.36.sin2 20 2px xdxp ∞∫=π 18.37.1 220−=∞∫cospx xdxpπDEFINITE INTEGRALS 111 18.38.cos coslnpx qx xdxq p−=∞∫0 18.39.cos cos ( )px qx xdxqp −=− ∞∫ 20 2π 18.40.cosmx xadxaema 220 2 +=−∞∫π 18.41.xm x xadx ema sin 220 2 +=−∞∫π 18.42.sin ()()mx xx adxaema 22 20 21+=−−∞∫π 18.43.dx ab x ab += −∫ sin2 22 02 π π 18.44.dx ab x ab += −∫ cos2 22 02 π π 18.45.dx ab xba ab += −∫− coscos ( / ) / 021 22π 18.46.dx ab xdx ab xa ab (s i n ) ( c o s ) ( )/+=+=− ∫ 202 22 2 3 22 π π 0 02π∫ 18.47.dx ax a aa122 1012202 −+=−<< ∫ cos,π π 18.48.xx d x ax aaa asin cos(/) l n ( ) , ln (1211 120−+=+< ∫ππ π ++>⎧ ⎨⎪ ⎩⎪ 11/) ,aa 18.49.cos cos,,, , ,mx dx ax aa aamm 12 110 1 2222 0−+=−<=π π… ∫ ∫ 18.50. sin cosax dx ax dxa22 0 01 22==∞ ∞∫ ∫π 18.51. sin ( / ) sin ,/ ax dxnannnn n =>∞∫112110Γπ 18.52. cos ( / ) cos ,/ ax dxnannnn n =>∞∫112110Γπ 18.53.sin cosx xdxx xdx ==∞ ∞∫ ∫0 0 2π 18.54.sin () s i n ( /),x xdxpppp =< <∞∫π π 2201 0 Γ 18.55.cos () c o s ( /),x xdxpppp =< <∞∫π π 2201 0 Γ 18.56. sin cos cos sin ax bx dxab ab a2 022 21 22∞∫=−⎛ ⎝⎜⎞ ⎠⎟πDEFINITE INTEGRALS 112 18.57. cos cos cos sin ax bx dxab ab a222 021 22=+⎛ ⎝⎜⎞ ⎠⎟∞∫π 18.58.sin3 303 8x xdx=∞∫π 18.59.sin4 40 3x xdx=∞∫π 18.60.tanx xdx=∞∫π 2 0 18.61.dx xm1 4 02 += ∫ tan/ π π 18.62.x xdxsin/= −+−+{} ∫21 11 31 51 7222202/midhorizellipsisπ 18.63.tan− ∫=−+−+1 01 22221 11 31 51 7x xdx /midhorizellipsis 18.64.sinln− ∫=1 01 22x xdxπ 18.65.1 01 1−−= ∫∫∞ cos cosx xdxx xdx γ 18.66.1 12 0+−⎛ ⎝⎜⎞ ⎠⎟=∞∫xxdx xcos γ 18.67.tan tanln−−∞ −= ∫11 0 2px qx xdxp qπ Definite Integrals Involving Exponential Functions Some integrals contain Euler’s constant g = 0.5772156 . . . (see 1.3, page 3). 18.68. eb x d xa abax−∞=+ ∫cos220 18.69. eb x d xb abax−∞=+ ∫sin220 18.70.eb x xdxb aax−∞−∫=sintan 01 18.71.ee xdxb aax bx−−∞ −= ∫0ln 18.72. ed xaax−∞= ∫2 1 2 0π 18.73. eb x d xaeax b a −−∞= ∫22 1 24 0cos/ πDEFINITE INTEGRALS 113 18.74. ed xaeb aax bx c b ac a −+ + −∞= ∫() ( ) /22 1 2 244 0πerfc where erfc (p) =−∞∫2 2 πed xx p 18.75. ed xaeax bx c b ac a −+ + − −∞∞= ∫() ( ) /2244 π 18.76. xe d xn ana x n− +∞=+∫Γ()1 10 18.77. xe d xm ama x m− +∞=+∫2 12 2120Γ[( )/ ] () / 18.78. ed xaeax b x ab −+∞−= ∫(/ )22 1 2 02π 18.79.xd x ex−=++++ =∞∫ 11 11 21 31 4 6 022222 /midhorizellipsisπ 18.80.x edx nn xn n n−∞ −=+ + +⎛ ⎝⎜⎞ ⎠⎟ ∫1 0 11 11 21 3Γ() /midhorizellipsis For even n this can be summed in terms of Bernoulli numbers (see pages 142–143). 18.81.xd x ex+=−+−+ =∞∫ 11 11 21 31 4 12 022222 /midhorizellipsisπ 18.82.x edx nn xn n n−∞ +=− + −⎛ ⎝⎜⎞ ⎠⎟ ∫1 0 11 11 21 3Γ() /midhorizellipsis For some positive integer values of n the series can be summed (see 23.10). 18.83.sincothmx edxm mx20 11 421 2π−=−∞∫ 18.84.1 10+−⎛ ⎝⎜⎞ ⎠⎟ =−∞∫ xedx xxγ 18.85.ee xdxxx−−∞ −= ∫2 01 2γ 18.86.1 1 0ee xdxxx −−⎛ ⎝⎜⎞ ⎠⎟ =−∞∫γ 18.87.ee xp xdxbp apax bx−−∞ −=+ +⎛ ⎝⎜⎞ ⎠⎟ ∫ secln 022 221 2 18.88.ee xp xdxb pa pax bx−−∞−− −=− ∫ csctan tan 011 18.89.ex xdx aaaax−∞− −=− + ∫(c o s )cot ln ( )1 212012DEFINITE INTEGRALS 114 Definite Integrals Involving Logarithmic Functions 18.90. xx d xn mmnmnn n (ln )()! (),, , , =− +>− =+ ∫1 110 1 2101… If n ≠ 0, 1, 2, … replace n! by Γ(n + 1). 18.91.lnx xdx11 22 01 +=− ∫π 18.92.lnx xdx162 01 −=− ∫π 18.93.ln ( )1 12 012+= ∫x xdxπ 18.94.ln ( )1 6 012−=− ∫x xdxπ 18.95. ln ln ( ) lnxx d x12 2 2122 01+= − − ∫π 18.96. ln ln ( )xx d x1262 01−= − ∫π 18.97.xx xdx p p pp−∞ +=− < < ∫1 02 101lncsc cotππ π 18.98.xx xdxm nmn−=+ + ∫lnln 01 1 1 18.99. ex d xx−∞=− ∫ln γ 0 18.100. ex d xx−∞=− + ∫2 422 0ln ( ln )πγ 18.101. lne edxx x+ −⎛ ⎝⎜⎞ ⎠⎟=∞∫1 1 4 02π 18.102. ln sin ln cos ln/ /xd x xd x== − ∫ ∫π π π 22 02 02 18.103. (ln sin ) (ln cos ) (ln )/xd x xd x22 23 02 0 2224== +∫ππ π π π/2∫ 18.104. xx d xln sin ln =− ∫π π2 0 22 18.105. sin ln sin ln/xx d x =− ∫21 02π 18.106. ln ( sin ) ln ( cos ) ln ( ) ab x d x ab x d x a a b+= + = + − 222 02ππ π π∫ ∫02 18.107. ln( cos ) lnab x d xaa b+=+−⎛ ⎝⎜⎞ ⎠⎟ ∫ππ22 0 2DEFINITE INTEGRALS 115 18.108. ln ( cos )ln , ln ,aa b x b d xaa b bb a22220 20−+ => >⎧π π/H11084/H11084⎨ ⎨⎪ ⎩⎪∫0π 18.109. ln ( tan ) ln/182 04+= ∫xd xπ π 18.110. sec lncos cos{(cos ) ( xbx axdx a1 11 212 ++⎛ ⎝⎜⎞ ⎠⎟=− −ccos ) }/−∫12 02bπ 18.111. ln sinsin sin sin22 12 23 322 2xdxaaa ⎛ ⎝⎜⎞ ⎠⎟=− + + +⎛/midhorizellipsis⎝ ⎝⎜⎞ ⎠⎟ ∫0a See also 18.102. Definite Integrals Involving Hyperbolic Functions 18.112.sin sinhtanhax bxdxba b=∞∫ππ 220 18.113.cos coshax bxdxba b=∞∫ππ 22 0sech 18.114.xd x ax a sinh02 24∞∫=π 18.115.xd x ax annn nn n n sinh() 01 11 121 211 11 2∞+ ++ + ∫=−++Γ +++ {} +1 31n/midhorizellipsis If n is an odd positive integer, the series can be summed. 18.116.sinhcscax edxba babx+=−∞∫ 1 21 2 0ππ 18.117.sinhcotax edxaba bbx−=−∞∫ 11 22 0ππ Miscellaneous Definite Integrals 18.118.fa x fb x xdx f fb a() (){() () } l n−=− ∞∞∫00 This is called Frullani’s integral. It holds if f ′(x) is continuous and fx f xdx() ()−∞ ∞∫0 converges. 18.119.dx xx=+ ++∫1 11 21 31201 3/midhorizellipsis 18.120. () () ( )()( ) ()ax ax d x amn mnmn m n+−=+−− + − −11 12ΓΓ Γ a aa∫DEFINITE INTEGRALS Section V: Differential Equations and Vector Analysis 19 BASIC DIFFERENTIAL EQUATIONS and SOLUTIONS DIFFERENTIAL EQUATION SOLUTION 19.1. Separation of variables fx fxdxgy gydy c1 22 1() ()() ()+=∫ ∫ f1(x) g1(y) dx + f2(x) g2(y) dy = 0 19.2. Linear first order equation ye Qe dx cPdx Pdx ∫∫=+∫dy dxpxy Qx+=() () 19.3. Bernoulli’s equation /H9271en Q e d x cnP d x nP d x () ()()111−−∫∫=− + ∫ where y = y1−n. If n = 1, the solution is ln ( )yQ P d x c=− +∫dy dxPxy Qxyn+=() () 19.4. Exact equation Mx NyMxd y c ∂+ −∂ ∂∂⎛ ⎝⎜⎞ ⎠⎟= ∫ ∫ ∫ where ∂x indicates that the integration is to be per- formed with respect to x keeping y constant.M(x, y)dx + N(x, y)dy = 0 where ∂M/∂y =∂N/∂x. 19.5 Homogeneous equation ln()xd Fc =−+ ∫/H9271 /H9271/H9271 where y = y/x. If F(y) = y, the solution is y = cx.dy dxFy x=⎛ ⎝⎜⎞ ⎠⎟ 116 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 117 19.6. ln() {() () }xGd GFc =−+ ∫/H9271/H9271 /H9271/H9271 /H9271 where y = xy. If G(y) = F(y), the solution is xy = c.y F(xy) dx + x G(xy) dy = 0 19.7. Linear, homogeneous second order equationLet m1, m2 be the roots of m2 + am + b = 0. Then there are 3 cases.Case 1. m 1, m2 real and distinct: yc e c emx mx=+1212 Case 2. m1, m2 real and equal: yc e c x emx mx=+1211 Case 3. m1 = p + qi, m2 = p − qi: yec q xc q xpx=+ (c o s s i n )12 where p = −a/2, qb a=−24/.dy dxady dxby2 20 ++ = a, b are real constants. 19.8. Linear, nonhomogeneous second order equationThere are 3 cases corresponding to those of entry 19.7 above. Case 1. yc e c e e mmeR x d x e mmx mx mx mx mx=+ +− +−∫12 1212 1 1 2() 2212 −−∫meR x d xmx() Case 2. yc e c x e xe e R x dx ex emx mx mx mx mx m=+ + −− −∫1211 11 1() 1 1xRx d x() ∫ Case 3. yec q xc q x eq x qeR xpx px px=+ +−∫(c o s s i n ) sin() c o12 s s cos() s i nqx dx eq x qeR x q x d xpx px−−∫dy dxady dxby R x2 2++ = () a, b are real constants.BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS 19.9. Euler or Cauchy equation Putting x = et, the equation becomes dy dtady dtby S et2 21 +− + =() ( ) and can then be solved as in entries 19.7 and 19.8 above.xdy dxaxdy dxby S x22 2++ = () 19.10. Bessel’s equation yc J x c Y xnn=+12() ()/H9261/H9261 See 27.1 to 27.15.xdy dxxdy dxxn y22 2220 ++ −= ()/H92612 19.11. Transformed Bessel’s equation yx c Jrxc Yrxp qrr qrr=⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨⎪ ⎩⎪+⎛ ⎝⎜⎞ ⎠⎟− 12 //αα ⎫ ⎫ ⎬⎪ ⎭⎪ where qp=−22β.xdy dxpxdy dxax yr 22 22221 0 ++ + + =() ( ) β2 19.12. Legendre’s equation yc P x c Q xnn=+12() () See 28.1 to 28.48.() ( )121 022 2−− + + =xdy dxxdy dxnn yBASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS 118 20 FORMULAS from VECTOR ANALYSIS Vectors and Scalars Various quantities in physics such as temperature, volume, and speed can be specified by a real number. Such quantities are called scalars. Other quantities such as force, velocity, and momentum require for their specification a direction as well as magnitude. Such quantities are called vectors. A vector is represented by an arrow or directed line segment indicating direction. The magnitude of the vector is determined by the length of the arrow, using an appropriate unit. Notation for Vectors A vector is denoted by a bold faced letter such as A (Fig. 20.1). The magnitude is denoted by |A| or A. The tail end of the arrow is called the initial point, while the head is called the terminal point. Fundamental Definitions 1. Equality of vectors. Two vectors are equal if they have the same magnitude and direction. Thus, A = B in (Fig. 20-1). 2. Multiplication of a vector by a scalar. If m is any real number (scalar), then m A is a vector whose magnitude is |m | times the magnitude of A and whose direction is the same as or opposite to A according as m > 0 or m < 0. If m = 0, then m A = 0 is called the zero or null vector. 3. Sums of vectors. The sum or resultant of A and B is a vector C = A + B formed by placing the initial point B on the terminal point A and joining the initial point of A to the terminal point of B as in Fig. 20-2b. This definition is equivalent to the parallelogram law for vector addition as indicated in Fig. 20-2c. The vector A − B is defined as A + (−B). Fig. 20-2 Extension to sums of more than two vectors are immediate. Thus, Fig. 20-3 shows how to obtain the sum E of the vectors A, B, C, and D.Fig. 20-1 Fig. 20-1 119 120 Fig. 20-3 4. Unit vectors. A unit vector is a vector with unit magnitude. If A is a vector, then a unit vector in the direction of A is a =A/A where A > 0. Laws of Vector Algebra IfA,B,C are vectors and m, n are scalars, then: 20.1. A +B= B+A Commutative law for addition 20.2. A + (B +C)= (A +B)+C Associative law for addition 20.3. m(nA) = (mn)A =n(mA) Associative law for scalar multiplication 20.4. (m+n)A=mA+nA Distributive law 20.5. m(A +B)=mA+mB Distributive law Components of a Vector A vector A can be represented with initial point at the origin of a rectangular coordinate system. If i ,j,k are unit vectors in the directions of the positive x, y, z axes, then 20.6. A =A1i+A2j+A3k where A1i,A2j,A3k are called component vectors of A in thei, j, k directions and A1,A2,A3 are called the components ofA. Dot or Scalar Product 20.7. A • B = AB cos q 0 /H11017 q/H11017 p where q is the angle between A and B. Fundamental results follow:A kz xy iA2jA3k A2ij Fig. 20-4A kz xy iA2jA3k A2ij Fig. 20-4FORMULAS FROM VECTOR ANALYSIS 20.8. A • B=B• A Commutative law 20.9. A •(B+C)=A• B+A• C Distributive law 20.10. A • B =A1B1+A2B2+A3B3 where A=A1i+A2j+A3k,B=B1i+B2j+B3k. Cross or Vector Product 20.11. A ×B=AB sin q u 0/H11017 q/H11017 p where q is the angle between A and B and u is a unit vector perpendicular to the plane of A and B such that A,B, u form aright-handed system (i.e., a right-threaded screw rotated through an angle less than 180° from A to B will advancein the direction of u as in Fig. 20-5). Fundamental results follow: 20.12. ABi jk i×= =− + −A AA BBB AB AB AB AB1 23 123 23 32 31 13() ( ) )( )jk+−AB AB12 21 20.13. A ×B=− (B×A) 20.14. A ×(B+C)=A×B+A×C 20.15. |A×B|= area of parallelogram having sides A and B Miscellaneous Formulas Involving Dot and Cross Products 20.16. AB Ci()×= = + +AAA BBB CCCABC ABC A123 123 1 23123 231 3 BBC ABC A BC ABC12 321 213 132−−− 20.17. |A• (B × C) |= volume of parallelepiped with sides A, B, C 20.18. A × (B ×C)=B(A • C)− C(A • B) 20.19. (A×B)×C=B(A • C)− A(B • C) 20.20. ( A×B)• (C × D) = (A • C)(B • D)− (A • D)(B • C) 20.21. (A×B)× (C ×D)=C{A •(B×D)}− D{A •(B×C)} =B{A •(C×D)}−A{B •(C×D)}BAu Fig. 20-5BAu Fig. 20-5FORMULAS FROM VECTOR ANALYSIS 121 122 Derivatives of Vectors The derivative of a vector function A(u) = A1(u)i + A2(u)j + A3(u)k of the scalar variable u is given by 20.22. d duuu u udA dudA dudA uAA Aij =+−=++ →lim() ( ) ΔΔ Δ 012 3 3 duk Partial derivatives of a vector function A(x, y, z) are similarly defined. We assume that all derivatives exist unless otherwise specified. Formulas Involving Derivatives 20.23. d dud dud du()AB ABAB ii i=+ 20.24. d dud dud du()AB ABAB ×= × + × 20.25. d duCd dud du{( ) } ( )ABABC ABCA B ii ii×= × + ×⎛ ⎝⎜⎞ ⎠⎟+×d d duC ⎛ ⎝⎜⎞ ⎠⎟ 20.26. AAid duAd du=A 20.27. AAA id duif =0 ||is a constant The Del Operator The operator del is defined by 20.28. ∇=∂ ∂+∂ ∂+∂ ∂ij kxy z In the following results we assume that U = U(x, y, z), V = V(x, y, z), A = A(x, y, z) and B = B(x, y, z) have partial derivatives. The Gradient 20.29. Gradient of U = grad UUxy zUU xU yU z=∇ =∂ ∂+∂ ∂+∂ ∂⎛ ⎝⎜⎞ ⎠⎟=∂ ∂+∂ ∂+∂ ∂ij k i j k k The Divergence 20.30. Divergence of A = div A = ∇ • A=∂ ∂+∂ ∂+∂ ∂⎛ ⎝⎜⎞ ⎠⎟++ =∂ ∂+∂ij k i j kxy zAA A A xi()12 3 1A A yA z2 3 ∂+∂ ∂FORMULAS FROM VECTOR ANALYSIS 123 The Curl 20.31. Curl of AAA ij k i j== ∇ × =∂ ∂+∂ ∂+∂ ∂⎛ ⎝⎜⎞ ⎠⎟×+ +curl xy zAA A(12 3 3 123 3 2k ij k i) =∂ ∂∂ ∂∂ ∂ =∂ ∂−∂ ∂⎛ ⎝⎜⎞ ⎠⎟+xyz AAA A yA z∂ ∂ ∂−∂ ∂⎛ ⎝⎜⎞ ⎠⎟+∂ ∂−∂ ∂⎛ ⎝⎜⎞ ⎠⎟A zA xA xA y1 3 21jk The Laplacian 20.32. Laplacian of UU UU xU yU z=∇ =∇ ∇ =∂ ∂+∂ ∂+∂ ∂22 22 22 2i() 20.33. Laplacian of AAAAA=∇ =∂ ∂+∂ ∂+∂ ∂22 22 22 2xyz The Biharmonic Operator 20.34. Biharmonic operator on UU U U xU yU zU x=∇ =∇ ∇ =∂ ∂+∂ ∂+∂ ∂+∂ ∂∂42 2 4 44 44 44 22() y yU yzU xz24 224 2222+∂ ∂∂+∂ ∂∂ Miscellaneous Formulas Involving ∇ 20.35. ∇+ = ∇ + ∇()UV U V 20.36. ∇+ = ∇ + ∇ii i()AB A B 20.37. ∇× + =∇× +∇×()AB A B 20.38. ∇= ∇ + ∇ii i() ( ) ( )UU UAA A 20.39. ∇× = ∇ × + ∇×() ( ) ( )UU UAA A 20.40. ∇× =∇ × −∇ ×iii()()()AB B A A B 20.41. ∇× × = ∇ − ∇ − ∇ + ∇( ) () () () ()A BB A BA AB AB ii i i 20.42. ∇ = ∇ + ∇ + ×∇ × + ×∇ ×() () () ( ) ( )AB B A A B B A A Bii i 20.43. ∇× ∇ =(),U 0 that is, the curl of the gradient of U is zero. 20.44. ∇∇ × =i() , A 0that is, the divergence of the curl of A is zero. 20.45. ∇× ∇× =∇ ∇ −∇() ( ) AA A i2FORMULAS FROM VECTOR ANALYSIS Integrals Involving Vectors If AB() () .ud duu = then the indefinite integral of A(u) is as follows: 20.46. AB c() () ,ud u u c =+ = ∫constant vector The definite integral of A(u) from u = a to u = b in this case is given by 20.47. AB B() () ()ud u b a ab=− ∫ The definite integral can be defined as in 18.1. Line Integrals Consider a space curve C joining two points P1(a1, a2, a3) and P2(b1, b2, b3) as in Fig. 20-6. Divide the curve into n parts by points of subdivision (x1, y1, z1), . . . , (xn−1, yn−1, zn−1). Then the line integral of a vector A(x, y, z) along C is defined as 20.48. Ar Ar A rii idd x y z c n PP ppp pn p== Δ ∫∫ ∑→∞=lim ( , , ) 12 1 where Δ= Δ + Δ + Δ Δ= − Δ= −++ ri j kpp p pp p p p p p xyz x x x y y y ,,,11 Δ= −+zz zpp p 1 and where it is assumed that as n → ∞ the largest of the magnitudes | /H9004rp| approaches zero. The result 20.48 is a generalization of the ordinary definite integral (see 18.1). The line integral 20.48 can also be written as 20.49. Arid A dx A dy A dz CC∫∫=+ +()123 using A = A1i + A2j + A3k and dr = dxi + dyj + dzk. Properties of Line Integrals 20.50. Ar Ariidd pp PP 12 21∫∫=− 20.51. Ar Ar Ariiiddd PP PP PP 12 13 32∫∫∫=+ Independence of the Path In general, a line integral has a value that depends on the particular path C joining points P1 and P2 in a region /H5118. However, in the case of A = ∇f or ∇ × A = 0 where f and its partial derivatives are continuous in /H5118, the line integral Arid C∫ is independent of the path. In such a case, 20.52. Ar Ariidd P P CPP∫∫== − φφ() ()21 12 where f (P1) and f (P2) denote the values of f at P1 and P2, respectively. In particular if C is a closed curve,P1P2 C (xp,yp, zp) yz x Fig. 20-6P1P2 C (xp,yp, zp) yz x Fig. 20-6124 FORMULAS FROM VECTOR ANALYSIS 125 20.53. Ar Arii/integralloopdd CC∫∫== 0 where the circle on the integral sign is used to emphasize that C is closed. Multiple Integrals Let F (x, y) be a function defined in a region /H5118 of the xy plane as in Fig. 20-7. Subdivide the region into nparts by lines parallel to the x and y axes as indicated. Let /H9004A p = /H9004xp /H9004yp denote an area of one of these parts. Then the integral of F(x, y) over /H5118 is defined as 20.54. Fxyd A Fx y A npp p pn ( , ) lim ( , ) =Δ →∞=∫ ∑/H51181 provided this limit exists. In such a case, the integral can also be written as 20.55. Fxyd y d x Fxyd yyf xfx xab yf x(,) (,)()() (= = =∫ ∫ =12 1) )()fx xabdx2∫ ∫{⎫⎬⎭= where y = f1(x) and y = f2(x) are the equations of curves PHQ and PGQ, respectively, and a and b are the x coordinates of points P and Q. The result can also be written as 20.56. F x y dx dy F x y dx dy xgygy ycd x(,) (,) ()()={}= = = ∫ ∫ 12 ggygy ycd 12 ()()∫ ∫= where x = g1(y), x = g2(y) are the equations of curves HPG and HQG, respectively, and c and d are the y coordinates of H and G. These are called double integrals or area integrals. The ideas can be similarly extended to triple or volume integrals or to higher multiple integrals. Surface Integrals Subdivide the surface S (see Fig. 20-8) into n elements of area Δ= = Sp n xy z xypp p p p p p,, , , , ( , , ) ( , , 12… Let whereAA z zp) is a point P in /H9004 Sp. Let Np be a unit normal to /H9004 Sp at P. Then the surface integral of the normal component of A over S is defined as 20.57. AN A NiidS S n Spp p pn =Δ →∞=∫ ∑lim 1 Fig. 20-7 Fig. 20-7 SΔSpγ Δxp ΔypNp y xz Fig. 20-8SΔSpγ Δxp ΔypNp y xz Fig. 20-8FORMULAS FROM VECTOR ANALYSIS 126 Relation Between Surface and Double Integrals If /H5118 is the projection of S on the xy plane, then (see Fig. 20-8) 20.58. AN A N Nkii idSdx dy S= ∫∫ ∫/H5118 The Divergence Theorem Let S be a closed surface bounding a region of volume V; and suppose N is the positive (outward drawn) normal and dS = N dS. Then (see Fig. 20-9) 20.59. ∇=∫∫iiAA SdV d VS The result is also called Gauss’ theorem or Green’s theorem. Stokes’ Theorem Let S be an open two-sided surface bounded by a closed non-intersecting curve C(simple closed curve) as in Fig. 20-10. Then 20.60. Ar A Sii/integralloopdd S C=∇ ×∫ ∫() where the circle on the integral is used to emphasize that C is closed. Green’s Theorem in the Plane 20.61. ()Pd x Qd yQ xP ydx dy CR+=∂ ∂−∂ ∂⎛ ⎝⎜⎞ ⎠⎟ ∫∫/integralloop where R is the area bounded by the closed curve C. This result is a special case of the divergence theorem or Stokes’ theorem.z xySdSN Fig. 20-9dSN S y xz C Fig. 20-10FORMULAS FROM VECTOR ANALYSIS 127 Green’s First Identity 20.62. {( ( ) ( )} ( )φψ φ ψ φ ψ∇+ ∇ ∇ = ∇∫∫2ii VdV dS where f and y are scalar functions. Green’s Second Identity 20.63. () ( )φψ ψ φ φ ψ ψ φ∇− ∇ = ∇ − ∇∫∫22dV d VSiS Miscellaneous Integral Theorems 20.64. ∇× = ×∫∫AAdV VSdS 20.65. φφdd CSrS∫∫=× ∇ Curvilinear Coordinates A point P in space (see Fig. 20-11) can be located by rectangular coordinates (x , y, z,) or curvilinear coordinates (u1, u2, u3) where the transformation equations from one set of coordinates to the other are given by 20.66. xx u u u yy u u u zz u u u= = =(, , ) (, , ) (, , )123 123 123 If u2 and u3 are constant, then as u1 varies, the position vector r = xi + yj + zk of P describes a curve called the u1 coordinate curve. Similarly, we define the u2 and u3 coordinate curves through P . The vectors ∂∂r/,u1∂∂ ∂∂rr/,/uu23 represent tangent vectors to the u1, u2, u3 coordinate curves. Letting e1, e2, e3 be unit tangent vectors to these curves, we have 20.67. ∂ ∂=∂ ∂=∂ ∂=rerereuhuhuh 111 222 333,, where 20.68. huhuhu1 12 23 3=∂ ∂=∂ ∂=∂ ∂rrr,, are called scale factors. If e1, e2, e3 are mutually perpendicular, the curvilinear coordinate system is called orthogonal. u3 curve u2 curveu1 curve yz Pe2u1 = c 1 u3 = c 3u2 = c2e3 e1 x Fig. 20-11u3 curve u2 curveu1 curve yz Pe2u1 = c 1 u3 = c 3u2 = c2e3 e1 x Fig. 20-11FORMULAS FROM VECTOR ANALYSIS 128 Formulas Involving Orthogonal Curvilinear Coordinates 20.69. duduuduudu h du h du rrrre =∂ ∂+∂ ∂+∂ ∂=+ 11 22 331 1 1 2 2eee23 3 3+hd u 20.70. d d d h du h du h dus2 12 12 22 22 32 32==++rri where ds is the element of are length. If dV is the element of volume, then 20.71. dV h du h du h du h h h du=× =||() ( ) ( )11 1 2 2 2 33 3 123ee e i1123 12 3123du du uu udu du duxyz=∂ ∂∂ ∂×∂ ∂=∂ rr ri(,, ) ) (, , )∂uu udu du du 12 3123 where 20.72. ∂ ∂=∂∂ ∂∂ ∂∂ ∂∂(,,) (, , )/// /xyz uuuxu xu xu yu 123123 1123 123∂∂ ∂∂ ∂∂ ∂∂ ∂yu yu zu zu z d u// /// sometimes written J(x, y, z; u1, u2, u3), is called the Jacobian of the transformation. Transformation of Multiple Integrals Result 20.72 can be used to transform multiple integrals from rectangular to curvilinear coordinates. For example, we have 20.73. F x y z dx dy dz G u u uxyz(,,) ( , , )(,,)=∂ ∂ ∫ ∫ ∫ ∫ ∫ ′123 /H5118 /H5118((, , )uuudu du du 123123 ∫ where /H5118′ is the region into which /H5118 is mapped by the transformation and G (u1, u2, u3) is the value of F (x, y, z) corresponding to the transformation. Gradient, Divergence, Curl, and Laplacian In the following, Φ is a scalar function and A = A1e1 + A2e2 + A3e3 is a vector function of orthogonal curvi- linear coordinates u1, u2, u3. 20.74. Gradient of Φ = grad Φ = ∇Φ =∂ ∂+∂+∂ ∂ee e1 112 223 33hu h d u huΦΦΦ 20.75. Divergence of AAA== ∇ =∂ ∂+∂ ∂div i1 123 123 1 231 2hhh uhhAuhhA ()() + +∂ ∂⎡ ⎣⎢⎤ ⎦⎥uhh A 312 3() 20.76. Curl of AAAeee == ∇ × =∂ ∂∂ ∂∂ ∂curl1 12311 2 2 33 123hhhhh h uuu hhA hA hA hh uhAuhA11 2 2 33 23 233 3221=∂ ∂−∂ ∂⎡ ⎣⎢ () ()⎤ ⎤ ⎦⎥+∂ ∂−∂ ∂⎡ ⎣⎢⎤ ⎦⎥ +ee1 13 311 133 21 1hh uhAuhA () () hhh uhAuhA 12 122 211 3∂ ∂−∂ ∂⎡ ⎣⎢⎤ ⎦⎥ () ( ) eFORMULAS FROM VECTOR ANALYSIS 129 20.77. Laplacian of ΦΦΦ=∇ =∂ ∂∂ ∂⎛ ⎝⎜⎞ ⎠⎟+∂ ∂2 123 123 11 231 1 hhh uhh hu uhh hhu uhh hu22 312 33∂ ∂⎛ ⎝⎜⎞ ⎠⎟+∂ ∂∂ ∂⎛ ⎝⎜⎞ ⎠⎟⎡ ⎣⎢ ⎢⎤ ⎦⎥ΦΦ ⎥ ⎥ Note that the biharmonic operator ∇= ∇ ∇42 2ΦΦ () can be obtained from 20.77. Special Orthogonal Coordinate Systems Cylindrical Coordinates (r, q,z) (See Fig. 20-12) 20.78. x=r cos q, y=r sin q, z=z 20.79. hh r h12 222 3211== =,, 20.80. ∇=∂ ∂+∂ ∂+∂ ∂+∂ ∂22 222 22 211ΦΦΦ Φ Φ rr r r z θ Spherical Coordinates (r, q, f) (See Fig. 20-13) 20.81. x= r sin q cos f, y=r sin q sin f, z=r cos q 20.82. hh r h r12 222 322 21===,, s i n θ 20.83. ∇=∂ ∂∂⎛ ⎝⎜⎞ ⎠⎟+∂ ∂∂ ∂⎛ ⎝2 22 211ΦΦΦ rrrdr r sinsinθθθθ⎜ ⎜⎞ ⎠⎟+∂ ∂1 222 2rsinθφΦ Parabolic Cylindrical Coordinates (u ,y,z) 20.84. xu y u z z=− = =1 222() , , /H9271/H9271 20.85. hhu h12 222 2 321 ==+ = /H9271, 20.86. ∇=+∂ ∂+∂ ∂⎛ ⎝⎜⎞ ⎠⎟+∂ ∂2 222 22 22 21ΦΦΦ Φ uu z /H9271/H9271 The traces of the coordinate surfaces on the xy plane are shown in Fig. 20-14. They are confocal parabolaswith a common axis.yyz xxzez eq P rer q(r,q, z) Fig. 20-12. Cylindrical coordinates.z yxz y xefer eq fP(r, q,f,) Fig. 20-13. Spherical coordinates.yyz xxzez eq P rer q(r,q, z) Fig. 20-12. Cylindrical coordinates.z yxz y xefer eq fP(r, q,f,) Fig. 20-13. Spherical coordinates. Fig. 20-14 Fig. 20-14FORMULAS FROM VECTOR ANALYSIS 130 Paraboloidal Coordinates (u, y, f) 20.87. xu yu z u== = −/H9271/H9271 /H9271cos , sin , ( )φφ1 222 where u/H11084/H11084/H1101700 0 2,,/H9271 φπ< 20.88. hhu hu12 222 2 322 2==+ = /H9271/H9271, 20.89. ∇=+∂ ∂∂ ∂⎛ ⎝⎜⎞ ⎠⎟++∂ ∂∂2 22 2211ΦΦΦ uu uuuu () () /H9271/H9271 /H9271 /H9271/H9271∂ ∂⎛ ⎝⎜⎞ ⎠⎟+∂ ∂ /H9271/H92711 222 2uΦ φ Two sets of coordinate surfaces are obtained by revolving the parabolas of Fig. 20-14 about the x axis which is then relabeled the z axis. Elliptic Cylindrical Coordinates ( u,y, z) 20.90. xa u ya u zz== =cosh cos , sinh sin , /H9271/H9271 where uz/H11084/H1101700 2,, /H9271<− ∞ < < ∞ π 20.91. hha u h12 222 2 2 321 == + = (sinh sin ), /H9271 20.92. ∇=+∂ ∂+∂ ∂⎛ ⎝⎜⎞ ⎠⎟+∂2 22 22 22 221ΦΦΦ au u(sinh sin ) /H9271/H9271Φ Φ ∂z2 The traces of the coordinate surfaces on the xy plane are shown in Fig. 20-15. They are confocal ellipses and hyperbolas. Fig. 20-15. Elliptic cylindrical coordinates.FORMULAS FROM VECTOR ANALYSIS 131 Prolate Spheroidal Coordinates (x, h, f) 20.93. xa ya za == =sinh sin cos , sinh sin sin , cosh cξη φ ξηφ ξ oosη where ξη π φ π/H11084/H11017 /H11017/H1101700 0 2,, < 20.94. hha ha12 222 2 2 322 2 2== = (sinh sin ), sinh sin ξη ξη 20.95. ∇=+∂ ∂∂ ∂⎛ ⎝⎜⎞ ⎠2 22 21ΦΦ a(sinh sin ) sinhsinhξη ξ ξξξ⎟ ⎟ ++∂ ∂∂ ∂⎛ ⎝⎜⎞ ⎠⎟+11 22 2a(sinh sin ) sinsinξη η ηηηΦ a a22 22 2sinh sin ξη φ∂ ∂Φ Two sets of coordinate surfaces are obtained by revolving the curves of Fig. 20-15 about the x axis which is relabeled the z axis. The third set of coordinate surfaces consists of planes passing through this axis. Oblate Spheroidal Coordinates (x, h, f) 20.96. xa ya za == =cosh cos cos , cosh cos sin , sinh s ξηφ ξη φ ξ iinη where ξπ η π φ π/H11084/H11017 /H11017 /H1101702 2 0 2,/ / ,−< 20.97. hha ha12 222 2 2 322 2 2== + = (sinh sin ), cosh cos ξη ξ η 20.98. ∇=+∂ ∂∂ ∂⎛ ⎝⎜⎞ ⎠2 22 21ΦΦ a(sinh sin ) coshcoshξη ξ ξξξ⎟ ⎟ ++∂ ∂∂ ∂⎛ ⎝⎜⎞ ⎠⎟+11 22 2a(sinh sin ) coscosξη η ηηηΦ a a22 22 2cosh cos ξη φ∂ ∂Φ Two sets of coordinate surfaces are obtained by revolving the curves of Fig. 20-15 about the y axis which is relabeled the z axis. The third set of coordinate surfaces are planes passing through this axis. Bipolar Coordinates ( u,y, z) 20.99. xa uyau uzz =−=−=sinh cosh cos,sin cosh cos,/H9271 /H9271/H9271 where 02/H11017uz<− ∞ < < ∞ − ∞ < < ∞ π,, /H9271 or 20.100. xy a ua u x a y a22 2 2 2 2 2+− = − + = (c o t ) c s c , (c o t h ) /H9271 csch2 2/H9271,zz=FORMULAS FROM VECTOR ANALYSIS 132 20.101. hha uh12 222 2 321 ==−=(cosh cos ),/H9271 20.102. ∇=−∂ ∂+∂ ∂⎛ ⎝⎜⎞ ⎠⎟+∂ ∂22 22 22 22 ΦΦΦ Φ (cosh cos ) /H9271 /H9271u au z z2 The traces of the coordinate surfaces on the xy plane are shown in Fig. 20-16. Fig. 20-16. Bipolar coordinates. Toroidal Coordinates ( u,y,f) 20.103. xa uya=−=−sinh cos cosh cos,sinh sin cosh co/H9271 /H9271/H9271 /H9271φφ s s,sin cosh cos uzau u=−/H9271 20.104. hha uha 12 222 2 3222 ==−=− (cosh cos ),sinh (cosh /H9271/H9271 /H9271ccos )u2 20.105. ∇=−∂ ∂−∂ ∂⎛ ⎝⎜⎞ ⎠⎟23 21ΦΦ (cosh cos ) cosh cos/H9271 /H9271u au u u + +−∂ ∂−∂ ∂(cosh cos ) sinhsinh cosh cos/H9271 /H9271/H9271/H9271 /H9271u au3 2Φ /H9271 /H9271/H9271 /H9271⎛ ⎝⎜⎞ ⎠⎟+−∂ ∂(cosh cos ) sinhu a2 222 2Φ f The coordinate surfaces are obtained by revolving the curves of Fig. 20.16 about the y axis which is relabeled the z axis. Conical Coordinates (l, m, v) 20.106. xvyaav a abzbb==−− −=− /H9261/H9261 /H9261 μμ μ ab,() ( ),()22 22 2222(()vb ba22 22− − 20.107. hhv abh12 2222 22 2 2 322 1==− −−= ,() () (),( /H9261/H926122μ μμμ− − −−v va vb2 22 22) () ()FORMULAS FROM VECTOR ANALYSIS 133 Confocal Ellipsoidal Coordinates (l, m,y) 20.108.x ay bz ccba x ay b2 22 22 2222 2 221−+−+−=< < < −+/H9261/H9261/H9261/H9261 , μ2 22 222 2 2 22 22 21−+−=< < < −+−+−μμμz ccb a x avy bvz c, v vcbv a =< < <⎧ ⎨⎪ ⎪ ⎪ ⎩⎪ ⎪⎪ 1 22 2, or 20.109. xaaa v ab ac yb222 2 22 2 2 22=−−− −− =−() () () () () (/H9261 /H9261μ ))( )( ) () () () ()bb v ba ac zcc22 22 2 2 222−− −− =−−μ μ /H9261 (() () ()cv ca cb2 22 22− −−⎧ ⎨⎪ ⎪⎪ ⎩⎪ ⎪⎪ 20.110. hv abc hv12 22 2 224=−− −−− =−() ( ) () () () () (μ μ/H9261/H9261 /H9261/H9261/H9261 /H9261 /H9261 /H9261− −−− =−− −μ μμμ μ) () () () () () (4 422 2 32 2abc hvv a vvb vc v)( )( )22−−⎧ ⎨⎪ ⎪⎪ ⎩⎪ ⎪⎪ Confocal Paraboloidal Coordinates (l, m,v) 20.111. x ah bzb x ay bz2 22 22 2 22 2−+−=− − ∞ << −+−=−/H9261/H9261/H9261/H9261, ,μμμ bba x avy bvzv a v22 2 22 22<< −+−=− << ∞⎧ ⎨⎪ ⎪ ⎪ ⎩⎪ ⎪⎪μ , or 20.112. . xaaa v ba ybbb222 2 22 222=−−− − =−−() () () () () (/H9261 /H9261μ μ2 2 22 22− − =++− −⎧ ⎨⎪ ⎪⎪ ⎩⎪ ⎪ ⎪v ab zv a b) /H9261μ 20.113. hv ab hv12 22 224 4=−− −− =−−() ( ) () () () () (μ μμ/H9261/H9261 /H9261/H9261 /H9261 aab hvv av bv22 32 2216−− =−− −−⎧ ⎨⎪ μμ μ)( ) () () () ()/H9261⎪ ⎪ ⎪ ⎩⎪ ⎪⎪FORMULAS FROM VECTOR ANALYSIS Section VI: Series 21SERIES of CONSTANTS Arithmetic Series 21.1. a a da d a nd n a nd++++ + ⋅ ⋅ ⋅ + +− = +−() ( ) { ( ) } {( ) 21 2 11 2 }}( )=+1 2na l where l /H11005 a /H11001 (n /H11002 1)d is the last term. Some special cases are 21.2. 123 11 2 + + +⋅⋅⋅+ = + nn n () 21.3. 135 2 12+ + +⋅⋅⋅+ − = ()nn Geometric Series 21.4. aa ra r a r a rar rar l rnn + + + +⋅⋅⋅+ =− −=− −− 23 1 1 11() where l /H11005 arn/H110021 is the last term and r≠1. If /H110021 < r < 1, then 21.5. aa ra r a ra r+ + + +⋅⋅⋅=−23 1 Arithmetic-Geometric Series 21.6. aa d ra d r an d rarnn ++ ++ + ⋅ ⋅ ⋅ + +− =−−() ( ) { ( ) }(21121 )){ ( ) } () 111 11 2−+−+ − −− rrd nr n r rnn where r≠1. If /H110021 < r < 1, then 21.7. aa d rd d ra rrd r+ + + + +⋅⋅⋅=−+−() ( )()21 12 2 Sums of Powers of Positive Integers 21.8. 123121 1 211 2 ppp pp pp nn pnBp n Bp+ + +⋅⋅⋅+ =+++ −+− !(ppp np−−+⋅⋅⋅−12 43)( ) ! where the series terminates at n2 or n according as p is odd or even, and Bk are the Bernoulli numbers (see page 142). 134 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 135 Some special cases are 21.9. 1231 2+ + +⋅⋅⋅+ =+nnn() 21.10. 12312 1 6222 2+ + +⋅⋅⋅+ =++nnn n() ( ) 21.11. 1231 4123333 322 2+ + +⋅⋅⋅+ =+= + + +⋅⋅⋅+ nnnn()() 21.12. 12312 13 3 1 30444 42 + + +⋅⋅⋅+ =++ + −nnn n n n() ( ) ( ) If Snkkkk k= + + +⋅⋅⋅+123 where k and n are positive integers, then 21.13. kSkSk kSk+⎛ ⎝⎜⎞ ⎠⎟++⎛ ⎝⎜⎞ ⎠⎟+⋅⋅⋅++⎛ ⎝⎜⎞ ⎠⎟=1 11 21 12 (()()nnk+− ++111 Series Involving Reciprocals of Powers of Positive Integers 21.14. 11 21 31 41 52 − + − + −⋅⋅⋅= ln 21.15. 11 31 51 71 94− + − + −⋅⋅⋅=π 21.16. 11 41 71 101 133 91 32 − + − + −⋅⋅⋅= +πln 21.17. 11 51 91 131 172 821 2 4− + − + −⋅⋅⋅= ++ π ln( ) 21.18.1 21 51 81 111 143 91 32 −+− + − ⋅ ⋅ ⋅ = +πln 21.19.1 11 21 31 4 6222 22 + + + +⋅⋅⋅=π 21.20. 1 11 21 31 4 90444 44 + + + +⋅⋅⋅=π 21.21.1 11 21 31 4 945666 66 + + + +⋅⋅⋅=π 21.22. 1 11 21 31 4 12222 22 − + − +⋅⋅⋅=π 21.23. 1 11 21 31 47 720444 44 − + − +⋅⋅⋅=π 21.24.1 11 21 31 431 30 240666 66 − + − +⋅⋅⋅=π , 21.25.1 11 31 51 7 822 2 22 + + + +⋅⋅⋅=π 21.26.1 11 31 51 7 9644 4 44 + + + +⋅⋅⋅=πSERIES OF CONSTANTS 21.27. 1 11 31 51 7 96066 6 66 + + + +⋅⋅⋅=π 21.28.1 11 31 51 7 3233 3 33 − + − +⋅⋅⋅=π 21.29. 1 11 31 51 732 12833 3 33 + − − +⋅⋅⋅=π 21.30.1 131 351 571 791 2 iiii+ + + +⋅⋅⋅= 21.31. 1 131 241 351 463 4 iiii+ + + +⋅⋅⋅= 21.32. 1 131 351 571 798 1622 22 22 222 iiii+ + + +⋅⋅⋅=−π 21.33.1 1231 2341 34543 9 16222 2 22 2222 ii i i ii+++ ⋅ ⋅ ⋅ =−π 21.34. 11 1 21 3 11 01 aa dadadud u ua d −+++−++⋅⋅⋅=+− ∫ 21.35. 1 11 21 31 42 2222 2212 ppp ppp pB p+ + + +⋅⋅⋅=−π () ! 21.36.1 11 31 51 721 2222 2 222 pp p ppp pB p+ + + +⋅⋅⋅=−() () !π 21.37. 1 11 21 31 421 2222 221 2 ppp ppp pB p− + − +⋅⋅⋅=−−() () !π 21.38.1 11 31 51 7221 21 21 2121 2 pp p pp p pE ++ + ++ + − + − +⋅⋅⋅=π 2 22() !p Miscellaneous Series 21.39. 1 2212 22+ + +⋅⋅⋅+ =+cos cos cossin( / ) sin(αα αα αnn /) ) 21.40. sin sin sin sinsin[ / ( )] siααα αα+ + +⋅⋅⋅+ =+2312 1nn nn/ sin( )12 2nα α/ 21.41. 12 31 1223+ + + +⋅⋅⋅=− −rr rr rcos cos coscos cosαα αα α+ +rr21 ,| | < 21.42. rr rr rrsin sin sinsin cos, αα αα α+ + +⋅⋅⋅=− +23 22312|| <r1 21.43. 12221 + + +⋅⋅⋅+ =−++ rr r nrn rnnn cos cos coscos cαα αα oos( ) cos cosnr rr+− + −+11 122αα α 21.44. rr r nrr nnn sin sin sinsin sin(αα αα+ +⋅⋅⋅+ =−++ 21 21))s i n cosαα α+ −++rn rrn2 212SERIES OF CONSTANTS 136 137 The Euler-Maclaurin Summation Formula 21.45. Fk Fkd k F Fn Fknn() () {() () } {(=− ∑ ∫=− + + ′11 01 20 1 12nnF F nF F) ( )} { ( ) ( )} ,{(−− ′′′ −′′′ +01 7200 1 30 240v))( ) ( )( )() () },,{( ) ( nF F nF−− −vv ii vii01 1 209 6000 0 12012 1 2 1)} ()() !{( ) (() ()+⋅⋅⋅ − −−− −p p ppB pFn F ))}+⋅⋅⋅ The Poisson Summation Formula 21.46. Fk e Fxd x kimx m() () =−∞∞ −∞∞ =−∞∞ ∑ ∫∑={}2πSERIES OF CONSTANTS 22 TAYLOR SERIES Taylor Series for Functions of One Variable 22.1. fx fa f a x afa xa fn () () () ( )() ( ) !( =+ ′ −+′′ −+⋅⋅⋅+2 2− −− −+−11 1)() ( ) () !ax a nRn n where Rn, the remainder after n terms, is given by either of the following forms: 22.2. Lagrange’s form: Rfx a nnnn =−()() ( ) !ξ 22.3. Cauchy’s form: Rfx x a nnnn =−− −− ()() ( ) ( ) () !ξξ1 1 The value x, which may be different in the two forms, lies between a and x. The result holds if f(x) has continuous derivatives of order n at least. Iflim , nnR →∞=0 the infinite series obtained is called the Taylor series for f(x) about x /H11005a. If a /H11005 0, the series is often called a Maclaurin series. These series, often called power series, generally converge for all values of x in some interval called the interval of convergence and diverge for all x outside this interval. Some series contain the Bernoulli numbers Bn and the Euler numbers En defined in Chapter 23, pages 142/H11002143. Binomial Series 22.4. ()() !() ( )ax a n axnnaxnn nnn n n+= + +−+−−−−12 2 1 212 3 3 1233 12!ax anaxnan nn n− −−+⋅⋅⋅ =+⎛ ⎝⎜⎞ ⎠⎟ +⎛ ⎝⎜⎞ ⎠⎟ x xnaxn 23 3 3+⎛ ⎝⎜⎞ ⎠⎟ +⋅⋅⋅− Special cases are 22.5. ()ax a a xx+= + +22 22 22.6. ()ax a a x a x x+= + + +33 2 2333 22.7. ()ax a a x a x a x x+= + + + +44 3 2 2 3 446 4 22.8. ()1112 3 4+ = − + − + −⋅⋅⋅−xx x x x /H110021 < x < 1 22.9. ()11 2 3 4 522 3 4+ = − + − + −⋅⋅⋅−xx x x x /H110021 < x < 1 22.10. ()1 1 3 6 10 1532 3 4+ = − + − + −⋅⋅⋅−xx x x x /H110021 < x < 1 138 139 22.11. ()111 213 24135 24612 2 3+ = − + − +⋅⋅⋅−xx xx/ i iii ii/H110021 < x /H11017 1 22.12. ()111 21 2413 24612 2 3+ = + − + −⋅⋅⋅xx xx/ ii ii/H110021 < x /H11017 1 22.13. ()111 314 36147 36913 2 3+ = − + − +⋅⋅⋅−xx x x/ i iii ii/H110021 < x /H11017 1 22.14. ()111 32 3625 36913 2 3+ = + − + −⋅⋅⋅xx xx/ ii ii/H110021 < x /H11017 1 Series for Exponential and Logarithmic Functions 22.15. exxxx= + + + +⋅⋅⋅12323 !!/H11002∞ < x < ∞ 22.16. ae x axa xaxx a= = + + + +⋅⋅⋅lnln(l n) !(l n) !12323 /H11002∞ < x < ∞ 22.17. ln ( )1234234 + = − + − +⋅⋅⋅xxxxx/H110021 < x /H11017 1 22.18.1 21 13 5 7357 ln+ −⎛ ⎝⎜⎞ ⎠⎟= + + + +⋅⋅⋅x xxxxx/H110021 < x < 1 22.19. lnxx xx xx x=− +⎛ ⎝⎜⎞ ⎠⎟+− +⎛ ⎝⎜⎞ ⎠⎟+− +⎛21 11 31 11 51 13 ⎝ ⎝⎜⎞ ⎠⎟+⋅⋅⋅⎧ ⎨⎪ ⎩⎪⎫ ⎬⎪ ⎭⎪5 x > 0 22.20. lnxx xx xx x=−⎛ ⎝⎜⎞ ⎠⎟+−⎛ ⎝⎜⎞ ⎠⎟+−⎛ ⎝⎜⎞ ⎠⎟+11 211 3123 ⋅ ⋅⋅⋅ x/H110841 2 Series for Trigonometric Functions 22.21. sin!!!xxxxxx =− + − + − ∞ << ∞357 357/midhorizellipsis 22.22. cos!!!xxxxx =− + − + − ∞ < < ∞ 1246246 /midhorizellipsis 22.23. tan()xxxx x Bxnn nn =+ + + + +−− 35 7 22 2 32 1517 31522 1/midhorizellipsis1 1 2() !n+/midhorizellipsis ||x<π 2 22.24. cot() !xxxx x Bx nn nn =−− − −− −−1 34 52 9452 235 22 1 /midhorizellipsis/midhorizellipsis 0<<||x π 22.25. sec() !xxx x Ex nnn =+ + + + + +125 2461 720 224 6 2 /midhorizellipsis/midhorizellipsis ||x<π 2 22.26. csc,()xxxx x Bn n=++ + ++−−1 67 36031 15 12022 135 21 /midhorizellipsisx x nn21 2− +() !/midhorizellipsis 0<<||x π 22.27. sin−=+ + + +135 71 2313 24 5135 246 7xxxx x i iii ii/midhorizellipsis ||x<1 22.28. cos sin−−=− =−+ + +⎛ ⎝⎜⎞1135 221 2313 24 5xx xxx ππ i i/midhorizellipsis⎠ ⎠⎟ ||x<1TAYLOR SERIES 140 22.29. tan|| −=−+−+ < ±−+ −1357 353571 211 31 5xxxxxx xx x/midhorizellipsis π+++ − −⎧ ⎨⎪ ⎩⎪ /midhorizellipsis (, )if ifxx/H11084/H1101711 22.30. cot tan|| −−=− =−− + −⎛ ⎝⎜⎞ ⎠⎟< 1135 223 51 xxxxxx pππ/midhorizellipsis π π+− + − = > = < −⎧ ⎨11 31 501 1 135xx xpx p x /midhorizellipsis (, ) if if⎪ ⎪⎪ ⎩⎪ ⎪ 22.31. sec cos ( / )−−== − + +⋅+11 351211 2313 245xxxx xπ ii i/midhorizellipsis⎛ ⎛ ⎝⎜⎞ ⎠⎟ ||x>1 22.32. csc sin ( / )−−== + +⋅+11 35111 2313 245xxxx xii i/midhorizellipsis ||x>1 Series for Hyperbolic Functions 22.33. sinh!!!xxxxxx = ++++ − ∞ < < ∞357 357/midhorizellipsis 22.34. cosh!!!xxxxx =+ + + + − ∞ < < ∞ 1246246 /midhorizellipsis 22.35. tanh() (xxxx xnn n =− + − +−−− 35 7 12 2 32 1517 31512 2 1/midhorizellipsis) ) () !Bx nnn21 2− +/midhorizellipsis ||x<π 2 22.36. coth() (xxxx x Bxnn nn =+− + +−−−1 34 52 9451235 12 21 /midhorizellipsis2 2n)!+/midhorizellipsis 0<<||x π 22.37. sech xxx x Ex nn nn =− + − +−125 2461 7201 224 6 2 /midhorizellipsis() () !+ +/midhorizellipsis ||x<π 2 22.38. csch xxxx xnn =−+ − +−−1 67 36031 15 12012 235 2 ,()(/midhorizellipsis112 11 2−+−) () !Bx nnn /midhorizellipsis 0<<||x π 22.39. sinh−=−+ − + 135 7 2313 245135 2467xxxx x ii iiii iii/midhorizellipsis || | ln | |x xxx< ±+ − +1 21 2213 244135 24624ii iiii iii 6 61 16xx x−⎛ ⎝⎜⎞ ⎠⎟+ −−⎡ ⎣⎢⎤ ⎦⎥⎧ ⎨⎪⎪ ⎩⎪ ⎪/midhorizellipsisif if/H11084 /H11017 22.40. cosh ln( )−=± − + +1 2421 2213 244135 24xxxxii iiii ii66601 61 i/midhorizellipsisxxx+⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨⎪ ⎩⎪⎫ ⎬⎪ ⎭⎪+> −−if cosh , /H11084 if cosh ,−<⎡ ⎣⎢⎤ ⎦⎥ 101xx /H11084 22.41. tanh−=+ + + +1357 357xxxxx/midhorizellipsis | x | < 1 22.42. coth−=+ + + +1 35711 31 51 7xx xxx/midhorizellipsis | x | > 1 Miscellaneous Series 22.43. exxxxxsin= + +−−+128 1 5245 /midhorizellipsis −∞< < ∞ x 22.44. eexx xxcos=− + − +⎛ ⎝⎜⎞ ⎠⎟ 12631 72024 6 /midhorizellipsis −∞< < ∞ xTAYLOR SERIES 141 22.45. exxx xxtan=+ + + + +1223 823 4 /midhorizellipsis ||x<π 2 22.46. ex x xxxx n xxnn sinsin( / )/ =+ + − − + +2356 2 33 0 9 024/midhorizellipsisπ n n!+/midhorizellipsis −∞< < ∞ x 22.47. ex xxx n x nxnn coscos( / ) !/ =+ − − + + + 1362434 2 /midhorizellipsis/midhorizellipsisπ −∞< < ∞ x 22.48. ln | sin | ln | | xxxx x Bxn n=− − − − −− 24 6 21 2 6 180 28352/midhorizellipsisn n nn() !2+/midhorizellipsis 0<<||x π 22.49. ln | cos |(xxxx xn =− − − − − −− 246 8 21 2 21 2 4 517 252022/midhorizellipsisn n nnBx nn−+1 22) () !/midhorizellipsis ||x<π 2 22.50. ln | tan | ln | |(xxxx xnn =+ + + + +24 6 22 37 9062 283522/midhorizellipsis− −−+121 2) () !Bx nnnn /midhorizellipsis 02<<||xπ 22.51. ln( )()( )1 1111 22 1 21 33 + +=−+ +++ −x xxx x /midhorizellipsis ||x<1 Reversion of Power Series Suppose 22.52. y C x C xC xC xC xC x= ++++++122 33 44 55 66/midhorizellipsis then 22.53. x C y C yC yC yC yC y= ++++++122 33 44 55 66/midhorizellipsis where 22.54. cC11 1= 22.55. cC c13 22=− 22.56. cC c c c15 322 13 2=− 22.57. cC c cc c cc17 41 2 3 23 12 4 55=− − 22.58. cC ccc cc cc c c cc19 512 24 12 32 13 524 1226 3 14 21 =+ − + −3 3 22.59. c C ccc ccc ccc ccc111 613 25 123 313 34 12 2 78 472 8=+ +−3 32 14 612 22 42528 42 −− −cc ccc c Taylor Series for Functions of Two Variables 22.60. f xy f ab x a f ab y b f abxy ( ,) ( ,) ( ) ( ,) ( ) ( ,) !=+ − + − +1 2{{( ) ( , ) ( )( ) ( , ) ( )xafa b xa yb fa b ybxx xy −+ − − + −222 ffa byy(, ) } +/midhorizellipsis where fa b fa bxy(, ) , (, ) , … denote partial derivatives with respect to x , y, … evaluated at x /H11005 a, y /H11005 b.TAYLOR SERIES 23 BERNOULLI and EULER NUMBERS Definition of Bernoulli Numbers The Bernoulli numbers BBB123,,, … are defined by the series 23.1. x exBx Bx Bxxx−=− + − + − <1122 4 6212 24 36 !!!|| /midhorizellipsis π 23.2. 1222 4 612 34 36 −= + + + <xx Bx Bx Bxx cot!!!|| /midhorizellipsis π Definition of Euler Numbers The Euler numbers E1, E2, E3, … are defined by the series 23.3. sech xEx Ex Exx =− + − + <1246 212 24 36 !!!|| /midhorizellipsisπ 23.4. sec!!!|| xEx Ex Exx =+ + − + <1246 212 24 36 /midhorizellipsisπ Table of First Few Bernoulli and Euler Numbers Bernoulli Numbers Euler Numbers B B B B BB1 2 3 4 5616 13 014 213 0 56 6691 27= =====/ /// / /330 763617 51043 867 798 174 61 7 89 10B B B B= = = =/ / ,/ ,11 330 854 513138236 364 091 2730 11 12/ ,/ ,, /B B= =E E E E EE E1 2 3 4 5671 561 1385 50 5212 702 765= =====, ,, = ===199 360 981 19 391 512 145 2 404 879 8 9,, ,,, ,,,E E 6675 441 370 371 188 237 525 69 348 810 11, ,,,, ,,E E= = 774 393 137 901 15 514 534 163 557 086 912,,, ,,, ,,, E= 005 142 143 Relationships of Bernoulli and Euler Numbers 23.5. 21 2221 4221 62 14 2nBnBn +⎛ ⎝⎜⎞ ⎠⎟ −+⎛ ⎝⎜⎞ ⎠⎟ ++⎛ ⎝⎜⎞ ⎠⎟221 2 1 2 26 312Bn B nnn n −− + =−/midhorizellipsis()( ) 23.6. EnEnEnEnn n n=⎛ ⎝⎜⎞ ⎠⎟ −⎛ ⎝⎜⎞ ⎠⎟ +⎛ ⎝⎜⎞ ⎠⎟ −− −2 22 42 6123 3 1 −−/midhorizellipsis()n 23.7. Bn nEnEn nn n =−−⎛ ⎝⎜⎞ ⎠⎟ −−⎛ ⎝⎜⎞ ⎠⎟ −2 22 121 121 322 1()nnnn nE−−−+−⎛ ⎝⎜⎞ ⎠⎟ −−⎧ ⎨ ⎩⎫ ⎬ ⎭231 21 51/midhorizellipsis() Series Involving Bernoulli and Euler Numbers 23.8. Bn n nn n n=+ + +⎧⎨⎩⎫⎬⎭−() !2 211 21 3212 2 2π/midhorizellipsis 23.9. Bn n nn n n =−+++⎧⎨⎩⎫⎬⎭22 2111 31 522 2 2() ! () π/midhorizellipsis 23.10. Bn n nn n n =−−+−⎧⎨⎩⎫⎬⎭−22 2111 21 321 2 2 2() ! () π/midhorizellipsis 23.11. En nn nn n =− + −⎧⎨⎩⎫⎬⎭+ ++ +2211 31 522 21 21 21() ! π/midhorizellipsis Asymptotic Formula for Bernoulli Numbers 23.12. Bn e nnnn~( )422ππ−BERNOULLI AND EULER NUMBERS 24 FOURIER SERIES Definition of a Fourier Series The Fourier series corresponding to a function f(x) defined in the interval cxcL/H11017/H11017 +2 where c and L > 0 are constants, is defined as 24.1.aanx Lbnx Lnn n0 12++⎛ ⎝⎜⎞ ⎠⎟ =∞ ∑ cos sinππ where 24.2.aLfxnx Ldx bLfxnx LdxnccL nc= =+∫1 12() c o s () s i nπ π ccL+∫⎧ ⎨⎪ ⎩⎪2 Iff(x) and f′(x) are piecewise continuous and f (x) is defined by periodic extension of period 2 L, i.e., f(x/H11001 2L)/H11005f(x), then the series converges to f (x) if x is a point of continuity and to1 2 00 {( ) ( ) }fx fx++ − if x is a point of discontinuity. Complex Form of Fourier Series Assuming that the series 24.1 converges to f (x), we have 24.3. fx c enin x L n()/= =−∞∞ ∑π where 24.4. cLfx e d xai b n ai bnin x Lnn n ==−> +− −1 201 2 1 2 ()() (/π − −+< =⎧ ⎨⎪ ⎩⎪∫ nccLn an)0 01 202 Parseval’s Identity 24.5.1 22 02 22 12 Lfx d xaabnn nccL{() } ( ) =+ + =∞+∑ ∫ Generalized Parseval Identity 24.6.1 2200 1Lfx g xd xacac bd ccL nn n n n()() ( )+ =∞ ∫ ∑ =+ + where an,bn and cn,dn are the Fourier coefficients corresponding to f (x) and g(x), respectively. 144 145 24.7. fxx x()=<< −− < <⎧⎨⎩10 10π π Fig. 24-1 4 13 35 5 πsin sin sinxxx+++⎛ ⎝⎜⎞ ⎠⎟/midhorizellipsis 24.8. fx xxx xx() | |==<< −− < <⎧⎨⎩0 0π π Fig. 24-2 π π24 13 35 522 2 −+ + +⎛ ⎝⎜⎞ ⎠⎟cos cos cosxxx/midhorizellipsis 24.9. fx x x() ,=− < < ππ Fig. 24-3 212 23 3sin sin sinxxx−+−⎛ ⎝⎜⎞ ⎠⎟/midhorizellipsis 24.10. fx x x() ,=< < 02 π Fig. 24-4 π−+ + +⎛ ⎝⎜⎞ ⎠⎟ 212 23 3sin sin sinxxx/midhorizellipsis 24.11. fx x x() | s i n | ,=− < < ππ Fig. 24-5 24 2 134 356 57 ππ−+ + +⎛ ⎝⎜⎞ ⎠⎟cos cos cos xxx iii/midhorizellipsisSpecial Fourier Series and Their GraphsFOURIER SERIES 146 24.12. fxxx x()sin=<< <<⎧⎨⎩0 02π ππ Fig. 24-6 11 222 134 356 57 ππ+− +++⎛ ⎝⎜ sincos cos cosxxxx iii/midhorizellipsis⎞ ⎞ ⎠⎟ 24.13. fxxx xx()cos cos=<< −− < <⎧⎨⎩0 0π π Fig. 24-7 82 1324 3536 57 πsin sin sinxxx ii i/midhorizellipsis +++⎛ ⎝⎜⎞ ⎠⎟ 24.14. fx x x() ,=− < <2ππ Fig. 24-8 π2 22 23412 23 3−− + −⎛ ⎝⎜⎞ ⎠⎟cos cos cosxxx/midhorizellipsis 24.15. fx x x x() ( ) ,=− < <ππ 0 Fig. 24-9 π2 22 262 14 26 3−+ + +⎛ ⎝⎜⎞ ⎠⎟cos cos cos xxx/midhorizellipsis 24.16. fx x x x x( ) () () ,=− + − < <ππ π π Fig. 24-10 1212 23 333 3sin sin sinxxx−+−⎛ ⎝⎜⎞ ⎠⎟/midhorizellipsisFOURIER SERIES 147 Miscellaneous Fourier Series24.17. fxx xx()=<<− −<<++<<⎧ ⎨⎪ ⎩⎪00 1 02 πα πα πα πα π Fig. 24-11 α ππαα α−−⎛ ⎝⎜ +−2 122 2 33 3sin cos sin cos sin cosxx x/midhorizellipsis⎞ ⎞ ⎠⎟ 24.18. fxxx x xx x()() ()=−< < −− − < <⎧⎨⎩ππ ππ0 0 Fig. 24-12 8 13 35 533 3πsin sin sinxxx+++⎛ ⎝⎜⎞ ⎠⎟/midhorizellipsis 24.19. fx x x x() s i n , , sin sin=− < < ≠ −μπ π μ μπ π μinteger 2 122 − −−+−−⎛ ⎝⎜⎞ ⎠⎟22 233 32222sin sinxx μμ/midhorizellipsis 24.20. fx x x() c o s , , sin cos=− < < ≠ +μπ π μ μμ π π μinteger 21 22xxx x 12 23 322 2222−−−+−−⎛ ⎝⎜⎞ ⎠⎟μμμcos cos/midhorizellipsis 24.21. fx a x a x x a a( ) tan [( sin ) / ( cos )], , | | =− − << <−111 ππ ssin sin sinxaxax +++23 2233/midhorizellipsis 24.22. fx a x a x a axa() l n ( c o s ) , , | | cos=− + − < < < −+12 1 22 2ππ 2 22333 cos cos xax ++⎛ ⎝⎜⎞ ⎠⎟/midhorizellipsis 24.23. fx a x a x a a( ) tan [( sin ) / ( )], , | |=− −<< <− 1 221 112ππ ssin sin sinxaxax +++35 3355/midhorizellipsis 24.24. fx a x a x a a() t a n [ ( c o s ) / ( ) ] , , | | =− −<< <− 1 221 112ππ ccos cos cosxaxax −+−35 3355/midhorizellipsisFOURIER SERIES 148 24.25. fx e x nx nx n() , sinh ( ) ( cos si=− < < +−−μππ μπ πμμ 21 21 nn)nx nnμ22 1+⎛ ⎝⎜⎞ ⎠⎟ =∞ ∑ 24.26. fx x x xx() s i n h , sinh sin sin=− < < +−μπ π μπ π μ2 122 222 2222233 3 +++−⎛ ⎝⎜⎞ ⎠⎟μμsinx/midhorizellipsis 24.27. fx x x x() c o s h , sinh cos c=− < < −++μπ π μμ π π μμ21 2122 2oos cos2 23 32222xx +−++⎛ ⎝⎜⎞ ⎠⎟μμ/midhorizellipsis 24.28. fx x x xx( ) ln | sin |, lncos cos cos=< < −+ + +1 2 0 212 23π x x 3+⎛ ⎝⎜⎞ ⎠⎟/midhorizellipsis 24.29. fx x x xx() l n | c o s | , lncos cos cos=− < < −− + −1 2 212 2ππ 3 3 3x+⎛ ⎝⎜⎞ ⎠⎟/midhorizellipsis 24.30. fx x x x xx() , cos cos co=−+ ++1 62 1 21 42 2202 12 2ππ π /H11017/H11017 ss3 32x+/midhorizellipsis 24.31. fx x x x x xx() ( ) ( ) , sin sin s=− − ++1 12 3320 2 12 2ππ π /H11017/H11017 iin 3 33x+/midhorizellipsis 24.32. fx x x x x x() , cos=− + −1 904 1 1222 1 123 1 48402 1ππ π π /H11017/H11017 444 42 23 3+++cos cos xx/midhorizellipsisFOURIER SERIES 149Section VII: Special Functions and Polynomials 25 THE GAMMA FUNCTION Definition of the Gamma Function /H9003(n) for n >0 25.1. Γ()nt e d t nnt=>−−∞∫1 00 Recursion Formula 25.2. ΓΓ() ( )nn n+=1 If n = 0, 1, 2, …, a nonnegative integer, we have the following (where 0! = 1): 25.3. Γ() !nn+=1 The Gamma Function for n <0 For n < 0 the gamma function can be defined by using 25.2, that is, 25.4. ΓΓ()()nn n=+1 Graph of the Gamma Function −1−11 123452345 −2−2−5 −4 −3−3 −4 −5nΓ(n) Fig. 25-1 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. Special Values for the Gamma Function 25.5. Γ()1 2=π 25.6. Γ()(), , , ... mmmm +=⋅⋅⋅ −=1 2135 2 1 2123iiπ 25.7. Γ()() (), , , ... −+ =− ⋅⋅⋅ −= mmmmm 1 212 135 2 1123π ii Relationships Among Gamma Functions 25.8. ΓΓ()( )sinppp1−=π π 25.9. 2221 1 2xxx x−+= ΓΓ Γ()( ) ( ) π This is called the duplication formula. 25.10. ΓΓ Γ Γ()xxmxmxm m+⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟⋅⋅⋅ +− ⎛ ⎝⎜⎞ ⎠⎟=12 1mmm xmx m 12 1 22/( ) /() ()−−π Γ For m = 2 this reduces to 25.9. Other Definitions of the Gamma Function 25.11. Γ( ) lim() ( ) ( )xk xx x kk kx+=⋅⋅⋅ + + ⋅⋅⋅ + →∞1123 12ii 25.12. 11 1 Γ()/ xxex mexx m m=+⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨⎪ ⎩⎪⎫ ⎬⎪ ⎭⎪− =∞ ∏γ This is an infinite product representation for the gamma function where g is Euler’s constant defined in 1.3, page 3. Derivatives of the Gamma Function 25.13. ′== −−∞∫Γ() l n1 0ex d xxγ 25.14. ′=− + −⎛ ⎝⎜⎞ ⎠⎟+−+⎛ ⎝⎜⎞ ⎠⎟+⋅⋅⋅+Γ Γ() ()x xx xγ1 111 21 1111 1 nx n−+−⎛ ⎝⎜⎞ ⎠⎟+⋅⋅⋅ Here again is Euler’s constant g.150 THE GAMMA FUNCTION Asymptotic Expansions for the Gamma Function 25.15. Γ(),xx x exx xxx+= + + − + ⋅ ⋅−12 11 121 288139 51 84023π ⋅⋅⎧ ⎨ ⎩⎫ ⎬ ⎭ This is called Stirling’s asymptotic series. If we let x = n a positive integer in 25.15, then a useful approximation for n ! where n is large (e.g., n > 10) is given by Stirling’s formula 25.16. nn n enn!~ 2 π− where ~ is used to indicate that the ratio of the terms on each side approaches 1 as n → ∞. Miscellaneous Results 25.17. |() |sinhΓixxx2=π π151 THE GAMMA FUNCTION 26 THE BETA FUNCTION Definition of the Beta Function B(m, n) 26.1. Bmn t t d t m nmn(, ) ( ) , =− > >−−∫1 01110 0 Relationship of Beta Function to Gamma Function 26.2. Bm nmn mn(,)()( ) ()=+ΓΓ Γ Extensions of B(m, n) to m < 0, n < 0 are provided by using 25.4. Some Important Results 26.3. Bm n Bn m(,) ( , ) = 26.4. Bm n dmn(,) s i n c o s/=−−∫221 2 1 02θθ θπ 26.5. Bm nt tdtm mn (,)()=+− +∞∫1 01 26.6. Bm n r rtt rtdtnmmn mn (,) ( )() ()=+− +−− + ∫1111 01 152 27 BESSEL FUNCTIONS Bessel’s Differential Equation 27.1. xy x y x n y nn22 200 +′+− =() /H11084 Solutions of this equation are called Bessel functions of order n. Bessel Functions of the First Kind of Order n 27.2. Jxx nx nx nnnn n()( )() () (=+−++++ 21122 2 2 42 2 224 Γ i 4 4 12 12 0) () ( / ) !( )−⋅⋅⋅⎧ ⎨ ⎩⎫ ⎬ ⎭ =− +++ =∞ ∑kn k kx kn kΓ 27.3. Jxx nx nx nnn n −− −=−−−+−()() () () ( 21122 2 2 42 224 Γ i 442 12 12−−⋅⋅⋅⎧⎨⎩⎫⎬⎭ =− +−− =n x kk nkk n k) () ( ) !( )/ Γ0 0∞ ∑ 27.4. Jx J x nnn n −=− = () ( ) () ,, , . . . 10 12 If nJ xn≠012, , , ..., ( ) and J–n (x) are linearly independent. If nJ xn≠012, , , ..., ( ) is bounded at x = 0 while J–n (x) is unbounded. For n = 0, 1 we have 27.5. Jxxx x 02 24 226 22 2 122 42 4 6()= − + − +⋅⋅⋅ii i 27.6. Jxxx x x 13 25 2227 2222 24 246 2468()= − + − +⋅⋅⋅ii i i i i 27.7. Jx Jx0/H11032()=− ()1 Bessel Functions of the Second Kind of Order n 27.8. YxJx n J x nn nnn p()() c o s () sin, , , ... lim=−≠−π π012 → →−− =⎧ ⎨⎪ ⎪ ⎩⎪ ⎪nppJx p J x pn() c o s () sin, , , ...π π012 This is also called Weber’s function or Neumann’s function [also denoted by Nn(x)]. 153 For n = 0, 1, 2, …, L’ Hospital’s rule yields 27.9. Yx x Jxnk knn kn () { l n ( ) } ()() ! !=+ −−− =−2211 01 πγπ/ ∑ ∑ ∑− =∞ −− + +() () { ( ) ( ) }()x kn kxkn k k/ /2 1122 02 πΦΦkkn kn k+ +!( )! where g = .5772156 … is Euler’s constant (see 1.20) and 27.10. ΦΦ() , ()pP= + + +⋅⋅⋅+ = 11 21 3100 For n = 0, 27.11. Yx x Jxxx 002 24 22222 22 411 2() { l n ( ) } ()=+ + − + (πγπ/ ) )++ + ( )−⋅⋅⋅⎧⎨⎩⎫⎬⎭x6 22224611 21 3 27.12. Yx Y x nnn n −=− = () ( ) () ,, , . . . 10 12 For any value nJ xn /H110840, ( ) is bounded at x = 0 while Yn(x) is unbounded. General Solution of Bessel’s Differential Equation 27.13. yA J x B Jx nnn=+ ≠−() () ,, , . . .012 27.14. yA J x B Y x nnn =+ () () a l l 27.15. yA J x B J xdx xJ xnnn n=+ ∫() ()()2 all where A and B are arbitrary constants. Generating Function for Jn(x) 27.16. eJ x txt t nn n()()− =−∞∞ =∑12// Recurrence Formulas for Bessel Functions 27.17. Jxn xJx J xnn n+− =−112() () () 27.18. ′=−−+Jx J x J xnn n() { () () }1 2 11 27.19. xJ x xJ x nJ xnn n′=−− () () ()1 27.20. xJ x nJ x xJ xnn n′=−+ () () ()1154 BESSEL FUNCTIONS 27.21. d dxxJ x xJ xn nn n {( ) } ( ) =−1 27.22.d dxxJx xJ xn nn n {( ) } ( )−− + =−1 The functions Yn(x) satisfy identical relations. Bessel Functions of Order Equal to Half an Odd Integer In this case the functions are expressible in terms of sines and cosines. 27.23. Jxxx122 /() s i n=π 27.26. Jxxx xx−=+⎛ ⎝⎜⎞ ⎠⎟ 322 /()cossinπ 27.24. Jxxx− =122 /() c o sπ 27.27. Jxxxxxx52 22313 /() s i n c o s =−⎛ ⎝⎜⎞ ⎠⎟−⎧ ⎨⎪ ⎩⎪⎫ ⎬⎪ ⎭⎪ π 27.25. Jxxx xx322 /()sincos =−⎛ ⎝⎜⎞ ⎠⎟π 27.28. Jxxxxxx−=+ −⎛ ⎝⎜⎞ ⎠⎟⎧ ⎨⎪ ⎩⎪⎫ ⎬⎪ ⎭⎪5223 31/() s i n c o sπ For further results use the recurrence formula. Results for Yx Yx12 32//( ), ( ), ... are obtained from 27.8. Hankel Functions of First and Second Kinds of Order n 27.29. Hx J x i Y xnnn()() () ()1=+ 27.30. HxJ x i Y xnn n()() () ()2=− Bessel’s Modified Differential Equation 27.31. xy x y x n y n22 200 ′′+′−+ =() /H11084 Solutions of this equation are called modified Bessel functions of order n. Modified Bessel Functions of the First Kind of Order n 27.32. I x i J ix e J ix x nxnn nni n n n( ) () () ()== =++−− π/2 211Γ224 22 2 2 42 2 2 42 () () ()() nx nnx +++++⋅⋅⋅⎧⎨⎩⎫⎬⎭=i/nnk kkn k+ =∞ ++ ∑2 01 !( )Γ 27.33. I x i J ix e J ix x nnn nni n n n−− − − −== =−( ) () () ()π/2 21Γ1 122 2 2 42 2 4 224 +−+−−+⋅⋅⋅⎧ ⎨ ⎩⎫ ⎬ ⎭=x nx nnx () () ()( i//2 12 0) !( )kn kkk n− =∞ +−∑Γ 27.34. Ix I xnnn− ==() () ,, , . . . 012155 BESSEL FUNCTIONS If n≠012, , , ..., then In(x) and I–n(x) are linearly independent. For n = 0, 1, we have 27.35. Ixxx x 02 24 226 22212 24 246()= + + + +⋅⋅⋅ii i 27.36. Ixxx x x 13 25 227 22 2224 2 46 246 8()= + + + +⋅⋅⋅ii i i i i 27.37. ′= Ix Ix01() () Modified Bessel Functions of the Second Kind of Order n 27.38. KxnIx I x n nnn p()sin{ ( ) ( )} , , , ... lim=−≠− →π π 2012 n npppIxI x nπ π 2012sin{ ( ) ( )} , , , ...−−=⎧ ⎨⎪⎪ ⎩⎪ ⎪ For n = 0, 1, 2, …, L’ Hospital’s rule yields 27.39. Kx x Ix n knn nk() ( ) { l n ( ) ) () ( )(=− + + − − −+121 2111/ γ ))!( ) () ( ) !( )!x x kn kkn kn nn k/ /2 1 222 01 2− =− +∑ +− +{{( ) ( ) }ΦΦkn k k++ =∞ ∑ 0 where Φ(p) is given by 27.10. For n = 0, 27.40. Kx x Ixxx 002 24 22222 411 2() { l n ( ) } ()=− + + + +⎛ ⎝⎜/ γi⎞ ⎞ ⎠⎟++ +⎛ ⎝⎜⎞ ⎠⎟+⋅⋅⋅x6 22224611 21 3 ii 27.41. Kx K x nnn−==() () ,, , . . . 012 General Solution of Bessel’s Modified Equation 27.42. yA I x B Ix nnn =+ ≠− () () ,, , . . .012 27.43. yA I x B K xnn =+ () () a l l n 27.44. yA I x B I xdx xI xnn n=+ ∫() ()()2 alln where A and B are arbitrary constants. Generating Function for In(x) 27.45. eI x txt t nn n(/ )()+ =−∞∞ =∑12/BESSEL FUNCTIONS 156 157 Recurrence Formulas for Modified Bessel Functions 27.46. IxIxn xIxnn n+− =−112() () () 27.52. KxKxn xKxnn n+− =+112() () () 27.47. ′=+−+ I x IxIxnn n() { () () }1 2 11 27.53. ′=− +−+ K x KxKxnn n() { () () }1 2 11 27.48. xI x xI x nI xnn n′=−−() () ()1 27.54. xK x xK x nK xnn n′=− −− () () ()1 27.49. xI x xI x nI xnn n′=++ () () ()1 27.55. xK x nK x xK xnn n′=−+ () () ()1 27.50. d dxxI x xI xn nn n {( ) } ( ) =−1 27.56. d dxxK x xK xn nn n {( ) } ( ) =−−1 27.51.d dxxIx xI xn nn n {( ) } ( )−− + =1 27.57.d dxxKx xK xn nn n {( ) } ( )−− + =−1 Modified Bessel Functions of Order Equal to Half an Odd Integer In this case the functions are expressible in terms of hyperbolic sines and cosines. 27.58. Ixxx122 /() s i n h=π 27.61. Ixxxx x− =−⎛ ⎝⎜⎞ ⎠⎟ 322 /() s i n hcosh π 27.59. Ixxx− =122 /() c o s hπ 27.62. Ixx xxxx52 22313 /( ) sinh cosh =+⎛ ⎝⎜⎞ ⎠⎟ −⎧⎨⎩⎫⎬⎭π 27.60. Ixxxx x322 /() c o s hsinh=−⎛ ⎝⎜⎞ ⎠⎟π 27.63. Ixx xxxx− =+⎛ ⎝⎜⎞ ⎠⎟ −⎧⎨⎩⎫⎬⎭52 22313 /( ) cosh sinhπ For further results use the recurrence formula 27.46. Results for K1/2(x), K3/2(x), … are obtained from 27.38. Berand Bei Functions The real and imaginary parts of Jx eni()/34πare denoted by Bern(x) and Bein(x) where 27.64. Ber/ nkn kxx kn knk()() !( )cos()=++++ =∞ ∑2 1322 0Γπ 4 4 27.65. Bei/ nkn kxx kn knk()() !( )sin()=++++ =∞ ∑2 1322 0Γπ 4 4 If n = 0. 27.66. Ber/ 2!/ 2 ()() () !xxx= − + −⋅⋅⋅122 448 2 27.67. Bei // 3!/ 2 () ( )() () !xxxx= − + −⋅⋅⋅ 222 5261 0 2BESSEL FUNCTIONS 158 Ker and Kei Functions The real and imaginary parts of eK x eni ni −ππ//24() are denoted by Kern(x) and Kein(x) where 27.68. Ker / Ber Beinn n xx x x() { l n ( ) } () ()=− + + 21 4 γπ +−− + +− =− ∑1 212 3 2 4 12 01() ! ( ) !cos() nk x knkkn kn/ π 2 2232 0() !( )!{( ) ( ) } c o s( x kn kkn knnk k/+ =∞ +++ ∑ ΦΦ+ +2 4k)π 27.69. Kei / Bei Bernn n xx x x() { l n ( ) } () ()=− + − 21 4 γπ −−− + +− =− ∑1 212 3 2 4 12 01() ! ( ) !sin() nk x knkkn kn/ π 2 2232 0() !( )!{( ) ( ) } s i n( x kn kkn knnk k/+ =∞ +++ ∑ ΦΦ+ +2 4k)π and Φ is given by 27.10. If n = 0, 27.70. Ker / Ber Bei/() { l n ( ) } () ()()xx x xx=− + + + − 24124 γπ 2 212121 28 21 21 31 4!()() !() + + + + + −⋅⋅⋅x/ 4 27.71. Kei / Bei Ber( ) / () { l n ( ) } () ( )(xx x x x=− + − + − 2422γπ x x/ 3!2216 1 21 3)()+ + +⋅⋅⋅ Differential Equation For Ber, Bei, Ker, Kei Functions 27.72. xy x y i x n y22 20 ′′+′−+ =() The general solution of this equation is 27.73. yA B e r x i x B x i xnn nn =+ ++{ ( ) ( )} { ( ) ( )} Bei Ker Kei Graphs of Bessel Functions Fig. 27-1 Fig. 27-2BESSEL FUNCTIONS 159 Fig. 27-3 Fig. 27-4 Fig. 27-5 Fig. 27-6 Indefinite Integrals Involving Bessel Functions 27.74. xJ x dx xJ x01() () = ∫ 27.75. xJ xd x xJ x x J x J xd x2 02 10 0 () () () () =+ − ∫ ∫ 27.76. xJ x d x xJx m x J x m xmm m m 011 02211 () () ( ) () ( ) =+ − − −−−JJx d x0() ∫ ∫ 27.77. Jx xdx J xJx xJx d x0 2 10 0()()()() =− − ∫∫ 27.78. Jx xdxJx mxJx mxmm m0 1 220 1111 () () ()() () (=−−−−−−m mJx xdxm−−∫ ∫ 120 2)() 27.79. Jx d x Jx10() () =− ∫ 27.80. x Jx d x x Jx Jx d x10 0() () () =− + ∫ ∫ 27.81. x Jx d x x Jx m x Jx d xmm m 101 0 () () () =− +−∫ ∫BESSEL FUNCTIONS 160 27.82. Jx xdx J x J x dx1 10()() () =− + ∫ ∫ 27.83. Jx xdxJx mx mJx xdxmm m11 10 11 () () ()=− +−− ∫ ∫ 27.84. xJ xd x xJ xn nn n − = ∫ 1() () 27.85. xJ x d x xJxn nn n− +−=− ∫ 1() () 27.86. x J xd x x J x m n x J xd xm nm nm n () () ( ) () =− + + −−− − ∫ ∫ 11 1 1 27.87. xJ x J x dxx J xJ x J xJ nnnn n() (){( ) ( ) ( )αβαβ α βα∫=′ − ′n nx() }β βα22− 27.88. xJ x dxxJxxn xnn22 222 22221 () { () }ααα= ′ +−⎛ ⎝⎜⎞ ⎠⎟ ∫{{( ) }Jxnα2 The above results also hold if we replace Jn(x) by Yn(x) or, more generally, AJn(x) + BYn(x) where A and B are constants. Definite Integrals Involving Bessel Functions 27.89. e J bx dx abax−∞= +∫ 022 01() 27.90. e J bx dxaba ba bnax nn n−∞=+− +>− ∫()()22 22 01 27.91. cos ( )ax J bx dx abab ab022 01 0= −> <⎧ ⎨⎪ ⎩⎪∞∫ 27.92. Jb x d xbnn() , 011∞∫=> − 27.93. Jb x xdxnnn(), , , , ... 01123∞∫== 27.94. eJ b x d xe aaxba −∞− ∫=0042 ()/ 27.95. xJ x J x dxJJ JJ nnnn nn() ()() () () (αβαβ αβα β 01∫=′− ′) ) β22−α 27.96. xJ x dx J n Jnn n2 1 22 1 222 011 ( ) {( ) } ( ) {( )αα α α = ′ +− ∫/} }2 27.97. xJ x I x dxJI JI 000100 00() ()() () () (αββα βαα β∫=′− ′ ) ) αβ22+BESSEL FUNCTIONS 161 Integral Representations for Bessel Functions 27.98. Jx x d001() c o s (s i n)=∫πθθπ 27.99. Jx n x d nn() c o s ( s i n)=− =∫1 0πθθ θπinteger 27.100. Jxx nxd nnn nn()()cos( sin )cos , =+>−21 22 1 20 πθθ θΓπ π∫ 27.101. Yx x u d u002( ) cos( cosh )=−∞∫π 27.102. Ix x d e dx 0002 11 2() c o s h (s i n)sin==∫∫πθθπθπθπ Asymptotic Expansions 27.103. Jxxxn n() ~ c o s2 24 πππ−−⎛ ⎝⎜⎞ ⎠⎟ where x is large 27.104. Yxxxn n() ~ s i n2 24 πππ−−⎛ ⎝⎜⎞ ⎠⎟ where x is large 27.105. Jxnex nnn () ~1 2 2 π⎛ ⎝⎜⎞ ⎠⎟ where n is large 27.106. Yxnex nnn () ~ −⎛ ⎝⎜⎞ ⎠⎟−2 2 π where n is large 27.107 Ixe xnx () ~2π where x is large 27.108 Kxe xnx () ~− 2π where x is large Orthogonal Series of Bessel Functions Let λλλ123, , , ... be the positive roots of RJ x SxJ x nnn() () , .+ ′=> −01 Then the following series expansions hold under the conditions indicated. SR=≠0012 , , , , ,... i.e.,3 λλλ are positive roots of JN(x)=0 27.109. fx A J x A J x A J xnnn() ( ) ( ) ( )= + + +⋅⋅⋅11 2 2 33λλλ where 27.110. AJxf x J x dxk nknk = +∫2 1201 ()() ( )λλ In psarticular if n = 0, 27.111. fx A J x A J x A J x() ( ) ( ) ( )= + + +⋅⋅⋅10 1 20 2 30 3λλλ where 27.112. AJxf x J x dxk kk = ∫2 12 001 ()() ( )λλBESSEL FUNCTIONS 162 R/S>−n 27.113. fx A J x A J x A J xnnn () ( ) ( ) ( )= + + +⋅⋅⋅11 2 2 33λλλ where 27.114. AJJ Jxf x J x dxk n k nk nknk =−−+2 2 110 () () ()() ( )λλ λλ1 1∫ In particular if n = 0. 27.115. fx A J x A J x A J x() ( ) ( ) ( )= + + +⋅⋅⋅10 1 20 2 30 3λλλ where 27.116. AJJxf x J x dxk kkk =+ ∫2 02 12 001 () ()() ( )λλλ The next formulas refer to the expansion of Bessel functions where S≠0. R/S=−n 27.117. fx A x A J x A J xn nn () ( ) ( ) = + + +⋅⋅⋅01 1 2 2 λλ where 27.118.An x f x d x AJJ Jn k nk n k01 01 2 121 2=+ =−+ −∫() ( ) () ()λλnnknk xf x J x dx +∫⎧ ⎨⎪⎪ ⎩⎪ ⎪ 101 ()() ( )λλ In particular if n = 0 so that R = 0 [i.e., l1, l2, l3, … are the positive roots of J1 (x) = 0], 27.119. fx A A J x A J x() ( ) ( )= + + +⋅⋅⋅01 0 1 2 0 2 λλ where 27.120.Ax f x d x AJxf x J x dxk kk001 02 0012 2= =⎧∫ ∫() ()() ( )λλ⎨ ⎨⎪⎪ ⎩⎪ ⎪ R/S<−N In this case there are two pure imaginary roots ± il0 as well as the positive roots l1, l2, l3, … and we have 27.121. fx A I x A J x A J xnnn () ( ) ( ) ( )= + + +⋅⋅⋅00 11 2 2λλλ where 27.122.AII Ixf x I x dx nn nn 0 2 01 0 1 0002=+−+ () () ()() ( )λλ λλ1 1 2 112∫ =−−+AJJ Jxf x J xk n k nk nknk() () ()() ( )λλ λλddx 01∫⎧ ⎨⎪⎪ ⎩⎪ ⎪BESSEL FUNCTIONS 163 Miscellaneous Results 27.123. c o s (s i n ) () () c o s () c o sxJ x J x J xθθ θ=+ + + ⋅ ⋅02 422 24 ⋅⋅ 27.124. sin ( sin ) ( ) sin ( ) sin ( ) sinxJ xJ x J xθθθ=+ +22 3 213 55 5θ+⋅⋅⋅ 27.125. Jx y Jx J y nnk knk( ) ( ) ( ) , , , ...+= =± ± =−∞∞ − ∑ 012 This is called the addition formula for Bessel functions. 27.126. 12 202 2= + +⋅⋅⋅+ +⋅⋅⋅Jx Jx J xn () () () 27.127. x J xJ xJ x n Jxn = +++ ⋅ ⋅ ⋅ + + ++ 23 5 2 1135 2 1 { ( )( )( ) () ( ) ⋅ ⋅⋅⋅} 27.128. xJ x J x J x n J xn2 2462 2 2 4 16 36 2 = + + +⋅⋅⋅+{( ) ( ) ( ) ( ) ( ) ++⋅⋅⋅} 27.129.xJ xJx Jx Jx1 246 423()() () ()= − + −⋅⋅⋅ 27.130. 1 22202 12 22 32= ++++ ⋅ ⋅ ⋅Jx Jx Jx Jx() () () () 27.131. ′′=− +−+ Jx J x Jx J xnn n n() { () () () }1 4 22 2 27.132. ′′′ =− + −−− + +Jx J x J x J x Jnn n n n() { () () () (1 8 311 333 x x)} Formulas 27.131 and 27.132 can be generalized. 27.133. ′ −′ =−− Jx J x JJxn xnn n n() () ()sin2 π π 27.134. Jx J x J x J xn xnn n n() () () ()sin −+ − − +=112 π π 27.135. J x Yx Jx Y x Jx Yx Jxn n nn nn n++ −= ′−′11() () () () () () ( ))( )Yxxn=2 π 27.136. s i n { () () () }xJ x J x J x= − + −⋅⋅⋅2135 27.137. cos ( ) ( ) ( )x J xJ xJ x= − + −⋅⋅⋅024 22 27.138. s i n h { ( )( )( )}x I xI xI x= +++ ⋅ ⋅ ⋅2135 27.139. cosh ( ) { ( ) ( ) ( ) } x I x I xI xI x= + + + +⋅⋅⋅0 246 2BESSEL FUNCTIONS 28 LEGENDRE and ASSOCIATED LEGENDRE FUNCTIONS Legendre’s Differential Equation 28.1. () ( )12 1 02− ′′− ′++= xy x y n n y Solutions of this equation are called Legendre functions of order n. Legendre Polynomials If n = 0, 1, 2, …, a solution of 28.1 is the Legendre polynomial Pn(x) given by Rodrigues’ formula 28.2. Pxnd dxxn nn nn()!() =−1 212 Special Legendre Polynomials 28.3. Px0 1 ()= 28.7. Px x x41 84235 30 3 () ( )=− + 28.4. Px x1()= 28.8. Px x x x51 85363 70 15 () ( )=− + 28.5. Px x21 2231 () ( )=− 28.9. Px x x x61 16642231 315 105 5 () ( )=− + − 28.6. Px x x31 2353 () ( )=− 28.10. Px x x x x71 16753429 693 315 35 () ( )= −+− Legendre Polynomials in Terms of U where x /H11549cosU 28.11. P0 1 (cos )θ= 28.12. P1(cos ) cosθθ= 28.13. P21 413 2 (cos ) ( cos )θθ=+ 28.14. P31 835 3 (cos ) ( cos cos )θθ θ=+ 28.15. P41 6492 0 2 3 5 4 (cos ) ( cos cos )θθ θ=+ + 164 165 28.16. P51 12830 35 3 63 5 (cos ) ( cos cos cos )θ θθθ=+ + 28.17. P61 51250 105 2 126 4 231 6 (cos ) ( cos cos cosθθ θ θ=+ + + ) ) 28.18. P71 1024175 189 3 231 5 42 (cos ) ( cos cos cosθθ θ θ=+ + + 997cos )θ Generating Function for Legendre Polynomials 28.19. 1 122 0 −+= =∞ ∑tx tPx tnn n() Recurrence Formulas for Legendre Polynomials 28.20. ( ) () ( ) () ()nP x nx P x n P xnn n +− ++ =+−12 1 011 28.21. ′ −′=++Pxx P x n P xnn n1 1 () () ( ) () 28.22. xP x P x nP xnn n′−′ =− () () ()1 28.23. ′ −′ =++−PxPx n P xnn n 11 21 () () ( ) () 28.24. ( ) () () ()x P xn x P xn Pxnn n2 1 1− ′−−− Orthogonality of Legendre Polynomials 28.25. Px P x d x m nmn() () =≠ −∫0 11 28.26. {( ) }Px d xnn2 11 2 21=+ −∫ Because of 28.25, Pm(x) and Pn(x) are called orthogonal in –1 /H11017 x /H11017 1. Orthogonal Series of Legendre Polynomials 28.27. fx A Px A Px A Px() () () ()=+ ++00 1 1 22 /midhorizellipsis where 28.28. Akfx Px d xkk=+ −∫21 211() ()LEGENDRE AND ASSOCIATED LEGENDRE FUNCTIONS 166 Special Results Involving Legendre Polynomials 28.29. Pn()11= 28.30. Pnn()()−= −11 28.31. Px P xnn n () ( ) ( )−= − 1 28.32. Pn n nnnn() ()()00 1135 1 2462= −−odd even/ii/midhorizellipsis ii/midhorizellipsis⎧ ⎧ ⎨⎪ ⎩⎪ 28.33. Px x x dnn() c o s = (+− ) ∫112 0πφφπ 28.34. Px d xPxPx nnnn()() ()=− ++−∫11 21 28.35. ||Pxn()/H110171 28.36. Pxiz zxdzn nn nc()() ()=− −++∫1 21 12 1π/integralloop where C is a simple closed curve having x as interior point. General Solution of Legendre’s Equation The general solution of Legendre’s equation is 28.37. yA U x B V xnn =+ () () where 28.38. Uxnnxnn n nxn()() !() ( ) () !=−++−+ +− 11 2213 424/midhorizellipsis 28.39. Vx xnnxnnn n n()() ( ) !() () ( ) ( )=−−++−−+ + 12 313243 5 55 !x−/midhorizellipsis These series converge for –1 < x < 1. Legendre Functions of the Second Kind If n = 0, 1, 2, … one of the series 28.38, 28.39 terminates. In such cases, 28.40. PxUx U n VxV nnnn nn()() ( ) , , , () ( ) , ,== =/ /10 2 4 11 3… 55,…⎧ ⎨⎪ ⎩⎪ where 28.41. Unnnnnn() ( ) ! ! , , ,11 2202422 =−⎛ ⎝⎜⎞ ⎠⎟⎡ ⎣⎢⎤ ⎦⎥ =/…LEGENDRE AND ASSOCIATED LEGENDRE FUNCTIONS 167 28.42. Vnnnnnn() ( ) ! !()1121 212 12 =−−⎛ ⎝⎜⎞ ⎠⎟⎡ ⎣⎢⎤ ⎦⎥ =−−/1135,,,… The nonterminating series in such a case with a suitable multiplicative constant is denoted by Qn(x) and is called Legendre’s function of the second kind of order n. We define 28.43. QxUV x n VU x nnnn nn()() ( ) , , , () ( ) , ,== −=10 24 11 35… ,,…⎧ ⎨⎪ ⎩⎪ Special Legendre Functions of the Second Kind 28.44. Qxx x01 21 1() l n=+ −⎛ ⎝⎜⎞ ⎠⎟ 28.45. Qxxx x1 21 11 () l n=+ −⎛ ⎝⎜⎞ ⎠⎟− 28.46. Qxxx xx 2231 41 13 2() l n=−+ −⎛ ⎝⎜⎞ ⎠⎟− 28.47. Qxxx x xx 33253 41 15 22 3() l n=−+ −⎛ ⎝⎜⎞ ⎠⎟−+ The functions Qn(x) satisfy recurrence formulas exactly analogous to 28.20 through 28.24. Using these, the general solution of Legendre’s equation can also be written as 28.48. yA P x B Q xnn =+ () () Legendre’s Associated Differential Equation 28.49. () ( )12 11022 2 − ′′− ′++ −−⎧⎨⎩⎫⎬⎭= xy x y n nm xy Solutions of this equation are called associated Legendre functions. We restrict ourselves to the important case where m, n are nonnegative integers. Associated Legendre Functions of the First Kind 28.50. Px xd dxPxx nd nmmm m nm nm () ( ) ()() !=− =−+ 11 22222 // n n mnn dxx+ −()21 where Pn(x) are Legendre polynomials (page 164). We have 28.51. Px P xnn0() ()= 28.52. Px mnnm()=>0i fLEGENDRE AND ASSOCIATED LEGENDRE FUNCTIONS 168 Special Associated Legendre Functions of the First Kind 28.53. Px x112 121 () ( )=−/ 28.56. Px x x31 3 222 1251 1 () ( ) ( )=− −/ 28.54. Px x x212 1 231 () ( )=−/ 28.57. Px x x32215 1 () ( )=− 28.55. Px x22231 () ( )=− 28.58. Px x332 3215 1 () ( )=−/ Generating Function for Pxnm() 28.59. () ! ( ) !( )()21 21 222 21 2mx t mt x tPx tmm mm nm − −+=+/ /n n nm=∞ ∑ Recurrence Formulas 28.60. ( ) () ( ) () ( ) ()nm P xn x P x n m P xnm nm nm+− − + + ++− 12 111 = =0 28.61. Pxmx xP x nm nmnm nm ++−+ −+− +2 21 21 21 1()() ()() ( ) (/ ++=10)( )Pxnm Orthogonality of Pxnm() 28.62. Px Px d x nllmm() ()1110=≠ −∫if 28.63. Pxd xnnm nmnm()() ! () !{} =++ − −∫2 11 2 21 Orthogonal Series 28.64. fx AP x A P x A P xmmm mmm mmm() () () ()=+ + +++ + +11 2 2 /midhorizellipsis where 28.65. Akk m kmfx P x d xkkm=+− + −∫21 2 11 () ! () !() () Associated Legendre Functions of the Second Kind 28.66. Qx xd dxQxnmmm m n () ( ) ()=−122 / where Qn(x) are Legendre functions of the second kind (page 166). These functions are unbounded at x = ±1, whereas Pxnm()are bounded at x = ± 1. The functions Qxnm()satisfy the same recurrence relations as Pxnm()(see 28.60 and 28.61). General Solution of Legendre’s Associated Equation 28.67. yA P x B Q xnm nm=+ () ()LEGENDRE AND ASSOCIATED LEGENDRE FUNCTIONS 29 HERMITE POLYNOMIALS Hermite’s Differential Equation 29.1. ′′− ′+= yx y n y22 0 Hermite Polynomials If n = 0, 1, 2, …, then a solution of Hermite’s equation is the Hermite polynomial Hn(x) given by Rodrigue’s formula. 29.2. Hx ed dxennxn nx() ( ) (=−−122) Special Hermite Polynomials 29.3. Hx0 1 ()= 29.7. Hx x x44216 48 12 ()=−+ 29.4. Hx x1 2 ()= 29.8. H x xxx55332 160 120 ()=− + 29.5. Hx x2242 ()=− 29.9. Hx x x x664264 480 720 120 ()=− + − 29.6. Hx x x3381 2 ()=− 29.10. Hx x x x x7753128 1344 3360 1680 ()=− + − Generating Function 29.11. eHx t ntx t nn n2 02− =∞ =∑() ! Recurrence Formulas 29.12. Hx x H x n Hxnn n+− =−11 22 () () () 29.13. ′=− Hx n H xnn() () 21 Orthogonality of Hermite Polynomials 29.14. eHx H x d x mnx mn− −∞∞∫=≠20 () () 29.15. eH x d x nx nn − −∞∞∫=222 {( ) } ! π 169 170 Orthogonal Series 29.16. fx A H x A H x A H x() () () ()=+ ++00 1 1 22 /midhorizellipsis where 29.17. Akef x H x d xk kx k =− −∞∞∫1 22 !() ()π Special Results 29.18. Hx xnnxnn n n nnn() ( )() !()() ( ) (=−−+−− −−21 121232 ) ) !()224xn−−/midhorizellipsis 29.19. Hx H xnn n () ( ) ( )−= − 1 29.20. Hn2100− =() 29.21. Hnnnn 201 2 1 3 5 2 1() ( ) ( )=− − iii/midhorizellipsis 29.22. Ht d tHx nH nnnnx()() ()() ()=+−+++∫11 0 210 21 29.23. d dxeH x eH xx nx n {( ) } ( )−− + =−22 1 29.24. eH t d t H eH xt nnx nx− −− − =− ∫22 1100 () ( ) ( ) 29.25. te H x td t nP xnt nn− −∞∞∫=2() ! ( ) π 29.26. Hx yn kHx H yn n kn kn k () ( )( )+=⎛ ⎝⎜⎞ ⎠⎟ =− ∑1 2222 0/ This is called the addition formula for Hermite polynomials. 29.27. Hx Hy kHx H yH x Hykk knn n n n() () !() () () () 2211=−++ +1 1 0nx ykn !( )−=∑HERMITE POLYNOMIALS 30 LAGUERRE and ASSOCIATED LAGUERRE POLYNOMIALS Laguerre’s Differential Equation 30.1. xy x y ny′′+− ′+= ()10 Laguerre Polynomials If n = 0, 1, 2, …, then a solution of Laguerre’s equation is the Laguerre polynomial Ln(x) given by Rodrigues’ formula 30.2. Lx ed dxxenxn nnx() ( )=− Special Laguerre Polynomials 30.3. Lx0 1 ()= 30.4. Lx x1 1 ()=− + 30.5. Lx x x2242 ()=−+ 30.6. Lx x x x33291 8 6 ()=− + − + 30.7. Lx x x x x443 216 72 96 24 ()=− + − + 30.8. Lx x x x x x554 3 225 200 600 600 120 ()=− + − + − + 30.9. Lx x x x x x x665 4 3 236 450 2400 5400 4320 720 ()=− + − + − + 30.10. Lx x x x x x776 5 4 349 882 7350 29 400 52 920 () , ,=− + − + − + xxx235 280 5040−+, Generating Function 30.11. e tLx t nxt t nn n−− =∞ −=∑/( )() !1 01 171 172 Recurrence Formulas 30.12. Lx n x L xn Lxnn n +− −+ − + =12 1 21 0 () ( ) () () 30.13. ′− ′ +=−−Lx n L x n L xnn n() () ()110 30.14. xL x nL x n L xnn n′=−− () () ()2 1 Orthogonality of Laguerre Polynomials 30.15. eLx Lx d x m nx mn−∞=≠ ∫() () 0 0 30.16. eL xd x nx n−∞∫= 022{( ) } ( ! ) Orthogonal Series 30.17. fx A L x A Lx A L x() () () ()=+ ++00 1 1 22 /midhorizellipsis where 30.18. Akef x L x d xkx k =−∞∫1 20 (! )() () Special Results 30.19. Lnn() !0= 30.20. Lt d t LxLx nnnnx() ( )()=−++∫1 0 1 30.21. Lx xnx n n x nnnnn () ( )!() !() =− − +−−−−− 111 2121 2 2 2 /midhorizellipsisn nn!⎧⎨⎩⎫⎬⎭ 30.22. xe L xd xpn np npx nn−∞=< −=⎧ ⎨⎪ ⎩⎪() () ( ! )0 12 0if if∫ ∫ 30.23.Lx Ly kLx L y L x Ly nkk nn n n() () (! )() () () () (211=−++ !!) ( )2 0xykn −=∑ 30.24.tL x keJ x tk k t k() (! )()2 0 02 = =∞ ∑ 30.25. Lx u e J x u d unnx u() ( )=−∞∫ 002LAGUERRE AND ASSOCIATED LAGUERRE POLYNOMIALS 173 Laguerre’s Associated Differential Equation 30.26. xy m x y n m y′′++ − ′+− = () ( ) 10 Associated Laguerre Polynomials Solutions of 30.26 for nonnegative integers m and n are given by the associated Laguerre polynomials 30.27. Lxd dxLxnmm m n () ()= where Ln(x) are Laguerre polynomials (see page 171). 30.28. Lx Lxnn0() ()= 30.29. Lx m nnm()=>0i f Special Associated Laguerre Polynomials 30.30. Lx111 ()=− 30.35. Lx336 ()=− 30.31. Lx x2124 ()=− 30.36. Lx x x x413 24 48 144 96 ()=− + − 30.32. Lx222 ()= 30.37. Lx x x42212 96 144 ()=− + 30.33. Lx x x31231 8 1 8 ()=− + − 30.38. Lx x4324 96 ()=− 30.34. Lx x3261 8 ()=− + 30.39. Lx4424 ()= Generating Function for Lxnm() 30.40.() ()() !/( ) − −=+−− =∞ ∑1 111mm mxt t nm n nmt teLx nt Recurrence Formulas 30.41.nm nLx x mn L xn Lxnm nm nm −+ +++− − ++−1 12112 1 () ( ) () () )=0 30.42. d dxLx L xnm nm{( } ( ) )=+1 30.43. d dxxe L x m n x e L xmx nmm x nm{( ) } ( ) ( )−− − −=− − 111 30.44. xd dxLx xm Lx mn L xnm nm nm{( }( )( )( ) ( ) )=− + − −−11LAGUERRE AND ASSOCIATED LAGUERRE POLYNOMIALS 174 Orthogonality 30.45. x eLx Lx d x p nmx nm pm −∞=≠ ∫() () 0 0 30.46. xe L x d xn nmmx nm −∞=− ∫{( }(! ) () !)23 0 Orthogonal Series 30.47. fx AL x A L x A L xmmm mmm mmm() () () ()=+ + +++ + +11 2 2 /midhorizellipsis where 30.48. Akm kxe L xfx d xkmx km=−−∞∫() ! (! )()()30 Special Results 30.49. Lxn nmxnn mxnn nm n nm nm() ( )! () !() !(=−−−−+−− −111 −−− − −+ {}−− 11 22 )( )( ) !nm nmxnm/midhorizellipsis 30.50. xeL xd xnm n nmmx nm +−∞=−+ − ∫123 021{( }() ( ! ) () !)LAGUERRE AND ASSOCIATED LAGUERRE POLYNOMIALS 31CHEBYSHEV POLYNOMIALS Chebyshev’s Differential Equation 31.1. ( ) , , , ... 10 01222−− ′+= = xy x y n y nn Chebyshev Polynomials of the First Kind A solution of 31.1 is given by 31.2. Tx n x xnxxn nnn( ) cos ( cos ) ( == −⎛ ⎝⎜⎞ ⎠⎟ −+−−12 2 214)⎛ ⎛ ⎝⎜⎞ ⎠⎟ −−−xxn42 21() /midhorizellipsis Special Chebyshev Polynomials of The First Kind 31.3. Tx0 1 ()= 31.7. T xxx442881 ()=−+ 31.4. Tx x1()= 31.8. Tx x x x55316 20 5 ()=−+ 31.5. Tx x2221 ()=− 31.9. Tx x x x664 232 48 18 1 ()=−+− 31.6. Tx x x3343 ()=− 31.10. Tx x x x x775 364 112 56 7 ()=− +− Generating Function for Txn() 31.11.1 122 0− −+= =∞ ∑tx tx tTx tnn n() Special Values 31.12. Tx T xnn n () ( ) ( )−= − 1 31.14. Tnn()()−= −11 31.16. Tn2100+ =() 31.13. Tn()11= 31.15. Tnn 201() ( )=− Recursion Formula for Txn() 31.17. Tx x T xTxnn n+− −+ =11 20 () () () 175 176 Orthogonality 31.18. Tx T x xdx m nmn() () 102 11 −=≠ −∫ 31.19. {( ) } , , ...Tx xdxn nn2 2 11 10 21 2 −== =⎧⎨−∫π πif /i f⎩ ⎩ Orthogonal Series 31.20. fx A T x A Tx A T x() () () ()=+ + +1 2 00 1 1 22 /midhorizellipsis where 31.21. Afx T x xdxkk= −−∫2 12 11 π() () Chebyshev Polynomials of The Second Kind 31.22. Uxnx x nn()sin{( ) cos } sin (cos )=+ =+⎛ ⎝⎜⎞ ⎠− −1 1 11 1 ⎟ ⎟−+⎛ ⎝⎜⎞ ⎠⎟ −++⎛ ⎝⎜⎞ ⎠⎟ −−−xnxxnxnn n 1 311 5122 4() ( x x22)−/midhorizellipsis Special Chebyshev Polynomials of The Second Kind 31.23. Ux0 1 ()= 31.27. U xxx44216 12 1 ()=−+ 31.24. Ux x1 2 ()= 31.28. Ux x x x55332 32 6 ()=−+ 31.25. Ux x2241 ()=− 31.29. Ux x x x664264 80 24 1 ()=−+− 31.26. Ux x x3384 ()=− 31.30. Ux x x x x775 3128 192 80 8 ()=−+ − Generating Function for Uxn() 31.31.1 122 0−+= =∞ ∑tx tUx tnn n() Special Values 31.32. Ux U xnn n () ( ) ( )−= − 1 31.34. Unnn()() ( )−= − +11 1 31.36. Un2100+ =() 31.33. Unn()11=+ 31.35. Unn 201() ( )=−CHEBYSHEV POLYNOMIALS 177 Recursion Formula for Uxn() 31.37. Ux x U x Uxnn n+− −+ =11 20 () () () Orthogonality 31.38. 102 11−= ≠ −∫xU x U xd x m nmn() () 31.39. 122 112−= −∫xUx d xn{( } )π Orthogonal Series 31.40. fx A U x A U x A U x() () () ()=+ ++00 1 1 22 /midhorizellipsis where 31.41. Ax f x U x d xkk=− −∫212 11 π() () Relationships Between Txn() and Uxn() 31.42. Tx Ux x U xnn n() () ()=−−1 31.43. ( ) () () ()12 11 −= −−+ xU x x Tx T xnn n 31.44. UxTd xnn()() ()= −−+ −∫1 11 2 11 π/H9271/H9271 /H9271/H9271 31.45. TxU xdnn()()=− −− −∫1 12 1 11 π/H9271/H9271 /H9271/H9271 General Solution of Chebyshev’s Differential Equation 31.46. yAT x B x U x n AB xnn=+− = +− −() () , , , sin11 232 1 1if i… f fn=⎧ ⎨⎪ ⎩⎪ 0CHEBYSHEV POLYNOMIALS 32 HYPERGEOMETRIC FUNCTIONS Hypergeometric Differential Equation 32.1. x x y c a b x y abyn(){ ( ) }11 0−+ − + + ′−= Hypergeometric Functions A solution of 32.1 is given by 32.2. Fabcxab cxaa bb cc(,;;)() () ()=+ +++ +1111 12 1i ii ix xaa a bb b cc c2 12 12 123 1 2+++ ++ ++() ( ) () ( ) () ( ) iiix x3+/midhorizellipsis If a, b, c are real, then the series converges for –1 < x < 1 provided that c – (a + b) > –1. Special Cases 32.3. Fp x xp(, ; ;) ( )−− = +11 1 32.8. Fx x x(, ; ; ) ( s i n )1 21 23 221=−/ 32.4. Fx x x(,; ; ) [ l n ( ) ]112 1 −= + / 32.9. Fx x x(,; ; ) ( t a n )1 23 2211−=−/ 32.5. lim ( , ; ; ) nxFnx n e →∞= 11 / 32.10. Fp p x x(, ; ; ) ( )11 1 =−/ 32.6. Fx x(, ; ; s i n ) c o s1 21 21 22−= 32.11. Fn n x P xn (, ; ; () )( )+− − =11 1 2 / 32.7. Fx x(,;; ) s e c1 2211s i n = 32.12. Fn n x T xn (, ; ; ( ) ) ()−− =1 212 / General Solution of The Hypergeometric Equation If c, a – b and c – a – b are all nonintegers, then the general solution valid for | x| < 1 is 32.13. y A F a b c x B xF ac bc c xc= + −+ −+ −−(,;; ) ( , ; ; )111 2 178 179 Miscellaneous Properties 32.14. Fabccc a b ca cb(,;;)() ( ) () ()1=−− −−ΓΓ ΓΓ 32.15.d dxFabcxab cFa b c x (,;;) ( , ; ;) =+ + + 111 32.16. Fabcxc bc buu ubc b(,;;)() () ( )() ( =−−−−− − Γ ΓΓ1111 xxd ua)−∫01 32.17. Fabcx x Fc ac bcxcab(,;;) ( ) ( , ;;) =− − −−−1HYPERGEOMETRIC FUNCTIONS Section VIII: Laplace and Fourier Transforms 33 LAPLACE TRANSFORMS Definition of the Laplace Transform of F(t) 33.1. /H5112{( ) } ( ) ( )Ft e Ftd t fsst==−∞∫0 In general f (s) will exist for s > a where a is some constant. /H5112 is called the Laplace transform operator. Definition of the Inverse Laplace Transform of f(s) If /H5112{F(t)} = f (s), then we say that F (t) = /H5112–1{f(s)} is the inverse Laplace transform of f (s). /H5112–1 is called the inverse Laplace transform operator. Complex Inversion Formula The inverse Laplace transform of f(s) can be found directly by methods of complex variable theory. The result is 33.2. Ftief s d sief s d sst Tst ci Tc() () l i m () == →∞ −1 21 2 ππ+ + −∞+∞∫ ∫iT cici where c is chosen so that all the singular points of f (s) lie to the left of the line Re{s} = c in the complex s plane. 180 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 181 Table of General Properties of Laplace Transforms f(s) F(t) 33.3. af s b f s12() ()+ aF t bF t12() ()+ 33.4. fs a()/ aF a t() 33.5. f(s – a) eatF(t) 33.6. e–asf(s) /H5121()()taFt a t a ta−=−> < {0 33.7. sf(s) – F (0) ′Ft() 33.8. sfs s F F200 () () ()−− ′ ′′Ft() 33.9. sfs s F s F Fnn n n() () () ()()−− ′−−−− −12 100 0 /midhorizellipsis F(n)(t) 33.10. ′fs() –tF(t) 33.11. ′′fs() t2F(t) 33.12. f(n)(s) (–1)ntnF(t) 33.13.fs s()Fud ut() 0∫ 33.14.fs sn()/midhorizellipsis Fud utu nFud unnt t t()() () !() =− −− ∫ ∫ ∫1 0 0 0 1 33.15. f(s)g(s) FuGt ud ut() ( ) − ∫0LAPLACE TRANSFORMS 182 f(s) F(t) 33.16. fu d u s()∞∫Ft t() 33.17.1 1 0 −−−∫eeF u d usTsuT() F(t) = F(t + T) 33.18.fs s() 1 24 0πteF u d uut−∞∫/() 33.19.11 sfs()Ju t F u d u002∞∫() ( ) 33.20.11 1sfsn+()tu J u t F u d unn n/2 2 02−∞∫/() ( ) 33.21.fs s s()+ +1 12/Ju t u F u d ut 002(() ) ( ) − ∫ 33.22.1 232 4 02 πue f u d usu −−∞∫//() F(t2) 33.23.fs ss(ln ) lntFu uduu() ()Γ+∞∫ 1 0 33.24.Ps Qs() ()P Qek k kn tk() ()α αα ′=∑ 1 P(s) = polynomial of degree less than n, Q(s) = (s – a1)(s – a2) … (s – an) where a1, a2, …, an are all distinct.LAPLACE TRANSFORMS 183 Table of Special Laplace Transforms f(s) F(t) 33.25.1 s1 33.26.1 2st 33.27.1123snn =,,,…t nn− −=1 101() !,! 33.28.10snn >t nn−1 Γ() 33.29.1 sa−eat 33.30.1123(),,,sann−= …te nna t− −=1 101() !,! 33.31.10()sann−>te nna t−1 Γ() 33.32.1 22sa+sinat a 33.33.s sa22+cos at 33.34.1 22()sb a−+ea t abtsin 33.35.sb sb a− −+()22 ea tbtcos 33.36.1 22sa−sinhat a 33.37.s sa22−cosh at 33.38.1 22()sb a−−ea t abtsinhLAPLACE TRANSFORMS 184 f(s) F(t) 33.39.sb sb a− −−()22 ea tbtcosh 33.40.1 () ()sa sbab−−≠ee babt at−− 33.41.s sa sbab() ()−−≠be ae babt at−− 33.42.1 22 2()sa+sin cosat at at a− 23 33.43.s sa()22 2+ta t asin 2 33.44.s sa2 22 2()+sin cosat at at a+ 2 33.45.s sa3 22 2()+cos sinat at at−1 2 33.46.sa sa22 22 2− +()ta tcos 33.47.1 22 2()sa−at at at acosh sinh − 23 33.48.s sa()22 2−ta t asinh 2 33.49.s sa2 22 2()−sinh coshat at at a+ 2 33.50.s sa3 22 2()−cosh sinh at at at+1 2 33.51.s sa2 22 3 2()/−ta tcosh 33.52.1 22 3()sa+() s i n c o s33 822 5−−a t at at at a 33.53.s sa()22 3+ta t a t a t asin cos −2 38 33.54.s sa2 22 3()+() s i nc o s1 822 3+−a t at at at a 33.55.s sa3 22 3()+3 82ta t a t a t asin cos +LAPLACE TRANSFORMS 185 f(s) F(t) 33.56.s sa4 22 3()+() s i n c o s35 822−+a t at at at a 33.57.s sa5 22 3()+() c o s s i n87 822−−a t at at at 33.58.322 22 3sa sa− +()ta t a2 2sin 33.59.sa s sa32 22 33− +()1 22ta tcos 33.60.sa s a sa42 2 4 22 46−+ +()1 63ta tcos 33.61.sa s sa32 22 4− +()ta t a3 24sin 33.62.1 22 3()sa−( ) sinh cosh33 822 5+−a t at at at a 33.63.s sa()22 3−at at t at a2 38cosh sinh − 33.64.s sa2 22 3()−at at a t at acosh ( ) sinh +−22 31 8 33.65.s sa3 22 3()−3 82ta t a t a t asinh cosh + 33.66.s sa4 22 3()−( ) sinh cosh35 822++a t at at at a 33.67.s sa5 22 3()−( ) cosh sinh87 822++a t at at at 33.68.322 22 3sa sa+ −()ta t a2 2sinh 33.69.sa s sa32 22 33+ −()1 22ta tcosh 33.70.sa s a sa42 2 4 22 46++ −()1 63ta tcosh 33.71.sa s sa32 22 4+ −()ta t a3 24sinh 33.72.1 33sa+e aat ateat at/ /2 232 333 23 2sin cos −+⎧ ⎨ ⎩⎫ ⎬ ⎭−LAPLACE TRANSFORMS 186 f(s) F(t) 33.73.s sa33+e aat ateat at/ /2 32 33 233 2cos sin +−⎧ ⎨ ⎩⎫ ⎬ ⎭− 33.74.s sa2 33+1 323 22eeatat at−+⎛ ⎝⎜⎞ ⎠⎟/cos 33.75.1 33sa−e aeat atat at− −−⎧ ⎨ ⎩⎫ ⎬ ⎭/ /2 232 33 233 2cos sin 33.76.s sa33−e aat ateat at− −+⎧ ⎨ ⎩⎫ ⎬ ⎭/ /2 32 333 23 2sin cos 33.77.s sa2 33−1 323 22eeatat at+⎛ ⎝⎜⎞ ⎠⎟−/cos 33.78.1 444sa+1 43aat at at at(sin cosh cos sinh ) − 33.79.s sa444+sin sinhat at a22 33.80.s sa2 444+1 2aat at at at(sin cosh cos sinh ) + 33.81.s sa3 444+cos coshat at 33.82.1 44sa−1 23aat at (sinh sin ) − 33.83.s sa44−1 22aat at (cosh cos ) − 33.84.s sa2 44−1 2aat at (sinh sin ) + 33.85.s sa3 44−1 2(cosh cos ) at at+ 33.86.1 sa sb++ +ee ba tbt at−−− −23() π 33.87.1 ss a+erfat a 33.88.1 ss a()−ea t aaterf 33.89.1 sab−+etbe b tat b t 1 2 π−⎧⎨⎩⎫⎬⎭erfc( )LAPLACE TRANSFORMS 187 f(s) F(t) 33.90.1 22sa+Ja t0() 33.91.1 22sa−Ia t0() 33.92.()sas sann 22 221+− +>− aJ a tn n() 33.93.()ss a sann−− −>−22 221 aI a tn n() 33.94.e sabs s a()−+ +22 22Jat t b0 2 (( ) ) + 33.95.e sabs a−+ +22 22Jat b t b tb022 0() −> <⎧⎨ ⎩ 33.96.1 22 3 2()sa+/tJ at a1() 33.97.s sa()/ 22 3 2+tJ at0() 33.98.s sa2 22 3 2()+/Ja t a t Ja t01() ()− 33.99.1 22 3 2()sa−/tI at a1() 33.100.s sa()22 3 2−/ tI at0() 33.101.s sa2 22 3 2()/−Ia t a t Ia t01() ()+ 33.102.1 11 see sess s() ( )−=−− − See also entry 33.165.Ft nn t n n() , , ,, ,=< + = /H11017 10 1 2 … 33.103.1 1 se re sr ess s() ( )−=−− −Ft rk kt ()[] = =∑ 1 where [t] = greatest integer /H11017 t 33.104.e se re sr es ss s− −=− −− −11 1 () ( ) See also entry 33.167.Ft r n t n nn() , , ,, ,=< + = /H11017 10 1 2 … 33.105.e sas−/cos 2 at tπLAPLACE TRANSFORMS 188 f(s) F(t) 33.106.e sas−/ /32sin 2 at aπ 33.107.e snas n− + >−/ 1 1t aJa tn n⎛ ⎝⎞ ⎠/2 2() 33.108.e sas−e tat−24 / π 33.109. eas−a teat 2342 π−/ 33.110.1−−e sas erf /()at2 33.111.e sas− erfc /()at2 33.112.e ssbas− +()eb ta tbb t a()++⎛ ⎝⎜⎞ ⎠⎟ erfc2 33.113.e snas n− + >−/ 1 112214 2022 πtaue J ud unnua t n +−∞∫/() 33.114. lnsa sb+ +⎛ ⎝⎞ ⎠ee tbt at−−− 33.115.ln[( ) ]sa a s22 2 2+ /Ci at() 33.116.ln[( ) ]sa a s+/Ei at() 33.117.−+(l n )γ s s γ= Euler’s constant = .5772156 …lnt 33.118. lnsa sb22 22+ +⎛ ⎝⎜⎞ ⎠⎟2(cos cos ) at bt t− 33.119.πγ22 6ss s++(l n ) γ= Euler’s constant = .5772156 …ln2t 33.120.lns s−+(ln )tγ γ= Euler’s constant = .5772156 … 33.121.ln2s s(ln )t+−γπ2 1 62 γ= Euler’s constant = .5772156 …LAPLACE TRANSFORMS 189 f(s) F(t) 33.122.′+− +>−+ΓΓ() () l nnn s snn1111 ttnln 33.123. tan ( )−1as/sinat t 33.124.tan ( )−1as s/Si at() 33.125.e sasas/ erfc /()e tat−2 π 33.126. es asa2242/erfc /()2 22 aeat π− 33.127.es a ssa2242/erfc /()erf( )at 33.128.ea s saserfc 1 π()ta+ 33.129. eE i a sas()1 ta+ 33.130.1 2 aas Si as asCi ascos ( ) sin ( )π−{}−⎡ ⎣⎢⎤ ⎦⎥1 22ta+ 33.131. sin ( ) cos ( )as Si as asCi asπ 2−{}+t ta22+ 33.132.cos ( ) sin ( )as Si as asCi as sπ 2−{}−tan ( )−1ta/ 33.133.sin ( ) cos ( )as Si as asCi as sπ 2−{}− 1 222 2 lnta a+⎛ ⎝⎜⎞ ⎠⎟ 33.134.π 22 2−⎡ ⎣⎢⎤ ⎦⎥+ Si as Ci as() ()122 2tta aln+⎛ ⎝⎜⎞ ⎠⎟ 33.135. 0 /H5114(t) = null function 33.136. 1 δ(t) = delta function 33.137. eas−δ()ta− 33.138.e sas− See also entry 33.163./H5121()ta−LAPLACE TRANSFORMS 190 f(s) F(t) 33.139.sinh sinhsx ss ax annx ant an n+− =∞ ∑21 1πππ ()sin cos 33.140.sinh coshsx ss a41 2121 221 21πππ ()sin()sin() − −−− =∞ ∑n nnnx ant a 33.141.cosh sinhsx sa st annx ant an n+− =∞ ∑21 1πππ ()cos sin 33.142.cosh coshsx ss a141 2121 221 21+− −−− =∞ πππ ()cos()cos()n nnnx ant a ∑ ∑ 33.143.sinh sinhsx ss a2xt aa nnx ant an n+− =∞ ∑21 22 1πππ ()sin sin 33.144.sinh coshsx ss a2xa nnx antn +− −−− 81 2121 221 222πππ () ()sin()cos() a an=∞ ∑ 1 33.145.cosh sinhsx ss a2t aa nnx ant an n2 22 12211 +−−⎛ ⎝⎞ ⎠=∞ ∑πππ ()cos cos 33.146.cosh coshsx ss a2ta nnx ann n+− −−− =∞ ∑81 2121 22 22 1ππ () ()cos()sin( 1 1 2)πt a 33.147.cosh coshsx ss a31 216 1 2121 22222 33 ()() ()cos()txaa nnxn +− −− −− ππ a ant an=∞ ∑− 121 2cos() π 33.148.sinh sinhxs as212 122 2 πππ anenx ann t a n() s i n−− =∞ ∑/ 33.149.cosh coshxs asππ anennn ta n212 1 4 112 12 22 2()( ) c o s(()−−−− − =∞ ∑/ − −1 2)πx a 33.150.sinh coshxs sa s2121 212 1 4 122 2 aenx ann t a n() s i n()()−−−− − =∞ ∑π π/ 33.151.cosh sinhxs sa s12122 2 1aaenx ann t a n+−− =∞ ∑() c o sπ π/ 33.152.sinh sinhxs sa sx anenx an nnt a+− =∞ −∑21 122 2 πππ ()sin/ 33.153.cosh coshxs sa s141 2121 121 422 2+− −− =∞ −−∑ππ ()cos()()n nnt a nen/ π πx a2 33.154.sinh sinhxs sa s2xt aa nenx an nt a n+−−− =∞ ∑2112 33 122 2 πππ ()() s i n/ 33.155.cosh coshxs sa s21 216 1 21222 332122()() ()()xa ta nen nt−+ −− −−− ππ/4 4 12 21 2a nnx acos()− =∞ ∑πLAPLACE TRANSFORMS 191 f(s) F(t) 33.156.Ji xs sJ ia s0 0() ()1222 0 1 1−− =∞ ∑eJ x a Jnta n nn nλλ λλ//() () where λl, λ2,… are the positive roots of J0(λ) = 0 33.157.Ji xs sJ i a s0 2 0() ()1 4222 2 0 3 122 ()() ()xa taeJ x a Jnta n nn n−+ +−λλ λλ// = =∞ ∑ 1 where λ1, λ2,… are the positive roots of J0(λ) = 0 33.158.1 22asastanh⎛ ⎝⎞ ⎠ Triangular wave function Fig. 33-1 33.159.1 2 sastanh⎛ ⎝⎞ ⎠ Square wave function Fig. 33-2 33.160.π πa asas 22 22 +⎛ ⎝⎞ ⎠coth Rectified sine wave function Fig. 33-3 33.161.π πa as eas() ( )22 21 +−− Half-rectified sine wave function Fig. 33-4 33.162.1 12ase seas as −−− −() Sawtooth wave function Fig. 33-5LAPLACE TRANSFORMS 192 f(s) F(t) 33.163.e sas− See also entry 33.138. Heaviside’s unit function /H5121(t – a) Fig. 33-6 33.164.ee sas s−−−()1/H9280 Pulse function Fig. 33-7 33.165.1 1seas()−− See also entry 33.102. Step function Fig. 33-8 33.166.ee sess s−− −+ −2 21() F(t) = n2, n /H11017 t < n + 1, n = 0, 1, 2, … Fig. 33-9 33.167.1 1− −− −e sr es s() See also entry 33.104. F(t) = rn, n /H11017 t < n + 1, n = 0, 1, 2, … Fig. 33-10 33.168.π πae asas()1 22 2+ +− Ftta t a ta()sin( )=>⎧⎨⎩π/0 0/H11017/H11017 Fig. 33-11LAPLACE TRANSFORMS 34 FOURIER TRANSFORMS Fourier’s Integral Theorem 34.1. fx A x B x d() {( ) c o s ( ) s i n }=+∞∫αα α α α 0 where 34.2. Af x x d x Bf x x d x() ( ) c o s () ( ) s i nαπα απα= =−∞∞ −∞∞∫1 1∫ ∫⎧ ⎨⎪ ⎩⎪ Sufficient conditions under which this theorem holds are: (i) f(x) and f ′(x) are piecewise continuous in every finite interval –L < x < L; (ii) |( ) |fx d x converges; −∞∞∫ (iii) f(x) is replaced by 1 2 00 {( ) ( ) }fx fx++ − if x is a point of discontinuity. Equivalent Forms of Fourier’s Integral Theorem 34.3. fx fu x ud u d u() () c o s ( ) =− =−∞∞ =−∞∞∫ ∫1 2παα α 34.4. fx e d fu e d u fu eix iu i() () ()= =−∞∞− −∞∞∫∫1 2 1 2πα παα α αα()xudu d− −∞∞ −∞∞∫ ∫ 34.5. fx x d fu u d u() s i n () s i n=∞∞∫∫2 00παα α where f (x) is an odd function [ f(−x) = −f(x)]. 34.6. fx x d fu u d u() c o s () c o s=∞∞∫∫2 00παα α where f (x) is an even function [ f(−x) = f (x)]. 193 194 Fourier Transforms The Fourier transform of f (x) is defined as 34.7. /H5106{() } ( ) ()fx F fx e d xix==− −∞∞∫αα Then from 34.7 the inverse Fourier transform of F(a ) is 34.8. /H5106− −∞∞== ∫1 1 2{() } () ()Ff x F e dixαπααα We call f (x) and F(a) Fourier transform pairs. Convolution Theorem for Fourier Transforms If F(a) = /H5106{f (x)} and G(a ) = /H5106{g(x)}, then 34.9. 1 2παα ααFGe d f u g x u d u f gix()() ( ) ( )* −∞∞ −∞∞∫∫=− = where f*g is called the convolution of f and g. Thus, 34.10. /H5106{ f*g} = /H5106{ f} /H5106{g} Parseval’s Identity If F(a) = /H5106{ f(x)}, then 34.11. | ( )| | ( )|fx d x F d22 1 2 −∞∞ −∞∞∫∫=παα More generally if F(a) = /H5106{ f(x)} and G(a) = /H5106 {g(x)}, then 34.12. fx g xd x F G d()() ( ) ( ) −∞∞ −∞∞∫∫=1 2παα α where the bar denotes complex conjugate. Fourier Sine Transforms The Fourier sine transform of f (x) is defined as 34.13. Ff x f x x d xSS() { ( ) } ( ) s i nαα==∞∫/H5106 0 Then from 34.13 the inverse Fourier sine transform of FS(a ) is 34.14. fx F F x dSS S () { ( ) } ( ) s i n==−∞∫/H51061 02απαα αFOURIER TRANSFORMS 195 Fourier Cosine Transforms The Fourier cosine transform of f (x) is defined as 34.15. Ff x f x x d xCC() { ( ) } ( ) c o sαα==∞∫/H5106 0 Then from 34.15 the inverse Fourier cosine transform of FC(a) is 34.16. fx F F x dCC C () { ( ) } ( ) c o s==−∞∫/H51061 02απαα α Special Fourier Transform Pairs f(x) F(a ) 34.17.1 0|| ||xb xb< > {2s in bα α 34.18.1 22xb+παe bb− 34.19.x xb22+−−iebπα 34.20. f(n)(x) inanF(a) 34.21. xnf(x) idF dnn nα 34.22. f(bx)eitx1 bFt bα−⎛ ⎝⎞ ⎠FOURIER TRANSFORMS 196 Special Fourier Sine Transforms f(x) FC(a ) 34.23.10 0<< > {xbxb 1−cosbα α 34.24. x–1π 2 34.25.x xb22+πα 2eb− 34.26. e–bxα α22+b 34.27. xn – 1e–bxΓ() s i n ( t a n /) ()/nn b bn− +1 22 2α α 34.28. xebx−2 παα 43242 beb //− 34.29. x–1/2 π α2 34.30. x–nπα πnn nn− <<12 202csc( / ) ()Γ 34.31.sinbx x1 2lnα α+−⎛ ⎝⎞ ⎠b b 34.32.sinbx x2πα α πα/ /2 2< > {b bb 34.33.cosbx x0 4 2α πα πα<=>⎧ ⎨⎪ ⎩⎪b bb/ / 34.34. tan ( / )−1xbπ αα 2eb− 34.35. csc bxππ α 22bbtanh 34.36.1 12ex−ππ α α 421 2coth⎛ ⎝⎞ ⎠−FOURIER TRANSFORMS 197 Special Fourier Cosine Transforms f(x) FC(a ) 34.37.10 0<< > {xbxb sinbα α 34.38.1 22xb+παe bb− 2 34.39. ebx− b b α22+ 34.40. xenb x−−1 Γ() c o s ( t a n /) ()/nn b bn− +1 22 2α α 34.41. ebx−2 1 224 πα beb −/ 34.42. x−12/ π α2 34.43. xn− πα πnn nn− <<12 201sec( / ) (),Γ 34.44. lnxb xc22 22+ +⎛ ⎝⎜⎞ ⎠⎟eecb−−−αα πα 34.45.sinbx xπα πα α/ /2 4 0< = >⎧ ⎨⎪ ⎩⎪b b b 34.46. sinbx2πα α 84 422 bb bcos sin −⎛ ⎝⎜⎞ ⎠⎟ 34.47. cosbx2πα α 84 422 bb bcos sin +⎛ ⎝⎜⎞ ⎠⎟ 34.48. sech bxππ α 22bbsech 34.49.cosh ( / ) cosh ( )π πx x2 ππ α πα 22 cosh ( / ) cosh ( ) 34.50.e xbx−π ααα222 {cos( ) sin( )} bb −FOURIER TRANSFORMS Section IX: Elliptic and Miscellaneous Special Functions 35ELLIPTIC FUNCTIONS Incomplete Elliptic Integral of the First Kind 35.1. uF kd kd kx== −= −−∫ ∫(, ) sin ( )( )φθ θφ 11 122 2 2 2 0 0/H9271 /H9271/H9271 where f = am u is called the amplitude of u and x = sin f, and where here and below 0 < k < 1. Complete Elliptic Integral of the First Kind 35.2. KF kd kd k== −= −−∫(, /) sin ( )( )πθ θ2 11 122 2 2 2 01 0/H9271 /H9271/H9271π π π/2 2 22 4 211 213 24135∫ =+⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟+ kki iii 22462 6 ii/midhorizellipsis⎛ ⎝⎜⎞ ⎠⎟+⎧ ⎨⎪ ⎩⎪⎫ ⎬⎪ ⎭⎪k Incomplete Elliptic Integral of the Second Kind 35.3. Ek k dkdx(, ) s i nφθ θφ=− =− −∫∫11 122 022 2 0/H9271 /H9271/H9271 Complete Elliptic Integral of the Second Kind 35.4. EE k k dkd == − =− −∫∫(, /) s i n/πθ θπ211 122 0222 2 01 /H9271 /H9271/H9271 ==−⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝⎜⎞ ⎠⎟−π 211 213 24 3135 242 224 kki iii ii i/midhorizellipsis6526⎛ ⎝⎜⎞ ⎠⎟−⎧ ⎨⎪ ⎩⎪⎫ ⎬⎪ ⎭⎪k Incomplete Elliptic Integral of the Third Kind 35.5. Π(,, ) ( sin ) sin ( ) (knd nkd nφθ θθ= +−= +− 11 1 122 2 2/H9271 /H9271 /H9271/H9271/H927122 2 0 01)( )−∫ ∫kx φ 198 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 199 Complete Elliptic Integral of the Third Kind 35.6. Π(,, /) ( sin ) sin ( ) (knd nkd nπθ θθ2 11 122 2 2= +−= +/H9271 /H9271 11122 2 01 02 −−∫ ∫/H9271/H9271)( )/ kπ Landen’s Transformation 35.7. tansin cosφφ φ=+2 21 1 kor ksin sin( )φφ φ=− 21 This yields 35.8. Fkd k kd k(, ) sin sinφθ θθ θφ φ= −=+ −∫12 1 1221 122 10 01∫ ∫ where kk k121=+ /( ). By successive applications, sequences kkk123,,, … and φφφ123,,, … are obtained such that kk k k<<<<<123 1/midhorizellipsis where lim . nnk →∞=1 It follows that 35.9. Fkkkk kd kkk k(, ) sinln tan ΦΦ= −= ∫123 2123 01 4…… θ θπ+ +⎛ ⎝⎜⎞ ⎠⎟Φ 2 where 35.10. kk k12 1=+, kk k nn 21 12 1=+ →∞, lim …and Φ φ The result is used in the approximate evaluation of F(k, f). Jacobi’s Elliptic Functions From 35.1 we define the following elliptic functions: 35.11. xu u==sin ( )am sn 35.12. 12−= =xu u cos ( ) am cn 35.13. 1122 2−= − =kx k u u sn dn2 We can also define the inverse functions sn , ,−− −11 1xxxcn dn and the following: 35.14. nssnuu=1 35.17. scsn cnuu u= 35.20. cscn snuu u= 35.15. nccnuu=1 35.18. sdsn dnuu u= 35.21. dcdn cnuu u= 35.16. nddnuu=1 35.19. cdcn dnuu u= 35.22. dsdn dnuu u= Addition Formulas 35.23. snsn cn dn cn sn dn sn s()uuu u ku+=+ −/H9271/H9271/H9271 /H9271 122n n2/H9271ELLIPTIC FUNCTIONS 35.24. cncn cn sn sn dn dn sn sn2()uuu u ku+=− −/H9271/H9271/H9271 /H9271 /H9271 122 35.25. dndn dn sn sn cn cn sn sn2()uuk u u ku+=− −/H9271/H9271/H9271 /H9271 /H92712 221 Derivatives 35.26. d duuu usn cn dn= 35.28.d duuk u udn sn cn=−2 35.27. d duuu ucn sn dn=− 35.29. d duuu usc=dc nc Series Expansions 35.30. sn ( )!()!( uu kukkuk =− + + + + − + + 1311 451 135 123 245 23357467 kku++)!/midhorizellipsis 35.31. cnuukukku= − ++ −+ + +1214414 4 1 662 24 246 !()!()!/midhorizellipsis 35.32. dnukukkukk ku=− + + − + +124416 44622 224 22 46 !()!()! !+/midhorizellipsis Catalan’s Constant 35.33.1 21 2 11 11 31 5915965 22222 Kd kdd k k= −=−+− =θ θsin./midhorizellipsis 5594 02 01 01… θπ = =∫ ∫ ∫/ k Periods of Elliptic Functions Let 35.34. Kd k= −∫θ θπ 122 02 sin,/′= −′∫Kd kθ θπ 122 02 sin/where ′=−kk 12 Then 35.35. sn u has periods 4K and 2iK ′ 35.36. cn u has periods 4K and 2K + 2iK ′ 35.37. dn u has periods 2K and 4iK ′200 ELLIPTIC FUNCTIONS 201 Identities Involving Elliptic Functions 35.38. sn221 uu+=cn 35.39. dn22 21 uk u+=sn 35.40. dn cn22 2 2uk uk−= ′ where ′=−kk 12 35.41. sncn dn212 12uu u=− + 35.42. cndn cn dn2 22 12uuu u=+ + 35.43. dndn cn dn22212 12uku k u u=−+ + + 35.44.12 12− +=cn cnsn dn cnu uuu u 35.45. 12 12− +=dn dnu ukuu usn cn dn Special Values 35.46. sn 0 = 0 35.47. cn 0 = 1 35.48. dn 0 = 1 35.49. sc 0 = 0 35.50. am 0 = 0 Integrals 35.51. sn dn cnud ukuk u ∫=−1ln( ) 35.52. cn dn ud uku ∫=− 11cos ( ) 35.53. dn sn ud u u =−∫sin ( )1 35.54. sc dc nc ud u kuk u = −+−( ) ∫1 11 22ln 35.55. cs nsud u u u =− ∫ln( ) ds 35.56. cd nd sd ud ukuk u =+ ∫1ln ( ) 35.57. dc nc cud u u u=+ ∫ln( ) s 35.58. sd cd ud u kkku =− −−∫1 121sin ( ) 35.59. ds ns csud u u u=− ∫ln( ) 35.60. ns ds csud u u u =− ∫ln ( ) 35.61. nc dcscud u kuu k= −+ −⎛ ⎝⎜⎞ ⎠⎟ ∫1 1122ln 35.62. nd cd ud u ku = −−∫1 121cos ( )ELLIPTIC FUNCTIONS 202 Legendre’s Relation 35.63. EK E K KK′+′− ′=π/2 where 35.64. Ek d=−∫122 02sin/θθπKd k= −∫θ θπ 122 02 sin/ 35.65. ′=− ′ ∫Ek d 122 02sin/θθπ′= −′∫Kd kθ θπ 122 02 sin/ELLIPTIC FUNCTIONS 36 MISCELLANEOUS and RIEMANN ZETA FUNCTIONS Error Function erf ( )2 2 0xe d uux=− π∫ 36.1. erf ( )!!!xxxxx= − +−+⎛ ⎝⎜⎞ ⎠⎟2 31 52 7335 7 π ii i/midhorizellipsis 36.2. erf ( ) ~()()xe x xx xx 111 213 2135 22 22 2 2 3 −− + −− πii i+ +⎛ ⎝⎜⎞ ⎠⎟/midhorizellipsis 36.3. erf erf() ( ) ,−= −xx erf ( ) ,00= erf ( )∞= 1 Complementary Error Function erfc ( ) 1 erf ( )2 2xx e d uu x=− =−∞ π∫ 36.4. erfc ( )!!!xxxxx= − − +−+⎛ ⎝⎜⎞ ⎠⎟ 12 31 52 7335 7 π ii i/midhorizellipsis 36.5. erfc ( ) ~() ()xe x xx xx− −+ − +2 11 213 2135 222 2 2 3πii i/midhorizellipsis /midhorizellipsis⎛ ⎝⎜⎞ ⎠⎟ 36.6. erfc ( ) , 01= erfc ( ) ∞= 0 Exponential Integral Ei( )xe uduu x=−∞∫ 36.7. Ei ( ) lnxxe uduux=− − +−− ∫γ1 0 36.8. Ei ( ) ln!!!xxxx x=− − + − + −⎛ ⎝⎜⎞ ⎠⎟ γ11 2 2 3323 iii/midhorizellipsis 36.9. Ei ( ) ~!!!xe xx xxx− −+ − +⎛ ⎝⎜⎞ ⎠⎟ 1123 23 /midhorizellipsis 36.10. Ei ( )∞= 0 Sine Integral Si( )sin 0xu udux=∫ 36.11. Si ( )!!!!xxx x x= −+−+11 33 55 77357 iiii/midhorizellipsis 36.12. Si ( ) ~sin ! ! cosxx xx xxx xπ 213 512 35 − −+−⎛ ⎝⎜⎞ ⎠⎟−−/midhorizellipsis!!! xx244+−⎛ ⎝⎜⎞ ⎠⎟/midhorizellipsis 36.13. Si Si() ( ) ,−= −xx Si ( ) ,00= Si ( ) /∞= π2 203 204 Cosine Integral Ci( )cosxu udu x=∞∫ 36.14. Ci ( ) lncosxxu udux=− − +−∫γ1 0 36.15. Ci ( ) ln!!! !xxxxxx=− − + − + − +γ246 8 22 44 66 88iiii/midhorizellipsis 36.16. Ci ( ) ~cos ! ! sin !xx xx xxx x x13 512 35 2 −+−⎛ ⎝⎜⎞ ⎠⎟−−/midhorizellipsis ++−⎛ ⎝⎜⎞ ⎠⎟4 4! x/midhorizellipsis 36.17. Ci( )∞= 0 Fresnel Sine Integral Sx ud ux()2sin2 0=π∫ 36.18. Sxxx x x()!! ! !=− + − +⎛ ⎝⎜2 31 73 1 15 1 573 7 11 15 πii i i/midhorizellipsis⎞ ⎞ ⎠⎟ 36.19. Sx xx xx() ~ ( c o s )1 21 211 3 21357 22 25 49 −− + −⎛ πi iii/midhorizellipsis⎝ ⎝⎜⎞ ⎠⎟+− +⎛ ⎝⎜⎞ ⎠⎟⎧⎨⎩⎫⎬⎭(sin )xxx2 33 71 2135 2ii/midhorizellipsis 36.20. Sx S x() ( ) ,−= − S() ,00= S()∞=1 2 Fresnel Cosine Integral Cx ud ux()2cos2 0=π∫ 36.21. Cxxx x x()!!! !=− + − +⎛ ⎝⎜⎞ ⎠⎟2 15 29 41 3 6591 3 π ii i/midhorizellipsis 36.22. Cx xx xx() ~ ( s i n )1 21 211 3 21357 22 25 49 +− + −⎛ πi iii/midhorizellipsis⎝ ⎝⎜⎞ ⎠⎟−− +⎛ ⎝⎜⎞ ⎠⎟⎧⎨⎩⎫⎬⎭(cos )xxx2 33 71 2135 2ii/midhorizellipsis 36.23. Cx C x() ( ) ,−= − C() ,00= C()∞=1 2 Riemann Zeta Function ζ()1 11 21 3xxxx=+++ /midhorizellipsis 36.24. ζ()(), xxu edu xx u =>− −∞∫111 10 Γ 36.25. ζπ π ζ( ) () c o s ( /)()12 21−=−−xx x xxxΓ (extension to other values) 36.26. ζπ()() !,,, 22 212321 2 kB kkkk k==− …MISCELLANEOUS AND RIEMANN ZETA FUNCTIONS 205205Section X: Inequalities and Infinite Products 37INEQUALITIES Triangle Inequality 37.1. || | | | | || | |aa a a aa12 1 2 12−+ + /H11017/H11017 37.2. || |||| | | aa a a a ann 12 1 2+++ + + + /midhorizellipsis/midhorizellipsis /H11017 Cauchy-Schwarz Inequality 37.3. () ( ) ( ab ab ab a a a b bnn n 11 2 22 12 222 12 22++ + + + + + /midhorizellipsis/midhorizellipsis /H11017 +++/midhorizellipsis bn2) The equality holds if and only if ab ab abnn 11 2 2// / .== = /midhorizellipsis Inequalities Involving Arithmetic, Geometric, and Harmonic Means If A, G, and H are the arithmetic, geometric, and harmonic means of the positive numbers a1, a2, ..., an, then 37.4. HG A/H11017/H11017 where 37.5. Aaa a nn=+++12/midhorizellipsis 37.6. Ga a ann=12… 37.7.11 11 1 12Hn aa an=+ + +⎛ ⎝⎜⎞ ⎠⎟/midhorizellipsis The equality holds if and only if aa an 12=== /midhorizellipsis . Holder’s Inequality 37.8. || (|||| ||)/ab ab ab a a annpp np 11 2 2 1 21++ + + + + /midhorizellipsis/midhorizellipsis /H11017ppq q nqqbb b(| | | | | | )/ 121++ + /midhorizellipsis where 37.9.1111 1pqpq += > > , The equality holds if and only if | | /| | | | /| | | | /| |.ab a b a bpp np n 11 121 21 −− −== = /midhorizellipsis For p = q = 2 it reduces to 37.3. Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 206 INEQUALITIES Chebyshev’s Inequality Ifaa a bb bnn 12 12/H11084/H11084/H11084 /H11084/H11084/H11084 /midhorizellipsis/midhorizellipsis and , then 37.10.aa a nbb b nab abnn 12 12 1 12+++⎛ ⎝⎜⎞ ⎠⎟+++⎛ ⎝⎜⎞ ⎠⎟+ /midhorizellipsis/midhorizellipsis/H110172 2++/midhorizellipsis ab nnn or 37.11. () () ( a aa b bb n a b a ba bnn nn 12 12 1 12 2+++ +++ + ++ /midhorizellipsis/midhorizellipsis /midhorizellipsis /H11017 ) ) Minkowski’s Inequality Ifa1,a2,…an,b1,b2,…bn are all positive and p > 1, then 37.12. {( ) ( ) ( ) } (/ab ab ab a app nnpp p p 11 2 21 12++ ++ + + + + /midhorizellipsis /H11017 /midhorizellipsis/midhorizellipsis/midhorizellipsis++ + + +ab b bnpp p p npp)( )//1 121 The equality holds if and only if ab ab abnn 11 2 2// / .== = /midhorizellipsis Cauchy-Schwarz Inequality for Integrals 37.13. fx g xd x fx d x g x d ab ab()() [() ] [() ]∫∫⎡ ⎣⎢⎤ ⎦⎥{}2 22/H11017 x x ab∫{} The equality holds if and only if f (x)/g(x) is a constant. Holder’s Inequality for Integrals 37.14. | ( ) ( )| | ( )| | ( )|/ f x g x dx f x dx g x dx abp abp q a ∫∫ {}/H110171b bq ∫{}1/ where 1/p + 1/q = 1, p > 1, q >1. If p = q = 2, this reduces to 37.13. The equality holds if and only if |() | / |() |fx g xp−1 is a constant. Minkowski’s Inequality for Integrals Ifp > 1, 37.15. |() () | |() | |// fx g x d x fx d x gp abp p abp + {} { } + ∫∫11 /H11017 (() |/ xd xp abp ∫{}1 The equality holds if and only if f (x)/g(x) is a constant. 38 INFINITE PRODUCTS 38.1. sinxxx xxx=−⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝⎜⎞ ⎠114192 22 22 2ππ⎟ ⎟/midhorizellipsis 38.2. cosxxx x=−⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝1414 914 252 22 22 2ππ π⎜ ⎜⎞ ⎠⎟/midhorizellipsis 38.3. sinhxxxx x=+⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞114192 22 22 2πππ ⎠ ⎠⎟/midhorizellipsis 38.4. cosh xxx x=+⎛ ⎝⎜⎞ ⎠⎟+⎛ ⎝⎜⎞ ⎠⎟+⎛1414 914 252 22 22 2ππ π ⎝ ⎝⎜⎞ ⎠⎟/midhorizellipsis 38.5.111122 Γ()/ xxexexexx x=+⎛ ⎝⎞ ⎠⎧⎨⎩⎫⎬⎭+⎛ ⎝⎞ ⎠⎧⎨⎩−− γ ⎫ ⎫⎬⎭+⎛ ⎝⎞ ⎠⎧⎨⎩⎫⎬⎭−133 xex//midhorizellipsis See also 25.11. 38.6. Jxxxx 02 122 222 32111 ()=−⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝⎜⎞ /H9261/H9261/H9261 ⎠ ⎠⎟/midhorizellipsis where /H92611,/H92612,/H92613,… are the positive roots of J0(x) = 0. 38.7. Jx xxxx 12 122 222 32111 ()=−⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝⎜/H9261/H9261/H9261⎞ ⎞ ⎠⎟/midhorizellipsis where /H92611,/H92612,/H92613,… are the positive roots of J1(x) = 0. 38.8.sincos cos cos cosx xxxx x=2481 6/midhorizellipsis 38.9.π 22 12 34 34 56 56 7=iiiiii /midhorizellipsis This is called Wallis’ product. 207 Section XI: Probability and Statistics 39 DESCRIPTIVE STATISTICS The numerical data x1, x2,… will either come from a random sample of a larger population or from the larger population itself. We distinguish these two cases using different notation as follows: n = number of items in a sample, N = number of items in the population, x = (read: x-bar) = sample mean, m (read: mu) = population mean, s2 = sample variance, s 2 = population variance, s = sample standard deviation, s = population standard deviation Note that Greek letters are used with the population and are called parameters, whereas Latin letters are used with the samples and are called statistics. First we give formulas for the data coming from a sample. This is followed by formulas for the population. Grouped Data Frequently, the sample data are collected into groups (grouped data). A group refers to a set of numbers all with the same value x i, or a set (class) of numbers in a given interval with class value xi. In such a case, we assume there are k groups with fi denoting the number of elements in the group with value or class value xi. Thus, the total number of data items is 39.1. nfi=∑ As usual, Σ will denote a summation over all the values of the index, unless otherwise specified. Accordingly, some of the formulas will be designated as (a) or as (b), where (a) indicates ungrouped data and (b) indicates grouped data. Measures of Central Tendency Mean (Arithmetic Mean) The arithmetic mean or simply mean of a sample x1, x2,…, xn, frequently called the “average value,” is the sum of the values divided by the number of values. That is: 39.2(a). Sample mean: xxx x nx nni=+++=12 /midhorizellipsis Σ 39.2(b). Sample mean: xfx fx fx ff ffx fkk kii i=++ + +++=11 2 2 12/midhorizellipsis /midhorizellipsisΣ Σ Median Suppose that the data x1, x2,…, xn are now sorted in increasing order. The median of the data, denoted by Mo r Median is defined to be the “middle value.” That is: 208 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 209 39.3(a). Medianwhen is odd and when==+ ++ +xn n k xxk kk1 121 2, nnn kis even and =⎧ ⎨⎪ ⎩⎪ 2. The median of grouped data is obtained by first finding the cumulative frequency function Fs. Specifically, we define Fff fss=+++12/midhorizellipsis that is, Fs is the sum of the frequencies up to fs. Then: 39.3(b.1). Medianwhen (odd) and ==+ < + ≤ ++ + xn k F k F xjj j j1 1 21 1 x xnk F kj j+==⎧ ⎨⎪ ⎩⎪1 22 when (even) and ,. Finding the median of data arranged in classes is more complicated. First one finds the median class m, the class with the median value, and then one linearly interpolates in the class using the formula 39.3(b.2). Median =+−−LcnF fmm m(/)21 where Lm denotes the lower class boundary of the median class and c denotes its class width (length of the class interval). Mode The mode is the value or values which occur most often. Namely: 39.4. Mode xm= numerical value that occurs the most number of times The mode is not defined if every xm occurs the same number of times, and when the mode is defined it may not be unique. Weighted and grand means Suppose that each xi is assigned a weight wi≥ 0. Then: 39.5. Weighted Mean xwx wx wx ww wwx wwkk kii i=++ + ++ +=11 2 2 12/midhorizellipsis /midhorizellipsisΣ Σ Note that 39.2(b.1) is a special case of 39.4 where the weight wi of xi is its frequency. Suppose that there are k sample sets and that each sample set has ni elements and a mean x.Then the grand mean, denoted by xi is the “mean of the means” where each mean is weighted by the number of ele- ments in its sample. Specifically: 39.6. Grand Mean xnx nx nx nn nnx nkk kii i=++ + +++=11 2 2 12/midhorizellipsis /midhorizellipsisΣ Σ Geometric and Harmonic Means Thegeometric mean (G.M.) and harmonic mean (H.M.) are defined as follows: 39.7(a). G.M. =xx xnn 12/midhorizellipsis 39.7(b). G.M. =xx xff kfn k 1212/midhorizellipsisDESCRIPTIVE STATISTICS 39.8(a). H.M. =++ +=n xx xn xni 11 1 112// / ( / ) /midhorizellipsis Σ 39.8(b). H.M. =++ +=n fx fx fxn fxkk ki 11 2 2// / ( / ) /midhorizellipsis Σ Relation Between Arithmetic, Geometric, and Harmonic Means 39.9. H.M. G.M. ≤≤ x The equality sign holds only when all the sample values are equal. Midrange The midrange is the average of the smallest value x1 and the largest value xn. That is: 39.10. midrange: mid =+xxn 1 2 Population Mean The formula for the population mean m follows: 39.11(a). Population mean: μ=+++=xx x Nx NNi 12/midhorizellipsis Σ 39.11(b). Population mean: μ=++ + +++=fx fx fx ff ffx fkk kii i11 2 2 12/midhorizellipsis /midhorizellipsisΣ Σ (Recall that N denotes the number of elements in a population.) Observe that the formula for the population mean m is the same as the formula for the sample mean x. On the other hand, the formula for the population standard deviation s is not the same as the formula for the sample standard deviation s. (This is the main reason we give separate formulas for m and x.) Measures of Dispersion Sample Variance and Standard Deviation Here the sample set has n elements with mean x. 39.12(a). Sample variance: sxx nxx n nii i 222 2 11=− −=− −ΣΣ Σ() ( ) / 39.12(b). Sample variance: sfx x ffx fx f fii iii ii i 222 2 1=− −=− Σ ΣΣΣΣ Σ() ()() / (i i)−1 39.13. Sample standard deviation: ss==Variance2 EXAMPLE 39.1: Consider the following frequency distribution: xi12 34 56 fi81 471 231 Then n = Σ fi = 45 and Σ fi xi = 126. Hence, by 39.2(b), Mean xxf fii i== =Σ Σ126 4528.DESCRIPTIVE STATISTICS 210 211 Also, n – 1 = 44 and Σfxii2430= . Hence, by 39.12(b) and 39.13, ss22430 126 45 4417 5 13 2 =−≈=() /.. and We find the median M, first finding the cumulative frequencies: FF F F F Fn1 2345682 2 2 9 4 1 4 4 4 5== = = = = =,, , , , Here n is odd, and (n + 1)/2 = 23. Hence, Median M==23 3rd value The value 2 occurs most often, hence Mode = 2 M.D. and R.M.S. Here M.D. stands for mean deviation and R.M.S. stands for root mean square. As previously, x is the mean of the data and, for grouped data, n =Σfi. 39.14(a). M.D. =−1 nxxi 39.14(b). M.D. =−1 nfx xii 39.15(a). R.M.S. =12 nxi()Σ 39.15(b). R.M.S. =12 nfxii()Σ Measures of Position (Quartiles and Percentiles) Now we assume that the data x1,x2,…,xn are arranged in increasing order. 39.16. Sample range: xn – x1. There are three quartiles: the first or lower quartile, denoted by Q1 or QL; the second quartile or median, denoted by Q2 or M; and the third or upper quartile, denoted by Q3 or QU. These quartiles (which essentially divide the data into “quarters”) are defined as follows, where “half” means n/2 when n is even and (n-1)/2when n is odd: 39.17. Q L(=Q1)= median of the first half of the values. M( =Q2 ) = median of the values. QU (= Q3)= median of the second half of the values. 39.18. Five-number summary: [L, QL, M, QU,H] where L =x1 (lowest value) and H =xn (highest value). 39.19. Innerquartile range: QU – QL 39.20. Semi-innerquartile range: QQQUL=− 2 The kth percentile, denoted by Pk, is the number for which kpercent of the values are at most Pk and (100–k) percent of the values are greater than Pk. Specifically: 39.21. Pk= largest xs such that Fs≤k/100. Thus, QL= 25th percentile, M = 50th percentile, QU= 75th percentile.DESCRIPTIVE STATISTICS 212 Higher-Order Statistics 39.22. The rth moment: (a) mnxrir=1Σ, (b) mnfxri ir=1Σ 39.23. The rth moment about the mean x: (a) μrir nxx =−1Σ() , (b) μri ir nfx x =−1Σ() 39.24. The rth absolute moment about mean x: (a) μrir nxx =−1Σ , (b) μri ir nfx x =−1Σ 39.25. The rth moment in standard z units about z = 0: (a) αrir nz =1Σ, (b) ασri ir ii nfz zxx==− 1Σ where Measures of Skewness and Kurtosis 39.26. Coefficient of skewness: γμ σα13 3 3== 39.27. Momental skewness: μ σ3 32 39.28. Coefficient of kurtosis: αμ σ44 4= 39.29. Coefficient of excess (kurtosis): αμ σ44 433−= − 39.30. Quartile coefficient of skewness: Qx Q QQQQ Q QQUL UL−+ −=−+ −2232 1 31ˆ Population Variance and Standard Deviation Recall that N denotes the number of values in the population. 39.31. Population variance: σ222 2 =−=− ΣΣ Σ() ( ) /xx Nxx n Nii i 39.32. Population standard deviation: σσ==Variance2 Bivariate Data The following formulas apply to a list of pairs of numerical values: ( , ), ( , ), ( , ), , ( , )xy xy xy xynn 11 2 2 33… where the first values correspond to a variable x and the second to a variable y. The primary objective is to determine whether there is a mathematical relationship, such as a linear relationship, between the data. The scatterplot of the data is simply a picture of the pairs of values as points in a coordinate plane.DESCRIPTIVE STATISTICS 213 Correlation Coefficient A numerical indicator of a linear relationship between variables x and y is the sample correlation coef- ficient r of x and y, defined as follows: 39.33. Sample correlation coefficient: rxx yy xx yyii ii=−− −−Σ ΣΣ() () () ()22 We assume that the denominator in Formula 39.33 is not zero. An alternative formula for computing r follows: 39.34. rxy x y n xx n yyii i i ii ii=− −−ΣΣ Σ ΣΣ ΣΣ() () / () / ()222 2 2/n Properties of the correlation coefficient r follow: 39.35. (1) –1 /H11088 r /H11088 1 or, equivalently, /H11341/H11341 /H33355r1. (2) r is positive or negative according as y tends to increase or decrease as x increases. (3) The closer |r| is to 1, the stronger the linear relationship between x and y.The sample covariance of x and y is denoted and defined as follows: 39.36. Sample covariance: sxx yy nxyii=−− −Σ() () 1 Using the sample covariance, Formula 39.33 can be written in the compact form: 39.37. rs ssxy xy= where sx and sy are the sample standard deviations of x and y, respectively. EXAMPLE 39.2: Consider the following data: x 50 45 40 38 32 40 55 y 2.5 5.0 6.2 7.4 8.3 4.7 1.8 The scatterplot of the data appears in Fig. 39-1. The correlation coefficient r for the data may be obtained by first constructing the table in Fig. 39-2. Then, by Formula 39.34 with n = 7, r=− −1431 8 300 35 9 7 13 218 300 7 218 672.() ( . ) / ,( ) / . + +≈− (. ) /. 35 9 70 95622 Here r is close to –1, and the scatterplot in Fig. 39-1 does indicate a strong negative linear relationship between x and y .DESCRIPTIVE STATISTICS 214 Fig. 39-1 Fig. 39-2 Regression Line Consider a given set of n data points Pi (xi, yi). Any (nonvertical) line L may be defined by an equation of the form y = a + bx Let yi* denote the y value of the point on L corresponding to xi; that is, let ya b xii* . =+ Now let dy yya b xii i i i=− =−+* () that is, di is the vertical (directed) distance between the point Pi and the line L. The squares error between the line L and the data points is defined by 39.38. Σddd din2 12 222=++ + /midhorizellipsis The least-squares line or the line of best fit or the regression line of y on x is, by definition, the line L whose squares error is as small as possible. It can be shown that such a line L exists and is unique. The constants a and b in the equation y = a + bx of the line L of best fit can be obtained from the following two normal equations, where a and b are the unknowns and n is the number of points: 39.39.na x b y xa x b x yii iii i+= +=⎧ ⎨⎪ ⎩⎪() ()( )ΣΣ ΣΣΣ2 The solution of the above normal equations follows: 39.40. bnx y x y nx xrs sayii i i iiy xi=− −==ΣΣ Σ ΣΣΣ () () ();22n nbx nyb xi−= −Σ The second equation tells us that the point (, )xy lies on L, and the first equation tells us that the point (, )xs yr sxy++ also lies on L. EXAMPLE 39.3: Suppose we want the line L of best fit for the data in Example 39.2. Using the table in Fig. 39-2 and n = 7, we obtain the normal equations 7 300 35 9 300 13 218 1431 8ab ab+= +=. ,. Substitution in 39.40 yields b=− −=−7 1431 8 300 35 9 7 13 218 30002(. ) ( ) ( . ) (, )( ).22959 35 9 70 2959300 717 8100 a=− − =.(. ) .DESCRIPTIVE STATISTICS 215 Thus, the line L of best fit is y = 17.8100 – 0.2959x The graph of L appears in Fig. 39-3. Fig. 39-3 Curve Fitting Suppose that n data points Pi (xi, yi) are given, and that the data (using the scatterplot or the correlation coefficient r) do not indicate a linear relationship between the variables x and y, but do indicate that some other standard (well-known) type of curve y = f(x) approximates the data. Then the particular curve C that one uses to approximate that data, called the best-fitting or least-squares curve, is the curve in the collection which minimizes the squares error sum Σddd din2 12 222=++ + /midhorizellipsis where di = yi – f(xi). Three such types of curve are discussed as follows. Polynomial function of degree m: ya a xa x a xmm=+ + ++01 22/midhorizellipsis The coefficients aaa am 012,,,, … of the best-fitting polynomial can be obtained by solving the following system of m + 1 normal equations: 39.41. na a x a x a x y axax aii m im i ii01 22 012++ + + = ++ΣΣ Σ Σ ΣΣ/midhorizellipsis 2 231ΣΣ Σxa xx yim im ii ++ =+/midhorizellipsis ..................... ........................................... .................... ax ax a xim im im 011 2 ΣΣ Σ ++++ +++ =22/midhorizellipsisax x ymim im i ΣΣ Exponential curve: ya b y a b xx== + or log log (log ) The exponential curve is used if the scatterplot of log y verses x indicates a linear relationship. Then log a and log b are obtained from transformed data points. Namely, the best-fit line L for data points P ′(xi, log yi) is 39.42.na x b y xa x b xii ii i′+ ′= ′+ ′=() ( l o g ) () ( ) ( l oΣΣ ΣΣΣ2gg)yi⎧ ⎨⎪ ⎩⎪ Then a = antilog a′, b = antilog b′. EXAMPLE 39.4: Consider the following data which indicates exponential growth: x 1 234 5 6 y 6 18 55 160 485 1460DESCRIPTIVE STATISTICS 216 Thus, we seek the least-squares line L for the following data: x 123456 log y 0.7782 1.2553 1.7404 2.2041 2.6857 3.1644 Using the normal equation 39.42 for L, we get ′= ′= ab 0 3028 0 4767., . The antiderivatives of a′ and b′ yield, approximately, a = 2.0, b = 3.0 Hence, y = 2(3x) is the required exponential curve C. The data points and C are depicted in Fig. 39-4. Fig. 39-4 Power function: y = axb or log y = log a + b log x The power curve is used if the scatterplot of log y verses log x indicates a linear relationship. The log a and b are obtained from transformed data points. Namely, the best-fit line L for transformed data points P ′(log xi, log yi) is 39.43.na x b y xa x bii ii′+= ′+ΣΣ ΣΣ(log ) (log ) (log ) (log )2= =⎧ ⎨⎪ ⎩⎪ Σ(log log )xyii Then a = antilog a′.DESCRIPTIVE STATISTICS 40 PROBABILITY Sample Spaces and Events Let S be a sample space which consists of the possible outcomes of an experiment where the events are subsets of S. The sample space S itself is called the certain event, and the null set ∅ is called the impossible event . It would be convenient if all subsets of S could be events. Unfortunately, this may lead to contradictions when a probability function is defined on the events. Thus, the events are defined to be a limited collection C of subsets of S as follows. DEFINITION 40.1: The class C of events of a sample space S form a σ -field. That is, C has the following three properties: (i) S ∈C. (ii) If A1, A2,… belong to C, then their union A1∪ A2∪ A3∪ … belongs to C. (iii) If A ∈C, then its complement Ac∈C. Although the above definition does not mention intersections, DeMorgan’s law (40.3) tells us that the complement of a union is the intersection of the complements. Thus, the events form a collection that is closed under unions, intersections, and complements of denumerable sequences. If S is finite, then the class of all subsets of S form a σ-field. However, if S is nondenumerable, then only certain subsets of S can be the events. In fact, if B is the collection of all open intervals on the real line R, then the smallest σ-field containing B is the collection of Borel sets in R. If Condition (ii) in Definition 40.1 of a σ-field is replaced by finite unions, then the class of subsets of S is called a field. Thus a σ-field is a field, but not visa versa. First, for completeness, we list basic properties of the set operations of union, intersection, and complement. 40.1. Sets satisfy the properties in Table 40-1. TABLE 40-1 Laws of the Algebra of Sets Idempotent laws: (1a) A ∪ A = A (1b) A ∩ A = A Associative laws: (2a) (A ∪ B) ∪ C = A ∪ (B ∪ C) (2b) (A ∩ B) ∩ C = A ∩ (B ∩C) Commutative laws: (3a) A ∪ B = B ∪ A (3b) A ∩ B = B ∩ A Distributive laws: (4a) A ∪ (B ∩C)= (A ∪ B) ∩ (A ∪ C) (4b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ B) Identity laws: (5a) A ∪ ∅ = A (5b) A ∩U= A (6a) A ∪U=U (6b) A ∩∅=∅ Involution law: (7) (AC)C= A Complement laws: (8a) A ∪ Ac=U (8b) A ∩ Ac=∅ (9a) Uc=∅ (9b) ∅c= U DeMorgan’s laws: (10a) (A ∪ B)c= Ac∩ Bc (10b) (A ∩ B)c= Ac∪ Bc 217 40.2. The following are equivalent: (i) A ⊆ B, (ii) A ∩ B = A, (iii) A ∩ B = B. Recall that the union and intersection of any collection of sets is defined as follows: ∪j Aj= {x | there exists j such that x ∈ Aj} and ∩j Aj= {x | for every j we have x ∈ Aj} 40.3. (Generalized DeMorgan’s Law) (10a)'(∪j Aj)c=∩j Ajc; (10b)'(∩j Aj)c=∪j Ajc Probability Spaces and Probability Functions DEFINITION 40.2: Let P be a real-valued function defined on the class C of events of a sample space S. Then P is called a probability function, and P(A) is called the probability of an event A, when the following axioms hold: Axiom [P1] For every event A, P(A) ≥ 0. Axiom [P2] For the certain event S, P(S) = 1. Axiom [P3] For any sequence of mutually exclusive (disjoint) events A1, A2,…, P(A1∪ A2∪… )= P(A1)+ P(A2)+… The triple (S, C, P), or simply S when C and P are understood, is called a probability space. Axiom [P3]implies an analogous axiom for any finite number of sets. That is: Axiom [P3'] For any finite collection of mutually exclusive events A1, A2,…, An, P(A1∪ A2∪…∪ An)= P(A1)+ P(A2)+…+ P(An) In particular, for two disjoint events A and B, we have P(A ∪ B ) = P(A) + P(B). The following properties follow directly from the above axioms. 40.4. (Complement rule) P(Ac)= 1 – P(A). Thus, P( ∅)= 0. 40.5. (Difference Rule) P(A\B) = P(A) – P(A ∩ B). 40.6. (Addition Rule) P(A ∪ B)= P(A) + P(B) – P(A ∩ B). 40.7. For n ≥ 2, P( Aj jn =1∪)≤ PAj jn()=∑1 40.8. (Monoticity Rule) If A ⊆ B, then P(A) ≤ P(B). Limits of Sequences of Events 40.9. (Continuity) Suppose A1, A2,… form a monotonic increasing (decreasing) sequence of events; that is, Aj⊆ Aj+1 (Aj⊇ Aj+1). Let A = ∪jjA (A = ∩j Aj). Then lim P(An) exists and lim P(An)= P(A) For any sequence of events A1, A2,…, we define lim inf An=k=+∞ 1∪jk=+∞∩ Aj and lim sup An=k=+∞ 1∩jk=+∞∪ Aj If lim inf An= lim sup An, then we call this set lim An. Note lim An exists when the sequence is monotonic.PROBABILITY 218 40.10. For any sequence Aj of events in a probability space, P(lim inf An)≤ lim inf P(An)≤ lim sup P(An)≤ P(lim sup An) Thus, if lim An exists, then P(lim An)= lim P(An). 40.11. For any sequence Aj of events in a probability space, P(∪j Aj)≤j∑ P(Aj). 40.12. (Borel-Cantelli Lemma) Suppose Aj is any sequence of events in a probability space. Furthermore, supposen=+∞∑1 P(An) < +∞ . Then P(lim sup An)= 0. 40.13. (Extension Theorem) Let F be a field of subsets of S. Let P be a function on F satisfying Axioms P1, P2, and P3¢. Then there exists a unique probability function P* on the smallest σ -field containing F such that P* is equal to P on F. Conditional Probability DEFINITION 40.3: Let E be an event with P(E) > 0. The conditional probability of an event A given E is denoted and defined as follows: P(A|E)=PA E PE() ()∩ 40.14. (Multiplication Theorem for Conditional Probability) P(A ∩ B) = P(A)P(B |A). This theorem can be genealized as follows: 40.15. P(A1∩ … ∩ An)= P(A1)P(A2|A1)P(A3|A1∩ A2)… P(An|A1∩ … ∩ An-1) EXAMPLE 40.1: A lot contains 12 items of which 4 are defective. Three items are drawn at random from the lot one after the other. Find the probabiliy that all three are nondefective. The probability that the first item is nondefective is 8/12. Assuming the first item is nondefective, the probability that the second item is nondefective is 7/11. Assuming the first and second items are nondefec- tive, the probability that the third item is nondefective is 6/10. Thus, p=8 127 116 1014 55⋅⋅= Stochastic Processes and Probability Tree Diagrams A (finite) stochastic process is a finite sequence of experiments where each experiment has a finite num- ber of outcomes with given probabilities. A convenient way of describing such a process is by means of a probability tree diagram, illustrated below, where the multiplication theorem (40.14) is used to compute the probability of an event which is represented by a given path of the tree. EXAMPLE 40.2: Let X, Y, Z be three coins in a box where X is a fair coin, Y is two-headed, and Z is weighted so the probability of heads is 1/3. A coin is selected at random and is tossed. (a) Find P(H), the probability that heads appears. (b) Find P(X|H), the probability that the fair coin X was picked if heads appears. PROBABILITY 219 The probability tree diagram corresponding to the two-step stochastic process appears in Fig. 40-1a. (a) Heads appears on three of the paths (from left to right); hence, P(H) =1 31 2⋅+1 31⋅+1 31 3⋅=11 18 (b) X and heads H appear only along the top path; hence P(X∩ H)=1 31 2⋅=1 6 and so P(X |H)=PX H PH() ()∩=16 11 18/ /=3 11 H T H H T 2/31/31/21/2 1/3X Y Zo1 1/3 1/3 (a)D D D 5%4%3% NNA B C30%50% 20%o N (b) Fig. 40-1 Law of Total Probability and Bayes’ Theorem Here we assume E is an event in a sample space S, and A1, A2,… An are mutually disjoint events whose union is S; that is, the events A1, A2,…, An form a partition of S. 40.16. (Law of Total Probability) P(E) = P(A1)P(E|A1)+ P(A2)P(E|A2)+…+ P(An)P(E|An) 40.17. (Bayes’ Formula) For k = 1, 2, …, n, P(Ak|E)=PA PE A PEkk() ( ) ()|=PA PE A PA PE A PA PE A Pkk() ( ) () ( ) () ( )| ||11 2 2+ +⋅⋅⋅+ (() ( )AP EAnn| EXAMPLE 40.3: Three machines, A, B, C, produce, respectively, 50%, 30%, and 20% of the total number of items in a factory. The percentages of defective output of these machines are, respectively, 3%, 4%, and 5%. An item is ran-domly selected. (a) Find P(D), the probability the item is defective. (b) If the item is defective, find the probability it came from machine: (i) A, (ii) B, (iii) C.(a) By 40.16 (Total Probability Law), P(D) = P(A)P(D |A)+ P(B)P(D |B)+ P(C)P(D |C) = (0.50)(0.03) + (0.30)(0.04) + (0.20)(0.05) = 3.7% (b) By 40.17 (Bayes’ rule), (i) P(A |D)= PAPD A PD()( ) ()|=(. ) (. ) .05 0 00 3 0 037= 40.5%. Similarly, (ii) P(B|D)=PBPD B PD()( ) ()|= 32.5%; (iii) P(C |D)=PCPD C PD()( ) ()|= 27.0%PROBABILITY 220 Alternately, we may consider this problem as a two-step stochastic process with a probability tree dia- gram, as in Fig. 40-1(b). We find P(D) by adding the three probability paths to D: (0.50)(0.03) + (0.30)(0.04) + (0.20)(0.05) = 3.7% We find P(A |D) by dividing the top path to A and D by the sum of the three paths to D. (0.50)(0.03)/0.037 = 40.5% Similarly, we find P(B |D)= 32.5% and P(C |D)= 27.0%. Independent Events DEFINITION 40.4: Events A and B are independent if P(A ∩ B)= P(A)P(B). 40.18. The following are equivalent: (i) P(A ∩ B) = P(A)P(B), (ii) P(A |B)= P(A), (iii) P(B |A)= P(B). That is, events A and B are independent if the occurrence of one of them does not influence the occur- rence of the other. EXAMPLE 40.4: Consider the following events for a family with children where we assume the sample space S is an equiprobable space: E= {children of both sexes}, F = {at most one boy} (a) Show that E and F are independent events if a family has three children. (b) Show that E and F are dependent events if a family has two children. (a) Here S = {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}. So: E = {bbg, bgb, bgg, gbb, gbg, ggb}, P(E) = 6/8 = 3/4, F = {bgg, gbg, ggb, ggg}, P(F) = 4/8 = 1/2 E ∩ F = {bgg, gbg, ggb}, P(E ∩ F)= 3/8 Therefore, P(E)P(F) = (3/4)(1/2) = 3/8 = P(E ∩ F). Hence, E and F are independent. (b) Here S = {bb, bg, gb, gg}. So: E= {bg, gb}, P(E) = 2/4 = 1/2, F = {bg, gb, gg}, P(F) = 3/4 E ∩ F = {bg, gb}, P(E ∩ F)= 2/4 = 1/2 Therefore, P(E)P(F) = (1/2)(3/4) = 3/8 ≠ P(E ∩ F). Hence, E and F are dependent. DEFINITION 40.5: For n > 2, the events A1, A2,…, An are independent if any proper subset of them is independent and P(A1∩ A2∩ …∩ An)= P(A1)P(A2)… P(An) Observe that induction is used in this definition. DEFINITION 40.6: A collection {Aj| j ∈ J} of events is independent if, for any n > 0, the sets Aj1, Aj2,…, Ajn are in- dependent. The concept of independent repeated trials, when S is a finite set, is formalized as follows. PROBABILITY 221 DEFINITION 40.7: Let S be a finite probability space. The probability space of n independent trials or repeated trials, denoted by Sn, consists of ordered n-tuples (s1, s2,…, sn) of elements of S with the probability of an n-tuple defined by P((s1, s2,…, sn))= P(s1)P(s2)… P(sn) EXAMPLE 40.5: Suppose whenever horses a, b, c race together, their respective probabilities of winning are 20%, 30%, and 50%. That is, S = {a, b, c} with P(a) = 0.2, P(b) = 0.3, and P(c) = 0.5. They race three times. Find the probability that (a) the same horse wins all three times (b) each horse wins once (a) Writing xyz for (x, y, z), we seek the probability of the event A = {aaa, bbb, ccc}. Here, P(aaa) = (0.2)3= 0.008, P(bbb) = (0.3)3= 0.027, P(ccc) = (0.5)3= 0.125 Thus, P(A) = 0.008 + 0.027 + 0.125 = 0.160. (b) We seek the probability of the event B = {abc, acb, bac, bca, cab, cba}. Each element in B has the same probability (0.2)(0.3)(0.5) = 0.03. Thus, P(B) = 6(0.03) = 0.18.PROBABILITY 222 41 RANDOM VARIABLES Consider a probability space (S, C, P). DEFINITION 41.1. A random variable X on the sample space S is a function from S into the set R of real numbers such that the preimage of every interval of R is an event of S. If S is a discrete sample space in which every subset of S is an event, then every real-valued function on S is a random variable. On the other hand, if S is uncountable, then certain real-valued functions on S may not be random variables. Let X be a random variable on S, where we let R X denote the range of X; that is, RX= {x | there exists s ∈ S for which X(s) = x} There are two cases that we treat separately. (i) X is a discrete random variable; that is, RX is finite or countable. (ii) X is a continuous random variable; that is, RX is a continuum of numbers such as an interval or a union of intervals. Let X and Y be random variables on the same sample space S. Then, as usual, X + Y, X + k, kX, and XY (where k is a real number) are the functions on S defined as follows (where s is any point in S): (X+ Y)(s) = X(s) + Y(s), (kX)(s) = kX(s), (X + k)(s) = X(s) + k, (XY)(s) = X(s)Y(s). More generally, for any polynomial, exponential, or continuous function h(t), we define h(X) to be the function on S defined by [h(X)](s) = h[X(s)] One can show that these are also random variables on S. The following short notation is used: P(X= xi) denotes the probability that X = xi. P(a ≤ X ≤ b denotes the probability that X lies in the closed interval [a, b]. μX or E(X) or simply μ denotes the mean or expectation of X. σX2 or Var(X) or simply σ2 denotes the variance of X. σX or simply σ denotes the standard deviation of X. Sometimes we let Y be a random variable such that Y = g(X), that is, where Y is some function of X. Discrete Random Variables Here X is a random variable with only a finite or countable number of values, say RX= {x1, x2, x3,…}where, say, x1 < x2, < x3 < …. Then X induces a function f(x) on RX as follows: f(xi)= P(X = xi)= P({s ∈ S | X(s) = xi}) The function f(x) has the following properties: (i) f(xi)≥ 0 and (ii) Σi f(xi)= 1 Thus, f defines a probability function on the range RX of X. The pair (xi, f(xi)), usually given by a table, is called the probability distribution or probability mass function of X. 223 Mean 41.1. μX= E(X) =Σ xif(xi) Here, Y = g(X). 41.2. μY= E(Y) =Σ g(xi) f(xi) Variance and Standard Deviation 41.3. σX2= Var(X) =Σ(xi – m)2f(xi)=E((X – m)2) Alternately, Var(X) =s2 may be obtained as follows: 41.4. Var(X) =Sxi2f(xi) – m2=E(X2) – m2 41.5. σX=Var X() =EX()22−μ REMARK: Both the variance Var(X) = s2and the standard deviation s measure the weighted spread of the values xi about the mean m; however, the standard deviation has the same units as m. EXAMPLE 41.1: Suppose X has the following probability distribution: x246 8 f(x) 0.1 0.2 0.3 0.4 Then: m= E(X) =Σ xif(xi)= 2(0.1) + 4(0.2) + 6(0.3) + 8(0.4) = 6 E(X2)=Σxi2f(xi)= 22(0.1) + 42(0.2) + 62(0.3) + 82(0.4) = 40 s2= Var(X) = E(X2)−m2= 40 − 36 = 4 s=Var X() =4=2 Continuous Random Variable Here X is a random variable with a continuum number of values. Then X determines a function f(x), called thedensity function of X, such that (i) f(x) ≥ 0 and (ii) −∞∞∫ f(x) dx = fxd x R()∫= 1 Furthermore, P(a≤ X ≤ b) = ab∫ f(x) dx Mean 41.6. μX= E(X) = −∞∞∫xf(x) dx Here, Y = g(X). 41.7. μY= E(Y) = −∞∞∫ g(x) f(x) dx RANDOM VARIABLES 224 Variance and Standard Deviation 41.8. σX2= Var(X) = −∞∞∫ (x − m)2f(x)dx =E((X −m)2) Alternately, Var(X) =s2 may be obtained as follows: 41.9. Var(X) = −∞∞∫ x2f(x)dx −m2=E(X2)−m2 41.10. sX=Var X() =EX()22−μ EXAMPLE 41.2: Let X be the continuous random variable with the following density function: f(x)=(/ )12 0 2 0xi fx elsewhere≤≤ ⎧⎨⎩ Then: E(X) = −∞∞∫xf(x) dx =1 202∫x2 dx =x3 02 6⎡ ⎣⎢⎤ ⎦⎥=4 3 E(X2)= −∞∞∫x2f(x) dx =1 202∫x3 dx =x4 02 8⎡ ⎣⎢⎤ ⎦⎥= 2 s2= Var(X) = E(X2)−m2= 2 −16 9=2 9 s=Var X() =2 9=1 32 Cumulative Distribution Function Thecumulative distribution function F(x) of a random variable X is the function F:R →R defined by 41.11. F(a) = P(X ≤ a) The function F is well-defined since the inverse of the interval (−∞, a] is an event. The function F(x) has the following properties: 41.12. F(a) ≤ F(b) whenever a ≤ b. 41.13. lim x→−∞ F(x) = 0 and lim x→+∞ F(x) = 1 That is, F(x) is monotonic, and the limit of F to the left is 0 and to the right is 1. If X is the discrete random variable with distribution f(x), then F(x) is the following step function: 41.14. F(x) = xxi≤∑ f(xi) If X is a continuous random variable, then the density funcion f(x) of X can be obtained from the cum- mulative distribution function F(x) by differentiation. That is, 41.15. f(x)=d dxF(x) =F′(x) Accordingly, for a continuous random variable X, 41.16. F(x) = −∞∫x f(t) dtRANDOM VARIABLES 225 Standardized Random Variable Thestandardized random variable Z of a random variable X with mean m and standard deviation s > 0 is defined by 41.17. Z =X−μ σ Properties of such a standardized random variable Z follow: μZ= E(Z) = 0 and σZ= 1 EXAMPLE 41.3: Consider the random variable X in Example 41.1 where μX= 6 and σX= 2. The distribution of Z = (X – 6)/2 where f(z) = f(x) follows: Z −2 −10 1 f(Z) 0.1 0.2 0.3 0.4 Then: E(Z) =Σ zif(zi)= (−2)(0.1) + (−1)(0.2) + 0(0.3) + 1(0.4) = 0 E(Z2)=Σ zi2f(zi)= (−2)2(0.1) + (−1)2(0.2) + 02(0.3) + 12(0.4) = 1 Var(Z) = 1 − 02= 1 and sZ=Var X() =1 Probability Distributions 41.18. Binomial Distribution: Φ(x) = tx≤∑n t⎛ ⎝⎜⎞ ⎠⎟ ptqn-t p > 0, q > 0, p + q = 1 41.19. Poisson Distribution: Φ(x) = tx≤∑λλ te t− ! 41.20. Hypergeometric Distribution: Φ(x) = tx≤∑ z r ts nt rs n⎛ ⎝⎜⎞ ⎠⎟−⎛ ⎝⎜⎞ ⎠⎟ +⎛ ⎝⎜⎞ ⎠⎟ 41.21. Normal Distribution: Φ(x) =1 222 π−∞−∫xte/dt 41.22. Student’s t Distribution: Φ(x) =11 2 21212 nn nt nxn πΓ Γ+⎛ ⎝⎜⎞ ⎠⎟ +⎛ ⎝⎜⎞ ⎠⎟−∞−+ ∫(/)() / ddt 41.23. c2(Chi Square) Distribution: Φ(x) =1 2220nx n/(/)Γ ∫ t(n - 2)/2e-t/2 dt 41.24. F Distribution: Φ(x) =Γ ΓΓnn nn nntnn x12 12 22 1202 2212+⎛ ⎝⎜⎞ ⎠⎟ ∫// (/ ) (/ )((/ ) ( ) /()nn nnn t d t11 221 212 −− ++RANDOM VARIABLES 226 Section XII: Numerical Methods 42 INTERPOLATION Lagrange Interpolation Two-point formula 42.1. px fxxx xxfxxx xx() ( ) ( )=− −+− −01 0110 10 where p (x) is a linear polynomial interpolating two points ( , ( )), ( , ( )),xf x xf x x x00 1 1 0 1 ≠ General formula 42.2. px fx L x fx L x fx Lnn n nn () ( ) () ( ) () ( ) (,, , =+ + +00 1 1 /midhorizellipsis x x) where Lxx xxnki ki ii kn , ,=− −=≠∏ 0 and where p (x) is an nth-order polynomial interpolating n + 1 points (, () ) , , , , ;xf x k n x x i jkk i j=≠ ≠ 01… and for Remainder formula Suppose fx abn() [, ] .∈+C1 Then there is a ξ() (, )xa b∈ such that: 42.3. fx pxfx nxxxx xxn () ()(() ) () !() () ( =++−− −+1 01 1ξ/midhorizellipsisn n) Newton’s Interpolation First-order divided-difference formula 42.4. fx xfx fx xx[,]() () 0110 10=− − Two-point interpolatory formula 42.5. px f x f x x x x( ) () [,] ( )=+ −00 1 0 where p(x) is a linear polynomial interpolating two points ( , ( )), ( , ( )),xf x xf x x x00 1 1 0 1 ≠ 227 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. 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Second-order divided-difference formula 42.6. fx x xfx x fx x xx[,,][, ] [,] 01212 01 20=− − Three-point interpolatory formula 42.7. px fx fx x x x f x x x x x( ) () [,] ( ) [,,] ( )=+ − + −00 1 0 0 1 20 (()xx−1 where p(x) is a quadrant polynomial interpolating three points ( , ( )), ( , ( )), ( , ( ))xf x xf x xf x00 1 1 23 General kth-order divided-difference formula 42.8. fx x xfx x x fx x x kkk[,, ,}[, , , ] [,, , ] 0112 01 1………=−− xxxk−0 General interpolatory formula 42.9. px fx f x x x x f x x x xn ( ) () [,] ( ) [,, ,] (=+ − + +00 1 0 0 1 /midhorizellipsis… −−− −− xxx xxn 01 1)( ) ( ) /midhorizellipsis where p(x) is an nth-order polynomial interpolating n + 1 points (, () ) , , , , ;xf x k n x x i jkk i j =≠ ≠ 01… and for Remainder formula Suppose fx a bn() [,] .∈+C1Then there is a ξ() (,)xa b∈ such that 42.10. fx p xfx nxxxx xxn () ()(() ) () !() () ( =++−− −+1 01 1ξ/midhorizellipsisn n) Newton’s Forward-Difference Formula First-order forward-difference at x0 42.11. Δfx fx fx() () ()01 0=− Second-order forward difference at x0 42.12. ΔΔ Δ2 01 0fx fx fx() () ()=− General kth-order forward difference at x0 42.13. ΔΔ Δkk kfx fx fx() () ()01 11 0 =−−− Binomial coefficient 42.14. s kss s k k⎛ ⎝⎜⎞ ⎠⎟=−− +() ( ) !11/midhorizellipsis Newton’s forward-difference formula 42.15. pxn kfx kn k() ( ) =⎛ ⎝⎜⎞ ⎠⎟ =∑ 00 Δ where p(x) is an nth-order polynomial interpolating n + 1 equal spaced points (, () ) , , , , xf x x x k hk nkk k=+ =001…INTERPOLATION 228 Newton’s Backward-Difference Formula First-order backward difference at xn 42.16. ∇= −− fx fx fxnn n() () ( )1 Second-order backward difference at xn 42.17. ∇= ∇ − ∇−2 1 fx fx fxnn n() () ( ) General kth-order backward difference at xn 42.18. ∇= ∇ − ∇−− −k nk nk n fx fx fx() () ( )11 1 Newton’s backward-difference formula 42.19. pxn kfxk kn k n () ( ) ( )=−−⎛ ⎝⎜⎞ ⎠⎟∇ =∑ 1 0 where p(x) is an nth-order polynomial interpolating n + 1 equal spaced points (, () ) , , , , xf x x x k h k nkk k=+ =001… Hermite Interpolation Two-point basis polynomials 42.20. Hxx xxxx xxH100 0112 012 11 12,,() (), =−− −⎛ ⎝⎜⎞ ⎠⎟− −==−− −⎛ ⎝⎜⎞ ⎠⎟− −121 1002 102xx xxxx xx() () ˆ ()() (),ˆ ()( ,,Hx xxx xxHx xx 10 012 012 11 1 =−− −=−−x x xx02 102) ()− Two-point interpolatory formula 42.21. H x f x Hf x Hf x Hf x3 0 10 1 11 0 10 1 ( ) () () () ˆ (,, , =+ + ′ +′) )ˆ ,H11 where H3(x) is a third-order polynomial, agrees with f (x) and its first-order derivatives at two points, i.e., Hx f x Hx fx Hx f x H30 0 30 0 31 1() () , () () , () () ,= ′ =′ = ′331 1() ()xf x=′ General basis polynomials 42.22. Hxx LxLx Hnjj nj jnj nj , ,,, ()() , ˆ ( =−− ′⎛ ⎝⎜⎞ ⎠⎟= 122xxx L xjn j−)( ),2 where Lxx xxnji ji ii jn , ,=− −=≠∏ 0INTERPOLATION 229 General interpolatory formula 42.23. H x f x Hx f x Hxnj nj j nj jn j21 0+ = ==+ ′∑ () ( ) () ( ) ˆ(),, 0 0n ∑ where Hxn21+()is a (2n + 1)th-order polynomial, agrees with f (x) and its first order derivatives at n + 1 points, i.e., Hx f x Hx f x k nnk k nk k21 2101++= ′ =′ = () () , () () , , , … Remainder formula Suppose fx a bn() [,] .∈+C22Then there is a ξ() (,)xa b∈ such that 42.24. fx H xfx nxx xxnn () ()(() ) () !() ( =++−−++ 2122 02 22ξ 1 122)( )/midhorizellipsisxxn−INTERPOLATION 230 43 QUADRATURE Trapezoidal Rule Trapezoidal rule 43.1. fxd xbafa fb ab() ~ [() () ]−+ ∫ 2 Composite trapezoidal rule 43.2. fxd xhfa fa i h fb in ab() ~ () ( ) ()22 11 ++ +⎛ ⎝⎜⎞ ⎠⎟ =− ∑ ∫ where hb a n=−() / is the grid size. Simpson’s Rule Simpson’s rule 43.3. fxd xbafa fabfb ab() ~ () ()−++⎛ ⎝⎞ ⎠+⎡ ⎣⎢⎤ ⎦⎥ ∫ 642 Composite Simpson’s rule 43.4. fxd xhfx fx fx fi in i () ~ ( ) ( ) ( )/ 32402 2 22 21 +++− =− ∑ (()/ xn in ab =∑ ∫⎛ ⎝⎜⎞ ⎠⎟ 12 where n even, hb a n xa i h i ni=− = + =() / , , , , , . 01… Midpoint Rule Midpoint rule 43.5. fxd x b afab ab() ~ ( ) −+⎛ ⎝⎞ ⎠ ∫ 2 Composite midpoint rule 43.6. fxd x h fx ab i in () ~ ( )/ 22 02 ∫ ∑ = where n even, hb a n xa i h i ni=− + = + − = − +() / () , ( ) , , , , . 21 1 0 1… 231 Gaussian Quadrature Formula Legendre polynomial 43.7. Pxnd dxxn nn nn()![( ) ] =−1 212 Abscissa points and weight formulas The abscissa points xkn()and weight coefficient ωkn()are defined as follows: 43.8. xkn()=the kth zero of the Legendre polynomial Pn(x) 43.9. ωkn nkn knPx x()() ()()=′ −2 12 2 Tables for Gauss-Legendre abscissas and weights appear in Fig. 43-1. Gauss-Legendre formula in interval (–1, 1) 43.10. fxd x fx Rkn kn n kn () ( )() ()=+ =− ∑ ∫ω 111 Gauss-Legendre formula in general interval (a, b) 43.11. fxd xbafabxba ab kn kn kn ()() ()=−++− ⎛ ⎝⎞ ⎠ ∫ ∑ =22 21ω + +Rn Remainder formula 43.12. Rba n nnfnn n=− ++() ( ! ) ( )[( )!]()()21 4 32 21 2ξ for some ab<<ξ . Fig. 43-1QUADRATURE 232 44 SOLUTION of NONLINEAR EQUATIONS Here we give methods to solve nonlinear equations which come in two forms: 44.1. Nonlinear equation: f(x) = 0 44.2. Fixed point nonlinear equation: x = g(x) One can change from 44.1 to 44.2 or from 44.2 to 44.1 by settting: gx f x x f x gx x() () () ()=+ =− or Since the methods are iterative, there are two types of error estimates: 44.3. || | |fx x xnn n() <− <+/H9280/H9280or1 for some preassigned /H9280 > 0. Bisection Method The following theorem applies: Intermediate Value Theorem: Suppose f is continuous on an interval [a, b] and f (a) f(b) < 0. Then there is a root x* to f (x) = 0 in (a, b). The bisection method approximates one such solution x*. 44.4. Bisection method: Initial step: Set a0 = a and b0 = b. Repetitive step: (a) Set ca bnn n=+() / . 2 (b) If fa fcnn() () , <0 then set aann+=1 and bcnn+=1 ; else set acnn+=1 and bbnn+=1 . Newton’s Method Newton method 44.5. xxfx fxnnn n+=−′1() () Quadratic convergence 44.6. lim() (() ) nn nxx xxfx fx →∞+− −=′′ ′|| ||∗ ∗∗ ∗1 222 where x* is a root of the nonlinear equation 44.1. 233 Secant Method Secant method 44.7. xxxxf x fx fxnnnn n nn+− −=−− −11 1() ( ) () ( ) Rate of convergence 44.8. lim() ( nn nnxx xxx xfx f →∞+ −− −−=′′ ′|| || | |∗ ∗∗∗ 1 12(() )x∗2 where x* is a root of the nonlinear equation 44.1. Fixed-Point Iteration The following definition and theorem apply: Definition: A function g from (a, b) to (a, b) is called a contraction mapping if || | |gx gy L x y x y a b() () , (, )−− ∈ /H11349 for any where L < 1 is a positive constant. Fixed-point theorem: Suppose that g is a contraction mapping on (a, b). Then g has a unique fixed point in (a, b). Given such a contraction mapping g, the following method may be used. Fixed-point iteration 44.9. xg xnn+=1 ()SOLUTION OF NONLINEAR EQUATIONS 234 45 NUMERICAL METHODS for ORDINARY DIFFERENTIAL EQUATIONS Here we give methods to solve the following initial-value problem of an ordinary differential equation: 45.1. dx dtfx t xt x= =⎧ ⎨⎪ ⎩⎪(,) ()00 The methods will use a computational grid: 45.2. ttn hn=+0 where h is the grid size. First-Order Methods Forward Euler method (first-order explicit method)45.3. xt h xt h f xt t() ( )( ( ) , )+= + Backward Euler method (first-order implicit method) 45.4. x th x t h f x th th() ( )( () , )+= + + + Second-Order Methods Mid-point rule (second-order explicit method) 45.5. xx thfx t t xt h xt h f x th* *() ( () , ) () ( ) ,=+ += + +⎛ ⎝⎜2 2⎞ ⎞ ⎠⎟⎧ ⎨⎪⎪ ⎩⎪ ⎪ Trapezoidal rule (second-order implicit method) 45.6. xt h xthfx t t fx t h t h ( ) () { ( () , ) ( ( ) , ) }+= + + + +2 Heun’s method (second-order explicit method) 45.7. xx t h f x t t xt h xthfx t t*() ( () , ) ( ) () { ( () , )=+ += + +2ffx t h(, ) }*+⎧ ⎨⎪ ⎩⎪ 235 236 Single-Stage High-Order Methods Fourth-order Runge–Kutta method (fourth-order explicit method) 45.8. xt h xt F F F F() ( )( ) += + + + +1 622123 4 where Fh f x t Fh f xFthFh f xF 121 32 22 2== + +⎛ ⎝⎜⎞ ⎠⎟ =+ (,) , , , ,, , ( , )thFh f x F t h +⎛ ⎝⎜⎞ ⎠⎟ =++243 Multi-Step High-Order Methods Adams-Bashforth two-step method 45.9. x t h x t h fx t t fx t h t h() ( ) ( ( ) , ) ( () , )+= + − − −⎛ ⎝⎞ 3 21 2 ⎠ ⎠ Adams-Bashforth three-step method 45.10. x t h x t h fx t t fx t h t h() ( ) ( ( ) , ) ( () , )+= + − − −+23 124 35 5 1222 f x th th(( ) , )−−⎛ ⎝⎞ ⎠ Adams-Bashforth four-step method 45.11. x t h x t h fx t t fx t ht h()( ) ( ( ) , ) ( () ,+= + − − −55 2459 24))( ( ) , )( ( ) , )+− − −− −⎛ ⎝37 24229 2433 f x th th f x th th⎜ ⎜⎞ ⎠⎟ Milne’s method 45.12. xt h xt h h f xt t f xt h t h() ( ) ( ( ) , ) ( () , )+= − + − − − 38 34 3++− −⎛ ⎝⎞ ⎠8 322 f x th th(( ) , ) Adams-Moulton two-step method 45.13. x th x t h f x th th f x tt( ) () ( ( ) , ) ( () , )+= + + ++ −5 122 31 112f x th th(( ) , )−−⎛ ⎝⎞ ⎠ Adams-Moulton three-step method 45.14. x th x t h f x th th f x t t( ) () ( ( ) , ) ( () ,)+= + + ++ −3 819 245 5 241 2423 f x th th f x t h t h(( ) , ) (( ) , )−− + − −⎛ ⎝⎜⎞ ⎠⎟NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 46 NUMERICAL METHODS for PARTIAL DIFFERENTIAL EQUATIONS Finite-Difference Method for Poisson Equation The following is the Poisson equation in a domain (a, b) × (c, d): 46.1. ∇= ∇ =∂ ∂+∂ ∂222 22 2 ufxy, Boundary condition: 46.2. uxy gxy(,) (,) = for x = a, b o r y = c, d Computation grid: 46.3. xa i x i n yc j y j mi j=+ Δ = =+ Δ =for for01 01,, , ,, ,… … where Δ= −xb a n() / andΔ= −yd c m() / are grid sizes for x and y variables, respectively. Second-order difference approximation 46.4. () ( , ) ( , )DD u x y f x yx y ij ij22+= where Dux yux y ux y ux y xi jij i j ij 2 11 2(, )(, ) ( , )(, )=−++− Δ Δ =−++x Dux yux y ux y ux y yi jij ij i2 2 12(, )(, ) (, ) (,j j y− Δ1 2) Computational boundary condition 46.5. ux y gay ux y gby jjj n jj(,) ( ,) , (,) ( ,) , ,012 == = for … …, (, ) (, ) , (, ) (,)m ux y gx c ux y gx d iii i m i 0== = for 1 12,, ,…n Finite-Difference Method for Heat Equation The following is the heat equation in a domain (, ) (, ) (, ) :ab cd T ×× 0 46.6. ∂ ∂=∇u tu2 237 238 Boundary condition: 46.7. uxyt gxy x ab y cd(,,) (,) , , == = for or Initial condition: 46.8. uxy u xy(,,) (,) 00= Computational grid: 46.9. xa i x i n yc j y j m ti j=+ Δ = =+ Δ =for for01 01,, , ,, ,… … k kkt k=Δ = for 0 1 ,, ,… where Δ= − Δ= −xb a n yd c m() / , () / , andΔtare grid sizes for x, y and t variables, respectively. Computational boundary condition 46.10. ux y gay ux y gby jjj n jj(,) ( ,) , (,) ( ,) , ,012 === for … …, (, ) (, ) ,(, ) (,)m ux y gx c ux y gx d iii i m i 0== = for 1 12,, ,…n Computational initial condition 46.11. ux y u x y i n j mij ij(, , ) (, ) ,, ,; , , , 01 2 0 10== = for …… Forward Euler method with stability condition 46.12. ux y t ux y t tD D ux yij k ij k x y i(, , ) (, ,) ( ) (,+=+ Δ +122 jjkt,) 46.13. 22122Δ Δ+Δ Δt xt y/H11349 Backward Euler method (unconditional stable) 46.14. ux y t ux y t tD D ux yij k ij k x y i(, , ) (, ,) ( ) (,+=+ Δ +122 jjkt,)+1 Crank-Nicholson method (unconditional stable) 46.15. ux y t ux y t tD D uxij k ij k x y i(, , ) (, ,) ( ) { (,+=+ Δ +122yyt u xytjk i jk,) (, , ) } /++12 Finite-Difference Method for Wave Equation The following is a wave equation in a domain (, ) (, ) (, ) :ab cd T ×× 0 46.16.∂ ∂=∇2 222 u tAu where A is a constant representing the speed of the wave.NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 239 Boundary condition: 46.17. uxy t gxy x a b y c d(,,) (,) , , == = for or Initial condition: 46.18. uxy u xyu tuxy u xy (,,) (,) , (,,) (,) 0001=∂ ∂= Computational grids: 46.19. xa i x i n yc j y j m ti j=+ Δ = =+ Δ =for for01 01,, , ,, ,… … k kkt k=Δ = − for 1 0 1 ,, ,… where Δ= − Δ= −xb a n yd c m() / , () / , andΔtare the grid sizes for x, y, and t variables, respectively. A second-order finite-difference approximation 46.20. ux y t ux y t ux y t tijk ijk ijk(, , ) (, , ) (, , )+−=−+ Δ112222 2 2AD Du xytxyi j k() ( , , )+ Computational boundary condition 46.21. ux y gay ux y gby jjj n jj(,) ( ,) , (,) ( ,) , ,012 === for … …, (, ) (, ) ,(, ) (,)m ux y gx c ux y gx d iii i m i 0== = for 1 12,, ,…n Computational initial condition 46.22. ux y t u x y i n jij ij(, ,) (, ) ,, , ; , , ,0012 01 == = for …… m m ux y t u x y tu x y iij ij ij(, , ) (, ) (, ) ,−=+ Δ =102 11 for 2 20 1,, ; , ,,……nj m= Stability condition 46.23. ΔΔ ΔtA x x/H11349min( , )NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 47 ITERATION METHODS for LINEAR SYSTEMS Iteration Methods for Poisson Equation The finite-difference approximation to the Poisson equation follows: 47.1. uuuu u f ii j i j ij ij ij ij+− +−+++−=11 114,, ,, , ,for , , , , , ,, ,,, ,jn uu j n ujn j i=− == = −12 1 01 2 10 0… … for === = −⎧ ⎨⎪⎪ ⎩⎪ ⎪ui nin,,, , 01 2 1 for … Three iteration methods for solving the system follow: Jacobi method 47.2. u uuuuijk ijk ijk ijk ijk ,, , , , (+ +− +− = +++−1 11 111 4f fij,) Gauss-Seidel method 47.3. u uuuuijk ijk ijk ijk ij ,, , , , (+ +−+ +− = +++1 111 111 4k k ijf+−1 ,) Successive-overrelaxation (SOR) method 47.4.uu u u u fij i jk ij i jk ij i ,* ,,* ,,*(=+ + + −+− +−1 411 11 , , ,, ,*) ()j ijk ijk ijuu u+=− +⎧ ⎨⎪ ⎩⎪11ωω Iteration Methods for General Linear Systems Consider the linear system 47.5. Ax = b where A is an n × n matrix and x and b are n-vectors. We assume the coefficient matrix A is partitioned as follows: 47.6. A = D – L – U where D = diag (A), Lis the negative of the strictly lower triangular part of A, and U is the negative of the strictly upper triangular part of A. 240 241 Four iteration methods for solving the system follow: Richardson method 47.7. xI A x bkk+=− +1() Jacobi method 47.8. Dx L U x bkk+=+ +1() Gauss-Seidel method 47.9. ()DL x U x bkk−= ++1 Successive-overrelaxation (SOR) method 47.10. () () ( )DL x U x b D xkk k−= + + −+ωω ω11ITERATION METHODS FOR LINEAR SYSTEMS This page intentionally left blank TABLESPART B Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. This page intentionally left blank Section I: Logarithmic, Trigonometric, Exponential Functions 1FOUR PLACE COMMON LOGARITHMS log10N or log N 245 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 1FOUR PLACE COMMON LOGARITHMS log10N or log N(Continued) 246 2Sin x (x in degrees and minutes) 247 3Cos x ( x in degrees and minutes) 248 4Tan x ( x in degrees and minutes) 249 5CONVERSION OF RADIANS TO DEGREES, MINUTES, AND SECONDS OR FRACTIONS OF DEGREES 250 6 CONVERSION OF DEGREES, MINUTES, AND SECONDS TO RADIANS 251 7NATURAL OR NAPIERIAN LOGARITHMS loge x or ln x 252 ln 10 = 2.30259 4 ln 10 = 9.21034 7 ln 10 = 16.11810 2 ln 10 = 4.60517 5 ln 10 = 11.51293 8 ln 10 = 18.42068 3 ln 10 = 6.90776 6 ln 10 = 13.81551 9 ln 10 = 20.72327 7NATURAL OR NAPIERIAN LOGARITHMS loge x or ln x(Continued) 253 8EXPONENTIAL FUNCTIONS ex 254 9EXPONENTIAL FUNCTIONS e–x 255 10EXPONENTIAL, SINE, AND COSINE INTEGRALS Ei Si Ci () , ()sin,( )cosxe udu xu udu xu uu xx== =−∞∫0∫ ∫ ∫∞ xdu 256 Section II: Factorial and Gamma Function, Binomial Coefficients 11FACTORIAL n nn!=123iii /midhorizellipsis i 257 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 12 GAMMA FUNCTION Γ()xt e d t xxt x=−−∞∫112for /H11017/H11017 [For other values use the formula Γ(x + 1) = x Γ(x)] 258 13BINOMIAL COEFFICIENTS n kn kn knn n k kn nk⎛ ⎝⎜⎞ ⎠⎟=−=−− +=−⎛ ⎝! !( )!() ( ) !11/midhorizellipsis ⎜ ⎜⎞ ⎠⎟ = ,!01 Note that each number is the sum of two numbers in the row above; one of these numbers is in the same col- umn and the other is in the preceding column (e.g., 56 = 35 + 21). The arrangement is often called Pascal’s triangle (see 3.6, page 8). 259 13BINOMIAL COEFFICIENTS n kn kn knn n k kn nk⎛ ⎝⎜⎞ ⎠⎟=−=−− +=−⎛ ⎝! !( )!() ( ) !11/midhorizellipsis ⎜ ⎜⎞ ⎠⎟ = ,!01 (Continued) 260For k > 15 use the fact that n kn nk⎛ ⎝⎜⎞ ⎠⎟=−⎛ ⎝⎜⎞ ⎠⎟. Section III: Bessel Functions 14BESSEL FUNCTIONS J0(x) 15BESSEL FUNCTIONS J1(x) 261 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 16BESSEL FUNCTIONS Y0(x) 17BESSEL FUNCTIONS Y1(x) 262 18BESSEL FUNCTIONS I0(x) 19BESSEL FUNCTIONS I1(x) 263 21BESSEL FUNCTIONS K1(x)20BESSEL FUNCTIONS K0(x) 264 22BESSEL FUNCTIONS Ber(x) 23BESSEL FUNCTIONS Bei(x) 265 24BESSEL FUNCTIONS Ker (x) 25BESSEL FUNCTIONS Kei(x) 266 26VALUES FOR APPROXIMATE ZEROS OF BESSEL FUNCTIONS The following table lists the first few positive roots of various equations. Note that for all cases listed the successive large roots differ approximately by π = 3.14159. . . . 267 Section IV: Legendre Polynomials 27LEGENDRE POLYNOMIALS Pn(x) [P0(x)=1, P1(x)=x] 268 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 28LEGENDRE POLYNOMIALS Pn(cos /H9258) [ P0(cos /H9258)=1] 269 Section V: Elliptic Integrals 29 COMPLETE ELLIPTIC INTEGRALS OF FIRST AND SECOND KINDS Kd kEk d k = −=− = ∫∫θ θθθππ 11 22 0222 02 sin, sin , sin//ψ ψ 270 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 30 INCOMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND Fkd kk (, ) sin,s i n φθ θψφ= −= ∫122 0 31INCOMPLETE ELLIPTIC INTEGRAL OF THE SECOND KIND Ek k d k ( , ) sin , sin φθ θ ψφ=− =∫122 0 271 Section VI: Financial Tables 32COMPOUND AMOUNT: (1 + r)n If a principal P is deposited at interest rate r (in decimals) compounded annually, then at the end of n years the accumulated amount A = P(1 + r)n. 272 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 33PRESENT VALUE OF AN AMOUNT: (1 /H11545 r)/H11546n The present value P which will amount to A in n years at an interest rate of r (in decimals) compounded annually is P = A(1 + r)/H11002n. 273 34AMOUNT OF AN ANNUITY:(1 1+-r rn) If a principal P is deposited at the end of each year at interest rate r (in decimals) compounded annually, then at the end of n years the accumulated amount is Pr rn()11−−⎡ ⎣⎢⎤ ⎦⎥. The process is often called an annuity. 274 35PRESENT VALUE OF AN ANNUITY:1( 1-+-r rn) An annuity in which the yearly payment at the end of each of n years is A at an interest rate r (in decimals) compounded annually has present value Ar rn11−+⎡ ⎣⎢⎤ ⎦⎥−(). 275 Section VII: Probability and Statistics 36AREAS UNDER THE STANDARD NORMAL CURVE from −∞ to x Φ()/xe d ttx=− −∞∫1 222 π NOTE: erf (x) = 2Φ(x 2)− 1 276 Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. Click here for terms of use. 37 ORDINATES OF THE STANDARD NORMAL CURVE yex=− 1 222 π/ 277 38 PERCENTILE VALUES (tp) FOR STUDENT'S t DISTRIBUTION with ndegrees of freedom (shaded area = p) 278 39PERCENTILE VALUES (/H92732 p) FOR /H92732 (CHI-SQUARE) DISTRIBUTION with n degrees of freedom (shaded area = p) 279 4095th PERCENTILE VALUES FOR THE F DISTRIBUTION n1 = degrees of freedom for numerator n2 = degrees of freedom for denominator (shaded area = .95) 280 4199th PERCENTILE VALUES FOR THE F DISTRIBUTION n1 = degrees of freedom for numerator n2 = degrees of freedom for denominator (shaded area = .99) 281 42RANDOM NUMBERS 282 283The following list show special symbols and notations together with pages on which they are defined or first appear. Cases where a symbol has more than one meaning will be clear from the context. Symbols Bern(x), Bein(x) Ber and Bei functions, 157 B(m, n) beta function, 152 Bb Bernoulli numbers, 142 C(x) Fresnel cosine integral, 204 Ci(x) cosine integral, 204 e1, e2, e3 unit vectors in curvilinear coordinates, 127 erf(x) error function, 203 erfc(x) complementary error function, 203 E = E(k, p /2) complete elliptic integral of the second kind, 198 E = E(k, f) incomplete elliptic integral of the second kind, 198 Ei(x) exponential integral, 203 E n Euler number, 142 E(X) mean or expectation of random variable X, 223 f[x0, x1, ..., xk] divided distance formula, 287, 288 F(a), F(x) cumulative distribution function, 209 F(a, b; c; x) hypergeometric function, 178 F(k, f ) incomplete elliptic integral of the first kind, 198 g, g−1 Fourier transform and inverse Fourier transform, 194 G. M. geometric mean, 209 h 1, h2, h3 scale factors in curvilinear coordinates, 127 Hn(x) Hermite polynomial, 169 Hn(1)(x), Hn(2)(x) Hankel functions of the first and second kind, 155 H. M. harmonic mean, 210 i, j, k unit vectors in rectangular coordinates, 120 I n(x) modified Bessel function of the first kind, 155 Jn(x) Bessel function of the first kind, 153 K = F(k, p /2) complete elliptic integral of the first kind, 198 Kern(x), Kein(x) Ker and Kei functions, 158 Kn(x) modified Bessel function of the second kind, 156 ln x or loge x natural logarithm of x, 53 log x or log10 x common logarithm of x, 53 Ln(x) Laguerre polynomials, 171 Lnm(x) associated Laguerre polynomials, 173 l, l−1 Laplace transform and inverse Laplace transform, 180 M.D. mean deviation P(A/E) conditional probability of A given E, 219 P n(x) Legendre polynomials, 164 Pnm(x) associated Legendre polynomials, 173 QU, M, QL quartiles, 211 Qn(x) Legendre functions of second kind, 167 Qnm(x) associated Legendre functions of second kind, 168 r sample correlation coefficient, 213 R.M.S. root-mean-square, 211 s sample standard deviation, 208 s 2 sample variance, 210 sxy sample covariance, 213 Si(x) Sine integral, 203 S(x) Fresnel sine integral, 204 T n(x) Chebyshev polynomials of first kind, 175Index of Special Symbols and Notations Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. 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INDEX OF SPECIAL SYMBOLS AND NOTATIONS 284 Un(x) Chebyshev polynomials of second kind, 176 Var(X) variance of random variable X, 224 xx, sample mean, grand mean, 208, 209 xk(n) kth zero of Legendre polynomial Pn(x), 232 Yn(x) Bessel function of second kind, 153 Z standardized random variable, 226 Greek Symbols ar rth moment in standard units, 212 p pi, 3 g Euler’s constant, 4 f spherical coordinate, 38 Γ(x) gamma function, 149 Φ (p) sum 11 21 310 0 154 ++++ = /midhorizellipsisp,( ) ,Φ ζ(x) Rieman zeta function, 204 m population mean, 208 Φ (x) probability distribution function, 226 q coordinate: cylindrical 37, s population standard deviation, 223 polar, 11, 24; spherical, 38 s 2 population variance, 223 Notations A ~ B A is asymptotic to B or A/B approaches 1, 151 |A| absolute value of A = Ai f A AifA≥ −<⎧⎨⎩0 0 n! f actorial n, 7 n k⎛ ⎝⎜⎞ ⎠⎟ binomial coefficients, 8 ydy dxfx ydy dxfx e t c/H11032/H11032 /H11032/H11032 /H11032/H11032== ==⎫ ⎬⎪ ⎭⎪() () , .2 2 derivatives of y or f(x) with respect to x, 62 Dd dxpp p= pth deriv ative with respect to x, 64 ∂ ∂∂ ∂∂ ∂∂f xf xf xyetc ,, , .2 partial derivatives, 65 ∂ ∂(,,) (, , )xyz uuu123 Jacobian, 128 fx d x()∫ indefinite integral, 67 fx d x ab()∫ definite integral, 108 Aidr C∫ line integral of A along C, 124 ABi dot product of A and B, 120 A × B cross product of A and B, 121 ∇ del operator, 122 ∇= ∇∇2i Laplacian operator, 123 ∇4 = ∇2 (∇2) biharmonic operator, 123 285Adams-Bashforth methods, 236 Adams-Moulton methods, 236Addition formula: Bessel functions, 163 Hermite polynomials, 170 Addition rule (probability) 208Addition of vectors, 119Algebra of sets, 217Algebraic equations, solutions of, 13Alphabet, Greek, 3Analytic geometry, plane, 22–33 solid, 34–40 Annuity table, 274Anti-derivative, 67 Anti-logarithms, 53 Arithmetic: mean, 208series, 134 Arithmetic-geometric series, 134Associated Laguerre polynomials, 173 (See also Laguerre polynomials) Associated Legendre functions, 164 (See also Legendre functions)of the first kind, 168of the second kind, 168 Asymptotic expansions or formulas: Bernoulli numbers, 143 Bessel functions, 160 Backward difference formulas, 228 Her and Bei functions, 157 Bayes formula, 220Bernoulli numbers, 142 asymptotic formula, 143series, 143 Bernoulli’s differential equation, 116Bessel functions, 153–164 graphs, 159integral representation, 161modified, 155recurrence formulas, 154, 157series, orthogonal, 161tables, 261–267 Bessel’s differential equation, 118, 153 general solution, 154modified differential equation, 155 Best fit, line of, 214Beta function, 152 Biharmonic operator, 123Binomial: coefficients, 7, 228, 259distribution, 226 formula, 7series, 136 Bipolar coordinates, 131Bisection method, 223 Bivariate data, 212 Carioid, 29 Cassini, ovals of, 32Catalan’s constant, 200 Catenary, 29Cauchy or Euler differential equation, 117 Cauchy’s form of remainder in Taylor series, 134 Cauchy-Schwarz inequality, 205 for integrals, 206 Central tendency, 208Chain rule for derivatives, 67 Chebyshev polynomials, 175 of the first kind, 175of the second kind, 176recurrence formula, 175 Chebyshev’s differential equation, 175 general solution, 177 Chebyshev’s inequality, 206 Chi-square distribution, 226 table of values, 279 Circle, 17, 25Coefficient: of excess (kurtosis), 212 of skewness, 212 Coefficients: binomial, 7 multinomial, 9 Complementary error function, 203Complex: conjugate, 10numbers, 10logarithm of, 55plane, 10 Components of a vector, 120 Compound amount, 262Confocal: ellipsdoidal coordinates, 133paraboloidal coordinates, 133 Conical coordinates, 129Conics, 25 (See also Ellipse, Parabola, Hyperbola) Conjugate, complex, 10Constant of integration, 67 Constants, 3 series of, 134 Continuous random variable, 224Convergence, interval of, 138.Conversion factors, 15Convolution theorem, Fourier transform, 194Coordinates, 127 bipolar, 131confocal ellipsoidal, 133 confocal paraboloidal, 133 conical, 132curvilinear, 127cylindrical, 129elliptic cylindrical, 130 oblate spheroidal, 131 paraboloidal, 130prolate spheroidal, 131 spherical, 129toroidal, 132 Correlation coefficient, 213 Cosine, 43 graph of, 46table of values, 245 Cosine integral, 203, 256 Cosines, law of, 51Covariance, 213Cross or vector product, 121 Cubic equation, solution of, 13 Index Copyright © 2009, 1999, 1968 by The McGraw-Hill Companies, Inc. 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INDEX 286 Cumulative distribution function, 225 Curl, 123Curve fitting, 215Curvilinear coordinates, 134 Cycloid, 28Cylindrical coordinates, 37, 129 Definite integrals, 108–116 approximate formula, 109definition of, 108 Degrees, conversion to radians, 251Del operator, 122 DeMoivre’s theorem, 11 Derivatives, 62–66 chain rule for, 62higher, 64Leibniz’s rule, 64of vectors, 122 Deviation: mean, 210standard, 210 Differential equations, numerical methods for solution: ordinary, 235–236partial, 237–240 Differentials, 65, 66Differentiation, 62–66 (See also Derivatives) Direction numbers, 34 cosines, 34 Discrete random variable, 223Distributions, probability, 226Divergence, 122, 128 theorem, 126 Divided-difference formula (general), 228Dot or scalar product, 120Double integrals, 125 Eccentricity, 25 Ellipse, 18, 25Ellipsoid, 39Elliptic cylinder, 41Elliptic cylindrical coordinates, 130Elliptic functions, 198–202 Jacobi’s, 199series expansion, 200 Elliptic integrals, 198–199 table of values, 270–271 Epicycloid, 30Equality of vectors, 119 Equations, algebraic, 13Error functions, 203Euler: constant, 4differential equation, 117methods, 235numbers, 142 Euler-Maclaurin summation formula, 137 Exact differential equation, 116Excess, coefficient of kurtosis, 212 Exponential curve (least-squares), 215 Exponential function, 53–54 series for, 139table of values, 254–255 Exponential integral, 203, 256Exponents, 53 F distribution, 226 table of values, 280–281 Factorial n, 7 table of values, 257 Factors, special, 5 Financial tables, 272–275 Finite-difference methods for solution of: heat equation, 237Poisson equation, 237wave equation, 238 First-order divided-difference formula, 227 Five number summary [L, Q L, M, QH,H], 211 Fixed-point iteration, 234Folium of Descartes, 31Forward difference formulas, 228Fourier series, 144–146 Fourier transform, 193 convolution of, 194 cosine, 194, 197Parseval’s identity for, 193sine, 194, 196tables, 195–199 Fourier’s integral theorem, 193Fresnel sine and cosine integral, 204Frullani’s integral, 115 Gamma function, 149, 150 relation to beta function, 152table of values, 258 Gauss’ theorem, 126 Gauss-Legendre formula, 232Gauss-Seidel method, 230Gaussian quadrature formula, 231Generating functions, 157, 165, 168, 169, 171, 173, 175, 176Geometric: mean (G.M.), 209 series, 134 Geometry, 16–21 analytic, 22–40 Gradient, 122, 128Grand mean, 209Greek alphabet, 3Green’s theorem, 126Griggsian logarithms, 53 Half angle formulas, 48 Half rectified sine wave function, 191Hankel functions, 155Harmonic mean, 209Heat equation, 237 Heaviside’s unit function, 192Hermite: interpolation,229polynomials, 169–170 Hermite’s differential equation, 169Heun’s method, 235Holder’s inequality, 205 for integrals, 206 Homogeneous differential equation, 116 linear second order, 117 Hyperbola, 25Hyperbolic functions, 56–61 graphs of, 59 inverse, 59–61series for, 140 Hyperboloid, 39 Hypergeometric: differential equation, 178distribution, 226 functions, 178 Hypocycloid, 28, 30 Imaginary part of a complex number, 10 Indefinite integrals, 67–107 definition of, 67 tables of, 71–107 transformation of, 69 Independent events, 221 INDEX 287 Inequalities, 205 Infinite products, 207Integral calculus, fundamental theorem, 108 Integrals: definite (see Definite integrals)improper, 108indefinite (see Indefinite integrals)line, 124multiple, 125 surface, 125 Integration, 64 (See also Integrals) constant of, 67general rules, 67–69 Integration by parts, 67 generalized, 69 Intercepts, 22Interest, 272–275Intermediate Value Theorem, 233Interpolation, 227 Hermite, 229 Interpolatory formula (general), 228Interquartile range, 211Interval of convergence, 138Inverse: hyperbolic functions, 59–61Laplace transforms, 180trigonometric functions, 49–51 Iteration methods, 240 for general linear systems, 240for Poisson equation, 240 Jacobi method, 240 Jacobi’s elliptic functions, 199 Jacobian, 128 Ker and Kei functions, 158–159 Kurtosis, 212 Lagrange: form of remainder, 138interpolation, 227 Laguerre polynomials, 172 generating function for, 173recurrence formula, 192 Laguerre’s associated differential equation, 170 Laguerre’s differential equation, 172Landen’s transformation, 199Laplace transform, 180–192 complex inversion formula for, 180definition of, 180inverse, 180tables of, 181–192 Laplacian, 123, 128Least-squares: curve, 215line, 214 Legendre functions, 164–168 of the second kind, 166 Legendre polynomial, 164–165, 232 generating function for, 164recurrence formula for, 166tables of values for, 269 Legendre’s associated differential equation, 168 Legendre’s differential equation, 118, 164Leibniz’s rule, 64Lemniscate, 28Limacon of Pascal, 32Line, 22, 35 of best fit, 214regression, 214 Line integral, 124Logarithmic functions, 53–55 (See also Logarithms) series for, 139table of values, 245–246, 252–253 Logarithms, 53–55 of complex numbers, 55Griggsian, 53 Maclaurin series, 138 Mean, 208 continuous random variable, 224 deviation (M.D.), 211discrete random variable, 223geometric, 209grand, 209harmonic, 209population, 212weighted. 209 Mean value theorem, for definite integrals, 108generalized, 109 Median, 208Midpoint rule, 231, 235 Midrange, 210Milne’s method, 236Minkowski’s inequality, 206 for integrals, 206 Mode, 209 Modified Bessel functions, 155–157 generating function for, 157graphs of, 159recurrence formulas for, 157 Modulus of a complex number, 11Moment, rth, 212Momental skewness, 212 Moments of inertia, 41 Monoticity Rule (Probability), 218Mutinomial coefficients, 9 Multiple , integrals, 125 Napier’s rules, 52 Natural logarithms and antilogarithms, 53 tables of, 252–253 Neumann’s function, 153Newton’s: backward-difference formula, 228 forward-difference formula, 228interpolation, 227method, 233 Nonhomogeneous differential equation, linear second order, 117 Nonlinear equations, solution of, 233Normal curve, 276–277 distribution, 226 Normal equations for least-squares line, 214 Null function, 189Numbers: Bernoulli, 142Euler, 142 Numerical methods for partial differential equations, 237–239 Oblate spheroidal coordinates, 131 Orthogonal curvilinear coordinates, 127–128 formulas involving, 128 Orthogonality: Chebyshev’s polynomials, 176Laguerre polynomials, 172Legendre polynomials, 165 Ovals of Cassini, 32 Parabola, 25 segment of, 18 Parabolic cylindrical coordinates, 129Paraboloid, 40Paraboloidal coordinates, 130Parallelepiped, 19Parallelogram, 7 Parameter, 208Parseval’s identity for: Fourier series, 144Fourier transform, 194 Partial: derivatives, 65differential equations, numerical methods, 237 Pascal’s triangle, 8Percentile, kth, 211Periods of elliptic functions, 200Plane analytic geometry, formulas from, 22–27 Plane, complex, 10Poisson: distribution, 226 equation, 237 summation formula, 137 Polar: coordinates, 24form of a complex number, 11 Polygon, regular, 17Polynomial function (least-squares), 214 Polynomials: Chebyshev’s, 175Laguerre, 171 Legendre, 164 Population, 208 mean 210standard deviation, 212variance, 212 Power function (least-squares), 214Power series, 138–141 reversion of 141 Powers, sums of, 134Present value, of an amount, 273 of an annuity, 275 Probability, 217 distribution, 223function, 218tables, 276 Products, infinite, 207 special, 5 Pulse function, 192 Pyramid, volume of, 20 Quadrants, 43 Quadratic convergence, 233Quadratic equation, solution of, 103Quadrature, 231–232Quartic equation, solution of, 13 Quartile coefficient of skewness, 212 Quartiles [Q L, M, QU], 211 Radians, 4, 44 table of conversion to degrees, 250 Random numbers table, 282Random variable, 223–226 standardized, 226 Range, sample, 210Real part of a complex number, 10 Reciprocals of powers, sums of, 135 Rectangle, 13Rectangular coordinate system, 120Rectangular coordinates, 24 transformation to polar coordinates, 24 Rectangular formula, 109 Rectified sine wave function, 191Recurrence or recursion formulas: Bessel functions, 154Chebyshev’s polynomials, 175gamma function, 149Hermite polynomials, 169 Laguerre polynomials, 171 Legendre polynomials, 165 Regression line, 214Regular polygon, 17Remainder: Cauchy’s form, 13Lagrange form, 138 Remainder formula: Gauss-Legendre interpolation, 232Hermite interpolation, 230Lagrange interpolation, 227 Reversion of power series, 141Richardson method, 240 Riemann zeta function, 204 Right circular cone, 20Rochigue’s formula: Laguerre polynomials, 171Legendre’s polynomials, 164 Root mean square (R.M.S.), 211 Roots of complex numbers, 11 Rose, 29Rotation, 24, 37Runge-Kutta method, 236 Sample, 208 covariance, 213 Saw tooth wave function, 191Scalar, 119 multiplication of vectors, 119 Scalar or dot product, 120 Scale factors, 127Scatterplot, 212Schwarz (Cauchy-Schwarz) inequality, 205 for integrals, 206 Secant method, 233Second-order differential equation, 117 Second-order divided-difference formula, 228 Sector of a circle, 17Segment: of circle, 18of parabola, 18 Semi-interquartile range, 211 Separation of variables, 116 Series, arithmetic, 134 arithmetic-geometric, 134binomial, 188of constants, 134Fourier, 144–148geometric, 134power, 138of sums of powers, 134 Taylor, 138–141 Simpson’s formula, 109, 231Sine, 43 graph of, 46table of values, 247 Sine integral, 88 table of values, 264 Sines, law of, 51Skewness, 212Solid analytic geometry, 34–40Solutions of algebraic equations, 13–14SOR (successive-overrelaxation) method, 240Sphere, equations of, 38 surface area, 19volume, 21 Spherical coordinates, 38, 129Spherical triangle, 51INDEX 288 Spiral of Archimedes, 33 Square wave function, 191 Squares error, 215 Standard deviation, 210 continuous random variable, 225discrete random variable, 224population, 212sample, 210 Standardized random variable, 215Statistics, 208–216 tables, 276–281 Step function, 192 Stirling’s formula, 150 Stochastic process, 219Stokes’ theorem, 126 Student’s t distribution, 226 table of, 298 Successive-overrelaxation (SOR) method, 240Summation formula: Euler-Maclaurin, 137 Poisson, 137 Surface integrals, 125 Tangent function, 43 graph of, 46table of values, 249 Tangents, law of, 51, 52 Taylor series, 138–141 two variables, 141 Three-point interpolatory formula, 228Toroidal coordinates, 132 Torus, surface area, volume, 18Total probability, Law of, 220Tractrix, 31Transformation: Jacobian of, 128of coordinates, 24, 36–37, 128of integrals, 70, 128 Translation of coordinates: in a plane, 24in space, 36 Trapezoid, area, perimeter, 16Trapezoidal rule (formula), 109, 231, 235Tree diagrams, Probability, 219 Triangle inequality, 205 Triangular wave function, 191 Trigonometric functions, 43–52 definition of, 43graphs of, 46inverse, 49–50series for, 139tables of, 247–249 Triple integrals, 125 Trochoid, 30Two-point formula, 228Two-point interpolatory formula, 228 Unit function, Heaviside’s, 192 Unit normal to the surface, 125 Unit vector, 120 Variance, 210 continuous random variable, 225 discrete random variable, 224population, 210 sample, 210 Vector analysis, 119–133 Vector or cross-product, 121 Vectors, 119 derivatives of, 122integrals involving, 124unit, 119 V olume integrals, 125 Wallis’ product, 207 Wave equation, 238 Weber’s function, 153 Weighted mean, 209 Witch of Agnesi, 31 x-intercept, 22y-intercept, 22Zero vector, 119 Zeros of Bessel functions, 267 Zeta function of Riemann, 204INDEX 289