[Daniel_Zwillinger]_The_Handbook_of_Integration(BookSee.org)

Handbook of Integration Daniel Zwillinger Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, New York -­'= Jones and Bartlett Publishers Boston London Editorial, Sales, ami Customer Service Offices Jones and Bartlett Publishers One Exeter Plal' ... Boston, MA 02116 Jones and Bartlett Publishers International P.O. Box 1498 London W6 7RS England Copyright © 1992 by Jones and Bartlett Publishers, Inc. All rights reserved. No part of the material protected by this (.'opyright notice may he reproduced or utilized in any fornl, electronic or mechanical, including photocopying, recording, or by any infornmtion storage and retrieval system, without written pernlission from the copyright owner. This book was typeset by the author using TEX. Figures 57.1 and 57.2 originally appeared on pages 79 and 80 in H. Piessens, E. de Doncker-Kapenga, C.W. Oberhuber, and D.K. Kalumer, Qruul}JIICk, Springer-Verlag, 1983. Reprinted courtesy of Springer-Verlag. Library of Congress Cataloging-in-Publication Data Zwillinger, Daniel, 1957- Handbook of integration / Daniel Zwillinger. p. cm. Includes bihiliographical references and index. ISBN 0-86720-293-9 1. Numerical integration. I. Title. QA299.3.Z85 1992 515' .43-de20 Printed in the United States of America 92-14050 CIP 96 95 94 93 92 10987654321 Table of Contents Preface Introduction . . . . How to Use This Book ix xi . xiii I Applications of Integration 1 2 3 4 5 6 7 8 9 10 11 Differential Equations: Integral Representations Differential Equations: Integral Transforms Extremal Problems .. 1 6 . . . .. 14 Function Representation Geometric Applications MIT Integration Bee ............ 20 Probability ........ . Summations: Combinatorial Summations: Other . . . Zeros of Functions Miscellaneous Applications . . . . . 24 ...... 28 ........ 30 31 ...... 34 .... 40 45 II Concepts and Definitions 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Definitions . . . . Integral Definitions . . . . . Caveats ......... . Changing Order of Integration Convergence of Integrals Exterior Calculus . . . Feynman Diagrams . . . Finite Part of Integrals Fractional Integration . . . . Liouville Theory Mean Value Theorems Path Integrals Principal Value Integrals Transforms: To a Finite Interval Transforms: Multidimensional Integrals Transforms: Miscellaneous . . . . . . . v ... 47 · 51 ........ 58 61 · 64 67 70 73 75 79 . . . . 83 .... 86 · 92 95 97 103 vi III Exact Analytical Methods 28 Change of Variable 29 Computer Aided Solution 30 Contour Integration 31 Convolution Techniques 32 Differentiation and Integration 33 Dilogarithms 34 Elliptic Integrals 35 Frullanian Integrals 36 Functional Equations 37 Integration by Parts 38 Line and Surface Integrals 39 Look Up Technique 40 Special Integration Techniques 41 Stochastic Integration 42 Tables of Integrals IV Approximate Analytical Methods 43 Asymptotic Expansions 44 Asymptotic Expansions: Multiple Integrals 45 Continued Fractions 46 Integral Inequalities 47 Integration by Parts 48 Interval Analysis 49 Laplace's Method 50 Stationary Phase 51 Steepest Descent 52 Approximations: Miscellaneous V Numerical Methods: Concepts 53 54 55 56 57 58 59 60 61 Introduction to Numerical Methods Numerical Definitions . . . . . . . . . . . . Error Analysis . . . . . . . . . . . . . . . . Romberg Integration / Richardson Extrapolation Software Libraries: Introduction ... . Software Libraries: Taxonomy .... . Software Libraries: Excerpts from G AMS Testing Quadrature Rules . . . . . . . . Truncating an Infinite Interval . . . . . . 109 117 129 140 142 145 148 157 160 162 164 170 181 186 190 195 199 203 205 215 218 221 226 230 240 243 244 246 250 254 258 260 272 275 Table of Contents vii VI Numerical Methods: Techniques 62 Adaptive Quadrature 277 63 Clenshaw-Curtis Rules 281 64 Compound Rules 283 65 Cubic Splines 285 66 Using Derivative Information 287 67 Gaussian Quadrature 289 68 Gaussian Quadrature: Generalized 292 69 Gaussian Quadrature: Kronrod's Extension 298 70 Lattice Rules 300 71 Monte Carlo Method 304 72 N umber Theoretic Methods 312 73 Parallel Computer Methods 315 74 Polyhedral Symmetry Rules 316 75 Polynomial Interpolation 319 76 Product Rules 323 77 Recurrence Relations 325 78 Symbolic Methods 329 79 Tschebyscheff Rules 332 80 Wozniakowski's Method 333 81 Tables: Numerical Methods 337 82 Tables: Formulas for Integrals 340 83 Tables: Numerically Evaluated Integrals 348 Mathematical Nomenclature 351 Index 353 Preface This book was begun when I was a graduate student in applied math­ ematics at the California Institute of Technology. Being able to integrate functions easily is a skill that is presumed at the graduate level. Yet, some integrals can only be simplified by using clever manipulations. I found it useful to create a list of manipulation techniques. Each technique on this list had a brief description of how the method was used and to what types of integrals it applied. As I learned more techniques they were added to the list. This book is a direct outgrowth of that list. In performing mathematical analysis, analytic evaluation of integrals is often required. Other times, an approximate integration may be more informative than a representation of the exact answer. (The exact repre­ sentation could, for example, be in the form of an infinite series.) Lastly, a numerical approximation to an integral may be all that is required in some applications. This book is therefore divided into five sections: • Applications of Integration which shows how integration is used in differential equations, geometry, probability and performing summa­ tions; • Concepts and Definitions which defines several different types of inte­ grals and operations on them; • Exact Techniques which indicates several ways in which integrals may be evaluated exactly; • Approximate Techniques which indicates several ways in which inte­ grals may be evaluated approximately; and • Numerical Techniques which indicates several ways in which integrals may be evaluated numerically. This handbook has been designed as a reference book. Many of the techniques in this book are standard in an advanced course in mathematical methods. Each technique is accompanied by several current references; ix x these allow each topic to be studied in more detail. This book should be useful to students and also to practicing engineers or scientists who must evaluate integrals on an occasional basis. Had this book been available when I was a graduate student, it would have saved me much time. It has saved me time in evaluating integrals that arose from my own work in industry (the Jet Propulsion Laboratory, Sandia Laboratories, EXXON Research and Engineering, and the MITRE Corporation) . Unfortunately, there may still be some errors in the text; I would greatly appreciate receiving notice of any such errors. Please send these comments care of Jones and Bartlett. No book is created in a vacuum, and this one is no exception. Thanks are extended to Harry Dym, David K. Kahaner, Jay Ramanthan, Doug Reinelt, and Michael Strauss for reviewing the manuscript. Their help has been instrumental in clarifying the text. Lastly, this book would have not been possible without the enthusiasm of my editor, Alice Peters. Boston, MA 1992 Daniel Zwillinger Introduction This book is a compilation of the most important and widely applicable methods for evaluating and approximating integrals. As a reference book, it provides convenient access to these methods and contains examples showing their use. The book is divided into five parts. The first part lists several appli­ cations of integration. The second part contains definitions and concepts and has some useful transformations of integrals. This section of the book defines many different types of integrals, indicates what Feynman diagrams are, and describes many useful transformations. The third part of the book is a collection of exact analytical evaluation techniques for integrals. For nearly every technique the following are given: · the types of integrals to which the method is applicable · the idea behind the method · the procedure for carrying out the method · at least one simple example of the method · notes for more advanced users · references to the literature for more discussion or examples. The material for each method has deliberately been kept short to simplify use. Proofs have been intentionally omitted. It is hoped that, by working through the simple example(s) given, the method will be understood. Enough insight should be gained from working the example(s) to apply the method to other integrals. References are given for each method so that the principle may be studied in more detail, or more examples seen. Note that not all of the references listed at the end of a section may be referred to in the text. The author has found that computer languages that perform symbolic manipulations (such as Macsyma) are very useful when performing the xi xii calculations necessary to analyze integrals. Examples of several symbolic manipulation computer languages are given. Not all integrals can be evaluated analytically in terms of elementary functions; sometimes an approximate evaluation will have to do. Other times, an approximate evaluation may be more useful than an exact eval­ uation. For instance, an exact evaluation in terms of a slowly converging infinite series may be laborious to approximate numerically. The same integral may have a simple approximation that indicates some characteristic behavior or easily allows a numerical value to be obtained. The fourth part of this book deals with approximate analytical solution techniques. For the methods in this part of the book, the format is similar to that used for the exact solution techniques. We classify a method as an approximate method if it gives some information about the value of an integral but will not specify the value of the integral at all values of the independent variable(s) appearing in the integral. The methods in this section describe, for example, the method of stationary phase and the method of steepest descent. When an exact or an approximate solution technique cannot be found, it may be necessary to find the solution numerically. Other times, a numerical solution may convey more information than an exact or approx­ imate analytical solution. The fifth part of this book deals with the most important methods for obtaining numerical approximations to integrals. From a vast literature of techniques available for numerically approximating integrals, this book has only tried to illustrate some of the more important techniques. At the beginning of the fifth section is a brief introduction to the concepts and terms used in numerical methods. This book is not designed to be read at one sitting. Rather, it should be consulted as needed. This book contains many references to other books. While some books cover only one or two topics well, some books cover all their topics well. The following books are recommended as a first source for detailed understanding of the integration techniques they cover: Each is broad in scope and easy to read. References [1] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scien­ tists and Engineers, McGraw-Hill, New York, 1978. [2] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984. [3] W. Squire, Integration for Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970. How to Use This Book This book has been designed to be easy to use when evaluating inte­ grals, whether exactly, approximately, or numerically. This introductory section outlines how this book may be used to analyze a given integral. First, determine if the integral has been studied in the literature. A list of many integrals may be found in the "Look Up Technique" section beginning on page 170. If the integral you wish to analyze is contained in one of the lists in that section, then see the indicated reference. This technique is the single most useful technique in this book. Special Forms [I] If the integral has a special form, then it may be evaluated in closed form without too much difficulty. If the integral has the form: (A) r R(x) dx, where R(x) is a rational function then the integral can be evaluated in terms of logarithms and arc-tangents (see page 183). (B) r P(x, JR) log Q(x, JR) dx, where P(, ) and Q(, ) are rational functions and R = A2 + Ex + ex2, then the integral can be evaluated in terms of dilogarithms (see page 145). (e) r R(x, vT(x)) dx where R(, ) is a rational function of its ar­ guments and T(x) is a third of fourth order polynomial , then the integral can be evaluated in terms of elliptic functions (see page 147). (D) J;' J(cosO,sinO) dO, then the integral may be re-formulated as a contour integral (see page 129). [2] If the integral is a contour integral, see page 129. [3] If the integral is a path integral, see page 86. [4] If the integral is a principal -value integral (i.e., the integral sign looks like f), then see page 92. xiii xiv How to Use This Book [5] If the integral is a finite-part integral (Le., the integral sign looks like f), then see page 73. [6] If the integral is a loop integral (i.e, the integral sign looks like f), or if the integration is over a closed curve in the complex plane, then see pages 129 or 164. [7] If the integral appears to be divergent, then the integral might need to be interpreted as a principal-value integral (see page 92) or as a finite-part integral (see page 73). Looking for an Exact Evaluation [1] If you have access to a symbolic manipulation computer language (such as Maple, Macsyma, or Derive), then see page 117. [2] For a given integral, if one integration technique does not work, try another. Most integrals that can be analytically evaluated can be evaluated by more than one technique. For example, the integral 1= f"'sinx dx Jo x is shown to converge on page 66. Then I is evaluated (using different methods) on pages 118, 133, 144, 145, and 185. Looking for an Approximate Evaluation [1] If A is large, C is an integration contour, and the integral has the form: (A) Ie e>.J(a;)g(x) dx, then the method of steepest descents may be used (see page 229). (B) Ie e>.J(a;)g(x) dx, where f(x) is a real function, then Laplace's method may be used (see page 221). (C) Ie ei>.J(a;)g(x) dx, where f(x) is a real function, then the method of stationary phase may be used (see page 226). There is a collection of other special forms on page 181. [2] Interval analysis techniques, whether implemented analytically or nu­ merically, permit exact upper and lower bounds to be determined for an integral (see page 218). How to Use This Book xv Looking for a Numerical Evaluation [1] It is often easiest to use commercial software packages when looking for a numerical solution (see page 254). The type of routine needed may be determined from the taxonomy section (see page 258). The taxonomy classification may then be used as the entry in the table of software starting on page 260. [2] If a low accuracy solution is acceptable, then a Monte Carlo solution technique may be used, see page 304. [3] If the integral in question has a very high dimension, then Monte Carlo methods may be the only usable technique, see page 304. [4] If a parallel computer is available to you, then see page 315. [5] If the integrand is periodic, then lattice rules may be appropriate. See page 300. [6] References for quadrature rules involving specific geometric regions, or for integrands with a specific functional form, may be found on page 337. [7] Examples of some one-dimensional and two-dimensional quadratures rules may be found on page 340. [8] A listing of some integrals that have been tabulated in the literature may be found on page 348. Other Things to Consider [1] Is fractional integration involved? See page 75. [2] Is a proof that the integral cannot be evaluated in terms of elementary functions desired? See page 77. [3] Does the equation involve a large or small parameter? See the asymp­ totic methods described on pages 195 and 199. I Applications of Integration 1. Differential Equations: Integral Representations A pplicable to Linear differential equations. Idea Sometimes the solution of a linear ordinary differential equation can be written as a contour integral. Procedure Let Lz;[·] be a linear differential operator with respect to z, and suppose that the ordinary differential equation we wish to solve has the form Lz;[u(z)] = O. (1.1) We look for a solution of (1.1) in the form of an integral u(zl = fc K(z, {lv({l de, (1.2) 1 2 I Applications of Integration for some function v( e) and some contour C in the complex e plane. The function K(z, e) is called the kernel. Some common kernels are: Euler kernel: Laplace kernel: Mellin kernel: We combine (1.2) and (1.1) for K(z, e) = (z -e)n K(z,e) = eF.z K(z,e) = zF. fc Lz[K(z, e)]v(e) de = o. (1.3) Now we must (conceptually) find a linear differential operator Ad,], oper­ ating with respect to e, such that Lz[K(z, e)] = AdK(z, e)]. After Ad,] has been found, then (1.3) can be rewritten as fc AdK(z,€)]v(€)d{ = o. (1.4) Now we integrate (1.4) by parts. The resulting expression will be a dif­ ferential equation to be solved for v( e) and some boundary terms. The boundary terms will determine the contour C. Knowing both v(e) and C, the solution to (1.1) is given by the integral in (1.2). Example Consider Airy's differential equation Lz[u] = u"(z) -zu(z) = o. (1.5) We assume that the solution of (1.5) has the form u(z) = fc eZ(v(€)d€, (1.6) for some v(e) and some contour C. Substituting (1.6) into (1.5) we find fc ev(€)ez( d€ -z fc v(€)ez( d{ = O. (1.7) The second term in (1.7) can be integrated by parts to obtain fc eve €)eZ ( d{ -[V( €)ezt + fc Vi (€)eZ ( d{ = 0, or -[v(€)eZ(L + fc eZ ( [eV(€) + Vl(€)] d{ = O. (1.8) 1. Differential Equations: Integral Representations 3 Figure 1. A solution to (1.5) is determined by any contour C that starts and ends in the shaded regions. All of the shaded regions extend to infinity. One possible contour is shown. We choose (1.9) and the boundary conditions (1.10) With these choices, equation (1.8) is satisfied. From (1.9) we can solve for v(e): v(el = exp( -~). (1.11) Using (1.11) in (1.10) we must choose the contour C so that (1.12) for all real values of z. The only restriction that (1.12) places on C is that the contour start and end in one of the shaded regions shown in Figure 1. Finally, the solution to (1.5) can be written as the integral (1.13) Asymptotic methods can be applied to (1.13) to determine information about u(z). 4 I Applications of Integration Notes [1] This method is also known as Laplace's method. [2] Loop integrals are contour integrals in which the path of integration is given by a loop. For example, the integral I~o;) indicates an integral that starts at negative infinity, loops around the origin once, in the clockwise sense, and then returns to negative infinity. Using the methods is this section, the fundamental solutions to Le­ gendre's differential equation, {I -x2)y" -2xy' + n{n + l)y = 0, can be written in the form of loop integrals: _ 1 1(1+'%+> {(2 -1)" PII{z) -21ri 2"{Z _ ()"+l d( 1 I(H.-H) «(2 -1)" QII(Z) = 4. . II 11+1 d( IsmV1r 2 (z -() (1.14.a-b) where the integration contour in (1.14.a) is a closed curve with positive direction passing through the (-plane, avoiding the half-line (-co, -1), and admitting 1 and Z as inner points of the domain it bounds. The contour in (1.14.b) is a closed co-shaped curve encircling the point 1 once in the negative direction and the point -1 once in the positive direction. Equation (1.14.a) is known as SchHifli's integral representation. [3] Since there are three regions in Figure 1, there are three different contours that can start and end in one of these regions; each corresponds to a solution of (1.5). The functions Ai{x) and Bi(x), appropriately scaled, are obtained by two of these three choices for the contour (see page 171). The third solution is a linear combination of the functions Ai(x) and Bi{x). [4] Sometimes a double integral may be required to find an integral represen­ tation. In this case, a solution of the form u{ z) = II K (Z; s, t )w( s, t) ds dt is proposed. Details may be found in Ince [8], page 197. As an example, the equation d2 d (x2 -1)~ + (a + b + l)xdY + aby = 0 dx x has the two linearly independent solutions y±(x) = 100 /.00 exp [±!tBt _ ~(.2 + t2)].4-'tO-' d.dt. [5] Poisson's integral formula is an integral representation of the solution to Laplace's differential equation. If u(r, 8) satisfies Laplace's equation V2u = Urr+r-1Ur+r-2u88 = 0 for 0 < r < R, and u{R, 8) = /(8) for 0::; 8 < 211', then u(r, 8) for 0 < r < R is given by 1 (21f R2 _ r2 u{r, 8) = 211' Jo R2 _ 2Rr cos(8 _ ¢) + r2 J{¢) d¢. (1.15) This is known as the Poisson formula for a circle. Integral solutions for Laplace's equation are also known when the geometry is a sphere, a half­ plane, a half-space, or an annulus. See Zwillinger [10] for details. 1. Differential Equations: Integral Representations 5 [6] Pfaffian differential equations, which are equations of the form [7] [8] [9] [10] [XnF (x !) + G (x !)] y = 0, can also be solved by this method. See Zwillinger [10]. An application of this method to partial differential equations may be found in Bateman [2], pages 268-275. The Mellin-Barnes integral representation for an ordinary differential equa­ tion has the form -1 ~ [ II;:l r (bj -e) II;=l r (1 -aj + e) 1 u(z)- v(e)z II~ r(l-bo+e)II~ r(ao-e) cLe· C J=m+l J J=n+l J In this representation, only the contour C and the constants {ai, bj, m, n, q, r} are to be determined (see Babister [1] for details). The ordinary integral 1% A(t) dt is a construction that solves the initial-:to value problem: y'(x) = A(x) with y(xo) = 0 (here 0 is the matrix of all zeros). The product integral is an analogous construction that solves the initial-value problem: y'(x) = A(x)y(x) with y(xo) = I (here I is the identity matrix). See Dollard and Friedman [6] for details. Given a linear differential equation (ordinary or partial): L[u] = f(x), the Green's function G(x,z) satisfies L[G] = «5(x-z), and a few technical condi- tions. (Here, «5 represents the usual delta function.) The solution to the orig­ inal equation can then be written as the integral u(x) = J f(z)G(x, z) dz. See Zwillinger [10] for details. References [1] A. W. Babister, Transcendental Functions Satisfying Nonhomogeneous Lin­ ear Differential Equations, The MacMillan Company, New York, 1967, pages 24-26. [2] [3] [4] [5] [6] [7] [8] [9] [10] H. Bateman, Differential Equations, Longmans, Green and Co., 1926, Chap­ ter 10, pages 260-264. R. G. Buschman, "Simple contiguous function relations for functions defined by Mellin-Barnes integrals," Indian J. Math., 32, No.1, 1990, pages 25-32. G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Vari­ able, McGraw-Hill Book Company, New York, 1966, pages 231-239. B. Davies, Integral Transforms and Their Applications -Second Edition, Springer-Verlag, New York, 1985, pages 342-367. J. D. Dollard and C. N. Friedman, Product Integration with Applications to Differential Equations, Addison-Wesley Publishing Co., Reading, MA, 1979. R. A. Gustafson, "Some Q-Beta and Mellin-Barnes Integrals with Many Parameters Associated to the Classical Groups," SIAM J. Math. Anal., 23, No.2, March 1992, pages 525-55l. E. L. Ince, Ordinary Differential Equations, Dover Publications, Inc., New York, 1964, pages 186-203 and 438-468. F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. D. Zwillinger, Handbook of Differential Equations, Academic Press, New York, Second Edition, 1992. 6 I Applications of Integration 2. Differential Equations: Integral Transforms Applicable to Linear differential equations. Idea In order to solve a linear differential equation, it is sometimes easier to transform the equation to some "space," solve the equation in that "space," and then transform the solution back. Procedure Given a linear differential equation, multiply the equation by a kernel and integrate over a specified region (see Table 2.1 and Table 2.2 for a listing of common kernels and limits of integration). Use integration by parts to obtain an equation for the transform of the dependent variable. You will have used the "correct" transform (Le., you have chosen the correct kernel and limits) if the boundary conditions given with the original equation have been utilized. Now solve the equation for the transform of the dependent variable. From this, obtain the solution by multiplying by the inverse kernel and performing another integration. Table 2.1 and Table 2.2 also list the inverse kernel. Example Suppose we have the boundary value problem for y = y(x) yxx + y = 1, y(O) = 0, y(l) = O. (2. La-c) Since the solution vanishes at both of the endpoints, we suspect that a finite sine transform might be a useful transform to try. Define the finite sine transform of y(x) to be z(e), so that z(e) = J.' y(x)sinexdx. (2.2) (See "finite sine transform-2" in Table 2.1). Now multiply equation (2.1.a) by sin ex and integrate with respect to x from 0 to 1. This results in /.' yzz(x) sinexdx + /.' y(x) sin ex dx = /.' sin ex dx. (2.3) If we integrate the first term in (2.3) by parts, twice, we obtain /.1 Yxx(x) sin ex dx = yx(x) sinexlx=1 -ey(x) cosexlX=1 o x=o x=o -e /.' y(x) sin ex dx. (2.4) 2. Differential Equations: Integral Transforms 7 Since we will only use e = 0, 7r, 27r, ... (see Table 2.1), the first term on the right-hand side of (2.4) is identically zero. Because of the boundary conditions in (2.1.b-c), the second term on the right-hand side of (2.4) also vanishes. (Since we have used the given boundary conditions to simplify certain terms appearing in the transformed equation, we suspect we have used an appropriate transform. If we had taken a finite cosine transform, instead of the one that we did, the boundary terms from the integration by parts would not have vanished.) Using (2.4), simplified, in (2.3) results in /.1 /.1 1 -cos e -e2 y(x)sin ex dx+ y(x)sinexdx= . o 0 e Using the definition of z(e) (from (2.2» this becomes or Now that we have found an explicit formula for the transformed function, we can use the summation formula (inverse transform) in Table 2.1 to determine that y(x) = L 2z(e) sin ex, (=0,7I',21r, ... = "" 1 -cose . L..J 2 2) sm ex, (1-e e (=0,71',271', ..• (2.5) 00 1-(-I)k . = L 2 ( 2 2) sm k7rX, k=O 1 -7r k 7rk _ "" 4 sin k7rx -L..J 2 2 ' k=1,3,5, ... (1 -'If' k )'If'k where we have defined k = e/'If'. The exact solution of (2.1) is y(x) = 1-cos x + COS.1 -1 sinx. If this sm1 solution is expanded in a finite Fourier series, we obtain the representation in (2.5). 8 I Applications of Integration Table 2.1 Different transform pairs of the form v(e.) = 1~ u(x)K(x,e.) <Ix, u(x) = L H(X,ek)V(ek). h Finite cosine transform -1, (see Miles [17], page 86) here I and h are arbitrary, and the {ek} satisfy ek tan ekl = h. V(ek) = 11 u(x) cos (Xek) dx, u(x) = ~ (2 -6(l:o)(e~ + h2 ) cos (ekX) (C ) o '7; h + l(d + h2) v ~k . Finite cosine transform -2, (see Butkov [3], page 161) this is the last transform with h = 0, I = 1, so that ek = 0, 1r, 21r, .... v(e.) = 11 u(x) cos (xe.) <Ix, u(x) = ~)2 -6(00) cos ({.x) v(e.). o h Finite sine transform -1, (see Miles [17], page 86) here I and h are arbitrary, and the {ek} satisfy ek cot(ekl) = -h. V(ek) = u(x) sin (Xek) dx, u(x) = ~ 2 k 2 2k V(ek)' 11 (e + h2) sin (e x) o '7; h + l(ek + h ) Finite sine transform -2, (see Butkov [3], page 161) this is the last transform with h = 0, 1= 1, so that ek = 0, 1r, 21r, .... v(e.) = 11 U(x) sin (xe.) <Ix, u(x) = ~)sin(e.x)v(e.). o (I: Finite Hankel transform -1, (see Tranter [24], page 88) here n is arbitrary and the {ek} are positive and satisfy In(ek) = O. V(ek) = 11 u(x)xJn(Xek) dx, u(x) = L 2 ~n(Xek)~ V(ek)' o (I: Jm+1(ek Finite Hankel transform -2, (see Miles [17], page 86) here nand hare arbitrary and the {ek} are positive and satisfy ekJ~(aek) + hJn(aek) = O. 14 ~ 2dJn(Xek) V(ek). V(ek) = u(x)xJn(Xek) dx, u(x) = L, {(h2 + d) a2 _ m2} J!(aek) o (I: 2. Differential Equations: Integral Transforms 9 Finite Hankel transform -3, (see Miles [17J, page 86) here b > a, the {ek} are positive and satisfy yn(aek)Jn(bek) = In(aek)Yn(bek), and Zn(Xek) := yn(aek)Jn(Xek) -In(aek)Yn(Xek). lb '"' 1r2 e~J~(hek)Zn(Xek) V(ek) = U(X)XZn(Xek) dx, u(x) = L...J"2 2 2 V(ek). a (I: In(aek) -In(bek) Legendre transform, (see Miles [17], page 86) here ek = 0,1,2, .... 11 '"' 2ek + 1 V(ek) = u(x)Pea: (x) dx, u(x) = L...J -2-Pea: (X)V(ek). -1 ~ Table 2.2 Different integral transform pairs of the form v(~) = LP K(x,~)u(x)<lx, u(x) = /.. H(x,~)v(~)d{. Fourier transform, (see Butkov [16], Chapter 7) v(e) = . rn= e1z ( u(x) dx, u(x) = . rn= e-1ze v(e) cLeo 1 100 . 1 100 . v 21r -00 V 21r -00 Fourier cosine transform, (see Butkov [16], page 274) vee) = VI f.~ cos(xe) u(x) <lx, u(x) = VI f.~ cos(xe) vee) d{. Fourier sine transform, (see Butkov [16], page 274) vee) = VI f.~ sin(x~)u(x)<lx, u(x) = VI f.~ sin(xe)v(e)d{· Hankel transform, (see Sneddon [20], Chapter 5) vee) = f.~ u(x)xJ.(xe) <lx, u(x) = f.~ U.(x~)v(~) d{. 10 I Applications of Integration Hartley transform, (see Bracewell [5]) 1 100 v(e) = rn= (cosxe + sinxe) u(x) dx, v21r -00 1 100 u(x) = rn= (cosxe + sinxe) v(e) df.. v21r -00 Hilbert transform, (see Sneddon [20], pages 233-238) 100 1 100 1 v(e) = 1- 00 u(x) 1r(X _ e) dx, u(x) = 1- 00 _I,. _\ v(e) df.. K-transform, (see Bateman [3]) 100 1 !.O'+iOO v(e) = 0 Kv(xe)~u(x) dx, u(x) = 1ri O'-ioo Iv(xe)~v(e) df.. Kontorovich-Lebedev transform, (see Sneddon [20], Chapter 6) v(e) = 100 Ki(X) u(x) dx, u(x) = 22100 e sinh(1re)Ki(x) v(e) df.. o x 1r 0 Kontorovich-Lebedev transform (alternative form), (see Jones [14]) 100 11~ v(e) = 0 H?>(x) u(x) dx, u(x) = -2x -ioo eJ(x) v(e) df.. Laplace transform, (see Sneddon [20], Chapter 3) 100 1 !.O'+iOO v(e) = e-:Z:( u(x) dx, u(x) = 21ri . e:Z:( v(e) df.. o 0'-100 Mehler-Fock transform of order ffl, (see Sneddon [20], Chapter 7) v( e) = 100 sinh( x )Pi; -1/2 (cosh x) u( x) d:!:, u(x) = 1.00 e tanh("e)p;(_1/2(COshx) v(e) d{. Mellin transform, (see Sneddon [20], Chapter 4) 100 1 !.O'+iOO v(e) = X(-l u(x) dx, u(x) = 21ri . x-( v(e) df.. o 0'-100 2. Differential Equations: Integral Transforms 11 Weber formula, (see Titchmarsh [23], page 75) vee) = J.~ Fx[J"(xe)Y"(ae) -Y"(xe)J"(ae)] u(x)dx, u(x) = ..;xlOO Jv(xe)Y~(ae) -Y~(xe)Jv(ae) v(e) de. o J" (ae) + Y" (ae) Weierstrass transform, (see Hirschman and Widder [11], Chapter 8) v(e) = -- e«(-X) /4 u(x) dx, 1 100 2 ~ -00 Notes [1] There are many tables of transforms available (see Bateman [3] or Magnus, Oberhettinger, and Soni [16]). It is generally easier to look up a transform than to compute it. [2] 'Iransform techniques may also be used with systems of linear equations. [3] 'Iransforms may also be evaluated numerically. There are many results on how to compute the more popular transforms numerically, like the Laplace transform. See, for example, Strain [22]. [4] The finite Hankel transforms are useful for differential equations that contain the operator LH [u] and the Legendre transform is useful for differential equations that contain the operator LL[U], where [5] [6] [7] [8] 2 [ ] Ur n LH U = Urr + - -"'2u T T and a ( 2 aU) LL[U] = aT (1 -r ) ar . For example, the Legendre transform of LL[U] is simply -ek(ek + 1)V(ek). Integral transforms are generally created for solving a specific differential equation with a specific class of boundary conditions. The Mathieu inte­ gral transform (see Inayat-Hussain [12]) has been constructed for the two­ dimensional Helmholtz equation in elliptic-cylinder coordinates. Integral transforms can also be constructed by integrating the Green's func­ tion for a Sturm-Liouville eigenvalue problem. See Zwillinger [25] for details. Note that many of the transforms in Table 2.1 and Table 2.2 do not have a standard form. In the Fourier transform, for example, the two V2-i terms might not be symmetrically placed as we have shown them. Also, a small variation of the K-transform is known as the Meijer transform (see Ditkin and Prudnikov [8], page 75). If a function f(x, y) has radial symmetry, then a Fourier transform in both x and y is equivalent to a Hankel transform of f(r) = f(x, y), where r2 = x2 + y2. See Sneddon [20], pages 79-83. 12 I Applications of Integration [9] Two transform pairs that are continuous in one variable and discrete in the other variable, on an infinite interval, are the Hermite transform 00 U(X) = L VnHn(x)e-x2/2, n=O 1 100 Vn = (2n)!J1T -00 u(x)Hn(x)e-X2 /2 dx, where Hn(x) is the n-th Hermite polynomial, and the Laguerre transform 00 I u(x) = L vnL~(x)~ . 1'-' n=O Vn = /.00 u(x)L~(x)xQe-X dx, where L~(x) is the Laguerre polynomial of degree n, and a ~ O. See Haimo [10] for details. [10] Classically, the Fourier transform of a function only exists if the function being transformed decays quickly enough at ±oo. The Fourier transform can be extended, though, to handle generalized functions. For example, the Fourier transform of the n-th derivative of the delta function is given by .r [6(n)(t)] = (iw)n. Another way to approach the Fourier transform of functions that do not decay quickly enough at either 00 or -00 is to use the one-sided Fourier transforms. See Chester [6] for details. [11] Many of the transforms listed generalize naturally to n dimensions. For example, in n dimensions we have: ! v(e) = (211")-n/2 I. e~'xu(x) dx, (A) Fourier transform: I.an . u(x) = (211")-n/2 e-~,xv(e) cLe. an (B) Hilbert transform (see Bitsadze [4]): 8f r(n/2) !. Yi -Xi • -8 . = ~ I _ rn 4>(Y) dy, ~ = 1,2, ... , n -1, X, 11" Rn-l Y X ,1,.( ) __ r(n/2) !. (y -x)· V f d 'I' Y -n/2 Iy _ xln y. 11" Rn-l [12] Apelblat [2] has found that repeated use of integral transforms can lead to the simplification of some infinite integrals. For example, let Fa (y) denote the Fourier sine transform of the function f (x), Fa (y) = Jooo f (x) sin yx dx. Taking the Laplace transform of this results in G(s) = f.oo e-" {f.oo f(x) sin yx dx } dy -/.00 xf(x) d - 2 2 X, o s +x (2.6) where the order of integration has been changed and then the inner integral evaluated. For some functions f(x) it may be possible to find the corre­ sponding Fs(Y) and G(s) using comprehensive tables of integral transforms. 2. Differential Equations: Integral Transforms 13 Equating this to the expression in (2.6) may result in a definite integral hard to evaluate in other ways. As a simple example of the technique, consider using 1 (x) = 2 1 2' x(a + x ) With this we can find Fa (y) = ~ (1 - e -all). Taking the Laplace transform of this, and equating the result to (2.6), we have found the integral /.00 dx 1r o (a2 + X2)(S2 + x2) = 2as(a + s)' [13] Carson's integral is the integral transformation O(P) = p Jooo e-pt I(t) dt. See Iyanaga and Kawada [8]. [14] A transform pair that is continuous in each variable, on a finite interval, is the finite Hilbert transform v(e) = !11 u(x) dx, 1rJ_l X -e 1 [ IiI ~ ] u(x) = ~ C - - c _ v(e) de , V 1 -x2 1r -1 ~ X where C is an arbitrary constant, and the integrals are principal value integrals. See Sneddon [20], page 467, for details. [15] Note that, for the Hilbert transform, the integrals in Table 2.2 are principal value integrals. References [1] M. Abramowitz and I. A. Stegun, Handbook 01 Mathematical Functions, National Bureau of Standards, Washington, DC, 1964, pages 1019-1030. [2] A. Apelblat, "Repeating Use of Integral Transforms-A New Method for Evaluation of Some Infinite Integrals," IMA J. Appl. Mathematics, 27, 1981, pages 481-496. [3] Staff of the Bateman Manuscript Project, A. Erdelyi (ed.), Tables of Integral '!'ranslorms, in 3 volumes, McGraw-Hill Book Company, New York, 1954. [4] A. V. Bitsadze, "The Multidimensional Hilbert Transform," Somet Math. Dokl., 35, No.2, 1987, pages 390-392. [5] R. N. Bracewell, The Hartley '!'ransform, Oxford University Press, New York, 1986. [6] C. R. Chester, Techniques in Partial Differential Equations, McGraw-Hill Book Company, New York, 1970. [7] B. Davies, Integral Transforms and Their Applications -Second Edition, Springer-Verlag, New York, 1985. [8] V. A. Ditkin and A. P. Prudnikov, Integral Transforms and Operational Calculus, translated by D. E. Brown, English translation edited by I. N. Sneddon, Pergamon Press, New York, 1965. [9] H.-J. Glaeske, "Operational Properties of a Generalized Hermite Transfor­ mation," Aequationes Mathematicae, 32, 1987, pages 155-170. [10] D. T. Haimo, "The Dual Weierstrass-Laguerre Transform," Trans. AMS, 290, No.2, August 1985, pages 597-613. [11] I. I. Hirschman and D. V. Widder, The Convolution '!'ransform, Princeton University Press, Princeton, NJ, 1955. 14 I Applications of Integration [12] A. A. Inayat-Hussain, "Mathieu Integral Transforms," J. Math. Physics, 32, No.3, March 1991, pages 669-675. [13] S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT Press, Cambridge, MA, 1980. [14] D. S. Jones, "The Kontorovich-Lebedev Transform," J. Inst. Maths. Ap­ plics, 26, 1980, pages 133-141. [15] O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables, translated by L. W. Longdon, Halstead Press, John Wiley & Sons, New York, 1983. [16] W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, New York, 1966. [17] J. W. Miles, Integral Transforms in Applied Mathematics, Cambridge Uni­ versity Press, 1971. [18] C. Nasim, "The Mehler-Fock Transform of General Order and Arbitrary Index and Its Inversion," Int. J. Math. & Math. Sci., 7, No.1, 1984, pages 171-180. [19] F. Oberhettinger and T. P. Higgins, Tables of Lebede'fJ, Mehler, and Gen­ eralized Mehler Transforms, Mathematical Note No. 246, Boeing Scientific Research Laboratories, October 1961. [20] I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill Book Com­ pany, New York, 1972. [21] I. Stakgold, Green's Functions and Boundary Value Problems, John Wiley & Sons, New York, 1979. [22] J. Strain, "A Fast Laplace Transform Based on Laguerre Functions," Iv/ath. of Comp., 58, No. 197, January 1992, pages 275-283. [23] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Clarendon Press, Oxford, 1946. [24] C. J. Tranter, Integral Transforms in Mathematical Physics, Methuen & Co. Ltd., London, 1966. [25] D. Zwillinger, Handbook of Differential Equations, Academic Press, New York, Second Edition, 1992. 3. Extremal Problems Applicable to integral. Yields Finding a function that maximizes (or minimizes) an A differential equation for the critical function. 3. Extremal Problems Procedure Given the functional J[u] = f f L(x, 8xj )u(x) dx, R 15 (3.1) where the operator L() is a linear or nonlinear function of its arguments, how can u(x) be determined so that J[u] is critical (i.e., either a maximum or a minimum)? The variational principle that is most often used is 8J = 0, which states that the integral J[u] should be stationary with respect to small changes in u(x). If we let hex) be a "small," continuously differentiable function, then we can form J[u + h] -J[u] = f f [L(X, 8., )( u(x) + h(x)) -L(x, 8., )u(X)] dx. (3.2) R By integration by parts, (3.2) can often be written as J[u + hJ -J[u] = f f N(x, 8xj )u(x) dx + O(llhI12) R plus some boundary terms. The variational principle requires that 8J := J[u + h] -J[u] vanishes to leading order, or that N(x, 8xj )u(x) = O. (3.3) Equation (3.3) is called the first variation of (3.1), or the Euler-Lagmnge equation corresponding to (3.1). (This is also called the Euler equation.) A functional in the form of (3.1) determines an Euler-Lagrange equation. Conversely, given an Euler-Lagrange equation, a corresponding functional can sometimes be obtained. Many approximate and numerical techniques for differential equations utilize the functional associated with a given system of Euler-Lagrange equations. For example, both the Rayleigh-Ritz method and the finite element method create (in principle) integrals that are then analyzed (see Zwillinger [5]). The following collection of examples assume that the dependent vari­ able in the given differential equation has natural boundary conditions. If the dependent variable did not have these specific boundary conditions, then the boundary terms that were discarded in going from (3.2) to (3.3) would have to be satisfied in addition to the Euler-Lagrange equation. 16 I Applications of Integration Example 1 Suppose that we have the functional J(y] = fR F (x, y, y', y") dx. Form­ ing (3.2) we find J[u + h] -J[u] = L [F(X, u + h, u' + hi, u" + h") -F( x, U, u', Ull)] dx = L [ (F(e) + F.(e)h + F •• (e)h' + F ... (e)h" + 0 (h2) ) -F(e) ] dx = L [F.(e)h + F •• (e)h' + F ... (e)hll ] dx + 0 (h2) , (3.4) where the bullet stands for (x, 1£, 1£', 1£"), and the 0 (h2) terms should really be written as 0 (h2, (h')2, (h")2). Now integration by parts can be used to find !. Fyi (e )h' dx = Fyi (e )hl -!. hi. (Fyi) R 8R R dx !. Fy" (e )h" dx = Fy" (e )h'l -!. h' i. (Fy" ) R 8R R dx = Fyll(e)h'l -i.(Fyl(e))hl +!. h d2 2 (Fy"). 8R dx 8R R dx If we take the natural boundary conditions Fyi (e)1 = 0 and Fy" (e)1 = 8R 8R 0, then (3.4) becomes J[u + h]-J[u] = L [F.(e)h -!F •• (e)h + :2 F ... (e)h] dx + 0 (h2) = L h [F.(e) -!F •• (e) + !2F ... (e)] dx + 0 (h2). (3.5) If (3.5) is to vanish for all "small" functions h, then it must be that d d2 Fy(e) --Fyi (e) + -2 Fyll(e) = o. dx dx This is the Euler-Lagrange equation associated with the original functional. 3. Extremal Problems Example 2 The Euler-Lagrange equation for the functional J[y] = L F (x,y,y', ... ,yIn») dx, where y = y(x) is 8F d (8F) d2 (8F) n~ (8F) 8y -dx 8y' + dx2 8y" -... + (-1) dxn 8y(n) = o. For this equation the natural boundary conditions are given by y(xo) = Yo, y(xd = Yb Example 3 y'(xo) = y~, y'(xd = y~, ... , ... , y(n-l)(XO) = y~n-l), y(n-l)(Xl) = y1n-1). The Euler-Lagrange equation for the functional 17 J[u] = ! ! F(x, y, u, ux, uy, Uxx, uxy, Uyy) dx dy, (3.6) R where u = u(x, y) is 8F 8 (8F) 8 (8F) 82 (8F) 8u -8x 8ux -8y 8uy + 8x2 8uxx 82 (8F) 82 (8F) +----+---=0 8x8y 8uxy 8y2 8uyy • Example 4 The Euler-Lagrange equation for the functional which is a special case of (3.6), is: :x (a::) + :y (b:) -cu = f. 18 I Applications of Integration Example 5 For the 2m-th order ordinary differential equation (in formally self­ adjoint form) f) _l)k dk k (Pk(X) dk~) = f(x), k=O dx dx u(a) = u'(a) = ... = u(m-l)(a) = 0, u(b) = u'(b) = ... = u(m-l)(b) = 0, a corresponding functional is b ( m (dk )2 ) J[ul = 1. ~Pk(X) dx~ -2f(x)u(x) dx. Notes [1] Note that two different functionals can yield the same set of Euler-Lagrange equations. For example, 6 I J dx = 6 I(J + y + xy') dx. The reason that 6 I(y + xy') dx = 0 is because the integrand is an exact differential (I(y + xy') dx = I d(xy)). Hence, this integral is path independent; its value is determined by the boundary conditions. The Euler-Lagrange equations for the two functionals I J Ux:tUyy dx dy and I J (U%y)2 dx dy are also the same. [2] Even if the boundary conditions given with a differential equation are not natural, a variational principle may sometimes be found. Consider J[u) = J.~' F(x,u,u')dx -9'(X,U>!.=., + 9.(X,U>!.=.; where 91(X,U) and 92(X,U) are unspecified functions. The necessary condi­ tions for 1.£ to minimize J[u] are (see Mitchell and Wait [3]) aF _.!!... aF = 0 au dx au' , aF + a911 = 0 au' au ' %=%1 aF + a921 = 0 au' au . %=%2 If 91 and 92 are identically zero, then we recover the natural boundary conditions. However, we may choose 91 and 92 to suit other boundary conditions. For example, the problem 1.£" + f(x) = 0, 1.£' +aul = 0, 1.£' +/31.£1 = 0, %=%1 %=%2 corresponds to the functional J.%2 [1 2 ] (31.£21 au21 J[u] = %1 2 (1.£') -f(x)u dx + 2 %=%2 -2 %=%1' 3. Extremal Problems 19 [3] This technique can be used in higher dimensions. For example, consider the functional J[u] = f f F(x,y,u,uz,uy,u:z;z,uzy,uyy)dxdy R + 1 G(x,y,u,u cr, ucrcr, Un) 00, DR where 8/ 8u and 8/ 8n are partial differential operators in the directions of the tangent and normal to the curve 8R. Necessary conditions for J[u] to have a minimum are the Euler-Lagrange equations (given in (3.6)) together with the boundary conditions: [ 8F 8 8F] [ 8F 8 8F] 8uz -8x 8uzz Ycr -8uy -8y 8uyy Xcr - ------ XcrYcr+- ---(Xcr-Ycr) [a ( 8F 8F )] 1 [8 8F 2 2] 8u 8uzz 8uyy 2 8u 8uzy 1 [( 8 8F) (8 8F) ] + 2' 8x 8uzy Xcr -8y 8uzy Ycr 8 8G 82 8G +Gu---+-2--= 0, 8u 8ucr 8u 8ucrcr 8G 8F 2 8F 2 8F -8 +-8 Ycr+-8 xcr+-8 XcrYcr =0, Un Uzz Uyy Uzy where Xcr = dx/du and Ycr = dy/du. See Mitchell and Wait [3] for details. References [1] E. Butkov, Mathematical Physics, Addison-Wesley Publishing Co., Reading, MA, 1968, pages 573-588. [2] L. V. Kantorovich and V.!. Krylov, Approximate Methods of Higher Anal­ ysis, Interscience Publishers, New York, 1958, Chapter 4, pages 241-357. [3] A. R. Mitchell and R. Wait, The Finite Element Method in Differential Equations, Wiley, New York, 1977, pages 27-31. [4] H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations, D. Van Nostrand Company, Inc., New York, 1966. [5] D. Zwillinger, Handbook of Differential Equations, Academic Press, New York, Second Edition, 1992. 20 I Applications of Integration 4. Function Representation Idea Certain integrals can be used to represent functions. Procedure This section contains several different representational theorems. Each has found many important applications in the literature. Bochner-Martinelli Representation Let f be a holomorphic function in a domain D C Cn, with piecewise smooth boundary aD, and let f be continuous in its closure D. Then we have the representation (n -1)! ( f«(;) ~ 17. _ z.) dl. = {f(z), if zED, (211"it laD I' -zl2n f:: \~J J~, 0, if z ¢ D, where lCi = del A d(l A··· A [dei] A··· A den A d(n, and [dei] means that the term dei is to be omitted. For n = 1, this is identical to the Cauchy representation. Another way to write this result is as follows: Let H (G) be the ring of holomorphic functions in G. Let G j be a domain in the zrplane with piecewise smooth boundary Cj. If f E H(G) (where G := II;=l Gj) is continuous on G, then ~ ( f«(;) d(l A ... A d(n = {f(z), for z E G, (211"~) lCI X ... XCn «1 -Zl) ... «n -Zn) 0, for z ¢ G. For details see Krantz [5] or Iyanaga and Kawada [6] (page 101). Cauchy Representation Cauchy's integral formula states that if a domain D is bounded by a finite union of simple closed curves r, and J is analytic within D and across r, then J(~) = ~ ( J(z) dz, 27r~ lr z -~ for e E D. (See page 129 for several applications of this formula.) If D is the disk Izi < R, then Cauchy's theorem becomes Poisson's integral formula 1 /.211'. R2 _ r2 J(z) = 27r 0 J(Re'tP) R2 + r2 _ 2Rrcos(O _ 4J) d4J. 4. Function Representation 21 There is an analogous formula, called Villat's integral formula, when D is an annulus. See Iyanaga and Kawada [6], page 636. There are also extensions of this formula when J(z) is not analytic. In terms of the differential operator 8z = ! (8x + 8y), the Cauchy-Green formula is J(~) = ~ ( J(z) dz -.! II 8J_l_ dxdy, 21rt lr z -e 1r az z -e D for ~ E D. This formula is valid whenever J is smooth enough for the derivative 8z to make sense. If J is analytic, then the Cauchy-Riemann equations hold; these equations are equivalent to 8z = O. See Khavinson [7] for details. Green's Representation Theorems • Three dimensions: If ¢ and V2¢ are defined within a volume V bounded by a simple closed surface S, P is an interior point of V, and n represents the outward unit normal, then 1 1 V2ljJ 1 J. 1 8¢ 1 J. 8 (1) ljJ(p)=-- -dV+- --dS-- ljJ--dS. 41r v r 41r S r 8n 41r s 8n r (4.1) Note that if ¢ is harmonic (Le., V2¢ = 0), then the right-hand side of (4.1) simplifies. • Two dimensions: If ¢ and V2 ¢ are defined within a planar region S bounded by a simple closed curve C, P is an interior point of S, and nq represents the outward unit normal at the point q, then ljJ(P) = 21 ( V2¢(q) log Ip -ql dS + 21 { ¢(q) 88 log Ip -ql dq 7Th 1r~ nq -~ ( log Ip -ql~¢(q) dq. 21r 10 8nq • n dimensions: If ¢ and its second derivatives are defined within a region n in R n bounded by the surface ~, and nq represents the outward unit normal at the point q, then for points P not on the surface E we have (if n > 3) 1 ( V2¢(q) ¢(P) = -(n -2)un in Ip _ qln-2 df!q 1 !. ( 1 8¢( q) _ ¢( ) ~ 1 ) d~ + (n -2)un E Ip -qln-2 8nq q 8nq Ip _ qln-2 q where Un = 21rn/2/r(n/2) is the area of a unit sphere in Rn. See Gradshteyn and Ryzhik [2], 10.717, pages 1089-1090 for details. 22 I Applications of Integration Herglotz's Integral Representation The Herglotz integral representation is based on Poisson's integral representation. It states: Let J(z) be holomorphic in Izl < R with positive real part. Then /.271' Reiq, + z J(z) = iq, dp(cP) , o Re -z for Izl < R, where p(cP) is a monotonic increasing real-valued function with total variation unity. This function is deter­ mined uniquely, up to an additive constant, by J(z). See Iyanaga and Kawada [6], page 161. Another statement of this integral representation is (see Hazewinkel [3], page 124): Let J(z) be regular in the unit disk D = {z Ilzl < I}, and assume that it has a positive real part (i.e., Re J(z) < 0), then J(z) can be represented as J(z) = I ~ ~ ~ dl'(~) + ie, where the imaginary part of e is zero. Here I' is a positive measure concentrated on the circle {~ I I~ I = I}. Parametric Representation of a Univalent Function From Hazewinkel [3], page 124, we have: Let J(z) be analytic in the unit disk D = {z Ilzl < I}, and assume that ImJ(x) = 0 for -1 < x < 1 and ImJ(z) Imz > 0 for 1m z :j:. O. Then J(z) can be represented as I zdl'(e) J(z) = (z -~)(z _ ~)' where I' is a measure concentrated on the circle {~ II~I = I} and normalized by 111'11 = J dl'(~) = 1. Pompeiu Formula From Henrici [4] we have the following theorem: Theorem: Let R be a region bounded by a system r of regular closed curves such that points in R have winding number 1 with respect to r. If J is a complex-valued function that is real-differentiable in a region containing R u r, then for any point z E R there holds J(z) = ~ 1 J(z) dt -! /1 8J(t) _1-dxd 211'1 r t - z 11' Oz t - z y, R where t = x + iy. Note that if fez) is analytic, then this reduces to Cauchy's formula. 4. Function Representation 23 Solutions to the Biharmonic Equation Some function representations require that the function have some specific properties. For example, if u is biharmonic in a bounded region R (that is, V4u = 0), and if f = V2u, then u(z) may be written as (for z E R): u(z) = 2~ / / f(t) log It ~ zl dxdy + v(z) R where t = x + iy and v is harmonic in R (that is, v satisfies Laplace's equation V2v = 0). See Henrici [4]. Notes [1] Schlafli's integral representation is an integral representation of the Legendre function of the second kind. [2] See also the section on integrals used to represent the solutions of differential equations (page 1). [3] If a domain D is simply connected, and the vector field V tends suffi­ ciently rapidly to zero near the boundary of D and at infinity, then we have Helmholtz's theorem: V = V tP + V x A, where tP = -/ / / ~~~ dV and A = / / / V4:rV dV. D D [4] If we define the one-form (n -I)! 1 ~ _-- w(<:, z) = (21rit 1<: _ zl2n f=: {(j -Zj) d(i then a generalization of the Bochner-Martinelli representation, which is analogous to the Cauchy-Green formula is given by (see Hazewinkel [3], page 404): If the function 1 is continuously differentiable in the closure of the domain D C en with piecewise-smooth boundary aD, then, for any point zED, I(z) = 1 I«()w«(,z) -181(<:) A w(<:, z). aD D 24 I Applications of Integration References [1] I. A. Ayzenberg and A. P. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis, Translations of Mathematical Mono­ graphs, Volume 58, Amer. Math. Soc., Providence, Rhode Island, 1983. [2] 1. S. Gradshteyn and 1. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York, 1980. [3] M. Hazewinkel (managing ed.), Encyclopaedia of Mathematics, Kluwer Aca­ demic Publishers, Dordrecht, The Netherlands, 1988. [4] P. Henrici, Applied and Computational Complex Analysis, Volume 3, John Wiley & Sons, New York, 1986, pages 290, 302. [5] S. Krantz, Function Theory of Several Complex Variables, John Wiley & Sons, New York, 1982. [6] S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT Press, Cambridge, MA, 1980. [7] D. Khavinson, "The Cauchy-Green Formula and Its Application to Prob­ lems in Rational Approximation on Sets with a Finite Perimeter in the Complex Plane," J. Funct. Anal., 64, 1985, pages 112-123. [8] R. M. Range, Holomorphic Functions and Integral Representations in Sev­ eral Variables, Springer-Verlag, New York, 1986. 5. Geometric Applications Idea Integrals and integration have many uses in geometry. Length If a two-dimensional curve is parameterized by x(t) = (x(t), y(t» for a ~ t ~ b, then the length of the curve is given by For a curve defined by y = y(x), for a ~ x ~ b, this simplifies to A curve in three-dimensional space {x( t), y( t), z( t) }, for a ~ t ~ b, has length 5. Geometric Applications 25 Area If a surface is described by z = !(x, y), for (x, y) in the region Rxy, then the area of the surface, S, is given by If, instead, the surface is described parametrically by x = (x, y, z) with x = x(u,v), y = y(u,v), z = z(u,v), for (u,v) in the region Ruv, then the area of the surface, S, is given by S= j j1xuxXv1dUdV= j jJEG-F2dUdV, Ruv Ruv where E = Xu • Xu = x; + y~ + z~, F = Xu • Xv = XuXv + YuYv + zuzv, (5.1) G = Xv • Xv = x; + y~ + z~. General Coordinate Systems In a three-dimensional orthogonal coordinate system, let {8i} denote the unit vectors in each of the three coordinate directions, and let {Ui} denote distance along each of these axes. The coordinate system may be designated by the metric coefficients {gIl, g22, g33}, defined by .. = (8Xl)2 (8X2)2 (8X3)2 g" 8 + 8 + 8 ' Ui Ui Ui (5.2) where {Xl! X2, X3} represent rectangular coordinates. Then an element of area on the UIU2 surface (i.e., U3 is held constant) is given by dSl2 = [ ..f9ll dUl] [v'022dU2]. Metric coefficients for some common orthogonal co­ ordinate systems may be found on page 113. Moon and Spencer [2] list the metric coefficients for 43 different orthogonal coordinate systems. (These consist of 11 general systems, 21 cylindrical systems, and 11 rotational systems.) Operations for orthogonal coordinate systems are sometimes written in terms of {hi} functions, instead of the {gii} terms. Here, hi = y'9ii, so that, for example, dS12 = [h1dul] [h2du2]. 26 I Applications of Integration Volume Using the metric coefficients defined in (5.2), we define 9 = 911922g33. An element of volume is then given by Moments of Inertia For a bounded set S with positive area A and a density function p(x, y), we have the following definitions: 1 J p(x,y) dA = M = total mass 1: J p( x, y)x dA = My = first moment with respect to the x-axis :Is J p(x, y)y dA = Mx = first moment with respect to the y-axis 1: J p(x, y)x2 dA = Iy = second moment with respect to the y-axis 1: J p(x, y)y2 dA = Ix = second moment with respect to the x-axis J: J p(x, y)(x2 + y2) dA = 10 = polar second moment with respect to the origin. Example 1 Consider a helix defined by x( t) = (a cos t, a sin t, bt) for t in the range [O,27r). The length of this curve is L = 102 " v' a2 8in2 t + a2 C082 t + b2 dt = 102 • v' a2 + b2 dt = 21fv' a2 + b2• Example 2 Consider a torus defined by x = (( b + a sin cP) cos (J, (b + a sin cP) sin (J, acoscP), where 0 ~ (J ~ 27r and 0 ~ cP ~ 27r. From (5.1), we can compute E = xe • Xe = (b + a sin cP)2, F = Xe • Xt/J = 0, and G = xt/J • xt/J = a2. Therefore, the surface area of the torus is {k{k {k{k S= Jo Jo VEG-F2d(JdcP= Jo Jo a(b+asincP)d(JdcP=47r2ab. Example 3 In cylindrical coordinates we have {Xl = r cos cP, X2 = r sin cP, X3 = z} so that {hr = 1, he = r, hz = 1}. Consider a cylinder of radius Rand height H. This cylinder has three possible areas we can determine: {H {21r {H {27r Sez = Jo Jo dSez = Jo Jo hehz d(Jdz = 27rRH, {R {21r {R {27r Ser = Jo Jo dSer = Jo Jo hehr d(J dr = 7r R2, B .. = IoH loR dB .. = IoH loR h,.h. dr dz = RH. 5. Geometric Applications 27 We can identify each of these: SOz is the area of the outside of the cylinder, SOr is the area of an end of the cylinder, and Srz is the area of a radial slice (that is, a vertical cross-section from the center of the cylinder). We can also compute the volume of this cylinder to be 1H 1R 1271' 1H 1R 1271' V = 0 0 0 hohr hz d(} dr dz = 0 0 0 r d(} dr dz = 1r R2 H. Notes [1] IfC is a simple closed curve, positively oriented, that is piecewise continuous, then the line integrals §c x dy and -§c y dx both have the same value, which is equal to the area enclosed by C. This is an application of Green's theorem, see page 164. [2] The quadratic form (see (5.1» I = dx . dx = E du2 + 2F dudv + G dv2 is called the first fundamental form of x = x( u, v). The length of a curve described by x(u(t), v(t», for t in the range [a, b) is [3] The Gauss-Bonnet formula relates the exterior angles of an object with the curvature of an object (see Lipschutz [2]): Let C be a curvilinear polygon of class C2 on a patch of a surface of class greater than or equal to 3. We presume that C has a positive orientation and that its interior on the patch is simple connected. Then L "gds+ J J K dS = 2r -L)" R ' where /;,9 is the geodesic curvature along C, K is the Gaussian curvature, R is the union of C and its interior, and the {Oi} are the exterior angles on C. For example, consider a geodesic triangle formed from three geodesics. Along a geodesic we have /;,9 = 0, so that Li Oi = 27r -J J K ds. For a R planar surface K = O. Hence, we have found that the sum of the exterior angles in a planar triangle is 27r. (This is equivalent to the usual conclusion that the sum of the interior angles of a planar triangle is 7r.) For a sphere of radius a, we have K = l/a2• Therefore, the sum of the exterior angles on a spherical triangle of area A is 27r -A/a2• 28 I Applications of Integration References [1] W. Kaplan, Advanced Calculus, Addison-Wesley Publishing Co., Reading, MA,1952. [2] M. M. Lipschutz, Differential Geometry, McGraw-Hill Book Company, New York, 1969. Schaum Outline Series. [3] P. Moon and D. E. Spencer, Field Theory Handbook, Springer-Verlag, New York,1961. 6. MIT Integration Bee Every year at the Massachusetts Institute of Technology (MIT) there is an "Integration Bee" open to undergraduates. This consists of an hour­ long written exam, with the highest scorers going on to a verbal exam run like a Spelling Bee. It is claimed that completion of first semester calculus is adequate to evaluate all of the integrals. In 1991 the written exam was given on January 15 and consisted of the following forty integrals that had to be evaluated: (1) ! e199b dx (3) ! logxdx (5) ! esin2 xecos2 x dx (7) !+-dX x +1 (9) ! xex sinxdx (11) ! dx sec x + tanxsinx (13) ! V-l+ 1: 3x dx (15) !SinX esecx dx cos2 x (17) ! dx 1rX2 + tan-1 x + x2 tan-1 x + 1r (19) ! dx x2 -lOx + 26 (21) ! dx 12 + 13cosx ! (1-4x4) -1/2 (23) (4X)-1 dx (2) !(sinx -cosx)2dx (4) ! dx 1rX (6) ! Xl:X (8) ! x + 1 dx Vx2 + 2x+2 (10) ! e(ez+x) dx (12) e e dx ! 5x + 7x eX + e-x (14) ! sinh x -coshxdx (16)! x2 + 1 dx x4 -x2 + 1 (18) ! sec3 x dx (20) ! dx x2 -llx -26 (22) !X3 + 1 dx x+l (24) ! e1991 dx 6. MIT Integration Bee (25) I (log x + l)xX dx I dx (27) vx (1 + vx) (29) I v'cscx -sinxdx (31) 142X dx (33) I xex2 dx (35) I eX :.:-x (37) I cos(sinx)cosxdx (39) I rrdx v'16 -e2 (26) I (cos 2x )( sin 6x) dx (28) I e1/xx-3 dx (30) Ix2 + 1 dx x3 -x (32) I x5ex dx I dx (34) (x2 + 1)2 (36) I tan x log I sec xl dx (38)/~ x2 -9 (40) I v'tanxdx. 29 On January 22, the top 11 scorers on the written exam participated in the Integration Bee. (These people had obtained between 26 and 35 correct answers to the above written exam.) The first few rounds were run with a fixed time in which to simplify a specific integral. The integrals, and the time allowed for each, were: • 1 minute for f sin-1 x dx . x2 -2x + 2 • 2 mInutes for J 2 dx, x +1 . sin2 x cos2 x • 2 mmutes for f 1 2 dx, +cos x • 2 minutes for f v'x + x2VX dx (since five people in a row did not obtain the correct answer, this integral was discarded and the people who could not integrate it were not penalized), • 2 minutes for f cos 4x cos 2x dx, v'x3 -1 • 2 minutes for f dx. x After these integrals, there were four finalists. The ranking of the finalists was achieved by four rounds of competitive integration (a pair was given the same integral; whoever obtained the correct answer first was the winner of that round). The integrals to be evaluated were leX(:; 2) dx I cotx d log(sinx) x Ixsec2xdx Ixsecx(xtanx+2) dx. 30 I Applications of Integration The 1991 title of "Grand Integrator of MIT" was awarded to Chris Teixeira. The second and third place winners were Belle Y seng and Trac Tran. 7. Probability Idea This section describes how integration is used in probability theory. Procedure If p( x) represents the density function of the random variable X then the expectation of the function g(X) is given by E[g(X)) = f p(x)g(x) dx, where the range of integration is specified by the density function. Expec­ tations of certain functions have special names and notations. For example, mean of X = J-t = J-tl = E[X), variance of X = Var(X) = 0"2 = JL~ = E[(X -J-t)2), n-th moment of X = J-tn = E[xn), n-th central moment of X = JL~ = E[(X -J-t)n), characteristic function of X = 4>( t) = E [eitX] , generating function of X = 1/J(s) = E [sX] . The random variable X, with density function /(x), has the distri­ bution function F(x) = J~oo /(t) dt. The probability that X < x is then given by F(x). Notes [1] The mean is sometimes called the "average." The skewness is defined to be to be ""3/U3, and the excess is defined to be ""4/u4 -3. [2] If the random variable X has the density function f(x), then the entropy of X is defined to be (see McEliece [3]) H(X) = -E[logf(X)[ = f.~ f(x)logf(x)dx. [3] If the random variable Xn (for n = 1,2, ... ) has the distribution IPn (for n = 1,2, ... ), respectively, and if lim 100 f(x) dlPn(x) = 100 f(x) dIP 00 (x) n-oo -00 -00 for every continuous function f with compact support, then the sequence {lPn} is said to converge in distribution to Xoo. 8. Summations: Combinatorial 31 [4] For a continuous parameter random variable {X(t)}, we can also define mean of X = J.'(t) = E[X], variance of X = Var(t) = E[(X(t) -1'(t))2], covariance of X = Cov(s, t) = E[(X(t) -l'(t»(X(s) -I'(s))], References [1] W. Feller, An Introdu.ction to Probability Theory and Its Applications, John Wiley & Sons, New York, 1968. [2] C. W. Helstrom and J. A. Ritcey, "Evaluation of the Noncentral F-Distri­ bution by Numerical Contour Integration," SIAM J. Sci. Stat. Compu.t., 6, No.3, 1985, pages 505-514. [3] R. J. McEliece, The Theory of Information and Coding, Addison-Wesley Publishing Co., Reading, MA, 1977. 8. Summations: Combinatorial Applicable to Evaluation of combinatorial sums. Procedure A combinatorial sum may sometimes be written as a summation over contour integrals. Interchanging the order of integration (when permitted), allows a different integral to be evaluated. Evaluating this new integral will then yield the desired sum. Finding the contour integral representation of the terms in the sum­ mation may be aided by Table 8. Example 1 Consider the sum where m is an integer. By use of Table 8 we make the identification If we choose PI and P2 appropriately (Le., in this case we require lyl2 > 4(1+x)(1+y) in the integration), then we may move the summation inside the integrals and evaluate the sum on k to obtain 32 I Applications of Integration Table 8. Representations of combinatorial objects as contour integrals. Here, resF(x) denotes the sum of the residues of F(x) at all poles within some region Z centered about the origin. That is: resF(x) = -21 _ J;r F(x) dx. Z 1rt Binomial Coefficients (where 0 < P < 1): (m) = res(l + x)mx-n-1 = -21 -J. (1 + x)mx-n-1dx. n Z 11"1 Izl=p Multinomial Coefficients (where rep) = {x = (Xl, .•. ,Xk) I Ixl = Pi, o < Pi < 1, i = 1, ... , k } ): Bernoulli numbers: En = n!res (eZ -1) -1 X -n • Z Euler numbers: En = n!res cosh-1 (x)x-n-1• Z Power terms Since m is an integer, the integral with respect to x may be evaluated by the residue theorem to obtain Evaluating this last integral, by another application of the residue theorem, we obtain our final form for the summation This is one of the so-called Moriety identities. 8. Summations: Combinatorial 33 Example 2 Consider the summation n m m+n+p- min(m,n) ( ) ( ) ( k) R(m,n.p):= ?; k k m+n ' (8.1) where m, n, and p are non-negative integers. Using Table 8, it is easy to show that The reason that the k summation can be extended to include large values of k is because there are no contributions from these values. By defining S:r; = {x = (X},X2,X3) Ilxll = IX21 = 2,lx31 = !}, this integral can be written as If we introduce the new variables h and t2 and define the curve St = {t = (tl, t2) I Itll = It21 = l~}' then this last three-dimensional integral and summation may be written as the following five-dimensional contour inte­ gral: 1 1 (1 + X3)P+1X3 R(m,n,p) = --:; m+l n+l dxdt, (21ri) St XS:r: f(h, xdf(t2, X2)(XIX2(1 + X3) -1)tl t2 where I(a, b) := X3 -a(l + b)(l + X3). If this five-dimensional integral is evaluated with respect to X}, X2, and X3, in that order, then we obtain Using Table 8, this two-dimensional integral is equal to (m+p) (n+p) R(m,n,p) = m n' (8.2) 34 I Applications of Integration The final result, equations (8.1) and (8.2), can be evaluated for differ­ ent choices of the parameters to obtain, for instance, and Notes [1] Both of the examples in this section are from Egorychev [2], pages 52 and 169. [2] In the paper by Gillis et al. [3] the following representation of the Legendre polynomials is used to evaluate the integral J~l Pn1 (x) ... Pnk (x) dx, where nl, ... ,nk are non-negative integers: [3] Bressoud [1] uses a combinatorial approach to evaluate integrals of the form J1r/2 J1r/2 n . 2k(u) d d h S' f l' -1r/2'" -1r/2 uES sm u al ••• ai, were IS a set 0 non mear sums of elements of the {ai}, and k is an integer-valued function. References [1] D. M. Bressoud, "Definite Integral Evaluation by Enumeration, Partial Re­ sults in the MacDonald Conjectures," Combinatoire enumerative, Lecture Notes in Mathematics #1234, Springer-Verlag, New York, 1986, pages 48- 57. [2] G. P. Egorychev, Integral Representation and the Computation of Combina­ torial Sums, Translations of Mathematical Monographs, 59, Amer. Math. Soc., Providence, Rhode Island, 1984. [3] J. Gillis, J. Jedwab, and D. Zeilberger, "A Combinatorial Interpretation of the Integral of the Product of Legendre Polynomials," SIAM J. Math. Anal., 19, No.6, November 1988, pages 1455-1461. [4] K. Mimachi, "A Proof of Ramanujan's Identity by Use of Loop Integrals," SIAM J. Math. Anal., 19, No.6, November 1988, pages 1490-1493. 9. Summations: Other Idea Some summations can be determined by simple manipulations of in­ tegrals. 9. Summations: Other 35 Procedure One technique for evaluating infinite sums is by use of the Watson transform (see page 44). Under suitable convergence and analyticity con­ straints, we have: Theorem: If g(z) is analytic in a domain D with a Jordan contour C, then -21 .1g(Z) cot 7rZ dz = L9(n) 7r' c for those integers n that are within c. Alternately (see Iyanaga and Kawada [1], page 1164): If an analytic func­ tion J(z) is holomorphic except at poles an (n = 1,2, ... , k) in a domain bounded by the simple closed curve C and containing the points z = m (for m= 1,2, ... ,lV), then f; f(m) = 2~i k 1r(cot1rZ)f(z) dz -t. Res [1r(cot 1rZ)f(z)ll.= ... · When the left-hand side is replaced by L:~=1 ( -1)m J (m), then the cot 1r z must be replaced by cosec 1r z. Another technique that can be used to evaluate summations is the Euler-Maclaurin summation formula. From Wong [3] (page 36) we find If f(t) is defined on 0 ~ t < 00, and if f(2n)(t) is absolutely integrable on (0,00) then, for n = 1,2, ... /(0) + ... + /(n) = [o/(X) dx + i [/(0) + /(n)] m-1 + L (~;)! [J'28-1)(n) -1'28-1)(0)] + RM(n), 8=1 (9.1) .. 1n B2m -B2m (X -l x J ) (2 ) where the remamdens Rm(n) = 0 (2m)! f m (x) dx. The remainder can be bounded by 1R.n(n) I ~ (2 -2'-2m) ~~':'i: 1" Il2m)(x)1 dx. In this theorem, the Bernoulli polynomials {Bs(x)} are defined by the generating function text 00 tS -t-= LBs(x),. (9.2) e -1 s=o s. The Bernoulli numbers {Bs} are given by Bs = Bs(O) and a generating function for them can be obtained from (9.2), by setting x = o. 36 I Applications of Integration 1\ I I I ---- ~ -N -1 N - --~ -:<3-~ ~1-1-~ -~-~ -~----: ~N-+l --> I -ia * I I I I Figure 9.1 Contour for the integral in (9,3). Example 1 As an example of the Watson transform, consider the sum S = L:::"=ll/(n' + a'). We define the integral I = ~ r 7r(cot7rz) dz 27ri leN Z2 + a2 (9.3) where CN is the contour shown in Figure 9.1. Note that the vertical and horizontal sides to CN are at the values -N -4 and N + 4· The contour integral in (9.3) can be evaluated by using Cauchy's theorem (see page 129). The poles within the contour are at z = ±ia, 0, ±1, ±2, ... , ±N. The residues at ±ia are 7r cot(±i7ra)/(±2ia), and the residue at z = n (for n = 0, ±1, ... , ±N) is l/(n' + a'). Hence, ~ 1 [7r cot( i7ra) 7r cot( -i7ra)] I= ~ , , + . + . . n +a 2za -2~a n=-N (9.4) As N ~ 00, it is easy to show from (9.3) that I ~ o. Indeed, since the cotangent function is bounded , we have I = 0 (N-') O(N) ~ 0 as N ~ 00. Taking the limit as N ~ 00, and combining (9.3) and (9.4), we find f 1 [7rCot(i7ra) 7rCot(-i7ra)] = 0 n' + a' + 2ia + -2ia n=-oo or 00 1 7r 1 " = -coth 7ra --,. L..., n2 + a2 2a 2a n=l (9.5) 9. Summations: Other 37 If the limit a --+ 0 is taken in this formula, then we obtain the well-known result (see also Example 3): E~=l n-2 = 7r2/6. Example 2 As an example of the Euler-Maclaurin summation formula, consider the harmonic numbers, defined by Hn = t + ~ + ... +~. Using (9.1) we find: 1 1 B2 28 -1 , ( ) m-l [ 1 Hn = logn+ 2" 1 + n + ~ (2.)! - ( n2B ). + (2. -I)! +Rm(n-l), (9.6) where D( -1) = /.n-l B2m -B2m (x -LxJ) d .L"m n ()2m+l x. o 1 +x Taking the limit of n --+ 00 in (9.6) results in an expression for Euler's constant '1: . . /.00 B2m -B2m (x -LxJ) where the error term IS gIven by Em = ()2m+l dx. o 1 +x Example 3 As an example of a different technique, consider the evaluation of the zeta function at an argument of two: ((2) := E~=1 n-2• We have 00 1 00 1 ((2) = ~ (2n + 1)2 + ~ (2n)2 00 1 1 = ~ (2n + 1)2 + 4((2) 4 00 1 = 3 ~ (2n + 1)2 . 38 I Applications of Integration Since J01 x2n dx = 1/(2n + 1), we can write (J01 y2n dy) (J01 x2n dX) = 1/(2n + 1)2. Therefore, we have (2) = ~ f 1 n=O 4 00 /.1/.1 = 3" L (xy)2n dxdy n=O 0 0 4 /.1 /.1 00 = 3" L(xy)2n dxdy o 0 n=O _ ~ /.1 /.1 dxdy - 2 2' 3001-xy Now make the change of variables from {x,y} to {u,v} via x = sinu/cosv, y = sin v / cos u. The Jacobian of the transformation is given by J = 18(x, y) I = I Xu Yu I 8(u, v) Xv Yv cosu sin usin v = I • cos y cos2 v smvsmu cos v cos2 u cosu _ 1 _ sin2 usin2 v -cos2 U cos2 V = l_x2y2. Continuing the calculation of the zeta function, we find (2) = ~ /.1 /.1 dx d~ _ 3001-xy 4 /.1 /.1 J = -3 2 .. dudv o 0 1-X Y (9.7) 41x=I/.1I=1 = - dudv. 3 x=o 11=0 The region of integration in the (u, v) plane becomes the triangle with vertices at (u = 0, v = 0), (u = 0, v = 1r/2), and (u = 1r/2, u = 0) (see Figure 9.2). Since this triangle has area 1r2/8 we finally determine 4 2 2 (2) = -~ =~. 3 8 6 9. Summations: Other 39 v 71" 2' ~: ~:~: ~: ~: ~: ~: ~: ~:~:::: .. Figure 9.2 Integration region in (9.7). Notes [1] Under some continuity and convergence assumptions, the Poisson summa­ tion formula states (see Iyanaga and Kawada [1], page 924) f: f(n) = f: [" .2w;n. f(t) dt. n=-oo n=-oo -00 (This formula can also be extended to functions of several variables.) For example, if we take f(t) = e-7I"t2 x (for some fixed x > 0), then we obtain ~ 1 (1rn2) = ~ v'xexp ---;-. n=-oo (9.8) For small values of x, the sum on the right-hand side of (9.8) converges much more quickly than the sum on the left-hand side. See also Smith [2]. We note in passing that the equation in (9.8) represents the following functional relationship of theta functions: 8( x) = .Jx 8 (~) . [2] For another example similar to Example 1, the summation ~ cosnt = 1rcosha(1r -t) __ 1_ ~ n2 + a2 a sinh 1ra 2a2 n=l can be derived from the integral frac121ri Ie 1r2COS Z;1r ~ t) dz. (z + a ) SlD1rZ References [1] [2) [3] S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT Press, Cambridge, MA, 1980. P. J. Smith, "A New Technique for Calculating Fourier Integrals Based on the Poisson Summation Formula," J. Statist. Comput. Simulation, 33, No.3, 1989, pages 135-147. R. Wong, Asymptotic Approximation of Integrals, Academic Press, New York, 1989. 40 I Applications of Integration 10. Zeros of Functions Applicable to Functions with zeros that we would like to characterize. Idea By evaluating certain integrals, information about the location of zeros of functions can be obtained. Procedure There are several theorems that can be used to determine the location of zeros of functions. We illustrate two such theorems. A standard theorem from complex analysis states (see Levinson and Redheffer [6], Theorem 6.1): Theorem: Let I{z) be a meromorphic function in a simply connected domain D containing a Jordan contour C. Suppose I has no zeros or poles on C. Let N be the number of zeros and P the number of poles of I in C, where a mUltiple zero or pole is counted according to its multiplicity. Then 1 r J'(z) 211'i 1c I{z) dz = N -P. The principle of the argument is the name given to the statement: N -P = 2~ arg/(z)lc' (10.1) (10.2) which is just a reformulation of (10.1). (Note that "arg" denotes the argument or phase of the following function.) The quantity N -P is also known as the index of f relative to the contour C. See Example 1 for an application of (10.2). Another useful theorem is (see Bharucha-Reid and Sambandham [1], Lemma 4.9): Theorem: If I(t) E C1 for a < t < band I(t) has a finite number of points in a < t < b with f'(t) = 0, then the number of zeros of I(t) in the interval a < t < b is given by 1 roo J.b n{a,b;f)= 211' 1-00 d( a cos[CI(t)1IJ'{t)ldt, where mUltiple zeros have been counted once. See Example 2 for an application of this theorem. 10. Zeros of Functions 41 Figure 10. The contour used in Example 1. Example 1 Consider the polynomial z3 -z2 + 2. How many roots does it have in the first quadrant? We will use the principle of the argument with J(z) = Z3 -z2 + 2 and the contour in Figure 10.1 as R ~ 00. To use (10.2), we must determine how the argument of J(z) changes on the three components of the contour C: • The horizontal component (y = 0 and 0 ~ x ~ R): We observe that J(O) > 0 and J(oo) > o. Since there is only one inflection point of J(z) on this component (at z = i), we conclude that J(z) is always positive on this segment. Hence, there is no change in arg J on this segment: arg J{X)IX=R = o. x=o (10.3) • The curved component z = Rei9 with 0 ~ () ~ ~: On this component, J(z) = R3e3i9(1 + w) where Iwl < 2/ R for large values of R. Hence, arg J (Rei9) = 3(} + arg(l + w). Therefore, 19=1r/2 3 argJ(Rei9) = ; + e1, 9=0 (10.4) where e1 ~ 0 as R ~ 00. • The vertical component (x = 0 and 0 ~ y ~ R): We observe that J(iy) = (_y2 + 2) + i{ _y3). As y decreases from R to 0, Re J(iy) changes its sign at y = v'2 from negative to positive, while 1m J{iy) remains negative. Hence, as y decreases from R to 0, J{iy) starts in the third quadrant and ends in the fourth quadrant. Therefore, Iy=o 3 argJ(iy) = 27T -; + e2 = ~ + e2, y=R (10.5) where e2 ~ 0 as R ~ 00. Combining the results in (10.3), (10.4), and (10.5) we find that argj(z)lc = 0 + C; +e,) + G + e2). As R ~ 00 we conclude that arg J{z)lc = 27T. From (10.2) we conclude, therefore, that there is exactly one root of J(z) in the first quadrant. 42 I Applications of Integration Example 2 In this example we answer the question: "What is the average number of real roots of the polynomial f(x; a) := ao + alX + ... + an_1Xn-1 when the {ai} are chosen randomly?" The analysis here is from Kac [5]; see also Bharucha-Reid and Sambandham [1]. For definiteness, we presume that the {ai} lie on the surface of the n-dimensional sphere of radius unity, Sn(l) (i.e., the {ail satisfy lal2 = L:;:~ a~ = 1). Define Nn(a) to be the number of real roots of f(x; a). A simple scaling argument shows that Nn(o:a) = Nn(a) for any non-zero constant 0:; this will be needed later. Define Mn to be the average number of real roots of the polynomial f(x; a) as a varies over Sn (1). That is, Mn:= ISn~l)IL.(ltn(a)du. (Here du represents the surface element on a sphere.) It is not difficult to show that Mn can also be represented in the form 1 100 100 1 2 Mn = / .. . e-2lal Nn(a) da (21t')n 2 -00 -00 (where da = dao dal ... dan-i), since this last integral can be rewritten as follows: Mn = 1 /2 1.00 e-r2/2 ( ( Nn(a) dUr) dr (21t')n 0 iSn(r) = 1 1.00 e-r2/2 (rn-1 [ Nn(a) dU) dr (21t')n/2 0 iSn(l) = ISn~l)1 (L'(l) Nn(a) dU) If we we use the notation 0(1) to denote the number (or fraction) of real roots in the interval (-1,1) and 0(2) to denote the number (or fraction) of real roots not in the interval (-1,1), then N~l)(a) = N~2)(a) because ~n-1 k (~n-l -k) n-1 (Th t' r. I' I L.Jk=O akX = L.Jk=O an-l-kX x. a 18, lor every po ynomla having x as a root, there is a corresponding polynomial with x-1 as a root.) This implies that M~l,<a) = M~2)(a) and Mn(a) = 2M~1)(a). Using the second theorem stated in the procedure, with a = -1 and b = 1, we find that M~l) = 1 n/21°O ... 100 e-laI2/2 da ~ 100 d( 11 cos[(f(t))lf' (t)1 dt. (21t') -00 -00 21t' -00 -1 (10.6) 10. Zeros of Functions 43 (Note that we write f(t) for the function /(t; a).) Interchanging the order of integration, and recognizing that the absolute value function has the . I' I . I I 1 100 1 -cos TJY d II ( 0 ) sImp e Integra representatIon Y = - 2 TJ, a ows 1 .6 to 11" -00 TJ be written as M~l) = 21 fl dt foo d( R«(, t), 11" J-I J-oo where 1 100 [ ] dTJ R«(, t) = - S(t, (, 0) -S(t, (, TJ) 2" 11" -00 TJ and 1 100 100 2 S(t, (, TJ) = nl2 ••• e-1al 12 dacos[(/(t)] cos[TJ!'(t»). (211") -00 -00 Writing the cosine function in complex exponential form allows the a­ integrals to be evaluated (recall that /(t) = /(t; a» to determine that 1 [ ( n-l ) S(t, e, '1) = 2 exp -~ {; (etk + k'1tk-l)' + exp ( -~ ~ (etk -k'1tk-l)') ]. Hence, R«(, t) = .!. fOO [e-«(2 12) L:::~ t2 A: _ e-(I/2) L:::~«(tk+k'1tA:-l )2] dTJ. 1T J-oo TJ2 If we define the functions An(t) = L:~:~ tk, Bn(t) = L:::~ kt2k-l, and en(t) = L:::~ k2t2k, then we can finally find M(l) =.!. fl v' An(t)Cn(t) -B~(t) n 11" J-I An(t) dt. Our final answer is therefore Mn = 2M~I) =.!. fl 11" J-I 1 _ (ntn-l(l -t2») 2 1_t2n 1-( dt. An asymptotic analysis then reveals that Mn ~ ~ log n as n -+ 00. 11" (10.7) 44 I Applications of Integration Notes [1] The same result in (10.7) is obtained in three different cases: (A) The {ail are chosen to be uniformly distributed on the interval (-1,1); (B) The {ail are chosen to be equal to +1 and -1 with equal probability; (C) The {ail are chosen to be uniformly distributed on the unit ball (as shown in Example 2). [2] If C· is the image of C under f in the first theorem in the Procedure, then N -P turns out to be equal to the number of times C· winds around the origin; i.e., it is the winding number of C· with respect to the origin. (3] In example 1, the polynomial J(z) = z3 -Z2 + 2 has the roots z = -1 and z = 1 ± i. [4] If g(z) is analytic in D, the zeros of fez) are simple and occur at the points {ail, the poles of fez) are simple and occur at the points {bi}, then the result in (10.1) may be extended to If we choose fez) = sin 1rZ, then (10.8) becomes -21 ·lg(Z)cot1rZdZ = Lg(n), 1r~ c (10.8) for those integers n that are within the contour C. This formula is very useful for evaluating infinite sums; it is known as the Watson transform. An example of its usage may be found in the section beginning on page 34. [5] Rouche's theorem compares the number of zeros of two related functions. It states (see Levinson and Redheffer [6], Theorem 6.2): Let f (z) and g( z) be analytic in a simple connected domain D containing a Jordan contour C. Let If(z)1 > Ig(z)1 on C. Then fez) and fez) + g(z) have the same number of zeros inside C. This theorem can be used to prove that a polynomial of degree n has n roots. For the polynomial h(z) = E~o aii of degree n, choose fez) = aozn and g(z) = h(z) -fez). References [1] A. T. Bharucha-Reid and M. Sambandham, Random Polynomials, Academic Press, New York, 1986, Chapter 4, pages 49-102. [2] M. P. Carpentier, "Computation of the Index of an Analytic Function," in O. Keast and G. Fairweather (eds.), Numerical Integration: Recent Devel­ opments, Software and Applications, Reidel, Dordrecht, The Netherlands, 1987, pages 83-90. [3] M. P. Carpentier and A. F. Dos Santos, "Solution of Equations Involving Analytic Functions," J. Comput. Physics, 45, 1982, pages 210-220. 11. Miscellaneous Applications 45 [4] N. 1. Ioakimidis, "Quadrature Methods for the Determination of Zeros of Transcendental Functions-a Review," in P. Keast and G. Fairweather (eds.), Numerical Integration: Recent Developments, Software and Applications, Reidel, Dordrecht, The Netherlands, 1987, pages 61-82. [5] M. Kac, "On the Average Number of Real Roots of a Random Algebraic Equation," Bull. Amer. Math. Soc., 49, pages 314-320, 1943. [6] N. Levinson and R. M. Redheffer, Complex Variables, Holden-Day, Inc., San Francisco, 1970, Section 4.6, pages 216-223. 11. Miscellaneous Applications Idea This section describes other uses of integration. Physics Let F be the force on an object in three-dimensional space. The work done in moving an object from point a to point b is defined by the line integral b w= fa F·ds, where s is an element of the path traversed from a to b. In a conservative force field, the force can be written as the gradient of a scalar potential field: F = V P. In this case, the amount of work performed is independent of the path and is given by W = PCb) -Pea). For example, since gravity is a conservative force field, g = (0,0, g) = V(gz), the work performed in moving an object from the location a = (ax,ay,a z) to the location b = (bx,by,b z) is just W = g(bz -az). Mechanics The momentum of a rigid body K is defined by Q = f(dr/dt) dm where dm is the mass of the volume element at a point rand (dr/dt) is its velocity. See Iyanaga and Kawada [6], page 454. The angular momentum of a rigid body, about an arbitrary point ro, is defined by H = f(r -ro) x (dr/dt) dm. See Iyanaga and Kawada [6], page 455. Mechanics If V2u = -1 in G, and u = ° along 8G, then the torsional rigidity of the domain G is defined to be P = 4 J J u dA. For a disk of radius R, we G have P = 7rR4/2. See Hersch [4]. 46 I Applications of Integration Notes [11 Using an integral representation of derivatives, Calio et al. [2] demonstrate how quadrature formulas can be used to differentiate analytic functions. Let Q be a bounded two-dimensional domain with a partly smooth curve of boundary. Aizenberg [11 contains an integral representation for the difference between the number of lattice points of Q and its volume. Then a similar result for a three-dimensional domain is given. [21 [31 Ioakimidis [51 uses contour integrals to find the location of branch points. References [1] L. A. Aizenberg, "Application of the Multidimensional Logarithmic Residue to Number Theory. An Integral Formula for the Difference Between the Number of Lattice Points in a Domain and its Volume," Ann. Polon. Math., 46, 1985, pages 395-40l. [2] F. Callo, M. Frontini, and G. V. Milovanovic, "Numerical Differentiation of Analytic Functions Using Quadratures on the Semicircle," Compo & Maths. with Appls., 22, No. 10, 1991, pages 99-106. [3] H. M. Haitjema, "Evaluating Solid Angles Using Contour Integrals," Appl. Math. Modelling 11, No.1, 1987, pages 69-7l. [4] J. Hersch, "Isoperimetric Monotonicity: Some Properties and Conjectures (Connections Between Isoperimetric Inequalities)," SIAM Review, 30, No.4, December 1988, pages 551-577. [5] N. I. Ioakimidis, "Locating Branch Points of Sectionally Analytic Functions by Using Contour Integrals and Numerical Integration Rules," Int. J. Compo Math., 41, 1992, pages 215-222. [6] S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT Press, Cambridge, MA, 1980. [7] 1. Vardi, "Integrals, an Introduction to Analytic Number Theory," Amer. Math. Monthly, 95, No.4, 1988, pages 308-315. II Concepts and Definitions 12. Definitions Asymptotic Expansion Let f (x) be continuous in a region R and let {<Pn(x)} be an asymptotic sequence as x -+ Xo. Then the formal series E~=o an<Pn(x) is said to be an infinite asymptotic expansion of f(x), as x -+ Xo, with respect to {<Pn (x)} if the equivalent sets of conditions m f(x) = L an<Pn(x) + O(<Pm+l(X)), as x -+ Xo, (12.1) n=O I· n=O ° [f(X) -t an<pn(x)] 1m = , %-%0 <Pm (x) (12.2) for each m = 0,1,2, ... are satisfied. If, instead, (12.1) only holds for m = 0, 1, 2, ... , N -1, then N-l f(x) "oJ L an<Pn(X), as x -+ Xo n=O 47 48 II Concepts and Definitions is said to be an asymptotic expansion 01 1 to N terms with respect to the asymptotic sequence {4>n(x)}. Note that 00 is defined on page 351. Asymptotic Sequence The sequence of functions {4>n(x)}, n = 0,1,2, ... is called an asymptotic sequence as x --. Xo in some region R if, for every n, 4>n(x) is defined and continuous in Rand 4>n+l(X) = o(4)n(x)), as x --. Xo. Auxiliary Asymptotic Sequence Let {4>n(X)} be an asymptotic se­ quence as x --. Xo. Then the formal series E~=o anln(x) is said to be an asymptotic expansion of I(x), with respect to the auxiliary asymptotic sequence {4>n (x)} if N I(x) = L In (x) + O(4)N(X)), as x --. Xo, n=O for each m = 0,1,2, .... This is denoted I(x) /'oJ E~=o In(x) when the auxiliary asymptotic sequence is understood. Borel Field of Sets Given a system of sets M, the Borel field of sets is the smallest system of sets containing M and closed with respect to the operations of countable union and taking complements. Bromwich Integral The Bromwich integral is a contour integral that has the shape of a semicircular region, extending to infinity, with the Bat portion vertical in the complex domain. Holomorphic Function A function is holomorphic in a domain if it is holomorphic at every point in the domain. A function is holomorphic at a point if it satisfies the Cauchy-Riemann equations at that point. Inner Product An inner product is a binary operation on a vector space which produces a scalar. The inner product must also satisfy the following properties (where a, b, and c are vectors, z is a scalar, and an overbar indicates complex conjugation): [1] (a, b) = (b, a) [2] (a, b + c) = (a, b) + (a, c) [3] (za, b) = z(a, b) [4] (a, a) > 0, unless a = 0, when (0,0) = 0 Integrand In the integral J: 1 (x) dx, the integrand is 1 (x). Jacobian The Jacobian, or the Jacobian determinant, is denoted by 8(F, G, H, ... )/8(u, v, w, ... ) and is defined to be the determinant Fu 8(F,G,H, ... ) I Gu - H 8(u,v,w, ... ) -u F1J G1J H1J Fw Gw Hw When used in a change of variable computation (see page 109), the absolute value of the Jacobian is used. 12. Definitions 49 L~ Functions A measurable function f (x) is said to belong to Lp if fo If(x)IP dx is finite. Lebesgue Measurable Set A set of RP is called a Lebesgue measurable set (or simply a measurable set) if it belongs to the smallest u-algebra containing the Borel sets and the sets of measure zero of RP. Leibniz's Rule Leibniz's rule states that d (1,9(t) ) 1,9(t) 8h -d h(t,~) de = g'(t)h(t, g(t» -f'(t)h(t, f(t)) + at (t,~) de· t f(t) f(t) Linear Independence Given the smooth functions {Yl, Y2, ... , Yn}, the Wronskian is the determinant Yl Y2 Yn Y~ Y~ Y~ (n-l) Yl (n-l) Y2 (n-l) Yn If the Wronskian does not vanish in an interval, then the functions are linearly independent. Lower Limit, Upper Limit In the integral f: f(x) dx, the lower limit is the value a, the upper limit is the value b. Measure A (positive) measure on a u-algebra A is a mapping p. of A into [0,00] such that if E is a disjoint union of a sequence of sets En E A, then p.(E) = E p.(En). Meromorphic Function A meromorphic function is analytic, except possibly for the presence of poles. Norms If f is a measurable function on R n then we define the Lp norm of f (for 0 < p < (0) by IIfllp = (fRR If(x)IP dX)l/P. Orthogonal Two functions f(x) and g(x) are said to be orthogonal with respect to a weighting function w(x) if the inner product vanishes, i.e., (f(x), g(x)) := f f(x)w(x)g(x) dx = 0 over some appropriate range of integration. Here, an overbar indicates the complex conjugate. Pole An isolated singularity of f(z) at a is said to be a pole if f(z) = g(z)/(z-a)m, where m ~ 1 is an integer, g(z) is analytic in a neighborhood if a and g(o) :F O. The integer m is called the order of the pole. Sigma-algebra A family A of subsets of a set X is called au-algebra if the empty set is in A, and if A is closed under complementation and countable union. 50 II Concepts and Definitions Set of Measure Zero A set E in RP is called a set of measure zero (or a negligible set) if there exists a Borel set A such that E C A and v( A) = O. Variations Let f(x) be a real bounded function defined on [a, b). Given the subdivision a = Xo < Xl < ... < Xn = b, denote the sum of positive (negative) differences f(Xi) -f(Xi-d by P (-N). The suprema of P, N, and P + N, for all possible subdivisions of [a, b), are called the positive variation, the negative variation, and the total variation of f(x) on [a, b). If anyone of these three values is finite, then they are all finite. In this case, f(x) is said to be of bou.nded variation. The continuous function X sin ~ is not of bounded variation, while the discontinuous function sgn X is of bounded variation. If g(x) is an increasing functiQn on [a, b) then the total variation of g(x) on [a, b), written var[a,b]g, is given by g(b) -g(a). Hence, by writing an arbitrary continuous function f(x) as the difference of two increasing functions f(x) = h (x) -f2(X) we find var[a,b]f = var[a,b]h + var[a,b]h = (h(b) -h(a» + (f2(b) -f2(a». One way to form the decomposition f(x) = h(x) -f2(X) is by "(x) = [[!'(t)]+ dt, h(x) = [[!'(t)r dt. (Note that the notation []-and []+ is defined on page 352.) Bounded Variation A function g(x) is of bounded variation in [a, b) if and only if there exists a number M such that E~l Ig(Xi) -g(xi-dl < M for all partitions a = Xo < Xl < X2 < ... < Xm = b of the interval. Alternately, g(x) is of bounded variation if and only if it can be written in the form g(x) = gl(X) -g2(X) where the functions gl(X) and g2(X) are bounded and nondecreasing in [a, b). Weyl's Integral Formula Let G be a compact connected semisimple Lie group and H a Cartan subgroup of G. If 1', (3, and .,\ are all ~ormalized to be of total measure 1, then f f(g) dJ.t(g) =.!. f f f (ghg-l) J(h) d.,\(g*) d{3(h) JG w JHJa/H for every continuous function f on G, where w is the order of the Weyl group of G. Here J is given by 2 J(expX) = II (ea(X)/2 -e-a(X)/2) , aEP where P is the set of all positive roots a of G with respect to H and X is an arbitrary element of the Lie algebra of H. 13. Integral Definitions 51 13. Integral Definitions Idea There are many different types of integrals of interest. These integrals include the following: Abelian (see below) contour (see page 129) fractional (see page 75) improper (see below) Lebesgue (see below) loop (see page 4) Riemann (see below) Stratonovich (see page 186) Properties of Integrals Cauchy (see page 92) Feynman (see page 70) Henstock (see below) Ito (see page 186) line (see page 164) path (see page 86) stochastic (see below) surface (see page 24) Lebesgue [16] defined six properties that the integral of a bounded function should have. These properties are: [1] J: f(x) dx = J::~ f(x -h) dx; [2] J: f(x) dx + IbC f(x) dx + Jca f(x) dx = 0; [3] J: [hex) + f2(X)] dx = J: hex) dx + J: f2(X) dx; [4] Jo1 1 dx = 1; [5] If f > 0 and b > a, then J: f(x) dx > 0; [6] If fn(x) ~ fn+l(X) and limn_co fn(x) = f(x) for all x, then limn_co J: fn(x) dx = J: f(x) dx. Squire [22] indicates that more than sixty kinds of integrals have been developed that satisfy the above criteria, in different degrees of generality. In the following sections we describe only a few of the different types of integrals. Pesin [20] has a very comprehensive review of many types of integrals. Abelian Integral Suppose that we have an algebraic curve whose equation is G(x, y) = o. Let y = f (x) be the algebraic function satisfying this equation, and define S to be the associated Riemann surface on which y is single-valued. Define the rational function R by R(x, y) = P(x, y)/Q(x, y), where both P and Q are polynomial functions. Note that R is single-valued on S and that the only singularities that R has on S are a finite number of poles. An Abelian integral has the form lex, y) = J::Y R(x, y) dx, where the path of integration is on the surface S. The value t~f this integral depends upon the integration path. Note that lex, y) is regular for all finite paths 52 II Concepts and Definitions that avoid the poles of the integrand. There are only three kinds of Abelian integrals; an Abelian integral is of [1] the first kind if it is regular everywhere, [2] the second kind if its only singularities are poles, [3] the third kind if it has logarithmic singularities. No other types of singularities are possible for an Abelian integral. Note that an Abelian integral can be of the first kind and not be constant; Liouville's theorem does not apply since I is defined on a Riemann surface, not the complex plane. If the limits of integration are fixed, then all possible values of an Abelian integral can be determined by considering the combinatorial topol- ogy of S. Fixing the points A and B on S, define J = I: R(x, y) dx. If Pis a specific path on S from A to B, then any other path from A to B is of the form P + r, where r is a closed path passing through A and B. Define K to be the Abelian integral associated with the path r: K = II' R(x, y) dx. Note that the value of K is not changed as r is continuously distorted, provided that r stays on S and does not cross any poles of R(x, y). As an example, elliptic integrals can be defined by: R(x, y) = l/y and G(x, y) = (1 -x2)(1 -k2x2) -y2 = o. ~uch research has been performed on the inversion of Abelian inte­ grals. For example, Theorem 6.2 of Bliss [1] (page 170) states that: Theorem: If an Abelian integral u = f«:I:,II» T/(x, y) dx on JI:l:O,1I0 the Riemann surface T of an irreducible algebraic equation f(x, y) = 0 defines a single-valued inverse function x(u), y(u), then the genus of the curve f = 0 must be either p = 0 or p = 1. In the case p = 0 the integral is either of the second kind with a single simple pole, or of the third kind with two simple logarithmic places and no other singularities. In the case p = 1 the integral is of the first kind. For details, see Hazewinkel [8] (pages 14-16), or Lang [15]. Henstock Integral Given the interval [a, b] and a positive function 0 : [a, b] --+ R, define a partition to be given by {(ti' [Xi-I,Xi])}~=l' where the intervals [Xi-I, Xi] are non-overlapping, their union is the interval [a, b], and the following con­ dition is satisfied: ti E [Xi-I, xd C (ti -O(ti), ti + O(ti))' for i = 1,2, ... , n. A function f : [a, b] --+ R is called Henstock-integrable if there exists a number I such that for every e > 0 there exists a positive function o : [a, b] --+ R such that every partition of the interval [a, b] results in n L !(ti) (Xi -xi-d -I < e. i=l 13. Integral Definitions 53 The number I, usually written as J: f(t) dt, is called the Henstock integral of f. For details, see Peng-Yee [19]. Improper Integrals An integral in which the integrand is not bounded, or the interval of integration is unbounded, is said to be an improper integral. For example, the following are improper integrals: 11 dx o x and 100 dx o 1 + x4' Suppose that f(x) has the singular point z in the interval (a, b), and suppose that f(x) is integrable everywhere in the interval, except at the point z. The integral J: f(x) dx is then defined to have the value li~ {f.z -O f(x) dx + [b f(x) dX} , 0-0 a Jz+p where the limits are to be evaluated independently. Lebesgue Integral Let X be a space with a non-negative complete countably-additive measure p" where p,(X) < 00. Separate X into {Xn} so that U~=IXn = X. A simple function 9 is a measurable function that takes at most a countable set of values; that is g(x) = Yn, with Yn ¥= Yk for n ¥= k, if x E Xn. A simple function 9 is said to be summable if the series E:=l Ynp.Xn converges absolutely; the sum of this series is the Lebesgue integral J x 9 dp.. A function f : X -t R is summable on X (denoted fELl (X, p.)) if there is a sequence of simple, summable functions {gn}, uniformly conver­ gent to f on a set of full measure, and if the limit limn_oo Jx gn dp. is finite. The number I is the Lebesgue integral of the function f; this is written 1= [f dl'. (13.1) A simple figure can clarify how the Lebesgue integral is evaluated. Given the function f(x) on (a, b), subdivide the vertical axis into n + 1 points: min f ~ Yo < Yl < ... < Yn ~ max f. Then form the sum n LYi-l x measure (x I Yi-l ~ f(x) < Yi) i=l in which the measure is the sum of the lengths of the subintervals on which the stated inequality takes place. (See Figure 13.1.) In the limit of n -t 00, as the largest length (Yi -Yi-l) tends to zero, this sum becomes the Lebesgue integral of f(x) from a to b. The Lebesgue integral is a linear non-negative functional with the following properties: 54 II Concepts and Definitions I(x) ~ ------------------- Y4 Y2 Yl a x Figure 13.1 A schematic of how the Lebesgue integral is to be evaluated. Regions with similar values are shaded in the same way. [1] If IE L1(X, Il) and if Il {x E X I I(x) :F h(x)} = 0, then h E LI(X, Il) and Ix I dll = Ix hdll· [2] If I E LI(X, Il), then III E LI(X, Il) and IIx I dill ~ Ix III dll· [3] If I E LI (X, Il), Ihl ~ I and h is measurable, then hELl (X, Il) and IIx hdlll ~ Ix I dll· [4] Ifm ~ I ~ M and if I is measurable, then IE LI(X,Il) and mllX ~ Ixldll~MIlX. For functions from Rn to R, if the measure used is the Lebesgue measure, then (13.1) may be written as J = IRn I(x) dx. For different measures, the functional J is called a Lebesgue-Stieltjes integral. Every function that is Riemann integrable on a bounded interval is also Lebesgue integrable, but the converse is not true. For example, the Dirichlet function (equal to 0 for irrational arguments, but equal to 1 for rational arguments) is Lebesgue integrable but not Riemann integrable. Alternatively, the existence of an improper Riemann integral does not imply the existence of a Lebesgue integral. For example, the integral 1000 sin xIx dx = 1r 12 is not Lebesgue integrable because Lebesgue inte­ grability requires that both I and III should be integrable. In this example 1000 I sin xlIx dx = 00. Riemann Integral Let I(x) be a bounded real-value function defined on the interval I = [a, b]. Denote a partition of I by D = {xo, ... , xn} where a = Xo < Xl < ... < Xn = band n is finite. Let Ii denote the sub-interval [Xi, Xi+l]. Define mi = inf I(x), :x;E1i Mi = sup I(x). :x;Eli 13. Integral Definitions 55 f(x) f(x) x x Figure 13.2 An illustration of the lower (left) and upper (right) sums of a function. The oscillation of f on Ii is defined to be Mi -mi. Now define the Darboux sums u(D) and Q:(D): n n u(D) = L Mi (Xi -Xi-I) , Q:(D) = L mi (Xi -Xi-I) . i=1 i=1 Considering all possible partitions of D, we define -b Riemann upper integral of f =!. f(x) dx = infu(D), a D Riemann lower integral of f =!.b f(x) dx = supQ:(D) . ..,g D (See Figure 13.2.) If the Riemann upper and lower integrals of f coincide, then the common value is called the Riemann integral of f on [a, b] and is denoted by J: f(x) dx. In this case, the function f is said to be Riemann integrable, or just integrable. Darboux's theorem states that: Theorem (Darboux): For each e > 0 there exists a positive 6 such that the inequalities -b O'(D) -!. fix) dx < E, IJ!:(D) -[ fix) dx < E, hold for any partition D with max(xi -XH1) < 6, for i = 1,2, ... ,n. From Darboux's theorem we conclude that necessary and sufficient conditions for a function f(x) to be integrable on [a, b] is that for each positive € there exists a 6 such that n b L f((i)(Xj -xj-d -!. f(x) dx < €, j=1 a where 6(D) = maxi(Xi -xi-d < 6 and (i is chosen arbitrarily from Ii. 56 II Concepts and Definitions Stochastic Integrals Let X(t) be an arbitrary random process defined in some interval a ~ t ~ b and let h( t, r) be an arbitrary deterministic function defined in the same interval. Define the integral /(r) = f: h(t, r)X(t) dt. If the integral exists, then /( r) is, itself, a random variable. The integral can be shown to exist for each sample function x(t) if t EUh(t, r)X(t) 11 dt = t Ih(t, r)IE[X(t)] dt < 00. Furthermore, when the integral exists, we can write E[/(r)] = E [t h(t, r)X(t) dt] = t h(t, r)E[X(t)] dt, and the interchange of the order of integration and the expectation opera­ tion is justified. Even if the integral does not exist in the usual sense for each sample function x(t) of X(t), it may be possible to define the equality in some stochastic sense. (See page 186.) Notes [1] Other properties of an integral can be inferred from the stated properties of integrals. Let I denote the set of all functions integrable on the interval I = [a, b]. If I and 9 belong to I, and a and f3 are arbitrary real numbers then (A) III E I, (B) al + f3g E I, (C) I*g E I, (D) min{/,g} E I, (E) max{/,g} E I, (F) I/g E I (assuming that Igl ~ A > 0 on I). [2] We also have the conventions faa I(x) dx = 0 and fba I(x) dx = -f: I(x) dx. [3] Botsko [3] describes a generalization of the Riemann integral that admits every derivative into the set of integrable functions. As an example, Botsko considers the function f' where f(x) = { x'sin :. ~fO < x:5 1, 1 If x = O. The function f' has an unbounded derivative and is not Lebesgue integrable, but can be integrated with Botsko's integral. [4] The Burkill integral [8] was originally introduced to determine surface areas, see Burkill [4]. In modern usage, it is used for integration of non-additive functions. The Burkill integral is less general than the subsequently intro­ duced Kolmogorov integral; any function that is Burkill-integrable is also Kolmogorov-integrable. The name of "Burkill integral" is also given to a number of generalizations of the Perron integral [20]. 13. Integral Definitions 57 [5] The Henstock integral is also known as the generalized Riemann integral. The Henstock integral and the Denjoy integral are equivalent. The restricted Denjoy integral includes the Newton integral and the Lebesgue integral. [6] The Perron integral, the Luzin integral, the gauge integral, the Kurzweil­ Henstock integral, and the special Denjoy integral are all equivalent. See Henstock [9]. In one dimension, the Perron integral is equivalent to the restricted Denjoy integral. A multi-dimensional Perron integral is described in Jurkat and Knizia [13]. [7] The Boks integral [8] is a generalization of the Lebesgue integral, first proposed by Denjoy, but studied in detail by Boks [2]. The definition starts by taking a real-valued function f defined on a segment [a, b] and periodically extending it to the entire real line (with period b -a). The A-integral [8] is more convenient to use than the Boks integral. [8] Other types of integrals not described in this book include: (A) Banach integrals, Birkhoff integrals, Bochner integrals, Denjoy inte­ grals, Dunford integrals, Gel'fand-Pettic integrals, and harmonic inte­ grals (see Iyanaga and Kawada [12], pages 12-15, 337-340, 627-629, 787). (B) Norm integrals, refinement integrals, gauge integrals, Perron integrals, absolute integrals, general Denjoy integrals, and strong variational in­ tegrals (see Henstock [9]). (C) Borel's integral, Daniell's integral, Denjoy integral, improper Dirichlet integrals, Harnack integrals, Holder's integrals, Khinchin's integrals, Radon's integrals, Young's integrals, and De la Vallee-Possin's integrals (see Pesin [20]). (D) Curvilinear integrals are better known as line integrals, see page 164. (E) Fuzzy integrals (see Ichihashi et al. [11]). (F) Kolmogorov integrals (see Hazewinkel [8], page 296). [9] It is also possible to define an integral over an algebraic structure. For example, integrals in a Grassman algebra are discussed in de Souza and Thomas [21]. [10] The Lommel integrals are specific analytical formulas for the integration of products of Bessel functions. See Iyanaga and Kawada [12], page 155. [11] The notion of stochastic integration was first introduced by Wiener in con­ nection with his studies of the Brownian motion process. Given a one­ dimensional path X(t) and any function f(t), Wiener wanted to be able to define the integral JoT f(t) dX(t). This integral makes no sense as a Stieltjes sum since X (t) is not a function of bounded variation. References [1] [2] [3] [4] G. A. Bliss, Algebraic Functions, Dover Publications, Inc., New York, 1966. T. J. Boks, "Sur les rapports entre les methodes de l'integration de Riemann et de Lebesgue," Rend. Circ. Mat. Palermo, 45, No.2, 1921, pages 211-264. M. W. Botsko, "An Easy Generalization of the Riemann Integral," Amer. Math. Monthly, 93, No.9, November 1986, pages 728-732. J. C. Burkill, "Functions of Intervals," Proc. London. Math. Soc., 22, No.2, 1924, pages 275-310. 58 [51 [61 [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] II Concepts and Definitions G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Vari­ able, McGraw-Hill Book Company, New York, 1966. P. J. Daniell, "A General Form of Integral," Ann. of Math., 19, 1918, pages 279-294. A. Denjoy, "Une extension de l'integrale de M. Lebesgue," C. R. Acad. Sci., 154, 1912, pages 859-862. M. Hazewinkel (managing ed.), Encyclopaedia of Mathematics, Kluwer Aca­ demic Publishers, Dordrecht, The Netherlands, 1988. R. Henstock, The General Theory of Integration, Oxford University Press, New York, 1991. W. V. D. Hodge, The Theory and Application of Harmonic Integrals, Cam­ bridge University Press, New York, 1989. H. Ichihashi, H. Tanaka, and K. Asai, "Fuzzy Integrals Based on Pseudo­ Additions and Multiplications," J. Math. Anal. Appl., 130, 1988, pages 354- 364. S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT Press, Cambridge, MA, 1980. W. B. Jurkat and R. W. Knizia, "A Characterization of Multi-Dimensional Perron Integrals and the Fundamental Theorem," Can. J. Math., 43, No.3, 1991, pages 526-539. A. Ya. Khinchin, "Sur une extension de l'integrale de M. Denjoy," C. R. Acad. Sci., 162, 1916, pages 287-291. S. Lang, Introduction to Algebraic and Abelian Functions, Addison-Wesley Publishing Co., Reading, MA, 1971. M. Lebesgue, "Le~ons sur l'integration," Gauthier-Villars, Paris, Second Edition, 1928, page 105. E. J. McShane, "Integrals Devised for Special Purposes," Bull. Amer. Math. Soc., No.5, September 1963, pages 597-627. R. M. McLeod, The Generalized Riemann Integral, Mathematical Associa­ tion of America, Providence, RI, 1980. L. Peng-Yee, "Lanzhou Lectures on Henstock Integration," World Scientific, Singapore, 1989. I. N. Pesin, "Classical and Modern Integration Theories," translated by S. Kotz, Academic Press, New York, 1970. S. M. de Souza and M. T. Thomas, "Beyond Gaussian Integrals in Grassman Algebra," J. Math. Physics, 31, No.6, June 1990, pages 1297-1299. W. Squire, Integration for Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970. 14. Caveats Idea There are many ways in which an integration "result" may be incor­ rect. 14. Caveats Example 1 Consider the integral /.1 dx J(a, b) = --b' o ax+ 59 (14.1) for real x and arbitrary nonzero complex a and b. The indefinite integral has the primitive log( ax + b) I a. Hence, a careless "direct" derivation would yield the result J(a, b) J: log(a + b) _ log b. a a (14.2) The problem, of course, is that the logarithm function has a branch cut. Hence, the two logarithms in (14.2) may not be on the same Riemann sheet. The correct way to evaluate (14.1) is to separate the region of integra­ tion into two sub-intervals, with the division point being the value where ax + b may vanish. An easier way, for this integral, is to first write the integral as 1 /.1 dx 1 ( b) 11 J(a, b) = - bl = -log x + - . a 0 x+ a a a 0 No matter what the sign is of Im(bla), the argument of the logarithm never crosses the cut (since x is real). Thus, the answer is J(a, b) = ~ [log (1 + D -log~] . Note that since 1 + b I a and bl a have the same imaginary part, we may combine them to obtain our final answer 1 a+b J(a, b) = ;; log -b-' Example 2 Consider the simple integral fdX? -;-== logx. (14.3) Clearly, the integrand (l/x) is an odd function, yet log x is neither an even nor an odd function. Hence, there must be an error in (14.3). The error in this case is simple, the integral should be written as fdX -;-= log Ixl. 60 II Concepts and Definitions Now the fact that the result is an even function is clearly indicated. As a similar example, consider the integral I / dx ? -1 X = = sln -. Ja2 -x2 a (14.4) when Ixl ~ a. The integrand is an even function of both x and a, so that the formula for the integral should be odd in x and even in a. However, the formula in (14.4) is odd with respect to both x and a. The correct evaluation of the integral in (14.4) can be written as I = sin-1(x/lal) or as 1= tan-1 x Ja2 -x2 Example 3 Consider the integral I = Jo47r dx/(2 + sinx). Since this integrand has an elementary indefinite integral we readily find X 4'1r 14'1r dx 2 2 tan -+ 1 ___ = -tan-1 2 o 2 + sin x Va Va o (14.5) _ 2 [ -1 2 tan 21l' + 1 -1 2 tan 0 + 1] -Va tan Va -tan Va . Now we must carefully consider what branch of the inverse tangent function to take in each of these two terms. In this case, we must take -1 2 tan 21l' + 1 -1 1 2 1l' tan Va = tan Va = 1l' + '6 -1 2 tan 0 + 1 -1 1 1l' tan Va = tan Va = '6 so that I = Ja [(2" + iD -(i) 1 = ~. Notes [1] Very few tables of integrals use absolute value signs, such as are required in Example 2. [2] If the principle branch of the inverse tangent function were chosen for both terms in (14.5), then the calculation would return the value zero. This should be identified immediately as being in error, since the integrand is always positive. Observe that contour integration techniques (see page 129) can quickly yield the correct evaluation of I in Example 3. [3] Particular care must be used when a symbolic manipulation package (see page 117) is used to compute a definite integral. Results from such packages may be incorrect for many reasons; choosing the wrong branch cut, as Example 3 illustrates, is a common error. 15. Changing Order of Integration 61 15. Changing Order of Integration Applicable to Multiple integrals. Yields A different multiple integral. Idea Sometimes the order of integration in a multiple integral can be changed. This may make it easier to evaluate the integral. Procedure When Fubini's theorem is satisfied, the order of integration in a mul­ tiple integral may be changed. Fubini's theorem states: Theorem: (Fubini) If f(x, y) is measurable and non-negative, and anyone of the integrals [dx [JdY, [dY [Jdx exists, then the other integrals exist, and all are equal. For principal-value integrals, the Poincare-Bertrand theorem may be useful (see Muskelishvili [3]): Theorem: (Poincare-Bertrand) If f(x, y) is analytic, then /(u) = 100 ~loo f(x,y) dy J-oo x-uJ_oo y-x (loo d ,100 f(x, y) dx) 2f( ) = -00 YJ-oo (x -u){y -x) -1r U,u. This theorem is also true with weaker conditions on f(x}. 62 II Concepts and Definitions Example Consider the double integral 1111 Y 1= ') dxdy. o 0 (xy2 + 1) The integrand is positive in the region of integration. In the process of evaluating the integral we will determine if it exists. Integrating first with respect to x we have 11 11 Y I = 0 dy 0 dx (xy2 + 1) - 1:1:=1 = l dy (y (X;.1+ 1)) 1.=0 11 y = -2-dy o y + 1 1 2 )IY=l = -log {y + 1 y-O 2 - = ! log 2. Since this integral exists, we know that integrating first with respect to y will give the same answer. We demonstrate this: 1= 11 dxl1 dy--y - o 0 (xy2 + 1) 111 11 1 = -dx dz----= 2 0 0 (xz + 1)2 111 ( -1 )1%=1 = 2 0 dx x( xz + 1) %=0 111 dx = 2 0 l+x 1 1:1:=1 = -log(l +x) 2 :1:=0 = ! log 2. 15. Changing Order of Integration 63 Notes [1] The order of integration can make a difference if Fubini's theorem is not satisfied. As an example, define the function f(x, y) = (x2 _ y2)J(X2 + y2)2, and consider integrating it over the unit square. Depending on the order of the integrations, we will obtain different answers. Integrating first with respect to y results in f(x,y)= dx ~-~2dy= dx~ J 11 11 2 2 11 ( ) 111=1 o 0 0 (x + y ) 0 x + Y 11=0 11 dx 7r = 0 x2 + 1 = '4. If, instead, we integrated first with respect to x we find f(x,y)= dy ~-Y22dx= dy-~ J 11 11 2 2 11 ( )1%=1 o 0 0 (x + y ) 0 x + Y %=0 --11 ---.!!:JL - -~ - 0 y2 + 1 -4· [2] Other examples where the requirements for Fubini's theorem are not satisfied are easy to find. For example, 11 11 2 2 X -y dx 2 2)2 dy = o 0 (x + y 11 dx o 1 + x2 = 7r 4' while d x -y dx=- ---.!!:JL-_~ 11 11 2 2 11 o y 0 (x2 + y2)2 0 1 + x2 -4 Another example is given by 11 dy 1"" (.-.• -2.-2.,) dx i-l"" dx 11 (.-., -2.-2'9) dy. [3] The technique of interchanging integration order can be used to derive analytical formulae for integrals. • For example, the integral I = J J XII dx dy can be evaluated in two 0<%<1 aSIISb different ways, 1= dy xlldx= -y-=log--lb 11 lb d ( 1 + b) a 0 a y+l l+a 11 lb 11 (II) III=b 11 b a = 0 dx a XII dy = 0 dx l:g x lI=a = 0 X lo~ : dx 64 II Concepts and Definitions /.lXb_Xa (l+b) to yield the result -I --dx = log -1 -• o ogx + a . J J J dxdydz • As a second example, the mtegral 2 2 2 2 0$%.11:9 (1 + x z )(1 + y z ) %>0 be used to derive the relation (see George [2], page 206) /.00 (tan -1 z) 2 o --z- dz = 1r log 2. [4] A more general statement of Fubini's theorem is: Theorem: (Fubini) Let X = RP and Y = RP; then the formula can j jf(X,Y)dxd Y= L dx I f(x,y)dy = I dy LJ(X,y)dx XxY is valid in each of the following two cases: [1] f is a measurable positive arithmetic function on X x Y; [2] f is an integrable function over X x Y. References [1] G. Fubini, "Sugli Integrali Multipli," Opere Scelte, Cremonese, Vol 2, 1958, pages 243-249. [2] C. George, "Exercises in Integration," Springer-Verlag, New York, 1984. [3] N. I. Muskelishvili, Singular Integral Equations, Noordhoff, Groningen, 1953, pages 56-61. 16. Convergence of Integrals Applicable to Single and multiple integrals. Yields Knowledge of whether an integral converges or diverges. Idea By comparing a given integral to a different integral, the convergence or divergence of the original integral may be determined. 16. Convergence of Integrals 65 Procedure Most techniques that indicate convergence or divergence use some sort of integral inequality (see page 205). In this section we use the following two theorems: Theorem (Comparison test for convergence): Let f(x) and g(x) be continuous for a < x ~ b, with 0 ~ If(x)1 ~ g(x). If I: g(x) dx converges, then I: f(x) dx converges and o ~ II: !(x) dxl ~ I: g(x) dx. Theorem (Comparison test for divergence): Let !(x) and g(x) be continuous for a < x ~ b, with 0 ~ g(x) ~ !(x). If J: g(x) dx diverges, then I: !(x) dx diverges. To use these theorems effectively, a knowledge base must be created of converging and diverging integrals. Some common integrals used for comparison include (assuming a < b): (A) lb (x ~a)P' which converges for p < 1 and diverges for p ~ 1. (B) lb (b ~xx)P' which converges for p < 1 and diverges for p ~ 1. (C) 11 dx, -IXP which converges for p < 1 and diverges for p ~ 1. (D) 1°Od~, 1 x which converges for p > 1 and diverges for p :5 1. (E) 100 (1 dx )p' which converges for p > 1 and diverges for p ::; 1. 2 X ogx Example 1 Consider the integral I = I~oo:l: :Ji t dx. For the given range of integration (i.e., for x E [0,00]) we can easily bound the trigonometric term: Ixsin2 xl :5 x. The integral J = I~oo :1:3:"1 dx will now be shown to converge, which then implies the convergence of I. First, we write /.1 X 100 x J = J1 + J2 = -3--dx + -3--dx. o x +1 1 X +1 The integrand in J1 is bounded above by x, and I; x dx = ! (this shows that J1 is convergent). The integrand in J2 can be bounded above by x-2 and J100 x-2 dx = 1 (this shows that J2 is convergent). Hence, J is convergent (it is bounded above by ~) and so is I. 66 II Concepts and Definitions Example 2 Consider the integral I = ['" (f sin(y2x3) dY) <lx. Changing variables in the inner integral to z = y2 x3 results in = ! 100 (/.G2 :Z;3 sin z ) ~ I 2 . r: dz 3/2 • 1 0 VZ x The inner integral is bounded for all a and x since 1000 s~ dz converges. The integral floo x-3/2 dx also converges, so we conclude that I converges. Notes [1] Note that we can analytically integrate J in Example 1. We find that J = 100 --!-dx -00 x + 1 = .!. 10 1 -x + x + ~ tan -1 2x -1 - ( 2 ) 1:1:-00 6 g (1 + X)2 V3 V3 :1:=0 7r = 3V3 ~ 0.605. [2] Another useful theorem for determining whether an integral converges is Chartier's test (see Whittaker and Watson [3]) Theorem (Chartier): If f(x) decreases to zero monotoni­ cally as x ..... 00, and If: <p(t) dtl is bounded as x ..... 00, then f: f(x)<p(x) dx converges. (A) For example, consider 1= fooo x-1 sinxdx. In this case, f(x) = x-1 is monotonically decreasing and fo:l: sin t dt = 1 -cos t, which is certainly bounded. We conclude that I converges. (B) For a more interesting example, consider J = f1°O xcos(x3 -x) dx. This integral can be written as J = [00 ---:-(dxd sin(x3 -x») dx. 1 3x -1 We recognize that x/(3x2 -1) decreases monotonically, and that [" (! sin(x3 -xl) dx = sin(x3 -xl, which is bounded. We conclude that J converges. [3] The following decomposition K -/.00 cos x sin ax d _ 1 /.00 sin(a + l)x d 1 /.00 sin (a -l)x d - x - - x+ - x, o x 20 x 20 x combined with the first example in the last note shows that K converges. [4] Divergent integrals can sometimes be regularized to obtain a finite value, see Wong and McClure [2]. 17. Exterior Calculus 67 References [1] W. Kaplan, Advanced Calculus, Addison-Wesley Publishing Co., Reading, MA,1952. [2] R. Wong and J. P. McClure, "Generalized Mellin Convolutions and their Asymptotic Expansions," Can. J. Math, 36, 1984, pages 924-960. [3] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cam­ bridge University Press, New York, 1962. 17. Exterior Calculus Applicable to Integration of differential forms. Idea The integral of a differential form is a higher-dimensional generaliza­ tion of such ideas as the work of a force along a path, or the flux of a fluid across a surface. Procedure A real-valued function of x, also known as a scalar field, is called a zero-form in exterior calculus. A one-form (in, say, three space) is an expression of the form FI dx + F2 dy + F3 dz, where FI, F2, and F3 are functions of x, y, and z. A two-form in "standard order" is an expression of the form FI dy " dz + F2 dz " dx + F3 dx " dy. The inverted v's are called carets or wedges. The wedges distinguish be­ tween a two-form, such as 4y3 dz "dx, and objects such as 4y3 dz dx, that occur in double integrals where the order of the differentials is not important. In n-dimensional space, p-forms exist for p = 0, 1, ... ,n. The addition of two p-forms, or the multiplication of a p-form by a scalar, is another p-form. The wedge product has the following properties: • The wedge product is associative: w" (9" () = (w" 9) "(. • The wedge product is anti-commutative: w" 9 = (_I)kl9" w, for w a k-form and 9 an 1-form. • The wedge product of repeated differentials vanishes: dx A dx = 0 • The wedge product of a p-form and a q-form is a (p + q)-form. When dealing with n-dimensional space, it is useful to deal with forms in a more formal way. If we choose a linear coordinate system {Xl, X2,"" xn} on R n, then each Xi is a I-form. If the {Xi} are linearly independent, then every I-form w can be written in the form w = alXl + ... + anxn, where 68 II Concepts and Definitions the {ai} are real functions. The value of w on the vector , is equal to we,) = aIXI{')+ ... +anxn{'), where XI{'), ... , xn{'), are the components of , in the chosen coordinate system. A 2-form W2 is a function on pairs of vectors which is bilinear and skew-symmetric: W2{AI'1 + A2'2, (3) = AIW2{'b (3) + A2W2{'2, (3), W2{'b (2) = -W2{'2, 'd· From this it is easy to derive that W2{', ') = 0 for every 2-form. Analo­ gously, an exterior form of degree k, or a k-form, is a function of k vectors which is k-linear and antisymmetric. A differentiable k-form wklx' at a point x of a manifold M, is an exterior k-form on the tangent space T Mx to M at the point x. That is, it is a k-linear skew-symmetric function of k vectors {' b ... , 'k} tangent to M at x. Every differentiable k-form on the space Rn with a given coordinate system {Xl, ... , xn} can we written uniquely in the form wk = L ah , ... ,it (x) fixi, 1\ ... 1\ dXit' il <···<it w here the {ail t ... tit (x)} are smooth functions on Rn. A chain of dimension n on a manifold M consists of a finite collection of n-dimensional oriented cells 0'1, ••• , O'r in M and integers ml, ... , mr, called multiplicities (the multiplicities can be positive, negative, or zero). A chain is denoted by Ck = mlO'I + ... + mrO'r. Let {Xl, ... , Xl} be an oriented coordinate system on R k. Then every k-form on Rk has the form wk = t/J{x) dXII\·· ·I\dxk, where t/J{x) is a smooth function. Let D be a bounded convex polyhedron in R k. We define the integral of the form wk on D to be where the integral on the right is understood to be the usual limit of Riemann sums. To integrate a k-form over an n-dimensional manifold, the role of the usual path of integration is replaced by a k-dimensional cell q of M represented by a triple 0' = (D, /, Or) where (A) D c Rk is a convex polyhedron; (B) /: D ~ M is a differentiable map; 17. Exterior Calculus 69 (C) Or represents an orientation of D on R k • Then the integral of the k-form w over the k-dimensional cell u is the integral of the corresponding form over the polyhedron D: fO' w = fD /"'w. Here the form /'" w is defined by where the {t;d are tangent vectors and /'" is the differential of the map /. The integral of the form wk over the chain Ck is the sum of the integrals on the cells, counting multiplicities: fCte wk = Ei mi fO', wk. Some classes of integrals can be immediately evaluated. Using the two definitions: • A differential form w on a manifold M is closed if its exterior derivative is zero: dw = 0, • A cycle on a manifold M is a chain whose boundary is equal to zero, we have the two theorems: • The integral of a closed form wk over the boundary of any (k + 1)- dimensional chain Ck+l is equal to zero: That is, fpc wk = 0 if v te+l dwk = 0; • The integral of a differential over any cycle is equal to zero: That is, 1 dwk = 0 if 8Ck+l = o. Cte+l Example The two-form T = (x dx + y dz) " (y dx -y2 dy) can be expanded into T = xy dx " dx + y2 dz " dx -xy2 dx " dy -y3 dz " dy. The first term vanishes because it contains a repeated differential. Writing the remaining terms in standard order results in Notes [1] As a matter of convention, the function identically equal to zero is called a p-form for every p. [2] Stokes' formula can be stated as fOe W = fe dw, where c is the (k + 1)­ dimensional chain on a manifold M and W is any k-form on M. 70 II Concepts and Definitions References [11 L. Arnold, Mathematical Methods of Classical Mechanics, translated by K. Vogtmann and A. Weinstein, Second edition, Springer-Verlag, New York, 1989, Chapter 7. [2] B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modem Geometry Methods and Applications, Springer-Verlag, New York, 1991. [3] D. G. B. Edelen, Applied Exterior Calculus, John Wiley & Sons, New York, 1985. [4] P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics #107, Springer-Verlag, New York, 1986. [5] H. Whitney, Geometric Integration Theory, Princeton University Press, Prince­ ton, NJ, 1957. 18. Feynman Diagrams Applicable to Feynman diagrams are used to denote a collection of ordinary or path integrals. Idea Some integrals equations have a "natural" expansion scheme in terms of integrals. The diagrams are used to keep track of the terms. Procedure If a given differential equation is only a "small" perturbation from a linear differential equation (with a known Green's function), then we may obtain an equivalent integral equation. This integral equation may be expanded methodically into a series of integrals. Diagrams are often used to keep track of the terms. Example Consider the nonlinear ordinary differential equation dz 2 dt = f(t) + g(t)z , z(O) = o. in which the nonlinear term (i.e., the g(t) function) is "small." This equation may be directly integrated to obtain z(t} = 1.' /(1") d1" + 1.' g(1"}z2(1"} d1". (18.1) 18. Feynman Diagrams 71 --+ H(t -r) ._{< x --+ / f(r)dr -< --+ / g(r)dr Figure 18.1 Rules for creating diagrams and rules for interpreting diagrams. If the value of z(t) from the left hand side of (18.1) is used in the right hand side, then z(t) = l' I(r)dr+ l' g(r) [f I(rddrlf dT + 2 l' g(r) [f I(rl) drl] [f g(r2)z2(r2)dr2] dr (18.2) + l' g(r) [1' g(r2)z2(r.)dr2f dr. A "natural" perturbation expansion would be to keep the first two terms in the right hand side of (18.2), and assume that the last two terms are "small". If Iz(t)1 « 1 then this may well be the case since the last two terms involve Izl2 while the first two terms involve constants. A functional iteration technique can be used to derive (18.2) and its higher order extensions from diagrams. We need two sets of rules: One set of rules describes how the diagrams may be computed; the other set of rules describes how the diagrams are to be turned into mathematical expressions. If we use the rules in Figure 18.1, (where H() denotes the Heaviside function), then the first two steps in the diagrammatic solution to z(t) (from (18.1)) are given by the diagrams in Figure 18.2. Note that the third and fourth diagrams in Figure 18.2 represent the same mathematical expression since they are topologically equivalent. The purpose of the Heaviside function is to restrict the range of integration. By careful inspection, the mathematical expressions associated with the last set of diagrams will be seen to be identical to (18.2). See Zwillinger [5] for more details. 72 II Concepts and Definitions z(t) = t --- = ~+ ~ = ~+ ~+t Figure 18.2 Two steps in the diagrammatic expansion of (18.1). Notes [1] Often an "algebra of diagrams" is created, so that diagrams can be added, subtracted and multiplied without recourse to the mathematical expression that each diagram represents. This would require amplification of the rules that were used in the example. [2] This technique is particularly important in problems in which there is no "small" parameter. In these cases, the formally correct diagrammatic expan­ sion may be algebraically approximated by exactly summing certain classes of diagrams. See Mattuck [4] for details. References [1] F. Battaglia and T. F. George, "A Rule for the Total Number of Topologi­ cally Distinct Feynman Diagrams," J. Math. Physics, 25, No. 12, December 1984, pages 3489-3491. [2] T. Kaneko, S. Kawabata, and Y. Shimizu, "Automatic Generation of Feyn­ man Graphs and Amplitudes in QED," Gomput. Physics Gomm., 43, 1987, pages 279-295. [3] J. Kiiblbeck, M. Bohm, and A. Denner, "Feyn Arts -Computer-Algebraic Generation of Feynman Graphs and Amplitudes," Gomput. Physics Gomm., 60, 1990, pages 165-180. [4] R. D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, Academic Press, New York, 1976. 19. Finite Part of Integrals 73 [5] D. Zwillinger, Handbook of Differential Equations, Academic Press, New York, Second Edition, 1992. 19. Finite Part of Integrals Applicable to Divergent integrals. Yields A convergent expression. Idea Many divergent integrals diverge in a common way. By removing the diverging piece of an integral, a finite term is left. Procedure Given the integral I = t(x-a)~-n/(X)dx, (19.1) we assume that f(a) =f. 0 and -1 < A ~ O. Clearly, this integral diverges for all integral value of n. However, when f E Cn[s, r] and A =f. 0, we define the finite part of I to be the value of lim [lb (x -a),\-n f(x) dx -g(t)(t -a),\-n+l] , (19.2) t-a+ t where g(t) is any function in Cn[s, r] such that the above limit exists. Denote the finite part of the integral I by f:(x -a),\-n f(x) dx. The limit in (19.2) can be explicitly evaluated to find (for A =f. 0): f.b _ '\-n _ n-l f(k)(a)(b -a),\-n+k+l (x a) f(x)dx -L (A _ k )kl a k=O n + + 1 . + 1 I f.b (x -a),\-n f.z (x -y)n-l f(n)(y) dy dx. (n -1). a a (19.3) For A = 0 we find: f.b _ -n _ n-2 f(k) (a)(b -a)-n+k+l log(b -a)f(n-l)(a) a (x a) f(x) dx -~ (-n + k + l)k! + (n -I)! + 1 I f.b (x -a)-n f.z (x -y)n-l f(n)(y) dydx. (n -1). a a (19.4) 74 II Concepts and Definitions Some special cases are easy to express in terms of a Cauchy principal value type limit. For example, if w(x) is a non-negative weight function integrable on the interval (a, b), c is in the interval (a, b), and f(x) is dif­ ferentiable in a neighborhood of c with its derivative satisfying a Lipschitz condition, then (see Paget [8]) f w(x)f(x) dx = lim [(/.c-e + rb ) w(x)f(x) dx _ 2 W(C)f(C)] . (x -C)2 e-O a Jc+e (x -c)2 C (19.5) Example as Consider the finite-part integral J = 13: f(:) dy. This can be written To y J = 13: f(y) -f(01-yf'(O) dy + 1(0)13: d~ + 1'(0)13: dy. To y Toy To y For finite-part integrals, we have the usual relations and i3: dy - = logx. o y (19.6) Combining these results, and assuming that 1 (y) possesses a continuous second derivative on the interval [0, xl, we have J = 13: I(y) -1(0) -yl'(O) dy _ 1(0) + f'(O)logx. o y2 x (19.7) Notes [1] This is also called Hadamard's finite part 0/ an integral. [2] The finite parts of integrals have different properties from usual integrals. Some of these properties are (from Davis and Rabinowitz [2]): (A) f is a consistent extension of the concept of regular integrals. (B) f is additive with respect to the union of integration intervals and is invariant with respect to translation. (C) f is a continuous linear functional of the integrand. (D) fooo x-a dx = 0 if a> 1. (E) fol x-1 dx = 0 (this is consistent with (19.6». (F) For a > 1, we have fol x-a dx = 1/(1 -a) < O. Hence, if /(x) > 0, it may be that f /(x) dx < O. [3] Kutt [7] has derived quadrature rules for the evaluation of finite-part inte- f.b x -s grals of the form k+l /(x) dx, when 8 E (a, b) and k = 1,2, .... a Ix -81 20. Fractional Integration 75 [4] It is possible to write some finite-part integrals as derivatives of principal­ value integrals. For example: Jb w(x)f(x) dx = .!!f.b w(x)f(x) dx. Ia (x -e)2 de a X -e This is the basis for the numerical technique in Paget [8]. [5] Davis and Rabinowitz [2] give the example 1.1 y-2 dy = 2V5 [log (10v'1O _ 30) _ 1] _ Vi. o V(y -2)2 + 1 25 5 [6] Other classes of finite-part integrals also exist. For example, there are theories developed for each of the integrals Jb f(x) logm(x -a) dx and Jb f(x) dx Ia (x -a)n->. L sinn->.(x -a) where m is a positive integer. [7] Finite part integrals arise naturally in fluid mechanics, solid mechanics, and electromagnetic theory. See, for example, Ioakimidis [4]. References [1] [2] [3] B. Bialecki, "A Sinc Quadrature Rule for Hadamard Finite-Part Integrals," Numer. Math., 57, 1990, pages 263-269. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, pages 11-15 and 188-190. N.1. Ioakimidis, "On the Gaussian Quadrature Rule for Finite-Part Integrals with a First-Order Singularity," Comm. Appl. Numer. Meths., 2, 1986, pages 123-132. [4] N. I. Ioakimidis, "Application of Finite-Part Integrals to the Singular Inte­ gral Equations of Crack Problems in Plane and Three-Dimensional Elastic­ ity," Acta. Mech., 45, 1982, pages 31-47. [5] N. Ioakimidis, "On Kutt's Gaussian Quadrature Rule for Finite-Part Inte­ grals," Appl. Num. Math., 5, No.3, 1989, pages 209-213. [6] N. I. Ioakimidis and M. S. Pitta, "Remarks on the Gaussian Quadrature Rule for Finite-Part Integrals with a Second-Order Singularity," Computer Methods in Appl. Mechanics and Eng., 69, 1988, pages 325-343. [7] H. R. Kutt, "The Numerical Evaluation of Principal Value Integrals by Finite-Part Integration," Numer. Math., 24, 1975, pages 205-210. [8] D. F. Paget, "The Numerical Evaluation of Hadamard Finite-Part Inte­ grals," Numer. Math., 36, 1981, pages 447-453. [9] G. Tsamasphyros, and G. Dimou, "Gauss Quadrature Rules for Finite Part Integrals," Internat. J. Numer. Methods Engrg., 30, No.1, 1990, pages 13- 26. 20. Fractional Integration Applicable to Fractional integrals. 76 II Concepts and Definitions Procedure The Riemann-Liouville fractional derivative of order v is defined by (20.1) (This is sometimes represented by cI;.) The fractional derivative has the following properties: [1] The operation of order zero leaves a function unchanged: cD~f(x) = f(x). [2] The law of exponents for integration of arbitrary order holds: cD~ cD: f(x) = cD:+v f(x). [3] The fractional operator is linear: cD~[af(x) + bg(x)] = acD~f(x) + bcD~g(x) (assuming, for v < 0, that f and 9 are analytic). [4] The operation cD~f(x) yields f(v)(x), the v-th derivative of f, when v is a positive integer. If v is a negative integer, say v = -n, then cD;n f(x) is the same as the ordinary n-fold integration of f(x). (The integration constants are chosen so that cD;n f(x) vanishes, along with its first n -1 derivatives, at x = c.) [5] If f(z) is an analytic function of the complex variable z, the function cD; f (z) is an analytic function of v and z. Many common functions can be written as fractional integrals of other functions. For example, we find J (v'U) = _u _ D-(p+l/2) cos v u -p/2 (t:::) p u 2P v:rr 0 U v'U F (b ) xl-gr(g) D-(g-b) (b-l(1 )-a) 2 1 a, ; g; x = f(b) 0 x x -x . It is also possible to represent some ordinary integrals as fractional integrals in non-obvious ways. For example, from Oldham and Spanier [3] we have (page 182) 1X1/P ( + 1) d-l/p f f (x -xzP) dz = r P_-x-l/P __ _ o P dx l/p or, when x = 1 rl f(l-zP) dz = r (p+ 1) d-~Pfl . 10 P dx IIp x=l 20. Fractional Integration 77 For example, we find /.1 sin ( ~) dz = r (~) d-_1 :;2 sin (v'X) I o dx x=1 Vi I = """2 (..jiXJ 1 (v'X)) x=1 1r = "2J1 (1) ~ .69122984 .... Notes [1] A semi-integral is a half integral, that is v = -i in (20.1). Table 20 contains a short table of semi-integrals. [2] Another definition of a fractional derivative is Weyl's integral .K~ = rtv) f.~ (t -x)"-' fit) dt for v > O. To compute fractional integrals, let m be the smallest positive integer such that v < m and define r = m -v. Then .K;'" = :;.. (rtr) f.~ (t -x)'-' fit) dt) . Note that we also have ,D;" = :;.. (rtr) [(X -1),-' fit) dt) when m is the smallest positive integer such that v < m and r = m -v. [3] Osler [4] has established the fractional integral generalization of Leibniz's rule: --- d>' dq[fg] 100 ( q ) ~-'Y->'f tJY+>'g dxq --00 ,\ + 'Y dxq--r->' dx -r+>' where'Y is arbitrary. References [1] A. Erdelyi, "Axially Symmetric Potentials and Fractional Integration," J. Soc. Indust. Appl. Math., 13, No.1, March 1965, pages 216-228. [2] F. G. Lether, D. M. Cline and O. Evans, "An Error Analysis for the Cal­ culation of Semiintegrals and Semiderivatives by the RL Algorithm," Appl. Math. and Comp., 17, 1985, pages 45-67. [3] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [4] T. J. Osler, "Leibniz Rule for Fractional Derivatives Generalized and an Application to Infinite Series," SIAM J. Appl. Math., 18, No.3, May 1970, pages 658-674. [5] B. Ross, Fractional Calculus and Its Applications, Proceedings of the Inter­ national Conference at the University of New Haven, June 1974, Springer­ Verlag, New York, Lecture Notes in Mathematics #457, 1975. 78 II Concepts and Definitions Table 20. A short table of semi-integrals. f d-1/2 f d,x-l/2 0 0 C, a constant 2Cj; X-1/2 ..fo 4x2/3 X 3Vi xn, n = 0,1,2, ... (n!)2(4xt+l/2 (2n + 1)!Vi xP, p> -1 r(p + 1) p+l/2 r(p+ ~)x VI +x j; (1 + x) tan-1 (v'X) -+ 7r Vi 1 2 Vl+x Vi arctan ( v'X) 1 2 sinh -1 (v'X) l+x y'7r(1 + x) e'Z e'Z erf (v'X) e'Z erf (v'X) e'Z -1 sin (v'X) ..j1rXJl (v'X) cos (v'X) y'1TXH-l (v'X) sinh (v'X) y'1TXh (v'X) cosh (v'X) ..j1rXL-l (v'X) sin (v'X) v;rHo (v'X) v'X cos (v'X) v;rJo (v'X) v'X log x 2j; [log(4x) -2] log x v;rlog (~) v'X 21. Liouville Theory 21. Liouville Theory Applicable to Indefinite one-dimensional integrals. Yields 79 Knowledge of whether the integral can be integrated in closed form in terms of "elementary functions." Idea The elementary functions are defined to be [1] rational functions, [2J algebraic functions (explicit and implicit), [3J exponential and logarithmic functions, [4J functions generated by a finite combination of the preceding classes. The derivative of an elementary function is an elementary function. The question that Liouville addressed is: "When is the integral of an elementary function an elementary function?" Procedure The theory underlying the evaluation of integrals is complex. We need the following definitions: • Let K be a field of functions. The function 0 is an elementary generator over K if o is algebraic over K, i.e., 0 satisfies a polynomial equation with coefficients in K; o is an exponential over K, i.e., there is a (in K such that 0' = ('0, which is an algebraic way of saying that 0 = exp (; o is a logarithm over K, i.e., there is a ( in K such that 0' = ('1(, which is an algebraic way of saying that 0 = log ( . • Let K be a field of functions. An over-field K(Ol,' .. ,On) of K is called a field of elementary functions over K if every OJ is an elementary generator over K. A function is elementary over K if it belongs to a field of elementary functions over K. Note that, with K = C (the complex numbers), trigonometric func­ tions (and their inverses) as well as rational functions are elementary. For example cos z = ~ (eiZ + e-iz). That is, the cosine function is made up of complex multiplications, complex exponentiations, and rational operations. The fundamental theorem in this area is Theorem: (Liouville's principle) Let / be a function from a function field K. If / has an integral elementary over K, it has an integral of the following form: / f = 'IJ() + t '" log Vi, i=l 80 II Concepts and Definitions where Vo belongs to K, the {Vi} belong to K, an extension of K by a finite number of constants algebraic over K, and the {Ci} belong to K and are constant. For example: [1] If an integral of an algebraic function y is elementary, then the integral must be of the form J ydx = Roex) + LCklogRkex) k where the {Rk} are all rational functions and the number of terms in the sum is finite but undetermined. [2] If y is given by the solution of an Nth-degree polynomial equation, and the integral of y is purely algebraic, then the integral must be of the form N-l J ydx = L RkeX)yk k=l where the {Rk} are all rational functions. [3] If the integral I = J J e9 dx is elementary, and J and g are elementary, then I is of the form I = Re9, where R is a rational function of x, J, and g. Risch proved an extension to the above theorem that incorporates logarithms and exponentials: Theorem: (Risch) Let K = C(x, (h, 82, ... , 8n) be a field of functions, where C is a field of constants and each (h is a loga­ rithm or an exponential of an element of C(x, (h, 82, ... , 8i-d and is transcendental over C(x, 81, 82, ... , 8i-1). Moreover, the field of constants of K must be C. There is an algorithm which, given an element f of K, either gives an elementary function over K that is the integral of f or proves that f has no elementary integral over K. Davenport proved an extension to the above theorem that incorporates some algebraic functions: Theorem: (Davenport) Let K = C(x, y, 81, 82, ... , 8n) be a field of functions, where C is a field of constants, y is algebraic over C(x), and each 8i is a logarithm or an exponential of an element of C(x, y, 81,82, ... , 8i-d, and is transcendental over C(x, y, 81, 82, ... , 8i-d. Moreover, the field of constants of K must be C. There is an algorithm that, given an element f of K, either gives an elementary function over K that is the integral of f or proves that f has no elementary integral over K. 21. Liouville Theory 81 There is, as yet, no theorem that will allow algebraic extensions that depend on logarithms or exponential functions. Note that the theorems above are for integrating elementary functions when the integrals also have to be elementary. There are two ways of expanding the above results: • Allow a larger or different class of functions to appear in the integrand (in this case the class of functions allowed in the evaluated integral may be the same or different). • Allow a larger or different class of functions to appear in the eval­ uated integral, such as those created from elementary functions and logarithmic integrals, error functions, or dilogarithms. See Baddoura [1], Cherry [3] and [4], and Davenport, Siret, and Tournier [6] for details. Example 1 Using the above statements, we can resolve the question "Is the integral I = J e:l:2 / x dx elementary?" If it is, then by statement number 3 (above), the integral must be of the form I = Re:l:2 where R is a rational function of x, x-I, and x2• That is, R is a rational function of x. Equating the two expressions for I, and differentiating, results in e:l:2 2 2 - = R' e:l: + 2xRe:l: or x{R' + 2xR) = 1. x (21.1) Since R is a rational function of x, it can be written in the form R = N(x)/D(x), where N{x) is a polynomial of degree nand D(x) is a polyno­ mial of degree d. Evaluating the differential equation for R in (21.1), using Nand D, results in xDN' -xD'N + 2x2ND = n2. (21.2) Each term in this equation is a polynomial, the degrees of the terms are d + n, d + n, d + n + 2, and 2d. The first two terms in (21.2) cannot equal each other unless N = n (which implies that R is a constant, which does not work). Clearly, the third term cannot have same degree as the first term. The fourth term cannot balance both the first and third terms, so no values of nand d will work. We conclude that I does not have an elementary integral. 82 II Concepts and Definitions Example 2 As another application of statement number 3, we investigate the integral 1= J e-x dx. If this integral is elementary, then the integral must be of the form I = Re-x2 where R is a rational function of x. Equating the two expressions for I and differentiating results in This can be written as 1 = R' -2Rx. If we write R(x) = P(x)/Q(x), where P(x) and Q(x) are relatively prime polynomials, then we obtain the relation Q(Q -P' + 2xP) = -PQ'. (21.3) Q(x) must have at least one root, f3. Hence, Q(x) can be written in the form Q(x) = (x-f3)TT(x), where T is a positive integer and T(f3) ¥-O. Since Q and P are relatively prime, P(f3) ¥-o. Now we have a contradiction: f3 is a root of the left-hand side of (21.3), with a multiplicity of at least T, but f3 is a root of the right-hand side of (21.3), with a multiplicity of at most T -1. Hence, the assumption that J e-x2 dx has an elementary integral must be incorrect. Example 3 Consider the integral I = J xe-x2 dx. Assuming the integral has the form 1= Re-x2 results in the following differential equation: R'-2xR = x. This has the solution R = -!, so f xe-x2 dx is integrable. Notes [1] The symbolic computer language MAPLE will optionally, in the course of trying to evaluate an indefinite integral, print out information similar to the steps in the above examples. This makes it easy to monitor the computation and verify the result. [2] The integration procedures described in this section are actually used in computer languages that can perform indefinite integration. See the section beginning on page 117. [3J Even if an indefinite integral is not elementary, it sometimes possible to evaluate a definite integral with the same integrand. For example, even though J e-:z:2 dx is not elementary (see Example 2), the definite integral J.:ooo e-:z:2 dx is easily shown to be equal to .;:i (see for example, page 115). [4] Sometimes, in practice, it seems as if most integrals of interest are not elementary. It is certainly true that many simple-looking integrals are not elementary, such as /ldX , /sinx dx, f sinx2 dx, J Jsinxdx, and ogx x J Jl-x3dx. 22. Mean Value Theorems 83 References [1] J. Baddoura, "Integration in Finite terms and Simplification with Diloga­ rithms: A Progress Report," in E. Kaltoflen and S. M. Watt (eds.), Comput­ ers and Mathematics, Springer-Verlag, New York, 1990, pages 166-181. [2] M. Bronstein, "Symbolic Integration: Towards Practical Algorithms," in Computer Algebra and Differential Equations, Academic Press, New York, 1990, pages 59-85. [3] G. W. Cherry, "An Analysis of the Rational Exponential Integral," SIAM J. Comput, 18, No.5, October 1989, pages 893-905. [4] G. W. Cherry, "Integration in Finite Terms with Special Functions: The Error Function," J. Symbolic Comp., 1, 1985, pages 283-302. [5] J. H. Davenport, "The Risch Differential Equation Problem," SIAM J. Com­ put, 15, No.4, November 1986, pages 903-918. [6] J. H. Davenport, Y. Siret, and E. Tournier, Computer Algebra: Systems and Algorithms for Algebraic Computation, Academic Press, New York, 1988, pages 165-186. [7] K. O. Geddes and L. Y. Stefanus, "On the Risch-Norman Integration Method and Its Implementation in Maple," Proceedings of ISSAC '89, ACM, New York, 1989, pages 218-227. [8] P. H. Knowles, "Integration of Liouvillian Functions with Special Func­ tions," SYMSAC '86, ACM, New York. [9] J. Moses, "Symbolic Integration, the Stormy Decade," Comm. A CM, 14, 1971, pages 548-560. [10] R. D. Richtmyer, "Integration in Finite Terms: A Method for Determining Regular Fields for the Risch Algorithm," Lett. Math. Phys. 10, No. 2-3, 1985, pages 135-141. [11] R. H. Risch, "The Problem of Integration in Finite Terms," 11rans. Amer. Math. Soc., 139, 1969, pages 167-183. [12] R. H. Risch, "The Solution of the Problem of Integration in Finite Terms," Bull. Amer. Math. Soc., 76, 1970, pages 605-608. [13] J. F. Ritt, "On the Integrals of Elementary Functions," 11rans. Amer. Math. Soc., 25, 1923, pages 211-222. [14] J. F. Ritt, Integration in Finite Terms: Liouville's Theory of Elementary Methods, Columbia University Press, New York, 1948. [15] W. Squire, Integration for Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970, pages 41-49. 22. Mean Value Theorems Applicable to Single and multiple integrals. Yields Information about a function integrated over a region. 84 II Concepts and Definitions Idea The integral of a function can sometimes be written as some value of that function times the integral of a simplier function. Procedure There are several theorems that yield information about the mean value of a quantity; we indicate only a few of them. First Mean Value Formula Assuming that a ~ b, we have (b -a)/m~ 2: 1.. I(x) dx 2: (b -a)/min. If /(x) is continuous then we have [ I(x) dx = (b -a)/(x), (22.1) where a ~ x ~ b. This theorem is sometimes written as (assuming, again, that f(x) is con­ tinuous) l I(x)g(x)tlx = I(x) l g(x)tlx. Second Mean Value Formula If / is a decreasing positive function on [a, b), and 9 an integrable function on this interval, then 11b /91 ~ /(a) sup 11% 91· a as%Sb a Mean Value Theorem for Double Integrals If /(x, y) is continuous on a compact region R, with area A, there exists a point «(, 1]) in the interior or R such that ~ ~ /(x,y)dA=/«(,1])A. R (22.2) 22. Mean Value Theorems 85 Example Consider the integral I = J: x sin x dx. The exact evaluation is I = sin x -xcosxl~ = 7r. From (22.1), we can write I = 7rxsinx, for at least one value of x in the range [0,7r]. In this case, we find that x ~ 1.1141 is one such value. Notes [1] There are two theorems that are also called the second mean value theorems (see Gradshteyn and Ryzhik [2], page 211): (A) If f(x) is monotonic and non-negative in the interval (a, b) (with a < b), and if g(x) is integrable over that interval, then there exists at least one point ( in the interval such that 1.' J(x)g(x) dx = J(a) 1.< g(x) dx. (B) If, in addition to the requirements in the last statement, f(x) is non­ decreasing, then there exists at least one point ( in the interval such that 1.' J(x)g(x) dx = J(b) I' g(x) dx. [2] There is also a mean value theorem from complex analysis (see Iyanaga and Kawada [3], page 624): Let u be a harmonic function (i.e., V2u = 0), let D be the domain of definition of u, and let S be the boundary of D. Then the mean value of u on the surface or the interior of any ball in D is equal to the value of u at the center of the ball. That is u(P) = ~ { UdT = +. ( udu. TnT J B(P,r) UnT J S(P,r) where Tn and Un are the volume and surface area of a unit ball in Rn, B(P,T) is the open ball with center at P and radius T, S (P, T) is the spherical surface with center at P and radius T, dT is the volume element, and du is an element of surface area. References [1] W. Kaplan, Advanced Calculus, Addison-Wesley Publishing Co., Reading, MA,1952. [2] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York, 1980. [3] S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT Press, Cambridge, MA, 1980. 86 II Concepts and Definitions 23. Path Integrals Idea Path integrals are integrals of functionals. Feynman path integrals are specific path integrals for the propagator of the Schrodinger equation; they have the form G = J eiS/Ii, where S is the action. Procedure 1 We illustrate the general procedure by considering a specific case. Consider a collection of particles that perform Brownian motion in one dimension and do not interact with one another. Define c(x, t) dx to be the number (or concentration) of particles in an interval of size dx about the point x at time t. It is well known that the concentration satisfies the partial differential equation (23.1) where D is the diffusion coefficient. If one particle starts out at Xo at time to, then the concentration has the initial condition c(xo, to) = 6(t-to). (The function 6() denotes Dirac's delta function.) The solution of (23.1) with this initial condition is (see Zwillinger [11]): 1 { (x -XO)2 } c(x, t) = exp . v' 411" D (t -to) 4D (t -to) (23.2) This gives the likelihood of finding the particle at the point x at any time t 2:: to. The likelihood of finding the particle near x, with an uncertainty of dx, is just c(x, t) dx. (In mathematics books, c(x, t) would be called the Green's function of (23.1); in physics books it would be called the propagator of a Brownian particle.) Now divide the interval (to, t) into N + 1 equal intervals of length c: tl < t2 < ... < tN (so that ti+l -ti = c). We would like to know the probability that the particle that started at (xo, to) is near the point Xl (with uncertainty dxd at time tI, near the point X2 (with uncertainty dX2) at time t2, ... , near the point x N (with uncertainty dx N) at time tN, and near the point x := XN+l (with uncertainty dx) at time t := tN+l' See Figure 23. Because the Brownian particle is memoryless, this probability is equal to the product of (23.2) taken over successive subintervals: Prob = 1 _ exp -4~ E (xj+l; Xj) II dXj. ( ) N+l {N 2} N+l v' 411" D( t to) j=O j=l (23.3) 23. Path Integrals I o c a t i o n 87 1 time Figure 23. A graphical illustration of the locational constraints on the particle. If the limit of infinite discretization is taken {Le., e -+ 0 and N -+ 00, while (N + l)e = t -to), then (23.3) can be interpreted as the probability that the particle follows a particular path X(T) from (xo, to) to (x, t), where x( Tj) = Xj. In this limit, the exponential in (23.3) can be written in the compact form: exp{ __ 1 {t (dX)2 dT}. 4D ito dT (23.4) Using (23.4) in (23.3) we obtain the path integral Prob = lx ,t exp{-4~ r (~x)2 dT} d[X{T)]. xo,to ito T (23.5) The value of this path integral is, however, given in (23.2): 1 { (x -xo)2 } Prob= ~p . J 47r D(t -to) 4D(t -to) (23.6) Note that the particular path taken, x( T), does not enter into the path integral in (23.5). This is because the integral is considering all possible paths from the initial point to the final point. It is worth emphasizing that (23.5) is only a notational way to indicate the limit of a discrete number of integrations. The terms in (23.5) can be individually interpreted as follows: (A) The symbol d[X{T)] denotes nf=l dXj and restricts the X{T) appearing in the integral to be constrained by x(to) = Xo; Xj < x{tj) < Xj + dXj (for j = 1,2, ... , N); and x(tN+d = x{t) = x. These limits for the x{ T) function are indicated by the limits on the first integral sign. (B) Each dXj is to be integrated from -00 to 00, and an implicit normal- ization factor is present ! ( 1 )N+ljOO JOO JOO d[X{T)] ~ "f4;fh dXl dXl··· dXN, 47r De -00 -00 -00 where e = (tj+l -tj)/N. 88 II Concepts and Definitions (C) The exponent in (23.5) is to be interpreted as [t (dX) 2 dT ==> t (Xj+l -Xj )2. to dT j=O c (D) After all the integrations have been performed, the limit N -. 00 (which corresponds to c -. 0) must be taken. Procedure 2 Consider a classical system that starts in state Xa at time ta and ends in state Xb at time tb. The evolution of the system can be described by a set of variables that are functions of time; these variables describe a "path" from state Xa to state Xb. The classical system will almost certainly have a Lagrangian L that depends on the path of the system L = L[path]. The action S is defined as the time integral of the Lagrangian Itb S[path] = L[path] dt. to (23.7) In quantum mechanics, the wave function 1jJ evolves according to 1jJ(b) = J P(b I a)1jJ{a) dXa where P(b I a) is the propagator from Xa to Xb. The propagator can be obtained by solving the Schrodinger equation iTiPt = HP, where H is the Hamiltonian and Ti denotes Planck's constant divided by 21i'. The propagator can also be obtained by the Feynman path integral 12:b,tb {i } P(b I a) = exp hS[path] d[path]. 2:0 ,to (23.8) This second method of obtaining the propagator is more useful in some circumstances, such as in dissipative systems that have a Lagrangian but not a Hamiltonian. See Wiegel [9] for more details. The path differen­ tial measure takes the form d[path] ==> AN rrf~l dXj, where AN is an appropriate normalization constant. Example 1 We indicate two ways in which the integral lO't {I /.t (dX)2 } 1= exp --D -d dT d[X{T)] 2:=O,t=O 4 ° T (23.9) may be evaluated. This integral is the one in (23.5), with Xo = x = to = 0, but here we presume that we do not know its value. The integral in (23.9) can be written, by the four rules given above, as ( 1 ) N+l100 100 100 I = lim.;;r:;rne dxo dXl . . . dx N N -00 41i'Dc -00 -00 -00 { } (23.10) x exp -4~e t. (X;+1 -x;)2 , 23. Path Integrals where e is related to N by e = t/(N + 1). Example 1.A 89 The quadratic terms appearing in the exponent in (23.10) can be written as (note that Xo = 0 and XN+l = x = 0) N N ~ (Xj+l -Xj)2 = ~ xkM~~)Xl' (23.11) j=O kJ=1 where M(N) is the tri-diagonal matrix 2 -1 o -1 2 -1 -1 2 -1 -1 2 -1 o -1 2 It is straightforward to show that det M(N} = N + 1. Since M(N} is Hermitian, its eigenvalues are real. Denoting its eigenvalues by {A j }, the sum in (23.11) may be written as 1:f=1 AjYI, where the {Yj} are linearly related to the {Xj}. Again, since M(N} is Hermitian, the Jacobian of this transformation is unity. Hence, (23.10) may be written as ( 1 )N+l1°O 100 100 I = lim r:-;::;- dyo dY1 . . . dy N N -00 V 47r De -00 -00 -00 x exp{-~e t),jyJ} 3=0 ( 1 )N+1 N ~ = NIi.!'oo.J4ifk )] V T (23.12) I, ( 1 ) I' ( 1) 1 = 1m = 1m =. N-oo V41t'DedetM(N} N-oo V47rDt V47rDt This is precisely the value in (23,6), when x = Xo = to = O. 90 II Concepts and Definitions Example l.B The integrations in (23.10) can also be performed one at a time. The formula 1"" e-a('-'L)'-b(.-.u)' dx = J " exp{-~(XL -XU)2} -00 a+ b a + b can be used to evaluate the integrals sequentially. First, choose XL = XO, X = Xl, and Xu = X2. Then the xl-integral appearing in (23.10) can be evaluated to obtain 1 100 {(Xl -xo)2 + (Xl -X2)2 } d J 41r D£ -00 exp 4D£ Xl = 2.. exp{-(xo -X2)2} v'2 8D£ This expression has the same form as a term in (23.10), but with £ replaced by 2£ throughout. Now the integration with respect to X3 is carried out, using XL = X2 and Xu = X4. Then the integrations with respect to X5,X7, ••• , are carried out. The variables left at this point are {XO,X2,X4,X6, ••• }. Now the integrals with respect to X2, X6, XlO, ... , are carried out, and again the same form is obtained, but now 2£ has been replaced by 4£. This operation is recursively applied. (Without loss of generality, we can assume that N + 1 is a power of two.) The end result is 1= lim 1 = 1 . N-oo V41rD(N + 1)£ V41rDt Notes [1] The main task of evaluating a path integral is to evaluate the multiple Riemann integrals and then take the N ---+ 00 limit. This is only possible in a few cases, such as when • the integrals involved are Gaussians or • a recursive formula is available to carry out the successive integrations. The propagators for a free particle and for the harmonic oscillator of constant frequency fall in the first category. We quote here the standard propagators: • For a free particle of mass m P(b I a) = (211"7nT) 3/2 exp{ ;;(Xb -Xa)2}, where T = tb -tao • For a free particle of mass m and frequency w P(b I a) = (211"i;::~WT) 3/2 exp{ 2",i:WWT ([x~ + x!] coswT -2Xb' Xa) }. [2] When a Feynman path integral is constructed, the action is rewritten, using the principle of canonical quantization, with the substitutions ..... "f;. 8 q --+ q = q p ---+ p = -tl& 8q , where q is a generalized coordinate, and p is a conjugate momentum. 23. Path Integrals 91 [3] When L = L(x,:ic,t), the action functional in (23.7) can be discretized using the midpoint rule to obtain N 8 [ ] -"L (Xj + Xj-1 Xj -Xj-1 .) N X -e L 2' 2e ,Je. j=1 For general L, the midpoint rule is required for the propagator as given by (23.8) to satisfy Schrodinger's equation. However, in some cases, other formulas for the derivatives may be used, see Khandekar and Lawande [4]. [4] When the classical Lagrangian has no explicit time dependence, the propa­ gator can be written in the form P(b I 0) = L exp{ -iEn(t~ -t.) } "'~(X.)"'n(X.), n where the {Ek} and the {tPk} are the complete set of energy eigenvalues and eigenfunctions: HtPk = EktPk. This expansion is known as the Feynman-Kac expansion theorem. [5] Note the oscillatory nature of the integral in (23.8) for small Ii. From the stationary phase technique (see page 226) it is clear that the major contributor to the propagator comes from paths for which 68 = 0 and 628 > O. The equation 68 = 0 is Hamilton's principle; its solution is the classical path of the system. This equation is also equivalent to the usual Euler-Lagrange equations. If the integrand is expanded about the classical path, and terms out to second order are kept, then the resultant propagator is equivalent to the WKB approximation of the Euler-Lagrange equations (see Zwillinger [11]). [6] Example I.B has the rudiments of a renormalization group calculation, see Zwillinger [11]. [7] Other terms used for path integrals are Feynman path integrals and func­ tional integrals. References [1] S. Albeverio and R. H~egh-Krohn, Mathematical Theory of Feynman Path Integrals, Springer-Verlag, New York, 1976. [2] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill Book Company, New York, 1965. [3] T. Kaneko, S. Kawabata, and Y. Shimizu, "Automatic Generation of Feyn­ man Graphs and Amplitudes in QED," Comput. Physics Comm., 43,1987, pages 279-295. [4] D. C. Khandekar and S. V. Lawande, "Feynman Path Integrals: Some Exact Results and Applications," Physics Reports, 137, No. 2-3, 1986, pages 115- 229. [5] P. K. MacKeown, "Evaluation of Feynman Path Integrals by Monte Carlo Methods," Am. J. Phys., 53, No.9, September 1985, pages 880-885. [6] L. S. Schulman, Techniques and Applications of Path Integration, John Wiley & Sons, New York, 1981. 92 II Concepts and Definitions [7] R. G. Stuart and A. G6ngora-T., "Algebraic Reduction of One-Loop Feyn­ man Diagrams to Scalar Integrals. II," Comput. Physics Comm., 56, 1990, pages 337-350. [8] G. J. van Oldenborgh, "FF - A Package to Evaluate One-Loop Feynman Diagrams," Comput. Physics Comm., 66, 1991, pages 1-15. [9] F. W. Wiegel, Introduction to Path-Integral Methods in Physics and Polymer Science, World Scientific, Singapore, 1986. [10] P. Zhang, "Simpson's Rule of Discretized Feynman Path Integration," J. Sci­ entific Comput., 6, No.1, March 1991, pages 57-60. [11] D. Zwillinger, Handbook of Differential Equations, Academic Press, New York, Second Edition, 1992. 24. Principal Value Integrals Applicable to A formally divergent definite integral. Yields A convergent integral. Idea Many integrals are improper because of the limiting process involved. By restricting the way limits are taken in a definite integral, a convergent expression may sometimes be obtained. Procedure Suppose that f(x) has the singular point z in the interval (a, b), and suppose that f(x) is integrable everywhere in the interval, except at the point z. By definition (see page 53), the integral J: f(x) dx has the value li~ {f.z -a f(x) dx + fb f(x) dX} , p-o a 1z+p (24.1) where the limits are to be evaluated independently. If the value in (24.1) does not exist, but the value li.!? {f.Z -7 f(x) dx + fb f(x) dX} "lOa 1Z+7 (24.2) does exist, then this value is the principal value, or the Cauchy principal value, of the improper integral I. This is denoted I = f: f(x) dx. Using the relation f: = f: +fbc, we can define the principal value of an integral with multiple singularities. 24. Principal Value Integrals 93 Example Consider the integral I = J: dx/(x -I}. Because of the singularity at x = 1, this is an improper integral. Hence, I does not exist in the usual sense. However, I does have a principal value which is easy to determine: Notes 13 ~ = lim {!.1-'Y ~ + 1,3 ~} To x-I "1-0 0 x-I 1 +"1 x-I = lim {lOg Ix -1111 -"1 + log Ix _ 1113 } "1-0 0 1+"1 = ~~ { (log bl-log 111) + (log 121-log 1'11) } = lim {log2} = log 2. "1-0 [1] Suppose that f(x) is continuous on the interval [a, b], and vanishes only at the point z in this interval. If the first and second derivatives of f exist in a region containing the point z, and /'(z) #< 0, then the integral J.. J7x) dx is improper, but exists in a principal value sense. For the example presented above, f(x) = x -1 and !,(x) = 1. [2] The variables in a principal value integral may be changed under some mild restrictions. We have (see Davis and Rabinowitz [4], page 22): Theorem: If z( Q) = a and z(f3) = b, and if, on the interval [Q, f3], z«() is monotonic, z'«() does not vanish, and z«() has a continuous second derivative, then f.b f(t) dt =f.{J f(z«(»z'«() d(, a t -X Q z«() -m(~) where x = m(~). [3] The integral J~oo f(x) dx may be written in two forms: (A) J~oo f(x) dx = limRl-OO J~Rl f(x) dx + limR2-oo JOR2 f(x) dx (B) J~oo f(x) dx = limR_oo (J~R f(x) dx + JoR f(x) dx) . The integral J~oo f(x) dx is convergent when the first set of limits converge. If the second formulation converges, but not the first, then the integral is convergent in the sense of Cauchy and it is denoted f~oo f(x) dx. For example, J~oo x dx converges in the sense of Cauchy, but the integral is not convergent. [4] Davies et al. [3] defines a higher-order principal value to be given by: 11.1.- dx- --- dx tOO f(x) 1 tOO 1 d!n-l) f(x) ( ) - -00 (x -ut -(n -1)! -00 (x -1.1.) dx(n-l) . 94 II Concepts and Definitions [5) Many methods for the numerical evaluation of principal value integrals may be found in the references. For example, in the work by Hunter and Smith [7), the location of the poles do not need to be known before the numerical algorithm is run. (A) Mastroianni [10] considers numerical methods for integrals of the form f f (x !~)(:)- t) Wl(X)W2(y) dxdy. (B) Rabinowitz [12) considers numerical methods for integrals of the form [', w~~~x) dx with w(x) = (1 -X2)"-',2 for 0 :s /J :s 2. References [1] G. Criscuolo and G. Mastroianni, "On the Uniform Convergence of Gaussian Quadrature Rules for Cauchy Principal Value Integrals," Numer. Math., 54, 1989, pages 445-461. [2] G. Criscuolo and G. Mastroianni, "On the Convergence of Product Formu­ las for the Numerical Evaluation of Derivatives of Cauchy Principal Value Integrals," SIAM J. Math. Anal., 25, No.3, June 1988, pages 713-727. [3] K. T. R. Davies, R. W. Davies, and G. D. White, "Dispersion Relations for Causal Green's Functions: Derivations Using the Poincare-Bertrand Theo­ rem and its Generalizations," J. Math. Physics, 31, No.6, June 1990, pages 1356-1373. [4] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, pages 11-15. [5] G. A. Gazonas, "The Numerical Evaluation of Cauchy Principal Value Inte­ grals via the Fast Fourier Transform," Int. J. Compo Math., 18, 1986, pages 277-288. [6] A. Gerasoulis, "Piecewise-Polynomial Quadratures for Cauchy Singular In­ tegrals," SIAM J. Numer. Anal., 23, No.4, 1986, pages 891-902. [7] D. B. Hunter and H. V. Smith, "The Evaluation of Cauchy Principal Value Integrals Involving Unknown Poles," BIT, 29, No.3, 1989, pages 512-517. [8] H. R. Kutt, "The Numerical Evaluation of Principal Value Integrals by Finite-Part Integration," Numer. Math., 24, 1975, pages 205-210. [9] J. N. Lyness, "The Euler-Maclaurin Expansion for the Cauchy Principal Value Integral," Numer. Math., 46, No.4, 1985, pages 611-622. [10] G. Mastroianni, "On the Convergence of Product Formulas for the Evalua­ tion of Certain Two-Dimensional Cauchy Principal Value Integrals," Math. of Comp., 52, No. 185, January 1989, pages 95-101. [11] P. Rabinowitz, "Numerical Evaluation of Cauchy Principal Value Integrals with Singular Integrands," Math. of Comp., 55, No. 191, July 1990, pages 265-276. [12] P. Rabinowitz, "A Stable Gauss-Kronrod Algorithm for Cauchy Principal­ Value Integrals," Comput. Math. Appl. Part B, 12, No. 5-6, 1986, pages ~49-1254. [13] P. Theocaris, N. I. Ioakimidis, and J. G. Kazantzakis, "On the Numerical Evaluation of Two-Dimensional Principal Value Integrals," Int. J. Num. Methods Eng., 15, 1980, pages 629-634. 25. Transforms: To a Finite Interval 95 25. Transforms: To a Finite Interval Applicable to Integrals that have an infinite limit of integration. Yields An exact reformulation. Idea Many transformations map an integral on an infinite domain to an integral on a finite domain. Such a transformation may be needed before a numerical approximation scheme is used. Procedure Given the integral 1= fO f(x)dx (25.1) a numerical approximation technique of the form N I ~ L, wjf(xj) (25.2) j=1 may not be appropriate because of a fundamental indeterminacy in (25.1). If S is any positive number, then 1= fO f(x)dx = S 10"0 f(Sx)dx. Hence, the region where most of the "mass" of the integrand lies may have been (inadvertently) scaled to be outside of the range of the {Xi}. Without some knowledge of the shape of f{x), it may be difficult to determine appropriate values for N and the {Xi} in (25.2). One approach is to transform I to a finite domain and then use a numerical integration scheme on the finite domain. It is generally easier to understand the form of an integrand on a finite domain, than it is on an infinite domain. Under the transformation t = t(x), we have I = 101 f[x(t)) Idx/dtl dt (see page 109). Table 25 contains several useful transformations of (25.1). 96 II Concepts and Definitions Table 25. Several transformations of the integral Jooo J(x) dx, with an infinite integration range, to an integral with a finite integration range. t(x) x(t) dx Finite interval integral dt e-z -Iogt 1 /.1 I( -log t) dt t o t x t 1 /.1 ( t) dt 1+x 1-t (1 -t)2 o I 1 -t (1 -t)2 tanh x II 1 + t 1 [I(!.IO 1+t) ~ -og--1-t2 2 I-t o 2 g 1 - t 1 -t2 Notes [1] Another useful transformation, for any positive value of S, is: f.~ I(x) dx = 8 [ [/(8t) + t~ I (f) 1 dt. [2] The transformation lb I(t) dt = 11/0 ~ I (.!.) dt, valid for ab > 0, is useful o lIb t t in the two cases: • b ~ 00 with a > OJ • a ~ -00 with b < O. [3) If an integral has a known power-law singularity at an endpoint, then a transformation can be made to remove it. For example, suppose that J(x) diverges as (x -a)Q near x = a (where -1 < a < 0). Then we can use where b > a. If, instead, J(x) diverges as (b -x)/3 near x = b (where -1 < (3 < 0), and b> a, then we can use lb /.(b_O)l-.a J(t) dt = ~ t/3/{l-/3) f (b -t1/(l-/3») dt. o 1 {3 0 [4] The change of variables in this section can be performed analytically, or a numerical quadrature routine can perform the change. 26. Transforms: Multidimensional Integrals 97 Reference [1] W. Squire, Integration Jor Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970, pages 174-176. 26. Transforms: Multidimensional Integrals Applicable to Definite multidimensional integrals. Yields A reformulation into a one-dimensional integral. Idea Some classes of multidimensional integrals can be written in t~rms of a single integral. Procedure The following listing contains many multidimensional integrals each involving a general function. These multidimensional integrals can be written in terms of a single integral. Two-Dimensional Integrals f.0 dw f.~!' (p cosh x + q cos w sinh x) sinh x <Ix = lI'[signp] J ([signp]VP2 _ q2) Vp2 -q2 if p2 > q2 and lim:z:-+oo J(x) = 0 (see Gradshteyn and Ryzhik [2], 4.620.1, page 618). f.20 dw f.~ !' (p cosh x + (q cos w + r sin w) sinh x ) sinh x <Ix = 211'[signp] J ([signp]VP2 -q2 -r2) Vp2 -q2 -r2 ifp2 > q2+r2 and lim:z:-+oo J(x) = 0 (see Gradshteyn and Ryzhik [2], 4.620.2, page 618). 98 II Concepts and Definitions 111' 111'/' (p -qcosx ) dxdy . . + rcoty o 0 smxsmy sinxsin2 y = 211"[signp] I ([signp]VP2 -q2 -r2) Vp2 -q2 -r2 ifp2 > q2+r2 and lim:z:_oo I(x) = 0 (see Gradshteyn and Ryzhik [2], 4.620.3, page 618). 1: dx 1:!' (p cosh x cosh y + q sinh x cosh y + r sinh Y ) cosh y dy = 211"[signp] I ([signp]VP2 _ q2 -r2) Vp2 -q2 -r2 ifp2 > q2+r2 and lim:z:_oo I(x) = 0 (see Gradshteyn and Ryzhik [2], 4.620.4, page 618). f.~ dx f.' / (pcoshx + qcos w sinh x) sinh' xsinwdw = 2 f.~ / ([Sign pI v' p' -q' cosh x ) sinh' x dx if lim:z:_oo I(x) = 0 (see Gradshteyn and Ryzhik [2], 4.620.5, page 619). f.~ f.~ / (a'x' + b'y') dxdy = 4:b f.~ x/ex) dx if lim:z:_oo I(x) = 0 (see Gradshteyn and Ryzhik [2], 4.623, page 619). f f xQ-1yP-l I [(;)P + (~) q] dxdy :z:~O, tI~O (;)P+(ffSl = aQbP r [alP, {3/ql11 l(t)tQ/P+P/q-l dt pq a/p + {3/q 0 if a,b,a,{3,p,q > 0 (see Prudnikov, Brychov, and Marichev [3], 3.1.2.1, page 565). f f /(ax + (Jy + 7) dxdy = 2 L y'i""=t2 / (tv' a' + (J' + 7) dt :z:2+tl2S1 if a, b > 0 and a, {3, and '1 are real (see Prudnikov, Brychov, and Marichev [3], 3.1.2.2, page 566). 26. Transforms: Multidimensional Integrals J J f(x + y)d:cdy = I: f(t)dt Izl+11I1$1 (see Prudnikov, Brychov, and Marichev [3], 3.1.2.3, page 566). 99 /.' /.' f(xy)(l -x)a-'ya(l -y)p-, d:cdy = B(a,p) /.' f(t)(l -t)aH-' dI if Rea > 0 and Re/3 > 0 (see Prudnikov, Brychov, and Marichev [3], 3.1.2.4, page 566). /."" /."" f(ax + by).-"-" d:cdy = 1 [aq/.OO f(t)e-pt/a dt _ bp/.oo f(t)e-qt/b dt] pq(aq -bp) 0 0 if a, b,p, q > 0 (see Prudnikov, Brychov, and Marichev [3], 3.1.3.1, page 567). /."" /."" f(lx -yl).-"-" d:cdy = pq(pl+ q) [p /."" f(t).-" dt + q /."" f(t).-.' dt] if p, q > 0 (see Prudnikov, Brychov, and Marichev [3], 3.1.3.2, page 567). /."" /."" f(xy).-"-·· d:cdy = 2 /."" Ko (2Jj>qt) f(t)dt if p, q > 0 (see Prudnikov, Brychov, and Marichev [3], 3.1.3.3, page 567). 100100 e-pz-q1l f (-=:JL.) dxdy= ..;:i(JP+y'q) 100 e-(v'P+vq)2 tf(t)dt o 0 ../x + y x + 1 .;pq 0 if p, q > 0 (see Prudnikov, Brychov, and Marichev [3],3.1.3.4, page 567). Three-Dimensional Integrals /."" d:c /.2< dw /.' f (PCOShX + (qcosw + r sin w) sin O.inh x) sinh' xsinOdO = 4 /."" f (ISignplv'p2 -q' -r' cosh x ) sinh' xd:c ifp2 > q2+r2 and limz_oo f(x) = 0 (see Gradshteyn and Ryzhik [2], 4.620.6, page 619). 100 II Concepts and Definitions f.~ dx f.2W duJ f.' I (p cosh X + [(qcosw + rsinw) sinO + B cosh 0] sinh X ) x sinh 2 x sin (J d(J = 4 .. f.~ I ([signp]v'P2 -q2 -r2 -B2 cosh X ) sinh2 xdx if p2 > q2 + r2 + 82 and lim:z:_ oc f(x) = 0 (see Gradshteyn and Ryzhik [2], 4.620.7, page 619 or Prudnikov, Brychov, and Marichev [3], 3.2.3.4, page 584). f f f XQ-ly~-lz7-1 f [(;)P + (i) q + (~) r] dxdydz :z:~O. lI~O. z~O (;)"+(fr+(~r = aQb~c7 r [ alp, {3lq, 'Ylr 111 f(t)tQ/p+Mq+7/r-l dt pqr alp + {3lq + 'Ylr 0 if a, b, C,p, q, r, a, {3, "I > 0 and the integral on the right-hand side converges absolutely (see Prudnikov, Brychov, and Marichev [3], 3.2.2.1, page 583). 1"1"-"1"-'-' xO-',/-' z ,-I I(x + y + z) dx dy d;; = r [ a, {3, "I 11a f(t)tQ+~+7-1 dt a+{3+'Y 0 if a, a, {3, "I > 0 and the integral on the right-hand side converges absolutely (see Prudnikov, Brychov, and Marichev [3], 3.2.2.3, page 583). Multi-Dimensional Integrals f f-.} I ('Lt,) IT (t~'-' dt,) = r~~~:) [I(X)X'E"'-' dx , ¥ ., L>i=:;l This is known as the Dirichlet reduction (see Squire [6], pages 82-83). 26. Transforms: Multidimensional Integrals 101 iLl / (~t,) IT ~(t,)dt, ti~O tl +t2+···+tn SI = /.' (f(U)[4>, (U) * q,,(U) * ... * 4>.-.(U))) * 4>.-.+1 * ... * 4>.(U) du where the stars represent convolutions (see Sivazlian [5]). /.' g(x.) <lx, /." g(x.) <lx ••.• /."-' g(x.) <lx. /." let) dt = ~! [ let) ([ 9(Y)dY)' dt (see Prudnikov, Brychov, and Marichev [3], 3.3.1.1, page 585). Ir·} / [(:: r + (::)"' + ... + (::)""] xr'-'~'-'···x:'-'dx R (PI) (P2) r (Pn) (PI P2 Pn) PI 2 Pn r - r - ... - 00 -+-+ ... +--1 = ql d ... qn al a2 an!. J(z)z al a2 an dz al a2 ... an r (PI + P2 + ... + Pn ) I al a2 an {Xl > 0 X2 > 0 . . . Xn > 0 - + - + ... + - > 1 where R is the region R: (Xl) Ql (X2) Q2 (xn) Qn ~ ft ~- when the integral on the right converges absolutely (see Gradshteyn and Ryzhik [2], 4.635.1, page 620). II ... I / [( ::) 0, + (::) 0, + ... + (::) on] xf'-'x~'-' ... x~n-' dx 5 r(PI)r(P2) r(Pn) I (PI P2 Pn_l) _ qfl d2 ... tfnn al a2'" an f. J(z)z al + a2 + ... + an dz -al a2 ... an r (PI + P2 + ... + Pn ) 0 al a2 an {Xl > 0 X2 > 0 . . . Xn > 0 Xl + X2 + ... + Xn < 1 whereSistheregionS: ( )Ql ()Q2 ()Qn ~ ft ~- when the integral on the right converges absolutely, and the numbers qi, ai, and Pi are positive. (see Gradshteyn and Ryzhik [2], 4.635.2, page 621). 102 II Concepts and Definitions where R is the region R: - - - , f(x) is con-{Xl > 0, X2 > 0, ... , Xn > 0 Xl + X2 + ... + Xn ~ 1 tinuous on (0,1) and qi ~ 0 and r > 0 (see Gradshteyn and Ryzhik [2], 4.637, page 622). if f(x) is continuous on the interval (0, R) (see Gradshteyn and Ryzhik [2], 4.642, page 623). /.' /.' ... /.' f(x,x •... xn)( 1 -x,)Pl -, (1 -x.)'" -, ... (1 -Xn)··-' X X~l X~l +P2 ••• X~l +P2+· .. +Pn-l dx = r(pt}r(p2) ... r(pn) /.1 f(x)(l- x)Pl+P2+ ... +Pn-1 dx r(p1 + P2 + ... + Pn) 0 when the integral on the right converges absolutely (see Gradshteyn and Ryzhik [2], 4.643, page 623). if R is the region R = x~ + x~ + ... + X~-l ~ 1, n ~ 3, and f(x) is continu­ ous on the interval [-{J,{J], with {J = v'p~ + p~ + ... + p~ (see Gradshteyn and Ryzhik [2],4.644, page 624). 27. Transforms: Miscellaneous 103 Notes See also the section on how to change variables on page 109. [1] [2] Schwartz [4] considers a numerical technique for integrals of the form (rr~::::l J gi(Xi) dXi) F (L:~=1 !i(Xi»). His technique is based on the use of integral transforms. References [1] R. Y. Denis and R. A. Gustafson, "An SU(N)N Q-Beta Integral Trans­ formation and Multiple Hypergeometric Series Identities," SIAM J. Math. Anal., 23, No.2, March 1992. [2] 1. S. Gradshteyn and 1. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York, 1980. [3] A. P. Prudnikov, Yu. A. Brychov, and O. 1. Marichev, Integrals and Series, Volume 1, translated by N. M. Queen, Gordon and Breach, New York, 1990. C. Schwartz, "Numerical Integration in Many Dimensions. I," J. Math. Physics, 26, No.5, May 1985, pages 951-954. [4] B. D. Sivazlian, "A Class of Multiple Integrals," SIAM J. Math. Anal., 2, No.1, February 1971, pages 72-75. [5] W. Squire, Integration for Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970. 27. Transforms: Miscellaneous Idea This section has a collection of transformations of integrals. Some Specific Forms for One-Dimensional Integrals When an integrand has a specific form, there are some common trans­ formations that might be useful. For example, if the integrand: (A) is a rational function of sinx and cosx, introduce u = tan(x/2) so that . 2u SlllX= ---2' l+u 1-u2 COS X = --2' l+u 2du dX=--2· l+u 104 II Concepts and Definitions . / sin2 x For example, the mtegral J = 4 • 3 dx becomes cos x + cosxsm x J- ~ / 8U2(U2 + I} -(u2 -l)(u2 -2u -l)(u4 + 2u3 + 2u2 -2u + I) _ / [ 4u3 + 6u2 + 4u - 2 + 2u - 2 __ 1 ___ 1_] du - u4 + 2u3 + 2u2 -2u + 1 3(u2 -2u -I} u + 1 u -1 = i log (u4 + 2u3 + 2u2 -2u + 1) + i log (3(u2 -2u -I}) -log(u + I} -log(u -I} (B) is a rational function of sinh x and cosh x, introduce u = tanh(x/2} so that . h 2u sm X=--2' l-u l+u2 cosh x = --2' l-u 2du dX=--2· l-u (C) is a rational function of x and v'f"='X2, introduce x = cos v to get to case (A). (D) is a rational function of x and ..jX2 -1, introduce x = cosh v to get to case (B). (E) is a rational function of x and ..j x2 + 1, introduce u = x +..j x2 + 1 so that (F) is a rational function of x and ..j ax2 + bx + c, then the substitu­ tion 2ax + b vv'14ac -b21-b v = {:=:} x = ---i.~ __ --'-_ v'14ac -b21 2a will reduce the integral to one of the above cases. (G) is a rational function of x and 1.1 = V:: : :' then use the substi­ tution 1.1= Vax+b <=*x= cx+d Integrals of the form f xm(a + bxn)p dx, where m, nand p are rational numbers, can be expressed in terms of elementary functions in each of the 27. Transforms: Miscellaneous 105 following cases (see Gradshteyn and Ryzhik [3] 2.202, page 71): p is an integer; (m + 1) / n is an integer; and p + (m + 1) / n is an integer. Three useful transformations of definite integrals from Gradshteyn and Ryzhik [3] (3.032.1-3.032.3) are {~/2 (~/2 10 f(sinx) dx = Jo f(cosx) dx {2~ (~ Jo f(pcosx + qsinx) dx = 2 Jo f ( Jp2 + q2 cos x ) dx {~/2 {~/2 10 f(sin2x)cosxdx= 10 f(cos2x)cosxdx. The Slobin transformation is (see Squire [6], page 95) 100 f (t -~) dt = 100 f(t) dt. -00 t -00 The Wolstenholme transformation is (see Squire [6], page 95) 10 ( a2 ) dt 1° ( a2 ) dt 1 f t2 + f.2 T = 1 f t + T T' Two unnamed transformations for definite integrals (from Squire [6] page 94) are: f~ 1r f~ J 0 x f (sin x, cos2 x) dx = 2" J 0 f (sin x, cos2 x) dx 100 f (n -n) tan-1 x dx 1r 11 f (n -n) dx x +x = -2 x +x -. o x 0 x Dealing with Singularities Often, integrable singularities can cause a numerical routine to have trouble approximating an integral. While there are no universal principles to be applied, some simple techniques can be useful. Consider the integral I = 10' lji dx. Assuming that 1(0) is finite, I has an integrable singularity. But a computer routine evaluating this integrand near x = 0 may have trouble. 106 II Concepts and Definitions (A) If the f(x) term was not present, then we would note the indefinite integral J dx / Vi = 2 Vi. Since the form Vx appears, it might be reasonable to transform the integral to be on this scale. Changing variables via u = Vi changes I to I = 2 J01 f(u2) duo This integral no longer has a singularity at the origin. (B) Another technique is to subtract out a singularity. For example, if f(O) has the finite value A, then I can be written as I -11 f(x) -A + A d -11 f(x) -A d 2A -c x-c x+ . o yX 0 yX This new integrand has more smoothness than the original integrand, thus should be easier to evaluate numerically. (C) Another technique is to break the integral up into several sub-integrals. For the above integral we might try where e is large enough so that a numerical routine does not have trouble with J1I say e = 10-4• Then the integral J2 has a narrow region of integration; expansion methods, such as Taylor series, may yield a good numerical approximation. Of course, commercial integration software can automatically detect power singularities, and apply an appropriate (numerical) transformation to avoid computational difficulties. Products of Special Functions Piquette [5] considers integrals of the form 1= J f(x) n~1 R~!(x) dx, where R~!(x) is of the i-th type of a special function of order Il, obeying the set of recurrence relations R(i) -a (X)R(i) + b (x)R(i) ",+1- '" '" '" ",-1' dR(i) . -"'-= c (x)R(i) + d (x)R(S) dx '" '" '" ",-1· (27.1) Here {a"" b"" c"" d",} are known functions. Most of the special functions of physics, such as the Bessel functions, Legendre functions, Hermite func­ tions, and Laguerre functions have recursion relations in the form of (27.1). The indefinite integral I is assumed to have the form 1 1 1 m I = L L ... L AplP2 ... Pm (x) II R~l+Pi (x). Pl=OP2=0 Pm=o i=l 27. Transforms: Miscellaneous 107 Then the values of the {Ap} are determined, sometimes with the help of a differential equation. Piquette [5] obtains, for example, the following integral involving Legendre functions ! x [Pl/3(X)]3 dx = C1225x4 -134x2 -152) [P1/3(X)]3 + (-4 + 20X2)P1/3(X) [P4/3(X)]3 + (9x -25x3) [P1/3(X)] 2 P4/3(X) -I:X [P4/3(X)]2 . A Jacobian of Unity Sometime a judicious change of variables in a multiple integral may make it easier to evaluate. Finding the correct change of variables may sometimes be done by solving a partial differential equation. For example, changing variables in the usual way in two dimensions results in (see page 109) 1= !! I(x, y) dxdy = !! F(u,v) I ~~:: :~ I dudv, (27.2) where F(u, v) := f (x(u, v), y(u, v)). If u is chosen so that F(u, v) = F(u), and v is chosen so that la(x, y) 1-1 = au Bv _ Bu Bv = 1, (27.3) B(u, v) Bx By By Bx then (27.2) becomes I = II F(u) dudv, which may be easier to integrate. Note that equation (27.3) for v is a partial differential equation. Example Suppose we have the integral I = ],2 [ log ( x2 ~ y2) dx dy. Choos- ing u = x2_y2 simplifies (27.3) to 2X: +2y:: = 1 which has as a solution v = ! log( x + y). Using the u and v variables defined above, I becomes where 1= {J.81V3 (U) + /.121V3 (U) + 1151V4 (U)} log! dv du, 5 VI (u) 8 V2(U) 12 V2(U) 4 Vl(U) = ! log (3 + v'9-U) V2(U) = ! log (1 + v'f+u) V3(U) = ! log (2 + J4+U) V4(U) = ! log (4 + J16 -u) . It is now straightforward to evaluate this integral. See Morris [4] or Squire [6] pages 12-13 for details on this method. 108 II Concepts and Definitions Notes [1] If R(sinx,cosx) satisfies the relation R(sinx,cosx) = -R(-sinx,cosx) then it is normally convenient to make the substitution z = cos x. This results in the formula ! R(SinX,C08X)dx=-! R(~,z) p' [2] If R(sinx,cosx) satisfies the relation R(sinx,cosx) = -R(sinx, -cos x) then it is normally convenient to make the substitution z = sin x. This results in the formula ! R(SinX,C08X)dx= ! R(Z,~) p' [3] If R( sin x, cos x) satisfies the relation R( sin x, cos x) = R( -sin x, -cos x) then it is normally convenient to make the substitution z = tanx. This results in the formula !R(SinX'COSX)dx=!R(~' ~)~. 1 + Z2 1 + Z2 1 + z For example, we find (compare to the first example in this section) References J = sm x dx ! .2 cos4 X + cos x sin3 x = dz ! Z2 (z + 1)( Z2 -Z + 1) ! [ 2z -1 1 1 d = 3{Z2 -Z + 1) + 3{z + 1) z = ~ log (Z2 -Z + 1) + ~ log (z + 1) = j log (1 + z3) = ~ log (1 + tan 3 x) . [1] M. Abramowitz and 1. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964 [2] M. S. Ashbaugh, "On Integrals of Combination of Solutions of Second-Order Differential Equations," J. Phys. A: Math. Gen., 19, 1986, pages 3701- 3703. [3] 1. S. Gradshteyn and 1. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York, 1980. [4] W. L. Morris, "A New Method for the Evaluation of ffA f{x,y) dydx," Amer. Math. Monthly, 43, 1936, pages 358-362. [5) J. C. Piquette, "Applications of a Technique for Evaluating Indefinite In­ tegrals Containing Products of the Special Functions of Physics," SIAM J. Math. Anal., 20, No.5, September 1989, pages 1260-1269. [6] W. Squire, Integration for Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970. III Exact Analytical Methods 28. Change of Variable Applicable to Definite integrals. Yields A reformulation. Idea A re-parameterization of an integral may make it easier to evaluate. Procedure Given the integral (28.1) if there exists a transformation of the form x = u(y), which can be uniquely inverted on the interval a ~ x ~ b to give y = vex), then lY=V(b) lY=V(b) 1= f(u(y» dx = f(u(y»u'(y) dy. y=v(a) y=v(a) (28.2) It is sometimes easier to evaluate (28.2) than to evaluate (28.1) directly. 109 110 III Exact Analytical Methods y v x u Figure 28.1 Pictorial representation of a mapping from the {x, y} plane to the { u, 1J} plane. Similar results hold in multiple dimensions. If there exists a one-to-one transformation from the region R in the xy-plane to the region S in the uv-plane, defined by the continuously differentiable functions x = x(u, v), Y = y(u, v); and if the Jacobian J = J(u, v) = 8(x,y) = Ixu Yu I (28.3) 8(u, v) Xv Yv is nonzero in the region of interest, then f f f(x, y) dAz,y = f f f(x(u, v), y(u, v))IJ(u, v)1 dAu,v (28.4) R S (see Figure 28.1.) A listing of some common two-dimensional transforma­ tions is in Table 28.1. In an orthogonal coordinate system, let {8i} denote the unit vectors in each of the three coordinate directions, and let {Ui} denote distance along each of these axes. The coordinate system may be designated by the metric coefficients {gn, g22, g33}, defined by .. = (8Xl)2 (8X2)2 (8x3)2 gn 8 + 8 + 8 ' Ui Ui Ui (28.5) where {Xl, X2, X3} represent rectangular coordinates. Using the metric coefficients defined in (28.5), we define g = gng22g33; the magnitude of the Jacobian of the transformation is then given by IJI = vg. Moon and Spencer [4] list the metric coefficients for 43 different orthogonal coordinate systems. (These consist of 11 general systems, 21 cylindrical systems, and 11 rotational systems.) Operations with orthogonal coordinate systems are sometimes written in terms of the {hi} functions, instead of the {gii} terms. Here, hi = vgii, so that IJI = ..;g = hlh2h3' For example, cylindrical polar coordinates are defined by Xl = r cos 4J, X2 = r sin 4J, and X3 = z. Therefore, using {Ul = r, U2 = 8, U3 = z}, we find hl = 1, h2 = r, and h3 = 1. From this we compute the magnitude of the Jacobian to be IJI = r. A listing of some common three-dimensional transformations, and their Jacobians, is in Table 28.2. 28. Change of Variable 111 Example 1 Given the integral /.'lr /2 sin x cos x 1= 2 dx, o } + sin x (28.6) we choose to change variables by x = U(y) = sin -1 y. This function can be uniquely inverted (on the range 0 ~ x ~ ~) via y = v (x) = sin x. Thus, '( ) dy dy dy dx = u y dy = r:;--:;; = = - V } -y2 V} -sin2 x cos x This could also have been obtained directly by dy = v'(x) dx = cos dx. Our change of variable then permits (28.6) to be written as lY=1 ycosx dy lY=1 Y 1 IY=1 1= --2 --= --2dy = -loge} + y2) = ~ log 2. y=o 1 + y cos x y=o 1 + Y 2 y=o (28.7) Example 2 Consider the integral I = J ~ 1 x2 dx. If we choose to change variables by y = vex) = x2, then we note that vex) cannot be uniquely inverted on the given interval of integration. Hence, we must break up the integral I into pieces, with vex) uniquely invertible on each piece. If we write I as I = It + 12 = 10 x2 dx + /.1 x2 dx, -1 0 then we can use the change of variables {y = x2, X = -y'Y} on It and the change of variables {y = x2, X = y'y} on 12 to obtain 10 /,0 dy 1 /.1 } (2 ) 11 1 It = x2 dx = y --= -v'Y dy = _ _y3/2 = - -1 1 -2v:Y 2 0 2 3 0 3 and /.1 /.1 ~ 1/.1 1 12= x2dx= y-=- v:Ydy=-. o 0 2JY 2 0 3 We conclude that I = It + 12 = 1 + 1 = i· 112 III Exact Analytical Methods y u=4 v= -2 v u=1 2 1 123 x -1 -2 Figure 28.2 The region of integration for Example 2, and the results of the coordinate transformation 'U = X + y, v = x -2y. Example 3 Suppose we wish to evaluate the integral I = J J (x + y)2 dx dy where R R is the parallelogram shown in Figure 28.2. The sides of R are straight lines with equations of the form x + y = CI, x -2y = C2, with the values C1 = 1, C1 = 4, C2 = -2, and C2 = 1. If we introduce the new coordinates u = x+y, v = x -2y then the region R corresponds to the rectangle 1 $ u $ 4, -2 $ v $ 1. This mapping is clearly one-to-one, and the Jacobian is given by J = 8(x, y) = 1 = 11 1 11 = --31. 8(u, v) 8(u, v) 8(x, y) 1-2 Therefore, we find 1411 1411~ I = (x + y)21JI dvdu = - dvdu = 21. 1 -2 1 -2 3 Table 28.1 Some common two-dimensional transformations. 10 1fJX/0 o dx 0 f(x,y)dy= y {j 1fJ dyl° f(x, y) dx. o oy/fJ . ... :::::: ~ ~ ~ ~ ~ ~ ~ ~ ~ j ~ j ~ ~ ~ j ~ j ~ ~ ~ j ~ ~ o a 28. Change of Variable 113 Assuming that a = fJ/({3 + -y), a> 0, {3 > 0, -y > 0 then we have 10 J.6-~Z y dx I(x, y) dy = o (Jz lO {J lJ/'{J o dy 0 I(x, y) dx+ 16 1(6-J/)h dy I(x,y)dx. 0{J 0 o fJ a=--{3+-y x Note this second representation has the integral separated into two pieces. lR 1~ o dx 0 I(x,y) dy = y lR 1~ o dy 0 I(x, y) dx = lTr/2 lR o d'I/J 0 I(r cos'I/J,rsin'I/J)r dr. o R x 12R 1~ o dx 0 I (x, y) dy = y lTr/2 lRcostJI o d'I/J 0 I(r cos'I/J,rsin'l/J)r dr. o R 2R x Table 28.2 Some common three-dimensional transformations (see Moon and Spencer [4]). (A) Bispherical Coordinates ~ J(I-u~)(l-un Ju~-1 Xl = aU3 ,X2 = a , X3 = .....:....--=--- UI -U2 UI -U2 UI -U2 a a a j§-U2 hl= ~,h2= ~,ha= --2. (UI-U2)Vu~-1 (UI-U2)Vl-u~ UI-U2 l-u3 (B) Cylindrical Polar Coordinates (here J = r) Xl = rcostP, X2 = r sin tP, Xa = Z hI = 1, h2 = r, h3 = 1. 114 III Exact Analytical Methods 222 (C) Elliptic Cylinder Coordinates (here J = UI - C :2) ,--,,....-___ -,1,....-U2 Xl = UIU2, X2 = J(u~ -c2)(l -U~), X3 = U3 K AM SA AX hi = h2 = h3 = 1. (D) Ellipsoidal Coordinates r-, (u-~---a2-)-( u-~---a-2-) -( u-~---a-2-) Xl = H a2(a2 _ b2) hi = h2 = (U~ -b2) (u~ -b2) (U~ -b2) b2(b2 _ a2) (u~-un (u~ (U~ -a2) (u~ -b2) (U~ -un (U~ -U~) (u~ -a2) (u~ -b2) , h3 = X3 = UIU2U3 ab (U~ -un (u~ -un (U~ -a2) (U~ -b2) , (E) Parabolic Cylinder Coordinates (here J = u~ + U~) Xl = ! (U~ -u~) , X2 = UIU2, X3 = U3 hi = h2 = Ju~ + u~, h3 = 1. (F) Paraboloidal Coordinates (u~ -a2) (u~ -a2) (u~ -a2) Xl = tl 2 2 a -b X2 = (u~ -b2) (U~ -b2) (u~ -b2) b2 _ a2 1 (2 2 2 2 b2) X3 = 2' UI + U2 + 1£3 -a - hi = (u~ -u~) (U~ -U~) (u~ -a2) (U~ -b2) h2 = U2 (U~ -un (u~ -un (u~ -a2) (U~ -b2) , h3 = U3 (u~ -u~) (u~ -u~) (U~ -a2) (U~ -b2) , (G) Rotational Parabolic Coordinates (here J = UIU2JU~ + u~) Xl = UIU2U3, X2 = ulU2Jl -ui, X3 = !(u~ -u~) h h v' 2 2 h UIU2 I = 2 = UI + U2' 3 = r;--::;j' Vl-u~ (H) Rotational Prolate Spheroidal Coordinates Xl = J(u~ -a2) (1 -u~), X2 = J(u~ -a2) (1-u~) (1 -u~), X3 = UIU2 u~ -a2u~ lu~ -a2u~ I (u~ -a2) (1 -un hi = "I h2 = " . ." h3 = t 2 1-U3 28. Change of Variable 115 (I) Rotational Oblate Spheroidal Coo~dinates Xl = U3 v' (u~ + a2) (1 -u~), X2 = v'r:"( u-:~:-+-a-=-:2 )~(:-1---u"'""':~~)--:(-1---u-:~~), X3 = U1 U2 (u~ + a2 ) (1 -un hI = h2 = h3 = 1-u~ Notes [1] For spherical coordinates in n dimensions, we have the mapping {Xi I i = 1, •.• ,n} --+ {r,8i Ii = 1, ... ,n-1} via Xl = r sin 81 sin 82 ... sin 8n-2 sin 8n-1 X2 = r sin 81 sin 82 ... sin 8n-2 cos 8n-l X3 = r sin 81 sin 82 ... cos 8n-2 Xn-l ~ r sin 81 cos 82 Xn = rcos81. Hence, the Jacobian is given by J(r, 8) = (_1)n(n-l)/2 rn-l sinn-2 81 sinn-3 82 ... sin28n_1 sin8n-2. [2] The following two changes of variables are frequently useful: lP !(x) dx = (P -a) I.' !(IP -aJt + a) dt /.00 (0: + f3t) dt = (f3 -0:) 0 f '"'l'+t (1 + t)2· [3] The easiest derivation of the value of I = J~oo e-z2 dx is obtained from a clever change of variables. Multiplying I by itself results in [2 = ([: .-.' dx) ([: .-.' dY) /.00 /.211' = 0 0 e-r2 rdOdr = (/.oc .-r' rdr) ([" dO) _ (_! _r2Ir=(0) ( 18=211') _ -2e 8 - 7r. r=O 8=0 where we have changed from rectangular coordinates to polar coordinates via {r2 = X2+y2, tan 8 = x/y}, which introduced the Jacobian J = r. From this we conclude that I = Vi. 116 III Exact Analytical Methods [4] As Flanders [1] shows, there is another application of the transformation in the last note to the Fresnel integrals. Define F(t) = f.~ ,-.z· co..,' dz, G(t) = f.~ ,-tz' sin",' dz so that the Fresnel integrals are given by Fo = F(O) = Jooo cosx2 dx and Go = G(O) = Jooo sin x2 dx . We now compute F'(t) -G'(t) = (f.~ ,-1%' cos",' dz ) (f.~ ,-•• ' COS 1/' dl/) -(f.~ ,-'z' sin ",' dz) (f.~ ,-•• ' sin y' dy ) = f f e-t(:r:2+fl2 ) [cosx2 cosy2 -sinx2 siny2] dxdy :r:~o.tI~O = f.~ f.<I', -.r' (cos r')r dr dO = - r coszdz = ---1r f.oo -tz 1r t 4 0 4 1 + t2' (28.8) where the substitutions z = r2 = x2 + y2 were used. Similarly, it is possible to derive that Taking the limit t --. 0 in both (28.8) and (28.9) results in the simultaneous equations {FJ -G~ = 0, 2FoGo = 1r/4}. It is easy to show that Go > 0, so we conclude that Fo = Go = J 1r /8. [5] A precise statement of when a change of variables can be performed is contained in the following theorem: Theorem: Let V be an open set of RP and u a diffeomor­ phism of V onto u(V) (Le., u is a bijection of V onto u(V) such that u and u -1 are continuously differentiable). Then the formula 1 J(x)dx= f J(u(x»IJ(x)1 dx u(V) Jv is valid in each of the following two cases (see George [2], page 22): (A) J is a measurable positive arithmetic function on u(V); (B) J is an integrable function over V. 29. Computer Aided Solution 117 References [1] H. Flanders, "On the Fresnel Integrals," Amer. Math. Monthly, 89, April 1982, pages 264-266. [2} C. George, Exercises in Integration, Springer-Verlag, New York, 1984. [3} R. G. Helsel and T. Rad6, "The Transformation ofOouble Integrals," funs. Amer. Math. Soc., 54,1943, pages 83-102. [4] P. Moon and O. E. Spencer, Field Theory Handbook, Springer-Verlag, New York, 1961. 29. Computer Aided Solution Applicable to Indefinite or definite integrals. Yields An exact evaluation of an indefinite or definite integral. Idea Some commercial computer algebra packages include a symbolic inte­ grator. Procedure Find an computer system that runs any of the following computer lan­ guages: Axiom [7], Derive [2], Macsyma [4], MAPLE [1], Mathematica [8], or REDUCE [5]. Then learn how to use the language, and find the routine that symbolically evaluates integrals. To compare some of these different languages, a test suite of indefinite and definite integrals was created, and the integration routine from each language was run on the test suite. The indefinite integrals tested were 11: JSinxdx 12: J v'tanxdx 13: j_x_ dx 14: j_x_ dx x3 -1 sin2 x 15: j logx dx 16' j x dx v' x + 1 . v'1 + x + v'l -x 17: J e-a:z:2 dx 18: j+dx log x 19: j si xn2x dx no: j 2 dx . + cos x Note that integrals 17, 18, and 19 are not elementary integrals; their evalua­ tion depends on the error function, the exponential integral, and the cosine integral. Integral 110 is discussed on page 192. The definite integrals tested were 01: 1411' dx o 2+cosx 02: 100 Si: x dx -00 118 III Exact Analytical Methods 100 -z 03: _e_dx oVx 100 2-z X e 04: -2:z: dx o l-e 05: fooo e-:z:21og2 x dx 07: _x_dx 100 2 o 1 +X3 09: floo e-zxl/3 dx. Observe that none of these definite integrals, except integrals D7 and D8, have elementary indefinite integrals. Additionally, note also that in­ tegral D7 is divergent. Note that integral D8, as written, is improper and has no well defined value. If integral D8 were written as a principle value integral, J = f~l x-2 dx, then J = 2. A routine that evaluates definite integrals by first determining the indefinite integral would return this value. Example 1 The following is the result of running the test suite on a 386 computer running Derive. In Derive the integration function has the name INT. The Derive results were obtained by creating an input file, running that file, and then storing the output. Using this procedure, each input line is followed by the corresponding output line. The author changed some of the output appearing below to make it fit on the page better. Derive is intended to be used interactively in a graphical mode, and the graphical display of the results below looks much better than the corresponding printed output. For example, in the graphics mode, the square-root function displays the square-root sign, not the function name SQRT. il=INT(SIN(x),x) U=-CDS(x) i2=INT(SQRT(TAN(x»,x) i2=SQRT(2)*ATAN(SQRT(2)*SQRT(TAN(x»+1)/2+SQRT(2)*ATAN(SQRT(2)* SQRT(TAN(x»-1)/2-SQRT(2)*LN«CDS(x)*(SQRT(2)*SQRT(TAN(x))+l)+SIN(x»/ CDS(x»/4+SQRT(2)*LN«SIN(x)-CDS(x)*(SQRT(2)*SQRT(TAN(x»-1»/CDS(x»/4 i3=INT(x/(xA3-1),x) i3=SQRT(3) *ATAN(SQRT(3) * (2*x+1)/3)/3-LN(xA2+x+l)/6+LN(x -1)/3 i4=INT(x/SIN(x)A2,x) i4=LN(SIN(x»-x*CDT(x) i6=INT(LDG(x)/SQRT(x+1),x) i6=-4*LN«SQRT(x+l)-1)/SQRT(x»+2*SQRT(x+l)*LN(x)-4*SQRT(x+1) i6=INT(x/(SQRT(1+x)-SQRT(1-x»,x) i6=«x+1)A(3/2)-(1-x)A(3/2»/3 i7=INT(#eA(-a*xA2),x) i7=SQRT(pi)*ERF(SQRT(a)*x)/(2*SQRT(a» i8=INT(x/LDG(x)A3,x) i8=-xA2/LN(x)-xA2/(2*LN(x)A2)+2*INT(x/LN(x),x) 29. Computer Aided Solution 119 i9=INT(SIN(x)/x-2,x) i9=INT(COS(x)/x,x)-SIN(x)/x il0=INT(1/(2+COS(x»,x) il0=-2.SQRT(3).ATAN(SIN(x)/(COS(x)+1»/3+2.SQRT(3).ATAN(SQRT(3).SIN(x)/ (3. (COS(x)+1»)/3+SQRT(3).x/3 dl=INT(1/(2+COS(x»,x.O.4.pi) dl=4.SQRT(3).pi/3 d2=INT(SIN(x)/x.x.-inf.inf) d2=INT(SIN(x)/x,x,-inf,inf) d3=INT(le-(-x)/SQRT(x),x,O,inf) d3=INT(le-(-x)/SQRT(x),x,O,inf) d4=INT(x-2.'e-(-x)/(1-le-(-2.x»,x,O,inf) d4=INT(x-2.'e-x/(le-(2.x)-1),x,O,inf) d5=INT(le-(-x-2).LOG(x)-2,x,O,inf) d5=INT(le-(-x-2).LN(x)-2,x,O,inf) d6=INT(x-3.#e-(-x-2).LOG(x)-2,x,1,inf) d6=INT(x-3.#e-(-x-2).LN(x)-2,x,1,inf) d7=INT(x-2/(1+x-3).x.O.inf) d7=inf d8=INT(1/x-2.x.-l.l) d8=-2 d9=INT(le-(-x).x-(1/3).x.l.inf) d9=INT(le-(-x).x-(1/3).x.l.inf) In the above the following special notation has been used: inf for 00 and #e is used to specify the base of natural logarithms e ~ 2.718. From the above we observe that Derive could not evaluate the following integrals from the test suite: D2, D3, D4, D5, D6, and D9. Derive did not recognize the cosine integral in 19, but left the answer in terms of this special function. Finally, Derive made an error on integral D8 by not identifying the singularity at x = O. Example 2 The following is the result of running the test suite on a SP ARCstation running Macsyma. In Macsyma the integration function has the name integrate. Note that (el), (C2), ... are input lines ("command" lines) and that (01), (D2), ... are output lines ("display" lines). The symbol Y. refers to the last expression. The author changed some of the output appearing below to make it fit on the page better. (Cl) 11: integrate( sin(x). x); (Dl) -eOS(x) 120 III Exact Analytical Methods (C2) 12: integrate( sqrt(tan(x», x); LOG (TAN (X) + SQRT(2) SQRT(TAN(X» + 1) (D2) 2 (---------------------------------------4 SQRT(2) LOG(TAN(X) -SQRT(2) SQRT(TAN(X» + 1) + --------------------------------------4 SQRT(2) 2 SQRT(TAN(X» + SQRT(2) 2 SQRT(TAN(X» -SQRT(2) ATAN(------------------------) ATAN(------------------------) SQRT(2) SQRT(2) + ------------------------------ + ------------------------------) 2 SQRT(2) 2 SQRT(2) (CS) IS: integrate( x/(x-S-1),x); 2 X + 1 2 ATAN(-------) LOG(X + X + 1) SQRT(S) LOG(X -1) (DS) ---------------- + ------------- + ----------6 SQRT(S) S (C4) 14: integrate( x/sin(x)-2, x); (04) x LOG(SIN(X» ---~~(X) (C6) 16: integrate( log(x)/sqrt(x+1), X)i LOG(SQRT(X + 1) + 1) (D6) 2 (SQRT(X + 1) LOG(X) - 2 (--------------------- 2 LOG(SQRT(X + 1) -1) + -------------------- + SQRT(X + 1») 2 (C7) 17: integrate( exp(-a*x-2), x); SQRT(y'PI) ERF(SQRT(A) X) (D7) 2 SQRT(A) (C8) 18: integrate( x/(log(x»-S, x); 1 [ X (D8) I ------- dX ] S 1 LOG (X) (C9) 16: integrate( x/(sqrt(1+x)+sqrt(1-x»,X)i 1 [ X (D9) 1 ------------------------- dX ] SQRT(X + 1) + SQRT(1 -X) 1 (C10) block([algebraic:true],ratsimp(first(y'»); SQRT(1 -X) -SQRT(X + 1) (D10) 2 29. Computer Aided Solution (Cll) integrate(Yo.x); 3/2 3/2 2 (X + 1) 2 (1 -X) 3 (011) 2 (C12) 110: integrate( 1/(2+cos(x».x); SIN (X) 3 2 ATAN(--------------------) SQRT(3) (COS(X) + 1) (D12) SQRT(3) (C13) 19: integrate( sin(x)/xA2. x); 1 [ SIN (X) (013) I ------dX ] 2 1 X (C14) Dl: integrate( 1/(2+cos(x».x.0.4*Yopi)j 4 SQRT(3) YoPI (014) 3 (C15) D2: integrate( sin(x)/x.x.-inf.inf)j (D15) YoPI (C16) 07: integrate( xA2/(1+xA3).x.0.inf)j Error: Integral is divergent (C17) D3: integrate( exp(-x)/sqrt(x).x.O.inf)j (D17) SQRT(Y~I) (C18) D5: integrate( exp(-xA2)*log(x)A2.x.0.inf)j 2 5/2 SQRT(YoPI) LOG (2) SQRT(YoPI) YoGAMMA LOG(2) YoPI (018) ----------------- + ----------------------- + ------ 2 2 ~ 2 YoGAMHA SQRT(YoPI) + ----------------- (C19) D6: integrate( exp(-x)*log(x)A2*xA3.x.l.inf); I~ 1 [ 3 - X 2 (019) I X Y.E LOG (X) dX ] 1 1 (C20) D9: integrate( exp(-x)*xA(1/3).x.1.inf)j 4 (D20) GAMMA(-. 1) 3 8 121 122 III Exact Analytical Methods (C21) 04: integrate( exp(-x)*x~2/(1-exp(-2*x».x.O.inf); 2 X X LOG(Y.E + 1) X X (021) (limit ---------------- -LI (y~ ) + X LI (Y.E ) X -> INF 2 3 2 2 X X X X LOG(l -Y.E ) 7 ZETA(3) + LI (-Y.E ) -X LI (-Y.E ) + ---------------) + --------- 3 2 2 4 (C22) 08: integrate( 1/xA2.x.-l.l); Error: Integral is divergent In the above the following special notation has been used: ERF for the error function, %GAMMA for Euler's constant, GAMMA 0 for the incomplete gamma function, LI for the logarithmic integral of different orders, %E for the base of natural logarithms e, and %PI for 1r. From the above we observe that Macsyma could not evaluate the following integrals from the test suite: 18, 19, and D6. The result for 110 is correct, but only for a limited range. Macsyma could evaluate integral 16 only after the integrand was simplified. Macsyma left the result for integral D4 in the form of a limit. Finally, Macsyma successfully identified the singularity at x = 0 in integral D8. Example 3 In MAPLE the integration function has the name into The MAPLE results were obtained by creating an input file, running that file, and then storing the output. In this case ali of the input is followed by all of the output. The author changed some of the output appearing below to make it fit on the page better. MAPLE is intended to be be used interactively, and the graphical display of the results below looks much better than the corresponding printed output. « INPUT » 11 := int( sin(x). x) 12 := int( sqrt(tan(x». x) 13 := int( x/(xA3-1). x) 14 := int( x/sin(x)A2. x) 15 := int( log(x)/sqrt(x+l). x) 16 := int( x/(sqrt(l+x)+sqrt(l-x». x) 11 := int( exp(-a*x~2). x) 18 := int( x/(log(x»~3. x) 19 := int( sin(x)/x~2. x) 110:= int( 1/(2+cos(x». x) 01 := int( 1/(2+cos(x». x=0 .. 4*Pi) 02 := int( sin(x)/x. x=-infinity .. infinity) 03 := int( exp(-x)/sqrt(x). x=O .. infinity) D4 := int( exp(-x)*x~2/(1-exp(-2*x». x=O .. infinity) 05 = int( exp(-x~2)*log(x)~2. x=O •. infinity) 06 = int( exp(-x)*log(x)A2*x~3. x=l .. infinity) 01 = int( x~2/(1+xA3). x=O .. infinity) 08 = int( 1/x~2. x=-l .. l) 08B = int( 1/x~2. x=-1 .. 1. continuous) 09 = int( exp(-x)*xA(1/3). x=l .. infinity) 29. Computer Aided Solution 123 «OUTPUT» 11 := -cos (x) / 1/2 1/2 1/2 1/2 \ I 2 tan(x) tan (x) + 2 tan(x) + 1 I 1/2 12 := 1/2
arctan(--------------) -In(--------------------------~)I 2 I 1 -tan(x) 2 1/2 I \ (1 + tan (x) ) / 2 1/2 1/2 13 := 1/3 In(x-l) -1/6 In(x + x + 1) + 1/3 3 arctan(1/3 (2 x + 1) 3 ) 14 := -x cot(x) + In(sin(x» 16 := 1/2 1/2 1/2 1/2 2 (x+1) In(x) - 4 (x+l) - 2 In((x+l) -1) + 2 In((x+1) + 1) 3/2 3/2 16 := 1/3 (x + 1) -1/3 (x -1) I 1/2 1/2 Pi erf(a x) 17 := 1/2 -----------------1/2 a 2 2 x x 18 := -1/2 ------ ------+ 2 Ei(2 In(x)) 2 In(x) In(x) sin (x) 19 := ------- + Ci(x) x 1/2 1/2 110 := 2/3 3 arctan(1/3 tan(1/2 x) 3 ) 4 Pi / I 01:= I 1 ---------- dx 02 := I 2 + cos (x) / o infinity / I sin (x) I ------dx I x / -infinity 1/2 03 := Pi 04 := 7/4 Zeta(3) 124 III Exact Analytical Methods 5/2 1/2 2 1/2 1/2 2 05 := 1/16 Pi + 1/8 Pi gamma + 1/2 Pi gamma In(2) + 1/2 Pi In(2) 06 := 2 MeijerG(4, 4, 1) 07 := infinity 1 I I 1 08 := I ----dx I 2 I x -1 D8B := -2 D9 := GAMMA(4/3, 1) In the above the following special notation has been used: Ci for the cosine integral, Ei for the exponential integral, erf for the error function, gamma for Euler's constant, GAMMA 0 for the incomplete gamma function, infinity for 00, MeijerG for Meijer's G function, Pi for 1(", and Zeta for the zeta function '(x). From the above we observe that MAPLE could not evaluate the follow­ ing integrals from the test suite: Dl and D2. The result for 110 is correct, but only for a limited range. For integral D8, MAPLE recognized that the integrand was discontinuous and so did not evaluate it. When told that the integrand was continuous (which is incorrect), MAPLE then obtained the answer that would be obtained by just following symbolic rules. Example 4 In Mathematica the integration function has the name Integrate. The following Mathematica results were obtained by creating an input file, running that file, and then storing the output. This output was then manipulated so that each input line is followed by the corresponding output line. Mathematica for windows, the version run for the test suite, is intended to be used interactively in a graphical mode, and the graphical display of the results below looks much better than the corresponding printed output. 11: Integrate[ Sin [x] , x] I1:-Cos[x] 12: Integrate[ Sqrt[Tan[x]], x] 12: Integrate [Sqrt [Tan [x]] , x] 13: Integrate[ x/(xA3-1) ,x] 1 + 2 x ArcTan [-------] 2 Sqrt[3] Log[-l + x] Log[l + x + x ] 13:--------------- + ----------- ----------------Sqrt[3] 3 6 29. Computer Aided Solution 14: Integrate[ x/Sin[x)~2. x) 14:-(x Cot [x) + Log[Sin[x)) 16: Integrate[ Log[x)/Sqrt[x+l). x) 16:-4 Sqrt[l + x) + 4 ArcTanh[Sqrt[l + x)) + 2 Sqrt[l + x) Log[x) 16: Integrate[ x/(Sqrt[l+x)+Sqrt[l-x).x) 1 x 1 x 16:Sqrt[1 -x) (---) + (-+ -) Sqrt[l + x) 11: Integrate[ Exp[-a.x~2). x) 3 3 3 3 Sqrt [Pi) Erf [Sqrt [a) x) 11:----------------------- 2 Sqrt[a] 18: Integrate[ x/(Log[x)~3. x) 2 2 x x 18:2 ExplntegralEi[2 Log[x)) ---------- - 19: Integrate[ Sin[x)/x~2, x) Sin [x) 19:Coslntegral[x) ------- x 110: Integrate[ 1/(2+Cos[x),x) x Tan[-) 2 2 ArcTan[-------) Sqrt[3) 110:----------------- Sqrt[3) 2 Log [x) 2 Log[x) Dl: Integrate[ 1/(2+Cos[x), {x,O,4 Pi}] 4 Pi Dl:------- Sqrt[3) D2: Integrate[ Sin[x)/x, {x.-Infinity,Infinity}] D2:Pi D3: Integrate[ Exp[-x)/Sqrt[x), {x.O.Infinity}] D3:Sqrt [Pi) D4: Integrate[ Exp[-x).x~2/(1-Exp[-2.x), {x,O,Infinity}] 1 Zeta[3] D4:--------- 4 D6: Integrate[ Exp[-x~2).Log[x)~2. {x,O,Infinity}] 2 2 D6:(Sqrt[Pi] (2 EulerGamma + Pi + 8 EulerGamma Log[2] + 2 8 Log[2) » / 16 125 126 III Exact Analytical Methods 06: Integrate[ Exp[-x]*Log[x]~2*x~3, {x,1,Infinity}] 06: Indeterminate 07: Integrate[ x~2/(1+x~3), {x,O,Infinity}] 07: Infini ty 08: Integrate[ 1/x~2, {x,-l,l}] 08:-2 09: Integrate[ Exp[-x]*x~(1/3), {x,1,lnfinity}] 4 4 09:Gamma[-] -Gamma[-, 0, 1] 3 3 In the above the following special notation has been used: Coslnte­ gral for the cosine integral, Erf for the error function erf, ExplntegralEi for the exponential integral E1(x), EulerGamma for Euler's constant '1, GammaO for the gamma function r(a) = fooo ta-1e-t dt and for the gen­ eralized incomplete gamma function r(a, b, c) = fbc ta-1e-t dt, Pi for 1f', and Zeta for the zeta function ((x). From the above we observe that Mathematica could not evaluate the following integrals from the test suite: 12 and D6. Additionally, the result for 110 is correct, but only for a limited range. Perhaps because of this error, the result for integral D1 is incorrect. Finally, Mathematica made an error on integral D8 by not identifying the singularity at x = o. Example 6 Since REDUCE cannot perform definite integration, that part of the test suite was not run in REDUCE. The following is the result of running the indefinite integrals in the test suite on a SPARCstation 2. In REDUCE the integration function has the name into In this listing, each input line is followed by the corresponding output line. i1 := int( sin(x), x); 11 := -COS (X) i2 := int( sqrt(tan(x», x); 12 := INT(SQRT(TAN(X»,X) i3 := int( x/(x~3-1),x); 13 := 1 2*X + 1 1 2 1 ---*SQRT(3)*ATAN(---------) ----*LOG(X + x + 1) + ---*LOG(X -1) 3 SQRT(3) 6 3 i4 := int( x/sin(x)-2, x); -1 X 2 X 14:= -SIN (X) *COS(X)*X -LOG(TAN(---) + 1) + LOG(TAN(---» 2 2 29. Computer Aided Solution is := int( log(x)/sqrt(x+l). x); IS := 2*(SQRT(X + l)*LOG(X) -2*SQRT(X + 1) -LOG(SQRT(X + 1) -1) + LOG(SQRT(X + 1) + 1» 1 16 := INT(-------,X) 2 i7 := int( x/(log(x»A3, x); X -1 2 1 -2 2 17 := 2*INT(--------,X) -LOG(X) *X ----*LOG(X) *X LOG(X) 2 i8 := int( x/(sqrt(l+x)+sqrt(l-x»,x); 1 1 18:= ----*SQRT( -x + l)*X + ---*SQRT( -X + 1) 3 3 1 1 + ---*SQRT(X + l)*X + ---*SQRT(X + 1) 3 3 i9 := int( 1/(2+cos(x».x); x TAN(---) 2 2 19 := ---*SQRT(3)*ATAN(----------) 3 SQRT(3) itO:= int( sin(x)/x-2, x); SIN (X) 110 := INT(--------.X) 2 x 127 In the above the following special notation has been used: E for the base of natural logarithms e. From the above we observe that REDUCE could not evaluate the following indefinite integrals from the test suite: 12, 17, and 19. 128 III Exact Analytical Methods Notes [1] The version numbers of the computer languages used above were Derive 2.08, MACSYMA 417.100, MAPLE 5.0, Mathematica 2.1, REDUCE 3.4. [2] All of the above computer packages can perform operations on integrals that the system cannot simplify. That is, if a system cannot simplify the integrals I = IX J(x) dx or J = I: g(x) dx, it will still be able to differentiate I and numerically evaluate J. [3] In DERIVE, presently, the only definite integrals that can be computed are those for which an indefinite integral can be computed. [4] Sometimes a symbolic algebra system can determine a specific integral, yet it does not do so without coaching because the result is a mess. For example, the integral I dx / (x3 + X + 1) can clearly be integrated (the roots of a cubic can be found analytically, then partial fraction decomposition can be used). Yet Macsyma will not, unaided, carry out these steps. See Golden [3]. [5] Packages that can handle a wider variety of integrals are constantly being created. The theory underlying the algorithms is described in the section on Liouville theory (see page 77). [6] One must be wary when using computer algebra packages, as errors can sometimes arise. For example, the integral I = I xk dx is problematic since there are two different forms for the answer depending on whether or not k is equal to -1. Some systems will return Xk+1/(k + 1) without any further information; others will ask the user whether or not k is equal to -1; still others will look to explicitly declared domains to determine if k could be equal to -1. If Derive cannot determine if k could be -1, then it returns the ex­ pression (xk+l -1) /(k + 1) for the evaluation of I (see Stoutemyer [6]). This expression tends to log x as k --+ -1. Hence, this expression gives the correct result for any specific value of k, provided that a limit is used instead of simple substitution. [7] Most of the commercial computer algebra packages also include a numeric integrator. [8] Thanks are extended to Jeffrey Golden for running the test suite in Mac­ syma, Tony Hearn for running the test suite in REDUCE, and Richard Pavelle for running the test suite in MAPLE. References [1] B. W. Char, K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan, and S. M. Watt, MAPLE V Library Reference Manual, Springer-Verlag, New York, 1991. [2] DERIVE, Soft Warehouse, Inc., 3660 Waialae Avenue, Suite 304, Honolulu, HI 96816. [3] J. P. Golden, "Messy Indefinite Integrals," MACSYMA Newsletter, 7, No.3, Symbolics, Inc., Burlington, MA, July 1990, pages 9-17. [4] VAX UNIX MACSYMA Reference Manual, Symbolics Inc., Cambridge, MA, 1985. [5] G. Rayna, REDUCE: Software for Algebraic Computation, Springer-Verlag, New York, 1987. 30. Contour Integration 129 {61 David R. Stoutemyer , "Crimes and Misdemeanors in t.he Computer Algebra Trade," Notices oj the AMS, 38, No.7, September 1991, pages 778-785. [7] R. Sutor (cd.), Axiom User Guide, Numerical Algorit.hms Group, Inc., 1400 Opus Place, Suite 200, Downers Grove, IL, 1991. [8} S. Wolfram, Mathernatica: A System oj Doing Mathemat ics by Computer, Second Edition, Addison-Wesley Publishing Co., Reading, MA, 1991. 30. Contour Integration Applicable to integrals. Contour integrals, or integrals reducible to contour Yields An exact evaluation in terms of residues. Idea A definite integral can sometimes be written as a contour integral. Cauchy's residue theorem may then be used to evaluate the contour integral in terms of residues. Procedure Often, a definite integral can be manipulated into a contour integral. A contour integral is an integral for which the integration path is a closed contour in the complex plane. Contom integrals can often be evaluated by using Cauchy's residue theorem. One statement of the theorem is Theorem : Let J(z) be analyt.ic in a simply connected do­ main D except for finitely many points {C'tj} at which J may have isolat.ed singularities. Let. C be a simple closed contour, traversed in the positive sense (see Notes, below), that lies in D and does not pass through any paint C'tj. Then 1 f(z) dz = 2".; I: residue of J at ai where the sum is extended over all the points {C'tj} that are inside the contour C (see Figure 30.1). (30.1) If J(z), when expanded in a Laurent series about the point a, has the form ( ) a_m a_m+1 a_I () J z = ( )'" + ( )m I + ... + ( ) + 9 a Z-Q z-a Z-Q (30.2) where g(z) is analytic at a, then the residue of J at the point a is the term a-I. A straightforward way to determine this term, when J has the expansion indicated in (30.2), is by 1 d"-l I a-I = (n _ 1)! dzn I [(z -a)n J(z)] '=0 (30.3) 130 III Exact Analytical Methods X A I X I X, c x x ----,---- -> Figure 30.1 A contour C that contains a singularit.y. The singularities are indicated by crosses. where n ~ m. For computational ease, though, it is best to take n = m if the value of m is known. If the contour C passes through a simple pole, then the contour can sometimes be deformed around the pole and then the following theorem can be used: Theorem: Let J(z) have a simple pole at z = a with residue a_I and let A = A(e,1» be an arc of the circle Iz -"I = e that sublends an angle ¢ at the center of the circle. Then lim 1. fez) dz = i1>a-l. c-o+ ~ If we had integrated over the entire circle (i.e., <P = 211') then the residue theorem states that the answer would be 27ria_l1 without any limiting process involved. This theorem indicates that integrating over a fraction of the circle yields a corresponding fraction of 211'ia_" provided the pole is simple (i.e., m = 1 in (30.2)) and the limit E -+ 0 is taken. There are many examples in this section, and each onc attempts to demonstrate a different method or technique that can be used when evaluating contour integrals. Example 1 Consider the integral 12" dO I -:;----;: -0 2+cosO' (30.4) If we make the change of variable z = eW then dz = ieiO dO or dO = -iz-1 dz. Hence, using cosO = ~ (eiO + e-iO) = ~ (z + Z-l), we can write (30.4) as I -J _iz-1 dz -2+i(z+z 1) 'J 2 = -, dz. Z2 + 4z + 1 (30.5) 30. Contour Integrat ion 0_ 0+' -~----------~r- - I 131 --i> d+ = -2 +-V3: _ ().u;a oC ~ _ '2 -{3 = -".-J;;2 Figure 30.2 Contour of integration for Example 1. The singularities of the integrand are indicated by crosses. I I I , (Awn te."cloGk{.jl~ Now we must determine what the new range of integr'iOn is. As () varies from 0 to 27r, z traces out a circle of radius one in the"':"clockwise sense (see Figure 30.2). Hence, we can use the Cauchy residue theorem. If we write las '1 2 1 = -, dz, c (z -",+)(z -"'_) (30.6) where C represent s the contour in Figure 30.2, and "'± = -2 ± J3, then we see that only "'+ is inside of the contour C (the "'± are indicated on Figure 30.2). The Laurent series of the integrand about the point "'+ is 2 = ( 2 ) _1_ + h(z), (z-",+)(z-",_) "'+ -"'_ z-",+ (30.7) 2 where h(z) = ( )( ) is analytic at z = "'+. Hence, the residue 0_ -0+ z-o_ of the integrand at "'+ is equal to 2/(",+ -",_). Using this in (30.1) results in I = -i x 27ri L residues of integrand = 2rr C+ ~ "'_) = 2rr (-2 + J3) ~ (-2 _ J3) (30.8) 132 '" I I I __ --..I~) __ - - - ~ ICR R -R I III Exact Analytical Methods '" I ~R I • - - - -;' - -~ -R ICR R I Figure 30.3 (a) Integration contour for (30.9); (b) integration contour for (30.10). Example 2 Consider the integral J = 100 ;i:e dx = lim f 2eiZ dz, -00 x + 1 R-oo lCa z + 1 (30.9) where the contour CR is shown in Figure 30.3.a. To have a closed contour, we define the new integral K = (f + f ) +-dz, lCa lva z + 1 (30.10) where the contour VR is shown in Figure 30.3.b. The only isolated singu­ larity inside of the combined contour CR U VR is at z = i, and the residue at this pole is e-I/2i. This residue is found by using (30.3) with n = 1: a_I = 01, lill1: (z -i) 2eiZ ) = lill1: ( eiz .) = e2-.I . (30.11) . z-, Z + 1 z-, z + ~ ~ eiz 1 (e-l) 2el+iz + i{z + i) That is, -2--= --. -2' + h{z) where h{z) = (2) is z + 1 z -~ ~ 2e z + 1 analytic at z = i. Therefore, using the residue theorem, we have K = 21ri (e-I/2i) = 1re-l. Now, how does K (what we have calculated) relate to J (what we want to find)? Observe that I f ~dzl < f I~dzl lDa z2 + 1 -lDa Z2 + 1 1 leizl ~ I 2 Ildzl Da z +1 () (30.12) ~ m~ I 2 1 I ( Ie" I Idzl z=Re,8 Z + 1 lDa ~ (.~::'·IZ2 ~ 11) ,,~ (IRL 11) ", 30. Contour Integration -R -e A I -'--~ __ -L __ ~ R Figure 30.4 Integration contour for (30.14). 133 where we have used Jordan's lemma (see Notes, below). Hence, as R --+ 00, the integral I'D R vanishes and we have J = lim f = lim (K -1 ) = K = 7re-1, R-oo lCR R-oo 'DR (30.13) where the integrand is not displayed when it is understood to be the same as in (30.9). Using (30.9) in (30.13), and taking the real and imaginary parts, results in the two integrals 100 sinx d -0 2 x-, -00 x + 1 100 cosx d _ -1 -2-- X -7re . -00 x + 1 The first of these two integrals could have been obtained immediately by symmetry considerations. Example 3 100 sin x 100 eiz Consider the integral I = --dx. If we define J = - dz, -00 x -00 z then I = 1m J. Now consider the following contour integral (see Fig- ure 30.4): (1 1 1) eiz K = lim + + - dz. ~::~ C(e,R) Cc CR Z (30.14) Here, CR is a semicircular arc of radius R, Ce is a semicircular arc of radius e, and C(e, R) is the union of two intervals: C = [-R, -e] U fe, R). Since there are no singularities of the integrand within the contour C(e, R) U Ce U CR, we find that K = O. Now we will determine each of the integrals in (30.14). We immediately recognize that J = lim :..-dz. By Jordan's lemma, it is easy to see liz ~::~ C(e,R) Z that limR_oo Ic = O. The integral lim ~c (eiz / z) dz can be evaluated R e-O c 134 III Exact Analytical Methods l' I z = rei(21r-c5) Figure 30.5 Contour used in (30.15). The thick line represents the branch cut, and the crosses indicate the location of the poles. by using the second theorem in the introductory remarks. We note that the angle is tP = -1r (the angle is negative since it is being traversed in the negative direction) and that there is a simple pole at z = 0 with residue a_I = 1. Hence, lim I.c (eiz /z) dz = i(-1r)(I) = -i1r. Combining €-o e everything we have, we find J = lim f = K -lim (f + f ) = 0 -(0 -i1r) = i1f', ~:.: lc(€,R) ~:.: lCe lCR where the integrand is not displayed when it is understood to be the same as in (30.14). Taking the imaginary parts of this expression, we determine that 1= ImJ = 1r. Example 4 Consider the integral I = 1.00 3dx . We first consider, instead, the o x + 1 integral K _110gZ d - 3 Z, C Z + 1 (30.15) and will then determine I from K. For K the integration contour is given by C = CR U C€ U Cud U Cld, where "ud" and "ld" stand for the upper diagonal branch and lower diagonal branch, respectively, in Figure 30.5. Since K involves a logarithm, a branch cut in the complex plane is needed; Figure 30.5 shows the location of the branch cut. With this branch cut we have log z = log Izl + i arg z where 0 < arg z < 21r. 30. Contour Integration 135 The only poles in C are at the the cube roots of -1, {ZI' Z2, Z3}, and it is easy to derive that 3 [ l~g Z dz = 21ri L residue at z; lc z + 1 ;=1 = 41r2iV3 = [ + [ + [ + [ , 9 lCn lCe lCud lCld (30.16) where the integrand is not displayed when it is understood to be the same as in (30.15). In the limits R -+ 00 and c -+ 0 the integrals along the two circular arcs vanish, observe: • lR = 0 C,,!3R ) O(R) --+ 0 as R --+ 00; • fcc = O(logc)O(c) -+ 0 as c -+ O. Taking the limits R -+ 00 and c -+ 0 in (30.16) results in If we now take the limit 0 -+ 0 in this last integral, we obtain ---= lim . 3 e'O dr + 3 e,(21r-0} dr 41r2i v'3 {I.OO log (reiO). 10 log (rei(21r-0}). } 9 0-0 0 1 + (re'O) 00 1 + (rei(21r-0}) 1.00 log r d 10 log r + 21ri d = --r+ r o 1 + r3 00 1 + r3 10 21ri = --3 dr = -21riI. ool+r From this we conclude that 1= 21r/3.../3. 136 III Exact Analytical Methods " 1 CR 1 R .. 1··································)1 1 1 - - ---> Figure 30.6 Contour used in (30.17). The thick line represents the branch cut, and the cross indicates the location of the pole. Example 5 Consider the integral I = l' dx~. We first consider the -1 (ax + b) 1 -x2 integral K-f dz -lc(az+b)~ (30.17) and will determine I from K. In K, the contour C is given by C = C R U Cr U Cel U C_ U Ce2 U C_ U C!, where the different components should be clear from Figure 30.6. (Note that C_ is along the bottom of the cut, and has two components.) Since K involves a square root, branch cuts are present and we must define how the square root is to be evaluated. We define the function (Z2 _1)1/2 to be real and positive for Re z > 1, and we take z + 1 = Tl ei61 and z -1 = T2ei62 with 0 < 8j < 211'; see Figure 30.7. With these definitions, along the real axis between -1 and 1 we have (Z2 _1)1/2 = VI -x2ei(61 +62)/2. On the top of the cut (i.e., along C_) fh = 0 and 82 = 11'; on the bottom of the cut (i.e., along C_) 81 = 211' and 82 = 11'. The only pole in C is at z = -bfa, so that K = { d~ = 2,..i (residue at z = -~) = lc (az + b) z2 -1 a -/b2 -a2 (30.18) In the limits R --+ 00 and ei --+ 0 the integrals along the bounding box and the two circular arcs vanish. Observe: 30. Contour Integration 137 I :~z ---A~-:Le~---1 I 1 Figure 30.7 How the square root function is defined in (30.17). · 1. = 0 U2 ) O(R) -+ 0 as R -+ 00; · 1 .. = 0 (~) 0(0;) -+ 0 as 0; -+ O. If we allow the two vertical contours to come together, then the integrals along Cr and C l cancel, since the integrand is continuous between these contours and the contours are going in opposite directions. We presume that this limit has been taken, so that C_ is now continuous along the bottom of the cut. Taking the R ~ 00 and Ci ~ 0 limits in (30.17), using (30.18), and disregarding the Cr and C l integrals results in 27ri l' {1 1} dz - = 1m + v'b2 -a2 ei-O c_ c_ (az + b).Vz2=1 11 dx r-1 dx = -1 (ax + b)Vl -x2e1ri/2 + J1 (ax + b)Vl -x2e31ri/2 = (11 dx ) (-2i) = -2iI. -1 (ax + b)Vl -x2 We conclude that I = 7r / ~b2--a2• Example 6 The Laplace transform of the function J(t) is the function F(s) = C[J(t)] = fooo e-Bt J(t) dt. As a rule, F(s) will converge for some region in the complex s-plane, typically Res> 0, where 0 is some real constant. The inverse Laplace transform is given by J(t) = C-1[F(s)] = ~ r etB F(s) ds, 27rzJc where the contour of integration is the vertical line z = u + iy (with -00 < y < 00), and u is "to the right" of any singularities of F(t). Often, the contour in this integral can be made into a closed contour by closing the contour to the left; this results in the Bromwich contour, C U CR, shown in 138 III Exact Analytical Methods l' (J + ioo CR C ---> (J -ioo Figure 30.8 Bromwich contour used in inverse Laplace transforms. Figure 30.B. With a closed contour, of course, Cauchy's theorem can be used. Consider the inverse Laplace transform of the function F( 8) = 1/ (82 + 4). This function has singularities at 8 = ±2i, so we must take (J > O. Defining a new integral with the Bromwich contour K(t) = (f + f) +-d8, JCR Jc 8 + 4 we note that the only poles are at 8 = ±2i. Therefore, the residue theorem yields K(t) = 27ri [ (residue at s = 2i) + (residue at s = -2i)] = ~ sin 2t. The difference between f(t) (what we want) and K(t) (what we have calculated) is the integral Ic R • As we now show, this integral vanishes as R --. 00. Let x = Re 8 along CR' Then, for positive t, we have letBI < etz < etu along CR. Hence, 1 etB (etU) -2-d8 = 0 2' O(R) -+ 0 CR8 +4 R as R -+ 00, since (J is fixed. We obtain the final result f(t) = ~ sin2t. 30. Contour Integration 139 Notes [I) (2) Jordan's lemma states: If Cn is the contour z = Rei9, 0 ::; 0 :5 71", then feR leiZljdzj < rr (see Levinson and Redheffer 17l). This is often useful for estimating integrals. The integral in Example 1 had the form 1= Io2 r. J(cosO,sinO) dO. Integrals of this form can be recast as contour integrals by setting z == Ci8. As 0 varies from 0 to 2rr, z traverses a unit circle C counterclockwise. Hence, I=lJ(~,~) dz. c 2z 2zz z [3] Loop integrals are contour integrals in which the path of integration is given by a closed loop. For example, Hankel's representation of the Gamma function when Rex> 0 is given by r(x) = . 1 l(-t)Z-le-t dt 21 Sill loX c where the contour C lies in the complex plane cut along the positive real axis, starting at 00, going around the origin once counterclockwise, and ending at 00 again. [4] In the above theorems , the contours were sometimes described as being "traversed in the positive sense. II Consider a closed region surroundin g the contour C(t). Consider the vector giving the direction of increasing tj if a counterclockwise rotation of rr /2 causes the vector to point into the interior of the closed region, t.hen the contour C is being traversed in the positive sense and the region is positively oriented. [5] Plaisted [91 shows that the following problem involving the evaluation of a contour integral is NP-hard: Given integers band N and a set of k sparse polynomial s with integer coefficients {Pi(Z)}, it is NP-hard to determine whether the following contour integral is zero: 1 n;~ll};(Z) (I -~N) zN -bN dz C z -1 z z-b where C is any contour including the origin in its interior. See also Garey and Johnson (5). !6] Gluchoff [61 presents the following interpretation of a contour integral: 1 J(z) dz = L(C) ,~Vc lJ(z)T(z)) (30.19) where L(C) is the length of C, T(z) is the unit tangent to C at z, and "av" denotes t.he averaging function. For example, tor the integral I == ~:::I=n dz/z we identify L(C) = 2nR and T(z) = izl R so that (30.19) becomes 1 dz [liZ]. -= 21rR av --R = 2". av [I) = 2"i. 1:::I=n z Izl=R Z Izl=R 140 III Exact Analytical Methods [7] Several of the references describe quadrature rules for numerically approxi­ mating a contour integral. [8] The numerical integrator in the language Mathematica allows an arbitrary path in the complex place to be specified. For example, consider the integral in (30.5). The integration contour may be deformed to the square with vertices at {±1, ±i} without changing the value of the integral. If Mathe­ matica is input "NIntegrate [-2I/(z"2+4z+1) ,z, 1, I, -1, -1,1]", then the result is 3.6276 + o. I, which is the correct numerical value. (Here I is used to represent i.) [9] The theory of multidimensional residues is based on the general Stokes formula and its corollary, the Cauchy-Poincare integral theorem. Serious topological difficulties arise for analytic functions of several complex vari­ ables because the role of the singUlar point is now played by surfaces in en. These surfaces generally have a complicated structure requiring tools such as algebraic topology. References [1] B. Braden, "Polya's Geometric Picture of Complex Contour Integrals," Math. Mag., 60, No.5, 1987, pages 321-327. [2] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, pages 168-171. [3] D. Elliot and J. D. Donaldson, "On Quadrature rules for Ordinary and Cauchy Principal Value Integrals over Contours," SIAM J. Numer. Anal., 14, No.6, December 1977, pages 1078-1087. [4] R. J. Fornaro, "Numerical Evaluation of Integrals Around Simple Closed Contours," SIAM J. Numer. Anal., 1, No.4, 1973, pages 623-634. [5] M. R. Garey and D. S. Johnson, Computers and Intractability, W. H. free­ man and Co., New York, 1979, page 252. [6] A. Gluchoff, "A Simple Interpretation of the Complex Contour Integral," Amer. Math. Monthly, 98, No.7, August-September 1991, pages 641-644. [7] N. Levinson and R. M. Redheffer, Complex Variables, Holden-Day, Inc., San Francisco, 1970, Chapter 5, pages 259-332. [8] J. N. Lyness and L. M. Delves, "On Numerical Integration Round a Closed Contour," Math. of Comp., 21, 1967, pages 561-577. [9] D. A. Plaisted, "Some Polynomial and Integer Divisibility Problems Are NP-Hard," in Proc. 17th Ann. Symp. on Found. of Compo Sci., IEEE, Long Beach, CA, 1976, pages 264-267. 31. Convolution Techniques Applicable to Some one dimensional integrals that contain a product of terms. Yields An exact evaluation in terms of other integrals. 31. Convolution Techniques 141 Idea If the original integral can be written as a convolution, then the value of the integral may be determined by a sequence of integrals. Procedure Given the functions I(x) and g(x), and an integral operator I[], define the functions F«() = I[/(x)] and G«() = I[g(x)]. For many common integral operators there is a relation that relates an integral of I and g, of a specific form, to an integral involving F and G. These relations are often of the form A [F«()G«()] = J J[/(a(x,())),g(b(x,()),(]h«()d( (31.1) where A[] is an integral operator. Such a relation is known as a convolution theorem. Convolution theorems may sometimes be used to simplify integrals. Given an integral in the form of the right-hand side of (31.1), it may be easier to evaluate the left-hand side of (31.1). In the following examples, upper case letters denote transforms of lower case letters. Example 1 For the Mellin transform, defined by the convolution theorem is 100 (X) du 1 lc+ioo I -g(u)-= -2 • F(s)G(s)x-S ds o u u 1n c-ioo where the integral on the right-hand side is a Bromwich integral. Example 2 For the Laplace transform pair, defined by F(s) = .c[/(t») = 1"0 e-t'/(t) dt, 1 lc +ioo I(t) = .c-1[F(s)] = -2 • ets F(s) ds, 1i'~ c-ioo the convolution theorem is l I(r)g(t -r) dr = .c-1 [F(s)G(s»). 142 III Exact Analytical Methods Example 3 For the Fourier transform pair, defined by 1 100 . F(t) = F[f(x)) = rn= e-stz f(x) dx, v21r -00 1 100 f(x) = F-1[F(t)) = . rn= eitz F(t) dt, v21r -00 the convolution theorem is 1 100 . rn= f(x)g(x -e) de = F-1[F(t)G(t)). v21r -00 Notes [1] Bouwkamp [2] uses the convolution theorem for Fourier transforms to show that 121f 100 re-ar drd8 = 211" [1-aDKo (~aD) 11 (~aD)]. o 0 v'r2 + D2 -2rDcos8 a References [1] A. Apelblat, "Repeating Use of Integral Transforms-A New Method for Evaluation of Some Infinite Integrals," IMA J. Appl. Mathematics, 27,1981, pages 481-496. (2) C. J. Bouwkamp, "A Double Integral," Problem 71-23 in SIAM Review, 14, No.3, July 1972, pages 505-506. (3) E. Butkov, Mathematical Physics, Addison-Wesley Publishing Co., Reading, MA,1968. (4) N. T. Khai and S. B. Yakubovich, "Some Two-Dimensional Integral Trans­ formations of Convolution Type," Dokl. Akad. Nauk BSSR, 34, No.5, 1990, pages 396-398. 32. Differentiation and Integration Applicable to Definite and indefinite integrals. Yields An alternative representation of the integral. Idea By differentiating or integrating an integral with respect to a param­ eter, the integral may be more tractable. This parameter may have to be introduced into the original integral. 32. Differentiation and Integration 143 Procedure Given an integral, try to differentiate or integrate that integral with respect to a parameter appearing in the integral. If there are no parameters appearing in the integral, insert one and then perform the differentiation or integration. After the integration is performed then either an integration or a differentiation remains. Example 1 Given the integral u(x) = Jooo e-t' cosxtdt, differentiation shows that u(x) satisfies the differential equation u' + lxu = O. (This was obtained by integration by parts.) This differential equation is separable (see Zwill­ inger [3]) and the solution is found to be u(x) = Ce-x'/4, for some con­ stant C. From the defining integral, u(O) = Jooo e-t' dt = lJ1T, so we have the final answer (32.1) Example 2 Given the integral I(P) = 100 xsin(x)e-P"" dx we might introduce a parameter a and consider instead J(a,p) = 100 xsin(ax)e-P"" dx. (32.2) Note that J(p) = J(1,p). Both sides of (32.2) may be integrated with respect to a to determine ja J(a,p) da = ja 100 xsin(ax)e-PZ' dxda = -100 cos(ax)e-P"" dx. (32.3.a-b) This last integral may be evaluated by a simple change of variable and using (32.1). The result is ja J(a,p)da = ~Jie-a'I'p. Differentiating (32.4) with respect to a results in J(a,p) = -~ t;e-a'/4P 4 V p3 and so (32.4) 144 III Exact Analytical Methods Example 3 Given the integral K(b) /.00 sine bx) _qx2 d ,q = ---e x o x (32.5) we differentiate with respect to b to obtain ! K(b, q) = fooo cos(bx)e-qX2 , which is the same as (32.3.b). Therefore :bK(b,q) = ~lfe-b2/4q. (32.6) From (32.5) we note that K(O, q) = O. Hence, we need to solve the differential equation in (32.6) with this initial condition. The solution is K(b,q) = l Glfe-b2/4q) db /.b/2.jq 2 =..ji 0 e-z dz (32.7.a-c) = ~erf(2~) where we have recognized (32. 7.b) as being an error function (see page 172). Notes [1] As another example of the technique illustrated in Example 3, consider the integral F(y) = 100 log(1 + X:y2) dx. o 1 +x Differentiating with respect to y produces F'( ) -100 2x2 y ~ y -0 1 + xy2 1 + x2 (32.8) = /~ 1 [0 c: ,,2 -1 + ~V ) dx (32.9) 1r =--1+y if y > O. From (32.8) we recognize that F(O) = O. Therefore, we can integrate (32.9) to determine F(y) = 111 -1 7r dy = 1r log(l + y). o +y [2] Squire [2] starts with the identity f01 xt-1 dt = t-1, and integrates with respect to t (from t = 1 to t = a) to obtain 11 xo-1 -1 I dx = loga. o ogx 33. Dilogarithms 145 [3J Many other clever manipulations based on integration and differentiation can be conceived. For example (again from Squire [2]), the identity X-I = Jooo e-:z;t dt can be used in the following way: /.00 sin x d _ /.00. /.00 -:z;t dt dx _ /.00 ~ _ ?!. x -smx e - 2 - • o x 0 0 0 1+t 2 References [1] J. Mathews and R. L. Walker, Mathematical Methods of Physics, W. A. Benjamin, Inc., NY, 1965, pages 58-59. [2] W. Squire, Integration for Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970, pages 84-85. [3] D. Zwillinger, Handbook of Differential Equations, Academic Press, New York, Second Edition, 1992. 33. Dilogarithms Applicable to Integrals of a special form. Yields An exact solution in terms of dilogarithms. Idea Integrals of the form pI: P(x, vR) log Q(x, vR) dx, where P(, ) and Q(, ) are rational functions and R = A2 + Bx + Cx2, can be transformed to a canonical form. Procedure Given the integral 1= r P(x, vR) iogQ(x, vR) dx, (33.1) make the change of variable vR = A + xt. This transformation results in v'A2 +BX+Cx2_A t = --------- 2At-B x = C _ t2 x dx = (C ~ t')' [AC -Bt + At'] dt VR= AC-Bt+At2 C _t2 146 III Exact Analytical Methods so that (33.1) becomes 1= l' P(t) log Q(t) dt (33.2) where P and Q are rational functions of t. In principle, partial fraction expansion can be used on P to obtain P = P(t) + L ({3 Umn )n mn+t m,n Q = K II ('Ymn + t)n (33.3) m,n where P(t) is a polynomial in t. Using (33.3) in (33.2), and expanding, leads to many integrals. Those integrals that have the form f P( t) log( 'Y + t) dt or the form / ({Jm:m; t)ft log(-y + t) dt, for n '" -1, can be evaluated by integrating by parts. The only integral that cannot be evaluated in this manner is / log(Q + t) dt = {IOg(a -')') log (')' ~ t) -Li2 (; ~!) + C, if a # ,)" -y + t llog2(-y + t) + C, if Q = -y, (33.4) where Li2 () is the dilogarithm function and C is an arbitrary constant. The dilogarithm function is defined by L· ( ) !.:I) log(1 -x) d 12 X = - x. o x (33.5) Hence, the integral in (33.1) can always be integrated analytically in terms of the dilogarithm and elementary functions. A few integrals involving dilogarithms are shown in Table 33. Example The integral /:1) 1 1= ~IOg(X+v'1+X2) dx x l+x2 (33.6) becomes under the change of variable .fR = VI + x2 = 1 + xt (which is the same as x = 2t/(l -t2» 1= l' ~ [log(l + t) -log(l- t)1 dt, = Li2 (t) -Li2 (-t) , (33.7) = Li2 ( ~ -1 ) -Lb e -~) . 33. Dilogarithms 147 Table 33. Some integrals involving dilogarithms. 1:1: log(1 + axn) dx = -.!.Lb (-axn) o x n 1:1: log(a+ bt) dx _ 1 I 2 (b( t») 1 I 2 (be) - -og -e + e - - og -o e + et 2e e 2e e lL. (be -ae) lL. (be -ae) + -12 - -12 ---e b( e + et) e be [ J£-;.dx = ~Lio (( Vl+x2 -xn + ~log2 (~ +x) 1% log(1 + x2 ) d -1 L' ( 2) + 1 L' ( 2x ) L' ( ) x - -12 -x -12 -- -12 X o VI -x 4 2 1 + x2 1 + 41og2(1 + x2) -log(1 -x) log(1 + x2) 1% log x log(x -1) dx L' () I L' ( ) = 13 X -og X 12 X o X 11 a-lL' ( ) dx 11"2 q,(1 + a) + "y X 12 X = - - 2 o 6a a Notes [1] Tables of the dilogarithm may be found in Lewin [2]. [2] An extension of the dilogarithm function of one argument is the dilogarithm function of two arguments: L· (8) l1z log VI -2xcos8 + x2 d 12 X = -- x , 2 0 x where, of course, Lb (x,O) = Li2 (x) for -1 ::; x ::; 1. The higher order logarithmic functions are defined by Lin (x) = fo% Lin-l (x)/x dx. All of these functions have been well studied; recurrence relations and other formulae have been determined for each (see Lewin [2].) References [1) M, Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964, Section 27.7, page 1004. [2] L. Lewin, Dilogarithms and Associated Functions, MacDonald & Co., Lon­ don, 1958. [3) L. Lewin, Polylogarithms and Associated Functions, North-Holland Publish­ ing Co., New York, 1981. 148 III Exact Analytical Methods 34. Elliptic Integrals Applicable to Integrals of a special form. Yields An exact solution in terms of elliptic functions. Idea Integrals of the form Ix R{x, vT{x)) dx where R{, ) is a rational func­ tion of its arguments and T{x) is a third of fourth order polynomial can be transformed to a canonical form. Procedure Given an integral of the form where 1 R(x, .jT(x)) dx R{x, JT) = /l{X) + h{x)JT, J3{X) + J4{X)JT (34.1) T = a4x4 + a3x3 + a2x2 + alX + ao, and each Ji{X) represents a polynomial in x, rewrite the integrand as (34.2) where R1{x) and R2{X) are rational functions of x. Clearly, the integral J Rl{X) dx can be evaluated in terms of logarithms and arc-tangents (see page 183). We can always write T in the form (see Whittaker and Watson [10]) T = [AI (x -0)2 + B1(x -,8)2] [A2(x -a)2 + B2(x -,8)2]. Then, chang­ ing variables by t = (x -a)/(x -,8) results in dx = ± (a -,8) -1 dt JT v's' where S := V(A1t2 + Bd (A2t2 + B2). Hence, we find J ~) dx= J RJ;) dt. 34. Elliptic Integrals 149 where R3 is a rational function. We can now write R3(t) = ~(t2) + tRs(t2), where ~ and Rs are also rational functions (by, say, ~(t2) == (R3(t) + R3( -t»/2 and Rs(t2) == (R3(t) -R3( -t»/2t). Clearly, the in­ tegral J t~ (t2) / ..;s dt can be evaluated in terms of logarithms and arc­ tangents by introducing the variable z = t2 (see page 183). By a partial fraction decomposition, we can write ~(t2) t2B 1 v'S = LOs -IS + ~ f3i,B (1 . _'h.. r= (34.3) s ',s + n, for some constants {os}, {f3i,s}, and {ni}. By integrations by parts, it turns out that we only need to be able to evaluate integrals of three canonical forms to determine (34.3) completely. Thus, knowledge of these canonical forms allow us to determine (34.1) completely. These three canonical forms are, in Legendre's notation: • Elliptic integrals of the first kind (Iyanaga and Kawada [8], page 1452): /.4> de /.BiD 4> dt F(</>, k) = = . (34.4) o VI -k2 sin2 e 0 V(1 -t2)(1 -k2t2) • Elliptic integrals of the second kind (Iyanaga and Kawada [8], page 1452): /.4> /.S;. ~ J 1 -k2t2 E(</>, k) = VI -k2 sin2 e de = 2 dt. o 0 I-t (34.5) • Elliptic integrals of the third kind (Iyanaga and Kawada [8], page 1452): /.4> de 11(</>, n, k) = o (1 + nsin2 e) VI -k2 sin2 e fBiD4> dt = 10 (1 + nt2) v{1-t2)(1-k2t2)' (34.6) 150 III Exact Analytical Methods Other Manipulations There are many ways in which to manipulate an integral to obtain an elliptic integral of the first, second, or third kinds. For example, by a partial fraction decomposition of (34.2), we can write R2(X) _ " A ~ + " B. 1 VT -L..J s 'T ~ *,s ( _ C.)SVT' s V.1 *,S X , for some constants {As}, {Bi,s}, and {Ci}. By writing T = allx4 + a3x3 + a2x2 + alX + ao = b4(X -e)4 + b3(x -e)3 + b2(x -e)2 + b1 (x -e) + bo, (34.7) we see that we only need to be able to evaluate integrals of the form and (34.8) in order to evaluate (34.1). The recurrence relations (s + 2)a4Is+3 + !a3(2s + 3)ls+2 + a2(s + 1)ls+1 + lal(2s + 1)ls + saOls-1 = xSVT for s = 0,1,2, ... (2 -s)b4Js-3 + lb3(3 -2S)Js-2 + b2(1 -s)Js-1 + lb3(1 -2s)Js -sbOJs+1 = (x -e)SVT for s = 1,2,3, ... (34.9) allow some manipulation of the {In} and the {In}. There are several different methods for calculating 10, depending on the exact form of T(x). For example, if T(x) = Ql(X)Q2(X), where Ql(X) = ax2 + bx + e, Q2(X) = dx2 + ex + /, (34.10) then the change of variable z = JQ1(X)/Q2(X) allows 10 to be written as jx dx 10 = JQ1(X)Q2(X) jz dz -± -J(e2 -4d/)Z4 + 2(2a/ + 2cd -be)z2 + (b2 -4ae) (34. l1.a-b) Note that the radical in (34.11.b) is the discriminant of Q2Z2 -Ql (i.e., what would appear in the radical if the quadratic formula were used to solve Q2Z2 -Ql = 0 for the variable x). 34. Elliptic Integrals 151 Example 1 Suppose we have the integral 1% dx 1= Jx(3x2+2x+1)' Since this has the form of (34.10), with Ql(X) = x and Q2(X) = 3x2+2x+1 we make the change of variable t= Ql(X) .f x Q2(X) =Y3x2+2x+1' This leads to (using (34.11» I = ± J 2/V1 -4t2 -8t4 dt. Defining b2 = (1 + vIa)/4 and -a2 = (1-vIa)/4, we observe that I can be written in the form (34.12) This standard form can be found in Abramowitz and Stegun [1], 17.4.49. If the lower limit on the integral in (34.12) is taken to be b, then the integral is equal to 1= ± ~ Ja21+ b2 F (Sin-' t, Ja2a+ b2) = ± ~ Ja21+b2 nc-' n la2: b2 ), where nc is a Jacobian elliptic function (see Table 34). Example 2 Given the integral /.% dx 1= , 1 J(5x2 -4x -1) (12x2 -4x -1) we change variables by y = 1/ x to obtain /.1/% -dy I = 1 J(5 -4y -y2) (12 -4y _ y2) . The change of variable t = k(y + 2) results in 1 /.(1+2%)/3% dt 1=-4: , V(l-t2) (1-, •• t2)' = -~ [F C;x2x,D -F (I,D]. 152 III Exact Analytical Methods Example 3 Given the integral 12 dx 1= 0 .j(2x -x2) (3x2 + 4) , we change variables by t = (2 -3x)/(6 + 3x) to obtain 1 1=-1613 dt o .j(1 -9t2) (1 + 3t2) Introducing v2 = 1 -9t2, we find Table 34. Some relationships between elliptic integrals, elliptic functions in Legendre notation, and Jacobian elliptic functions (from Abramowitz and Ste­ gun [1], page 566). In the following, x = sintP. If a> b and k2 = (a2 -b2)/a2, then F(tP,k) = a1:1: dt = sc-1 (~I k2) o .j(t2 + a2)(t2 + b2) b with tP defined b'Y tan tP = ~ i F(tP,k)=aloo dt =cs-1 (~lk2) :I: .j(t2 + a2)(t2 + b2) a with tP defined by tan tP = ! i x F(tP, k) = aJ.:I: dt = nd-1 (~I k2) b .j(a2 -t2)(t2 -b2) b a2(x2 _ b2) with ,I,. defined bv sin2,1,. = . If' "If' x2(a2 _b2)' F(tP, k) = ala dt = dn-1 (~I k2) :I: .j(a2 -t2)(t2 -b2) a 2 2 with tP defined by sin2 tP = a 2 -X 2 • a -b 34. Elliptic Integrals 153 2 b2/ 2 then If a > b and k = a, -1 (~I k 2) 1" dt = sn b x F(~, k) = a 0 v' (a2 -t2 )(b2 -t2 ) with ~ defined by sin ~ = /j; lb dt F(cp, k) = a :z: J(a2 _ t2)(b2 1:z: dt F(cp, k) = a a J(t2 _ a2)(t2 100 dt F(cp, k) = a:z: J(t2 _ a2)(t2 If k2 = a2/(a2 + b2), then 2 = cd-1 (~lk2) t ) 2(b2 _ x2) • 2 a . with cp defined by Sin cp = b2(a2 _ x2) , = dc-1 (~I k2) b2) a x2 _ a2 with cp defined by sin2 cp = -2--2; X -b = ns-1 (~I k2) b2) a with cp defined by sin cp = ~. x F(~, k) = "rar+bf [ ".,. ~~~,.? b2) = nc-1 (il k2 ) 100 dt F(cp, k) ="rar+bf :z: J(t2 + a2)(t2 with cp defined by cos cp = !!.. ; _ ds-1 ( X k2) X b2) -.Ja2 + b2 2 b2 with cp defined by sin2 cp = a 2 + 2' a +x If k2 = b2 /(a2 + b2 ), then -1 (x.;;;;+b2 k 2) :z: dt = sd ab F(~, k) = "rar+bf 1 v'(t2 + a2)(b2 -t2 ) • 2 _ x 2(a 2 + b:); with cp defined by Sin cp -b2(a2 + x ) 1· dt = cn-l (~I k2 ) x F(~, k) ="rar+bf " v' (t2 + a2 )(b2 t2 ) with ~ defined by cos ~ = /j' 154 III Exact Analytical Methods Notes [1] For the elliptic integrals in Legendre's notation, k is called the parameter or modulus, k' = Vf='k2 is called the complementary modulus, fjJ is called the amplitude, and for k = sin a we define a to be the modular angle. In other representations of the elliptic functions, the variables m = k2 and ml = 1 -m = k,2 = 1 -k2 are sometimes used. [2] The elliptic integrals of the first and second kind are said to be complete when the amplitude is fjJ = Tr /2, and so x = 1. The following special notation is then used: The complementary values are defined by K' = K'(k) = K(k') = K( VI -k2) and E' = E'(k) = E(k') = E( Vl-k2). These values are related by Legendre's relation: EK' + E'K -KK' = Tr/2. [3] The 9 Jacobian elliptic functions, {cd, cs, dc, ds, nc, nd, ns, sc, sd}, can be defined in terms of the three "basic" Jacobian elliptic functions: sn, cn, and dn. We have the standard relationships (from Abramowitz and Stegun [1], 16.3) cnu cdu= -d ' nu snu sdu= -d ' nu 1 ndu= -d ' nu dnu dcu= --, cnu 1 ncu = --, cnu snu scu = --, cnu The "basic" functions may be calculated from snu = sn(ulk2) = sinfjJ 1 nsu = --, snu dnu dsu= --, snu cnu csu = --. snu cn u = cn( ulk2) = cos fjJ = VI -sn2 u dn u = dn(ulk2) = VI -k2 sn2 u (34.13) where u is determined by an elliptic function of the first kind: u = F(fjJ, k). (That is, sn is the inverse function to F. Observe: sn(F(fjJ, k)lk2) = sin fjJ.) Tables of the Jacobian elliptic functions may be found in Abramowitz and Stegun [1]. A superscript of -Ion anyone of these twelve functions denotes the inverse function. Table 34 relates the Legendre elliptic functions to the Jacobian elliptic functions. 34. Elliptic Integrals 155 [4] The differential equation (y')2 = 1 -y2 with y(O) = 0 has the solution y(x) = sin(x), with a period of 11 dx 11 dx P = 2 = 4 = 211'. -1~ o~ Analogously, the differential equation (y')2 = (l_y2)(1_k2y2) with y(O) = 0 has the solution y(x) = sn(x) = sn(xlk2), with a period of 11 dx P = 4 = 4K(k). o V(1 -x2)(1 -k2x2) [5] The Jacobian elliptic function sn(u) satisfies (see (34.13» sn( u) = sn( ulk2) = sn (14) dO I k2) = sin <p. o VI -k2 sin2 0 Under the transformation t = sin 0 and x = sin <p this becomes (see (34.4» sn k = x. (1% dt 2) o V(1 -t2)(1 -k2t2) Since sn(zIO) = sin z, the limit of k -+ 0 produces (1% dt ) = x sin 0 ~ or sin (sin -1 x) = x. [6] Technically, any doubly periodic meromorphic function is called an elliptic function. All of the usual elliptic functions are doubly periodic; for example, snx = sn(x + 2iK') = sn(x + 4K). [7] Carlson [4]-[7] has introduced a new notation for the elliptic functions that preserves certain symmetries. Define four functions: 1100 dt RF(X,y,Z) = 2 0 v(t+x)(t+y)(t+z) 3100 dt RJ(x,y, z,w) = 2 0 v(t + x)(t + y)(t + z)(t + w) Rc(x, y) = RF(X, y, y) RD(X,y,Z) = RJ(x,y,z,z). Each function has the value unity when all of its arguments are unity, and Rc and RJ are interpreted as Cauchy principal values when the last argument is negative. These functions may be calculated by the numerical routines presented in Carlson [5]-[6]. 156 III Exact Analytical Methods The relationship between Legendre's notation and Carlson's notation is as Collows: RF(X,y,Z) = . ~F (C08-' f!, JZ -X) VZ-X V-;, z-y F(tj), k) = sintj)RF(cos2 tj), 1-k2 sin2 tj), 1) E(tj),} = sintj)RF(cos2 tj), 1 -k2 sin2 tj), 1) -lk2 sin3 tj)RD(COS2 tj), 1 -k2 sin2 tj), 1) II( tj), n, k) = sin tj)RF( cos2 tj), 1 -k2 sin2 tj), 1) + ~nsin3 tj)RJ(cos2 tj), 1 -k2 sin2 tj), 1, 1 -nsin2 tj) From the above we find E(k) = RF(O,I- k2,1) -~k2RD(O,1- k2,1) and K(k} = RF(O,l -k2, 1). Use of Carlson's notation for elliptical integrals dramatically reduces the number of separate cases that need to be tabulated in integral tables. [8] The Weierstrass P function is an elliptic function defined by [9] P(u) = P(UjWl,W2) = ~ + t ( 1 2 -~) U n= ...• -l.O.l.... (u -O) 0 m= .... -l.O.l .... O=2mwl +2nw2 where the prime means that the term n = m = 0 is not present in the sum. The function z = P(u) is the inverse function of the elliptic integral (see Iyanaga and Kawada [8], page 483) {Z dz u = J-oo J4Z3 -92Z -93' (34.14) Weierstrass showed that any elliptic integral could be transCormed into terms oC the Corm of (34.14) by a linear fractional transformation. It is not hard to see that an integral of the Corm I = f Rl (x, J tj)( x» dx, where Rl is a rational function and tj)(x} is a quartic polynomial, can be mapped to the form oC (34.14). With a bilinear transformation, x = ay + ~ 'YY + u (with I ~ ~ I # 0), I takes the form I = J R2(U, Vt/J(y» dy, where R2 is a rational function and 1/J(y} is a quartic polynomial. By an appropriate choice of coefficients in the transformation, 1/J can have the form 1/J(y} = 4y3 -92Y -93. To understand this transformation, let c be a root of tj)(x) , and presume that tj)(x} does not have multiple roots. Then the transformation x = c+1/z results in ,1,( } _ tj)'(C}z3 + itj)"(C}Z2 + ... ." X - 4 • z That is, it yields a new elliptic integral that only involves a cubic polynomial. If we further let z = ay + b, then, Cor a suitable choice of b the coefficient of y2 in the cubic polynomial will vanish. Finally, by choice of a we can Coree the coefficient of y3 to be 4. For more details, see Akhiezer [2], pages 15-16. 35. Frullanian Integrals 157 [10] Integrals of the form I = J R(x, V3(x)) £ix, where R is a rational function and B is polynomial of degree n > 4 are called hyperelliptic integrals. If n = 2p + 2 (where p is an integer) then, by rational transformations, one can obtain an equivalent integral in which the polynomial is of degree 2p+ 1 (see the previous note). See Byrd and Friedman [3] for details. [11] Consider an ellipse described by the parametric equations x = acos8 and y = bsin 8, with b > a > O. The length of arc from 8 = 0 to 8 = q, is given by 8 = b Jo4> VI -k2 sin2 8 d8, where k2 = (b2 -a2)/b2• It is because of this application that the integrals in this section are called elliptic integrals. [12] Since there are many representations of elliptic functions, and many of them do not indicate the full functional dependence, symbolic and numerical tables of elliptic functions should be used carefully. References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964, Chapter 17, pages 587-626. [2] N. I. Akhiezer, Elements of the Theory of Elliptic Functions, Translations of Mathematical Monographs, Volume 79, Amer. Math. Soc., Providence, Rhode Island, 1990. [3] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, New York, 1954, pages 252-271. [4] B. C. Carlson, "Elliptic Integrals of the First Kind," SIAM J. Math. Anal., 8, No.2, April 1977, pages 231-242. [5] B. C. Carlson, "A Table of Elliptic Integrals of the Second Kind," Math. of Comp., 49, 1987, pages 595-606 (Supplement, ibid., SI3-S17). [6] B. C. Carlson, "A Table of Elliptic Integrals of the Third Kind," Math. of Comp., 51, 1988, pages 267-280 (Supplement, ibid., SI-S5). [7) B. C. Carlson, "A Table of Elliptic Integrals: Cubic Cases," Math. of Comp., 53, No. 187, July 1989, pages 327-333. [8) S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT Press, Cambridge, MA, 1980. (9) D. F. Lauden, Elliptic Functions and Applications, Springer-Verlag, New York, 1989. [10] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cam­ bridge University Press, New York, 1962. 35. Frullanian Integrals Applicable to Integrals of a special form. Yields An analytic expression for the integral. 158 III Exact Analytical Methods Idea A convergent integral can sometimes be written as the difference of two integrals that each diverge. If these two integrals diverge in the same way, then the difference may be evaluated by certain limiting processes. Procedure A special case will illustrate the general procedure. Consider the convergent integral I_/.oo sin3 x d _!. /.00 3sinx -sin3x d - 2 X - 2 X. o x 4 0 x It is improper to write this integral as 1= ! /.00 3sinx dx _ ! /.00 sin3x dx 4 0 x2 4 0 x2 ' since both of these integrals diverge. It is proper, however, to write 1-1 l' (J.oo 3sinx d J.oo sin3x d ) --1m --x- --x, 40-0 0 x2 0 x2 which can be written as (using y = 3x in the second integral) I = ~ lim (J.oo sin2 x dx -100 si~ Y dY) 40-0 0 X 30 Y = -lim smx dx 3 J.30' 40-0 0 x2 3 J.30 1 =-lim -dx 40-0 0 x = ~ log 3, . sin x 1 SInce -2--+ -as x -+ O. x x Example The above procedure can be used to derive the general rule f" I(ax): I(bx) dx = [/(00) -/(O)]IOg~. (35.1) (35.2) The integral J = /.00 tanh ax -tanhbx dx is in the form of (35.2), o x with J(z) = tanhz. Since tanh(oo) = 1 and tanh(O) = 0 we find that J = log(a/b). 35. Frullanian Integrals Notes [1] A modification of the formula in (35.2) is (see Ostrowski [3]) a [" f(ax): f(bx) dx = (M[f] -m[fD log b' where M[J] = limt_oo ~ J: J{z) dz and m[J] = limt_o t hl J{z) dz. [2] A generalization of the formula in (35.2) is 1~ [uf(u) -ilf(v)] dx = M[xf(x)] log ~~:~ -m[xf(x)] log ~~~~ 159 when u and v are positive absolutely continuous functions and the limits involving u/v are positive, and M[xJ{x)] and m[xJ{x)] exist. [3] A different generalization of the formula in (35.2) is /.00 J (axP) -J (bxq ) dx o x [log a log b] P _ q /.00 = [J{oo) -J{O)] - - - + - J(x)logxdx. p q pq 0 This formula can be extended (see Hardy [2]) to 1~ [f(a"') -f(b:J;Q)]logN xdx N+l _ 1 ~( n (N + 1) [IOgn a logn b] /.00 - N + 1 ~ -1) n N+l -N+l J(x)logN-n+l xdx. n=O p q 0 References [1] E. B. Elliot, "On Some (General) Classes of Multiple Definite Integrals," Proc. London Math. Soc., 8,1877, pages 35-47 and 146-158. [2] G. H. Hardy, "A Generalization of Frullani's Integral," Messenger Math., 34, 1905, pages 11-18. [3] A. Ostrowski, "On Some Generalizations of the Cauchy-Frullani Integral," Proc. Natl. Acad. Sci. USA, 35, 1949, pages 612-616. [4] W. Squire, Integration for Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970, pages 99-105. 160 III Exact Analytical Methods 36. :Functional Equations Applicable to Definite integrals. Yields Sometimes an integral can be formulated as the solution to a functional equation. (That is, an algebraic equation relating the unknown function at different values of the dependent variable.) Idea By manipulating an integral into a functional equation, it may be possible to evaluate the integral or to obtain information about the integral. Procedure There are many types of functional equations that may exist. There are no general rules on how best to proceed. Example Consider the integral I(a) = loW log (1 + 2a cos x + a2) dx. (36.1) Breaking up the region of integration, Jo1r = Jo1r/2 + J:/2' shows that 1(a) = 1( -a). Therefore: 2I(a) = 10· [log (1 + 2acosx + a2) + log (1 -2a cos x + a2)) dx (36.2) = 10· log (1 + a4 + 2a2(1-2 cos 2 x») dx. By performing many manipulations, such as l1r log (1 + 2a cos x + a2) dx = 2131r '4 log (1 + 2asinx + a2) dx, 1r/2 1r/2 the last integral in (36.2) can be shown to be equal to 2I(a) = loW log (1 + 2a2 sin2 x + a4) dx = I(a2). Hence, our functional equation is 21(a) = 1(a2). This is equivalent to (36.3) Now note that 1(0) = O. When a2 < 1, both terms on the right-hand side of (36.3) are approaching zero as n -. 00. Therefore, we conclude that 1(a) = 0 for a2 < 1. 36. Functional Equations 161 Notes [1] The above example is from Squire [2]. Squire shows that the complete evaluation of (36.1) is given by I(a) = {O I 2 for a2 < 1 11' og a for a2 ; 1 . [2] In Book [1] the problem was to show that 100 {I _ -q(t)}!!.._ e 3/2 -11', o t where 1100 ds q(t) = - 10g(1 + st)--2' 11' 0 1 +s The first solution given in Book [1] started from the two functional equations for q(t): q( t) = q ( ~) + ~ log t and q(t) + q( -t) = log(1 -it), where each functional equation has a different domain of applicability. [3] Consider the integral I = fo1r/2 sin2 X dx. This integral can be manipulated into I = fo1r/2 cos2 X dx. Adding these two representations of I results in f''Ir /2 (2 . 2) r1r /2 11' 11' 21 = Jo cos x + sm x dx = Jo dx = 2" so that I = '4' [4] Consider the integral 1-100 logx d - 2 X. o 1 +x Using the change of variable z = l/x we obtain We conclude that 1=0. [5] Consider the integral 100 logz 1= ---2 dz= -I. o l+z Using z = 11'/2 -X we find l1r/2 cos" z 1=. dz. o sm" z + cos" z Adding these two expressions for I results in 21 = fo1f /2 dy = 11'/2. Hence, 1= 11'/4 (independent ofn). 162 III Exact Analytical Methods References [1] D. L. Book, Problem #6575, Amer. Math. Monthly, 97, No.6, June-July 1990, pages 537-540. [2] W. Squire, Integration for Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970, pages 85-87. 37. Integration by Parts Applicable to Single and multiple integrals. Yields A reformulation of the integral. Idea There is a simple integration by parts formula; it enables many inte­ grals to be evaluated exactly. Procedure The single integral 1= J: f(x) dx may often be cast into the form 1= t u(x) dv(x). (37.1) If this is the case, then (37.1) may be evaluated by the integration by parts formula to obtain Ib b I = u(x)v(x) 4 -f. v(x)du(x). (37.2) Example 1 If we have the integral J = I." xC08xdx we recognize that u = x and v = sin x (since dv = cosxdx). Hence, (37.2) may be used to obtain J = XSiDXI: -I." sinxdx (37.3) since du = dx. The last integral appearing in (37.3) is elementary and so J = XSiDX[ + COSX[ = ysiny + cosy-1. (37.4) 37. Integration by Parts 163 Example 2 The Gamma function is defined by the integral r(z) = fo"" t,-le-t dt. If we choose x = e-t and u = t' / z (so that du = 1'-1 dt), then this integral has the form of (37.1). Hence, from (37.2) we have t' I"" 1"" t' r{z) = e-t_ - -(_e-t) dt zoo Z Ii"" = -t' (_e-t) dt z 0 1 = -r(z+ 1) z or r(z + 1) = zr{z). Since we can easily determine that r{l) = 1, we conclude that, when n is a positive integer, r{n+l) = n·{n-l)· ... ·2·1 = n!. Hence, the Gamma function is the generalization of the factorial function. Notes [11 The integration by parts formula may be re-applied to (37.2). For example, if f n stands for the n-th derivative of f and 9n stands for the n-th integral of g then (see Brown [Ij) , I' I' I' I' f. fgdx=fg," -/l92" +1293 a -/3g, a + (37.5) [21 Green's theorem is essentially a multidimensional generalization of the usual integration by parts formula, since it relates the value of an integral to the values of some functions on the boundary of the region. See the section on line and surface integrals (page 164) for more details. 13] Henrici 14] uses Cauchy's theorem to relate contour integrals to area inte­ grals. For example, for the region R bounded by the curve r we have [41 1 f(x) dz = 2i J J 8{).zl dx dy, n ,,-=rec.& "! =-r:; ," &­ Integration by parts may also be used to obtain an asymptotic expansion of an integral, see page 215. 164 III Exact Analytical Methods References [1] J. W. Brown, "An Extension ofIntegration by Parts," Amer. Math. Monthly, 67, No.4, April 1960, page 372. [2] P. S. Bullen, "A Survey of Integration by Parts for Perron Integrals," J. Austral. Math. Soc. Ser. A, 40, No.3, 1986, pages 343-363. [3] U. Das, and A. G. Das, "Integration by Parts for Some General Integrals," Bu.ll. Austral. Math. Soc., 37, No.1, 1988, pages 1-15. [4] P. Henrici, Applied and Compu.tational Complex Analysis, Volume 3, John Wiley & Sons, New York, 1986, pages 289-290. [5] W. Kaplan, Advanced Calculus, Addison-Wesley Publishing Co., Reading, MA,1952. [6] G. B. Thomas, Jr., and R. L. Finney, Calculus and Analytic Geometry, 7th Edition, Addison-Wesley Publishing Co., Reading, MA, 1988. 38. Line and Surface Integrals Applicable to Line and surface integrals. Yields A reformulation as an ordinary integral. Idea Using a parameterization, line integrals and surface integrals can be written as ordinary integrals. Procedure: Line Integrals A line integral is an integral whose path of integration is a path in n­ dimensional space. For example, in two dimensions, if f(x, y) is continuous on the curve C, then the integrals Ie f(x, y) dx and Ie f(x, y) dy both exist. Here, C is either continuous, or piece-wise continuous (in which case the above integrals are interpreted to be the sum of many integrals, each one of which has a smooth contour). Line integrals can be evaluated by reducing them to ordinary integrals. For example, if f(x, y) is continuous on C, and the integration contour is parameterized by (4)(t),1/J(t)) as t varies from a to b, then fc /(x, y) do: = l / (4)(t), 7/>(t)) 4>'(t) dt, fc /(x, y) dy = l / (4)(t), 7/>(t)) 7/>' (t) dt. In many applications, line integrals appear in the combination J = fc P(x,y)dx+ fc Q(x,y)dy, 38. Line and Surface Integrals 165 which is often abbreviated as J= kP(X,Y)dX + Q(x,y)dy, (38.1) where the parentheses are implicit. IT the vector u is defined by u = P(x, y)i + Q(x, y)j, then the integral in (38.1) can be represented as J = Ie UT ds, where ds is an element of arc-length, and UT = U· T denotes the tangential component of u (that is, the component of u in the direction of the unit tangent vector T, the sense given by increasing s). Alternately, if the vector v is defined by v = Q(x, y)i -P(x, y)j, then the integral in (38.1) can be represented as J = Ie Vn ds, where Vn = V· n denotes the normal component of v (that is, the component of v in the direction of the unit normal vector n which is 90° behind T). Path Independence Let X, Y, Z be continuous in a domain D of space. The line integral I = I X dx + Y dy + Z dz will be independent of the path in D • if and only if u = (X, Y, Z) is a gradient vector: u = gradF, where F is defined in D (that is, Fx = X, Fy = Y, and Fz = Z throughout D), • if and only if Ie X dx + Y dy + Z dz = 0 on every simple closed curve C inDo Green '8 Theorems Green's theorem, in its simplest form, relates a two-dimensional line integral to an integral over an area (see Kaplan [1]): Theorem: (Green's) Let D be a domain of the xy plane and let C be a piece-wise smooth simple closed curve in D whose interior R is also in D. Let P(x, y) and Q(x, y) be functions defined in D and having continuous first partial derivatives in D. Then i (Pdx+ Qdy) = f f (~~ -'Z) dxdy. (38.2) R Green's theorem can be written in the two alternative forms (using u = P(x, y)i + Q(x, y)j and v = Q(x, y)i -P(x, y)j, as above): i uTd. = f f curludxdy R i "nd. = f f divvdxdy. R (38.3a-b) The second relation in (38.3) is also known as Stokes' theorem. This theorem is sometimes stated as 166 III Exact Analytical Methods Theorem: (Stokes) Let S be a piecewise smooth oriented surface in space, whose boundary C is a piecewise simple smooth simple closed curve, directed in accordance with the given orientation of S. Let u = Li+Mj+Nk be a vector field, with continuous and differentiable components, in a domain D of space including S. Then, Ie 'UT ds = I I (curl u· n) dO', s where n is the chosen unit normal vector on S. That is L Ld:e+ Mdy+Ndz = II e:: -0:) dydz s + --- dzdx+ --- dxdy (8L 8N) (8M 8L) 8z 8x 8x 8y Green's theorem can be extended to multiply connected domains as follows: Theorem: Let P(x,y) and Q(x,y) be continuous and have continuous derivatives in a domain D of the plane. Let R be a closed region in D whose boundary consists of n dis­ tinct simple closed curves {Cl,C2, ... ,Cn}, where C1 includes {C2, ... ,Cn} in its interior. Then 1 [Pdx + Qdy] + 1 [Pdx + Qdy] + ... + 1 [Pdx + Qdy] ~ ~ ~ =11 (8Q -8P) dxd. 8x 8y y R Specifically, if ~~ = ~: in D, then 1 [Pclx + Qdy] + 1 [Pdx + Qdy] + ... + 1 [Pdx + Qdy] = O. ~ ~ ~ Procedure: Surface Integrals If a surface S is given in the form z = f(x, y) for (x, y) in Rz1l' with normal vector n, then the surface integral II Ldydz+Mdzdx+Ndxdy = ± I I (-L:~ -M: +N) dxdy, S Rq with the + sign when n is the upper normal, and the -sign when n is the lower normal. If we define v = Li + Mj + Nk then we may also write II Ldydz+Mdzdx+Ndxdy= II v·ndt; S S 38. Line and Surface Integrals Figure 38. The contour C for (38.4). l' I I 161 --> where dO' is an element of surface area. Here, n = ±( -fxi -fyj + k)/ JI + f~ + f~, with the + or -sign used according to whether n is the upper normal or lower normal. The generalization of (38.3.a) to 3 dimensions is known as the diver- gence theorem, or as Gauss' theorem: Theorem: (Divergence) Let v = Li + Mj + Nk be a vector field in a domain D of space. Let L, M, and N be continuous and have continuous derivatives in D. Let S be a piecewise smooth surface in D that forms the complete boundary of a bounded closed region R in D. Let n be the outer normal of S with respect to R. Then ! ! Vn du = !!! divvdxdydz; 5 R that is Example 1 !! Ldydz + M dzdx + N dxdy 5 jr If (aL aM aN) = 1. ax + ay + 8z dxdydz. R Consider the integral (38.4) where C is the semicircle y = VI -x2 shown in Figure 38. The contour C can be represented parametrically by x = cos t and y = sin t for 0 ~ t ~ 1r. Hence, the integral can be evaluated as (38.5) 168 III Exact Analytical Methods Alternatively, the integral could have been evaluated by using the x­ parameterization throughout I = f.-I (x3 -(1 _ X2)3/2) ( -x ) dx. 1 V1-x2 This integral, which looks more awkward, is equivalent to (38.5) under the substitution x = cos t. Example 2 Here are a few examples of Green's theorem: (A) Consider the integral K = Ie [(y2 + sin x2) dx + (cos y2 -x) dy], where C is the boundary of the unit square (R:= {0:5 x:5 1,0:5 y:5 I}). A direct evaluation of this integral by parameterizing C is quite difficult. For example, using x as the parameter on the bottom piece of C, {y = 0, 0 :5 x :5 I}, necessitates the evaluation of the integral fol sinx2 dx, which is not elementary. However, using Green's theorem in (38.2) (with P = y2 +sinx2 and Q = cosy2 -x) we may write this integral as K = f J (-1 -2y) dx dy. As a set of iterated integrals, we R readily find that K = -2. (B) Let C be the circle x2 + y2 = 1. Then, using (38.2) i [4xy3 dx + 6x2y2 dy] = J J (12xy2 -12xy2) dxdy = O. R (C) Let C be the ellipse x2 + 4y2 = 4. Then, using (38.2) Notes i [(2x -y)dx + (x + 3y)dy] = J J (1+ 1) dxdy R = 2(area of ellipse) = 411". [1] When the contour of integration in a line integral is closed, then we often represent the integral by the symbol J, rather than the usual f. [2] When the contour in a two-dimensional line integral is closed, and the integrand is analytic, then contour integration techniques may be used (see page 129). Sometimes a two-dimensional line integral can be extended to a closed contour and be evaluated in this manner. (Note, however, that the integrand in (38.4) is not analytic.) [3] The evaluation of the integral in Example 2.C required that an area be known. Using Green's theorems, we can write the following integral expres­ sions for the area bounded by the contour C: area = iXdV=-iVdx. 38. Line and Surface Integrals 169 [4J If D is a three-dimensional domain with boundary B, let dV represent the volume element of D, let ds represent the surface element of B, and let dS = ndS, where n is the outer normal vector of the surface B. Then Gauss's formulas are (see Iyanaga and Kawada [3J, page 1400) IIIVAdV= II dS·A= II(n'A)dS D B B IIIVXAdV= II dSxA= Il(n X A)dS D B B III V4>dV= II 4>dS D B where 4> is an arbitrary scalar and A is an arbitrary vector. [5J There are also Green's theorems that relate a volume integral to a surface integral. Let V be a volume with surface S, which we assume to be simple and closed. Define n to be the outward normal to S. Let 4> and 1/J be scalar functions which, together with V24> and V21/J, are defined in V and on S. Then (see Gradshteyn and Ryzhik [2J, page 1089) (A) Green's first theorem states that (B) Green's second theorem states that References [IJ W. Kaplan, Advanced Calculus, Addison-Wesley Publishing Co., Reading, MA,1952. [2] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York, 1980. [3J S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT Press, Cambridge, MA, 1980. [4J G. B. Thomas, Jr., and R. L. Finney, Calculus and Analytic Geometry, 7th Edition, Addison-Wesley Publishing Co., Reading, MA, 1988. 170 III Exact Analytical Methods 39. Look Up Technique Applicable to Integrals of certain forms. Yields An exact evaluation, an approximate evaluation, or a numerical tech­ nique. Idea Many integrals have been named and well studied. If a given integral can be transformed to a known form, then information about the evaluation may be obtained from the appropriate reference. Procedure Compare the integral of interest with the lists on the following pages. If the integral of interest appears, see the reference cited for that integral. There are four lists of integrals, those with no parameters (i.e., con­ stants), those with one parameter, those with two parameters, and those with three or more parameters. Notes [I} The integrals in this section cannot be evaluated, in closed form, in terms of elementary functions (see page 77). [2] Some of the integrals are only defined for some values of the parameters; these restrictions are not listed in the tables. For example, the gamma function r(x) is not defined when x is a negative integer. [3] Realize that the same integral may look different when written in different variables. A transformation of your integral may be required to make it look like one of the forms listed. [4] In this section, the references follow the listings of integrals. [5] If the integral desired is not of a common form, then it will not appear in this section. However, it might be tabulated in one of the tables of integrals, see page 190. 39. Look Up Technique Constants Defined by Integrals Catalan's constant (see Lewin [21], page 34) G = T12(1) = --dt = '" ft ~ 0.91596559. . 11tan-lt 00 (_I)k o t ~(2k+ 1) Euler's constant (see Gradshteyn and Ryzhik [15], 8.367.4, page 946) 'Y = -1~ .-'!ogtdt '" 0.577215. Unnamed -related to random permutations (see Goh and Schmutz [14]) Z = 100 log log (~) dt ~ 1.11786. o l-e Integrals with One Parameter Airy function (see Spanier and Oldham [26], 56:3:1, page 555) Ai(x) = ; 1~ cos Gt3 + xl) dt. 171 Related to Airy's function (see Abramowitz and Stegun [1], 10.4.42, page 448) Gi(z) = ; 1~ sin Gt3 + zt) dt. Related to Airy's function (see Abramowitz and Stegun [1], 10.4.44, page 448) Hi(z) = ; 1~ exp Gt3 + zt) dt. Bairy function (see Abramowitz and Stegun [1], 10.4) 1100 (t3 ) (t3 ) Bi(x) = -; 0 cos "3 +xt exp -"3 +xt dt. Binet integrals (see Van Der Laan and Temme [19], page 122) S(z) = 2100 ta~:tl(t/z) dt. o e -1 Bloch-Gruneisen integral (see Deutsch [9]) 1:1: t5 z(x) = 0 (et _ 1)(1 _ e-t) dt. ~ 172 III Exact Analytical Methods Clausen's integral (see Lewin [21], Chapter 4, pages 91-105) Ch (8) = -f.' log (2 sin Ddt. Cosine integral (see Abramowitz and Stegun [1], 5.2.2, page 231) . f.% cost -1 Cl(Z) = 'Y + log Z + dt. o t Auxiliary cosine integral (see Spanier and Oldham [26], 38:13:2, page 371) gi(x) = -;.-dt. f.oo t -zt o t + 1 Hyperbolic cosine integral (see Abramowitz and Stegun [1], 5.2.4, page 231) . f.% cosh t -1 Chl(Z) = 'Y + log Z + dt. o t Dawson's integral (see Spanier and Oldham [26], Chapter 42, pages 405-410) daw(x) = f.~ e"-" dt. Dilogarithm (see Lewin [21]) L· () 1z log t dt 12 x = -t _ 1 . 1 Error function (see Abramowitz and Stegun [1], 7.1.1, page 297) 2 f.% 2 erf(z) = ..ji 0 e-t dt. Complementary error function (see Abramowitz and Stegun [1], 7.1.2, page 297) 2 1.00 2 erfc(z) = ..ji % e-t dt. Complete elliptic integral of the first kind (see page 154) K(k) = f.7r/ 2 --;:::=d=8 == o ";1 -k2 sin2 () Complete elliptic integral of the second kind (see page 154) f.7r/2 E(k) = 0 ";1 -k2 sin2 () d(}. Exponential integral (see Abramowitz and Stegun [1], 5.1.1, page 228) 1.00 -t E1(z) = % Tdt. 39. Look Up Technique Exponential integral (see Abramowitz and Stegun [1], 5.1.2, page 228) foo -t Ei(x) = -Tdt. -:1: Fresnel integral (see Abramowitz and Stegun [1], 7.3.1, page 300) C(z) = [ cos (it') dt. Fresnel integral (see Abramowitz and Stegun [1], 7.3.2, page 300) S(z) = [Sin Cit') dt. Gamma function (see Abramowitz and Stegun [1], 6.1.1, page 255) r(z) = J."" t%-I.-. dt. 173 Product of Gamma functions (see Abramowitz and Stegun [1], 6.1.17, page 256) 100 e-l r(z)r(1 -z) = -00 t + 1 dt. Inverse tangent integral (see Lewin [21], Chapter 2, pages 33-60) T;'(x) = J.z taD~I(t) dt. Lebesgue constants (see Wong [27], page 40) Ln = ~J.1r/2Isin (2~ + 1) tl dt. 1r 0 sm t Legendre's Chi function (see Lewin [21], page 17) 1 J.:1: ( 1 + t) dt X2 (x) = '2 0 log 1 -t t' Logarithmic integral (see Abramowitz and Stegun [1], 5.1.3, page 228) li(x) =f.:1:~. o logt Psi (Digamma) function (see Abramowitz and Stegun [1], 6.3.21, page 259) 1/1(z) = J.oo [e-t _ e-%~t] dt. o t 1-e Phi function (normal probability function) (see Abramowitz and Stegun [1], 26.2.2, page 931) ~(x) = . ~1:1: e-t2 /2 dt. v21r -00 174 III Exact Analytical Methods Riemann's zeta function (see Abramowitz and Stegun [1], 27.1.3, page 998) 1 /.00 tn dt 00 1 (n + 1) = , -t--= L n+r. n. 0 e -1 k Ie=l Sine integral (see Abramowitz and Stegun [1], 5.2.1, page 231) Si(z} = /.Z sin t dt. o t Auxiliary sine integral (see Spanier and Oldham [26], 38:13:1, page 370) fi(x} = /.00 sin x dt. o t+x Hyperbolic sine integral (see Abramowitz and Stegun [1], 5.2.3, page 231) Shi(z} = /.Z sinh t dt. o t Trilogarithm (see Lewin [21], Chapter 6, pages 136-168) L. () /.Z Li2(t} dt 13Z= -t-. o Integrals with Two Parameters a-order Green's function (see Iyanaga and Kawada [16], page 165B) 1 /.00 -at (2 ) Ga(x} = 2" 0 (2:t}d/2 exp -~t dt. Anger function (see Abramowitz and Stegun [1], 12.3.1, page 498) 1/.1r JII(z} = -cos(vO -zsinO} dO. 7r 0 Bessel function (see Abramowitz and Stegun [1], 9.1.22, page 360) J, () 1/.1r ( . 0 O) dO sin V7r /.00 -z sinh t-llt dt II Z = - COS Z SID -V --- e . 7r 0 7r 0 Bessel function (see Abramowitz and Stegun [1], 9.1.22, page 360) YII(z} = .!/.1r sin(zsinO-vO} dO_'!/.oo {ellt + e-llt cos (V7r) } e-zsinht-llt dt. 7r 0 7r 0 Beta function (see Abramowitz and Stegun [1],6.2.1, page 258) B(z, w) = /.\.-. (1 -t)W-' dt. 39. Look Up Technique Bickley function (see Amos [2]) K. ( ) - { loo Kin-l (t) dt In X -Ko(x) for n = 1,2, ... for n = 0 Debye function (see Abramowitz and Stegun [1], 27.1.1, page 998) 1:1: tndt z(x, n) = -t--. o e -1 Dnestrovskii function of index q (see Robinson [25]) 1 lOOtQ-le-t Fq(z) = r( ) dt. q 0 t+ z Elliptic integral of the first kind (see page 147) F(q"k) = lBiD '" dt o ~(1 -t2)(1 -k2t2) Elliptic integral of the second kind (see page 147) l"n. J 1 -k't' E(q" k) = 2 dt. o I-t 175 Repeated integrals of the error function (see Abramowitz and Stegun [1], 7.2.3, page 299) in erfc(z) = - e-t dt . 2 100 (t -z) n 2 ..;:i % n! Exponential integral (see Abramowitz and Stegun [1], 5.1.4, page 228) En(z) = ~dt. 1,00 -%t 1 t Generalized exponential integral (see Chic coli et al. [7]) 1,00 -%t E,,(z) = 1 e ttl dt. Fermi-Dirac integral (see Fullerton and Rinker [11]) 100 tlJ FIJ(a) = ~ dt. o l+e Logarithmic Fermi-Dirac integral (see Fullerton and Rinker [11]) 100 tlJ log (1 + eQ -t) GIJ(a) = ~ dt. o l+e Generalized Fresnel integral (see Abramowitz and Stegun [1],6.5.7, page 262) C(x,a) = l~t.-lc08tdt. 176 III Exact Analytical Methods Generalized Fresnel integral (see Abramowitz and Stegun [1], 6.5.8, page 262) S(x, a) = J.~ t"-' sintdt. Hubbell rectangular-source integral (see Gabutti et al. [12]) /.b -1 ( 1 ) dx J(a, b) = tan ~ ~. o V 1 + x2 V 1 + Z2 Incomplete Gamma function (see Abramowitz and Stegun [1], 6.5.3, page 260) r(a,x) = J.~ t"-'.-' dt. Generalized inverse tangent integral (see Lewin [21], Chapter 3, pages 61-90) . /.:1: tan-1(t) TI2(X, a) = dt. o t+a Hurwitz function (see Spanier and Oldham [26], 64:3:1, page 655) 1 100 t"-1e-ut «v; u) = r( ) -t dt. v 0 1-e Repeated integrals of Ko (see Abramowitz and Stegun [1], 11.2.10, page 483) 100 -zcosht Kir(z) = z e ... _ dt. Kummer's function (see Lewin [21], page 178) A (z) = /.Z logn-l It I dt. n l+t o Legendre function of the first kind (see Spanier and Oldham [26], 59:3:1, page 583) P.(x) = ~ /.~ [x + "';x2 -leostr dt. Legendre function of the second kind (see Spanier and Oldham [26], 59:3:2, page 583) Q.(x) = 1~ [x+ ~eoshtr·-' dt. Pearcey integral (see Kaminski [18]) P(x,y) = 1: exp{i (~ +x; +yt)} dt. Polygamma function (see Abramowitz and Stegun [1], 6.4, page 260) /.00 n -:l:t w(n)(x) = (_I)n+l 0 \ e_ t dt. 39. Look Up Technique Log-sine integral of order n (see Lewin [21], page 243) Lsn(B) = -f.. IOgn-'12Sin ~I dO. 177 Extended log-sine integral of the third order of argument 8 and parameter a (see Lewin [21], page 243) LS3(B,a) = -I." log 12sin ~Ilog 12 sin (~+~) I dB. Sievert integral (see Abramowitz and Stegun [1], 27.4, page 1000) z(z,B) = f.' e-z"",. dt/>. Polylogarithm of order n (see Lewin [21], page 169) Lin(z) = f.z Lin-l(t) dt. o t Struve function (see Abramowitz and Stegun [1], 12.1.7, page 496) 2 (~)" f.1r/2 H,,(z) = ~ (2 1) sin(z cos 8) sin21/ 8 d8. 7rr II + 2 0 Modified Struve function (see Abramowitz and Stegun [1], 12.2.2, page 498) 2 (~) 1/ f.7r/2 LI/(z) = ~ / 1) sinh(zcos8)sin21/ 8d8. 7rr II + 2 0 Voigt function (see Lether and Wens ton [20)) 00 _t2 y f e ft dt. V(x,y) = ;: J-oo (x _ t)2 + y Weber function (see Abramowitz and Stegun [1], 12.3.3, page 498) EI/(z) = .!.f.7r sin(1I8 -zsin8)d8. 7r 0 Unnamed integral (see Bonham [5)) hn(z) = (_l)n f.' Pon(t) sinzt dt. Unnamed integral (see Bonham [5)) hn+l(z) = (_l)n+l f.' P2n+,(t) coutdt. 178 III Exact Analytical Methods Unnamed integral (see Chahine and Narasimha [6]) z(x, u) = /."" tn e -(I-u)' -.'1 dt. Unnamed integral (see Glasser [13]) Iv(x) = /.' tV(l -t)"1 sinxtl dt. Unnamed integral (see Nagarja [23]) lp _t2 z(x,p) = _e_ dt. o t +x Integrals with Three or More Parameters Incomplete Beta function (see Abramowitz and Stegun [1], 6.6.1, page 263) B.(a,b) = /.' to-'(l -t)·-1 dt. Elliptic integral of the third kind (see page 147) /.Sin q, dt n(t/>, n, k) = . o (1 + nt2) V(l -t2)(l -k2t2) Double Fermi-Dirac integral (see Fullerton and Rinker [11]) 100 t" 100 8'" H .... I'(a) = dt t-Q ds--__ . o l+e t l+e Generalized Fermi-Dirac integral (see Pichon [24]) oot" VI + ~8t Fk(11, 8) = 1 t-dt. o 1 +e '1 Gauss function (see Spanier and Oldham [26], 60:3:1, page 600) r(c) 11 tb-1 dt F(a, b, Cj x) = n/L\nL L\ 0 Incomplete hyperelliptic integral (see Loiseau, Codaccioni, and Caboz [22]) H(a,Xj~2,~n) = lX -;:.==dx=== o va -~2X2 -~nxn 39. Look Up Technique Hypergeometric function (see Abramowitz and Stegun [1], 15.3.1, page 558) F(a,b,c; z) = ~",~~c! n /,\0-'(1- t)<-0-'(1 -tz)-· dt. 179 Confluent hypergeometric function (see Abramowitz and Stegun [1], 13.2.1, page 505) M(a b c) = reb) 11 e.dta-l(l -t)b-a-l dt , , reb -a)r(a) 0 • Confluent hypergeometric function (see Abramowitz and Stegun [1], 13.2.5, page 505) U(a b c) -_1_/.00 e-%tta-1(1 + t)b-a-l dt " -r(a) 0 • Lerch's function (see Spanier and Oldham [26], 64:12:2, page 661) 1 100 t,,-le-ut ~(Xj IIj u) = r( ) ~ dt. II 0 1 -xe Generalized log-sine integral of order n and index m (see Lewin [21], page 243) L8~m)(B) = -/." Bm !ogn-m-' 128in ~I dB. Selberg Integral (see Aomoto [3» N S(a,/3,"() = i. II xj(1 -Xj)13 II IXi -xjl'Ydx 3 A .•• A dXN A dXN. (O,l)N-2 j=3 3~i<j~N Complex Selberg Integral (see Aomoto [3» . Ni. N Sc(a,/3,"() = (~) II IZjl2alzj -112P II IZi -Zj12'Y (O,I)2(N-2) j=3 3~i<j~N dZ3 /\ az3/\ ••• A dZN /\ azN. Shkarofsky functions (see Robinson [25» .1".(z, a) = -i /.~ (1 ~tit)' exp (izt -1 ~2it) . Wu's integral (see Fettis [10» J •• ,(~) = /.~ .-(',2)(%H%,)z' dz. Unnamed integral (see Anderson and Macomber [4» Hn(p,q) = /.~ dttn-2exp( -~(t -pl' -n· 180 III Exact Analytical Methods Unnamed integral (see Cole [8]) I(y, z,s) = 1"" exp[-s(t + I/cost -zsin t)l dt. Unnamed integral (see Kolbig [17]) z(v,.\, m) = l' :>:"-'(1 -:>:)-> logm :>:dx. References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. [2] D. E. Amos, "Algorithm 609: A Portable FORTRAN Subroutine for the Bickley Functions Kin(x)," ACM Trans. Math. Software, 9, No.4, December 1983, pages 480-493. [3] K. Aomoto, "On the Complex Selberg Integral," Quart. J. Math. Oxford, 38, No.2, 1987, pages 385-399. [4] D. G. Anderson and H. K. Macomber, "Evaluation of ~lOO du un-2 v21r 0 exp{ -~(u -p)2 -~ }," J. Math. and Physics, 45, 1966, pages 109-120. [5] R. A. Bonham, "On Some Properties of the Integrals Jo1 P2n (t) sin xt dt and Jo1 P2n+1(t) cosxt dt," J. Math. and Physics, 45,1966, pages 331-334. [6] M. T. Chahine and R. Narasimha, "The Integral Jooo vn exp[-(v -U)2 -x/v] dv," J. Math. and Physics, 43, 1964, pages 163-168. [7] C. Chiccoli, S. Lorenzutta, and G. Maino, "Recent Results for Generalized Exponential Integrals," Compo & Maths. with Appls., 19, No.5, 1990, pages 21-29. [8] R. J. Cole, "Two Series Representations of the Integral Jooo exR-s(.,p+y cos.,p -z sin .,p)] d.,p," J. Comput. Physics, 44, 1981, pages 388-396. [9] M. Deutsch, "An Accurate Analytic Representation for the Bloch-Gruneisen Integral," J. Phys. A: Math. Gen., 20, 1987, pages L8ll-L813. [10] H. E. Fettis, "On the Calculation of Wu's Integral," J. Comput. Physics, 53, 1984, pages 197-204. [11] 1. W. Fullerton and G. A. Rinker, "Generalized Fermi-Direc Integrals -FD, FDG, FDH," Comput. Physics Comm., 39, 1986, pages 181-185. [12] B. Gabutti, S. L. Kalla, and J. H. Hubbell, "Some Expressions Related to the Hubbell Rectangular-Source Integral," J. Comput. Appl. Math., 37, 1991, pages 273-285. [13] M. L. Glasser, "A Note on the Integral Jo1 sV(l -s)VI sinxsl ds with an Application to Schlomilch Series," J. Math. and Physics, 43, 1964, pages 158-162. [14] W. M. Y. Goh and E. Schmutz, "The Expected Order of a Random Permu­ tation," Bull. London Math. Soc., 23, 1991, pages 34-42. [15] 1. S. Gradshteyn and 1. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York, 1980. [16] S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT Press, Cambridge, MA, 1980. 40. Special Integration Techniques 181 (17) K. S. Kolbig, "On the Integral fol xV-1(1 -x)-.x logm xdx," J. Comput. Appl. Math., 18, No.3, 1987, pages 369-394. [18] D. Kaminski, "Asymptotic Expansion of the Pearcey Integral Near the Caustic," SIAM J. Math. Anal., 20, No.4, July 1989, pages 987-1005. [19] C. G. van der Laan and N. M. Temme, Calculation of Special Functions: The Gamma function, the Exponential Integrals and Error-like Functions, Centrum voor Wiskunde en Informatica, Amsterdam, 1984. [20) F. G. Lether and P. R. Wenston, "The Numerical Computation of the Voigt Function by the Corrected Midpoint Quadrature Rule for ( -00,(0)," J. Comput. Appl. Math., 34, 1991, pages 75-92. [21] L. Lewin, Dilogarithms and Associated Functions, MacDonald & Co., Lon­ don, 1958. [22] [23] [24] [25] [26] [27] J. F. Loiseau, J. P. Codaccioni, and R. Caboz, "Incomplete Hyperelliptic Integrals and Hypergeometric Series," Math. of Comp., 53, No. 187, July 1989, pages 335-342. K. S. Nagarja, "Concerning the Value of /.P e-u2 du," J. Math. and Phys-o u+x ics, 44, 1965, pages 182-188. B. Pichon, "Numerical Calculation of the Generalized Fermi-Direc Inte­ grals," Comput. Physics Comm., 55, 1989, pages 127-136. P. A. Robinson, "Relativistic Plasma Dispersion Functions: Series, Integrals, and Approximations," J. Math. Physics, 28, No.5, 1987, pages 1203-1205. J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere Publishing corporation, New York, 1987. R. Wong, Asymptotic Approximation of Integrals, Academic Press, New York, 1989. 40. Special Integration Techniques Applicable to There are many specialized integration techniques, each of which works on a special class of integrals. Integrands involving solutions of second order differential equations Consider the integral (40.1) where Yl and Y2 are linearly independent solutions to the ordinary differ­ ential equation: Y" = Q(x)y. (Note that the Wronskian of Yl and Y2 is a constant; W(Yl, Y2) = c ~ 0.) For this method to work, we require that / be homogeneous of degree -2, that is J(aYb aY2) = a-2f(Yb Y2). In this 182 III Exact Analytical Methods case we find where u = Y2/Yl' Example 1 The following example is from Ashbaugh [3]. The integral J = sm x dx / ·2 cos4 X + cos X sin3 X (40.2) can be evaluated by identifying Yl = cos x and Y2 = sin x as being solutions to y" + Y = O. Hence, we write J as J = / 4 y~ 3 dx = / 4 y~ 3 YIY; -Y~Y2 dx Yl + YIY2 Yl + YIY2 1 =/~dU l+u = i log 11 + u31 + C = ~ log 1 + (:: r + C I (COSX)31 = i log 1 + sin x + C = ilogll+tan3xl +C where C is an arbitrary constant. Example 2 The following example is from Ashbaugh [3]. The integral !.CX> dx K = 0 (Ai (x) -i Bi(x»2 (where Ai (Bi) is the Airy (Bairy) function) has the form of (40.1), with Yl(X) = Ai(x) -i Bi(x) and Y2(X) = Bi(a;). We can write K as (recall that 40. Special Integration Techniques Ai and Bi satisfy Airy's equation: y" = xy) K = x = Y1Y2 -Y1Y2 X 100 d 100 " d o (Ai(x) -i Bi(x))2 0 71"-1 y~ = 71" [x=oo 1 du }x=o ( )Ix=oo = 71" U x=o ( Bi(x) ) IX=oo = 71" Ai(x) -i Bi(x) x=o i-va =71"-4-' Integration of Rational Functions 183 A rational function can always be integrated into a sum of rational functions and logarithmic (or arctangent) functions. Using D(X) and N(x) to represent polynomials, a rational function can be written using partial fractions as N(x) "N(ri) f(x) = D(x) = (polynomial in x) + L: (x -riHli~i(ri -rj where the {ri} are the roots of D(x) = 0 (assumed here to be distinct). This integrand can be directly integrated to obtain J J . . J" N(ri) f(x) dx = (polynomIal m x) dx + ~ ( _ o)IT (0 _ : dx i x r, i~i r, r3 " N(ro) log(x -ro) = (different polynomial in x) + ~ IT ' (0 _ 0) , • i ii:i r, r3 (40.3) This is formally correct, even when the roots are complex. Since logarithms and arctangents are related by log(u -iv) = 2i tan-1(u/v), the expression in (40.3) is always applicable. Example 3 J dx J dx x4 -1 = (x -1)(x + 1)(x -i)(x + i) = ! log(x -1) + ! log(x + 1) + ii log(x -i) + -li log(x + i) 1 x-I 1 -1 = -log-- --tan x. 4 x+ 1 2 For the case when some of the roots are repeated, see Stroud [6]. 184 III Exact Analytical Methods Use of Infinite Series Sometime an integral may be evaluated by expanding the integrand in a series, integrating term by term, and then re-summing the result. This technique may also be used to obtain an asymptotic expansion, see Wong (7) for details. Example 4 /.00 sin ax . Consider the integral I = ~1 dx. Formally expandmg the o e - denominator, and interchanging integration and summation results in 1= f /.00 e-nx sinaxdx n=1 0 00 = L a2:n2 n=1 = ~ (cot a1r --.!..) 2 a1r where we used Jolley [4] to recognize the cotangent sum. Example 5 Consider the integral 1= /.00 .-os' J,(fjx) do: (40.4) where J1 is a Bessel function. From Abramowitz and Stegun [1) 9.1.10, we note that _ {3x 00 (-i{32x2)k J1({3x) -2" ~ k!(k + 1)! . (40.5) Using (40.5) in (40.4), and interchanging orders of integration (see page 109), results in where we had to recognize the series for the exponential function. 40. Special Integration Techniques 185 Series Transpositions Squire [5] presents an elegant technique using series expansions and re-summing. An example will demonstrate the general ideas. 100' .. sin x ... . . ConsIder the mtegral I = --dx. The mfimte IntegratIOn regIon o x can be written as an infinite sum of finite integration regions, and then each finite region of integration can have the variables changed: 00 1(k+l)1I'/2 sinx 00 111'/2 (k1r ) du 1= L --dx= L sin -+u "':"1--- k=O k1l'/2 X k=O 0 2 '2k1r + u 00 111'/2 sm ucos""2 cosusm""2 ( . k1r . k1r) = L 1 + 1 du k=O 0 '2k1r + u '2k1r + u 00 111'/2 smucos -smusm- ( . k1r . . k1r ) = L 1 2 + 1 2 du k=O 0 '2k1r + u '2(k + 1)1r -U 111' /2 00 (cos k1r sin k1r ) = du sin u L 1 2 + 1 2 O -k1r + u -(k + 1)1r -U k=O 2 2 = 111'/2 du sin u{i( [~ -_1 + ... j + [_1 --2 1 + ... j) o u 1r+U 1r-U 1r-U 111'/2 . (=01 ~)~1--~~----- ~J'\/ Yo/~[4J = du slnu --= du = - ry~,,-e.t o sin u 0 2 where the changes of variables used were u = x-k1r/2 and then u = 1r/2-u. The final summation was found in Jolley [4], summation #769. Note [1] Arora et al. [2] notice that the two properties (A) Joa f(x) dx = Joa /2 f(x) dx + Joa /2 f(a -x) dx (B) Joa f(x) dx = Joa f(a -x) dx, if used cleverly, can evaluate some sophisticated integrals. They evaluate the integra11°O log X3 dx = -~4 to demonstrate this. o 1 +x 186 III Exact Analytical Methods References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. [2] A. K. Arora, S. K Goel, and D. M. Rodriguez, "Special Integration Tech­ niques for Trigonometric Integrals," Amer. Math. Monthly, February 1988, Vo195, No.2, pages 126-130. [3] M. S. Ashbaugh, "On Integrals of Combination of Solutions of Second-Order Differential Equations," J. Phys. A: Math. Gen., 19, 1986, pages 3701- 3703. [4] L. B. W. Jolley, Summation of Series, Second Edition, Dover Publications, Inc., New York, 1961. [5] W. Squire, Integration for Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970, page 92. [6] A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall Inc., Englewood Cliffs, NJ, 1971, Section 2.4, pages 49-52. [7] R. Wong, Asymptotic Approximation of Integrals, Academic Press, New York, 1989, Chapter 4, pages 195-240. 41. Stochastic Integration Applicable to Integrals in which the measure involves Brownian noise. Yields Information on how to evaluate Ito and Stratonovich integrals. Idea Different types of stochastic integrals exist, depending on how points are chosen in a limiting process. Background Suppose that wet) is the Wiener process and G = G(t) = G(t, wet»~ is an arbitrary function of wet) and the time t. (See the Notes section for information about the Wiener process.) The stochastic integral I = it G(s) dw(s) to ( 41.1) is defined as a kind of Riemann-Stieltjes integral. That is, first divide the interval [to, t) into n sub-intervals: to ::; tl $ t2 $ ... ::; tn-l ::; tn = t. Then choose the points {Ti}, for i = 1,2, ... , n, such that Ti lies in the i-th sub-interval: ti-l $ Ti $ ti. The stochastic integral I is now defined as a limit of partial sums, I = limn_oo Sn, with n Sn = L G(Ti, W(Ti»[W(ti) -W(ti-l»). (41.2) i=l 41. Stochastic Integration 187 Note that this limit depends on the particular choice of the intermediate points {Ti}. Consider, for example, the special case of G(t) = w(t). Then we find the expectation of Sn to be n = L[min(Ti, ti) -min(Ti' ti-l)] i=1 n = L(Ti -ti-l). i=1 If, for example, we take Ti = ati + (1-a)ti-b where 0 < a < 1, then E[Sn] becomes n E[Sn] = L(ti -ti-l)a = (t -to)a. (41.3) i=1 Clearly, the value of Sn (and hence J), in this example, depends on a. For consistency, some specific choice must be made for the points {Ti}. For the Ito stochastic integral we choose Ti = ti-l (i.e., a = 0 in the above). We show this by use of the notation if. That is I.: G(s, w(s)) dw(s) = ~\!.m {t. G(ti-l, W(ti-l))[W(ti) -W(ti_1)]} , (41.4) where ms-lim refers to the mean square limit. 188 III Exact Analytical Methods Example Suppose we would like to evaluate the Ito stochastic integral Jt W(s) dw(s). Jto If we write Wi for W(ti) then (41.4) becomes (using G(s) = w(s)) 1ft: w(s) dw(s) = ms-liIlln_oo Sn with n Sn = L Wi-t[Wi -Wi-I] i=1 n = LWi-lAWi i=1 n = ~ L [(Wi-l + AWi)2 -(wi_d2 -(dWi)2] i=1 n n = ~ L [(Wi)2 -(wi_d2] -~ L(AWi)2 i=1 i=1 n = ! [w2(t) -w2(to)] -! L(AWi)2 i=1 (41.5) where AWi = Wi -Wi-I. Now the mean-square limit of ! E;=1 (AWi)2 can be shown to be ~ (t -to). Hence, I.: w(s) dw(s) = ~J.,m { ! [w2(t) -w2(to»)-! ~(dWi)2} = ! [w2(t) -w2(to) -(t -to)] . ( 41.6) Note that the result is not the same result that we would have obtained by the usual Riemann-Stieltjes integral (in which the last term would be absent). Note that the expectation of (41.6) yields the value 0, which is the same value as given by (41.3). 41. Stochastic Integration 189 Notes [1] The Wiener process is a Gaussian random process that has a fixed mean given by its starting point, E[w(t)] = Wo = w(to), and a variance of E[(w(t) -WO)2] = t -to. From this we can compute that E[w(t)w(s)] = min(t, s). The sample paths of wet) are continuous, but not differentiable. [2] We define the Stratonovich stochastic integral (indicated by use of the notation i) to be (see Schuss [2]) It G(w(s), x) dw(s) ho . {~ (W(t) + W(ti-l) ) [ () ( )]} =rr::l~m ~G 2 ,ti-l wti-Wti-l . [3] It can be shown that the Stratonovich integral has the usual properties of integrals. In particular, we have the fundamental theorem of integral calculus, It J'(w(s» dw(s) = f(w(t» -f(w(to», ho integration by parts, etc. Taking the Stratonovich integral of the integrand in (41.5) results in it: w(s) dw(s) = ~ [w2(t) -w2(to)]. [4] For arbitrary functions G, there is no connection between the Ito integral and the Stratonovich integral. However, when x(t) satisfies the stochastic differential equation dx(t) = a[x(t), t] dt + b[x(t), t] dw(t), it can be shown that (see Gardiner [1]) It b[x(s), s] dw(s) =i.t b[x(s), s] dw(s) + ~ it b[x(s), s] ab[X~;)' s] ds. ho to to This relates, in a way, the Stratonovich integral and the Ito integral. [5] Stochastic integration can also refer to the (ordinary) integration of ran­ dom variables. Since the linear operations of integration and expectation commute, the following results are straightforward to derive. Let {X(t)} be a continuous parameter stochastic process with finite second moments, whose mean (m(t) = E[X(t)]) and covariance (K(s, t) = Cov[X(s), X(t)]) are continuous functions of sand t. Then E [[ X(t) dt] = [m(t) dt E [It X(t)dti'] = [ [ E[X(s)X(t») dt ds Var [[ X(t)dt] = [ [K(S,t)dtd" 190 III Exact Analytical Methods [6] Suppose that x(t) is a random process that satisfies the stochastic differential equation dx(t) = a[x(t), t] dt+b[x(t), t] dw(t). If we define a(x, t) = a(x, t)­ ~b(x, t)8b(x, t)/8x, then the solution to the stochastic differential equation, x(t), can be shown to satisfy (see Gardiner [1]) x(t) = x(to) + it a[x(s), s] ds +£.t b[x(s), s] dw(s). to to References [1] C. W. Gardiner, Handbook of Stochastic Methods, Springer-Verlag, New York, Second Edition, 1985. [2] Z. Schuss, Theory and Applications of Stochastic Differential Equations, John Wiley & Sons, New York, 1980. [3] K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, Birkhauser, Basel, 1990. 42. Tables of Integrals Applicable to Specific definite and indefinite integrals. Yields An exact evaluation. Idea The evaluation of many integrals has been tabulated. Procedure Compare the integral of interest with the tables in the references. These tables typically include indefinite and definite integration of both elementary and special functions (including, for instance, Bessel functions and hypergeometric functions). Example Suppose that we would like to determine the value of the definite integral I = J:/2 tana xdx. Some tables have I tabulated more or less as written, some tables give only the indefinite integral, and other tables give information that can be manipulated to yield the value of I. We find: • Beyer [3] has a recursion for the indefinite integral in number 423: j tann-lax j (tann ax) dx = a{n _ 1) -(tann-2 ax) dx . • Bois [4] lists several recursions for an integrand similar to that in I. One of these (on page 133) is: jSinm x sinm-1 x m -ljSinm-1 x --xdx = -- xdx. cosn x (n -1) cosn-1 X n -1 cosn-1 X 42. Tables of Integrals 191 • Dwight [5] has a recursion for the indefinite integral in number 480.9: /tannXdx= ~:~X -/tann-2Xdx. • In Gradshteyn and Ryzhik [7], the definite integral appears as number 3.622.1: J.1r /2 7r JL7r tg±1' X dx = -sec -[when] I Re JLI < 1. 0 22 • In Gradshteyn and Ryzhik [7], the indefinite integral appears as numbers 2.527.2, 2.527.3, and 2.527.4: tgP x dx = g - tgP-2 x dx [when] p ::/= 1 / t p-l X / p-l / n ( l)k-l t 2n-2k+2 tg2n+l xdx = L - g x (-l)n lncosx k=l 2n -2k + 2 / n t 2n-2k+l tg2n xdx = ~(_l)k-l g X _ (-l)nx. ~ 2n-2k+ 1 k=l • In Grobner and Hofreiter [8] the definite integral appears as num- ber 331.31: J.~/2 7r tg~ xdx = A7r [when] -1 < A < 1. o 2cosT • In Prudnikov, Brychov, and Marichev [12], the integral appears as number 2.5.26.7: J.~/2 7r o tgl' X dx = 2 cos(JL7r /2) [when] I Re JLI < 1. • In Prudnikov, Brychov, and Marichev [12], the indefinite integral appears as numbers 1.5.8.1, 1.5.8.2, and 1.5.8.3: /(tgx)P dx = ±_l_(tgX)P-l -/(tgX)P-2 dx p-l / n 1 (tgx)2n dx = =FL(-l)k (tgx)2n-2k+l k=l 2n-2k+l +(-l)nx / n 1 (tgx)2n+l dx = =F ~(-l)k (tgx)2n-2k+2 k=l 2n -2k + 2 =F( -l)n log I cos xl. 192 III Exact Analytical Methods Notes [1] Realize that the same integral may look different when written in different variables. A transformation of your integral may be required to make it look like one of the forms listed. [2] It is not always clear that having a symbolic evaluation of an integral is more useful than the original integral. For example, it is straightforward to show that I{x) = J.x ~ =_1_10g (x: + xV2 + 1) o 1 + t 4V2 x -xV2 + 1 1 [ -1 ( x) -1 ( X )] + 2V2 tan V2 + x + tan J2 -x . Given a specific value of x, numerically determining I{x) by using this formula requires the computation of logarithms and inverse tangents. In some cases it might be easier to approximate I{x) numerically directly from its definition. [3] It is an unfortunate fact that a not insignificant fraction of the tabulated integrals are in error. See, for example, Klerer and Grossman [9]. A common error is to produce a discontinuous antiderivative when a continuous integral is available. For example, the symbolic computer language REDUCE produces (see page 117): l_jX_dx_ - 2 + cos x 2V3 (SinX) 2V3 (Vasinx) x = --3-arctan cos x + 1 + -3-arctan 3{cosx + 1) + Va' This (correct) antiderivative is continuous, yet Abramowitz and Stegun [1], 4.3.133, report the discontinuous result 7 = .5a arctan tanjf2). These antiderivatives agree on the interval -1f' < X < 1f', but 7 is periodic while I is not. [4] If an integral is recognized to be of a certain form, then appropriate tables may be used. For example, an integral of the form I = J~oo J{x)e-xt dx represents a Fourier transform of the function J (x). Hence, a table of Fourier transforms (such as Oberhettinger [11]) might be an appropriate place to look for an evaluation of I. [5] Note that Oberhettinger [11] has tables of Fourier transforms, Fourier sine transforms, and Fourier cosine transforms. [6] All of the integral evaluations in Gradshteyn and Ryzhik [7] are referenced, so that one level of checking against typographic errors can be performed. 42. Tables of Integrals 193 References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. [2) A. Apelblat, Table of Definite and Indefinite Integrals, American Elsevier Publishing Company, New York, 1983. [3] W. H. Beyer (ed.), CRC Standard Mathematical Tables and Formulae, 29th Edition, CRC Press, Boca Raton, Florida, 1991. [4] G. P. Bois, Tables of Indefinite Integrals, Dover Publications, Inc., New York, 1961. [5] [6] [7] [8] H. B. Dwight, Tables of Integrals and Other Mathematical Data, The MacMil­ lan Company, New York, 1957. Staff of the Bateman Manuscript Project, A. Erdelyi (ed.), Tables of Integral Transforms, in 3 volumes, McGraw-Hill Book Company, New York, 1954. 1. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York, 1980. W. Grabner and N. Hofreiter, Integralyafel, Springer-Verlag, New York, 1949. [9] M. Klerer and F. Grossman, "Error Rates in Tables of Indefinite Integrals," Indust. Math., 18, Part 1, 1968, pages 31-62. [10] G. F. Miller, Mathematical Tables: Volume 3. Tables of Generalized Expo­ nential Integrals, Her Majesty's Stationery Office, London, 1960. [11] F. Oberhettinger, Tables of Fourier Transforms and Fourier Transforms of Distributions, Springer-Verlag, New York, 1990. [12] A. P. Prudnikov, Yu. A. Brychov, and O. I. Marichev, Integrals and Series, Volumes 1, 2, and 3, translated by N. M. Queen, Gordon and Breach, New York, 1990. [13] A. D. Wheelon, Tables of Summable Series and Integrals Involving Bessel Functions, Holden-Day, Inc., San Francisco, 1968. IV Approximate Analytical Methods 43. Asymptotic Expansions Applicable to Definite integrals that depend on a parameter. Yields An asymptotic expansion. Idea When a parameter in an integral tends to some limit, it may be possible to find an asymptotic expansion of the integral that is valid in that limit. Procedure There are several general asymptotic expansion theorems that can be used to determine the asymptotic nature of an integral; we enumerate only a few. Theorem (Bleistein and Handelsman [2), page 71): Define [(,\) = J: h(tj '\)/(t) dt, where /(n}(t) is continuous for n = 0,1, ... , N + 1, and J<N+2}(t) is piecewise continuous in the interval [a, b]. If Ih(-n-l}(t; '\)1 ~ Qn(t)t/Jn('\), for n = 0,1, ... ,N + 1, 195 196 IV Approximate Analytical Methods where the functions {Qn(t)} are continuous on the interval [a, b) and the functions {CPn (x)} form an auxiliary asymptotic sequence as ~ -+ ~o, then N I(~) "-I L Sn(~), as ~ -+ ~o, n=O where the Sn(~) are defined by Special Case 1 (43.1) Under appropriate smoothness and boundedness conditions, the inte- gral J(A) = f: h(At)J(t) dt has the asymptotic expansion J(A) ~ t <;2~ [/<n)(b)h<-n-l)(Ab) -/<n)(a)h<-n-l) (Aa)] n=O as A -+ 00. Special Case 2 Under appropriate smoothness and boundedness conditions, the inte­ gral J(A) = f: ei~t J(t) dt (which is a special case of Special Case 1), has the asymptotic expansion as A -+ 00. Special Case 3 Under appropriate smoothness and bounded ness conditions, the inte­ gral J(A) = f: e-~t J(t) dt (which is a special case of Special Case 1), has the asymptotic expansion N -~a J(A) ,..., L ~n+l J(n)(a) n=O as A -+ 00. Watson's Lemma (Bleistein and Handelsman [2], page 103, or Wong [5], page 20): If j(t) is locally absolutely integrable on (O,oo), as t -+ 00, j(t} = O(eQt} for some real number Q, and, as t -+ 0+, j(t) "-I E:=o cmtam, where Re(am} 43. Asymptotic Expansions increases monotonically to +00 as m --+ 00, and Re(ao) > -1, then, as ..\ --+ 00, Theorem (Bleistein and Handelsman [2], page 120): Let h(t) and f(t) be sufficiently smooth functions on the infinite interval (0,00) having the asymptotic forms 00 N(m) h(t) "J exp(-dt") L L Cmnt-rm(logtt, m=O n=O 00 N(m) f(t) "J exp (-qtl') L L Pmnt-am (log t)n, m=O n=O with some conditions on the range of the parameters appear­ ing in the expansion. Let the Mellin transforms of hand f be denoted by M[h; z] and M[f; z] (see the Notes). If some technical conditions are satisfied, then 197 /.00 h(..\t)f(t) dt "J -L res (..\-Z M[h; z] M[!j 1 -z]) (43.2) o z represents a finite asymptotic expansion as ..\ --+ 00 with respect to the asymptotic sequence {..\-aj(log..\)nj-m}. The expression in (43.2) represents a sum of the residues over all of the poles in a specific region of the complex plane. Bleistein and Handelsman [2] simplify the expression appearing in (43.2) in five different cases, depending on the values of the parameters. Example If J(A) = J: t~ f(t) dt, then h(tj A) = t~f(t) and so h(-n-l)(t. \) _ t~+n+l ,/\ -IIj=o(A + j + 1) for n = 0, 1, ... where we have chosen the limit of integration in the repeated integrals of h to be zero. From (43.1) this results in the following asymptotic expansion: N t~+n+l [ J(A) "J L n . f(n)(b)b~+n+l -f(n)(a)a.\+n+l] (43.3) n=O IIj=o(A + J + 1) whenever f satisfies the hypotheses of the theorem. If f were a polynomial, then the expansion in (43.3) would be the exact evaluation for J(A). 198 IV Approximate Analytical Methods Notes [I} The Mellin transform of the function f(t} is M[f; z} = Jooo tz-1 f(t} dt which can be interpreted as the (z -1}-st moment of f(t). The so-called bilateral Laplace transform of g(t} is the Mellin transform of f(t} = g( -log t}. [2} Consider the integral 1 = J: e>.J(z) g(x} dx in the limit .\ -+ 00. An asymp­ totic expansion methodology for 1 is given by (A) the method of steepest descents (page 229) when f(x} is complex, (B) Laplace's method (page 221) when f(x} is real, (C) the method of stationary phase (page 226) when f(x} is purely imagi­ nary (no real component). [3] In asymptotic formulas, it is important to describe fully the region in which a parameter is tending to a limit. Error estimates should also be supplied with an asymptotic formula. For example, the complementary error function has the following expansions as x -+ 00: erfc(x} t"V 00 e-z2"'"'(_I}k (2k -I)!! L, 2k+1X2k+l k=O 00 erfc(x) t"V v'i+e-:z:'L( _1}k (~!1-_9!! k=O 2 x 311" 311" for - -< arg x < -4 4' for 11" '4 < argx < 711" 4' (43.4a-b) Note that these expansions both apply in the region of overlap, 11"/4 < argx < 311"/4. In this overlap region equation (43.4.a) has a large error while equation (43.4.b) has a small error. [4] Wong [5) (page 22) has a generalized Watson's lemma: rooei-r At Define 1(,\) = Jo f(t)e-dt, and assume that 1(.\0) ex- ists. If f(t} t"V E:=o emtam as t -+ 0 along argt = ,,(, where () ",,00 cmr(am + 1) Reao > 0 and Ream+l > Ream, then 1 .\ t"V L.Jm=o .•. [5] Salvy [4J has created a package for automatically determining the asymptotic expansion of some classes of integrals, using the symbolic computer language MAPLE. Salvy gives the following sample outputs from his program: 100 sin (~) dx 11" "( - 1 log x 11" 0 ( 1 ) • 0 t(t+x) t"V 2x + ~ -~ -4x3 + X4 100 e -zt 1 1 2 6 • El(X) = -t-dt t"V --z -2"'i' + ""'3'i' -4Z + ... 1 xe x e x e x e • t t"V + --+ ....:.....-.....:.....---=..----:~ 100 Vte-t log t d J7r("( + 2 log 2) 11" log x J7r(4 -2"( - 4 log 2) o 1 + tx X X3/2 x2 +o(~tn as x -+ 00. Here, "( is Euler's constant. 44. Asymptotic Expansions: Multiple Integrals 199 References [1] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scien­ tists and Engineers, McGraw-Hill, New York, 1978. [2] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, Inc., New York, 1986. [3] F. W. J. Olver, "Uniform, Exponentially Improved, Asymptotic Expansions for the Generalized Exponential Integral," SIAM J. Math. Anal., 22, No.5, September 1991, pages 1460-1474. [4] B. Salvy, "Examples of Automatic Asymptotic Expansions," SIGSAM Bul­ letin, ACM, New York, 25, No.2, April 1991, pages 4-17. [5] R. Wong, Asymptotic Approximation of Integrals, Academic Press, New York, 1989. 44. Asymptotic Expansions: Multiple Integrals Applicable to parameter. Multidimensional definite integrals that depend on a Yields An asymptotic expansion. Idea When a parameter in an integral tends to some limit, it may be possible to find an asymptotic expansion of the integral that is valid in that limit. Procedure It is difficult to state concisely very much about the different asymp­ totic behaviors that are possible in multiple integrals. In this section we will only focus on the integral ( 44.1) when 4> is a real function and A is real with IAI ..... 00. Integrals of this form are known as integrals of Laplace type. (The other interesting case that occurs in applications is when A is purely imaginary with IAI ..... 00; this leads to integrals of Fourier type.) Here, D is some (not necessarily bounded) domain in n-dimensional x space. 200 IV Approximate Analytical Methods Laplace Type Integrals We presume that A is real and that D is a bounded simply connected domain. The boundary of D, denoted by r, is an (n -1 )-dimensional hypersurface. We assume it can be represented as r : X(CT), CT = (O't, ••• ,O'n-d where CT takes values in the set P. We presume that 4>, 90, and Xi(CT) are sufficiently differentiable for what follows. There are now two cases, depending on where the maximum of 4> appears in D. Maximum on the Boundary We presume that the maximum of 4> appears on the boundary D at the unique point x = Xo. Let N be the outward normal to r, and let dE be the differential element of "surface area" on r. Now define the gradient operation, V = (CJz1, ... ,8zn), the functions .,p(CT) = 4>(X(CT)), Hj = 9jV4>/IV4>12, and 9j+l = V . Hj (for j = 0,1, ... ). Then an exact representation of the integral in (44.1) is given by (see Bleistein and Handelsman [2], page 332) M-l ( )M J(A) = -L (_A)-j-l {(Hj. N)e'\q, dE + -~ {9Me'\q, dx (44.2) j=O ir A in for M = 2,3, .... The first terms in this expression are lower dimensional integrals for which asymptotic expressions may be found (recursively, if necessary). The last term can be bounded and will become the "error term." For the particular case of n = 2, we can parameterize the boundary by r : (Xl(S), X2(S)) where s = 0 corresponds to the maximum at x = Xo. The leading order term in (44.2) (e.g., the j = 0 term) can then be written as I(A) ~ el~(Xo) A31~~(O)1 (Ho . N)I.=o' (44.3) if the maximum of \II at S = 0 is simple (so that \11"(0) < 0). If ~(O) is the curvature of r at x = Xo, then (44.3) can be further simplified to I(A) ~el~(Xo)go(Xo)~ x l4>zlz14>~2 -24>zIZ24>zl4>z2 + 4>z2z24>~1 =F ~ IV4>131-1 /2 1 ' X=Xo (44.4) where the minus (plus) sign holds when r is convex (concave) at x = Xo. 44. Asymptotic Expansions: Multiple Integrals 201 Maximum Not on the Boundary We presume that the maximum of fjJ appears in the interior of D at the unique point x = Xo. Near x = Xo we can expand <P in the form <p(x) -<p(xo) ~ ~(x­ xo)A(x -Xo)T, where the matrix A = (aij) is defined by aij = <PXiXj (xo). Let Q be an orthogonal matrix that diagonalizes A, Le., QTAQ= (~l ". 0). o '\n Then define the variable z by (x-xo) = QRzT where the matrix R = (rij) is defined by rij = Oij l'\il-1/2• Now the functions {hi} are chosen so that (i = hi(z) = Zi + o(lzl) (as Izl --+ 0) and E:=l h1 = 2 (fjJ(xo) -<p(x(z))). With these functions we define the Jacobian J(C) = ~~~:::::: ~=? and then 90(X(,))J(,) = Go(') = Go(O) +,. Ho. Then we have the recursive definitions Gj(,) = Gj(O) + , . Hj(,) and Gj+1(,) = V . Hj(,). (Note that there is an ambiguity in the {Hn}, this is not important.) Finally, the approximation to J('\) is given by (see Bleistein and Handelsman [2], page 335) ( 44.5) where Z [k(,)] = JD k(,) exp( -~,\'. ,) d,. Note that (44.5) is not an exact representation, since M exponentially small boundary integrals have been discarded. The leading order term in (44.5) can be written as e~q,(Xo) (211") n/2 J('\) '" T 90(Xo). Vldet (fjJxiXj(xo))1 (44.6) For the particular case of n = 2, the result in (44.6) can be written as J('\) '" 211" go(xo)e~q,(Xo) (44.7) ,\ V fjJX1Xl (Xo)<PX2X2 (xo) -fjJ~lx2 (Xo) Example 1 Consider the two-dimensional integral J('\) = J J e~(x-y2) dx dy where D D is the unit circle. This integral has 90 = 1 and <P = x -y2. In D, the maximum of fjJ is at Xo = (1,0). Since this a boundary point of D, (44.4) is the appropriate formula to use. The only nonzero terms that appear in (44.4) are: go(Xo) = 1, fjJ(xo) = 1, <Px(xo) = 1, K(Xo) = 1, an<Jv<P(Xo)1 = 1. Using these values results in the approximation J('\) '" e~ 211"/,\3. 202 IV Approximate Analytical Methods Example 2 Consider the two-dimensional integral K{,x) = I I e~{2-x2_y2) dx dy D where D is the unit circle. This integral has 90 = 1 and 4> = 2 -x2 -y2. In D, the maximum of 4> is at Xo = (O,O). Since this an interior point of D, ( 44. 7) is the appropriate formula to use. The only nonzero terms that appear in (44.7) are Yo{xo) = 1, 4>{xo) = 2, 4>xx{xo) = -2, and 4>yy{xo) = -2. Using these values results in the approximation K{,x) '" 7re2~ /,x. For this example, the integral can be computed exactly. We find K{,x) = e~(2-r >rdrd{} = __ e~(2-r > = 7r---1271' 11 2 7r 211 e2~ -e~ o 0 ,x 0 ,x as ,x --+ 00. Notes [1] Bleistein and Handelsman [2] also describe asymptotic results for multidi­ mensional Fourier integrals. [2] Consider the integral J(A) = II f(x)ei>.q,(X) dx, where A is a large parameter. R2 At caustic points (also known as turning points), defined by Vt/>(xo) = 0 and ( 82t/> ) det 8Xi8x j = 0, the classical stationary phase techniques do not apply. For caustic points where the Hessian determinant vanishes, but the Hessian is not identically the zero mat~, there are several canonical forms of physical interest. These include the following possibilities for t/>(x, y): cp(x, y) = x3 + y2 fold cp(x, y) = x4 + l cusp cp(x, y) = x5 + y2 swallowtail cp(x, y) = x6 + y2 butterfly. For caustic points where the Hessian is identically the zero matrix, the canonical forms for cp(x, y) (for co-dimension less than 5) are cp(x, y) = x3 -xy2 elliptic umbilic cp( x, y) = x3 + y3 or x3 + xy2 hyperbolic umbilic t/>(x, y) = X4 + xy2 parabolic umbilic. See Gorman and Wells [4] for details. [3] Bruning and Heintze [3] derive an asymptotic expansion for the integral 1. 9 (xa) xIJ log'Y xf(x) dx, as s --+ 0+. [o,l]n S 45. Continued Fractions 203 [4] McClure and Wong [5] derive an asymptotic expansion for the integral f g (xQ) xfJ f(x) dx, as s --+ 0+. 1[0.1)" s References [1] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scien­ tists and Engineers, McGraw-Hill, New York, 1978. [2] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, Inc., New York, 1986, Chapter 8, pages 321-366. [3] J. Briining and E. Heintze, "The Minakschisundaram-Pleijel Expansion in the Equivariant Case," Duke Math. J., 51, 1984, pages 959-980. [4] A. D. Gorman and R. Wells, "The Asymptotic Expansion of Certain Canon­ ical Integrals," J. Math. Anal. Appl., 152, 1984, pages 566-584. [5] J. P. McClure and R. Wong, "Asymptotic Expansion of a Multiple Integral," SIAM J. Math. Anal., 18, No.6, November 1987, pages 1630-1637. [6] F. W. Olver, Asymptotics and Special FUnctions, Academic Press, New York, 1974. [7] R. Wong, Asymptotic Approximations of Integrals, Academic Press, New York, 1989, Chapters 8 and 9, pages 423-515. 45. Continued :Fractions Applicable to Integrals for which a continued fraction can be found. Yields A numerical approximation of an integral. Idea If a continued fraction can be found for an integral, then it may be used to approximate the value of that integral. Procedure A continued fraction is an expression of the form c = i ai = ____ a_l ___ _ i=l bi a2 b1+----- b2 + a3 b3+ .... Here, ak is called the k-th partial numerator and bk is called the k-th partial denominator, for k = 1,2, ... ,00. For typographical convenience, and also to save space, continued fractions are sometimes represented as or as al a2 a3 b1+ b2+ b3+ .... 204 IV Approximate Analytical Methods We define k ai al Ck = ~ - = ---------­ i=l bi a2 b1+------­ a3 b2+----- b3 + ... ... +ak to be the k-th convergent of the continued fraction I. The continued fraction is said to converge if the sequence {en} converges. In many cases, an integral may be written as a continued fraction. Partial convergents of the continued fraction then yield approximations to the original integral. Notes [1] Given {ak} and {bk}, define the sequences {Pk} and {qk} by the recurrence relations qk = bkqk-l + akqk-l and Pk = bkPk-1 + akPk-1 for k = 1,2, .... The initial values are given by P-l = 1, po = 0, q-l = 0, and qo = 1. Then Ck = Pk/qk for k = 1,2, .... Note that this relates continued fractions to recurrence relations. It might be easier to observe that the three-term recurrence relation, Yn + anYn+1 + bnYn+l = 0, is formally equivalent to the continued fraction Yn bn bn+l --=-a n-----···· Yn+l an+l-an+2- [2] From a Taylor series in the form F(z) = E:'o diZ-(Hl), a z-fraction, which . . d f t' f th Ii eol /l I ell hi '1 IS a contmue rac Ion 0 e orm"j;" -11 -"j;" -11 -... , may easl y be constructed. By defining Fk(z) = E:'o dHkZ-(HI) = Ck(Z), where Ck(Z) = dkl _ /l,kl _ el,kl _ h,kl _ e2,kl _ fa,kl _ Iz 11 Iz 11 Iz 11 ... , we obtain the recurrence relations ej-I,k+1 + li,k+1 = Ii,k + ej,k for e li,k+1ej,k+1 = ej,kli+l,k for f with eO,k = ° and /l,k = dk+1/dk. After determining the {ej,k,Ii,k}, we find ej = ej,O and Ii = Ii,o for j ~ 1, and eo = do. This is known as the QD (for quotient difference) algorithm. See van der Laan and Temme [3] for details. U sing this algorithm and the asymptotic formula 2 100 _t2 2z _%2 ( 1 1 3 ) erfc(z) = t= e dt"'. t=e -2 --4 + -6 -... , V 7r % V 7r 2z 4z 8z we can derive a continued fraction approximation to the erfc function: .;:i %2 Ii () 11 II ~ I ~ I ~ I --:;-e er c z = j?" + 11 + j?' + 11 + Iz2 + .... 46. Integral Inequalities 205 References [1] B. Char, "On Stieltjes Continued Fraction for the Gamma Functions," Math. of Comp., 34, 1980, pages 547-551. [2] D. Dijkstra, "A Continued Fraction Expansion for a Generalization of Daw­ son's Integral," Math. of Comp., 31, 1977, pages 503-510. [3] C. G. van der Laan and N. M. Temme, Calculation of Special Functions: The Gamma function, the Exponential Integrals and Error-like Functions, Centrum voor Wiskunde en Informatica, Amsterdam, 1984. 46. Integral Inequalities Idea Some integrals may be easily bounded by known theorems. Procedure Given an integral that is to be bounded, a formula should be located that has the desired form. This is not a straightforward process. Example 11 e-Z Suppose we would like to bound the integral I - dx -0 VI +x2 • If we write this integral as I = J~ f(x)g(x) dx, with f(x) = e-z and g(x) = 1/\1'1 + X2, then we note that both f and 9 are decreasing functions on the interval [0,1]. Hence, Tschebyscheff's inequality can be used to derive a lower bound (see the table at the end of this section for an exact statement of the inequality). We have To obtain an upper bound, we can use HOlder's inequality with p = q = 2 (see the table at the end of this section for an exact statement of the inequality). We have 206 IV Approximate Analytical Methods I ~ J/.l e-2% dx /.1 ~ dx o 0 1 +x = J ( -e~2·)I: (tan-1 x )1: = J e:~ 1 ~ 0:< 0.583. Hence, we have found a fairly tight bound for I (that is, 0.556 < I < 0.584), without having to perform much computation. One Dimensional Inequalities -Named [1] Carleman's inequality (see Iyanaga and Kawada [9], page 1422) /.00 exp G!." \Ogf(t)dt) dx < e /.00 f(x)dx when J(x) > O. [2] Cauchy-Schwartz-Bunyakowsky inequality (see Squire [19], page 21) (t f(x)g(x) dx ) 2 ~ (t /,(x) dx ) (t g2(X) dx ) . Equality occurs only when J(x) = kg(x), with k real. [3] Hardy's inequality (see Iyanaga and Kawada [9], page 1422) /.00 (F~X) r dx ~ (1 ~ p r /.00 r(x)dx when p > 1 and J(x) > O. Equality is achieved only if J(x) = O. [4] Modified Hardy's inequality (see Izumi and Izumi [8]) [x-m (f.~/(t)dtr dx < (m~ J [x-m If (~) -f(x)l" dx when m > 1, p > 1, and J(x) > O. [5] Hardy-Littlewood supremum theorem (see Hardy, Littlewood, and Polya [7], page 298 (#398» /.4 (sup ~ c[:I: J(t)dt)k dx ~ (k ~ l)k /.4 Jk(t)dt o 0::;;«:1: x I:. ( 0 if k > 1 and J(x) is non-negative and integrable. 46. Integral Inequalities 207 [6] Holder's inequality (see Squire [19], page 21) I b I ( b ) IIp (lb ) 1/q 1. I(x)g(x) dx $ 1. I/(xll" dx • Ig(x)I'dx when p and q are positive and l/p + l/q = 1. Equality occurs only when alf(x)IP = f3lg(x)IQ, where a and {3 are positive constants. [7] Backward HOlder's inequality (see Brown and Shepp [1]) .~p / [/(x -y)g(y)] dy $ (/I/(X)IP dx) l/p (/lg(X)I' dx ) 1/, when f and 9 have compact support, p and q are positive, and l/p+ l/q = 1. [8] Backward HOlder's inequality (see Brown and Shepp [I)) / .~p [/(x -y)g(y)] dx?: (/I/(X)IP dx ) l/p (/lg(X)I' dx ) 1/, when f and 9 have compact support, p and q are positive, and l/p+l/q = 1. [9] Minkowski's inequality (see Squire [19], page 21) (b) IIp ( b ) IIp ( b ) IIp 1. I/(x) + g(x)IP dx $ 1. I/(x)IP dx 1. Ig(x)IP dx for p > 1. Equality occurs only when f(x) = kg(x), with k non-negative. [10] Ostrowski inequality (see Gradshteyn and Ryzhik [6], page 1100) I[ I(x)g(x) dxl $ I/(X)I.'m.lt g(x) dxl when f(x) is monotonic decreasing and f(a)f(b) ~ O. [11] Tschebyschef£ inequality (see Squire [19], page 22) [1(X)g(X) dx ?: b ~ a ([ I(x) dx ) ([ g(x) dx ) when f(x) and g(x) are both increasing or both decreasing functions. (12) Tschebyschef£ inequality (see Squire (19), page 22) [1(X)g(X) dx $ b ~ a ([ I(x) dx ) ([ g(x) dx ) when f(x) is an increasing function and g(x) is a decreasing function (or vice-versa). [13] Wirtinger's ineqUality (see Hardy, Littlewood, and Polya (7), page 185 (#257» [ 12 (x) dx $ 1'(I')2(X) dx If f(O) = few) = 0 and!' is L2. Equality is obtained only if f(x) = Csinx. [14] Generalized Wirtinger's inequality (1 ~ k < 00), (see Tananika (20)) ( 1 ) 11k /¥ (1 ) 1/2 llu,k dt ~ !2(1-k)/k(k + 2)(k-2)/2k r «k + 2)/2k) llu,2 dt . o w r(l/k) 0 208 IV Approximate Analytical Methods [15] Young's inequality (see Hardy, Littlewood, and Polya [7], page 111 (#156» ab $ f." f(x) dx + f.. r1(x) dx when I(x) is continuous, strictly monotone increasing in x ~ 0, 1(0) = 0, a ~ 0, and b ~ o. Equality occurs only if b = I(a). One Dimensional Inequalities: Arbitrary Intervals -Unnamed [16] If p(x) > 0 and f p(x) dx = 1 (see Hardy, Littlewood, and Polya [7], page 137 (#184», then (unless 1 is a constant) exp (1 p(x)logf(x)dx ) < 1 p(x)f(x)dx. [17] If 0 < r < s, p(x) > 0, and fp(x)dx = 1 (see Hardy, Littlewood, and Polya [7], page 143 (#192», then (unless 1 is a constant) (J p(x)r<x) dx ) l/r < (J p(x)f'(x) dx ) 1/. [18] If 0 < a ~ I(x) ~ A < 00 and 0 < b ~ g(x) ~ B < 00 (see Hardy, Littlewood, and Polya [7], page 166 (#230», then (J f2(X) dx ) (J o'(x) dx ) $ (~ [fi + g) 1 f(x)g(x) dX) 2 • [19] If a, b, a, f3 are positive and I(x) is an increasing positive function (see Hardy, Littlewood, and Polya [7], page 297 (#397», then f.a+Q (a) /.b+fJ (b) f.a+b+Q+fJ (a + b) 1 -dx + 1 -dx ~ 1 -dx. a X b X a+b X [20) If 1 ~ r < p and I(x) and g(x) are positive functions in LP (see Potze and Urbach [21)), then lexp (-1 rrr dx) -exp (-1 r-1gP+1-r dx)1 $ Cr,p (1 If -glP) l/p dx where Cr,p ~ O. 46. Integral Inequalities 209 One Dimensional Inequalities: Finite Intervals -Unnamed [21] If a ~ 0, b ~ 0, a =1= 1, I(x) is non-negative and decreasing, and I(x) =1= C (see Hardy, Littlewood, and Polya [7], page 166 (#229», then ([XO+bl dx r $ [1-(a:~! J] ([X2°ldx) ([X2·ldx). [22] If I(x) has period 211', f027r I dx = 0, !' is L2, and I(x) -::fi Asinx + B cosx (see Hardy, Littlewood, and Polya [7], page 185 (#258», then /.2. 12(x)dx < /."(t(X))2 dx. [23] If 0 :::; !' :::; 1, 0 ~ g(x) < x, and k > 1 (see Hardy, Littlewood, and Polya [7], page 298 (#400», then 11 (/(X) -l(g(X») k dx < kl(l) -Ik(l) . o x -g(x) -k -1 [24] If 0 ~ I' ~ 1 and 0 :::; g(x) < x (see Hardy, Littlewood, and Polya [7], page 298 (#400», then 11 I(x) -I(g(x» dx ~ 1(1) (1 -log 1(1» . o x -g(x) One Dimensional Inequalities: Infinite Intervals -Unnamed [25] If m > 1, n > -1, I is positive (see Hardy, Littlewood, and Polya [7], page 165 (#226», then /.~ xnr(x)dx $ n: 1 (/.~ xm(n+1)/(m-.) r (x) dx ym-.)Im (/.~ It (x )Im dx )'Im Equality occurs only when I = Bexp (_cx(m+n)/(m-1»), where B ~ 0 and C>O. [26] If a ~ 0, b ~ 0, a =1= b, and I is non-negative and decreasing (see Hardy, Lit- tlewood, and Polya [71, page 166 (#228)), then (unless I(x) = {~ : ~~:~) with C > 0) (/.~ xo+b I dx r $ [1 -(a: ~! J] (1~ x20 I dx ) (/.~ x2• I dx ) . 210 IV Approximate Analytical Methods [27] If f and f" are in L 2 [0, 00] (see Hardy, Littlewood, and Polya [7], page 187 (#259)), then (f.~(1'($»2dx) 2 ~ 4 (f.~ J'($)dx) (f.~(1"($»2dx). Equality occurs only when f{x) = Ae-B:r:/2 sin (Bxsin i-i) [28] If f and f" are in L2[0, 00] {see Hardy, Littlewood, and Polya [7], page 188 (#260», then f.~ (J2($) -(1'($»2 + (1"($»2) dx ~ O. Equality occurs only when f{x) = Ae-B:r:/2 sin (Bx sin i -i) [29] If f and f" are in L2[-00,oo] {see Hardy, Littlewood, and Polya [7], page 193 (#261», then (unless f{x) = 0) ([: (1'($»2 dx r < ([:f2($) dx ) u: (1"($»2 dx ) . [30] If p > 1 and f{x) ~ 0 {see Hardy, Littlewood, and Polya [7], page 240 (#327)), then (unless f{x) = 0) f.~ (~[ f(t)dt)" dx < (p ~ 1)" [f"($)dx. [31] If p > 1, 0 ~ a < IIp, and p ~ q ~ pl(l -ap) (see Hardy, Littlewood, and Polya [7], page 298 (#402», then f.~ $-<'-0+,,0)/. (r(~) [ f(Y)($ -y)O-l dY) 0 ~ K (f.~ f'($) dx ) 01 •. This result is also true if a ~ IIp, P > 1, and p ~ q. In both cases K = K(p,q,a) > O. [32] Under some continuity requirements, with a > 0 (see Mingarelli [10]) [:If($)12•o.' dx ~ 2~[:1!,($)12.0,' dx. 46. Integral Inequalities 211 Two Dimensional Inequalities [33J If p > 1, q> 1, p-l + q-l ~ 1, ~ = 2 -p-l -q-l, h < 1-p-l, k < 1-q-l, h + k ~ 0, and h + k > 0 if p-l + q-l = 1 (see Hardy, Littlewood, and Polya [7J, page 298 (#401», then 100100 J(x)g(y)_ _ dx dy ~ K (100 JP(x) dx) IIp (100 gq(x) dx) l/q . o 0 xhylelx -YI~ h Ie 0 0 Here K = K(p,q,h,k) > O. [34J If J(x), g(x) and h(x) are non-negative, and J*(x), g*(x) and h*(x) are the equi-measurable symmetrically decreasing functions (see Hardy, Littlewood, and Polya [7J, page 279, (#379», then 1:1: I(x )g(y )h( -x-v) dx dy $ 1:1: r<x)g· (y)h· ( -x-y) dx dy. [35J If p > 1, p' = p/(P -1), Jooo JP(x) dx ~ F and Jooo gP' (x) dx ~ G (see Hardy, Littlewood, and Polya [7J, page 226 (#316», then (unless J(x) = 0 or g(x) = 0) 100 100 J(x)g(y) dxdy < . 1T Fl/PGl/p'. -00 -00 x + y sm(1T/p) Other Inequalities [36J If the function J, g, ... ,h are linearly independent functions (i.e., there do not exist constants A,B, ... , C, some not equal to zero, such that AJ+Bg+ .. ·+Ch = 0) (see Hardy, Littlewood, and Polya [7J, page 134 (#182», then J J2(X) dx J J(x)g(x) dx J J(x)h(x) dx > o. J h(x)J(x) dx J h(x)g(x) dx Notes [IJ In this section, when no further explanation is given, functions with upper case letters are assumed to be integrals of functions with lower case letters. For example, F(x) = Jo:l: J(x) dx and G(x) = Jo:l: g(x) dx. Also, all the integrals in this section are assumed to exist. [2J If J(x) is a real continuously differentiable function that satisfies the bound­ edness constraints J~oo x2IJ(x)12 dx < 00 and J~oo IJ'(x)12 dx < 00, then for x ~ 0 we have xl/(x)12 $ 4 f.~ t21/(t)12 dt f.~ 1/'(t)12 dt. 212 IV Approximate Analytical Methods This, in turn, can be used to derive the inequality i: I/(x)I' dx ~ 2 i: t'l/(t)I' dt i: 1/'(t)I' dt. This last inequality is known as Heisenberg's uncertainty principle in quan­ tum mechanics. [3] Evans et al. [3] contains a complete analysis of the inequality u {11'(x>!' + (x' -r)l/(x)I'} dx ) 2 ~ K(r) U I/(x)I' dx ) U II"(x) -(x' -r)f(x)I' dx ) . [4] Pachpatte [18] derives generalizations of the inequalities with suitable constraints on f, m, and p. (Here, F. is related to the integral of f.) [5] The inequality J J Ifl2 dx dy ~ 41 (Jr If I dlzl) 2 for functions f holomor- G 1r phic in G Uris referenced in Gamelin and Khavinson [5]. [6] Gronwalls' inequality states (see Gradshteyn and Ryzhik [6], page 1127): Theorem: Let the three piecewise continuous, nonnegative functions {u,v,w} be defined in the interval [0, a] and satisfy the inequality w(t) ~ u(t) + l' v(r)w(r) dr, except at points of discontinuity of the functions. Then, except at these same points, w(t) ~ u(t) + l' u(r)v(r) exp (1' v(u) du ) dr. 46. Integral Inequalities 213 [7) Opial [12) showed that Joh I/(x)1 I' (x) dx :S !h Ioh(f'(x»2 dx, with certain conditions on I. A comprehensive survey of Opial-type inequalities may be found in Mitrinovic [11]. Yang [22] proved the generalization Theorem: If I(s, t), IIJ, and 1st are continuous functions on [a,b] x [c,d] and if I(a,t) = I(b,t) = Is(s,e) = Is(s,d) = ° for a :S s :S band e :S t :S d, then J." /." 1/(·, t) 11/ •• (s, t) I dt ds 5 (6 -a)!d -c) l1" 1/ •• (., t)I' dtd •. Two other generalizations of Opial's inequality are in Pachpatte [16]. One of these generalizations is (the other is similar): Theorem: Suppose the functions p, q are positive and con­ tinuous on a = [a,X] x [e,Y]. Let I = I(s,t), fs, 1st be continuous functions on a with I(a, t) = Is(s, e) = ° for a :S s :S X and e :S t :S Y. If m and n are positive integers, with m + n > 1, then J.X 1Y pilim 11 .. 1" dtds 5 K(X,Y,m,n) f.x 1Y ql/ •• lm+" dtds (46.1) where K(X, Y, m, n) is a finite constant that depends on the functions p and q. If m < 0, n > 0, and m + n > 1, then (46.1) holds with :S replaced with ~. [8] Assume that I(t) and cp(t) are nonnegative and measurable on R+, and that both a and b are in the range (0,00). Define 4>(x) = f: cp(t) dt, FL(X) = foz f(t)cp(t) dt, Fu(x) = fzoo I(t)cp(t) dt, and M = (PIle -Il)p. Then (see Copson [2]): 1" Ff~-C4>dz 5 M 1" fP~'-C4>dz f.~ Ff~-C4>dz;>: M f.~ f'~'-C4>dz if p 2: 1, c> 1 if 0 < p :5 1, c > 1 and ~(x) -+ 0 as x -+ 00 f.~ F&~-c4>dz S; M f.~ f,~,-c4>dz if p;>: 1, c < 1 1" F&~-c4>dz;>: M 1" fP~'-C4>dz if 0 < p S; 1, c < 1 l Ff~-l4>dz S; pi' l fP~·-l (log :~~r 4>dz ifp;>: 1 f.~ F&~-l4> dz ;>: pi' f.~ f·~·-l (log :~:~ r .; dz if 0 < p S; 1. 214 IV Approximate Analytical Methods [9] The HELP (Hardy, Everitt, Littlewood, Polya) inequalities are of the form (see Evans and Everitt [4)) where p, q, and ware real-valued functions on [a, b] (with -00 < a < b::; 00), and M[·] denotes the second order differential expression M[f] = -(pf')' + qf. There are some technical conditions on p, q, and w. References [1] G. Brown and L. A. Shepp, "A Backward Holder's Inequality," Problem number E 3370 in Amer. Math. Monthly, 98, No.7, August-September 1991, pages 650-652. [2] E. T. Copson, "Some Integral Inequalities," Proc. Roy. Soc. Edinburgh, 75A, No. 13, 1975/76, pages 157-164. [3] W. D. Evans, W. N. Everitt, W. K. Hayman, and S. Ruscheweyh, "On a Class of Integral Inequalities of Hardy-Littlewood Type," J. Analyse Math., 46, 1986, pages 118-147. [4] W. D. Evans and W. N. Everitt, "HELP Inequalities for Limit-Circle and Regular Problems," Proc. R. Soc. London A, 1991, 432, pages 367-390. [5] T. W. Gamelin and D. Khavinson, "The Isoperimetric Inequality and Ra­ tional Approximation," Amer. Math. Monthly, January 1989, page 22. [6] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York, 1980. [7] H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge Mathematical Library, Second Edition, New York, 1988. [8] M. Izumi and S. Izumi, "On Some Inequalities for Fourier Series," J. Anal. Math., 21, 1968, pages 277-291. [9] S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT Press, Cambridge, MA, 1980. [10] A. B. Mingarelli, "A Note on Some Differential Inequalities," Bull. Inst. Math. Acad. Sinica, 14, No.3, 1986, pages 287-288. [11] D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, New York, 1970. [12] Z. Opial, "Sur une inegalite," Ann. Polon. Math., 8, 1960, pages 29-32. [13] K. Ostaszewski and J. Sochacki, "Gronwall's Inequality and the Henstock Integral," J. Math. Anal. Appl., 127, 1987, pages 370-374. [14] B. G. Pachpatte, "On Opial-Type Integral Inequalities," J. Math. Anal. Appl., 120, No.2, 1986, pages 547-556. [15] B. G. Pachpatte, "On Some Variants of Hardy's Inequality," J. Math. Anal. Appl., 124, 1987, pages 495-501. [16] B. G. Pachpatte, "On Two Independent Variable Opial-Type Integral In­ equalities," J. Math. Anal. Appl., 125, 1987, pages 47-57. [17] B. G. Pachpatte, "On Some New Integral Inequalities in Two Independent Variables," J. Math. Anal. Appl., 129, No.2, 1988, pages 375-382. [18] B. G. Pachpatte, "On Some Integral Inequalities Similar to Hardy's Inequal­ ity," J. Math. Anal. Appl., 129, 1988, pages 596-606. 47. Integration by Parts 215 [19] W. Squire, Integration for Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970. [20] A. A. Tananika, "A Generalization ofWirtinger's Inequality," Differentsial'­ nye Uravneniya, 22, No.6, 1986, pages 1074-1076. [21] W. Potze and H. P. Urbach, "An Inequality in LP," Appl. Math. Lett., 3, No.3, 1990, pages 95-96. [22] G. S. Yang, "Inequality of Opial-type in Two Variables," Tamkang J. Math., 13, 1982, pages 255-259. 47. Integration by Parts Applicable to A single integral. Yields An asymptotic expansion of the integral. Idea By using integration by parts an asymptotic expansion may sometimes be obtained. Procedure Repeatedly using the process of integration by parts (see page 161) often allows an asymptotic expansion to be obtained. The remainder term is needed to determine the error at any stage of the approximation. Several theorems are available that can be used to state an asymptotic expansion of an integral iJIlmediately, see the Notes. Example 1 The exponential integral is defined by El(X) = roo e-t dt. Jx t ( 47.1) We make the identification (dv = e-tdt, u = t-1), and then use the integration by parts formula, J u dv = uv I -J v du, on (47.1) to obtain E 1(x) = _ e-t It=oo _ (JO e~t dt t t=x Jx t e-x (:to e-t = -;--Jx 7dt. (47.2) Making the identification (dv = e-tdt, u = t-2) in (47.2), and using integration by parts again, results in 216 IV Approximate Analytical Methods Integrating by parts a total of N times results in ( 47.3) Integrating by parts infinitely many times results in e-x E~=o (~~>;r! . This series diverges for all values of x (since the absolute value of the ratio of successive terms in the sum is nix, which increases as n increases), and so is not a good representation of El(X). However, it is not hard to bound the remainder term RN(X). We have IRN(X)I = N! ~+l dt 100 -t x t where we have used Stirling's approximation for N!, which is asymptotically valid for large values of N. From this rough approximation, we conclude that, for some values of x, smaller values of N may give a smaller remainder than larger values of N. Example 2 The complementary error function is defined by the integral erfc(x) = * J; e-t2 dt. By repeated integration by parts, we can obtain the asymp­ totic expansion 2e-X2 [ 1 1 3 1 erfc(x) f"V -- - - -+ - -... Vi 2x 4x3 8x5 2e-x2 ~ k (1)(3) ... (2k -1) = '-L) -1) 2k+1 2k+l . y7r k=O x Once again, this asymptotic series diverges for all values of x. However, for a fixed number of terms, the approximation becomes better as x increases. A numerical illustration of this asymptotic expansion is in Table 47. 47. Integration by Parts 217 Table 47. A numerical comparison of the complementary error function with the first term and first two terms in its asymptotic expansion. e-Z2 e-z2 1 1 erfc(x) xVi Vi [; -2x3] x=1 0.15730 0.41510 0.20755 x=2 0.00468 0.00517 0.00452 x=3 0.0000221 0.0000232 0.0000219 Notes [1] The approximation in (47.3) indicates a common trait of asymptotic se­ quences: For fixed x, the approximation gets worse as N increases, for fixed N, the approximation gets better as x increases. [2] Note that, from a numerical point of view, the integral in (47.2) is more rapidly converging than the integral in (47.1). [3] Wong [4] presents the following example where integration by parts does not lead to an asymptotic expansion. The integral I (x) = 100 1~! o (1 + t) (x + t) can be integrated by parts to obtain N-l 3n(n -I)! 1 I (x) = -L 2 . 5 ... (3n _ 1) xn + c5 N (x), n=1 (47.4) 3N-1(N -1)' 100 (1 + t)N-4/3 where c5N(X) = (N~) N dt. Approximations aris-2·5··· 3 - 0 (x+t) ing from (47.4) are not useful, since I(x) is positive but every term in (47.4) is negative. [4] Bleistein and Handelsman [2] have several general results about integration by parts. (A) Consider the integral 1(,\) = J: h(tj ,\)f(t;'\) dt. Assuming sufficient continuity, we have 1(,\) = E:=o Sn('\) + RN('\) (see Bleistein and Handelsman [2], Theorem 3.1) where Sn (>.) = (-W [In) (b; >')hC -n-I) (b; >.) -In) (B; >')hC -n-I) (B; >.) 1 RN(>') = (_l)N+1 f.' IN+1)(t; >.)hC-N-1)(t; >.) dt where g(n) denotes the n-th derivative of 9 if n is positive, and it denotes the Inl-th integral of 9 if n is negative. If the functions {</>n('\)} form an asymptotic sequence as ,\ -+ '\0, and if Ih(-n-1)(tj '\)1 ~ On (t)</>n('\), 218 IV Approximate Analytical Methods where the {an} are continuous, then as ..\ -+ ..\0 we have 1(,,\) '" E:=o Sn(..\). As an example, if f is sufficiently differentiable, and if 0 ~ a < b, then we have as..\ -+ 00. (B) Consider the integral 1(,,\) = J: h(..\t)f(t) dt. Assuming • sufficient continuity for f; • b -a is finite; • ..\-It/>n+l(''\) = O(t/>n(..\» as ..\ -+ 00; • IhC-n)(..\t)1 ~ an{t)t/>n(..\), where the {an} are continuous, then, as ..\ -+ 00 (see Bleistein and Handelsman [2], Theorem 3.2) As an example, we have the result (if f is sufficiently smooth) as..\ -+ 00. References [1] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scien­ tists and Engineers, McGraw-Hill, New York, 1978 [2] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, Inc., New York, 1986, Chapter 3, pages 69-10l. [3] A. Erdelyi, Asymptotic Expansions, Dover Publications, Inc., New York, 1956. [4] R. Wong, Asymptotic Approximation of Integrals, Academic Press, New York, 1989, pages 14-19. 48. Interval Analysis Applicable to expressions. Yields Ordinary integrals, or integrals containing interval An analytical approximation with an exact bound on the error. 48. Interval Analysis 219 Idea In interval analysis, quantities are defined by intervals with maximum and minimum values indicated by the endpoints. Definite integrals can often be approximated by an interval; intervals are better than ordinary numerical approximations since an exact bound on the error is obtained. Procedure We use the interval notation [a, b] to indicate some number between the values of a and b. We will allow coefficients of polynomials to be intervals. For example, the interval polynomial ( 48.1) evaluated at the point x = y, means that min 1 + TJy2 + (y3 ~ Q(y) ~ max 1 + TJy2 + (y3. (48.2) 2~11~3 2~11~3 -1~C~4 -1~C~4 There exists an algebra of interval polynomials. For example (x + [2, 3]x3) + ([1, 2]x + [1, 4]x3) = [2, 3]x + [3, 7]x3, ([1,3] + [-1, 2]X)2 = [1,9] + [-6, 12]x + [-2, 4]x2. If P(y) and Q(y) are interval polynomials, then at any point y we can write P(y) E [PL, Pu], Q(y) E [QL, Qu]. We say that P(x) contains Q(x) on some interval [e, d] if PL ~ QL, Qu ~ Pu for all y E [e, d]. This is denoted by Q(x) c P(x). We now use capital letters to denote intervals; i.e., F(X) denotes the interval [FL' FH] where FL = minxEx f(x) and FH = maxxEX f(x). If we define x1n) by (n) [ i-I i 1 Xi = a + ~(x -a), a + ;;(x -a) , i = 1,2, ... ,n, (48.3) then, if f (x) is sufficiently smooth, ( 48.4) Define Q to the right-hand side of (48.4). The width of the interval Q, w( Q), can be shown to satisfy w(Q) ~ K(x -a)2 (48.5) n where K is a positive constant independent of n. The quadrature formula in (48.4) is essentially a first order integration formula for J f(t) dt. Higher order formulae are also available. See, for example, Corliss and Rall [4]. 220 IV Approximate Analytical Methods Example 1 This example illustrates the use of (48.4). Consider the integral I = fo1 sin 1rX2 dx. Using (48.4) with n = 2, we have (48.6) where F(X) = sin(7rX2). The expression in (48.6) can be evaluated to yield Example 2 Ie (F ([o,sin2 tlH + F ([0, sin' iD !) = [O,!] ! + [0,1] ! = [O,~] . Consider the integral I = f~l f(x) dx, where f(x) = 1 + [1, 2]x + x2. It is straightforward to show that x _ { [1 + 2x + x2, 1 + x + X2] for -1 ~ x ~ 0, f( ) -[1 + x + x2, 1 + 2x + X2] for ° ~ x ~ 1. Hence, we have Notes [1] The techniques presented in this section can be evaluated numerically. In­ terval arithmetic packages are available in Algol (see Guenther and Mar­ quardt (5)), Pascal (see Wolff von Gudenberg (10)), and FORTRAN (see ACRITH [1], but see also Kahan and LeBlanc (6)). [2] Corliss and RaIl [4] describe an interval analysis program that evaluates integrals by implementing Newton-Cotes rules, Gauss rules, and Taylor series. They also consider problems in which the limits of integration are intervals. 49. Laplace's Method 221 References [1] ACRlTH High Accuracy Subroutine Library: General Information Manua~ IBM publication # GC33-6163, Yorktown Heights, NY, 1985. [2] O. Caprani, K. Madsen, and L. B. RaU, "Integration of Rational Functions," SIAM J. Math. Anal., 12, 1981, pages 321-341. (3) G. Corliss and G. Krenz, "Indefinite Integration with Validation," ACM 7rans. Math. Software, 15, No.4, December 1989, pages 375-393. [4] G. F. Corliss and L. B. Rall, "Adaptive, Self-Validating Numerical Quadra­ ture," SIAM J. Sci. Stat. Comput., 8, No.5, 1987, pages 831-847. [5] G. Guenther and G. Marquardt, "A Programming System for Interval Arith­ metic," in K. Nickel (ed.), Internal Mathematics 1980, Academic Press, New York, 1980, pages 355-366. [6] W. Kahan and E. LeBlanc, "Anomalies in the IBM ACRlTH Package," IEEE Proc. 7th Symp. on Computer Arithmetic, 1985. [7] R. E. Moore, Interval Analysis, Prentice-Hall Inc., Englewood Cliffs, NJ, 1966, Chapter 8, pages 70-80. [8] L. B. RaU, "Integration of Rational Functions II. The Finite Case," SIAM J. Math. Anal., 13, 1982, pages 690-697. [9] J. M. Yohe, "Software for Interval Arithmetic: A Reasonable Portable Pack­ age," ACM Trans. Math. Software, 5, No.1, March 1979, pages 50-63. [10] J. Wolff von Gudenberg, Floating-Point Computation in PASCAL-SC with Verified Results, in B. Buchberger and B. F. Caviness (eds.), EUROCAL '85, Springer-Verlag, New York, 1985, pages 322-324. 49. Laplace's Method Applicable to Integrals of the form leA) = J: g(x)e>.J(x) dx, where I(x) is a real-valued function. Yields An asymptotic approximation when A» 1. Idea For A --+ 00 the value of leA) is dominated by the contributions at those points where I(x) is a local maximum. 222 IV Approximate Analytical Methods Procedure Given the integral J(A), consider the term e~J(:I:) for fixed A. This term will have a stationary point, (i.e., a local maximum or minimum) when de~J(:I:) /dx = 0, or f'(x) = o. The behavior of J(A) is dominated at the local maximums of Ii points where I'(x) = 0 and (usually) I"(x) < O. If the stationary point Xi is an interior point (i.e., a < Xi < b) then I(A) may be approximated, in the neighborhood of this point, as (49.1) which is valid as A -+ 00, when 1"(Xi) < O. If 1"(Xi) > 0, then the point Xi is a local minimum of I, not a local maximum, and this point does not contribute to leading order. (For the case II/(Xi) = 0, see the Notes, below.) For each point where I has a local maximum there will be a term in the form of (49.1). To find the asymptotic approximation to I(A), these terms must be summed up. For the stationary boundary points (i.e., those stationary points that are on the boundary of the domain, either Xi = a or Xi = b) there is a term in the form of (49.1), but with half the magnitude (if II/() is negative at that boundary point). This is because the integral in the fourth line of (49.1) becomes an integral from 0 to 00 or -00 and not from -00 to 00. If either of the boundary points is not a stationary point, (Le., I'(a) :/: o or f'(b) :/: 0), then these boundary points points contribute g(a)e~J(a) AI'(a) or g(b)e~J(b) AI'(b) (49.2) to the sum forming the the asymptotic approximation of I(A). The contri­ butions from non-stationary boundary points will always be asymptotically 49. Laplace's Method 223 smaller than the contribution from the points at which I is a local max­ imum. Hence, if the region of integration contains any points at which I is a local maximum, and if only the leading order behavior is desired, then the non-stationary boundary points can be ignored. Example 1 For the integral J(A) = Jaw e->'cosx dx we identify I(x) = -cos x, g(x) = 1, a = 0 and b = 10. The stationary points are where I'(x) = sin x = 0, or x = {O, 1f', 21f', 31f', 41f', ... }. We are only interested in those stationary points in the range of integration, that is x = 0, x = 1f', X = 21f', and x = 31f'. Including the boundary points x = 0 and x = 10 we have four points that can potentially contribute to the leading order term in the asymptotic expansion. x = 0 This is a stationary boundary point. However, since 1"(0) = 1 > 0, this point does not contribute to leading order. x = 1f' This is an interior stationary point. Since I" (1f') = -1 < 0, this point contributes a term of the form in (49.1): _ ) >.f(7r) 1f' _ ffi >. I7r -g(1f' e A I 1" (1f')I -V Te . x = 21f' This is an interior stationary point. However, since 1"(21f') = 1 > 0, this point does not contribute to leading order. x = 31f' This is an interior stationary point that is a local maximum since 1"(31f') = -1 < o. Hence, this point contributes a term of the >'/(3 ) 31f' _ ffi >. form in (49.1): I3n = g(31f')e 7r A I I" (31f')I -V Te . x = 10 This is a non-stationary boundary point. Hence, this point con- . g(lO)e'\/(1O) e->'CoslO trIbutes (from (49.2)): 110 = AI' (10) = -A sin 10 . We can combine all of the leading order contributions we have to find However, since we have only kept the leading order term in the asymptotic expansion near each point, we can only keep the leading order term in the final answer (assuming no cancellation of terms has occurred). Our final result is therefore: as A -t 00. 224 IV Approximate Analytical Methods Example 2 Consider the integral J(,\) = f01 e~:z;2 dx, so that J(x) = x2 and g(x) = 1. In this case the stationary points are given by I'(x) = 2x = 0, or x = o. The point x = 0 is a stationary boundary point, but it does not contribute to leading order since J"(O) = 2 > o. The point x = 1 is a boundary point and the leading order asymptotic approximation is given by (49.2): g(1)e~/(1) e~ J('\) ""J \ ~'/1 \ = 2,\ as ,\ ~ 00. Example 3 Consider the integral K('\) = f01 e-~:z;2 dx, so that J(x) = -x2 and g(x) = 1. In this case the stationary points are given by J'(x) = -2x = 0, or x = O. The point x = 0 is a stationary boundary point, and it contributes to leading order since J"(O) = -2 < o. The leading order asymptotic approximation is given by (49.1) with a factor of ~ (since x = 0 is a boundary point): 1 K('\) ""J -g(l)e~/(O) 2 211" 1 fir ,\ IJ"(O)I = "2V A as ,\ ~ 00. ( 49.3) For this integral, we recognize that K (A) = ~ Vi erf ( VX), where erf is the error function. Use of the asymptotic expansion of the error function for large arguments also results in (49.3). Example 4 An integral representation of the gamma function, for ,\ > 0, is r(A) = I."" x~-le-'dx = I."" x-le-·e~log·dx. (49.4) If ,\ is an integer, then r('\) = (,\ -I)! (see page 163). From (49.4) we have J(x) = -logx, but J(x) has no finite stationary point about which to apply our above expansions. The change of variable x = ,\y transforms (49.4) to r('\) = ,\~ 1.00 e~(-Y+logy) dy. (49.5) o y Now J(y) = -y + logy and g(y) = y-1. The minimum of J(y) is at l' (y) = 0, or y = 1. The value y = 1 is an interior stationary point of (49.5), so we have (from (49.1)) ""J ~ ~/(1) I 211" _ ~ -~ (2; r('\) ,\ g(l)e t \ 1 ~"/1 \1 -,\ e V T· ( 49.6) From (49.6), and a little manipulation, we obtain the leading term in Stirling's approximation to the factorial: r(n + 1) = n! ""J V211"nnne-n. 49. Laplace's Method 225 Example 5 Consider the integral J(A) = 1000 e,xx-(x-l) log x dx, for A » 1. If we make the obvious identification, f(x) = x, then the region of maximum contribution will be around x = 00. To determine this contribution, some re-scaling of the problem is required. Looking at the whole integrand, the stationary point is given by .!!:... (e,xX-(X-l) log X ) = 0, dx x-I or A=--+logx. x As suggested above, the stationary point occurs at a large value of x. If x is large then, approximately, the stationary point is given by x = e,x-l. Making the change of variable t = x / e,x-l, we are led to consider I( A) = e2(~-1) 100 te( e'-I ) (.-lIog .) dt = e2(~-1) J (e~-l) 1 where J(() = I:' tet;(t-tlogt) dt. Since we want A» 1, this corresponds to ( » 1. Now it is a simple matter to show that J(() ,..., et; J27r/( (since the only stationary point is at t = 1). Hence, we obtain our final answer: J(A) ,..., J2;e3(,x-l)/2ee~-1 as A --+ 00. Notes [1] Laplace's method is an application of the method of steepest descents, see page 229. In the method of steepest descents, the function f(x) can be complex valued. In the method of stationary phase (page 226), the function 1 (x) is purely imaginary. [2] If more terms are kept in (49.1), then we obtain the approximation ~,x [ 1 (g" gf"" g' fill 5g (Jill) 2) ] 1('\) '" V =>J"e / 9 + ~ -2f" + 8 (1,,)2 + 2 (1',)2 -24 (1',)3 + ... where all the functions are evaluated at x = Xi (see Bender and Orszag [IJ, page 273). If the asymptotic expansion in Example 4 were continued to higher order, then we would find rCA) ~ A>e->J¥- [1 + l!A + 28~A2 -... J. This yields a better approximation to the factorial function than the one­ term Stirling's approximation. [3] Watson's lemma (page 197) applies to integrals of the form I: e,xtl F(y) dy. By an appropriate change of variable, 1(,\) can be changed to this form. For example, we can use the transformation y = f(x) and then (assuming monotonicity) A = f(a), B = I(b), F(y) = g(x)/ J'(x). 226 IV Approximate Analytical Methods [4] If the second derivative vanishes at an interior stationary point, so that the leading order behavior at a stationary points is given by [5] [6] [7] for x near Xi, then the first term in the asymptotic approximation of I(A) becomes (assuming that n is even, and j<n)(Xi) < 0, both of which are required for Xi to be a local maximum): (49.7) Multidimensional analogues of this technique are described on page 199. Skinner 013] considers uniform approximations for integrals of the form e~h(t) J: e-~h(:Z:)g(x)xo-l dx as A ~ 00. Temme [4] considers uniform approximations for integrals of the form (l/r(A» Jooo t~-le-%t f(t) dt as A ~ 00. References [1] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scien­ tists and Engineers, McGraw-Hill, New York, 1978. [2] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, Inc., New York, 1986. [3] L. A. Skinner, "Uniformly Valid Composite Expansions for Laplace Inte­ grals," SIAM J. Math. Anal., 19, No.4, July 1988, pages 918-925. [4] N. M. Temme, "Incomplete Laplace Integrals: Uniform Asymptotic Expan­ sion with Application to the Incomplete Beta Function," SIAM J. Math. Anal., 18, No.6, November 1987, pages 1638-1663. 50. Stationary Phase Applicable to Integrals of the form J(A) = J: g(x)ei>.f(x) dx, where f(x) is a real-valued function. Yields An asymptotic approximation when A» 1. Idea For A --+ 00 the value of J(A) is dominated by the contributions at those points where f(x) is a local minimum. 50. Stationary Phase 227 Procedure The Riemann-Lebesgue lemma states that lim lb h(t)ei"t dt = 0, "-00 a provided that f: I h( t) I dt exists. In simple terms, if the integrand is highly oscillatory, then the value of the integral is "small." Now consider the integral I(A) = lb ei~J(·)g(x)dx. (50.1) Through an appropriate change of variables, it can be shown (in non­ degenerate cases) that I(A) -. 0 as A -. 00. The maximum contributions to (50.1) will come from regions where the integrand is less oscillatory. These regions are specified by the stationary points of the integrand, that is, where I'(x) = o. Let the stationary points in the interval [a, b) be {Ci}. Following a derivation similar to that given for Laplace's method (see page 221), we find that the leading order contribution to I(A), due to the stationary point c, is (assuming that f(c) :f: 0, f"(c) :f: 0, and g(c) :f: 0): ( ) 211' [ . () i 1r II ( )] lc "'-J g c Alf"(c)1 exp 'tAf c -4" sgnf c (50.2) where sgn denotes the signum (or sign) function. The leading order asymptotic behavior of I(A) is then given by Ei ICi. It is difficult to obtain a better approximation than just the leading order approximation, because it requires delicate estimation of integrals. If a higher order approximation is desired, then the method of steepest descents (see page 229) should be used. Example The Bessel function In(x), for integral values of n, has the integral representation (see Abramowitz and Stegun [1], 9.lo22.b) 1171' In(A) = -cos(nt -Asint)dt 11' 0 (50.3) = ~ L 111' e±inteTi"sint dt. 211' ± 0 Each integral in the sum has the form of (50.1), with f±(t) = =Fsint and g±(t) = e±int. On the interval [0,11'] the only stationary point for f± is at t = 11'/2. Hence, we find f± (i) = =Fl and f~ (i) = =Flo Therefore, (50.2) can be evaluated to yield In(A) "'-J ~ L exp( ±in 211') exp( ±i [-A + !4]) V211'A ± 228 IV Approximate Analytical Methods for A » 1. Notes [1] Determining the asymptotic behavior of an integral by plugging into the above formulas is a dangerous approach. A simple example where this naive approach could go wrong is with the integral I = J~oo ei(3~t2 -3t3 ) dt. Use of the above formulas would result in a stationary point at t = 0, which leads to the incorrect approximation J 1r /3>..ei•tr /4. For this integral, there are stationary points at both t = 0 and t = >... Using the contributions from both of these stationary points results in the approximation I '" 2 -e cos - - -~ i~3 (>..3 1r) 3>" 2 4 . [2] If there are no stationary points in the interval of integration, then the leading order asymptotic behavior is determined by the contribution near the limits of integration. The leading order behavior, in this case, can be determined by integration by parts. [3] If the leading order expansion of I{t), near the critical point t = c, is given j<n)(c) by I{t) = I{c) + --I -{t -c)n + ... , then the leading order behavior of I n. is given by (assuming, again, that g{c) i-0): Equation (50.2) is just this formula evaluated at n = 2. Note that if the stationary point is a boundary point, then the factor of 2 in (50.4) does not appear. As an example, consider the Bessel function at large order and large ar­ gument. That is, consider the integral (see (50.3» Jm{m) = 1r-1 Jo1r cos{mt­ m sin t) dt, for m » 1. Writing this as Jm(m) = (21r)-1 L± Jo1f e±im(t-sin t) dt, we identify I{t) = t -sin t. For this integral, the only stationary point is at c = 0 (since I'{c) = 0). At this stationary point the second derivative vanishes: I" CO) = o. Directly keeping terms of the next order results in Jm(m) = Re (; [ .,m(.;D'-') dt) ~ Re (; [ .'n"'O dt) = .!.2-2/33-1/6r (~) m -1/3. 1r This result could also have been obtained from using (50.4). We have get) = 1/1r, I{t) = sin t -t, f'{t) = cos t -1, I"{t) = -sin t, and IIII{t) = -cos t. Hence, we find that c = 0, n = 3, and I/UI(c)1 = 1. Using (50.4) and 50. Stationary Phase 229 removing the factor of 2 since the stationary point is a limit of the integral we find Jm{m) ~ Re (r ~!) (~)"3 exp (_ i;) ) ~ ; r ~~) (~) 1/3 cos (-i) = .!.2-2/33-1/6r (~) m-1/3• 1T [4] Stationary phase is an application of the method of steepest descents, see page 229. In the method of steepest descents, the function !(x) can be complex valued. In Laplace's method (page 221), the argument to the exponential is purely real. [5] This method was first applied in Stokes [9]. A rigorous justification of the method is presented in Watson [10]. References [1] M. Abramowitz and 1. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. [2] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scien­ tists and Engineers, McGraw-Hill, New York, 1978. [3] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, Inc., New York, 1986, Chapter 6, pages 219-251. [4] R. A. Handelsman and N. Bleistein, "Asymptotic Expansions of Integral Transforms with Oscillatory Kernels; A Generalization of the Method of Stationary Phase," SIAM J. Math. Anal., 4, No.3, 1973, pages 519-535. [5] N. Chako, "Asymptotic Expansions of Double and Multiple Integrals Oc­ curring in Diffraction Theory," J. Inst. Maths. Applies, 1, December 1965, pages 372-422. [6] J. C. Cooke, "Stationary Phase in Two Dimensions," IMA J. Appl. Math., 29, 1982, pages 25-37. [7] F. De Kok, "On the Method of Stationary Phase for Multiple Integrals," SIAM J. Math. Anal., 2, No.1, February 1971, pages 76-104. [8] J. P. McClure and R. Wong, "Two-Dimensional Stationary Phase Approx­ imation: Stationary Point at a Corner," SIAM J. Math. Anal., 22, No.2, March 1991, pages 500-523. [9] G. G. Stokes, "On the Numerical Calculation of a Class of Definite Integrals and Infinite Series," Camb. Philos. 1rans., 9, 1856, pages 166-187. [10] G. N. Watson, "The Limits of Applicability of the Principle of Stationary Phase," Proc. Camb. Philos. Soc., 19, 1918, pages 49-55. 230 IV Approximate Analytical Methods 51. Steepest Descent Applicable to Integrals of the form Ie e~f(x)g(x) dx, as A tends to infinity, where C is a contour in the complex plane and J(x) and g(x) may be complex. Yields An asymptotic approximation when A » 1. Idea Given an integral in the form of Ie e~f(z)g(z) dz deform the contour of integration so that it is in the form of a Laplace integral (see page 221) and apply the method described there. Procedure Some definitions are needed before the method can be described. Given the complex analytic function J(z), of the complex variable z (i.e., z = x + iy), let JR and JJ denote the real and imaginary parts of that function (i.e., J(z) = JR(Z) + iJJ(z)). Given a point zo, a directed curve from Zo along which JR(Z) is decreasing is called a path of descent. The path of steepest descent is the curve whose tangent is given by -'\1 JR. This path is also one of the curves along which JI(Z) is constant. That is, a curve of steepest descent is also a curve of constant phase. Suppose that J(z) and its first n -1 derivatives vanish at z = Zo: dq~1 =0, forq=I,2, ... ,n-l, dz Z=%o a,n J I = aeict, with a > o. dzn %=Zo (51.1) That is, J(z) = J(zo) + J(n)(zo)(z-zo)n/n!+ .... If z = zo+cei9, then the directions of constant phase (i.e., JI is constant), from the point z = zo, are given by 1r a Op = (2p + 1)-- -n n (51.2) for p = 0,1, ... , n -1. Furthermore, JR decreases where cos(nOp + a) is positive, and increases where cos(nOp + a) is negative. Using a = 0 for illustrative purposes, Figure 51.1 shows the regions of increasing and decreasing values of JR. (A nonzero value for a would just rotate the shaded regions in Figure 51.1.) The method of steepest descents can be succinctly described by the following steps: 51. Steepest Descent 231 Figure 51.1 For a = 0, the regions of increasing JR are shown clear, the regions of decreasing JR are shown shaded. The paths of steepest descent are shown dashed, the paths of steepest ascent are shown dotted for (a) n = 2 and (b) n=3. [1] Identify the possible critical points of the integrand. These are the endpoints of integration, singular points of J(z) and g(z) and saddle points of J(z). (A saddle point is a point at which dJ /dz vanishes.) [2] Determine the paths of steepest descent from each of the critical points. A picture of the complex plane with curves drawn is usually very helpful. [3] Justify, via Cauchy'S theorem, the deformation of the original contour of integration C onto one or more of the steepest descent paths that were determined that connect the endpoints of integration. [4] The original integral is then equal to any contributions determined from the application of Cauchy's theorem and an integral along every steepest descent contour that has been used. Each of these integrals has the appropriate form so that Laplace's method may be used to determine the asymptotic value (see page 221). Special Case In the special case that n = 2, then we can formally proceed as follows. The function J (z) can be expanded about the point z = Zo to find J (z) - J(zo) = i J"(zo)(z -zO)2 + .... If a = arg J"(zo) and z = Zo + eei8, then this expansion can be written J(z) -J(zo) = i IJ"(zo)1 eiQe2e2i8 + ... = i IJ"(zo)1 (cos(a + 29) + i sin(a + 29» + .... The steepest paths are given by sin( a + 29) = o. The paths of steepest descent are given by cos( a + 29) < O. For n = 2, the complex plane has the appearance shown in Figure 51.1.a. If the contour can be made to pass through the saddle point, via curves of steepest descent, then the integral 232 IV Approximate Analytical Methods will have the form J(A) = 1 g(z)e"f(z) dz = f g(z)e"f(Z) dz -f g(z)e"f(z) dz. Jpath into saddle Jpath out of saddle Let A represent a point from which the contour enters the saddle, and let B represent a point that the contour approaches after leaving the saddle. Then the last integral can be written as J(A) = JoA g(z)e"f(z) dz- JoB g(z)e"f(z) dz. Changing variables by -r = I(z) -I(zo), these integrals can be written as lA dz lA dz J(A) = e"f(zo) e-"r g(z(r))- dr -e"f(zo) e-Mo g(z(r))- dr. o dr 0 dr (51.3) The Laplace type integrals in (51.3) will be dominated by the contributions near r = 0, which corresponds to z = zoo Near this point we have r = -(/(z) -/(zo)) = -!/"(zo)(z -Zo)2 + .... Inverting this relation we find (z -zo) = V 11"(%0)1' ! + ~e-ia/2rl/2 _ ~e-ia/2rl/2 V 1l"(Zo)1' This can be used to evaluate the dz/dr in (51.3). Substituting for this term, and extending both A and B to 00 (to get the leading order approximation) results in 2 (100 -"r 100 -"r ) 1/"(zo)1 e-ia/2g(zo) 0 ~JT dr + 0 ~JT dr 2 e"f(zo)-ia/2 1/"(zo)1 Al/2 "J g(zo) (51.4) This result should be used with caution because of the many assumptions in its derivation. It is usually best to work out each problem from scratch, without resorting to formulas like (51.4). 51. Steepest Descent 233 Imz'"T' ..... -_ ~ ···:~~~11S~·:·:·· .... , , Zo •... , .... i':;"$.t~:::... ' , , ;r Co Coo " I ' .... I .... .................... > .... -::-......... :::.:. C Rez • -zo Figure 51.2 The geometry of the complex plane for the integral in (51.5). Example 1 Consider the integral I(~) = 100 eA(z+iz-z3 ) dz (51.5) as A ~ 00. The integration contour, C, consists of the positive real axis. Identifying I(z) = z + iz -z3, we find that the saddle points are given by !' = 0 or z = ±zo = ±21/43-1/2ei1l'/8. Near the point z = Zo we can expand I(z) -I(zo) = ~/"(ZO)(Z -ZO)2 + .... Using z = Zo + eeiO, this expansion can be written as I(z) -I(zo) = _21/431/2e2 [cos (i + 20) a + i sin (i + 20)] . A path of constant phase has Im(/(z) -I(zo)) = 0, or sin (7r 18 + 20) = o. A path of descent (as opposed to a path of ascent) has Re(1 (z) -I (zo)) decreasing, therefore cos (7r/8 + 20) > O. Figure 51.2 shows the original contour of integration (C), the two saddle points (±zo), and the structure of the saddle around the point z = zo0 The steepest descent contours Co and Coo are also shown in Figure 51.2. The immediate problem is that the given integration contour C is not a steepest descent contour. However, using Cauchy's theorem, we can deform the integration contour as long as we account for any singularities that we may cross. Because there are no singularities of the integrand in the region bounded by the contours, Cauchy'S theorem tells us that (l + i. -lJ eA/(z) dz = O. 234 IV Approximate Analytical Methods Therefore, if we can determine the integrals along the steepest decent contours then we can determine J ('\). Changing variables by -r = J(z) -J(zo), we find (51.6) At this point we have reduced the problem to calculation of Laplace type integrals. Evaluating these integrals is straightforward. Near z = zo, which is where the dominant contribution of the integrals in (51.6) comes from, we have r = -!J"(zo)(z -zO)2 + .... This relation can be inverted to find _ {+{3r1/2 z -zo --{3r1/2 along Co, along Coo, where (3 = 2-1/83-1/4e-i7f/16. Therefore, (51.6) becomes Example 2 as A ~ 00. Consider the integral representation of the Airy function 2100 (t3) 1 f ( w3 ) J(s) = Ai(s) = -; -00 cos 3" + st dt = 27ri lc exp sw -3"" dw as s tends to positive infinity. The contour of integration is shown in Figure 51.3 (see page 3). Since this integral is not in the form we require, we change variables by z = wVs and ,\ = s3/2 to obtain A 1/3 f ([ z3] ) J(A) = J(s) = 27ri lc exp A z -3" dz. (51.7) 51. Steepest Descent 235 Figure 51.3 Contour of integration for (51.7). i'-. C4 tt:::: .. ..::::t~~~ C 3 Figure 51.4 The geometry of the complex plane for the integral in (51.7). Note that the contour remains unchanged. We can now proceed with the steps described above. In this case, I(z) = z -z3/3 and g(z) = 1. Since both I(z) and g(z) are analytic, the only critical points are the saddle points of I(z) and, possibly, the endpoints of the contour of integration. The saddle points of I(z) are given by I'(zo) = 1 -z6 = 0, or Zo = ±1. Expanding I (z) into real and imaginary components results in I(z) = z -~z3 = IR(Z) + ill(Z) = x (1 -ix2 + y2) -iy (x2 -iy2 -1) . The curves of steepest ascent and descent at the points Zo = ± 1, satisfy 11(Z) = 11(Z = ±1) = 11(X = ±1, y = 0) = O. That is, y (x2 -iy2 -1) = o. These curves are shown in Figure 51.4. We can now use (51.1) and (51.2) to determine exactly what the steepest descent paths are from the saddle points. Since I" (z) = -2z we find that I"(z = -1) = -2 = 2ei1l' and I"(Z = 1) = 2 = 2eiO. Hence, (tz=l = 1r and (tz=-l = O. We conclude, therefore, that: 236 IV Approximate Analytical Methods ). Imt:"( ~ B : c ------>-(---D .................. :I~············~~ ... :.". ~ A Figure 51.5 The geometry of the complex plane for the integral in (51.8). [1] The directions of steepest descent from z = -1 lie along the hyperbolas x2 -iy2 = 1 (these are labeled C1 and C2 in Figure 51.4). [2] The directions of steepest descent from z = 1 lie along the line y = 0 (these are labeled C3 and C4 in Figure 51.4). Now that we have identified the curves of steepest descent, we would like to move our given contour to one or more of these steepest descent contours (justified, of course, by Cauchy's theorem). Since the integrand is analytic (it has no poles or branch cuts) the original contour C can be deformed into the new contour to obtain C = -C1 -C4 + C3 (the plus and minus signs are needed since the contours shown in Figure 51.4 are oriented). We conclude that I(A) = ~~: (-1. -1. + 1,) exp (A [Z -Z:]) dz. These integrals are easily evaluated by Laplace's method. We find (see Bleistein and Handelsman [3], page 267 for details) I(A) "-J _ (Al/6 e-2>./3) _ (_ ~1/6 e2>'/3) + (~1/6 e2>'/3) 4y'7f 4z y'7f 4z y'7f "-J A 1/6 e2>./3 as A --. 00. 2iy'7f -1/4 This leads to Ai(s) "-J _s _e-(2s2 /3/3) as s --. 00. 2y'; Example 3 Consider the integral for the reciprocal of the Gamma function (see Copson [5]) () 1 1 r t->.t I A = r(A) = 27ri lc edt, (51.8) where A may be real or complex, the principal value of the logarithm is taken, and the contour C is shown in Figure 51.5. (The contour C starts at -00, circles the origin once in the positive sense, and returns to -00.) We are interested in (51.8) as A --. 00. Changing variables by t = AZ results in the representation 51. Steepest Descent 237 I(A) = 1 f e'\(z-logz) dz. 27riA,\-1 lc From this expression we have 9 (z ) = and I (z ) = z -log z. The only stationary point is at f' (zo) = 0 or Zo = O. At the stationary point we find I" (zo) = 1. A simple computation shows that the paths of steepest descent are as indicated in Figure 51.5. The contour C in Figure 51.5 can be trivially moved to go through the stationary point. Hence, the asymptotic behavior of I(A) is determined by the two paths of steepest descent indicated in Figure 51.5. Using the substitution z = 1 + is, and then the expansion log(l + x) = x -!X2 + lx3 -... , we find If the above Laplace type integrals were expanded to higher order, then we would obtain the expansion 1 ( e ) ,\ {T ( 1 1 139 ) I(A) = r(A) = X V 2; 1 -12A + 288A2 + 51840A 3 -• • • • (51.9) Example 4 Consider the integral I(A) = Jo1log zei,\z dz, for A »1. For this integrand, the half-plane with 1m z > 0 is a region of descent. The steepest descent contours are the lines with Re z equal to a constant. In this case the integration contour C is deformed into the 3 contours C1, C2, and C3 (see Figure 51.6). By Cauchy's theorem we have I(A) = Jc = JCI + JC2 + JC3' where the integrand is the same for each integral. In the limit of the rectangle extending vertically to infinity, we find that I(A) = J~oo + Jl1+iOO' The first of these integrals can be evaluated exactly /.'00 log zeu, dz = i [ log ( is )e -As ds .log A i-r + 7r /2 = -t-A---.,;.....-A~-(51.10) 238 IV Approximate Analytical Methods Imz it\ ..................... 1. .............................................................. . :::::::::::::::::::::t;:: ~: ~: ~: ~ >.: ~: ~: ~: ~: ~;::::::::::::::::::::: ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~f:~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ : ~ : ~ : ~ ~ :::::::::::::::::::: :,::::::::::::::::::::::::::::::::::::::::::;::::::::::::::::::::: .!l_.~l!l!lll:_~c'!!! I! I! I! I: I;: i! I! I!: i I! I! I! I! II ~]_jl._jl._l> I Rez Figure 51.6 Integration contour and steepest descent contours for Jo1}og zei>.z dz. where 'Y is Euler's constant. The second integral in the above can be evaluated by Watson's lemma. Using log(l + is) = -E~=1 (-is)" In, we find /.1 logzei>..z dz = -i /.00 log(l + is)ei>..(l+is) ds l+ioo 0 • i>.. ~ (-i)"(n -I)! as \ --. 00. f'oI ,e L..J A ,,+1 1\ ,,=1 (51.11) Combining the results in (51.10) and (51.11) results in an asymptotic expansion of I(A). Full details may be found in Bender and Orszag [1], pages 281-282. Program 51 s: if mod(capN.2)=O then 1 else ( if tpos then 1 else -1) $ vp[k]:= block([temp]. if k=O then vex) else (temp:vp[k-1], ratsimp(diff(temp.x»»$ phip[k]:= block([temp]. if k=O then phi (x) else (temp:phip[k-1]. ratsimp(diff(temp.x»»$ vpz[k]:=block([temp]. temp:vp[kl. ev(temp.x=zO) )$ phipz[k]:=block([templ. temp:phip[k]. ev(temp.x=zO) )$ alpha[n]:= (vpz[capN+n]/(capN+n)!) / (vpz[capN]/capN!) $ 51. Steepest Descent term(m,k,capN,kp):=«m-k-kp)*(m+l)/capN+kp) * (m-kp) !/k! * alpha [m-k-kp) *D [m,m-kp) $ D[m,k):= block([kp), if m=k then gamma( (m+l)/capN )/(capN*m!) 239 else factor( -l/(m-k)*sum( term(m,k,capN,kp), kp,O,m-k-l) »$ hold:-s*wpz[capN)*name/capN!$ specialsum3(mp):=sum( D[2*mp,k)*phipz[k), k, 0, 2*mp)$ Xe**(name*v(zO» * sum( 2* hold**(-(2*mp+l)/capN) * specialsum3(mp), mp,O,numberofterms) ; Notes [1] This is also called the saddle point method. [2] Bleistein and Handelsman [3] treat the more general problem 1(,\) = Ie g(x)H(.\w(x» dx where g(x) and w(x) are analytic in the region containing the contour C, and.\ -+ 00. Here H(z) is an entire transcendental function, such as sin z, cos z, or Ai( z ). [3] In many specific cases, it is often impractical to construct paths of steepest descent. In some problems it may be easier to use Perron's method (see Perron [6]), which avoids the explicit construction of steepest descent paths. See Wong [7] for details. [4] Campbell, Froman, and Walles [4] give explicit series for the asymptotic approximation of I e->"w(z)cp(z) dz as .\ -+ 00 for a well isolated singular point zoo A computer program written in REDUCE is also given. If w(j)(zo) = 0 for j = 1,2, ... , N -1 and w(N)(zo) =f:. 0 then the only inputs required for the program are w(z) and 4>(z) (and their derivatives), N, Zo, the number of terms desired in the expansion, and the sign of the parameter in the reparametrization of w(z) near the singular point. That is, near Zo we write w(z) = w(zo) -stN where { + 1 when N is even, s = +1 when N is odd and t > 0, -1 when N is odd and t < O. The author has re-written the program in Macsymaj it is shown in Program 51. As an illustration of the Macsyma program, consider the calculation in Example 3 (with .\ replace by -v). If the following program segment is placed before the code shown in Program 51: capN:2$ zO:l$ numberofterms:3$ tpos:false$ v(x):=x-log(x)$ phi(x):=l$ name:nu$ then the following output is obtained: 240 IV Approximate Analytical Methods sqrt(2) sqrt(~pi) sqrt(2) sqrt(~pi) sqrt(2) sqrt(~pi) (----------------- + ----------------- + -----------------sqrt(-nu) 3/2 5/2 12 (-nu) 288 (-nu) 139 sqrt(2) sqrt(~pi) nu ----------------------) 1.e 7/2 61840 (-nu) This result must be multiplied by _ .~ 1._1 to obtain the answer in (51.9). References [1] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scien­ tists and Engineers, McGraw-Hill, New York, 1978. [2] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, Inc., New York, 1986. [3] N. Bleistein and R. A. Handelsman, "A Generalization of the Method of Steepest Descent," J. Inst. Math. Appls., 10, 1972, pages 211-230. [4] J. A. Campbell, P. E. Froman, E. Walles, "Explicit Series Formulae for the Evaluation of Integrals by the Method of Steepest Descents," Stud. Appl. Math., 77,1987, pages 151-172. [5J E. T. Copson, Asymptotic Expansions, Cambridge University Press, New York, 1965. [6] O. Perron, "Uber die naherungsweise Berechnung von Funktionen groBber Zahlen," Sitzungsber. Bayr. Akad. Wissensch., Miinch. Ber., 1917, pages 191-219. [7] R. Wong, Asymptotic Approximation of Integrals, Academic Press, New York, 1989. 52. Approximations: Miscellaneous Idea This section contains some miscellaneous integral approximation tech­ niques. Procedure Consider the integral 1(s) = 1000 e-st f(t)g(t) dt. If f(t) is analytic, then it can be expanded in a Taylor series. Hence, 1 can be re-written as 1(8) = ~ f(:~O) f' e-"g(t)tn dt 00 f(n)(o) ~ = ~ --(-I)n-'O(s) ~ n! dsn n=O (52.1) where 'O(s) = 1000 e-stg(t) dt is the Laplace transform of get). 52. Approximations: Miscellaneous 241 Table 52. Some infinite series expansions for integrals of the form Jooo f(t)g(t) dt. 100 00 f(n)( ) e-ot f(t) dt = ""' __ 0 L-, an+1 o n=O cosatf(t)dt= I)-I)" 2n+20 100 00 1'2n+1)( ) o n=O a 100. f( ) 00 f(2n-1)(0) sm at t dt = ~ f( 0) -""' ( -1)" ....:.....-_..,...:........ o t 2 L-, (2n -l)an n=O Setting s = 0 in (52.1) allows an infinite series expansion for 1(0) = Jooo f(t)g(t) dt to be obtained. Each choice of g(t) results in a different expansion, some are given in Table 52. These results are only formally correct; in practice, the resulting expressions may be asymptotically valid. Example Using the fourth expansion in Table 52, with f(t) = sin t, we readily determine that J '-100 sin mt sin t d _ ~ 1 .- t -L-, . o t n=O (2n + ~)mn (52.2) A table of integrals shows that J = ~ log (: ~ ~), when m > 1. If this result is expanded around m = 00, then the result in (52.2) is obtained. Note [1] This technique is from Squire [1], who credits Willis [2]. References [1] W. Squire, Integration for Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970, pages 105-107. [2] H. F. Willis, "A Formula for Expanding an Integral as a Series," Phil. Mag., 39, 1948, pages 455-459. v Numerical Methods: Concepts 53. Introduction to Numerical Methods The last sections of this book are concerned with numerical meth­ ods for approximating definite integrals. Section V defines some of the terms used in numerical methods, describes different software libraries, and displays integrals that have appeared in the literature as test cases for numerical routines. Section VI describes a selection of many possible numerical techniques. Note that a definite integral, say I = J: J(x) dx, can be interpreted as the solution of a differential equation. In this interpretation, I = y(b) where y' = J(x) and y(a) = O. Hence, any numerical integration technique for differential equations can be adapted to numerically evaluate an integral. (See Zwillinger [4] for some numerical techniques for differential equations.) In fact, for an integrand that has.an integrable singularity, at an unknown location in the integrand, it is probably preferable to use a differential equation technique. One of the most comprehensive books on numerical integration tech­ niques is Davis and Rabinowitz [1]. It contains a comprehensive bibliog- 243 244 V Numerical Methods: Concepts raphy, up to 1984. The book by Stroud [2] contains extensive tables of numerical methods. The most popular computer library that the author is aware of is Quadpack (see Piessens at al. [3]). This library implements quadrature rules with the following desirable features: • All nodes are within the integration interval (sometimes the nodes include the end-points). • All weights are positive. References [I} P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984. [2} A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall Inc., Englewood Cliffs, NJ, 1971. [3} R. Piessens, E. de Doncker-Kapenga, C. W. Uberhuber, and D. K. Kahaner, Quadpack, Springer-Verlag, New York, 1983. [4} D. Zwillinger, Handbook of Differential Equations, Academic Press, New York, Second Edition, 1992. 54. Numerical Definitions Adaptive Quadrature For the integral I = fol f(x) dx, traditional quadrature rules use estimates of the form I ~ f = E~=l wif(xi) for some specific choice of the weights {Wi} and nodes {Xi}. An adaptive quadrature technique uses an algorithm to choose the weights {Wi} and nodes {Xi} during the computation. Thus, an adaptive algorithm dynamically adapts the integration rule to the particular properties of the integrand. Automatic Quadrature Routine This is a piece of software, designed to provide an approximation of specified tolerance to a given definite inte­ gral. Closed Rule Consider an approximation to the integral 1= f: I(x) dx. A quadrature rule that uses the values of f(a) and feb) in the numerical approximation of I is called a closed rule or a closed formula. Composite Rule If a numerical quadrature rule for integrating over a small parallelpiped region is repeatedly applied to find a rule applicable over a larger parallelpiped region, the new rule is called a compound or composite rule. See page 282. Fully Symmetric A region B is said to be fully symmetric if x = (Xl! X2,"" Xd) E B implies that (±Xp(I), ±Xp(2) , ••• , ±Xp(d») E B, where the set {p(I),p(2), ... ,p(d)} is any permutation of {I, 2, ... , d}. 54. Numerical Definitions 245 General Purpose Integrators A numerical quadrature rule is said to be general purpose if the routine works for generic integrands. If a routine requires that the integrand have the form w(x)f(x), for some special weight function w(x), then the routine is called special purpose. Nodes In the quadrature rule f: f(x) dx ~ Li wif(Xi), the distinct numbers {Xi} are called the nodes (also called abscissas, or points). Null Rule The numerical quadrature rule f: f(x) dx ~ Li wif(Xi) is said to be a null rule if Li Wi = 0 and at least one of the {Wi} values is non-zero. Furthermore, a null rule is said to have degree d if it integrates to zero all polynomials of degree less than or equal to d and fails to do so for f(x) = xd+1. Numerically Stable A numerical quadrature rule is said to be numer­ ically stable if all of the weights are positive. Open Rule Consider an approximation to the integral I = f: f(x) dx. A quadrature rule that does not use the values f(a) or f(b) in the numerical approximation of I is called an open rule or an open formula. This would be needed, for instance, if a = 0 and f(x) = X-1/2. In this case, I has an integrable singularity, but f(O) is infinite and should not be computed by a quadrature routine. Order Given a quadrature rule of the form b N 1 f(x) dx = L anf(xn) + E[f], a n=1 there exists a largest integer k such that E[P] vanishes for every polynomial p of degree less than k. The number k is usually called the order and k -1 is called the degree of precision of the rule. We have the bound k ::; 2N, where equality only holds for the Gauss-Legendre rule. Panel Rules Given an integral to evaluate 1= f: f(x) dx, the interval of integration [a, b] can be sub-divided into m equal length intervals, called panels. (The number m is called the mesh ratio.) Then one integration rule can be used on each panel, and the resulting approximation to I is called an m-panel rule. See also "composite rules," page 282. Positive Quadrature Rule The quadrature rule fOI f(x) dx ~ L~=1 Wi f(Xi) is called a positive quadrature rule if all of the weights are positive, Wi> O. Weights In the quadrature rule f: f(x) dx ~ Li wif(Xi), the numbers { Wi} are called the weights. 246 V Numerical Methods: Concepts 55. Error Analysis Applicable to Integration rules of the fprm t I(x)th mo ml mn ~ prj] := L akoJ (XkO) + L aklJ' (xkd + ... + L aknJ(n) (Xkn) k=O k=O k=O Yields An estimate of the error in using prj] to approximate J: J(x) dx. Idea An exact formulation of the error can sometimes be obtained. Procedure Define the error made when using prj] to approximate J: J(x) dx to be EIIl = PI/] -t I(x) th. A theorem by Peano states: Theorem (Peano): Suppose that Ell] = 0 for all polyno­ mials I with degree no greater than n (Le., polynomials of low degree are integrated exactly). Then, for all functions f E cn+1 la, b] the error can be represented as Elfl = /." ,cn+1l(t)K(t) dt where K(t) is the Peano kernel of E and is defined by 1 K(t) = ,E[g(xj t)], n. g(x; t) = (x -t)+ = {o(x -t)n if x ~ t, if x < t. (55.1) (55.2) (55.3) Note that t is just a parameter in the 9 function and that E operates only with respect to the x variable. (When evaluating the E function in (55.3), the representation in (55.1) should be used.) This theorem can be used to determine the error in integration rules explicitly. For many integration rules, the Peano kernel has a constant sign on [a, b]. Using the mean value theorem (see page 83) on (55.2) then allows the error to be written as Ell] = l(n+1l(€) t K(t) dt, (55.4) 55. Error Analysis 247 for some ~ E [a, b]. Hence, by determining J: K(t) dt, a simple, exact representation of the error is obtained. There is an easy way to determine the integral of K(t). Since (55.4) is presumed to be valid for all f E Cn+1[a, b] we can use f(x) = xn+l in (55.4) to obtain E[xn+l] = (n + 1)! J: K(t) dt so that for some e E [a, b]. Example E[f] = E[xn+l] f(n+l)(e) (55.5) (n + 1)! Consider using the Peano theorem to find the error in Simpson's rule. In Simpson's rule we have the approximation l, f(x) dx .., if( -1) + U(O) + V(1) and hence, E[jj = [V(-I) + U(O) + V(I)]-l, f(x)dx. For this integration rule, all polynomials of degree less than or equal to three are integrated exactly. Therefore, we can apply the theorem with n = 3. The Peano kernel becomes K(t) = 1E[g(xj t)] = 1E[(x -t)t] = 1 { [iH- t)~ + ~(O -t)~ + i(l-t)~]-l, (x -t)~ dx}. When t is in the range [-1, 1] this expression can be simplified to yield K(t) = 1 { [1.03 + ~(-t)~ + W -t)3]_ J.' (x -t)3 dx} = 1 {[~(-t)t + 1(1-t)3] -!(1-t)4}. The term (-t)t depends on whether or not t ~ O. Specifically, (-t)3 = {O if t ~ 0 + -t3 if t < 0 . Using this in the formula for K(t) we finally arrive at K(t) = { ;2 (1 -t)3(1 + 3t) if 0 ~ t ~ 1 K ( -t) if -1 ~ t ~ O. 248 V Numerical Methods: Concepts With Peano's kernel explicitly determined, we observe that it has a constant sign (that is, positive) on the interval [-1,1]. Hence, by carrying out the integration in (55.4) we find E[!l = j(4)<el [" K(t) dt = gI0f(4)(e) for some e in the range [-1, 1]. We could also have obtained the factor of g10' using (55.5): Notes Et4] = ;4 ([1(-1)4 + ~04 + 114]-t x4dx) = 2~ (~ + ~ -~) = g~. [1] The usual way of estimating the error using a quadrature routine is to use two different rules, say A and B, and then estimate the error by IA -BI, or some scaling of this. In the case of adaptive quadrature (see page 277), the subdivision process provides additional information that can be used. See, for example, Espelid and Sorevik [3]. [2] Interval analysis is a technique in which an interval which contains the numerical value of an integral is obtained, see page 218. [3] For most automatic quadrature routines, an absolute error Ca and a relative error er are input. For the integral I = J: f(x) dx, the routine will compute a sequence of values {Rnle, Enle }. Here, Rnle is an estimate of I using nk values of the integrand, and Enle is the associated error estimate. The routine will terminate (and return a value), when the error criteria is achieved. [4) Piessens at al. [10), Section 2.2.4.1, has a summary of the asymptotic ex­ pansion of integration errors. Piessens at al. [10] (page 40) also has the (pessimistic) error bound: Let If(k+1)(x)1 ~ Mk+l for a ~ x ~ b. Then the absolute error of the positive quadrature rule of precision d > k satis­ fies: 55. Error Analysis 249 [5] Some integration rules have errors that can be represented as sums of terms with each term involving derivatives of the integrand at an endpoint of integration. For example, the composite trapezoidal rule has this property (see page 342). Given an integrand, suppose that a transformation can be found such that the derivatives of the new integrand vanish at the endpoints. Then, using this transformation with a rule of the above type should result in small errors. See Davis and Rabinowitz [2] for details. [6] Some quadrature rules have the error decreasing exponentially with the number of nodes. For example, the "tanh rule" of Schwartz is given by 11 N f (tanh 1 hk) ( ) f(x) dx = h L 2 ; + 0 e-c../N -1 k=-N 2 cosh '2hk for some c > 0, where h = h(N) is given by a certain formula (and h rv CN-1/2). See Kahaner et ale [6]. References [1] J. Berntsen and T. O. Espelid, "Error Estimation in Automatic Quadrature Routines," ACM 7rans. Math. Software, 17, No.2, 1991, pages 233-252. [2] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, Chapter 4, pages 142-144 and 271-343. [3] T. O. Espelid and T. Sorevik, "A Discussion of a New Error Estimate for Adaptive Quadrature," BIT, 29, 1989, pages 283-294. [4] K.-J. Forster and K. Petras, "Error Estimates in Gaussian Quadrature for Functions of Bounded Variation," SIAM J. Math. Anal., 28, No.3, June 1991, pages 880-889. [5] W. Gautschi, E. Tychopoulos, and R. S. Varga, "A Note on the Con­ tour Integral Representation of the Remainder Term of a Gauss-Chebyshev Quadrature Rule," SIAM J. Numer. Anal., 27, No.1, 1990, pages 219-224. [6] D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Pren­ tice-Hall Inc., Englewood Cliffs, NJ, 1989. [7] C. Schneider, "Error Bounds for the Numerical Evaluation of Integrals with Weights," in Numerical Integration III, Birkhauser, Basel, 1988, pages 226- 236. [8] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, translated by R. Bartels, W. Gautschi, and C. Witzgall, Springer-Verlag, New York, 1976, pages 123-127. [9] A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall Inc., Englewood Cliffs, NJ, 1971, Chapter 5, pages 137-192. [10] R. Piessens, E. de Doncker-Kapenga, C. W. Uberhuber, and D. K. Kahaner, Quadpack, Springer-Verlag, New York, 1983. 250 V Numerical Methods: Concepts 56. Romberg Integration / Richardson Extrapolation Applicable to Numerical techniques for integrals. Yields A procedure for increasing the accuracy. Procedure Suppose that a grid with a characteristic spacing of h is used to approximate the integral 1 = f: I(x) dx numerically. Then the numerical approximation to the integral, 1 (h), can be represented by (56.1) where m is the order of the method, and the other terms represent the error (see page 245). For example, the trapezoidal rule has the parameters m = 3 and R3 = 11 2/"«(), with a < ( < b. If the numerical integration scheme is kept the same, but the charac­ teristic width of the grid is changed from h to k, then (56.2) Equations (56.1) and (56.2) can be combined to yield the approximation Note that 1(h, k) is one more order accurate than either 1(h) or 1(k). This process, known as Richardson extrapolation, may be repeated as often as is desired. If we write k = rh, then (56.3) may be written in the alternative form 1(h, k) := 1(h) -[1(h) -1:h)] + .... In this formulation it is clear l-r that if r is close to one, there may be roundoff problems. When Richardson extrapolation is used with the composite trapezoidal rule, and the number of nodes (nearly) doubles with each iteration (i.e., r ~ !), then we obtain the Romberg integration technique. For the integral 1 = f: I(x) dx, the m-th approximation is given by using 2m + 1 nodes: b -a [1 1 ] 1 ~ 10,m = 2fi' 2/0 + II + ... + 12m-l + 212m , 56. Romberg Integration / Richardson Extrapolation 251 Table 56.1. Results of using Romberg integration on (56.5). nodes 10,m him 12,m 13,m 14,m 15,m 3 0.905330 5 0.964924 0.984789 9 0.987195 0.994619 0.997895 17 0.995372 0.998097 0.999257 0.999710 33 0.998338 0.999327 0.999737 0.999897 0.999960 65 0.999406 0.999762 0.999907 0.999964 0.999986 0.999994 where Ii = !(Xi) and Xi = a + 2~ (b -a). The error in this approximation is given by (see page 342) where the {Bi} are the Bernoulli numbers and a < .,., < b. Note that only even powers of h appear in the error formula (56.4). Because of this, each extrapolation step increases the accuracy by two orders. The extrapolated values of the integral are then given by (1 )2i 1--1i-1,m-l -2 1i-1,m-1 ',m - (1)2 1-2 for i = 1,2, .... It should be noted that the values {11,m} are identical to the values obtained from using the composite Simpson's rule. Example Given the integral 311 1= -Vidx= 1 2 0 (56.5) we might choose to approximate the value by using Romberg integration. Table 56.1 has the result of using Romberg integration on this integral. Note that the initial data has (at best) about 11 bits of precision (that is, -log2 0.000594), yet the fully extrapolated result has more than 17 bits of precision (that is, -log2 0.000006). 252 V Numerical Methods: Concepts Table 56.2. Results of using Richardson extrapolation on (56.5). nodes m:!) R~) R~) m!) m!) R~) 3 0.90533 5 0.96492 1.02452 7 0.98052 0.99611 0.98664 9 0.98720 0.99387 0.99312 0.99355 11 0.99076 0.99433 0.99449 0.99440 0.99436 13 0.99293 0.99510 0.99536 0.99521 0.99515 0.99512 15 0.99437 0.99580 0.99603 0.99590 0.99585 0.99582 17 0.99537 0.99638 0.99657 0.99646 0.99642 0.99640 Notes [1] This method also works for non-uniform grids if every interval is subdivided. [2] While Romberg is intrinsically a recursive computation, it is possible to write explicit formulas for the successive approximations. See Davis and Rabinowitz [3] for details. [3] A benefit of doubling the number of nodes at each iteration of Romberg integration is that the integrand values at the old nodes can be re-used. Press et al. [6] present a modification of Romberg integration where the number of nodes must be tripled before the values at the old nodes can be re-used. (This is because the integration rule they use for lZN I(x) dx ZI depends on the values Im+l/2' where m is an integer.) [4] A liability of doubling the number of nodes at each iteration of Romberg integration is that the number of nodes increases exponentially. Hence, some modifications of Romberg integration use a different sequence of node numbers. The sequences of node numbers {I, 2, 3, 6, 9, 18, 27, 54, ... }, {I, 2, 3, 4, 6, 8, 12, ... }, and {I, 2, 3, 4, 5, ... } are discussed in Davis and Rabinowitz [3] and Fairweather and Keast [4]. Richardson extrapolation was used on problem (56.5) with the sequence of nodes {3,5, 7,9, ... }; the numerical results are in Table 56.2. [5] In some cases, the order of the method, and hence the value of m in (56.1), will be unknown. Richardson extrapolation method may still be used by first estimating the value of m numerically, or the Aitken /:!12 transformation may be used. The Aitken /:!12 transformation uses three successive terms of the form 8n = 800 + ahn to estimate 800 via S -Sn+1Sn-l -S! 00 -Sn+1 + Sn-l -2Sn (56.6) This transformation may be repeatedly applied; see Bender and Orszag [1] for details. If 2n + 1 terms are used, not just three, then this transformation is known as the Shanks transformation. Table 56.3 has the results of applying the Aitken transformation to the integral in (56.5), when the trapezoidal rule has been used with n nodes (with n = 3,5,7, ... ). The superscripts refer to the number of times the 56. Romberg Integration / Richardson Extrapolation 253 Table 56.3. Results of using the Aitken transformation on (56.5). nodes S~O) S~l) S~2) S~3) 3 0.90533 5 0.96492 0.98604 7 0.98052 0.99220 0.99690 9 0.98720 0.99486 0.99795 0.99919 11 0.99076 0.99629 0.99851 13 0.99293 0.99716 15 0.99437 Aitken transformation has been applied; the initial quadrature result has a superscript of zero. [6] Bulirsch and Stoer [2] have a modification of this method. They compute trapezoidal sums for several values of h, fit a rational function of h2 to this data (a Pade approximant), and then extrapolate to h = O. On a practical level, nonlinear transformations such as those using Pade approximants are much more useful that Romberg integration. See also Press et al. [6]. [7] Note that the term Rm appearing in both (56.1) and (56.2) might con­ ceivably not represent the same quantity. However, when an explicit error formula is available, such as the one in (56.4), then it is seen that the error terms are completely specified at every order. [8] Richardson extrapolation is sometimes called deferred approach to the limit. In more detail, Richardson extrapolation proceeds as follows. If I is any function of h which can be represented as m 1= I(h) + L aihi + cm(h)hm+l i=k and the {Ci} are known, then we may define bi = ai(ri -rk)/(1 -rk) and dm(h) = (Cm(rh)rm+l -cm(h)rk)/(1 -rk). This results in References [1] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scien­ tists and Engineers, McGraw-Hill, New York, 1978, page 369. [2] R. Bulirsch and J. Stoer, "Handbook Series Numerical Integration: Numer­ ical Quadrature by Extrapolation," Numer. Math., 8, 1966, pages 93-104. [3] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, pages 45-47 and 436-446. [4] G. Fairweather and P. Keast, "An Investigation of Romberg Quadrature," ACM TI-ans. Math. Software, 4, 1978, pages 316-322. 254 V Numerical Methods: Concepts [5] D. C. Joyce, "Survey of Extrapolation Processes in Numerical Analysis," SIAM Review, 13, 1971, pages 435-490. [6] W. H. Press, B. P. Flannery, S. Teukolsky, and W. T. Vetterling, Numerical Recipes, Cambridge University Press, New York, 1986, pages 123-127. [7] A. Sidi, "The Numerical Evaluation of Very Oscillatory Infinite Integrals by Extrapolation," Math. of Comp., 38, 1982, pages 517-529. 57. Software Libraries: Introduction When approximating the value of an integral numerically, it is best to use software prepared by experts whenever possible. Many commercial computer libraries and isolated computer routines for evaluating integrals are readily available. No endorsement of the software referenced in this book is intended, nor does it necessarily imply that unnamed integrators are not worth trying. However, if you are using a integrator on anything bigger than a pocket calculator, you should consider using one of the cited packages. In the literature there now appears to be commercially available "software" for integration with no error control, a user-specified step size, and no warning messages. We strongly advise against using such programs, even on a personal computer. For all but trivial integrals, such programs cannot be sufficiently reliable for accurate computational results. A taxonomy and description of mathematical and statistical routines in use at the National Institute of Standards and Technology (NIST) has been constructed, see Boisvert et al. [1] and [2]. The NIST guide, called G AMS (for Guide to Available Mathematical Software), was assembled to help users of the NIST computer facilities. The GAMS taxonomy for integration routines is reproduced on page 258. The integration routines listed in GAMS are reproduced on page 260. There are many computer routines that are not described in the NIST guide, some of which are described in the Notes. Notes [1] Appendix 2 of Davis and Rabinowitz [3] contains a collection of FORTRAN integration routines. [2] Appendix 3 of Davis and Rabinowitz [3] contains a bibliography of papers describing ALGOL, FORTRAN, and PL/1 integration routines. [3] In the books by Press et al. (see, for example, [10]), there are a collection of C, FORTRAN, and PASCAL integration routines. 57. Software Libraries: Introduction 255 [4] Quadpack, a library of FORTRAN routines for one-dimensional integration by Piessens at al. [9], has the following system for naming its routines: • The first letter is always "Q"; • The second letter is "N" for a non-adaptive integrator or "A" for an adaptive integrator; • The third letter is "G" is for a general (user-defined) integrand or "W" for one of several weight functions; • The fourth letter is "S" if routine handles singularities well, "P" if the location of points of special difficulty (such as singularities) can be input by the user, "I" if the region of integration is infinite, "0" if the weight function is oscillatory (Le., is a sine or cosine func- tion), "F" for a Fourier integral, "C" for a Cauchy principal-value integral, "E" for an extended parameter list (allows more user control). Figures Figure 57.1 and Figure 57.2 show decision trees from Piessens at al. [9] that lead the user to the correct QUadpack routines. [5] Quadlib [11] is a PC package derived from Quadpack. It contains the high-level double precision Quadpack subroutines along with an extensive collection of sample programs, test programs, and skeleton files. [6] Many scientific software routines, including those for integration, may be obtained for free (via electronic mail) from a variety of computer networks. For example, all of the Quadpack [9] routines are available. To receive instructions on how to obtain this software, send the mail message "send index" to one of the following five addresses: [email protected] [email protected] [email protected] [email protected] uunet!research!netlib See the article by Dongarra and Grosse [5] for details. [7] Many of the symbolic manipUlation languages (see page 117) can also eval­ uate integrals numerically. [8] Note that there are times when it is not wise to use an automatic quadrature routine, see Lyness [8]. References [1] R. F. Boisvert, S. E. Howe, D. K. Kahaner, and J. L. Springmann, Guide to Available Mathematical Software, NISTIR 90-4237, Center for Computing and Applied Mathematics, National Institute of Standards and Technology, Gaithersburg, MD, March 1990. [2] R. F. Boisvert, S. E. Howe, and D. K. Kahaner, "GAMS: A Framework for the Management of Scientific Software," ACM Trans. Math. Software, 11, No.4, December 1985, pages 313-355. 256 v Numerical Methods: Concepts b J f(x)dx, a and b finite. a Can you factorize the integrand as w(x).f(x) Where f is lIlDooth on [a,b], J Use QAWO I and w(x)-coe(~x) or ein(~x)? YES NO Can you factorize the integrand as w(x)f{x) Where f is lIlDooth on [a,b], J Use QAWS or QAWSE I w(x)_(x_a)c(b-x)8 1og1(x_a) logk(b-x) YES with k,laQ or 1 and a,8>-1, b>a ? NO Can you factorize the integrand a. w(x)f(x) Where f i. lIlDooth I Uee QAWC or QAWCE I on [a,b], and w(x)-l/(x-c)? YES NO Do you care about computer time, and are you willing to do some Use QAGS I analy.i. of the problem? NO YES Is the integrand smooth? I Use QNG or QAG I YES NO Split the integration I Are there discontinuities or singu- range at the points Where larities of the integrand or of its YES difficulties occur,and do I derivative within the interval, and your analysis for each sub- I do you know Where they are? interval separately. You can NO also use QAGP, Which is to be provided with the abscis- sae of the points involved. Has the integrand end-point singu- larities? I Use QAGS NO YES Has the integrand an oscillatory behaviour of non-specific type, and Use QAG with KEY -6 no singularities? YES NO Use QAGS. If this routine returns an error flag you are advised to do more analysis in order to reduce the degree of difficulty of your problem. QAGE may be of help for locating possible difficult spots of the integrand. ----- Figure 57.1. Decision tree for finite-range integration from Piessens at al. [9]. 57. Software Libraries: Introduction 257 a f f(x)dx, J f(x)dx, J a Does the integrand decay rapidly to zero for x·-, x--? NO Does the integrand oscillate over the entire infinite range? NO Do you care about computer time and are you willing to do some analysis of the problem? NO Use QAGI for integration over the infinite range f(x)dx, a finite Truncate the interval and go to -ws Section 4.2 for integration over the remaining finite range. ~IS the integral a Fourier YES transform? f YES NO I Use QAWP Sum auccessive positive and nega- tive contributione by integrating YES between zeros of the integrand (go r--to Section 4.2). Apply convergence acceleration by means of routine QEXT (Section 3.4.2) Which implements the c-algorithm. Has the integrand a non-smooth behaviour in the integration range, and do you know Where it occurs? -A singularity at the finite integration limit is allowed.- NO YES (1 ) For integration over the infinite Split off finite subrangee at the range pointe Where non-smooth behaviour -truncation of the interval, occurs, and use appropriate fi- -or application of a suitable nite-range integration on these transformation for reducing subranges (go to Section 4.2). the problem to integration For integration over the over a finite interval infinite sUbinterval go to (1). may be possible. You can also apply QAGI. Figure 57.2. Decision tree for infinite-range integration from Piessens at al. [9]. 258 V Numerical Methods: Concepts [3] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, Appendices 2 and 3, pages 480-517. [4] L. M. Delves and B. G. S. Doman "The Design of a Generic Quadrature Library in ADA," Numerical Integration, P. Keast and G. Fairweather (eds.), D. Reidel Publishing Co., Boston, 1987, pages 307-319. [5] J. J. Dongarra and E. Grosse, "Distribution of Mathematical Software Via Electronic Mail," Comm. of the ACM, 30, No.5, May 1987, pages 403-407. [6] D. Kahaner, "Sources of Information on Quadrature Software," in W. R. Cowell (ed.), Sources and Development of Mathematical Software, Prentice­ Hall Inc., Englewood Cliffs, NJ, 1984, Chapter 7, pages 134-164. [7] D. K. Kahaner, "Development of Useful Quadrature Software, with Partic­ ular Emphasis on Microcomputers," in P. Keast and G. Fairweather (eds.), Numerical Integration: Recent Developments, Software and Applications, D. Reidel Publishing Co., Boston, 1987, pages 343-369. [8] J. N. Lyness, "When Not to Use an Automatic Quadrature Routine," SIAM Review, 25, No.1, January 1983, pages 63-87. [9] R. Piessens, E. de Doncker-Kapenga, C. W. Uberhuber, and D. K. Kahaner, Quadpack, Springer-Verlag, New York, 1983. [10] W. H. Press, B. P. Flannery, S. Teukolsky, and W. T. Vetterling, Numerical Recipes, Cambridge University Press, New York, 1986. [11] Quadlib, McGraw-Hill Book Company, New York. 58. Software Libraries: Taxonomy Applicable to Definite integrals. Yields The software category code appropriate for a specific numerical inte­ gration problem. Procedure A taxonomy for integration software has been developed as part of the GAMS project at the National Institute of Standards and Technology [1]. This section contains the taxonomy of integration routines from GAMS [2]. Example Suppose that a multidimensional integration must be performed nu­ merically. If the integrand is known analytically, and the integration region is a hyper-rectangle, then software in the category code H2bl is required. Most modern software comes with the appropriate category codes, which makes it easy to determine if some software is appropriate for a spe­ cific problem. This category code can be used when searching electronic databases, such as MathSci [3] or Netlib [4]. 58. Software Libraries: Taxonomy 259 Taxonomy H2. H2a. H2al. H2ala. H2alal. H2ala2. H2alb. H2albl. H2alb2. H2a2. H2a2a. H2a2al. H2a2a2. H2a2b. H2a2bl. H2a2b2. H2a3. H2a3a. H2a3al. H2a3a2. H2a4. H2a4a. H2a4al. H2a4a2. H2b. H2bl. H2bla. H2blal. H2bla2. H2blb. H2blbl. H2blb2. H2b2. H2b2a. H2b2al. H2b2a2. H2b2b. H2b2bl. H2b2b2. H2c. Note Quadrature (numerical evaluation of definite integrals) One dimensional integrals Finite interval (general integrand) Integrand available via user-defined procedure Automatic (user need only specify desired accuracy) Nonautomatic Integrand available only on a grid Automatic (user need only specify desired accuracy) Nonautomatic Finite interval (specific or special type integrands including weight functions, oscillating and singular integrands, principal values inte­ grals, splines, etc.) Integrand available via user-defined procedure Automatic (user need only specify desired accuracy) Nonautomatic Integrand available only on a grid Automatic (user need only specify desired accuracy) Nonautomatic Semi-infinite interval (including e-z weight function) Integrand available via user-defined procedure Automatic (user need only specify desired accuracy) Nonautomatic Infinite interval (including e-z2 weight function) Integrand available via user-defined procedure Automatic (user need only specify desired accuracy) Nonautomatic Multidimensional integrals One or more hyper-rectangular regions (includes iterated integrals) Integrand available via user-defined procedure Automatic (user need only specify desired accuracy) Nonautomatic Integrand available only on a grid Automatic (user need only specify desired accuracy) Nonautomatic n dimensional quadrature on a nonrectangular region Integrand available via user-defined procedure Automatic (user need only specify desired accuracy) Nonautomatic Integrand available only on a grid Automatic (user need only specify desired accuracy) Nonautomatic Service routines (compute weight and nodes for quadrature formu­ las) [1] GAMS [2] does not list software for all of the above categories. 260 V Numerical Methods: Concepts References [1] R. F. Boisvert, S. E. Howe, and D. K. Kahaner, "GAMS: A Framework for the Management of Scientific Software," ACM 7Tans. Math. Software, 11, No.4, December 1985, pages 313-355. [2) R. F. Boisvert, S. E. Howe, D. K. Kahaner, and J. L. Springmann, Guide to Available Mathematical Software, NISTIR 90-4237, Center for Computing and Applied Mathematics, National Institute of Standards and Technology, Gaithersburg, MD, March 1990. [3) MathSci Disc, CD-ROM product, Amer. Math. Soc., Providence, Rhode Island, 1992. [4) J. J. Dongarra and E. Grosse, "Distribution of Mathematical Software Via Electronic Mail," Comm. of the ACM, 30, No.5, May 1987, pages 403-407. 59. Software Libraries: Excerpts from G AMS Applicable to Definite integrals. Yields Software appropriate for a specific integration problem. Procedure A taxonomy and description of mathematical and statistical routines in use at the National Institute of Standards and Technology (NIST) has been constructed, see Boisvert et al. [1]. The NIST guide, called GAMS [2], was assembled to help users of the NIST computer facilities. Since NIST has many software packages, many people outside of NIST have found the guide useful despite its limited coverage. This section contains the GAMS listing of integration routines, by software category code (see page 258). The following computer libraries (mostly in FORTRAN) are referred tot. Their inclusion does not constitute an endorsement. Nor does it necessarily imply that unnamed packages are not worth trying. • CMLIB [3) • Collected Algorithms of the ACM • IMSL [4] • NAG [5] • NMS [6] • PORT [7] • Scientific Desk [8] • SCRUNCH [9] t Identification of commercial products does not imply recommendation or endorsement by NIST. 59. Software Libraries: Excerpts from GAMS 261 Excerpts from GAMS H2a1a1: A614 Q1DA Q1DAX Q1DB QAG QAGE QAGS QAGSE QNG DCADRE QDAG QDAGS QDNG Automatic 1-D finite interval quadrature (user need only specify required accuracy), integrand available via user-defined procedure Collected Algorithms of the A CM INTHP: a Fortran subroutine for automatic numerical integration in Hp. The functions may have singularities at one or both endpoints of an interval. Each of finite, semi-infinite, and in­ finite intervals are admitted. (See K. Sikorski, F. Stenger, and J. Schwing, ACM TOMS 10 (1984) pp. 152-160.) CMLIB Library (Q1DA Sublibrary) A utomatic integration of a user-defined function of one variable. Special features include randomization and singularity weaken­ ing. Flexible subroutine for the automatic integration of a user-defined function of one variable. Special features include randomization, singularity weakening, restarting, specification of an initial mesh (optional), and output of smallest and largest integrand values. Automatic integration of a user-defined function of one variable. Integrand must be a Fortran FUNCTION but user may select name. Special features include randomization and singularity weakening. Intermediate in usage difficulty between Q1DA and Q1DAX. CMLIB Library (QUADPKS Sublibrary) Automatic adaptive integrator, will handle many non-smooth integrands using Gauss-Kronrod formulas. Automatic adaptive integrator, can handle most non-smooth functions, also provides more information than QAG. Automatic adaptive integrator, will handle most non-smooth in­ tegrands including those with endpoint singularities, uses extrap­ olation. Automatic adaptive integrator, can handle integrands with end- point singularities, provides more information than QAGS. Automatic non-adaptive integrator for smooth functions, using Gauss-Kronrod-Patterson formulas. IMSL Subprogram Library Numerical integration of a function using cautious adaptive Rom­ berg extrapolation. IMSL MATH/LIBRARY Subprogram Library Integrate a function using a globally adaptive scheme based on Gauss-Kronrod rules. Integrate a function (which may have endpoint singularities). Integrate a smooth function using a nonadaptive rule. 262 QDAGS DEFINT DOIAHF DOIAJF DOIARF DOIBDF QIDA ODEQ QUAD RQUAD H2Al H2AIU SIMP H2ala2: V Numerical Methods: Concepts IMSL STAT/LIBRARY Subprogram Library Integrate a function (which may have endpoint singularities). JCAM Software Library Uses double exponential transformation of Mori to compute def­ inite integral automatically to user specified accuracy. NAG Subprogram Library Computes a definite integral over a finite range to a specified relative accuracy using a method described by Patterson. Is a general-purpose integrator which calculates an approxima­ tion to the integral of a function F(x) over a finite interval (A,B). Computes definite and indefinite integrals over a finite range to a specified relative or absolute accuracy, using a method described by Patterson. Calculates an approximation to the integral of a function over a finite interval (A,B). It is non-adaptive and as such is rec­ ommended for the integration of smooth functions. These ex­ clude integrands with singularities, derivative singularities or high peaks on (A,B), or which oscillate too strongly on (A,B). NMS Subprogram Library Automatic integration of a user-defined function of one variable. Special features include randomization and singularity weaken­ ing. PORT Subprogram Library Finds the integral of a set of functions over the same interval by using the differential equation solver ODES!. For smooth functions. Finds the integral of a general user defined EXTERNAL function by an adaptive technique to given absolute accuracy. Finds the integral of a general user defined EXTERNAL function by an adaptive technique. Combined absolute and relative error control. Sciep,tijic Desk PC Subprogram Library Automatically evaluates the definite integral of a user defined function of one variable. Automatically evaluates the definite integral of a user defined function of one variable. SCRUNCH Subprogram Library Calculates an estimate of the definite integral of a user supplied function by adaptive quadrature. In BASIC. Nonautomatic I-D finite interval quadrature, inte­ grand available via user-defined procedure 59. Software Libraries: Excerpts from GAMS 263 QK15 QK21 QK31 QK41 QK51 QK61 DOIBAF QK15 H2alb2: DOIGAF PCHQA CSPQU H2AIT CMLIB Library (QUADPKS Sublibrary) Evaluates integral of given function on an interval with a 15 point Gauss-Kronrod formula and returns error estimate. Evaluates integral of given function on an interval with a 21 point Gauss-Kronrod formula and returns error estimate. Evaluates integral of given function on an interval with a 31 point Gauss-Kronrod formula and returns error estimate. Evaluates integral of given function on an interval with a 41 point Gauss-Kronrod formula and returns error estimate. Evaluates integral of given function on an interval with a 51 point Gauss-Kronrod formula and returns error estimate. Evaluates integral of given function on an interval with a 61 point Gauss-Kronrod formula and returns error estimate. NAG Subprogram Library Computes an estimate of the definite integral of a function of known analytical form, using a Gaussian quadrature formula with a specified number of abscissae. Formulae are provided for a finite interval (Gauss-Legendre), a semi-infinite interval (Gauss­ Laguerre, Gauss-Rational), and an infinite interval (Gauss-Hermite). NMS Subprogram Library Evaluates integral of given function on an interval with a 15 point Gauss-Kronrod formula and returns error estimate. Nonautomatic I-D finite interval quadrature, inte­ grand available only on a grid NAG Subprogram Library Integrates a function which is specified numerically at four or more points, over the whole of its specified range, using third­ order finite-difference formulae with error estimates, according to a method due to Gill and Miller. NMS Subprogram Library Integrates piecewise cubic from A to B given N-arrays X,F,D. Usually used in conjunction with PCHEZ to form cubic, but can be used independently, especially if the abscissae are equally spaced. PORT Subprogram Library Finds the integral of a function defined by pairs (x,y) of input points. The x's can be unequally spaced. Uses spline interpola­ tion. Scientific Desk PC Subprogram Library Computes the integral of the array f between x(i) and xU), given n points in the plane (x(k),f(k)), k=I, ... ,n. 264 H2a2al: BFQAD PFQAD QAGP QAGPE QAWC QAWCE QAWO QAWOE QAWS QAWSE QMOMO QDAGP QDAWC QDAWO QDAWS D01AKF V Numerical Methods: Concepts Automatic l-D finite interval quadrature (user need only specify required accuracy) (special integrand including weight functions, oscillating and singular integrands, principal value integrals, splines, etc.), integrand available via user-defined procedure CMLIB Library (BSPLINE Sublibrary) Integrates function times derivative of B-spline from Xl to X2. The B-spline is in B representation. Computes integral on (XI,X2) of product of function and the ID-th derivative of B-spline which is in piecewise polynomial representation. CMLIB Library (QUADPKS Sublibrary) Automatic adaptive integrator, allows user to specify location of singularities or difficulties of integrand, uses extrapolation. Automatic adaptive integrator for function with user specified endpoint singularities, provides more information that QAGP. Cauchy principal value integrator, using adaptive Clenshaw-Curtis method (real Hilbert transform). Cauchy principal value integrator, provides more information than QAWC (real Hilbert transform). Automatic adaptive integrator for integrands with oscillatory sine or cosine factor. Automatic integrator for integrands with explicit oscillatory sine or cosine factor, provides more information than QAWO. Automatic integrator for functions with explicit algebraic and/or logarithmic endpoint singularities. Automatic integrator for integrands with explicit algebraic and/or logarithmic endpoint singularities, provides more information than QAWS. Computes integral of k-th degree Chebyshev polynomial times one of a selection of functions with various singularities. IMSL MATH/LIBRARY Subprogram Library Integrate a function with singularity points given. Integrate a function F(x)/(x-c) in the Cauchy principal value sense. Integrate a function containing a sine or a cosine. Integrate a function with algebraic-logarithmic singularities. NAG Subprogram Library Is an adaptive integrator, especially suited to oscillating, non­ singular integrands, which calculates an approximation to the integral of a function F(x) over a finite interval (A,B). 59. Software Libraries: Excerpts from GAMS 265 D01ALF D01ANF D01APF D01AQF BQUAD H2a2a2: QC25C QC25F QC25S QK15W H2a2b1: BSQAD PPQAD Is a general purpose integrator which calculates an approximation to the integral of a function F(x) over a finite interval (A,B), where the integrand may have local singular behavior at a finite number of points within the integration interval. Calculates an approximation to the cosine or the sine transform of a function Gover (A,B), i.e., the integral of G(x)cos(wx) or G(x)sin(wx) over (A,B) (for a user-specified value of w). Is an adaptive integrator which calculates an approximation to the integral of a function G(x)W(x) over (A,B) where the weight function W has end-point singularities of algebraic-logarithmic type (see input parameter KEY). Calculates an approximation to the Hilbert transform of a func­ tion G(x) over (A,B), i.e., the integral of G(x)/(x-c) over (A,B), for user-specified values of A,B,C. PORT Subprogram Library Adaptively integrates functions which have discontinuities in their derivatives. User can specify these points. Nonautomatic 1-D finite interval quadrature (spe­ cial integrand including weight functions, oscillating and singular integrands, principal value integrals, splines, etc.), integrand available via user-defined procedure CMLIB Library (QUADPKS Sublibrary) Uses 25 point Clenshaw-Curtis formula to estimate integral of F(x)W(x) where W(x)=l/(x-c). Clenshaw-Curtis integration rule for function with cos or sin factor, also uses Gauss-Kronrod formula. Estimates integral of function with algebraic-logarithmic singu­ larities using 25 point Clenshaw-Curtis formula and gives error estimate. Evaluates integral of given function times arbitrary weight func­ tion on interval with 15 point Gauss-Kronrod formula and gives error estimate. Automatic 1-D finite interval quadrature (user need only specify required accuracy) (special integrand including weight functions, oscillating and singular integrands, principal value integrals, splines, etc.), integrand available only on a grid CMLIB Library (BSPLINE Sublibrary) Computes the integral of a B-spline from Xl to X2. The B-spline must be in B representation. Computes the integral of a B-spline from Xl to X2. The B-spline must be in piecewise polynomial representation. 266 DCSQDU BSITG E02AJF E02BDF BSPLI SPLNI E3HIN E3INT H2a3al: A614 A639 QAGI QAGIE V Numerical Methods: Concepts IMSL Subprogram Library Cubic spline quadrature. IMSL MATH/LIBRARY Subprogram Library Evaluate the integral of a spline, given its B-spline representation. NAG Subprogram Library Determines the coefficients in the Chebyshev series representation of the indefinite integral of a polynomial given in Chebyshev series form. Computes the definite integral of a cubic spline from its B-spline representation. PORT Subprogram Library Obtains the integrals of basis splines, from the left-most mesh point to a specified set of points. Integrates a function described previously by an expansion in terms of B-splines. Several integrations can be performed in one call. Scientific Desk PC Subprogram Library Evaluates the definite integral of a piecewise cubic Hermite func­ tion over an arbitrary interval. Evaluates the definite integral of a piecewise cubic Hermite func­ tion over an interval whose endpoints are data points. Automatic I-D semi-infinite interval quadrature (user need only specify required accuracy) (including e-X ) weight function), integrand available via user-defined procedure Collected Algorithms of the ACM INTHP: a Fortran subroutine for automatic numerical integration in Hp. The functions may have singularities at one or both end-points of an interval. Each of finite, semi-infinite, and in­ finite intervals are admitted. (See K. Sikorski, F. Stenger, and J. Schwing, ACM TOMS 10 (1984) pp. 152-160.) OSCINT: a Fortran subprogram for the automatic integration of some infinitely oscillating tails. That is, the evaluation of the integral from a to infinity of h(x)j(x), where h(x) is ultimately positive, and j(x) is either a circular function (e.g., cosine) or a first-kind Bessel function of fractional order. (See J. Lyness and G. Hines, ACM TOMS 12 (1986) pp. 24-25.) CMLIB Library (QUADPKS Sublibrary) Automatic adaptive integrator for semi-infinite or infinite inter­ vals. Uses nonlinear transformation and extrapolation. Automatic integrator for semi-infinite or infinite intervals and general integrands, provides more information than QAGI. 59. Software Libraries: Excerpts from GAMS 267 QAWF QAWFE QDAGI QDAWF DEHINT DOIAMF QAGI H2a3a2: QK15I DOIBAF H2a4al: A614 Automatic integrator for Fourier integrals on (a,oo) with factors sin(wx), cos(wx) by integrating between zeros. Automatic integrator for Fourier integrals, with sin(wx) factor on (a,oo), provides more information than QAWF. IMSL MATH/LIBRARY Subprogram Library Integrate a function over an infinite or semi-infinite interval. Compute a Fourier integral. JCAM Software Library Uses double exponential transformation of Mori to compute semi­ infinite range integral automatically to user specified accuracy. NAG Subprogram Library Calculates an approximation to the integral of a function F(x) over an infinite or semi-infinite interval (A,B). NMS Subprogram Library Automatic adaptive integrator for semi-infinite or infinite inter­ vals. Uses nonlinear transformation and extrapolation. Nonautomatic I-D semi-infinite interval quadrature) (including e-z weight function), integrand available via user-defined procedure CMLIB Library (QUADPKS Sublibrary) Evaluates integral of given function on semi-infinite or infinite interval with a transformed 15 point Gauss-Kronrod formula and gives error estimate. NAG Subprogram Library Computes an estimate of the definite integral of a function of known analytical form, using a Gaussian quadrature formula with a specified number of abscissae. Formulae are provided for a finite interval (Gauss-Legendre), a semi-infinite interval (Gauss­ Laguerre, Gauss-rational), and an infinite interval (Gauss-Hermite). Automatic I-D infinite interval quadrature (user need only specify required accuracy) (including e-Z2 ) weight function), integrand available via user-defined procedure Collected Algorithms of the ACM INTHP: a Fortran subroutine for automatic numerical integration in Hp. The functions may have singularities at one or both end-points of an interval. Each of finite, semi-infinite, and in­ finite intervals are admitted. (See K. Sikorski, F. Stenger, and J. Schwing, ACM TOMS 10 (1984) pp. 152-160.) 268 QAGI QAGIE D01AMF QAGI H2a4a2: QK151 D01BAF H2blal: ADAPT DBLIN DMLIN V Numerical Methods: Concepts CMLIB Library (QUADPKS Sublibrary) Automatic adaptive integrator for semi-infinite or infinite inter­ vals. Uses nonlinear transformation and extrapolation. Automatic integrator for semi-infinite or infinite intervals and general integrands, provides more information than QAGI. NAG Subprogram Library Calculates an approximation to the integral of a function F(x) over an infinite or semi-infinite interval (A,B). NMS Subprogram Library Automatic adaptive integrator for semi-infinite or infinite inter­ vals. Uses nonlinear transformation and extrapolation. Nonautomatic l-D infinite interval quadrature (in­ cluding e-X2 ) weight function), integrand available via user-defined procedure CMLIB Library (QUADPKS Sublibrary) Evaluates integral of given function on semi-infinite or infinite interval with a transformed 15 point Gauss-Kronrod formula and gives error estimate. NAG Subprogram Library Computes an estimate of the definite integral of a function of known analytical form, using a Gaussian quadrature formula with a specified number of abscissae. Formulae are provided for a finite interval (Gauss-Legendre), a semi-infinite interval (Gauss­ Laguerre, Gauss-rational), and an infinite interval (Gauss-Hermite). Automatic n-D quadrature (user need only specify required accuracy) on one or more hyper-rectangular regions, integrand available via user-defined proce­ dure CMLIB Library (ADAPT Sublibrary) Computes the definite integral of a user specified function over a hyper-rectangular region in 2 through 20 dimensions. User specifies tolerance. A restarting feature is useful for continuing a computation without wasting previous function values. IMSL Subprogram Library Numerical integration of a function of two variables. Numerical integration of a function of several variables over a hyper-rectangle (Gaussian method). 59. Software Libraries: Excerpts from GAMS 269 QAND TWODQ DOIDAF DOIEAF DOIFCF DOIGBF H2bla2: DOIFBF DOIFDF DOIGCF H2blb2: DBCQDU BS2IG BS3IG IMSL MATH/LIBRARY Subprogram Library Integrate a function on a hyper-rectangle. Compute a two-dimensional iterated integral using internal calls to a one-dimensional automatic integrator. NAG Subprogram Library Attempts to evaluate a double integral to a specified absolute accuracy by repeated applications of the method described by Patterson. Computes approximations to the integrals of a vector of similar functions, each defined over the same multi-dimensional hyper­ rectangular region. The routine uses an adaptive subdivision strategy, and also computes absolute error estimates. Attempts to evaluate a multidimensional integral (up to 15 di­ mensions), with constant and finite limits, to a specified relative accuracy, using an adaptive subdivision strategy. Returns an approximation to the integral of a function over a hyper-rectangular region, using a Monte-Carlo method. An ap­ proximate relative error estimate is also returned. This routine is suitable for low accuracy work. Nonautomatic n-D quadrature on one or more hyper­ rectangular regions, integrand available via user­ defined procedure NAG Subprogram Library Computes an estimate of a multidimensional integral (from 1 to 20 dimensions), given the analytic form of the integrand and suitable Gaussian weights and abscissae. Calculates an approximation to a definite integral in up to 30 dimensions, using the method of Sag and Szekeres. The region of integration is an n-sphere, or by built-in transformation via the unit n-cube, any product region. Calculates an approximation to a definite integral in up to 20 di­ mensions, using the Korobov-Conroy number theoretic method. Nonautomatic n-D quadrature on one or more hyper­ rectangular regions, integrand available only on a grid IMSL Subprogram Library Bicubic spline quadrature. IMSL MATH/LIBRARY Subprogram Library Evaluate the integral of a tensor-product spline on a rectangular domain, given its tensor-product B-spline representation. Evaluate the integral of a tensor-product spline in three dimen­ sions over a three-dimensional rectangle, given its tensor-product B-spline representation. 270 H2b2al: A584 A612 TWODQ D01JAF H2B2A H2b2a2: DTRIA D01PAF V Numerical Methods: Concepts Automatic n-D quadrature on a nonrectangular re­ gion (user need only specify required accuracy), in­ tegrand available via user-defined procedure Collected Algorithms of the ACM CUBTRI: a Fortran subroutine for adaptive cubature over a tri­ angle. (See D. P. Laurie, ACM TOMS 8 (1982) pp. 210-218.) TRIEX: a Fortran subroutine for integration over a triangle. Uses an adaptive subdivisional strategy with global acceptance criteria and incorporates the epsilon algorithm to speed convergence. (see E. de Doncker and 1. Robinson, ACM TOMS 10 (1984) pp. 17- 22.) CMLIB Library (TWODQ Sublibrary) Automatic (adaptive) integration of a user specified function f(x,y) on one or more triangles to a prescribed relative or abso­ lute accuracy. Two different quadrature formulas are available within TWODQ. This enables a user to integrate functions with boundary singUlarities. NAG Subprogram Library Attempts to evaluate an integral over an n-dimensional sphere (n=2, 3, or 4), to a user specified absolute or relative accuracy, by means of a modified Sag-Szekeres method. The routine can handle singularities on the surface or at the center of the sphere, and returns an error estimate. Scientific Desk PC Subprogram Library Computes the two-dimensional integral of a function f over a region consisting of n triangles. Nonautomatic n-D quadrature on a nonrectangular region, the integrand available via user-defined pro­ cedure JCAM Software Library Computes an approximation to the double integral of f( u, v) over a triangle in the uv-plane by using an n2 point, generalized Gauss­ Legendre product rule of polynomial degree precision 2n-2. From "Computation of Double Integrals over a 'Triangle," by F. G. Lether, Algorithm 007, J. Compo Appl. Math. 2(1976), pp. 219- 224. NAG Subprogram Library Returns a sequence of approximations to the integral of a function over a multi-dimensional simplex, together with an error estimate for the last approximation. 59. Software Libraries: Excerpts from GAMS 271 H2c: A647 A655 A659 FQRUL GQRCF GQRUL RECCF RECQR DOIBBF DOIBCF GAUSQ GQOIN GQMll Service routines for quadrature (compute weight and nodes for quadrature formulas) Collected Algorithms of the ACM Fortran subprograms for the generation of sequences of quasiran­ dom vectors with low discrepancy. Such sequences may be used to reduce error bounds for multidimensional integration and global optimization. (See B. L. Fox, ACM TOMS 12 (1986) pp. 362- 376.) IQPACK: Fortran routines for the stable evaluation of the weights and nodes of interpolatory and Gaussian quadratures with pre­ scribed simple or mUltiple knots. (See S. Elhay and J. Kautsky, ACM TOMS 13 (1987) pp. 399-415.) A Fortran implementation of Sobol's quasirandom sequence gen­ erator for multivariate quadrature and optimization. (See P. Brat­ ley and B. L. Fox, ACM TOMS 14 (1988) pp. 88-100.) IMSL MATH/LIBRARY Subprogram Library Compute a Fejer quadrature rule with various classical weight functions. Compute a Gauss, Gauss-Radeau or Gauss-Lobatto quadrature rule given the recurrence coefficients for the monic polynomials orthogonal with respect to the weight function. Compute a Gauss, Gauss-Radeau or Gauss-Lobatto quadrature rule with various classical weight functions. Compute recurrence coefficients for various monic polynomials. Compute recurrence coefficients for monic polynomials given a quadrature rule. NAG Subprogram Library Returns the weights and abscissae appropriate to a Gaussian quadrature formula with a specified number of abscissae. The formulae provided are Gauss-Legendre, Gauss-rational, Gauss­ Laguerre and Gauss-Hermite. Returns the weights (normal or adjusted) and abscissae for a Gaussian integration rule with a specified number of abscissae. Six different types of Gauss rule are allowed. PORT Subprogram Library Finds the abscissae and weights for Gauss quadrature on the interval (a,b) for a general weight function with known moments. Finds the abscissae and weights for Gauss-Laguerre quadrature on the interval (0,+00). Finds the abscissae and weights for Gauss-Legendre quadrature on the interval (-1,1). 272 V Numerical Methods: Concepts Notes [1] In the excerpts section, ACM TOMS stands for ACM Trans. Math. Soft­ ware. [2] Software is not listed for all the taxonomy classes that have been established. [3] The author thanks Dr. Ronald Boisvert of NIST for making part of GAMS available electronically. References [1] R. F. Boisvert, S. E. Howe, and D. K. Kahaner, "GAMS: A Framework for the Management of Scientific Software," ACM Trans. Math. Software, 11, No.4, December 1985, pages 313-355. [2] R. F. Boisvert, S. E. Howe, D. K. Kahaner, and J. L. Springmann, Guide to Available Mathematical Software, NISTIR 90-4237, Center for Computing and Applied Mathematics, National Institute of Standards and Technology, Gaithersburg, MD, March 1990. [3] CMLIB, this is a collection of code from many sources that NIST has combined into a single library. The relevant sublibraries are (A) CDRIV and SDRIV, see D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Prentice-Hall Inc., Englewood Cliffs, NJ, 1989. (B) DEPAC: Code developed by L. Shampine and H. A. Watts. (C) FISHPAK: Code developed by P. N. Swartztrauber and R. A. Sweet. (D) VHS3: Code developed by R. A. Sweet. [4] IMSL Inc., 2500 Park West Tower One, 2500 City West Blvd., Houston, TX 77042. [5] NAG, Numerical Algorithms Group, Inc., 1400 Opus Place, Suite 200, Down­ ers Grove, IL, 60515. [6] NMS, this is an internal name at NIST. The code is from D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Prentice-Hall Inc., Englewood Cliffs, NJ, 1989. [7] PORT, see P. Fox, et al., The PORT Mathematical Subroutine Library Manual, Bell Laboratories, Murray Hill, NJ, 1977. [8] Scientific Desk is distributed by M. McClain, NIST, Bldg 225 Room A151, Gaithersburg, MD 20899. [9] SCRUNCH, these are old, unsupported codes in BASIC. The codes are translations of Fortran algorithms from G. Forsythe, M. Malcom, and C. Moler, Computer Methods for Mathematical Computations, Prentice-Hall Inc., Englewood Cliffs, NJ, 1977. 60. Testing Quadrature Rules Applicable to Numerical approximations to integrals. Idea Many integrals have been used as examples to test quadrature rules. 60. Testing Quadrature Rules 273 Procedure As new quadrature rules are developed, they are compared to existing quadrature rules in terms of accuracy and efficiency. Many authors have in­ troduced example integrals to indicate the performance of their algorithms and implementations. We tabulate some of those integrals. • Lyness [4] uses the test integral J(.\) = 12 ( ~;1 <lx. 1 1 -.\ + 0.01 • Piessens et al. [5] uses the test integrals (the numbers correspond to their original numbering, numbers 4-6 represent previous integrals with different parameters): 1) 2) 3) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 11 XO log (!) dx = 1 2 o x (1 + a) l' 4-20 dx = tan-' (4 -.. )40-') + tan-' ( .. 40-') o (x -i) + 16-0 274 V Numerical Methods: Concepts 17) • Berntsen et al. [1) uses the test integrals: 1) fo11x -AlaI dx 2) fo1 12 (X) dx 3) f01 e-a3Ix-~1 dx /,2 10a4 4) 1 (X _ A)2 + 102a4 dx 1 + 4 Q feature: singularity feature: discontinuous feature: Co function feature: one peak 5) /,2 4 lOa!) 1 ~ (X _ Ai)2 + 102a!) dx feature: four peaks 6) fo1 2B(x -A) cos{B{x -A)2) dx feature: nonlinear oscillation 10a6 { 0 if X < A where B = max(A2, (1-A)2) and f2(x) = exp(a2x) othe~ise· • Hunter and Smith [3) use the principal-value integrals: f.7r /2 cos( cos t) 1) ° k2 _ sin2 t dt 2) f' t2e=':2 dt where 0 < k < 1 and A > o. • Corliss and RaIl [2) have a collection of test problems that exercise their interval analysis integration package: 1.[3.1,3.2) 1) sinxdx [0,0.1) 2) f01(Bsin(Bx) -Asin(Ax» dx 3) /.0.7 ~ 0.6 1-X 4) f: ..;xdx 5) f; f(x) dx 6) fO~31 dx 7) fo\x -2) dx /.1 dx 8) 0 1 + x4 61. Truncating an Infinite Interval {Ox < 0.3 where A = [0.0.1] and B = [3.1,3.2], and f(x) = 1: x ~ 0.3 References 275 [1] J. Berntsen, T. O. Espelid, and T. S0revik, "On the Subdivision Strategy in Adaptive Quadrature Algorithms," J. Comput. Appl. Math., 35, 1991, pages 119-132. (2) G. F. Corliss and L. B. RaIl, "Adaptive, Self-Validating Numerical Quadra­ ture," SIAM J. Sci. Stat. Comput., 8, No.5, 1987, pages 831-847. . [3] D. B. Hunter and H. V. Smith, "The Evaluation of Cauchy Principal Value Integrals Involving Unknown Poles," BIT, 29, No.3, 1989, pages 512-517. [4] J. N. Lyness, "When Not to Use an Automatic Quadrature Routine," SIAM Review, 25, No.1, January 1983, pages 63-87. [5] R. Piessens, E. de Doncker-Kapenga, C. W. Uberhuber, and D. K. Kahaner, Quadpack, Springer-Verlag, New York, 1983, pages 83-84. [6] I. Robinson, "A Comparison of Numerical Integration Programs," J. Com­ put. Appl. Math., 2, 1979, pages 207-223. 61. Truncating an Infinite Interval Applicable to Integrals that have an infinite limit of integration. Yields An approximating integral, with a bound on the error. Idea By truncating an infinite integral, a numerical routine may have an easier computation. Procedure An infinite integral can always be truncated to a finite interval. Esti­ mating the error made in the truncation process establishes the usefulness of the truncation. Example Consider the integral 1= /.00 -1 x e-.z2 dx. If we truncate the upper o +x limit of integration to be, say, a, then we have /.Q X 2 I ~ Ja = -l--e-.z dx. o +x 276 V Numerical Methods: Concepts In this case we can estimate the error made in the truncation process: where we have used the approximation _x_ < x for x > O. l+x (61.1) If we were to approximate I by numerically approximating J a, then we would need a ~ 4.3 to insure that I -Ja ::; 10-8• Reference [1] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, page 205. VI Numerical Methods: Techniques 62. Adaptive Quadrature Applicable to Integrals in any number of dimensions. Yields A numerical quadrature scheme. Idea If a numerical quadrature scheme does not result in a sufficiently accurate numerical approximation, then sampling the integrand at more nodes should increase the accuracy. However, the additional sampling only needs to be performed in problem areas (where the error estimates are large). Procedure Adaptive quadrature is an automatic procedure for increasing the accuracy of a numerical approximation to an integral by increasing the number of samples of the integrand. Additional samples only need to be taken where the quadrature scheme is having numerical difficulties. Hence, the overall scheme is given by the following steps: 277 278 VI Numerical Methods: Techniques a Figure 62. The geometry of an adaptive integration computation. The func­ tion f(x) lies in the shaded triangles; the area of these triangles provide error estimates. [1] Start with a given integral to be integrated over a given interval. [2] Use a quadrature rule to approximate the integral over the entire interval; this is the global approximation. Estimate the error in this approximation; this is the global error. Place the interval, and the estimated error on that interval, onto a list. [3] If the global error is not small enough, then: (A) Choose an interval from the list of intervals (presumably, choose the one with the largest estimated error). (B) Subdivide the chosen interval. (C) Approximate the integral over each of the new sub-intervals, and estimate the error in these approximations. (D) Update the list of intervals and the estimated error on each sub­ interval. (E) Update the global approximation to the integral and the estimate of the global error. Go to step [3]. [4] Done. Figure 62 shows an early stage in an adaptive integration computation. The adaptive strategy is to subdivide the largest shaded triangle. For the figure shown, the left-most region is the next to be sub-divided. Rice [12] determines that there are between 1 and 10 million adaptive algorithms that are potentially interesting and significantly different from one another. This number arises from the following: • There are at least six different processor components. That is, the integration rules chosen for each sub-interval can be the same, or the sub-intervals can have different rules of the same order, or many other possibilities. • For each choice of how the rules are to be chosen, there are at least five possibilities for each choice of rule. For example, using only three nodes per interval, there are the following methods: Simpson's,3-point 62. Adaptive Quadrature 279 Gauss, 3-point Tschebyscheff-Gauss, 3-point Tschebyscheff, and open Newton-Cotes. • There are at least three ways in which to determine a bound on the error. • There are many types of integrands that the routines could identify and handle specially. For instance, power-law singularities or discontinuous integrands might be identified. Note, though, that sometimes singu­ larities at the endpoints of the region of integration can be ignored and the computation will still converge (see, for example, Myerson [10]). • There are at least six different data structures for maintaining the list of intervals used at a given stage in an adaptive algorithm, including ordered lists, stacks, queues, and "boxes." For each data structure, there are several significant variations. Rice [12] arrives at the pessimistic conclusion that there are potentially 1 million research papers to be written, each with a novel algorithm that, for some test cases, is superior to the other 999,999 algorithms. Example Consider numerically evaluating the integral I = J; 4x3 dx = 1 using adaptive quadrature. We must choose an integration rule and a measure of the error. Using the trapezoidal rule in the form t f(x)dx = b; a (I(a) + f(b)) +E (b -a)3 we find the error estimate E = 12 f(2)«() for some ( E [a, b]. In this example f(2}(X) = 24x so we can bound the error by lEI ~ 2(b -a)3b. An adaptive numerical computation can then proceed as follows: [1] We start with one interval equal to the entire range of integration: interval {[O, In integral estimates {2} error estimates {2} I ~ 2 total error ~ 2. [2] We choose to subdivide the interval with the largest estimated error. Subdividing [0,1] we find: intervals {[O, ~], [~, In integral estimates {0.125,1.125} error estimates {0.125,0.250} I ~ 1.25 total error ~ 0.375. [3] Now the largest error is in the interval [~, 1], so we subdivide it to find: intervals {[O, ~], [~, 1], [1, In integral estimates {0.125, 0.2734, 0.7109} error estimates {0.125, 0.0234, 0.0313} I ~ 1.1093 total error ~ 0.1797. 280 VI Numerical Method.s: Techniques [4] Now the largest error is in the interval [0, ~], so we subdivide it to find: intervals {[O, ~], U' ~], [~, ~], [~, I]} integral estimates {0.0078, 0.0703, 0.2734, 0.7109} error estimates {0.0078, 0.0156, 0.0234, 0.0313} I ~ 1.0624 total error::; 0.0781. [5] Now the largest error is in the interval [~, 1], so we subdivide it to find: intervals {[O, !], [!, !], [!, ~], [~, ~], [~, I]} integral estimates {0.0078, 0.0703, 0.2734, 0.2729, 0.4175} error estimates {0.0078, 0.0156, 0.0234, 0.0034, 0.0039} I ~ 1.0419 total error::; 0.0541. At this point we can conclude that I lies in the range [0.99,1.10]. Notes [1] The subdivision procedure used in most adaptive quadrature codes is a simple bisection of the chosen interval. Berntsen et al. [2] present an al­ gorithm in which a subdivision strategy results in three non-equally sized sub-intervals. [2] When an adaptive algorithm is used, the nodes at which the integrand is evaluated cannot be determined beforehand. Therefore, adaptive techniques are inappropriate for tabulated integrands. [3] Most of the routines in Quadpack (see Piessens at al. [11]) are based on adaptive algorithms. [4] Corliss and RaIl [4], who combine interval analysis methods (see page 218) with adaptive integration, estimate that the increase in CPU time for modest accuracy requests is about a factor of 3-5, and for stringent accuracy re­ quests the factor is about 3-15. However, with a stringent accuracy request, the width of the final interval is only a few units in the last place (ULP). References [1] J. Berntsen, "Practical Error Estimation in Adaptive Multidimensional Quad­ rature Routines," J. Comput. Appl. Math., 25, No.3, 1989, pages 327-340. [2] J. Berntsen, T. O. Espelid, and T. S0revik, "On the Subdivision Strategy in Adaptive Quadrature Algorithms," J. Comput. Appl. Math., 35, 1991, pages 119-132. [3] J. Berntsen, T. O. Espelid, and A. Genz, "An Adaptive Algorithm for the Approximate Calculation of Multiple Integrals," ACM Trans. Math. Software, 17, No.4, December 1991, pages 437-451. [4] G. F. Corliss and L. B. RaIl, "Adaptive Self-Validating Numerical Quadra­ ture," SIAM J. Sci. Stat. Comput., 8, No.5, 1987, pages 831-847. [5] M. C. Eiermann, "Automatic, Guaranteed Integration of Analytic Func­ tions," BIT, 29, 1989, pages 270-282. [6] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, 418-434. [7] T. O. Espelid and T. Sorevik, "A Discussion of a New Error Estimate for Adaptive Quadrature," BIT, 29, No.2, 1989, pages 283-294. 63. Clenshaw-Curtis Rules 281 [8J D. K. Kahaner and O. W. Rechard, "TWODQD an Adaptive Routine for Two-Dimensional Integration," J. Comput. Appl. Math., 17 No. 1-2, 1987, pages 215-234. [9J W. M. McKeeman, "Algorithm 145, Adaptive Numerical Integration by Simpson's Rule," Comm. ACM, 5, No. 12, December 1962, page 604. [10] G. Myerson, "On Ignoring the Singularity," SIAM J. Numer. Anal., 28, No.6, December 1991, pages 1803-1807. [11] R. Piessens, E. de Doncker-Kapenga, C. W. Uberhuber, and D. K. Kahaner, Quadpack, Springer-Verlag, New York, 1983. [12] J. R. Rice, "A Metalgorithm for Adaptive Quadrature," J. ACM, 22, No.1, January 1973, pages 61-82. [13J H. D. Shapiro, "Increasing Robustness in Global Adaptive Quadrature through Interval Selection Heuristics," ACM 1Tans. Math. Software, 10, No.2, 1984, pages 117-139. 63. Clenshaw-Curtis Rules Applicable to One-dimensional definite integrals. Yields A numerical quadrature scheme. Idea A function can be approximated by a (finite) linear combination of basis functions. If the integrated value of the basis functions is known, then an approximation of the integral is obtained. Procedure Let {<Pn (x)} represent a set of functions for which the integrals Sn : = f: <Pn(x) dx are known. To approximate the value of I = f: f(x) dx, we approximate the integrand by Knowing the {ak} values, we find that I::::: alSl + a2s2 + ... + aNSN. For the Clenshaw-Curtis rules, we take {a = -1, b = I} and use the Tschebyscheff polynomials <p(x) = Tn(x) = cos (ncos-l x). The first few such polynomials are Tl(X) = 1, T2(x) = x, and T3(X) = 2X2 -1. In this case we can write the {ak} analytically as 211r ak = -f(cos9)cos(k9)d9. 1r 0 (63.1) To determine the value of ak numerically, as defined by (63.1), a technique may be used that is exact for trigonometric polynomials (see page 322). (That is, if f is a polynomial, then the exact value of ak will be returned.) 282 VI Numerical Methods: Techniques Example Consider the numerical evaluation of the integral 1= 4/1r f; VI -x2 dx. (The exact value is I = 1.) For the Tschebyscheff polynomials we can compute 81 = 101 T1(x) dx = 101 1 dx = 1, /.1 /.1 1 82 = 0 T2{x)dx = 0 xdx = 2' /.1 /.1 1 82 = 0 T3{X)dx = 0 (2x2 -1)dx = -'3' /.1 /.1 1 83 = 0 T4{X) dx = 0 (4x3 -3x) dx = -2' For our specific integrand, we can evaluate the integral in (63.1) to obtain ak = ~ 10· (~Vl -cos2 /I ) cost kIJ) d/l 8 /.1r = "2 sin 8 cos{ k8) d8 1r 0 { 0 if k is odd, = 16 if k is even. 1r2{1 -k2) Therefore, we can approximate I by the series 1 16 1 1 16 I~ 1·0+ -. - - -·0+ -. - + .... 2 1r2 3 2 31r2 The partial sum of this series after 2 terms is I ~ ~ ~ 0.81. After 4 terms 32 1r we obtain I ~ ~ ~ 1.08. Note [1] This method can be modified to account for integrals with weight functions; see Piessens at ale [4]. References [1] C. W. Clenshaw and A. R. Curtis, "A Method for Numerical Integration on an Automatic Computer," Numer. Math., 2,1960, pages 197-205. [2] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, pages 86-87, 193-196, and 446-449. [3] R. B. Kearfott, "A Sinc Approximation for the Indefinite Integral," Math. of Comp., 41, 1983, pages 559-572. [4] R. Piessens, E. de Doncker-Kapenga, C. W. Uberhuber, and D. K. Kahaner, Quadpack, Springer-Verlag, New York, 1983, Section 2.2.3, pages 28-39. 64. Compound Rules 64. Compound Rules Applicable to Definite integrals in any number of dimensions. Yields A numerical quadrature scheme. Idea 283 If a numerical quadrature scheme is known for integration over a small parallelpiped region, then the rule may be repeatedly applied to find a rule applicable over a larger parallelpiped region. Procedure Suppose we seek to approximate an integral numerically over a large region. We may subdivide the region into smaller regions (often called ''panels''), and then apply a quadrature scheme in each of the smaller regions. The reSUlting quadrature rule, in the larger region, is called a compound rule. Example Simpson's rule, in its most elementary form, approximates a one di­ mensional integral using three nodes via f.b b-a [ (a+b) ] a f(x) dx ~ -6-f(a) + 4f -2-+ f(b) . (64.1) To apply Simpson's rule on a large interval, say from c to d, the larger region can be subdivided into smaller regions and Simpson's rule applied in each of the smaller regions. If the interval [c, d) is subdivided into two equally spaced intervals, [c, e) and [e, d) (with e = (d + c)/2), then 1= f.d I(x) dz = [ I(x)dz+ [ I(x)dz, e-c [ (c+e) ] d-e [ (e+d) ] ~ -6-f(c) + 4f -2-+ fee) + -6-fee) + 4f -2-+ fed) , d -c [ (d -C) ( d -C) ( d -C) ] = 12 fCc) + 4f c + -4-+ 2f c + 2-4-+ 4f c + 3-4-+ fed) , = ~ [/0 + 4/1 + 2/2 + 4fa + I.] , (64.2) where fn = f(c + nh) and h = (e -c)/2 = (d -c)/4. 284 VI Numerical Methods: Techniques When manipulating integration rules, as above, it is often easier to just use subscripted variables. For example, the above derivation could be written as 1= ld f(x)dx = 1" f(x) dx + t f(x) dx, h h = 3 (/0 + 4h + 12) + 3 (/2 + 4/3 + 14) h = 3 (fo + 4h + 212 + 4/3 + 14] . If the original interval [c, d) had been divided into three equally sized intervals, instead of two, then the compound rule obtained would have been 1= t f(x)dx k k k ~ 3 (/0 + 4h + 12) + 3 (/2 + 4/3 + 14) + 3 (/4 + 4/5 + 15) (64.3) = ~ [fo + 4ft + 210 + 4fa + 2f4 + 4f5 + f6] where 1m = I(c+ mk) and k = {d -c)/6. The extended form of Simpson's rule, that is, the rule applied to n equal-width intervals, is: ld f(x)dx <::!. ~ [fo + 4(ft + fa + ... + hn-d + 2 (/2 + f4 + ... + hn-2) + Ion ] (64.4) where 1m = I{c + mk) and k = (d -c)/{2n). Notes [1] A compound rule is sometimes called a composite rule. [2] Compound rules also exist for simplices. See, for example, Lyness [3] or De Donker [2]. 65. Cubic Splines 285 References [1] P. J. Davis and P. Rabinowitz, Methods oj Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, pages 379-384. [2] E. De Doncker, "New Euler-Maclaurin Expansions and Their Application to Quadrature Over the a-Dimensional Simplex," Math. oj Comp., 33,1978, pages 1003-1018. [3] J. M. Lyness, "Quadrature over a Simplex: Part 1. A Representation for the Integrand Function," and "Quadrature over a Simplex: Part 2. A Represen­ tation for the Error Functional," SIAM J. Numer. Anal., 15, 1978, pages 122-133 and 870-887. [4) J. N. Lyness and A. C. Genz, "On Simplex Trapezoidal Rule Families," SIAM J. Numer. Anal., 17, 1980, pages 126-147. [5) A. Ralston, "A Family of Quadrature Formulas which Achieve High Accu­ racy in Composite Rules/' J. ACM, 6, 1959, pages 384-394. 65. Cubic Splines Applicable to One-dimensional definite integrals. Yields A numerical quadrature scheme. Idea Given data values of a function (not necessarily equally spaced), a cubic spline can be fit to those values, and then the integral of the cubic spline can be determined. Procedure Let [a, b) be a finite interval and assume that we have the points {Xi} with a :5 Xl < X2 < ... < xn+l :5 band n 2:: 2. Given the data values {!(Xi)}, let S(x) be the natural cubic spline which interpolates those data values. That is, • S(x) is a cubic polynomial in every interval [Xi, Xi+l] (say this poly­ nomial is Si (x)). • S(x) matches the data values at the nodes Si(Xi) = !(Xi) for i = 1, ... , n, and Sn(xn+d = !(Xn+l)' • At the nodes, S and its first and second derivatives are continuous: Si-l (Xi) = Si(Xi), S:_l (Xi) = SHXi), and S;~l (xd = S;'(Xi) for i = 2, ... ,n. • There is no curvature as the ends of the spline: S"(a) = S"(b) = 0 (this is what makes the spline "natural"). (In this definition we have implicitly assumed that a = Xl and b = Xn+b if this is not the case, then the range for i changes.) See Figure 65. 286 VI Numerical Methods: Techniques 81 82 8n-1 8n )( )( )( )( )( )( )( Xn-l xn Xn+l Figure 65. Location of the nodes {Xi} and the cubic polynomials {Si}. We can now approximate the integral of f by the integral of 8 and introduce an error E in doing so: t /(x)dx= t S(x)dx+E. (65.1) Example Consider approximating the integral I = f; sin 7rX dx. We choose to use the equally-spaced points {Xi} = {O, 1, ~, ~, I}. Hence, we have the data values {(O,O), (1, ~), (~,1), (~, ~), (I,O)}. We choose to represent the cubic on the interval [Xi, Xi+l] in the form 8i(x) = ai(x -Xi)3 + bi(X -Xi)2 + Ci(X -Xi) + di• Using this notation the first few equations for the unknowns {ai, bi, Ci, d; I i = I, ... , 4} are Sl(O) = 1(0) = 0 ===* dl = 0 Sf' (0) = 0 => bl = 0 Sl (~) = I (~) = yt. ===* 6~al + hbl + ~Cl + dl = yt. Sl (~) = S2 (~) => iial + hbl + ~Cl + dl = d2 Sf (~) = S~ (~) => 136a1 + ~bl + Cl = C2 Sf' (~) = sq (~) ===* ~al + b1 = 2b2 S2 (~) = S3 (~) ===* ===* Completing this list of equations, and solving the resulting linear system, results in the approximation interval [O,~]: -4.8960(x -0)3 + 3.1340(x -0) interval U' ~]: -2.0288(x -~)3 -3.6720(x -~)2 + 2.2164(x -U + 0.7071 interval [~,~]: 2.0288(x -~)3 -5.1936(x _ ~)2 + 1 interval [~, 1]: 4.8960(x -~)3 -3.6720(x -~)2 -2.2164(x -~) + 0.7071. (65.2) Now we can determine the approximation to the integral: (65.3) Evaluating (65.3), when the coefficients are given in (65.2), results in the approximation I ~ 0.6362. (Note that the exact value is I = : ~ 0.6366.) 66. Using Derivative Information 287 Notes [1] Among all functions S that are of class 02 [a, b] which interpolate the data values, it is the cubic spline approximation that minimizes the "total curva- ture": C = J: IS"(x)12 dx. [2] Cubic splines may be numerically computed by the software in Press et al. [5]. [3] Use of cubic splines does not result in a conventional quadrature rule. If the numerical approximation is written in the form J: !(x) dx ~ Ei Wd(Xi), then the weights {Wi} depend on all of the function values {!(Xi)}. References [1] [2] [3] [4] [5] [6] [7] G. H. Behforooz and N. Papamichael, "End Conditions for Interpolatory Cubic Splines with Unequally Spaced Knots," J. Compl£t. Appl. Math., 6, 1980, pages 59-65. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, 62-70. P. Dierckx, "Algorithm 003. An Algorithm for Smoothing, Differentiation and Integration of Experimental Data Using Spline Functions," J. Comput. Appl. Math., 1, 1975, pages 165-184. D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Pren­ tice-Hall Inc., Englewood Cliffs, NJ, 1989. W. H. Press, B. P. Flannery, S. Teukolsky, and W. T. Vetterling, Numerical Recipes, Cambridge University Press, New York, 1986. P. Rabinowitz, "Numerical Integration Based on Approximating Splines," J. Comput. Appl. Math., 33,No. 1,1990, pages 73-83. M. N. EI Tarazi and S. Sallam, "On Quartic Splines with Application to Quadratures," Computing, 38, 1987, pages 355-361. 66. Using Derivative Information Applicable to Definite integrals. Yields A numerical quadrature scheme. Idea A quadrature rule can use the value of the integrand at the nodes, and it can also use the value of the derivative of the integrand at the nodes. Procedure A quadrature rule in the form b N M !. /(x) dx ~ L: Wj/(Xj} + L: Vj!'(Xk) a ;=1 k=l can often be determined by making the rule exact for the polynomials {1,x,x2, ••• }. 288 VI Numerical Methods: Techniques Example Consider an integration rule of the form t I(x) dx '" 0/(0) + PI(l) + "11'(0) + 61'(1). Making this rule be exact for I (x) = {I, x, x2, x3}, we obtain the following set of simultaneous equations: I{x) = 1 ==> l=a+,B I(x) = x ==> ~=,B+'Y+6 I{x) = x2 ==> i =,8 + 26 I{x) = x3 ==> i =,8 + 36. These equations can be solved to obtain the approximation: [,(X) dx '" ! [/(0) + 1(1)] + .'2 [/,(0) -/'(1)]. This rule is known as the corrected trapezoidal rule. Notes [1] Some of the common quadrature rules can be improved by including deriva­ tive terms. The corrected trapezoidal rule is given in the example. The corrected midterm rule is ['(X)dx '" IW + t. [1'(1) -/'(0»). [2] When either the corrected midterm rule or the corrected trapezoidal rule is compounded (see page 282), the derivative at the nodes in common cancel, so that only the derivatives at the end points remain. For example, the compounded corrected trapezoidal rule is lb h ~ Q I(x) dx = 2" [/0 + 2/1 + 212 + ... + 2/n-l + In]+ 12 [/'(a) -/,(b)] +E where Ii = I(a + ih) and h = (b -a)/n. It can be shown that the error is bounded by lEI ~ 7210h4(b -a) max 1J<4)(X)I (Davis and Rabinowitz [2], Q<z<b page 132). - - [3] The trapezoidal rule, corrected by using both f' and I" terms, takes the form lb 2 h3 Q I(x) dx = ~ [/(a) + I(b)]+ ~o [/'(a) -/,(b)] + 120 [/"(a) + I" (b)] +E where E = -10~:00/(6)(,) and a < , < b (Davis and Rabinowitz [2], page 133). 67. Gaussian Quadrature 289 References [1] R. A. Cicenia, "Numerical Integration Formulae Involving Derivatives," J. Inst. Math. Appls., 24, 1979, 347-352. [2] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, pages 132-134. [3] J. D. Lambert and A. R. Mitchell, "The Use of Higher Derivatives in Quadra­ ture Formulae," Comput. J., 5, 1962-1963, pages 322-327. 67. Gaussian Quadrature Applicable to One-dimensional definite integrals. Yields Integration rules on a finite interval using non-uniformly spaced nodes. Idea A general expression for approximating an integral is proposed. The unknown constants in this expression are determined by making the quadra­ ture rule exact for polynomials of low degree. Procedure Given the integral 1= J: g(x) dx we first map it to the interval [-1,1]; this will minimize the algebra later. We have 1= lb g(x) dx = b -a rl 9 (b -a)t + (b + a)) dt = rl f(t) dt, a 2 J-l 2 J-l so that we can now focus on I = J~l f(t) dt. Now we search for an approximate integration rule of the form n I ~ f = L oif(ti). (67.1) i=l Since this integration rule has 2n unknown constants (the weights {Oi} and the nodes {ti}), we can hope to choose these constants so that the integration rule is exact for all polynomials of degree less than or equal to 2n -1. It turns out this is always possible. For example, with n = 2 we have If this formula is to be exact for polynomials of degree less than or equal to 3, then {o 11 02} and {t 1! t2} must satisfy the simultaneous algebraic equations: 290 VI Numerical Methods: Techniques Table 67. Values used in Gaussian quadrature formulas. Number of terms Nodes {til 2 -0.57735027 0.57735027 3 -0.77459667 0 0.77459667 4 -0.86113631 -0.33998104 0.33998104 0.86113631 f(t) = 1 => f(t) = t f(t) = t2 => Valid for polynomials Weights {Qi} up to degree 1 3 1 0.55555855 5 0.88888889 0.55555555 0.34785485 7 0.65214515 0.65214515 0.34785485 11 1 dt = 2 = al + a2 -1 11 tdt = 0 = altl + a2t2 -1 f(t) = t3 => 11 t3 dt = 0 = alt~ + a2t~. -1 These equations have the unique solution {al = a2 = 1, h = -t2 = 1/.f3 ~ 0.5773}. Hence, we have the approximation 1: I(t) dt "" 1(-0.5773) + 1(0.5773). To obtain Gaussian quadrature formulas for larger values of n, we must find the solutions to a large set of nonlinear algebraic equations. The results of such a calculation are shown in Table 67. Example Consider the integral J = 2 ~ 111 cos x dx = 1. We represent the sm -1 numerical approximation obtained by using Gaussian quadrature with n nodes by In. Using 2, 3, and 4 nodes we obtain the approximations J ~ J2 = 0.9957702972 J ~ J3 = 1.0000365711 J ~ J4 = 0.9999998148. 67. Gaussian Quadrature 291 Notes [1] The values of the {til in (67.1) turn out to be roots of the Legendre polynomial Pn{x). These polynomials are defined by the recurrence relation (n + I)Pn+1(X) = (2n + l)xPn(x) -nPn-1(x) with the initial conditions: Po{x) = 1 and P1(X) = x. Then, for example, we can calculate P2{X) = ~X2 -~. The roots of P2{X) are at x = ±1/v'3 ~ ±0.5773. The next Legendre polynomial is P3(x) = ~x3 -~x; its roots are at x = 0 and x = ±v'f ~ ±0.7746. [2] The values of the weights {oil in (67.1) are also functions of Legendre polynomials. If Xi is the i-th root of Pn{x), then the corresponding weight, Oi, is given by 0i = 2 2 2. For example, when n = 2 we find (1 -Xi) (P~(Xi» 01 = ( 2) 2' When XI = ±1/../3 this result. in 01 = 1. 1 -Xi (3Xi) [3] See the section on generalized Gaussian quadrature (page 291) for the anal­ ogous technique applied to integrals of the form J: w(x)f(x) dx, when w(x) is a positive weighting function. [4] Newton-Cotes rules (see page 319) are also interpolatory, but the nodes are chosen to be equidistant from one another. [5) Several modifications of the Gaussian principle have been developed, in which some of the nodes or weights, or both, are specified in advance. The Radeau formulas use one of the endpoints, the Lobatto formulas use both of the endpoints. The simplest Radeau formula has the form Jo1 f{x) dx ~ W1 f{xt} + W2 f{O), for some unknown {X1,W1,W2}. Making this quadrature rule exact for f(x) = xk (for k = 0,1,2) results in the quadrature rule J: f{x) dx ~ ~ f{~) + ~ f{O). [6) The Tschebyscheff weight function w(x) = (1 -x2)-1/2 is the only weight function (up to a linear transformation) for which all the weights in an-point Gauss quadrature formula are equal. See Peherstorfer [5] and the section on Tschebyscheff rules (page 331). References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964, Table 25.4, page 919. (2) W. Cheney and D. Kincaid, Numerical Mathematics and Computing, Second Edition, Brooks/Cole Pub. Co., Monterey, CA, pages 193-197. [3] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, Section 2.7, pages 95-132. [4) W. Gautschi, E. Tychopoulos, and R. S. Varga, "A Note on the Contour Integral Representation of the Remainder Term for a Gauss-Chebyshev Quadrature Rule," SIAM J. Numer. Anal., 27, No.1, 1990, pages 219-224. [5] F. Peherstorfer, "Gauss-Tschebyscheff Quadrature Formulas," Numer. Math., 58, 1990, pages 273-286. 292 VI Numerical Methods: Techniques 68. Gaussian Quadrature: Generalized Applicable to Integrals that have a positive weighting function. Yields Integration rules using non-uniformly spaced nodes. Idea A general expression for approximating an integral is suggested. The unknown constants in this expression are determined by making the quadra­ ture rule exact for polynomials of low degree. Procedure Consider the integral I[f) = t w(x)f(x) dx where w(x) is a specified positive function. To numerically approximate I[!] for different functions !(x), we choose to use a quadrature rule of the form n Jl!] ~ L Wi!(Xi) (68.1) i=l where the weights {Wi} and the nodes {Xi} are to be determined. These values are determined by requiring that (68.1) be exact when !(x) is a polynomial of low degree. In the section on Gaussian quadrature (page 289) the same formulation was used when w(x) = 1 and a and b were finite. In that section it was shown that the nodes {Xi} and the weights {Wi} are related to the roots of Legendre polynomials. In the more general case considered here, it can be shown that the weights {Wi} satisfy a polynomial specified by the weight function w(x). In particular, if the inner product of two functions ! and 9 is defined by (!,g) := J: w(x)!(x)g(x) dx, and j is any positive integer, then there exist j polynomials {Pk I k = 1,2, ... ,j} with the k-th polynomial being of degree k, such that (Pkl'Pk2) = 0 when kl ::j= k2• Such a set of polynomials can be constructed via Po(x) = 1, Pi+l(X) = (x -oi+dpi(X) --yl+lPi-l(X) for i 2:: 0, (68.2) 68. Gaussian Quadrature: Generalized 293 where P-l{X) = 0 and (68.3) We can now state what the values of {Wi} and {Xi} are. The {Xi I i = 1, 2, ... , n} are the roots of the polynomial Pn (x) and the {Wi I i = 1,2, ... , n} are the unique solution of the (nonsingular) linear system of n simultaneous equations L,.=n 1 Pk{Xi)Wi = {o(po,PO) for k = 0, for k = 1,2, ... , n -1. The nodes and weights appearing in (68.1) may also be found by finding the eigenstructure of a specific matrix. If the tri-diagonal matrix J n is defined by o o then the eigenvalues of In will be the nodes in the quadrature formula, { Xi}. Corresponding to each of these eigenvalues is an associated eigen- {(I) (2) (n»T ( h J ) Th . h . vector, Vi = Vi 'Vi , ... , Vi so t at n Vi = XiVi . e welg ts m the quadrature formula are then given by Wi = (v~I») 2. See Stoer and Bulirsch [3] for details. The roots, {Xi Ii = 1,2, ... , n}, turn out to be real, simple, and to lie in the interval (a, b). The weights satisfy Wi ;::= 0 and the relation b n 1 w{x)p{x) dx = L WiP{Xi) a i=1 holds for all polynomials p{x) of degree 2n -1 or less. For some specific weight functions, w{x), and intervals [a, b], the polynomials {Pk I k = 1,2, ... , n} turn out to be classical polynomials. For example, we find 294 VI Numerical Methods: Techniques interval w( x) Orthogonal polynomials [-1,1] [0,00] [-00,00] Tk(X), Tschebyscheff polynomials Lk(X), Laguerre polynomials Hk(X), Hermite polynomials In Table 68 may be found a more complete list of orthogonal polynomials. That table also describes the corresponding nodes and weights of each quadrature rule. The error term in such an approximation is given by the following theorem. Theorem: If j E C2n [a, b] then for some e E (a,b). Table 68. Some specific generalized Gaussian quadrature rules. 68.1 Gauss' formula Approximation: 1: /(x) dx = t. wd(x,) + R. Nodes: Xi is the i-th root of Pn(X) (Legendre polynomial) Weights: Wi = 2 2 (1 -x~) (P~(Xi)) 22n+l(n,)4 Error term: Rn = . j(2n)(e) with -1 < e < 1 (2n + 1)[(2n)!]3 Reference: Abramowitz and Stegun [1], 25.4.29 68.2 Radeau's integration formula fl n-l Approximation: 1-. /(x) dx = :2 f( -1) + ~ wd(x,) + R. d . h . f Pn-l(X) + Pn(X) No es: Xi IS t e I-th root 0 1 X+ Weights: Wi = 1 2 (1 -Xi) (P~-l(Xi)) 22n-ln[(n _ 1),]4 Error term: R = . j(2n-l)(e) with -1 < e < 1 n [(2n _ 1)!]3 . Reference: Abramowitz and Stegun [1], 25.4.31 68. Gaussian Quadrature: Generalized 68.3 Lobatto's integration formula 11 n-1 Approximation: -1 /(x) rb: = n( n 2_ 1) 1/(1) + /( -1) I + ~ w;f(x,) + R. Nodes: Xi is the (i -I)st root of P~-l(X) Weights: Wi = 2 2 n(n -1) (Pn-1 (Xi)) n(n -I)322n-1 [en -2)!J4 . Error term: Rn = - f(2n-2)(e) wIth -1 < e < 1 (2n -1)[(2n -2)!J3 Reference: Abramowitz and Stegun [IJ, 25.4.32 68.4 Weight function xlt 11 n Approximation: xlt f(x) dx = Lwi/(x i) + Rn -1 i=l Polynomials: qn(X) = ../k + 2n + IP~It,O)(I -2x) Nodes: Xi is the i-th root of qn(X) ( 1 )_1 Weigbts: w, = ~qJ(x,) 1=0 j<2n)(e) [nICk + n)!]2 . Error term: Rn = (k + 2n + I)(2n)! (k + 2n)! wIth 0 < e < 1 Reference: Abramowitz and Stegun [IJ, 25.4.33 68.5 Weight function Jr=X Approximation: 11 f(x)Jr=X dx = t Wi/(Xi) + Rn o i=l Nodes: Xi = 1 -el where ei is the i-th positive root of P2n+1(X) 295 Weights: Wi = 2elw~2n+l) where {w~2n+1)} are the Gaussian weights of order 2n+ 1 24n+3[(2n + 1),]4 Error term: Rn = . f(2n)(e) with 0 < e < 1 (2n)!(4n + 3)[(4n + 2)!]2 Reference: Abramowitz and Stegun [1], 25.4.34 68.6 Weight function ~ vI-x Approximation: 11 ~ dx = t wi/(Xi) + Rn o vI -X i=l Nodes: Xi = 1 -el where ei is the i-th positive root of P2n(X) Weights: Wi = 2w~2n) where {w~2n)} are the Gaussian weights of order 2n 24n+1 [(2n)'J3 Error term: R = ----·-f(2n)(e) with 0 < e < 1 n 4n + 1 [(4n)!J2 Reference: Abramowitz and Stegun [1], 25.4.36 296 1 68.7 Weight function ---=== ~ VI Numerical Methods: Techniques Approximation: (I ~ dx = t Wd(Xi) + Rn J -I Vi -X2 i=1 (2i -1)11" Nodes: Xi = cos -=-----~ 2n Weights: Wi = ! n Error term: Rn = 11" j(2n)(e) with -1 < e < 1 (2n)122n-1 Reference: Abramowitz and Stegun [1], 25.4.38 68.8 Weight function "'1 -X2 Approximation: I: f(x)~ dx = t. w;J(x.) + Rn (i + 1)11" Nodes: Xi = cos """"""-----"'­n+l 'IT • h 11". 2 (i + 1)11" vvelg ts: Wi = --sm n+l n+l Error term: Rn = 11" 2n+1 j(2n) (e) with -1 < e < 1 (2n)12 Reference: Abramowitz and Stegun [1], 25.4.40 68.9 Weight function -1 2 -x Approximation: /.' f(X)V 1 ~ x dx = t. w;J(x,) + Rn Nodes: Xi = cos2 (:~ ~ ~ ~) 'IT· h 211" vvelg ts: Wi = 2n + 1 Xi Error term: Rn = 11" 4n+1 j(2n)(e) with 0 < e < 1 (2n)12 Reference: Abramowitz and Stegun [1], 25.4.42 68.10 Weight function e-x Approximation: /.00 e-x j(x) dx = t Wd(Xi) + Rn o i=1 Nodes: Xi is the i-th root of Ln(x) (Laguerre polynomial) • (nl)2xi WeIghts: Wi = 2 2 ) (n + 1) Ln+I(Xi Error term· Rn = (nl)2 j<2n) (e) with 0 < e < 00 . (2n)! Reference: Abramowitz and Stegun [1], 25.4.45 68. Gaussian Quadrature: Generalized 68.11 Weight function e-z2 Approximation: i: e-" f(x) dx = t. w;f(x.) + R,. Nodes: Xi is the i-th root of Hn{x) (Hermite polynomial) . 2n-1n!y'1r WeIghts: Wi = ---=-2--=2~";"'--n Hn-1(Xi) ED n!y'1r j(2n){c) • h C rror term: .an = 2n{2n)! ~ WIt -00 < ~ < 00 Reference: Abramowitz and Stegun [1], 25.4.46 Notes 297 [I) The technical conditions required on w{x) for the orthogonal polynomials to exist are (see Stoer and Bulirsch [3J): • w{x) ~ 0 is measurable on the (finite or infinite) interval [a,b]; • All the moments J: xkw(x) dx for k = 0,1, ... exist and are finite; • If s{x) is a polynomial which is nonnegative on the interval [a,b), then J: w{x)s{x) dx = 0 implies that s(x) is identically zero. [2) Press and Teukolsky [2] discuss the numerical development of Gaussian quadrature rules when the weight function desired is not one of those in Table 68. [3] Depending on which polynomials are used to obtain the quadrature rule, the rules obtained by this technique are called Gauss-Hermite rules, Gauss­ Jacobi rules, Gauss-Laguerre rules, Gauss-Legendre rules, etc. [4] Frequently, Gaussian integration rules of successively higher order are tried when approximating an integral. Unfortunately, it is most often the case that information about the function cannot be re-used when using a higher order rule; that is, the roots of the polynomials at each order do not overlap. See the section on Kronrod rules (page 298) for a solution to this problem. References [1] M. Abramowitz and 1. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. [2] W. H. Press and S. A. Teukolsky, "Orthogonal Polynomials and Gauss­ ian Quadrature with Nonclassical Weight Functions," Compo in Physics, Jul/ Aug 1990, pages 423-426. [3] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, translated by R. Bartels, W. Gautschi, and C. Witzgall, Springer-Verlag, New York, 1976, pages 142-151. [4] A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall Inc., Englewood Cliffs, NJ, 1966. 298 VI Numerical Methods: Techniques 69. Gaussian Quadrature: Kronrod's Extension Applicable to One-dimensional definite integrals. Yields A sequence of integration rules starting with a Gaussian rule. Idea When an integral is to be evaluated by different Gaussian integration rules, none of the values of the integrand, except possibly for the x = 0 value, can be re-used. It is possible to devise interpolatory rules that re-use all of the nodes in a Gaussian rule. Procedure Consider the integral I = J: f(x) dx. Suppose the Gaussian n-point rule, Gn, is used to approximate I numerically. Later, it may be of interest to approximate I numerically using m nodes (with m > n). If the Gaussian m-point rule, Gm, is used, then the only values of f that were obtained using Gn that can be re-used is, possibly, the value x = O. (The value x = 0 can be re-used only if both nand m are odd). If f is an expensive function to compute, then it would be useful to re-use many values from the Gn computation. It is possible to start with the nodes from the rule Gn, {Xi I i = 1,2, ... , n}, and add new nodes {Yi I i = 1,2, ... , n + I} so that all polynomials of degree 3n + 1 are integrated exactly, if n is even (degree 3n + 2, if n is odd). The Kronrod rule, which uses the nodes {Xi,Yj} to evaluate polynomials of maximal degree exactly, will be called K2n+1. Note that the weights corresponding to the nodes {Xi} in K2n+1 will not be the same as the weights corresponding to the nodes {Xi} in Gn. As an example, Table 69 shows the rules G7 and K 15. Notes [1) Piessens et al. (11) contains the numerical values of the nodes and weights for the following rules: {G7,K1S}, {GlO,K21}, {Gls,K3t}, {G2o,K41}' {G25,Ksl}, and {G3o,K6t}. (2) The technique in this section may be continued; after the n-point Gaussian rule (Gn) is used, and the 2n + i-point Kronrod rule (K2n+1) is used, additional nodes may be added to interpolate higher order polynomials. Nodes and weights for the sequence of rules {G3,K7,P 15,P31, ... ,P25S} are given in Patterson (8). (Here, the rule Pk is exact for polynomials of degree (3k + 1)/2.) Similarly, the sequence of rules {GI0, K21, P43, Pad is given in Piessens et al. (11). (3) Kronrod extensions also exist for Gaussian rules with the weight function (1 -x2)" (for -~ ~ J.£ ~ ~). 69. Gaussian Quadrature: Kronrod's Extension 299 Table 69. Values used in 7-point Gaussian and 15-point Kronrod quadrature formulas. The formulas are symmetric, only the positive nodes are shown. (That is, if Wi!(ti) appears, then so does wi/( -ti).) Name 7-point Gaussian I5-point Kronrod Nodes 0.94910 0.74153 0.40584 o 0.94910 0.94910 0.86486 0.74153 0.58608 0.40584 0.20778 o Weights 0.12958 0.27970 0.38183 0.41795 0.02293 0.06309 0.10479 0.14065 0.16900 0.19036 0.20443 0.20948 [4] Instead of using a higher order rule, the same rule can be re-applied with a smaller interval size. For even more accuracy, an extrapolation technique can be used, see page 249. [5] Favati et al. [3] derive a set of symmetric, closed, interpolatory quadra­ ture formulas on the interval [-1,1] with positive weights and increasing precision. These formulas re-use previously computed functional values. They obtain a tree of quadrature rules having 74 elements, 27 leaves, and a maximum height of 14. That is, the sequence of height 14 is a collection of quadrature rules that use (2,3,5,7,9,13,19,27,41,57,85,117,181,249) nodes, and each rule re-uses all the nodes from the previous rule. Favati et al. [3] performed extensive numerical tests using their new rules in place of the Gauss-Kronrod rules in the routines QAG and QAGS, in the computer library Quadpack. For one-dimensional and two-dimensional integrals, the resulting programs appear to be faster, more reliable, and to require fewer function evaluations. [6] Rabinowitz [12] considers the numerical evaluation of integrals of the form 11 w~>.!ix) dx with w(x) = (1 -X2)#'-1/2 for 0 :5 p. :5 2. '-I References [1] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, pages 106-109. [2] F. Calia and W. Gautschi, "On Computing Gauss-Kronrod Quadrature Formula," Math. of Comp., 47, No. 176, 1986, pages 639-650, S57-S63. [3] P. Favati, G. Lotti, and F. Romani, "Interpolatory Integration Formulas for Optimal Composition," and "Algorithm 691: Improving QUADPACK Automatic Integration Routines," ACM Trans. Math. Software, 17, No.2, June 1991, pages 207-217 and 218-232. 300 VI Numerical Methods: Techniques (4] W. Gautschi and S. E. Notaris, "An Algebraic Study of Gauss-Kronrod Quadrature Formulae for Jacobi Weight Functions," Math. oj Comp., 51, No. 183, 1988, pages 231-248. (5] A. S. Kronrod, Nodes and Weights oj Quadrature Formulas, Consultants Bureau, NY, 1965. (6] G. Monegato, "Stieltjes Polynomials and Related Quadrature Rules," SIAM Review, 24, 1982, pages 137-158. (7] S. E. Notaris, "Gauss-Kronrod Quadrature Formulae for Weight Functions of Bernstein-Szego Type. II," J. Comput. Appl. Math., 29, No.2, 1990, pages 161-169. (8] T. N. L. Patterson, "The Optimal Addition of Points to Quadrature For­ mulae," Math. oj Comp., 22, 1968, pages 847-856. (9] F. Peherstorfer, "Weight Functions Admitting Repeated Positive Kronrod Quadrature," BIT, 30, No.1, 1990, pages 145-151. (10] F. Peherstorfer, "On Stieltjes Polynomials and Gauss-Kronrod quadrature," Math. oj Comp., 55, No. 192, 1990, pages 649-664. (11] R. Piessens, E. de Doncker-Kapenga, C. W. Uberhuber, and D. K. Kahaner, Quadpack, Springer-Verlag, New York, 1983. (12] P. Rabinowitz, "A Stable Gauss-Kronrod Algorithm for Cauchy Principal­ Value Integrals." Comput. Math. Appl. Part B, 12, No. 5-6, 1986, pages 1249-1254. 70. Lattice Rules Applicable to One-dimensional and multidimensional integrals on the unit cube. Yields A numerical approximation scheme. Idea A lattice rule uses all the nodes on a lattice that lie within and on the boundary of the unit cube. Procedure A lattice rule is a numerical scheme for approximating the value of an integral over the multidimensional unit cube. The integral is assumed to have the form I = lOB I(x) dx, where CS is the closed s-dimensional unit cube CS = {(Xl! ... , xs) I 0 ~ Xi ~ 1, i = 1,2, ... , s} . Every lattice rule can be written in the form I ~ 1 ~ ~ -I (i1Z1 imZm) ~ L...J .•• L...J + ... n1n2· .. nm . -1 . -1 n1 nm 3m- 31-(70.1) 70. Lattice Rules 301 where 1 is the periodic extension of 1 (see next paragraph), {Z I, Z2, ••• , Zm} are vectors with integer components, and {nl' n2, ... , nm} are given inte­ gers called invariants. (It is generally assumed that ni+l divides ni for i=I,2, ... ,m-1.) Let {x} denote the fractional part of x (e.g., {3.4} = 0.4), and let {x} = ({Xll X2,'''' xs}) = ({xt}, {X2}, ... , {xs}). Then the periodic ex­ tension of 1 is defined by I(x) := I( {x}), when {Xj};f; 0, for all j = 1,2, ... ,8. Hence, 1 coincides with 1 in the interior of the unit cube. At the points on the boundary of the unit cube, 1 is generally not continuous. At these points, 1 is defined by I(XI, X2,··., xs) = 8-s lim ~ ~ ... ~I (Xl + C-Tllx2 + C-T2,··· ,Xs + C-Ts) ~_o+LL L rl r2 rll with each Ti taking only the values ±l. If 1 is continuous on CS, then this limit exists and is a symmetrical average of the values of 1 at corre­ sponding points on opposite faces of the boundary. For example, in the one-dimensional case we find 1(0) = 1(1) = !(/(O) + 1(1)). (70.2) Sloan and Lyness [8] consider quadrature rilles for the 8-dimensional hypercube of the form which cannot be expressed in an analogous form with a single sum. These rules are called rank-2 lattice rules. If m summations are required to represent the rule, as shown in (70.1), then the rule has rank m. The number of nodes used by such a rule is II:'1 ni, and the rule may be expressed in a canonical form with m indepen­ dent summations. Under this classification, an N-node number-theoretic rule (see page 312) is a rank m = 1 rule with {nil = {N, 1, 1"", I}, and the product trapezoidal rille (see page 323) using NS nodes is a rank m = 8 rule with {nil = {N,N, .. ·,N}. 302 VI Numerical Methods: Techniques Example In the case of one dimension, (70.1) becomes (where the single sub­ script has been suppressed) 1 n _ (iZ) I~J=-L/ - . n ;=1 n Combining this with (70.2) and using Z = 1, we are led to the trapezoidal rule in the more familiar form J=-EJ .L =--/(0)+ L/ .L +-/(1) . 1 N (.) 1 (1 N-l (.) 1 ) N ;=1 N N 2 ;=1 N 2 The only N -point one-dimensional lattice rule is the trapezoidal rule. Notes [1] The s-dimensional product-trapezoidal rule is defined by I", ~. :t ... :t :t 7 ((jl'i2'~ .. ,j.») 3.::::::1 )2::::::131:::::: 1 n n n = :8 L'" E E 7 (;~1 + h:2 + ... + ;8;8) (70.3) ;.::::::1 ;,::::::1il=1 where e; is a unit vector with a one in the X; direction, and zeros elsewhere. [2] For the one-dimensional integration of a periodic function, the trapezoidal rule is an efficient choice. However, for s-dimensional integration of a periodic function over a hypercube, the s-dimensional product trapezoidal rule is not generally cost effective. Other lattice rules can be more effective. [3] Lattice rules often have many representations. For example, the two-dimen­ sional product of the 3-point and 4-point trapezoidal rules may be written in the two ways ~ ~ ~-I (. (1,0) . (0,1») = ~ ~-I (.(3,4») 12 L.J L.J 31 4 + 32 3 12 L.J 3 12 il::::::1;,::::::1 ;=1 (70.4) Figure 70.a shows this lattice rule. As another example, all of the following represent the same rule: ~ ~ ~ -I (. (1,2) . (3,1») =! ~-I (.(1,2») 25 L.J L.J 31 5 + 32 5 5 L.J 3 5 ;1::::::1 h=1 ;::::::1 = ~ tl (; (2;4») 3::::::1 (70.5) = ~ ~-I (.(2,4») 10 L.J 3 10 . ;=1 Figure 70.b shows this lattice rule. 70. Lattice Rules " Y I (a) 1 1 2 I ....................... I i ,. . . . . . . ~ ......•......•......• ! . . . . o ~.-.---.- ... -.-... --> o 1 '3 2 '3 1 x y 1 (b) 1 4 5' 3 5' 2 5' 1 5' o I ! ... '.' ... : .... ,' ... : .... I : : : : i··· .: .... ~ .... : .... : .... : 1····)····\····)···+···) ! ... ! ... : .... : .... : .... : I : : : : : I····:····:···~···:····: . . . . . . ~.-:.-.~.-:-.~.-.=- --> o 1 234 5' 5' 5' is 1 x 303 Figure 70. The nodes appearing in some lattice rules: (a) the lattice rule in (70.4); (b) the lattice rule in (70.5). [4] Sloan and Walsh [10] consider lattice rules of rank 2, i.e., rules of the form n nr I ~ _I-I: "1 (jlZI + hZ2), n2r ~ nr n h=I;'=1 where n > 1 and r ~ 1. According to a criterion introduced for number­ theoretic rules (see page 312) they find the "best rules" of the above form. Lyness and S0revik [4] use a different measure to determine optimal lattices. Both papers contain numerical results and identify useful rules. [5] Lyness and S0revik [3] investigate the number of distinct s-dimensional lattice rules that employ precisely N nodes, vs(N). They give many results for vs(N) including: (A) vs(MN) ~ vs(M)vs(N), (B) Equality holds in (A) if and only if M and N are relatively prime. (C) If the prime factorization of Lis L = IIjP~i, then vs(L) = IIj Vs (p~i) s-1 (iH 1) where Vs (pt) = II p i --1 for s ~ 1 and t ~ O. i=1 P [6] For lattice rules of the form ~ E~~1 ! ({ ~}), Haber [1] calculates several "good" choices for p of the form p = (1, b, b2, ••• , bS-1) for various values of Nand s. Here, goodness is defined by how well the integral rule performs on the test function II:::::l !(Xi), with !(Xi) = 1 + 1/'2(X2 -X + i). [7] Zinterhof [12] presents a fast method for generating lattices that are nearly as good as the optimal lattices. The lattices so generated are tested on the integrands s HI (Xl, ... , Xs) = II (1 - 2 log (2 sin 1/'Xi» i=l s ( 2 2 ) H2(Xl, ... ,XS)=D. 1-~ +~ (1-2{xd)2 304 VI Numerical Methods: Techniques in CS • These integrands are chosen because their Fourier transforms are ....... -1....... -2 H1{m) = R(m) and H2(m) = R(m) ,where m = (m1,'" ,ms) and R{m) = II;=1 max (1, Imi\). [8] Sloan and Lyness [9] give some of the properties of the projections of rules into lower dimensions. [9] Worlet [11] introduces some new families of integration lattices. These have a better order of convergence that previously known constructions. [10] The results in Niederreiter [5] suggest that in the search for efficient lattice rules, one should concentrate on lattice rules with large first invariant. References [1) S. Haber, "Parameters for Integrating Periodic Functions of Several Vari­ ables," Math. of Comp., 41, 1983, pages 115-129. [2) J. N. Lyness, "An Introduction to Lattice Rules and Their Generator Ma­ trices," IMA J. Num. Analysis, 9, No.3, 1989, pages 405-419. [3) J. N. Lyness and T. S~revik, "The Number of Lattice Rules," BIT, 29, No.3, 1989, pages 527-534. [4) J. N. Lyness and T. S~revik, "A Search Program for Finding Optimal Integration Lattices," Computing, 47, 1991, pages 103-120. [5) H. Niederreiter, "The Existence of Efficierit Lattice Rules for Multidimen­ sional Numerical Integration," Math. of Comp., 58, No. 197, January 1992, pages 305-314. [6) I. H. Sloan, "Lattice Methods for Multiple Integration," J. Comput. Appl. Math., 12-13, 1985, pages 131-143. [7) 1. H. Sloan and P. J. Kachoyan, "Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples," SIAM J. Numer. Anal., 24, No.1, 1987, pages 116-128. [8) 1. H. Sloan and J. N. Lyness, "The Representation of Lattice Quadrature Rules as MUltiple Sums," Math. of Comp., 52, January 1989, pages 81-94. [9] 1. H. Sloan and J. N. Lyness, "Lattice Rules: Projection Regularity and Unique Representations," Math. of Comp., 54, No. 190, April 1990, pages 649-660. [10) 1. H. Sloan and L. Walsh, "A Computer Search of Rank-2 Lattice Rules for Multidimensional Quadrature," Math. of Comp., 54, No. 189, January 1990, pages 281-302. [11) R. T. Worlet, "On Integration Lattices," BIT, 31, 1991, pages 529-539. [12) P. Zinterhof, "Gratis Lattice Points for Multidimensional Integration," Com­ puting, 38, 1987, pages 347-353. 71. Monte Carlo Method Applicable to Definite integrals, especially multidimensional integrals. Yields A numerical approximation derived from random numbers. 71. Monte Carlo Method 305 Idea Random numbers may be used to approximate the value of a definite integral. Procedure 1 Suppose we wish to approximate numerically the value of the definite integral 1= L9(X)dX, (71.1) where B is some bounded region. Since B is bounded, it may be enclosed in some rectangular parallelepiped R. Let 1B(X) represent the indicator function of B, that is 1 ( ) _ {I if x E B, B X -0 if x f/. B. Then the integral I may be written in the form I = fa (9(X)lB(X») dx = VtR) fa (9(X)lB(X)V(R») do: (71.2) where V(R) represents the volume of the region R. Equation (71.2) may be interpreted as the expectation of the function h(X) = g(X)lB(X)V(R) of the random variable X, which is uniformly distributed in the parallelepiped R (Le., it has the density function l/V(R)). The expectation of h(X) can be obtained by simulating random devi­ ates from X, determining h at these points, and then taking the average of the h values. Hence, simulation of the random variable X will lead to an approximate numerical value of the integral I. If N trials are taken, then the following estimate is obtained: (71.3) where each Xi is uniformly distributed in R. Another way to think about (71.3) is that g(ei), where ei is chosen uniformly in B, is an independent random variable with expectation I. Averaging several of these estimates together, which is what (71.3) does, results in an unbiased estimator of I. 306 VI N urnerical Methods: Techniques Procedure 2 Importance sampling is the term given to sampling from a non-uniform distribution so as to minimize the variance of the estimate for /. Consider writing (71.1) as / = Eu[g(x)] (71.4) where Eu [.) denotes the expectation taken with respect to the uniform distribution on B. In other words, / is the mean of g( x) with respect to the uniform distribution. Associated with this mean is a variance, defined by (71.5) Approximations to / obtained by sampling from the uniform distribution will have errors that scale with uu. If lex) represents a different density function to sample from, then we may write I = fa (~~:D /(x)dx = E, [~~:~] where E,[,) denotes the expectation taken with respect to the density lex). In other words, / is the mean of g(x)/l(x) with respect to the distribution lex). Associated with this mean is a variance; defined by 2. E ({ g(x) /}2) E [g2] /2 [g2(x)d /2 u 1'= I l( x) -= I 12 -= J B 1 (x) x- Approximations to / obtained by sampling from lex) will have errors that scale with u I' A minimum variance estimator may be obtained by finding the lex) such that u1 is minimal. Using the calculus of variations the density function for the minimal estimator is determined to be Ig(x)1 lopt(x) = Glg(x)1 = fBlg(x)1 dx (71.6) where the constant G has been chosen so that lopt(x) is appropriately normalized. (Since lopt(x) is a density function, it must integrate to unity.) Clearly, finding lopt(x) is as difficult as determining the original integral I! However, (71.6) indicates that lopt(x) should have the same general behavior as Ig(x)l. As Example 2 shows, sometimes an approximate lex) ~ lopt(x) can be chosen. 71. Monte Carlo Method 307 Procedure 3 Another type of Monte Carlo method is the hit-or-miss Monte Carlo method (see Hammersley and Handscomb [6]). It is very inefficient but is very easy to understand; it was the first application of Monte Carlo methods. Suppose that 0 ~ I(x) ~ 1 when 0 ~ x ~ 1. If we define { 0 if I (x) < y, g(x, y) = 1 if f(x) > y, then we may write I = fol I(x) dx = f; f; g(x, y) dy dx. This integral may be estimated by (71.7) where the {~i} are chosen independently and uniformly from the interval [0,1]. The summation in (71.7) counts the number of points in the unit square which are below the curve y = I(x) (this defines n·), and divides by the total number of sample points (Le., n). We emphasize again that the hit-or-miss method is computationally very inefficient. Example 1 We choose to approximate the integral 1= fol 3x2 dx, whose value is 1. To implement the method in (71.3), for Xi uniformly distributed on [0, 1] the FORTRAN program in Program 71 was constructed. The program takes the results of many trials and averages these values together. Note that the program uses a routine called RANDOM, whose source code is not shown, which returns a random value uniformly distributed on the interval from zero to one. The result of the program is as follows: AFTER AFTER AFTER AFTER AFTER AFTER AFTER AFTER AFTER AFTER 100 TRIALS, THE AVERAGE IS 200 TRIALS, THE AVERAGE IS 300 TRIALS, THE AVERAGE IS 400 TRIALS, THE AVERAGE IS 600 TRIALS, THE AVERAGE IS 600 TRIALS, THE AVERAGE IS 700 TRIALS, THE AVERAGE IS 800 TRIALS, THE AVERAGE IS 900 TRIALS, THE AVERAGE IS 1000 TRIALS, THE AVERAGE IS 1.006 1.084 1.046 1.033 0.996 1.028 1.036 1.029 1.032 1.038 We can also approximate I by using hit-or-miss Monte Carlo. (First, we scale the integrand by a factor of 3, to be Is (x) = x2, so that it is in the range [0, 1].) Now random deviates Xi and Yi (both obtained uniformly from 308 VI Numerical Methods: Techniques Figure 71. The 323 points (out of 1000) below the curve y = x2• the interval [0, 1]) are obtained. For each pair of values, n is incremented by one. If, for that pair of values, Yi ::; /s(Xi) = x~, then n* is also incremented by one. Use of (71.7) then results in an estimate for [. Performing this algorithm 1000 times, we obtained 323 instances when the Yi was less than yl (Figure 71 shows the locations of these points). Hence, the estimate of [ becomes ~ 1 N 3 [ = 3· N L 1B(Xi) = 1000 ·323 = 0.969. i=1 Example 2 Consider the integral J = [cos (";) dx = ~. If we let ei represent a sample from the uniform distribution from [0,1] then J may be approximated by Ju N 1 ~ ?re· J ~ Ju = N L.,cos-;f. i=1 The variance of this estimator for J is 2 /.1 2 (?rX) 2 1 4 Uu = 0 cos 2 dx -J = 2 -?r2 ~ .0947 .... Now we want to obtain a density function that more closely approxi- 2 mates the integrand. Since cos (?r2X) = 1-~ x2 + O(x4) for small values of x, we choose a /(x) that has a similar form. We take (71.8) 71. Monte Carlo Method 309 (The factor i arises from the normalization Jo1(1-x2) dx = j.) Using this new density function we find (see the Notes for how to generate deviates from this distribution) J = 11 (2 cos (~)) f(x) dx. o 3 I-x (71.9) If we let (i represent a random variable coming from a distribution that has the density f(x), then (71.9) may be sampled to yield an approximation to J: The variance of this second estimator for J is (7rX) 2 2 11 (2 cos 2 ) 2 0' / = 0 3 1 _ x2 f (x) dx -J ~ .00099 .... Since O'u is approximately 10 times larger than 0'/, the errors in using J / to approximate J will be about 10 times smaller than using Ju to approximate J, for the same number of trials. Of course, in practical cases it will not generally be possible to exactly determine the variances O'u and 0'/. However, estimates can be obtained for the variances by approximating the defining integrals. Program 71 SUM=O DO 10 J=1.1000 X=RANDOH{T) VAL=3.*X**2 SUM=SUM+VAL IF{ HOD{J.100) .NE. 0 ) GOTO 10 AVERAG=SUM/FLOAT{J) WRITE{6.5) J .AVERAG 5 FORHAT{' AFTER' .16. ' TRIALS. THE AVERAGE IS' .F1.3) 10 CONTINUE END 310 VI Numerical Methods: Techniques Notes [1] This method is of particular importance when multi-dimensional integrals are to be approximated numerically. For multi-dimensional integrals, Monte Carlo techniques may be the only techniques that will obtain an estimate in a reasonable amount of computer time. This is because the error in a Monte Carlo computation scales with u/...fN, where N is the number of samples of the integrand (trials), independent of dimension. For traditional methods, the number of samples of the integrand varies exponentially with the dimension (i.e., scales as aN for some a). [2] While the classical Monte Carlo method converges with order 1/...fN, where N is the number of samples, the quasi-Monte Carlo method can achieve an order of (log N)Q /N for some a > O. See Niederreiter [10)-[U} and Wozniakowski's method on page 333. [3} For some integrals, the variance in (71.5) may not exist. For example, with 1= J01 dx/ Vx, the variance is computed to be uL = J01 dx/x -12, and the first term is infinite. Use of importance sampling can result in a new integral that has a finite variance. See Kalos and Whitlock [7}. [4} Masry and Cambanis [8} discuss how the trapezoidal rule can be used in the Monte Carlo computation of the integral I = J01 f (x) dx. Choose n random deviates independently and uniformly on the interval [0, I}. Numerically order these deviates to form the sequence tn,l < tn,2 < ... < tn,n, and then add the points tn,o := 0 and tn,n+1 := 1. The sequence of {tn,i}' used in the trapezoidal rule, produces an estimate of I: [0, In = ~ t [f(tn'i) + f(tn'i+d] (tn,i+l -tn,i) . 1=0 If f has a continuous second derivative on the interval [0, 1}, then it can be shown that 2 [I' (1) -1'(0)] 2 + 0(1) E [I -In] = 4(n + l)(n + 2)(n + 3)(n + 4)' Hence, the error varies as 0 (n -4) for large n. [5} The integral 1= Jo1 g(x) dx may be written as 1= Jo1 ~ (g(x) + g(l -x» dx. Hence, the estimator 7 = ~ t. ~ (g(X;) + g(l -X;») , (71.10) where the Xi are chosen from the uniform distribution, can be used to approximate I. When g(x) is linear, this approximator gives the exact answer. (See Siegel and O'Brien [15) for techniques that are exact for other polynomials.) In cases where the function is nearly linear, the variance can be substantially reduced. This is known as the method of antithetic variates. For example, consider the integral I = Jo1 eZ dx = e -1. Using a straightforward Monte Carlo evaluation the variance is found to be u2 = Jo1 fez -(e -1)}2 dx = (3 -e)(e -1)/2 ~ 0.242 .... Using (71.10) reduces the variance to 0.0039, a substantial reduction. 71. Monte Carlo Method 311 [6] Error estimates are available for many different approximation schemes (see Cambanis and Masry [2]). Consider the mtegral l(g) = 101 g(t) dt, and let each U. below represent an independent random variable, uniformly distributed over the interval [0,1]. For the approximation l(g) R: 11(g) = .! "~-1 g(Ui), the mean-square n ~,- error is given by: E [(1(g) -11 (g»2] = [1(g2) -12(9)/ In. In the stratified sampling scheme the interval [0,1 is partitioned into n subintervals of equal length and the point Un,i is chosen uniformly in the i-th interval. For the approximation l(g) R: h(g) = .! "~-1 g(Un i), the mean-n~'- , square error is given by: lim n3 E [(1(g) -12 (g»2] = f2 fol [g'(t)]2 dt. (See n ..... oo also Press and Farrar [13].) In the stratified and symmetrized scheme the interval [0, 1] is partitioned into n subintervals of equal length and the point Un,i is chosen uniformly in the i-th interval. Let U~,i represent the symmetrically opposite point to Un,i in the i-th interval. If 9 has a continuous second derivative, then for the approximation l(g) R: 13(g) = 2~ E:=1 [g(Un,i) + g(U~,i)]' the mean­ square error is given by: lim (2n)5 E [(I(g) -13(g»2] = is fol [gll(t)]2 dt. n ..... oo [7] There are many other variance reduction techniques that are sometimes used in Monte Carlo calculations. These include the use of (see Hammers­ ley and Handscomb [6]): control variates, regression methods, orthonormal functions, and group sampling. [8] Suppose that I = I f(x) dx is approximated by a Monte Carlo computation. Suppose also that the integrals of some "reference functions" (functions which can be analytically integrated) are also approximated using the same set of Monte Carlo points. Then the accuracy of the estimate of I can be improved by using the estimated integrals of the reference functions. See Eberhard and Schneider [5] for details. [9] Ogata [12] investigates the two test integrals: fol fol ... fol eX1 eX2 ••• eXJ; dx and f~oo f~oo··· f~oo e-XBxT dx where B is a specific Toeplitz matrix. [10] Details on how to simulate a random variable from different distributions may be found in Devroye [3]. For example, random deviates from the density in (71.8), f(x) = i(1 -x2) for x in the range [0,1], may be obtained as follows: • Generate (1 and (2 independently and uniformly on the interval [0,1] . • If (2 ~ (1(3; (.>, then x = 1 -(1; otherwise, x = l ("'9 -8(2 -1). References [1] G. Bhanot, "The Metropolis Algorithm," Rep. Progr. Phys., 51, No.3, 1988, pages 429-457. [2] S. Cambanis and E. Masry, "Trapezoidal Stratified Monte Carlo Integra­ tion," SIAM J. Numer. Anal., 29, No.1, February 1992, pages 284-301. [3] L. Devroye, Non-Uniform Random Variate Generation, Springer-Verlag, New York, 1986. 312 VI Numerical Methods: Techniques [4] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, pages 389-415. [5] P. H. Eberhard and O. P. Schneider, "Reference Functions to Decrease Errors in Monte Carlo Integrals," Comput. Physics Comm., 67, 1992, pages 363-377. [6] J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods, John Wiley & Sons, New York, 1965. [7] M. H. Kalos and P. A. Whitlock, Monte Carlo Methods. Volume I: Basics, John Wiley & Sons, New York, 1986, Chapter 4, pages 89-116. [8] E. Masry and S. Cambanis, "Trapezoidal Monte Carlo Integration," SIAM J. Numer. Anal., 27, No.1, February 1990, pages 225-246. [9] B. J. T. Morgan, Elements of Simulation, The University Press, Cambridge, United Kingdom, 1984. [10] H. Niederreiter, "Quasi-Monte Carlo Methods and Pseudo-random Num­ bers," Bull. Amer. Math. Soc., 84, No.6, 1978, pages 957-1041. [11] H. Niederreiter, "Quasi-Monte Carlo Methods for Multidimensional Numer­ ical Integration," Numerical Integration, III (Oberwoljach, 1987), Internat. Schriftenreihe Numer. Math., #85, Birkhiiuser, Basel, 1988, pages 157-171. [12] Y. Ogata, "A Monte Carlo Method for High Dimensional Integration," Numer. Math., 55, No.2, 1989, pages 137-157. [13] W. H. Press and G. R. Farrar, "Recursive Stratified Sampling for Multi­ dimensional Monte Carlo Integration," Comput. in Physics, Marchi April 1990, pages 191-195. [14] Y. A. Shreider (ed.), Method of Statistical Testing, American Elsevier Pub­ lishing Company, New York, 1964. [15] A. F. Siegel and F. O'Brien, "Unbiased Monte Carlo Integration Methods with Exactness for Low Order Polynomials," SIAM J. Sci. Stat. Comput., 6, No.1, January 1985, pages 169-181. 72. Number Theoretic Methods Applicable to Definite integrals, especially multidimensional integrals. Yields Quadrature rules. Idea Using points from a deterministic sequence might be useful when numerically approximating an integral. 72. Number Theoretic Methods 313 Procedure There are two procedures in this section, both for the simple numerical integration rule J 1 N f(x) dx ~ N L f(Xi). i=l (72.1) The question is: how are the nodes {Xi} to be determined? In the first method, a different set of nodes is chosen for each value of N; for the second method, the first N nodes from an infinitely long sequence are chosen. For both methods we will use the notation {z} to denote the fractional part of z. For example, {1r} = 0.14159 .... Procedure 1 In the general d-dimensional case of interest, we would like to use a formula of the form (72.2) for some integer values {(Ji}, which are relatively prime to N. What are "good" values of {(Ji}? The usual definition for a good set of values base the decision on an error bound obtained by using (72.2) in (72.1). With this definition, there are several theorems (see Stroud [5]) on how to determine "optimal" values for the {(Ji}, when N is a prime number or the product of two prime numbers. For some values of N, there is a constant aN such that the values (Jl = 1 and (Ji = ak yield an "optimal" vector. Stroud [5] includes several tables of values. We have, for example, the following recommended values: -for d = 3: • for N = 101, {(Ji} = {I, 40, 85} • for N = 1069, {(Ji} = {I, 136, 323} • for N = 10007, {(Ji} = {I, 544, 5733} • for N = 100063 = 47·2129, {(Ji} = {I, 53584, 37334} -for d = 4: • for N = 307, {(Ji} = {1,42,229,101} • for N = 1069, {(Ji} = {I, 71, 765, 865} • for N = 10007, {(Ji} = {I, 1784, 430, 6588} • for N = 100063 = 47·2129, {(Ji} = {I, 92313,24700, 95582} -for d = 5: • for N = 1069, {(Ji} = {1,63, 762,970, 177} • for N = 10007, {(Ji} = {I, 198,9183,6967, 8507} • for N = 100063 = 47·2129, {(Ji} = {I, 90036,77477,27253, 6222} 314 VI N urnerical Methods: Techniques Procedure 2 A deterministic sequence of points {Xi,X2,"'} in the interval [a,b] is said to be equidistributed, or uniformly distributed on the interval [a, b], if for all bounded Riemann integrable functions f(x). It is not difficult to construct uniformly distributed sequences. For example, if ( is an irrational number, then Xn = {n(} is a uniformly distributed sequence. These sequences, when used in a simple quadra­ ture formula, have very good error estimates. For example, Davis and Rabinowitz [2] (page 400) have the estimate Theorem: Let /(x) be periodic in [0,1] and be of class C3[0,l] so that we have /(0) = /(1), /'(0) = /'(1), and /"(0) = /"(1). Let ( be a quadratic irrational number. Then for some constant c. Multi-dimensional uniformly distributed sequences can also be con­ structed. Suppose that (I, (2, ... , (d are irrational numbers that are linearly independent over the rational numbers. (That is, 1 + al (I + a2(2+ ... , +ad(d :f. ° for any rational numbers {aj}.) Then the d-dimensional nodes are equidistributed over the hypercube ° ~ Xi ~ 1, i = 1,2, ... , d. (That is, lim N1 E~1 f(Pi) = Jol ..• J; f(x) dx for any bounded Riemann-integrable N-oo function f(x).) Example Using the values N = 101 and {Oil = {1,40,85} the sequence of 1013 ~ 106 nodes is: {( 1~1' 1~1' ~tl)' (1~1' 1~01' 16~1)' (1~1' 1~01' 15 031)' ( 1 40 37) (1 80 85) (1 19 85) (1 59 85) 101' 101' ill! ' ... , TIIT' 101' 101' 101' 101' 101' 101' 101' 101 , ... , (1~1' 1~01' 18tl)' (1~1' 1~°1' 18 051)' ... , (O,O,O)}. 73. Parallel Computer Methods 315 Notes [1] Using the first method, the number of nodes at which the integrals are to be evaluated grows very quickly. For example, for the (relatively) modest values of N = 100063 and d = 3, there are 1015 nodes required! [2] Davis and Rabinowitz [2] give an example of the numerical evaluation of a 4-dimensional integral in which the irrational numbers chosen were: V2, va, lV6, and v'lO. [3] There is an common criterion by which a "best rule" may be determined (see Sloan and Walsh [7]). Let m be a fixed even positive integer (i.e., m = 2) and then define the function fm(x) = cPm(X1)cPm(X2) •.• cPm(Xn), on Rn where cPm(X) = 1 _ (_1)m/2 (211')m Bm(x). m! Here, Bm(x) is a Bernoulli polynomial of degree m. Among all quadrature rules Qi on the n-dimensional unit hypercube, the "best" one is the one that minimizes Qi[fm] -f fm = Qi[fm] -1. [4] In Wozniakowski's method (page 333), a different infinite sequence of values is used. [5] The number theoretic methods are generalized in the lattice rules, see page 300. References [1] R. Cranley and T. N. L. Patterson, "Randomization of Number Theoretic Methods for Multiple Integration," SIAM J. Numer. Anal., 13, No.6, De­ cember 1976, pages 904-914. [2] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, Section 5.9.3, pages 396- 410. [3] L. K. Hua and Y. Wang, Application of Number Theory to Numerical Anal­ ysis, Springer-Verlag, New York, 198!. [4] N. M. Korobov, "The Approximate Computation of Multiple Integrals," Dokl. Akad. Nauk SSSR, 124, 1959, pages 1207-1210. (Russian) [5] A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall Inc., Englewood Cliffs, NJ, 1971, Section 6.3, pages 198-208. 73. Parallel Computer Methods Applicable to Definite integrals. Yields Ways in which a parallel computer may be used. Idea Parallel computers can sometimes speed up the numerical computation of an integral. 316 VI Numerical Methods: Techniques Procedure There are many procedures that can be used, depending on the type of parallel computer under consideration. The most common use of a parallel computer is to partition an integration interval into many sub-intervals, and have the integration on each sub-interval performed in parallel. Notes [1] Genz [2] studied the implementation of a globally adaptive algorithm for single integrals on a SIMD computer. [2] Rice [6] considers algorithms for single one-dimensional integrals on a MIMD computer. [3] Burrage [1] considers a single one-dimensional integral being evaluated on a linear chain of transputers of arbitrary length. (A transputer is a local memory MIMD computer.) For some classes of problems, numerical ev­ idence suggests that linear speed-ups are achievable with any number of transputers. [4] Monte Carlo methods (see page 304) are generally the easiest to implement on parallel machines. While these methods are robust, they are generally the least efficient. References [1] [2] [3] [4] [5] [6] [7] K. Burrage, "An Adaptive Numerical Integration Code for a Chain of 'Trans­ puters," Parallel Computing, 16, 1990, pages 305-312. A. C. Genz, "Numerical Multiple Integration on Parallel Computers," Com­ put. Physics Comm., 26, 1982, pages 349-352. A. Genz, "Parallel Adaptive Algorithms For Multiple Integrals," in Mathe­ matics for Large Scale Computing, Lecture Notes in Pure and Appl. Math., #120, Marcel Dekker, New York, 1989, pages 35-47. J. M. Lemme and J. R. Rice, "Speedup in Parallel Algorithms for Adaptive Quadrature," J. ACM, 26, 1979, pages 65-71. J. R. Rice, "Parallel Algorithms for Adaptive Quadrature II. Metalgorithm Correctness," Acta Inf., 5, 1973, pages 273-285. J. R. Rice, "Adaptive Quadrature: Convergence of Parallel and Sequential Algorithms," Bull. Amer. Math. Soc., 80, 1974, pages 1250-1254. J. R. Rice, "Parallel Algorithms for Adaptive Quadrature III. Program Correctness," ACM 1'rans. Math. Software, 2, 1976, pages 1-30. 74. Polyhedral Symmetry Rules Applicable to Multidimensional integrals. Yields A numerical quadrature scheme. Idea Some quadrature rules are invariant under a symmetry group operating on the nodes. 74. Polyhedral Symmetry Rules 317 Procedure Consider the construction of a quadrature rule of the form N / / w(x)f(x) dx ~ L wkf(Xk), B k=i (74.1) where w(x) is a specific weight function. Generally, it is required that a quadrature rule be exact for all polynomials below some degree. Forcing (74.1) to be exact for low degree polynomials results in a large system of nonlinear algebraic equations that must be solved for the nodes {Xk}, the weights {Wk}, or both. To reduce the number of equations, we can impose full polyhedral symmetry on the formula. This may result in a formula that is not of the highest order, but it sometimes allows a formula to be quickly developed. Example Consider devising a 4-node quadrature rule on the square: (74.2) To completely specify the quadrature rule we require 12 values, {Wi,Xi,Yi I i = 1, ... , 4}. To force (74.2) to be exact for all polynomials of degree 0, 1, and 2, the unknowns must satisfy f(x,y) = 1: Wi + W2 + W3 + W4 =4 f(x,y)=x: WiXi + W2X2 + W3X3 + W4X4 =0 f(x, y) = y: WIYi + W2Y2 + W3Y3 + W4Y4 =0 (74.3) f(x,y) = X2: 222 2 =~ WIXi + W2X2 + W3X3 + W4X4 f(x,y) = y2: 2 + 2 2 2 WIYi W2Y2 + W3Y3 + W4Y4 =~ f(x, y) = xy: WIXIYI + W2X2Y2 + W3X3Y3 + W4X4Y4 =0. Solving these nonlinear algebraic equations is a non-trivial task. Even determining the number of solutions to these equations is difficult. Suppose, however, that we require the quadrature formula to be sym­ metric under rotation. That is, whenever the node (Xi, Yi) is in (74.2), then the nodes (-Xi, Yi), (Xi, -Yi), and (-Xi, -Yi) should also be in (74.2). This constraint adds the following restrictions to (74.3): WI = w2 = W3 = W4, Xl = X2 = -X3 = -X4, and Yi = -Y2 = -Y3 = Y4· With these constraints, the equations in (74.3) become: 318 VI Numerical Methods: Techniques f(x,y) = 1: f(x,y) = x: f(x,y) = y: f(x,y) = x2: f(x,y) = y2: f(x,y) = xy: 4WI =4 o =0 o =0 4WIX~ = ~ 4WlY~ = ~ o =0. (74.4) The equations in (74.4) can be readily solved to determine that WI = 1 and Xl = YI = l/y'?,· Notes [1] The quadrature rule found in the example is also exact for polynomials of degree 3 (i.e., the rules are exact for the functions f(x, y) = x3, f(x, y) = x2y, f(x, y) = xy2, and f(x, y) = y3). [2] The groups under which the nodes are mapped into themselves are the reflection groups of polyhedra. In three dimensions there only exist three such groups: (A) the extended tetrahedral group of order 24 (the orbits of this group can have 24, 12, 6, 4, or 1 point(s»; (B) the extended octahedral group of order 48 (the orbits of this group can have 48, 24, 12, 8, 6, or 1 point(s»; (C) the extended icosahedral group of order 120 (the orbits of this group can have 120, 60, 30, 20, 12, or 1 point(s». [3] Cools and Haegemans [3] consider quadrature formulas that are invariant with respect to a transformation group and prove sufficient conditions for such formulas to have positive weights. References [1] M. Beckers and A. Haegemans, "The Construction of Three-Dimensional Invariant Cubature Formulae," J. Comput. Appl. Math., 35, 1991, pages 109-U8. [2] R. Cools and A. Haegemans, "Construction of Fully Symmetric Cubature Formula of Degree 4k -3 for Fully Symmetric Planar Regions," J. A CM, 17, No. 1-2, 1987, pages 173-180. [3] R. Cools and A. Haegemans, "Why Do So Many Cubature Formulae Have So Many Positive Weights?," BIT, 1988, 28, No.4, pages 792-802. [4] D. A. Dunavant, "Efficient Symmetrical Cubature Rules for Complete Poly­ nomials of High Degree over the Unit Cube," Int. J. Num. Methods Eng., 23, 1986, pages 397-407. [5] T. O. Espelid, "On the Construction of Good Fully Symmetric Integration Rules," SIAM J. Numer. Anal., 24, No.4, 1987, pages 855-881. [6] A. C. Genz and A. A. Malik, "An Imbedded Family of Fully Symmetric Numerical Integration Rules," SIAM J. Numer. Anal., 20,1983, pages 580- 588. 75. Polynomial Interpolation 319 [7] P. Keast and J. N. Lyness, "On the Structure of Fully Symmetric Multi­ dimensional Quadrature Rules," SIAM J. Numer. Anal., 16, 1979, pages 11-29. [8] S. L. Sobolev, "Cubature Formulas on the Sphere Invariant Under Finite Groups of Rotations," Soviet Math. Dokl., 3, 1962, pages 1307-1310. [9] T. S0revik and T. O. Espelid, "Fully Symmetric Integration Rules for the 4-Cube," BIT, 29, No.1, 1989, pages 148-153. 75. Polynomial Interpolation Applicable to Definite integrals. Yields Integration rules on a finite interval using uniformly spaced nodes. Idea When values of a function at a discrete set of points are known, an interpolating polynomial can be passed through those points. The integral . of the original function will approximate the integral of the interpolating polynomial. Procedure Given the interval [a, b], discretize it into n segments of equal length by inserting the n + 1 nodes: {Xi I Xi = a + ih, i = 0, 1, ... , n} where h = (b -a)Jn. If the function f(x) is known at the n + 1 nodes (Le., Ii = f(Xi)), then the interpolatory polynomial Pn(x), of degree n or less, that goes through all n+ 1 pairs (Xi, fd is given by Lagrange's interpolation formula n Pn(x) = L fiLi(X), i=O (since Li(Xj) = 8ij, where 8ij is the Kronecker delta). Writing X = a + th, and using Xi = a + ih, this can be written as Li(X) = Ki(t) = IT ~ = kk. 1:=0 ~ Integrating Pn(x) from X = a to X = b leads to (75.1) n = h LWifi i=O 320 VI Numerical Methods: Techniques Table 75.1. Newton-Cotes rules obtained from polynomial interpolation. n Wi Error Name 1 1 1 h3 12 1(2) (e) trapezoidal rule 2 2 2 1 4 1 h5/o/(4)(e) Simpson's rule 3 3 3 3 3 9 9 3 h5 ioJ<4)(e) Simpson's 3lB-rule 8 8 8 8 4 14 64 24 64 14 h7 9:5/(6)(e) Milne's rule* 45 45 45 45 45 5 95 375 250 250 375 95 h7 275 1(6)(e) 288 288 28ii 28ii 288 288 12096 6 41 216 27 272 27 216 41 h9 lioo/(8)(e) Weddle's rule i4ii 140 140 140 i4ii 140 840 where Wi = Jon Ki(t) dt. Since the {Wi} do not depend on the function f(x), they can be pre-computed. For example, for n = 2 we find: Wo = -=--=-dt = -(t2 -3t + 2) dt = - 12 (t 1) (t 2 ) 1 12 1 o 0-1 0-2 2 0 3 12 (t -0) (t -2 ) 12 4 Wl = ----dt = -(t2 -2t) dt = -o 1-0 1-2 0 3 12 (t -0) (t -1 ) 112 1 W2 = 0 2 _ 0 2 _ 1 dt = 2 0 (t2 -t) dt = 3' This gives rise to the integration rule: f.b f.b h a f(x) dx ~ a P2(X) dx = 3 (fo + 411 + h) where h = (b -a)/2 and fk = f(a + kh). This is known as Simpson's rule. The error in using the integration rules in (75.1) can be shown to be given by t Pn(x)dx -t !(x)dx = hPn+1 En/(Pnl({) for some e E (a, b) where Pn and En are functions of n and not of f(x). The approximations given in (75.1) are known as the Newton-Cotes rules. Some tabulated values of the {Wi}, as well as the corresponding error term, are presented in Table 75.1. As indicated in that table, some of the N ewton-Cotes rules also have other names. * Also known as Boole's rule. 75. Polynomial Interpolation 321 Table 75.2. Open Newton-Cotes rules obtained from polynomial interpolation. n Wi Error Name 2 2 h3l f(2)(e) midpoint rule 3 3 3 h3 ~ f(2)(e) 2 2 4 8 4 8 h5~f(4)(e) 3 -3 3 5 55 5 5 55 h5 19;4 j<4)(e) 24 24 24 24 6 33 21 39 21 33 h7 14fof(6)(e) 10 -5' 5' -5' 10 Example Consider the integral I = ~3 /.3 cos X dx = 1. Define In to be sm 0 the result of using the n-node rule from Table 75.1. Then we obtain the following approximations to I: Notes I ~ 12 = 0.1063722664 I ~ 13 = 1.0379687263 I ~ 1'4 = 1.0163528101 I ~ Is = 0.9994771775 I ~ 16 = 0.9997082294 I ~ 17 = 1.0000059300. [1] Newton-Cotes rules for large values of n are not often used since some of the weights {Wi} become negative and numerical cancellation occurs in the computation. [2] In the above we discretized the interval [a, b] into n + 1 nodes that included the endpoints a and b. Hence, the above formulas are sometimes called closed Newton-Cotes rules. If we only consider the interior nodes, {Xi I Xi = a+ih,i = 1, ... ,n-l} (where, as before, h = (b -a)/n), and then approximate the given integral by the integral of the interpolating polynomial, then we will have derived on open formula. These formulas are sometimes called open Newton-Cotes rules. The first few such formulas are in Table 75.2. [3] Instead of just interpolating the value of f(x) at the nodes {Xi}, the values of f(x) and f'(x) may be used. For example, if values for the derivatives at the endpoints of the interval are given, then the approximate formula f.b h ~ Q f(x) dx ~ Z2(h) = 2" (f(a) + f(b» + 12 (f'(a) -!'(b» 322 VI Numerical Methods: Techniques may be used (here, h = (b -a». It can be shown that the error is given by with e E (a, b). See page 287. [4J Gaussian quadrature rules (see page 289) are also interpolatory, but the nodes are not equidistant from one another. Instead, the node locations are chosen to make the rule have as high a degree as possible. [5] Sometimes a quadrature formula is desired that integrates trigonometric polynomials, not ordinary polynomials, exactly. A trigonometric polynomial of degree m is a linear combination of the functions {1,cosx, sinx, cos2 x, cosxsinx, sin2x, ... , cosmx, cosm-1xsinx, ... sinmx}. Equivalently, a trigonometric polynomial of degree m is a linear combination of the functions {1, cos x, sin x, cos 2x, sin 2x, ... cos mx, sin mx}. The approximation [' I(x) dx "" t hI (fH (k -I)h) o k=l where h = 21r In and {3 is any real number satisfying 0 ~ {3 < h, is exact for all trigonometric polynomials of degree n -1 or less (see Mysovskikh [5J). Vanden Berghe et al. [7] consider quadrature rules that exactly inte­ grate ordinary polynomials and trigonometric polynomials. [6] Given data values defined on a set of nodes, one polynomial can be fit to all of the data values, as shown above. Alternatively, the region of integration may be broken into smaller regions, with an interpolatory polynomial fit to the data values in each sub-region. Kohler [3] considers the case when the interpolatory polynomial on a sub-region uses data values from outside that sub-region. [7] There are many interpolatory formulas, other than polynomials, that can be used to interpolate data. For example, De Meyer et al. [4] interpolate a set of values using the function f (x) = ekz E~=o aixi. This interpolating function can then be integrated to obtain, for example, their modified trapezoidal rule: lz0+h {eB -1-(J e-B -1 + (J } f(x)dx ~ h 2 f(xo) + 2 f(xo + h) , zo (J (J where (J = hk and k is an arbitrary parameter. This parameter is chosen in practice, by minimizing the error term. For the above rule, the leading h3 [4sinh2 «(J12)] 2 order error term has the form E = (J2 1 - (J2 (Dz -k) f(1]), where 1] is in the range Xo < 1] < Xo + h. 76. Product Rules 323 References [1] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, pages 74-81. [2] F. B. Hiderbrand, Introduction to Numerical Analysis, McGraw-Hill Book Company, New York, 1974. [3] P. Kohler, "On a Generalization of Compound Newton-Cotes Quadrature Formulas," BIT, 31, 1991, pages 540-544. [4] H. De Meyer, G. Vanden Berghe, and J. Vanthournout, "Numerical Quadra­ ture Based on an Exponential Type of Interpolation," Int. J. Compo Math., 38, 1991, pages 193-209. [5] I. P. Mysovskikh, "On Cubature Formulas that Are Exact for Trigonometric Polynomials," Soviet Math. Dokl., 36, No.2, 1988, pages 229-232. [6] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, translated by R. Bartels, W. Gautschi, and C. Witzgall, Springer-Verlag, New York, 1976, pages 118-123. [7] G. Vanden Berghe, H. De Meyer, and J. Vanthournout, "On a Class of Modified Newton-Cotes Quadrature Formulae Based Upon Mixed-type In­ terpolation," J. Comput. Appl. Math., 31, 1991, pages 331-349. 76. Product Rules Applicable to Multidimensional integrals. Yields A numerical quadrature scheme. Idea Suppose two numerical quadrature schemes are known, one for r­ dimensional Euclidean space and one for s-dimensional Euclidean space. The "product" of these two rules can be used to formulate a numerical quadrature scheme for (r + s )-dimensional Euclidean space. Procedure Let R (8) be a region in r-dimensional (s-dimensional) Euclidean space. Suppose we have the n-node and m-node quadrature rules L I(x) dx '" t, w;l(x;), f g(y) dy ~ t vkg(Yk)' 1s k=l (76.1.a-b) Then an (n + m)-node quadrature rule for the region B = R x 8, in an (r + s )-dimensional space, is given by L h(x,y) dxdy '" t. E W;Vkh(X;,Yk)' (76.2) 324 VI Numerical Methods: Techniques Example On the interval [a, b), Simpson's rule with three nodes approximates an integral by t J(x) dx '" ~ (10 + 4" + h) (76.3) where In = I(a + nh} and h = (b -a}/2. Likewise, on the interval [e, d), Simpson's rule with five nodes approximates an integral by f.d g(y) dy '" ~ (gO + 4g1 + 2g2 + 4g. + g.) where gm = g(e + mk} and k = (d -e}/4. (76.4) Taking the product of the rules in (76.3) and (76.4) results in the following approximation of a two-dimensional integral lb f.d h(x, y) dx dy '" ~k (ho•o + 4ho•1 + 2ho.2 + 4ho •• + ho •• + 4h1,o + 16h1,1 + 8h1,2 + 16h1,3 + 4h1,4 + h2,o + 4h2,1 + 2h2,2 + 4h2,3 + h2,4) where hn,m = h(a + nh, e + mk}. Notes [1] If (76.1.a) exactly integrates f(x), and if (76.1.b) exactly integrates g(y), and h(x, y) = f(x)g(y), then (76.2) will exactly integrate h(x, y). [2] This technique can be used for general Cartesian product regions, not just parallelpipeds; for instance, circular cylinders, circular cylindrical shell, and triangular prisms. [3] Stroud [3] analyzes product rules by use of transformations. Suppose the region of integration is S, and the integrals of interest have the weight function w(x). If the quadrature rule is to be exact for polynomials, then I = II ... I w(x) xr1 X~2 ••• x~n dx must be integrated exactly for some set s of {Oi}' If there exists a transformation of the form x = x(u) that turns I into the product I = (J Wl(Ut)gl(Ut} dUl)'" (J Wn(Un)gn(U n) dun) , and if suitable formulas are known for these single integrals, then one has obtained a product rule. [4] Using product rules, the number of nodes at which the integrand must be evaluated grows exponentially with the dimension of the integration. If a one-dimensional quadrature rule that uses 19 nodes is the basis for a 7-dimensional quadrature rule, then 197 ~ 109 integrand evaluations are required. [5] The rules devised by this technique are often not the most efficient in terms of number of integrand evaluations. 77. Recurrence Relations 325 [6) Often, a more computationally efficient quadrature rule for a multidimen­ sional integral can be found. For example, Acharya and Mohapatra [1) give the two-dimensional quadrature rule for analytic functions: l%o+h l%o+hl I(z, z') dz dz' ~ hh' [ -256/00 + 25 (/11 + 113 + /31 + /33) %o-h %o-h' + 40 (/20 + f02 + /40 + f04) 1 (76.5) where 10/3 = I(zo, z~), Zo = Zo + hkio-1, and z~ = Zo + h'kio-1• When k = 1/ V15, the rule in (76.5) has degree of precision 5. [7] A quadrature rule for the n-cube Cn can sometimes be used as the basis for a quadrature rule for Cm (with m > n). The rule for Cm is then called an extended rule. Product methods, described in this section, are only one way in which a rule can be extended. For other methods, see Stroud [3] (Chapter 4). References [1] B. P. Acharya and T. Mohapatra, "Approximations of Double Integrals of Analytic Functions of Two Complex Variables," Computing, 37, 1986, pages 357-364. [2] P. J. Davis and P. Rabinowitz, Methods 01 Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, pages 354-363. [3] A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall Inc., Englewood Cliffs, NJ, 1971, Chapter 2, pages 23-47. 77. Recurrence Relations Applicable to Integrals for which a recurrence relation can be found. Yields An asymptotic approximation, or a numerical computation scheme. Idea If a recurrence relation can be found for an integral, then it may be used to determine an asymptotic approximation, or it can form the basis of a numerical computation. 326 VI Numerical Methods: Techniques Procedure Often, an integral can be written in terms of a recurrence relation and some initial (or boundary) condition(s). (This is frequently accomplished by integration by parts.) This recurrence relation can be used to obtain asymptotic information about the integral. Or, the recurrence relation can be iteratively applied to determine numerical values for the integral. These numerical computations should only be performed after an asymptotic analysis has been performed, to prevent roundoff errors from ruining the numerical accuracy. Consider the three term recurrence relation Yn+l + anYn + bnYn-l = 0, n= 1,2, ... , (77.1) where {an} and {bn} are given sequences of real or complex numbers, and bn .:p O. From Van der Laan and Temme [6] we have the following theorem: Let an and bn have the asymptotic behavior with ab f:. 0 and both of a and f3 real. Let hand t2 be roots of the characteristic polynomial t2 + at + b = 0, with Itll ~ It21· Then there are three cases: (1) If a > ~f3 then (77.1) has two linearly independent solutions Un and Vn for which Un+l Q --'" -an Un Vn+l b fJ-Q --'" --n as n -. 00. Vn a (2) If a = ~f3 then (77.1) has two linearly independent solutions Un and Vn for which provided that Itt I > It21· If Itll = It21 then lim sup (IYnl (nl)-a)l/n = Ittl n-+oo for all non-trivial solutions of (77.1). (3) If a < ~f3 then lim sup (IYnl (n!)-fJI2) lIn = Ibll/2 n-+oo for all non-trivial solutions of (77.1). (77.2) (77.3) (77.4) (77.5) (77.6) The solution of the recurrence relation in (77.1) can be written in the form Yn = Attn + Bvn, where A and B are constants. 77. Recurrence Relations 327 Table 77.1. A recursion computation of the Bessel functions {Yn} using (77.7). This computation is stable. Y1 = 0.1478631434 Y2 = 0.3676628826 ys = -0.4536948225 Y10 = -25.129110 Y12 = -382.9821416 Y14 = -8693.938814 Y16 = -272949.0350 Y20 = -593396529.7 Y30 = -4.028568418 X 1018 Example Yn(5) Y1(5) = 0.1478631434 Y2(5) = 0.3676628826 Ys(5) = -0.4536948225 Ylo(5) = -25.129110 Y12(5) = -382.9821416 Y14(5) = -8693.938814 Y16(5) = -272949.0350 Y2o(5) = -593396529.7 Y3o(5) = -4.028568418 x 1018 The Bessel functions Yn{z), when n is an integer, are defined by (see page 174): From this, or otherwise (see Abramowitz and Stegun [1] 9.1.27.a), it can be shown that these Bessel functions satisfy the recurrence relation (using Yn = Yn{z)) 2n Yn+l --Yn + Yn-l = O. Z (77.7) The above theorem can be used on this recurrence relation with a = -2/ z, a = 1, b = 1, and f3 = O. We find that case (I) applies and results in Un+l/U n '" 2n/z and Vn+l/Vn '" z/2n. This implies that as n -. 00, for some values of A and B. Note that if A f:: 0, then the first term dominates the asymptotic expansion. It can be shown that Yn{z) '" n! {2/zt as n -. 00 (see Abramowitz and Stegun [1], 9.1.8). This asymptotic expansion agrees with the results of the theorem; here we have A f:: o. If (77.7) is used to compute {Yn(z)}, then roundoff errors will result in {vn} terms. Since the evolution of these terms is much smaller than the {Yn{z)} terms, this will be stable computation. Table 77.1 shows the computation of {Yn(5)} using (77.7). The values of Y1(5) and Y2(5) were used to initialize the recurrence relation. For large value of n, the computation is accurate to all decimal places. 328 VI Numerical Methods: Techniques Table 77.2. A recursion computation of the Bessel functions {In} using (77.7). This computation is not stable. Y1 = -0.3275791376 Y2 = 0.04656511628 Ys = 0.2611405461 Y10 = 0.00146780258 Y12 = 0.0000762771 Y14 = 0.000002778 Y16 = -0.00000065 Y20 = -0.0016 Y30 = -107 J1 (5) = -0.3275791376 h(5} = 0.04656511628 Js(5} = 0.2611405461 J1o(5} = 0.00146780265 J12(5} = 0.0000762781 J14(5} = 0.000002801 J16(5} = -0.000000077 ho(5} = 2.7 X 10-11 J30(5} = 2.7 x 10-21 There are other Bessel functions that satisfies the recursion in (77.7), the {In(z)}. It can be shown that In(z) f'>J ~! (~) n as n --+ 00 (see Abramowitz and Stegun [1], 9.1.7), which indicates that In(z) f'>J B (z/2nt. If (77.7) is used to compute {In(z)}, then roundoff errors will result in { un} terms. Since the evolution of these terms is much greater than the {In(z)} terms, this will be an unstable computation. Table 77.2 shows the computation of {In(5)} using (77.7). The values of Jl(5) and J2(5) were used to initialize the recurrence relation. For a few values of n, the computation is accurate. For n above about 10, however, the computational values are not meaningful. Notes [1] Van der Laan and Temme [6] indicate the results of applying the above theorem to Bessel functions (as we have), confluent hypergeometric functions (two different recursions), incomplete beta functions, Legendre functions (recursion with respect to order and with respect to degree), Jacobi polyno­ mials. and repeated integrals of the error function. All cases of the theorem are illustrated. [2] The exponential integrals En(z} = It:> t-ne-t;; dt have the recurrence re­ lation nEn+1(z} = e-Z: -zEn(z),n = 1,2, .... This relation is studied in Gautschi [3]. [3] For the integrals In, In, and Kn I c := -- dt 1.00 te-ct (t2) n n ( ) 0 (1 + t2) v'i 1 + e ' J c .- -- dt 1.00 e-ct (t2) n n( ).-0 (1 + t2)v'i 1 + t2 ' 1.00 e-ct ( t2 ) n Kn(c} := . G --2 dt, o vt 1 + t 78. Symbolic Methods 329 with c ~ 0 and n ~ 0, Acton [2] finds the recurrence relations: I _ 4nIn + 2cKn n-l -4n -1 ' J _ 4nJn + 2cIn-l n-l -4n -3 ' Kn-1 = Kn + In-l. References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. . 100 exp( -ct) dt [2] F. S. Acton, "Recurrence RelatlOns for the Fresnel Integral ..Ji, 2 o t(1 + t ) and Similar Integrals," Comm. ACM, 17, No.8, 1974, pages 480-481. [3] W. Gautschi, "Recursive Computation of Certain Integrals," J. ACM, 8, 1961, pages 21-40. [4] W. Gautschi, "Computational Aspects of Three-terms Recurrence Rela­ tions," SIAM Review, 9, 1967, pages 24-82. [5] W. Gautschi, "Recursive Computation of the Repeated Integrals of the Error Function," Math. of Comp., 15, 1961, pages 227-232. [6] C. G. Van der Laan and N. M. Temme, Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-like Functions, Centrum voor Wiskunde en Informatica, Amsterdam, 1984. [7] E. W.-K. Ng, "Recursive Formulae for the Computation of Certain Integrals of Bessel Functions," J. Math. and PhysiCS, 46, 1967, pages 223-224. [8] J. Wimp, Computations with Recurrence Relations, Pitman Publishing Co., Marshfield, MA, 1984. 78. Symbolic Methods Applicable to Definite integrals. Yields A numerical quadrature scheme. Idea Using symbolic operators, quadrature rules can be devised. 330 VI Numerical Methods: Techniques Procedure Define the following operators: • The forward-differencing operator A: Af(xo) = f(xo + h) -f(xo) A2 f(xo) = A[Af(xo)] = Af(xo + h) -Af(xo) • The backward-differencing operator V: V f(xo) = f(xo) -f(xo -h) V2 f(xo) = V[V f(xo)] = V f(xo) -V f(xo -h) • The stepping operator E: Ef(xo) = f(xo + h) E2 f(xo) = E[Ef(xo)] = f(xo + 2h) En f(xo) = f(xo + nh). The obvious relationships among these operators are A = E -1 and V = 1 -E-1. Since the above operators are linear operators, the usual laws of algebra can be applied. Hence, several other relationships between the operators can be developed, such as: EV = A, Envn = An, and An fo = vn fn (where fn = f(xn) = f(xo + nh». Since Yn = Enyo and x = nh, we have y~ = d~ (Enyo) = ~ d~ (Enyo) = ~(IOgE)(Enyo), or D = ~ log E, where D denotes the differentiation operator. This is equivalent to E = ehD• The integration operation, which is the inverse operator to D, can be represented as f = D-1 = hI log E. Example Using the above operators, we can derive many quadrature rules. Start with fs = f(xs) = ES fo, multiply by dx = h ds, and then integrate from Xo to Xl (i.e., from s = 0 to s = 1) to obtain /.%1 /.1 (h) IS=l h(E -1) f(x)dx = h ESfods = lEEsfo = I r.1 fo. %0 0 og s=o og Using E = 1 + A, the expression log ( 1 + A) may be formally expanded in a power series to obtain log(1 + A) = A -!A2 + iA3 -~A4 + .... Dividing A = E -1 by this last expression results in /.%1 hA f(x)dx = A _ !A2 + !A3 _ !A4 + fo %0 2 3 4 ••• = h (fo + !Afo -{2A2 fo + 2~A3 fo -... ) (78.1) = h (fo + l (/1 -fo) -1~ (12 -2/1 + fo) + ... ) . 78. Symbolic Methods 331 If n terms of this formula are used, then a polynomial of degree n is being fit to the points xa, Xl, •.• , xn; this interpolating polynomial is then integrated between Xa and Xl. If only the first two terms are used, then the trapezoidal rule is obtained in the form 1Xl 1 1 I(x) dx ~ la + 2(/1 -la) = 2(/a + Id· Xo Notes [1] Analogous to the result in the example, we can obtain 1X2 _ (hES ) IS =2 _ h(E2 -1) f(x)dx -10 Efo -10 E fo Xo g s=o g = h (2fo + 2~fo + !~2 fo + ... ). [2] If we define the central-differencing operator 6 by 6f(xo) = f(xo + ih) - f(xo -~h), then the following integration rules can be found (see Beyer [1]): 11/2 f(x) dx = (1 + -fi62 -5~~064 + 9637~O 66 -••• ) fo -1/2 I: f(x) dx = 2 (1 + *6' -1~64 + 151 1266 -... ) f. E f(x)dx = 4 (1 + ~6' + f.~ -9!566 + ... ) fo. [3] The operators defined above can also be used in some clever manipulations . 11 Rm(x)Rn(x) of mtegrals. For example, Ullah [4] evaluates I = I 2 kp+l dx, -1 (1 -x ) (with some restrictions on I, m, k, n, and p), in closed form, by first writing it in the form: 1= 11 (1 -x2)-n/2-p-l p,m(x)e(l-x2)D~ >..n/2 pr (v'l'="1) 1 ' -1 A=O where DA = 8/8>... References [1] W. H. Beyer (ed.), CRC Standard Mathematical Tables and Formulae, 29th Edition, CRC Press, Boca Raton, Florida, 1991. [2] C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, Addison­ Wesley Publishing Co., Reading, MA, 1984, pages 256-257. [3] F. B. Hiderbrand, Introduction to Numerical Analysis, McGraw-Hill Book Company, New York, 1974. [4] N. Ullah, "Evaluation of an Integral Involving Associated Legendre Poly­ nomials and Inverse Powers of (1 -x2)," J. Math. Physics, 4, No.4, April 1984, pages 872-873. 332 VI Numerical Methods: Techniques 79. Tschebyscheff Rules Applicable to One-dimensional definite integrals. Yields A numerical quadrature rule. Idea In Tschebyscheff rules the weight associated with each node is con­ stant. Procedure Tschebyscheff rules are quadrature rules of the form (79.1) Note that the weight function on the right-hand side is constant for all nodes in the summation. This formula can be made to be exact for all polynomials f(x) = xn for n = 0,1, ... , N. Setting up the necessary equations, and then solving them, we find that w = 2/N, and the {Xk} are the roots of the polynomial part of N (81 82 83 ) PN = u exp --; -2u2 -3u3 -••• where 8i = Njl ui du = {O N 2 -1 i + 1 for odd i, r . . The polynomial part, ITN, lor even ~. has degree N. Example For N = 2 we find Therefore, IT2 = x2 -1/3 and so Xk = ±1/ va ::::: ±0.5773. For N = 3 we find the polynomial part of P3 to be IT3 = X3 -x/2. Hence, for N = 3, the nodes are located at Xk = {O, ±1/v'2}. 80. Wozniakowski's Method 333 Notes [1] The reason that Tschebyscheff rules are of interest is that sometimes the integrand values, the f(Xi), are observed with errors. In this case the variance of the quadrature rule, f: f(x) dx ~ E~=1 wi/(Xi), is given by (T2 = 2::=1 wl. The value of (T2 is minimized when all the weights are equal. (Presuming that the integration rule is going to integrate constants exactly, which results in the constraint Ei Wi = 1.) [2] The equations TIN (x) = 0 have complex roots for N = 8 and for N 2:: 10, so that there do not exist useful Tschebyscheff rules in these cases. [3] Tschebyscheff rules have been extended to quadrature rules with weight functions, J: w(x)f(x) dx ~ 2::=1 Wf(Xi), see Forster [2]. [4] Guerra and Vincenti [6] consider quadrature rules of the form 11 r n w(x)f(x) dx = L A2h L j<2h) (Xj) + R(f)· -1 h=O j=l References [1] S. Iyanaga and Y. Kawada, Encyclopedic Dictionary oj Mathematics, MIT Press, Cambridge, MA, 1980, page 929. [2] K.-J. Forster, "On Chebyshev Quadrature for a Special Class of Weight Functions," BIT, 26, No.3, 1986, pages 327-332. [3] K.-J. Forster, "On Weight Functions Admitting Chebyshev Quadrature," Math. of Comp., 49, No. 179, 1987, pages 251-258. [4] K.-J. Forster, "On Chebyshev Quadrature and Variance of Quadrature For­ mulas," BIT, 28, No.2, 1988, pages 360-363. [5] K.-J. Forster and G. P. Ostermeyer, "On Weighted Chebyshev-type Quadra­ ture Formulas," Math. oj Comp., 46, No. 174, 1986, pages 591-599, S21- 827. [6] 8. Guerra and G. Vincenti, "The Chebyshev Problem for Quadrature For­ mulas with Derivatives of the Integrand," 22, No.3, 1985, pages 335-349. [7] D. K. Kahaner, "On Equal and Almost Equal Weight Quadrature Formu­ las," SIAM J. Numer. Anal., 6,1968, pages 551-556. [8] W. Squire, Integration for Engineers and Scientists, American Elsevier Pub­ lishing Company, New York, 1970, pages 132-135. 80. Wozniakowski's Method Applicable to Multidimensional integrals on a hypercube. Yields A numerical approximation scheme which uses the smallest number of nodes for a specified average error. 334 VI Numerical Methods: Techniques Idea Wozniakowski has devised a way to choose nodes optimally to approx­ imate a multidimensional integral numerically. Procedure Wozniakowski's technique applies to an integral over the d-dimensional unit hypercube I = l' l' '" l' J(xl dx , T ' d where x = (Xl, X2, ••• , Xd). Given a (small) value of c, we would like to approximate this integral by the formula I ~ ~ ~:=l !(Xk), and have the "average error" be less than c. This requires determining the number of nodes n and determining the location of the nodes {Xk}. If we restrict ourselves to integrands that are real and continuous (some class must be specified for an average error to make sense), then Wozniakowski finds that as c --+ 0, the number of nodes needed is n = 0 GI10gel(d-ll/2) . (80.1) Note that the number of nodes required to obtain the same accuracy using Monte Carlo techniques (nMonteCarlo, see page 304) or using a uniform grid (nuniform, see page 323) are nMo ... c .... o = 0 (e12)' n •• iform = 0 (e1d ) • Hence, the number of integrand evaluations in (80.1) is far less than the number of evaluations needed by these other two methods. Wozniakowski does not give formulae on how to determine the optimal nodes {Xk} exactly, but he does give formulae for determining nodes {Xk} and {ik} that are "nearly as good." That is, more of these nodes are required for the same average accuracy, but the number of nodes required only increases a little (the exponent (d -1)/2 in (80.1) changes to (d -1) or d). Even with this many nodes, it represents a substantial improvement over using Monte Carlo methods or a uniform grid. The procedure for determining the node locations is straightforward. For each value of k, start by writing k in base 2, then in base 3, then base 5, etc., using the first d -1 primes as the bases. As an example, we choose d = 6 and k = 42. Then we find: 42 = (101010h = (1120h = (132)5 = (60h = (39)11. 80. Wozniakowski's Method 335 Now, for each of these primes p, calculate the "radical inverse" for k, 4>p (k), which is obtained by reversing the digits of the base-p representation of k, then dividing the result by pi, where i is the number of digits in that representation. We have: ,A, (42) = (010101h = 21 ,A, (42) = (0211)J = 22 0/2 26 64' 0/3 34 81 ' 66 6 102 4>5(42) = 125' 4>7(42) = 49' 4>11(42) = 121' (80.2) Finally, the {Xk} are determined by (k+t ) Xk = (1,1, ... , 1) --:;;:-' 4>2(k), 4>3(k), ... , 4>Pd-l (k) where t is some constant which Wozniakowski does not evaluate. Since this constant is unknown, we could delete it and use instead i\ = (1,1, ... ,1) -(~, tP2(k), tPa(k), ... ,tPPd-l (k») . (80.3) If we do this, then the number of nodes needed varies as ! I log clq with e q = d -1 rather than the optimal q = (d -1)/2. Observe that, for d = 6, this results in (using the values in (80.2» ....... _ (n-42 43 59 59 43 19) X42 --n-' 64' 81' 125' 49' 121 . Note that we will have to recalculate the first component of each Xk if we decide to increase n. We can avoid this recalculation by using the first d primes (rather than the first d -1) and instead of the Xk or the Xk above, choose the nodes to be ik = (1,1, ... ,1) -(4)2(k), 4>3(k), ... , 4>Pd(k». This results in the error estimate ! I log elq with q = d. Observe that, for e d = 6, this produces the 42nd node :::::: _ (43 59 59 43 19 127 ) X42 -64' 81' 125' 49' 121 '169 . 336 VI Numerical Methods: Techniques y ~ --:-r.--:-., • ":.' ' •• ' '.1 •• , .' '.' .'1 • • .".. • 1 .: .. , ..... : ':' ..... :.: .. : ." ·'1 .. ' x Figure 80. The first 100 i nodes in two dimensions. Example If 100 nodes are be used in an integration routine, then the (nearly) "optimal nodes," the {ik} nodes, are shown Figure 80. Notes [1] The technique presented in this section comes with two obvious caveats: (A) There is no estimate of the overall multiplicative constants needed to find the requisite n from the desired error e, nor do we know how small e must be before we get into the asymptotic regime where the expressions for n are valid. (B) This technique is based on the average error, not the worst-case error, so we could be unlucky and do very badly for the integrand we are really interested in. [2] In one dimension, the number of nodes needed to obtain a minimal average error has a more concise answer. For the class of r times continuously differ- entiable functions equipped with a(sP(ecific type of )pr~~~J>:}l)'ty distribution, the number of nodes required is 0 (~2:+~~! ! ' where B2r+2 is a Bernoulli number. See Traub et al. [4] for details. [3] Writing the nodes in (80.3) as Xk = (1,1, ... , 1) -Zk, we recognize the {Zk} to be Hammersley points (see Davis and Rabinowitz [2]). Removing the first component from {Zk} results in the Halton points. Berblinger and Schlier [1] used the Halton points as a "quasi-random" sequence of nodes in a Monte Carlo type computation. [4] In this section we have chosen nodes to minimize the average error. Many results are known for the number of nodes needed to minimize the maximum error. For example, considering the Sobolev class of real functions defined on the d-dimensional unit hypercube whose r-th distribution derivatives exist and are bounded by one in the Lp norm, the number of nodes required is o (e-d1r) when pr > d. See Novak [3] for details. 81. Tables: Numerical Methods 337 References [1] M. Berblinger and C. Schlier, "Monte Carlo Integration with Quasi-random Numbers: Some Experience," Comput. Physics Comm., 66, 1991, pages 157- 166. [2] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984, pages 350-354. [3] E. Novak, Deterministic and Stochastic Error Bounds in Numerical Anal­ ysis, Lecture Notes in Mathematics #1349, Springer-Verlag, New York, 1988. [4] J. F. Traub, G. W. Wasilkowski, and H. Wozniakowski, Information-based Complexity, Academic Press, New York, 1988. [5] H. Wozniakowski, "Average Case Complexity of Multivariate Integration," Bull. Amer. Math. Soc., January 1991, 24, No.1, pages 185-194. 81. Tables: Numerical Methods Applicable to Definite integrals. Yields Quadrature rules. Idea Many tables of numerical methods for integrals have been created. Procedure As indicated in this book, there are many ways in which to derive a scheme that will evaluate an integral numerically. Many books and papers have tabulated schemes for numerically evaluating integrals. While these are not adaptive schemes, they are useful because they may be directly entered into a computer. In this section we merely reference where collections of quadrature rules may be found. Some one-dimensional and two-dimensional quadra­ ture rules may be found starting on page 340. Example 1 The book by Stroud [8] is perhaps the most comprehensive book on quadrature rules. It contains rules for the following regions: [1] the n-dimensional cube: Cn [2] the n-dimensional cubical shell: ~hell [3] the n-dimensional sphere: Sn [4) the n-dimensional spherical shell: s~hell (5) the surface of Sn: Un (6) the n-dimensional octahedron: Gn [7] the n-dimensional simplex: Tn [8] entire n-dimensional space with weight function exp (-x~ -... -x~) 338 VI Numerical Methods: Techniques [9] entire n-dimensional space with weight function exp ( -J x~ + ... + x~ ) [10] the two-dimensional hexagon: H2 [11] the two-dimensional ellipse with weight function [(x _ C)2 + y2] -1/2 [(X+C)2+y2]-1/2: ELP [12] the parabolic region bounded by x = a2 -y2 /4a2 and x = y2 /4b2 -b2: PAR [13] the parabolic region bounded by y = b -bx2/a2 and the x-axis.: PAR2 [14] the parabolic region bounded by y = b -bx2/a2 and y = bx2/a2 -b: PAR3 [15] a three dimensional pyramid: eN :C2 [16J a three-dimensional cone: CN:82 [17] a three-dimensional torus with circular cross section: TOR3 : 82 [18] a three-dimensional torus with square cross section: TOR3 : C2 where Stroud's notation for some of the regions has been used. For each of these regions, quadrature rules of varying degree and with a varying number of nodes are given. For example, Stroud reports the following quadrature rules for the n-dimensional cube (the numbering of the methods is his): 1-1 Degree 1, 1 Node (Centroid formula) Node (0,0, ... ,0) Weight V where V represents the volume of en, that is V = 2n. 1-2 Degree 1, 2n Nodes (Product trapezoidal rule) Nodes Weights (±1, ±1, ... , ±1) V/2n where the symbol (±s, ±s, ... , ±s) denotes a set of 2n nodes, the n signs ± are assumed independent. 2-2 Degree 2, 2n + 1 Nodes Nodes Weights (2r,2r, ... ,2r,2r) V (1, r, ... , r, r)s -rV (-I,r, ... ,r,r)s rV where r = 13/6 and a subscript of S indicates a symmetric set (Le., all permutations) of nodes. For example: (1, r, ... , r, r)s denotes the n nodes: {(I, r, r, ... , r), (r, 1, r, ... , r), (r, r, 1, ... , r), ... , (r,r,r, ... ,I)}. 3-4 Degree 3, 2n Nodes (Product Gauss rule) Nodes Weights (±r, ±r, ... , ±r) V/2n where r = 1/13. 81. Tables: Numerical Methods 339 5-9 Degree 5, 3n Nodes (Product Gauss formula) Nodes Weights (Til' Ti2' .•• , Tin) (Ail Ai2 ••. Ain) where each of the subscripts iI, i2, ... , in ranges independently over the integer 1,2,3 and TI = -J3/5, T2 = 0, T3 = J3/5, Al = 5/9, A2 = 8/9, A3 = 5/9. Example 2 The book by Stroud and Secrest [9] contains tables of Gaussian quadra­ ture rules for the following types of integrals: [1] f~l lex) dx ~ E~l Ai/(Xi) [2] f~l (1 -X2)Q I(x) dx ~ E~l Ai/(Xi) [3] J~l (1 + x)fJ I(x) dx ~ E~l Ai/(Xi) [4] J~l IxlQ I(x) dx ~ E~l Ai/(Xi) [5] J~oo e-z2 I(x) dx ~ E:l Ai/(Xi) [6] foOO e-Z I(x) dx ~ E~l Ai/(Xi) [7] f~oo IxIQe-Z2 I(x) dx ~ E:l Ai/(Xi) [8] f~oo IxlQe-1zl lex) dx ~ E~l Ai/(Xi) [9] J: log (I/x) I(x) dx ~ E~l Ai/(Xi) [10] 2~i fCc~ii: p-1eP F(P) dp ~ E~l Ai/(Xi) [11] f~l lex) dx ~ AI( -1) + E~l Ai/(Xi) + AI( +1) [12] J~l I(x) dx ~ AI( -1) + E~l Ai/(Xi) [13] J~l I(x) dx ~ E~l Ai/(Xi) + E~o B2k/(2k) (0) [14] J~oo e-z2 I(x) dx ~ E:l Ai/(Xi) + E~o B2kJ<2/C) (0). Example 3 The book by Krylov and Pal'tsev [5] contains tables of quadrature rules for the following types of integrals: [1] /.' XO log; J(x) dx '" t A;J(x,) [2] /.' xPlog; log 1 ~ xJ(x) dx '" tA;J(X') [3] 1.' log G) J(x) dx '" tA;J(X') [4] /.~ xPe-Zlog (1 +~) J(x)dx '" tAd(x,), 340 VI Numerical Methods: Techniques Notes [1] Stroud [8] includes in his tables all quadrature rules known to him that have, in his opinion, some major importance. It is an extensive list of quadrature rules. The ones that he deems to be particularly useful are specifically indicated. (Of course, there are infinitely many product rules for some regions; only representative samples of these rules are given.) [2] Appendix 4 of Davis and Rabinowitz [2] contains a comprehensive bibliog­ raphy of tabulated quadrature rules. References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. [2] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, Orlando, Florida, 1984. [31 A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables, Addison-Wesley Publishing Co., Reading, MA, 1962. [4] A. S. Kronrod, Nodes and Weights of Quadrature Formulas, Consultants Bureau, New York, 1965. [5] V. I. Krylov and A. A. Pal'tsev, Tables for Numerical Integration of Func­ tions with Logarithmic and Power Singularities, Israel Program for Scientific Translations, Jerusalem, 1971. [61 V. I. Krylov and N. S. Skoblya, Handbook of Numerical Inversion of Laplace Transforms, Israel Program for Scientific Translations, Jerusalem, 1969. [7] C. H. Love, Abscissas and Weights for Gaussian Quadrature for N = 2 to 100, and N = 125, 150, 175, 200, National Bureau of Standards Monograph 98, December 1966, National Bureau of Standards, Washington, DC. [8] A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall Inc., Englewood Cliffs, NJ, 1971. [9] A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall Inc., Englewood Cliffs, NJ, 1966. 82. Tables: Formulas for Integrals Applicable to One-dimensional and two-dimensional definite integrals. Yields A numerical scheme. Idea There exist standard quadrature rules for numerically approximating integrals over intervals and different geometric shapes. 82. Tables: Formulas for Integrals 341 Procedure There exist standard quadrature rules for numerically integrating dif­ ferent types of integrands. In this bection are quadrature rules for nu­ merically integrating one-dimensional and two-dimensional integrals, these formulas are organized by geometric shape. Most of the two-dimensional quadrature rules are from Abramowitz and Stegun [1]. Notes [1] For Gaussian quadrature rules, see the tables on pages 290 and 298. [2] For Newton-Cotes rules, see the table on page 320. For open Newton-Cotes rules, see the table on page 321. One Dimensional Integration Rules In the following, e is some number between Xo and Xn, and 1; = I(xj) = I(xo + jh) where h = (Xn -xo)/n. (A) Trapezoidal Rule (B) Modified Trapezoidal Rule lzn (1 1 ) I(x)dx =h 2/0 + /1 + ... + In-l + 21m Zo h lIn 5 (4) + 24 (-1-1 + /1 + In-l -In-I) + 720h I (e)· (C) Simpson's rule (D) Extended Simpson's rule lz2n h [ zo f(x) dx = '3 10 + 4 (/1 + h + ... + hn-d 1 nh5 (4) + 2 (12 + 14 + ... + hn-2) -90 I (e)· 342 VI Numerical Methods: Techniques (E) Euler-Maclaurin summation formula (also known as the composite trape­ zoidal rule) /.:l:n J(x) dx :1:0 = h [~Jo + /1 + h + ... + In-l + ~Jn] _ B2 h2 (I' _ I') _ _ B2k h2k (/(2k-l) _ 1(2k-l») + R 2! n JO ... (2k)! n JO 2k 9nB h2k+3 where R2k = (22~+2 2)1 max 1J'2k+2)(x)l, with -1 :::; 9 ::::; 1. Here, + . xo!fx!fzn the Bk'S are Bernoulli numbers (see [1], 25.4.7). (F) Five-point rule for analytic functions (J must be analytic) /.ZO+h h [ J(z) dz = 15 24J(zo) + 4J(zo + h) + 4J(zo -h) zo-h -4/(zo + ih) + 4/(zo -ih) ] + R where IRI ::::; {:l:~:; 1J'6)(z)1 and S is the square with vertices at {zo + ikh I k = 0,1,2,3}. Integration Formulae for Different Geometric Shapes Circumference of a circle If r represents the circumference of the circle x2 + y2 = h2, then we have the approximate integration rules (see [1], equation 25.4.60) where m ~ 1. The following figure indicates the location of the 12 points when m=6. h 82. Tables: Formulas for Integrals 343 Circular region If C represents the circle with radius h (x2 + y2 ::; h2), then we have the approximate integration rules (see [1], equation 25.4.61) ,,~2 f f f(x,y)dzdy = t w;f(x"y,) + R: c 1=1 (A) n = 4, R = O(h4) i i (Xi, Yi) Wi (±~h,±~h) 1 4 (B) n = 5, R = O(h4) (Xi,Yi) Wi (0,0) 1 '2 (±h,O) 1 8 (O,±h) 1 8 (C) n = 7, R = O(h4) (Xi, Yi) (0,0) (±h, 0) (±lh ±v'3h) 2' 2 (D) n = 7, R = O(h6) (Xi, Yi) (0,0) (±Jfh,O) (±~h,±~h) Wi 1 '2 1 12 1 12 Wi 1 4 1 8 1 8 h h h h 344 VI Numerical Methods: Techniques (E) n = 9, R = O(h6) (Xi, Yi) Wi (0,0) 1 6 h (±h, 0) 1 24 (0, ±h) 1 24 (±!h,±!h) 1 6 Squares If S represents a square with each side oflength 2h (ixi :5 h, lyl :5 h), then we have the approximate integration rules (see [1], equation 25.4.62 and Stroud [3]) 4~2 f f !(x,y) dxdy = t w;f(x.,y.) + R: s i=l (A) n = 3, R = O(h3) (Xi, Yi) Wi 1 • 3 h 1 .... ,. . . . . . .... 3 1 3 • (B) n = 3, R = O(h3) (Xi, Yi) Wi (h,h) 1 "1 • h (lh, -~h) 3 8 I (-&h, ~h) 27 I 56 • 82. Tables: Formulas for Integrals (C) n = 3, R = O(h3) (Xi, Yi) Wi I (h, th) 3 I 28 (~h, 0) 3 4 (h, -h) 1 "7 (D) n = 4, R = O(h4) (Xi, Yi) (h,h) (-h, -h) (-?sh, -Jsh) (7sh'-7sh) (E) n = 4, R = O(h4) (Xi, Yi) Wi (±h, 0) 1 6 (O,±~h) 1 3 (F) n = 4, R = O(h4) (Xi, Yi) (±~h'±7ah) Wi 1 12 1 12 5 12 5 12 Wi 1 4 345 • h ....• • h • • h • • • h • • 346 VI Numerical Methods: Techniques (G) n = 9, R = O(h4) (Xi, Yi) I Wi I (0,0) 4 I 9 h (±h,±h) 1 36 ......•..... (±h,O) 1 9 (0, ±h) 1 9 - --- (H) n = 9, R = O(h6) (Xi, Yi) Wi (0,0) 16 8I • ! • (±/fh,±/f h) 25 324 h •••• , ••••••••••••• 1 ••• (o,±/Ih) 10 8I • ~ • (±v'Th,o) 10 8I Equilateral triangle If T represents an equilateral triangle, then we have the approximate integration rules (see [1], equation 25.4.63) !~h2 J J f(x,y)dxdy = t wd(x"y,) +R: 4 T i=1 (A) n = 4, R = O(h3) (Xi, Yi) Wi (0,0) 3 4 (h,O) 1 12 (-!h ±:ilh) 1 2' 2 12 .. C> ........•.. h • • •••• .o 82. Tables: Formulas for Integrals (B) n = 7, R = O(h4) (Xi,Yi) Wi (0,0) 27 60 (h,O) 3 60 (-!h ±~h) 3 2' 2 60 (-~h,O) 8 60 (!h ±V3h) 8 4' 4 60 (C) n = 7, R = O(h6) (Xi, Yi) Wi (0,0) 270 1200 (~h,O) 155-v'f5 ~ ( =:illi! h ±.ill±! V3h ) 155-y5S 14 ' 14 1200 ( -~±lh,O) 155±v'f5 1200 (.Ji5-1 h ± ./15-1 V3h) ~ 14 , 14 1200 ~h ·V 8>-"'," ,."~"'., ,h" .. • 347 Regular hexagon If H represents a regular hexagon, then we have the approximate integration rules (see [1], equation 25.4.64) ~ ~ 2 11 f(x,y) dxdy = t Wi/(Xi,Yi) + R: 2v3h '-I H 1- (A) n = 7, R = O(h4) (Xi, Yi) Wi (0,0) 21 36 h (±h,O) 5 72 (±!h ±~h) 5 2' 2 72 348 VI Numerical Methods: Techniques (B) n = 7, R = O(h6) (Xi, Yi) Wi (0,0) 258 TIiii8 h (±VjAh,O) 125 1008 (±~h ±illh) 125 10 ' 10 1008 References [1] M. Abramowitz and 1. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. [2] F. B. Hiderbrand, Introduction to Numerical Analysis, McGraw-Hill Book Company, New York, 1974. [3] A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall Inc., Englewood Cliffs, NJ, 1971, Chapter 3, pages 53-126. 83. Tables: Numerically Evaluated Integrals Applicable to Specific definite integrals. Yields An exact evaluation. Idea The numerical evaluation of many integrals has been tabulated. Procedure When computers were not as common as they are now, it was useful to have tabulated values of certain definite integrals. Tables were published containing these values. These tables are less useful now that computers are readily available to perform numerical integrations as needed. They are still of occasional use, though, for testing new computational routines. Example 1 (A) For tables of Anger functions, see Abramowitz and Stegun [2] or Bernard and Ishimaru [6]. (B) For tables of Bessel functions, see Abramowitz and Stegun [2] or Haberman and Harley [9]. (C) For tables of the cosine integral, see Abramowitz and Stegun [2], or reference [15]. 83. Tables: Numerically Evaluated Integrals 349 (D) For tables of elliptic functions, see Abramowitz and Stegun [2], Belyakov et al. [4], Fettis and Caslin [8], or Selfridge and Maxfield [13]. (E) For tables of the Fresnel integral, see Abramowitz and Stegun [2], Martz [10], or Pearcey [11]. (F) For tables of the Gamma function, see Abramov [1] or Pearson [12]. (G) For tables of the sine integral, see Abramowitz and Stegun [2], or reference [15]. 1x ezzndz (H) For a table of the transport integral Z 2' see Rogers and o (e -1) Powell [14]. (I) For a table of the function 1% "Y(~ {) d{, see Anker and Gafarian [3]. Example 2 (A) For a computation method for computing the polygamma function, see DiMarzio [7]. Notes [1] Abramowitz and Stegun [2] also have tables of Clausen's integral, Debye function, dilogarithm, exponential integral, Sievert integral, and Struve func­ tions. [2] Many of the tables referenced in this section are now superfluous as the numerical values of the integrals can be readily computed. For example, Mathematica [16] has special commands for computing elliptic integrals, Fresnel integrals, dilogarithms, and many other functions. Also, for those functions for which a single command does not exist, the numerical integra­ tion routine can be used. References [1] A. A. Abramov, Tables ofLn r[z] for Complex Arguments, translated by D. G. Fry, Pergamon Press, New York, 1960. [2] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. [3) C. J. Anker, Jr. and A. V. Gafarian, The FUnction J(x, y) = 1" -Y(~' e) ~; Some Properties and a Table, System Development Corporation, 2500 Col­ orado Ave, Sanata Monica, CA, April 1962. [4] V. M. Belyakov, P. I. Kravtsova, and M. G. Rappoport, Tables of Elliptic Integrals, translated by P. Basu, The MacMillan Company, New York, 1965. [5] O. S. Berlyand, R. I. Gavrilova, and A. P. Prudnikov, Tables of Integral Error Functions and Hermite Polynomials, translated by P. Basu, The MacMillan Company, New York, 1962. [6] G. D. Bernard and A. Ishimaru, Tables of the Anger and Lommel- Weber Functions, University of Washington Press, Seattle, 1962. [7] F. DiMarzio, "An Improved Procedure for the Accurate Evaluation of Poly gamma Functions with Integer and Half-Integer Argument," Comput. Physics Comm., 39, 1986, pages 343-345. 350 VI Numerical Methods: Techniques [8] H. E. Fettis and J. C. Caslin, A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I, ARL 69-0172, 1969, A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. /I, ARL 69-0173, 1969, A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus: /II. Auxiliary Tables, ARL 70-0081, 1970, United States Air Force, Wright-Patterson Air Force Base, OH. [9] W. L. Haberman and E. E. Harley, Numerical Evaluation of Integrals Con­ taining Modified Bessel F"ILnctions, Hydromechanics Laboratory, Research and Development Report 1580, March 1964, Department of the Navy, Wash­ ington, DC. [10] C. W. Martz, "Tables of the Complex Fresnel Integral," NASA SP-3010, NASA, Washington, DC, 1964. [11J T. Pearcey, "Table of the Fresnel Integral to Six Decimal Places," Cambridge University Press, New York, 1956. [12] K. Pearson, Tables of the Incomplete r-Function, Cambridge University Press, 1934. [13J R. G. Selfridge and J. E. Maxfield, A Table of the Incomplete Elliptic Integral of the Third Kind, Dover Publications, Inc., New York, 1958. [14J W. M. Rogers and R. L. Powell, Tables of Transport Integrals In(x) = 1% e%zndz ( % 2' National Bureau of Standards Circular 595, July 3, 1958, o e -1) National Bureau of Standards, Washington, DC. [15] Staff of the Computation Laboratory, Tables of Generalized Sine-and Cosine­ Integral Functions: Part I, Harvard University Press, Cambridge, MA, 1949. [16J S. Wolfram, Mathematica: A System of Doing Mathematics by Computer, Second Edition, Addison-Wesley Publishing Co., Reading, MA, 1991. Mathematical Nomenclature GP[a, b] The class of functions that are continuous and have p continuous derivatives, on the interval [a, b]. E the expectation operator, see page 30. E the stepping operator, see page 329 . .r[u(x)] the Fourier transform of the function u(x), defined by ~ (X> u(x)eixw dx. v21r }-oo The inverse transform, denoted by F-1 [U (w )], is defined by _1_ roo U(w)e-ixw dw. yf2;}-00 H (x) the Heaviside or step function, defined by {o ifx<O H(x) := I~oo o(x) dx = 1/2 if x = 0 . 1 if x> 0 11 u(x) 1£ [u(x)] the Hilbert transform of the function u(x), defined by --dx. _IX -t £[u(x)] the Laplace transform of the function u(x), defined by 1000 u(x)e-SX dx. The inverse transform, denoted by £-I[U(S)], is defined by the Bromwich integral Ie U(s)eSX ds, where C is Bromwich contour (a contour that is to the right of all of the singularities of U(s), and is closed in the left half plane). Lp[a,b] The class of functions {u(x)} that satisfy I~oo lu(x)IPdx < 00. M[f(t)] the Mellin transform of the function f(t), defined by 1000 tz-1 f(t) dt. o We say that f(x) = O(g(x)) as x --+ Xo if there exists a positive constant G and a neighborhood U if Xo such that If(x)1 ~ Glg(x)1 for all x in U. o We say that f(x) = o(g(x)) as x --+ Xo if, given any J.L > 0 there exists a neighborhood U of Xo such that If(x)1 < J.Llg(x)I for all x in U. 'P the Weierstrass function (an elliptic integral). See page 156. 351 352 VI Numerical Methods: Techniques sgn the signum function, this returns the sign of its argument. If x > 0, then sgnx is +1. If x < 0, then sgnx is -1. If x = 0, then sgnx is indeterminate. o the central-differencing operator, see page 329. Oij the Kronecker delta, it has the value 1 if i = j and the value ° if i i: j. 6(x) the delta function, it has the properties that o(x) = ° for x i: 0, but J~oo o(x) dx = 1. !:1 the forward-differencing operator, see page 329. e generally a small number, usually assumed to be much less than one in magnitude. V the backward-differencing operator, see page 329. as If S is a region or volume, then as denotes its boundary. II·· ·11 the norm of the argument. f This is a Hadamard finite part integral, see page 73. if' This is an Ito integral, see page 187. § This is a line integral around a closed curve, see page 50 or page 168. f This is a principle value integral, see page 92. i This is a Stratonovich integral, see page 189. []-The negative part of a function: [f(x)]-:= -sup{ -f(x), o} = -21 (If(x)l- f(x)) = {Of(X) if f(x) ~ 0, if f(x) < 0. []+ The positive part of a function: [f(x)]+ := sup{f(x), O} = -21 (If(x)1 + f(x)) = {of(X) if f(x) ~ 0, if f(x) < 0, Index A A-integral, 57 a-order Green's function, 174 Abelian integral, 51 absolute convergence, 53 absolute error, 248 absolute integral, 57 absolute value function, 43 ACM,260 action, 86, 88 adaptive quadrature, 244, 277 Airy differential equation, 2 Airy function, 171, 183, 234 Aitken transformation, 252 algebraic topology, 140 ALGOL programs, 220, 254 amplitude, 154 analytic functions, 325, 342 Anger function, 174, 348 angular momentum, 45 annulus, 4, 21 antithetic variates, 310 applications, 1, 6, 14, 20, 24, 30, 31,34,40,45 arc-length, 157, 165 area, 25, 27, 168 asymptotic expansions, 48, 184, 195, 199 asymptotic sequence, 48, 217 automatic quadrature routine, 244 353 auxiliary asymptotic sequence, 48, 196 auxiliary cosine integral, 172 auxiliary sine integral, 174 average error, 333 average, 30, 139 Axiom, 117 B backward Holder inequality, 207 backward-differencing operator, 330 Bairy function, 171, 183 Banach integral, 57 BASIC programs, 262 basis functions, 281 Bernoulli numbers, 31, 35, 251, 336, 342 polynomials, 35, 315 Bessel functions, 57, 106, 174, 184, 227, 228, 348 integral, 327 best rule, 303, 315, 333 Beta function, 174, 178 Bickley function, 175 biharmonic equation, 23 bilateral Laplace transform, 198 Binet integral, 171 Birkhoff integral, 57 354 bispherical coordinates , 113 bits of precision, 251 Bloch-Gruneisen integral, 171 Bochner integral, 57 Bochner-Martinelli representation, 20,23 Boks integral, 57 BoDle's rule, 320 Borel integral, 57 Borel set, 48 boundary conditions, 6 boundary point, 222 boundary value problem, 6 bounded variation, 50, 57 branch cut, 59, 134, 136 branch point, 46 Bromwich contour, 138 Bromwich integral, 48, 141, 351 Brownian motion, 86 Brownian noise, 186 Burkill integral, 56 butterfly, 202 c C programs, 254 CPla, bl, 351 calculus of variations, 306 canonical form, 145 canonical quantization, 90 caret, 67 Carleman inequality, 206 Carlson notation , 156 Carson integral, 13 Cartan subgroup, 50 Cartesian product regions, 324 Catalan's constant, 171 Cauchy integrals, see integral: principal-value representation, 20 theorem, 22, 36, 129, 163, 231 Cauchy-Green formula, 21, 23 Cauchy-Poincare integral theorem, 140 Cauchy-Riemann equations, 21, 48 Cauchy-Schwartz -Bunyakowsky inequality , 206 caustic point, 202 caveats, 58, 228, 336 cd function, 154 cells, 68 Index central moment, 30 central-difrer encing operator, 331 centroid formula, 338 chain, 68 change of variable, 38, 93, 109 changing order of integration, 61 characteristi c function, 30 character istic polynomial s, 326 Chartier theorem, 66 Chebyshev, see Tschebyscheff Clausen's integral, 172, 349 Clenshaw-Curtis rule, 264, 265, 281 closed rule, 244, 321 CMLIB,260 cn function, 154 co-dimension, 202 combinatoria l sums, 31 combinatorial topology, 52 compact support, 30 comparison test, 65 complementary error function inte- gral, 216 complementary error function, 172 complementary modulus, 154 complete elliptic integral, 154, 172 complex Selberg integral, 179 composite rule, 244, 284 composite Simpson's rule, 251 composite trapezoidal rule, 249, 250, 342 compound rule, 244, 283, 288 computer aided solution, 117, 192 algebra, 117 libraries, 260 MIMD,316 programs , see programs SIMD,316 confluent hypergeometric functions, 179 conjugate momentum , 90 conservative force field, 45 Index constant Catalan, 171 Euler, 37,171,198,238 Lebesgue, 173 continued fractions, 203 contour Bromwich , 138 integrals, sec integral: contour Jordan, 35 control variates, 311 converge in distribution, 30 convergence test, 64, 65 convergent integral, 93 convergent series, 204 convolution theorem, 141 convolutions, 101, 140 coordinates bispherical , 113 cylindrical, 26, 113 ellipsoidal, 114 elliptic cylinder, 11, 114 oblat.e spheroidal, 115 ort.hogonal, 25, 110 parabolic cylinder, 114 parabolic, 114 paraboloid al, 114 polar, 115 prolate spheroidal, 114 rotational oblate spheroidal, 115 rotational parabolic, 114 rotational prolate spheroidal , 114 rotational, 114, 115 sphcrical, 115 spheroidal, 114, 115 corrected midterm rule, 288 corrected trapczoidal rulc, 288 cosine integral, 117, 172,348 cosine transform, 8 covariance, 31, 189 critical point, 231 cs function , 154 cubic splines, 266, 285 curl, 165, 166 curvature, 27, 200, 287 curvilinear integral, 57 cusp, 202 cylindrical coordinates, 26, 113 t>, 330 6,331 D Danicll integral, 57 Darboux sums, 55 Darboux theorem, 55 Davenport theorem, 80 Dawson's integral, 172 de function, 154 355 De la Vallee-Possin's integral, 57 Dcbye function, 175, 349 deferred approach to the limit, 253 definitions, 47, 51, 244 degree of precision , 245 delta function, 5, 86, 352 Denjoy integral, 57 density function, 3D, 305 derivative information, 287, 322 Derive, 117 determinant , 48 differential equations, 1,2,4,5,6, 15,23,70,86, 107, 181, 190, 243 differential forms, 67 differentiation by int.egration , 46 differentiation operator, 330 digamma function, 173 dilogarithms, 145, 172, 349 Dirac delt.a function, see delta funct.ion Dirichlet. funct.ion, 54 Dirichlet integral, 57 Dirichlet reduction, 100 discriminant., 150 dissipative systems, 88 distribution derivatives, 336 distribut.ion function, 30 div, 165, 167 divergence test, 65 divergence theorem, 167 divergent int.egral, 73 dn function, 154 Dnestrovskii funct.ion, 175 double Fermi-Dirac integral, 178 ds funct.ion, 154 Dunford int.egral, 57 356 E E,30, 189,246,330 eigenfunctions, 91 eigenstructure, 293 eigenvalue, 11, 91, 293 eigenvector, 293 electronic mail, 255 elementary functions, 79, 170 elementary generator, 79 elementary integral, 168 elementary over a field, 79 ellipsoidal coordinates, 114 elliptic cylinder coordinates, 11, 114 elliptic functions, 349 elliptic integral, 52, 148, 154, 172, 175, 178 elliptic umbilic, 202 entropy, 30 equidistributed sequence, 314 error analysis, 246, 294 error function integral, 216 error function, 81, 144, 172, 224 Euler equation, 15 Euler kernel, 2 Euler's constant, 37, 171, 198, 238 Euler-Lagrange equation, 15, 91 Euler-Maclaurin formula, 35, 342 exact bounds, 218 excerpts from GAMS, 260 excess, 30 expansion theorem, 91, 196 expectation operator, 30, 189, 306 exponential integral, 81, 117, 172, 173,175,198,215,328,349 extended log-sine integral, 177 rule, 325 Simpson's rule, 284, 341 exterior calculus, 67 extrapolation, 250 extremal problems, 14 F factorial function, 163, 224 Fejer rule, 271 Fermi-Dirac integral, 175, 178 Index Feynman diagrams, 70 Feynman path integrals, see inte- gral: path Feynman-Kac theorem, 91 field of elementary functions, 79 Figures, 3, 36, 38, 41, 53, 55, 71, 87, 110, 112, 129, 131, 132, 133, 134, 136, 138, 167, 230, 233, 235, 236, 237, 255, 278, 285, 302, 308, 336 finite element method, 15 finite transform cosine, 8 Fourier cosine, 8 Hankel, 8 Hilbert, 13 sine, 8 finite-part integrals, see integral: finite-part finite-part rule, 74 first fundamental form, 27 first kind Abelian integral, 52 first kind elliptic integral, 172, 175 first moment, 26 five-point rule, 342 fold, 202 force, 45 FORTRAN programs, 220,254, 255, 260 forward-differencing operator, 330 Fourier cosine transform, 8, 9, 192 series, 7 sine transform, 8, 9, 192 transform, 9, 12, 142, 192, 304, 351 fourth order polynomials, 148 fractional integration, 75 fractional part, 313 fractional transformation, 156 free software, 255 Fresnel integral, 116, 173, 175, 176, 349 Frullanian integral, 157 Fubini theorem, 61, 64 fully symmetric, 244 Index functional equations, 160 integral, 86 fundamental form, 27 fuzzy integral, 57 G Gamma function, 139, 163, 173, 176, 224, 236, 349 GAMS, 254, 258, 260 gauge integral, 57 Gauss formula, 169 Gauss function, 178 Gauss rule, 220, 261, 263, 265, 267, 268, 271, 294, 297, 298, 322 Gauss theorem, 167 Gauss-Bonnet formula, 27 Gauss-Hermite rule, 263, 267, 268, 271, 297 Gauss-Jacobi rule, 297 Gauss-Laguerre rule, 263, 267, 268, 271, 297 Gauss-Legendre rule, 245, 263, 267, 268, 270, 271, 297 Gauss-Lobatto rule, 271 Gauss-Radeau rule, 271 Gaussian curvature, 27 Gel'fand-Pettic integral, 57 general Denjoy integral, 57 general purpose integrators, 245 generalized coordinate, 90 Fermi-Dirac integral, 178 Fresnel integral, 175, 176 functions, 12 inverse tangent integral, 176 log-sine integral, 179 Wirtinger inequality, 207 generating function, 30, 35 genus of a curve, 52 geodesic curvature, 27 geodesic triangle, 27 geometric applications, 24 global approximation, 278 global error, 278 gradient vector, 165 grand integrator, 30 357 Grassman algebra, 57 Green's function, 5, 11, 70, 86 Green's representation theorem, 21 Green's theorem, 27, 163, 165, 169 Gronwalls inequality, 212 group icosahedral, 318 octahedral, 318 reflection, 318 renormalization, 91 sampling, 311 symmetry, 316 tetrahedral, 318 H Hadamard finite-part integrals, see integral: finite-part half-plane geometry, 4 half-space geometry, 4 Halton point, 336 Hamilton's principle, 91 Hamiltonian, 88 Hammersley point, 336 Hankel transform, 8, 9, 11 Hardy inequality, 206 Hardy-Littlewood supremum theo- rem, 206 harmonic function, 21, 23, 85 harmonic integral, 57 harmonic numbers, 37 Harnack integral, 57 Hartley transform, 10 Heaviside function, 71, 351 Heisenberg principle, 212 helix, 26 Helmholtz equation, 11 Helmholtz theorem, 23 Henstock integral, 52, 57 Herglotz representation, 22 Hermite functions, 106 polynomials, 294, 297 transform, 12 Hermitian matrix, 89 Hessian, 202 higher order logarithmic functions, 147 358 Hilbert inequality, 211 transform, 10, 12, 13, 264, 265, 351 hit-or-miss, 307, 308 holomorphic function, 20, 35, 48, 212 Hubbell rectangular-source inte- gral, 176 Hurwitz function, 176 hyperbolic cosine integral, 172 hyperbolic sine integral, 174 hyperbolic umbilic, 202 hypercube, 334 hyperelliptic integral, 157, 178 hypergeometric functions, 179 Holder inequality, 205, 207 Holder integral, 57 I icosahedral group, 318 importance sampling, 306 improper integral, 53, 57, 92 IMSL, 260 incomplete Beta function, 178 Gamma function, 176 hyperelliptic integral, 178 index, 40, 353 inequality, 65, 205, 276 backward Holder, 207 Carleman, 206 Cauchy-Schwartz-Bunyakowsky, 206 generalized Wirtinger, 207 Gronwalls, 212 Hardy, 206 Hilbert, 211 HOlder, 205, 207 Minkowski, 207 Ostrowski, 207 Tschebyscheff, 205, 207 Wirtinger, 207 Young, 208 inertia, 26 infinite asymptotic expansion, 48 infinite series, 184, 185 information theory, 30 inner product, 48, 292 integrable, 55 integral Abelian, 51 absolute, 57 auxiliary cosine, 172 auxiliary sine, 174 Banach, 57 Binet, 171 Birkhoff, 57 Bloch-Gruneisen, 171 Bochner, 57 Boks, 57 Borel, 57 Index Bromwich, see Bromwich integral Bromwich, 48, 141, 351 Burkill, 56 Carson, 13 Clausen's, 349 complementary error function, 216 complete elliptic, 172 complex Selberg, 179 contour, 1, 31 cosine, 117, 172, 348 curvilinear, 57 Daniell, 57 Dawson's, 172 De la ValIee-Possin's, 57 definitions, 51 Denjoy,57 Dirichlet, 57 divergent, 73 double Fermi-Dirac, 178 Dunford, 57 elementary, 168 elliptic, 52, 148, 154, 172, 175, 178 equation, 70 error function, 216 exponential, 81,117,172,173, 175, 198, 215, 328, 349 extended log-sine, 177 Fermi-Dirac, 175, 178 Feynman, 86 finite-part, 73, 352 Index integral (continued) first kind elliptic, 172, 175 Fresnel, 116, 173, 175, 176, 349 Frullanian, 157 functional, 86 fuzzy, 57 gauge, 57 Gel'fand-Pettic, 57 general Denjoy, 57 generalized Fermi-Dirac, 178 generalized Fresnel, 175, 176 generalized inverse tangent, 176 generalized log-sine, 179 harmonic, 57 Harnack, 57 Henstock, 52, 57 Hubbell rectangular-source, 176 hyperbolic cosine, 172 hyperbolic sine, 174 hyperelliptic, 157, 178 Holder, 57 improper, 53, 57, 92 incomplete hyperelliptic, 178 inequality, 65, 205, 276 interval, 274 inverse tangent, 173, 176 Ito, 187, 352 Khinchin, 57 Kolmogorov, 56, 57 Lebesgue, 53, 57 Lebesgue-Stieltjes, 54 line, 27,45, 57,164 log-sine, 177, 179 logarithmic Fermi-Dirac, 175 logarithmic, see logarithmic integral logarithmic, 81, 173 Lommel,57 loop, 4, 139, 352 multidimensional, 199, 300, 304, 316, 323 multiple, 61, 83, 107, 199 norm, 57 path, 70, 86 Pearcey, 176 Perron, 56, 57 359 integral (continued) principal-value, 13, 61, 74, 75, 92, 155, 264, 274, 352 product, 5 Radon, 57 refinement, 57 repeated, 175, 176 representation, 1, 4, 5, 22, 43, 224, 246 Riemann, 54, 57 second kind elliptic, 172, 175 Selberg, 179 Sievert, 177, 349 sine, 174, 177, 349 special form, 145, 148, 157 stochastic, 56, 186 Stratonovich, see Stratonovich integral Stratonovich, 189, 352 strong variational, 57 surface, 164 tabulated, 190, 348 test, 272,303, 304,311 third kind elliptic, 178 transform, 6 transport, 349 variational, 57 Wu's,179 Young,57 integrand, 48 integration and differentiation, 142 bee, 28 by parts, 6, 15, 143, 146, 150, 162, 189, 215, 228, 326 contour, 129, 163 fractional, 75 operator, 330 over a simplex, 284 over circular region, 343 over circumference of circle, 342 over hexagon, 347 over square, 344 over triangle, 346 over unit cube, 300 Romberg, 250 stochastic, 186 360 interpolating polynomials, 319, 331 interval analysis, 218, 248, 274, 280 interval integral, 274 introduction, 243 invariants, 301 inverse Laplace transform, 138 inverse tangent integral, 173, 176 Ito integral, 187, 352 J Jacobian elliptic functions, 152, 154 Jacobian, 38, 48, 110, 201 Jordan contour, 35, 40 Jordan lemma, 133, 139 K K -transform, 10 kernel, 2, 6 Khinchin integral, 57 Kolmogorov integral, 56, 57 Kontorovich-Lebedev transform, 10 Kronecker delta, 319, 352 Kronrod rule, 261, 263, 265, 267, 268, 297, 298 Kummer's function, 176 L Lp norm, 49 Lp function, 49 Lp[a, b], 351 Lagrange interpolation formula, 319 Lagrangian, 86, 88 Laguerre functions, 106 polynomials, 294, 296 transform, 12 Laplace differential equation, 4, 23 kernel, 2 method, 4, 198, 221, 230 transform, 10, 138, 141, 198, 240, 351 lattice rule, 300, 315 Index Laurent series, 129 least number of nodes, 333 Lebesgue constants, 173 Lebesgue integral, 53, 57 Lebesgue measurable set, 49 Lebesgue-Stieltjes integral, 54 Legendre chi function, 173 differential equation, 4 elliptic functions, 152 functions, 23, 106, 107, 176 notation, 149 polynomials, 34, 291, 294 relation, 154 transform, 9 Leibniz rule, 49, 77 lemma Jordan, 133, 139 Eliemann-Lebesgue, 227 Watson, 197, 198, 225, 238 length, 24 Lerch's function, 179 library of software, 254 Lie algebra, 50 Lie group, 50 line integrals, see integral: line linear differential equations, 1, 6 linearly independent, 49, 211 Liouville principle, 79 theorem, 52 theory, 79 Lipschitz condition, 74 Lobatto rule, 291, 295 log-sine integral, 177, 179 logarithmic Fermi-Dirac integral, 175 logarithmic integral, 81, 173 Lommel integral, 57 look up technique, 170 loop integrals, see integral: loop lower limit, 49 M Macsyma, 117, 239 MAPLE, 82,117,198 Mathematica, 117, 140, 349 Index maximizing an integral, 14 mean square limit, 187 mean value theorem, 83, 246 mean, 30, 31, 306 measurable function, 53 measurable set, 49 measure, 49, 53, 186 Mehler-Fock transform, 10 Meijer transform, 11 Mellin kernel, 2 transform, 11, 141, 197, 198, 351 Mellin-Barnes integral representa- tion, 5 meromorphic function, 40, 49, 155 mesh ratio, 245 method Laplace, 4, 198, 221, 230 Monte Carlo, 269, 304, 334 number theoretic, 269, 312 saddle point, 239 Sag-Szekeres, 270 stationary phase, 198, 226 steepest descent, 198, 225, 229, 230 Wozniakowski, 333 metric coefficients, 25, 110, 113 midpoint rule, 91, 321 midterm rule, 288 Milne's rule, 320 MIMD computer, 316 minimizing an integral, 14 minimum variance estimator, 306 Minkowski inequality, 207 MIT, 28 modified Struve function, 177 trapezoidal rule, 322, 341 modular angle, 154 modulus, 154 moments of inertia, 26 moments, 30, 198 momentum, 45, 90 Monte Carlo method, 269, 304, 334 Moriety identity, 32 multidimensional Fourier transform, 12 Hilbert transform, 12 integral, 199, 300, 323 transformation, 97 361 multiple integral, 61, 83, 107, 199 multiplicities, 68 multiply-connected domains, 166 N V,330 natural boundary conditions, 15 natural cubic spline, 285 nc function, 154 nd function, 154 negative part of a function, 352 negative variation, 50 negligible set, 50 Newton-Cotes rule, 220, 291, 320, 321 NMS, 260 nodes, 245, 319 nomanclature, 351 non-uniform grid, 289, 292, 298 non-unique representation, 302 norm integral, 57 normal probability function, 173 normal vector, 165 norms, 49, 352 NP-hard, 139 ns function, 154 null rule, 245 number of zeros, 40 number theoretic method, 269, 312 number theoretic rule, 301, 312 numerical cancellation errors, 321 numerically stable, 245 0,351 0,351 o oblate spheroidal coordinates, 115 octahedral group, 318 one-dimensional rule, 340 one-form, 23, 67 one-half integral, 77 one-sided Fourier transforms, 12 362 open rule, 245, 321 order of a pole, 49 order, 245, 250 orthogonal coordinates, 25, 110 orthogonal functions, 49 orthogonal polynomials, 292 orthonormal functions, 311 oscillation of a function, 55 Ostrowski inequality, 207 p P function, 156, 351 Pade approximant, 253 panel rule, 245, 283 parabolic coordinates, 114 parabolic cylinder coordinates, 114 parabolic umbilic, 202 paraboloidal coordinates, 114 parallel methods, 315 parameter, 154 partial differential equations, 4, 5, 23,86,107 partial fractions, 146, 149, 150, 183 PASCAL programs, 220, 254 path independence, 45 path integrals, see integral: path path of descent, 230 Patterson rule, 261 Peano kernel, 246 Peano theorem, 246 Pearcey integral, 176 periodic extension, 301 permutations, 171 Perron integral, 56, 57 perturbation series, 71 Pfaffian differential equations, 5 phi function, 173 PLll programs, 254 planar triangle, 27 Planck's constant, 88 Poincare-Bertrand theorem, 61 point boundary, 222 branch, 46 caustic, 202 critical, 231 Halton, 336 point (continued) Hammersley, 336 saddle, 231 stationary, 222, 227 turning, 202 Poisson integral formula, 4, 20 summation formula, 39 polar coordinates, 115 polar moment, 26 pole, 49 Index polygamma function, 176, 349 polyhedral symmetry rule, 316 polyhedral symmetry, 316 poly logarithm, 177 polynomials Bernoulli, 35, 315 characteristic, 326 fourth order, 148 Hermite, 294, 297 interpolating, 319, 331 Laguerre, 294, 296 Legendre, 34, 291, 294 orthogonal, 292 third order, 148 trigonometric, 322 Tschebyscheff, 294 Pompeiu formula, 22 PORT, 260 positive part of a function, 352 positive rule, 245 positive variation, 50 positively oriented, 139 potential field, 45 prime number, 313, 334 principal-value integral, 61, 74, 75, 92, 155, 264, 274, 352 principle of canonical quantization, 90 principle of the argument, 40 probability, 30 product Gauss formula, 339 Gauss rule, 338 integral, 5 of gamma functions, 173 rule, 270, 323 Index product (continued) trapezoidal rule, 301, 302, 338 programs ALGOL, 220, 254 BASIC, 262 C,254 FORTRAN, 220, 254, 255, 260 in text, 238, 309 Macsyma, 239 PASCAL, 220, 254 PL/1, 254 REDUCE, 239 prolate spheroidal coordinates, 114 propagator, 86 psi function, 173 Q QD algorithm, 204 Quadpack, 244, 255, 280, 299 quantum mechanics, 88, 212 quasi:'Monte Carlo method, 310 quotient difference algorithm, 204 R Radeau rule, 291, 294 radical inverse, 335 Radon integral, 57 random variable, 30, 189, 305 rank, 301 rational functions, 145, 148, 183 Rayleigh-Ritz method, 15 recurrence relations, 106, 150, 204, 325 REDUCE, 117, 192, 239 refinement integral, 57 reflection group, 318 reformulation of an integral, 142, 160, 162 regression methods, 311 regularization, 66 relative error, 248 relatively prime, 313 renormalization group, 91 repeated integrals of Ko, 176 repeated integrals of the error function, 175 representation theorem, 21 residue theorem, 32, 129 Richardson extrapolation, 250 Riemann integrable, 55, 314 integral, 54, 55, 57, 68 lower integral, 55 sheet, 59 surface, 51 upper integral, 55 zeta function, 174 363 Riemann-Lebesgue lemma, 227 Riemann-Liouville fractional deriva- tive, 76 rigidity, 45 Risch theorem, 80 Romberg extrapolation, 261 Romberg integration, 250 rotational coordinates, 114, 115 oblate spheroidal coordinates, 115 parabolic coordinates, 114 prolate spheroidal coordinates, 114 Rouche theorem, 44 rule best, 303, 315, 333 Boole's, 320 Clenshaw-Curtis, 264, 265, 281 closed, 244, 321 composite Simpson's, 251 composite trapezoidal, 249, 250, 342 composite, 244, 284 compound, 244, 283, 288 corrected midterm, 288 corrected trapezoidal, 288 extended Simpson's, 284, 341 extended, 325 Fejer,271 finite-part, 74 five-point, 342 Gauss, 220, 261, 263, 265, 267, 268,271,294,297,298,322 Gauss-Hermite, 263, 267, 268, 271,297 Gauss-Jacobi, 297 364 rule (continued) Gauss-Laguerre, 263, 267, 268, 271, 297 Gauss-Legendre, 245, 263, 267, 268, 270, 271, 297 Gauss-Lobatto, 271 Gauss-Radeau, 271 Kronrod, 261, 263, 265, 267, 268, 297, 298 lattice, 300, 315 Lobatto, 291, 295 midpoint, 321 midterm, 288 Milne's, 320 modified trapezoidal, 322, 341 Newton-Cotes, 220, 291, 320, 321 null, 245 number theoretic, 301, 312 one-dimensional, 340 open, 245, 321 panel, 245, 283 Patterson, 261 polyhedral symmetry, 316 positive, 245 product Gauss, 338 product trapezoidal, 301, 302, 338 product, 270, 323 Radeau, 291, 294 Simpson's, 247, 251, 283, 320, 324, 341 tanh, 249 trapezoidal, 249, 250, 251, 288, 301, 302, 310, 320, 322, 331, 341, 342 Tschebyscheff, 291, 332 two-dimensional, 325, 340, 343, 344, 346, 347 Weddle's, 320 s O'-algebra, 49 saddle point, 231 Sag-Szekeres method, 270 sample paths, 189 sc function, 154 Index Schlafli's integral representation, 4, 23 Schrodinger equation, 86, 88 Scientific Desk, 260 SCRUNCH, 260 sd function, 154 second kind Abelian integral, 52 second kind elliptic integral, 172, 175 second mean value formula, 84, 85 second moment, 26, 189 . Selberg integral, 179 self-adjoint, 18 semi-integral, 77 set of measure zero, 50 Shanks transformation, 252 Shkarofsky functions, 179 Sievert integral, 177, 349 sigma-algebra, 49 signum function, 227, 352 SIMD computer, 316 simple closed curve, 27, 129 simple function, 53 simple pole, 130 simplex, 284 simply connected domain, 129 Simpson's 3/8-rule, 320 Simpson's rule, 247, 251, 283, 320, 324, 341 sine integral, 174, 177, 349 sine transform, 8, 265 singular point, 53, 92 singularities, 105 skewness, 30 Slobin transformation, 105 sn function, 154 Sobolev functions, 336 software libraries, 254 special forms, 97 special purpose integrators, 245 special techniques, 181 spherical coordinates, 115 spheroidal coordinates, 114, 115 splines, 263, 264, 265, 266, 269, 285 standard order, 67 stationary phase method, 198, 226 stationary point, 222, 227 Index steepest descent method, 198, 225, 229,230 step function, 351 stepping operator, 330 Stirling's approximation, 216, 224, 225 stochastic differential equation, 190 integral, 56, 186 integration, 186 Stokes formula, 69, 140 Stokes theorem, 166 stratified and symmetrized scheme, 311 stratified sampling, 311 Stratonovich integral, 189, 352 strong variational integral, 57 Struve function, 177, 349 Sturm-Liouville problem, 11 summable function, 53 summation, 31, 34, 39, 44, 184 surface integral, 164 swallowtail, 202 symbolic methods, 329 symmetries, 155 symmetry group, 316 T tables of integrals, 190, 348 tables of methods, 337 Tables, 8, 9, 31, 77, 95, 112, 113, 146, 152, 206, 216, 241, 251, 252,253,290,294,298,320, 321, 327, 328 tangent vector, 165 tanh rule, 249 taxonomy, 258, 260 Taylor series, 204, 220 test integral, 117, 272, 303, 304, 311 testing quadrature rules, 272 tetrahedral group, 318 theorem, 52, 65, 79, 93, 116, 130, 248, 294 Cauchy, 22, 36, 129, 163, 231 Chartier, 66 convolution, 141 theorem (continued) Darboux,55 Davenport, 80 divergence, 167 expansion, 91, 196 Feynman-Kac, 91 Fubini, 61, 64 Gauss, 167 365 Green's representation, 21 Green's, 27, 163, 165, 169 Hardy-Littlewood supremum, 206 Helmholtz, 23 Liouville, 52 mean value, 83, 246 Peano, 246 Poincare-Bertrand, 61 representation, 21 residue, 32, 129 Risch, 80 Rouche,44 Stokes, 166 theta function, 39 third kind Abelian integral, 52 third kind elliptic integral, 178 third order polynomials, 148 three-dimensional transformation, 110, 113 topology, 52, 140 torsional rigidity, 45 torus, 26 total curvature, 287 total measure, 50 total variation, 50 transform finite Fourier sine, 8 Fourier cosine, 9, 192 Fourier sine, 9, 192 Fourier, 9, 12, 142, 192, 304, 351 Hankel, 8, 9, 11 Hartley, 10 Hermite, 12 Hilbert, 10, 12, 13, 264, 265, 351 integral, 6 inverse Laplace, 138 K,lO Kontorovich-Lebedev, 10 366 transform (continued) Laguerre, 12 Laplace, 10, 138, 141, 198, 240, 351 Legendre, 9 Mehler-Fock, 10 Meijer, 11 Mellin, 11, 141, 197, 198, 351 one-sided, 12 sine, 8, 265 to finite interval, 95 Watson, 35, 44 Weierstrass, 11 transformation Aitken, 252 fractional, 156 multidimensional, 97 Shanks, 252 Slobin, 105 three-dimensional, 110, 113 two-dimensional, 110, 112 unnamed, 105 Wo~tenholme, 105 transport integral, 349 transputers, 316 trapezoidal rule, 249, 250, 251, 288, 301, 302, 310, 320, 322, 331, 341, 342 traverse in the positive sense, 139 trigonometric polynomials, 322 trilogarithm, 174 truncating an infinite interval, 275 Tschebyschef£ inequality, 205, 207 polynomia~, 281, 294 rule, 291, 332 turning point, 202 two-dimensional rule, 325, 340, 343, 344, 346, 347 two-dimensional transformation, 110, 112 two-form, 67 ULP, 280 umbilic, 202 u unbiased estimator, 305 uncertainty principle, 212 uniform approximation, 226 uniform convergence, 53 Index uniform spaced nodes, 319 uniformly distributed sequence, 314 unit cube integration formula, 300 unit tangent, 139 univalent function, 22 unnamed transformation, 105 upper limit, 49 uses of integration, 45 v vaf[a,bJ, 50 variance, 30, 31, 306 variational equations, 15 variational integral, 57 variational principle, 15 Villat integral formula, 21 Voigt function, 177 w Watson lemma, 197, 198, 225, 238 Watson transform, 35, 44 wave function, 88 Weber formula, 11 Weber function, 177 Weddle's rule, 320 wedge product, 67 Weierstrass transform, 11 Weierstrass l' function, 156, 351 weight function, 245, 292,317 weights, 245 Weyl group, 50 integral formula, 50 integral, 77 white noise, 186 Wiener process, 189 winding number, 22, 44 Wirtinger inequality, 207 WKB,91 Wo~tenholme transformation, 105 work, 45 Wronskian, 49, 182 Wu's integral, 179 Index y ,Y9ung inequality, 208 Young integral, 57 z z-fraction, 204 zero-form, 67 zeros, 40 zeta function, 37abscissae, see nodes 367