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500 INTEGRALS OF ELEMENTARY AND SPECIAL FUNCTIONS ⊕ Francis J. O’Brien, Jr. GAMMA FUNCTION 500 Integrals of Elementary and Special Functions Library of Congress Cataloging-in-Publication Data O’Brien, Francis Joseph, Jr. 500 Integrals of Elementary and Special Functions p. cm. Includes bibliographical references and index. ISBN: 1-4392-1981-8 ISBN-13: 978-1439219812 1. Mathematics. 2. Differential and integral calculus. 3. Proofs and derivations. Library of Congress Catalog Card Number: Copyright © 2008 by Francis Joseph O’Brien, Jr. All rights reserved. No part of this publication may be reproduced, stored in retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the written permission of the author, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act. Printed in the United States of America 500 Integrals of Elementary and Special Functions Francis J. O’Brien, Jr. Naval Undersea Warfare Center Naval Sea Systems Command Newport, Rhode Island For my daughter, Miss Lily-Rae O’Brien ⊗ In gratitude to The Department of Defense PREFACE ▲ ● ▼ This book is a listing of solved formulas for about 500 integrals, sums, series, and products. The intended audience is students and practitioners in the fields of mathematics, pure and applied science, and engineering. The fundamental purpose of the book is to present a modest listing of formulas with worked out solutions structured on the widely used desk reference, Gradshteyn & Ryzhik’s Table of Integrals, Series, and Products (2007). We provide “new” formulas in a limited number of areas—exponential and logarithmic functions and selected special functions with emphasis on the gamma and related functions. The level of difficulty of the material in this book ranges from easy to moderately difficult, and covers selected topics in first year calculus through advanced calculus. The focus is on understanding how to evaluate the unsolved indefinite and definite integrals covered in the text. Throughout the computer is used as an answer- checker rather than the primary evaluator or prover. In this age of computer technology, one may wonder why books such as Table of Integrals, Series, and Products are even necessary. One of the many sophisticated computer programs used in the scientific, engineering and mathematical fields is Mathematica (http://www.wolfram.com/). This computer algebra system can display solutions to derivatives & integrals in a matter of seconds. It has been asserted that Mathematica can solve approximately 80% of the formulas in Table of Integrals, Series, and Products. Yet the Gradshteyn & Ryzhik book is still widely purchased, and is now available in the 7th edition. Computer calculators such as Mathematica can solve the majority of integrals in Gradshteyn & Ryzhik and this book. In addition, comprehensive mathematics websites such as The Wolfram Functions Site at http://functions.wolfram.com/ provide thousands of formulas, yet very little exists in the way of proofs, derivations, and justifications. The only significant effort I am aware of is a series of journal articles by Prof. V.H. Moll of Tulane University. He is in the process of deriving and verifying the formulas in Gradshteyn & Ryzhik. See the website, http://www.math.tulane.edu/~vhm/. Recent papers that amplify the derivations given in this book may be found at the docstoc.com website. They derive relations for the Pochhammer Symbol, double factorials, and gamma transformations, identities and special values. See http://www.docstoc.com/profile/waabu. Given the significant decline in enrollments in mathematics, science and engineering in the United States, it is important to supplement major mathematical reference works such as the Gradshteyn & Ryzhik handbook. Other writers may be motivated to assist in this effort. Structure of the Book As mentioned, 500 Integrals is based on the architecture of Gradshteyn & Ryzhik’s Table of Integrals, Series, and Products, 7th Edition (GR for short). In essence, the book contains previously published formulas in GR as well as candidate formulas for the next edition. The material is presented in the order of appearance in GR using their section nomenclature and formula numbering system. This format is used instead of conventional chapter and sub-chapter headings. The topics covered reflect the author’s own interests in theoretical fundamentals and potential applications in education, science and engineering. The first section, Mathematical and Graphical Summary of Selected Elementary and Special Functions, contains an abbreviated tutorial on all the notations, definitions and properties of the functions used in this book. This material is based on various standard sources including GR (Sections 0. and 1.) and new results. Key references are provided for additional background reading and study. The Math. Summary should be consulted for guidance in the solution of the formulas presented. The Notations section also contains useful information. The first major section of results consists of Indefinite Integrals of selected Elementary Functions (Sections 2.3 & 2.7 in GR). This is followed by Definite Integrals of Elementary Functions (Sections 3.3–3.4 & 4.2.–4.4 in GR), and lastly, selected Special Functions (Sections 8.2 & 8.3 in GR). A number of integrals are left as exercises. In addition, certain related integrals are solved “from scratch” so that readers can search for a simplified solution based on earlier material (or their own creation) while other calculations are stated as “similar to above” without solution, inviting readers to supply details. Speed of identification of a formula is a major concern for many users of mathematical reference handbooks. At the rear of the book is an Index of Formulas which lists all of the formulas presented in the book, arranged by section and page numbers corresponding to the Table of Contents. An Index of Symbols, Functions and Concepts may also be found in the rear of the book. In the author’s opinion the most useful formulas presented in this volume are those involving three-parameter algebraic-exponential functions expressed in terms of gamma functions (see Mathematical Summary [Incomplete Gamma Function—Indefinite Integrals], and Sections 2.32, 3.326, 3.381, 3.462). I view them as “reduction” formulas which help solve and create a number of useful integrals. Some formulas developed here have been used in the derivation of new elementary probability models; see “Summary of Four Generalized Exponential Models (GEM) For Continuous Probability Distributions,” Jan. 18, 2008, arXiv:0801.2941v2 [math.GM]. Notes on the entries in the book— • All integrals in this book omit the constant of integration • In most cases, derived solutions may be verified by ordinary differentiation or by differentiation under the integral sign using Leibnitz’s rules provided in Mathematical Summary o Computer verification is suggested when this is possible • Variables are considered to be real quantities, unless otherwise indicated 1 • A square root x or x 2 or π π is taken to be positive unless otherwise specified • Logarithms are natural logarithms, denoted ln x (base e), unless otherwise indicated • Errata for the 6th and 7th editions of Gradshteyn and Rhyzik can be obtained online at http://www.mathtable.com/gr/ • “GR” or “G & R 7e” refer to Gradshteyn and Rhyzik’s Table of Integrals, Series and Products, 7th edition, unless otherwise indicated • math.com refers to the public Internet website, http://www.quickmath.com/, which calculates indefinite and definite one-dimensional integrals, and performs other services including derivatives, partial fraction expansion, graphical plotting, matrix inversion, etc. o math.com displays the solutions for the lower incomplete gamma function as Γ(a ) − Γ(a, x ) vice γ (a, x ) • The Mathematica Integrator calculates indefinite integrals at http://integrals.wolfram.com/index.jsp o Sometimes math.com unable but Mathematica able to do an indefinite integral o Sometimes an indefinite integral can be calculated while the definite one cannot o Sometimes computer provides a solution at higher level of complexity than dγ (a, x ) needed; e.g., the derivative, (see Section 8.356) da o Sometimes computer returns a “different” solution compared to paper and pencil answer Computer always provides only a single solution when multiple ln (a + bx )dx e ax − 1 answers are possible; e.g., ∫ or ∫ ax dx (see Section x e +1 2.32) • “verified on math.com”, “verified on mathematica” means author’s answer confirmed by computer when possible o Post-solution simplification is often required for final form o Logarithmic indefinite integrals: computer calculation usually returns an incomplete gamma function solution vice desired reduction formula by integration by parts/change of variable or combination solution • “not verifiable on math.com” & “not verifiable on mathematica” means computer unable to calculate answer directly from the input, providing evidence that the human calculator is still useful in this electronic age. ——— I would appreciate communications regarding misprints and any helpful suggestions for improvement of presentation of the material. I can be reached by e-mail at either francis.j.obrien@navy.mil —or— francis28@cox.net. Acknowledgements The author would like to acknowledge and thank all those who have funded, assisted, and encouraged him over the years. These include the Office of Naval Research, the Base Commander, my research colleagues in the USW Combat Systems Department, and the attorneys and paralegals in Office of Patent Counsel of the Naval Undersea Warfare Center, Newport, Rhode Island. Professor Alan Jeffrey of the University of Newcastle Upon Tyne (England), Editor of Table of Integrals, Series and Products, has been most helpful in his encouragement. I also thank his co-editor, Dr. Daniel Zwillinger of Rensselaer Polytechnic Institute, for correspondence regarding errata and related matters. I also want to acknowledge the assistance of Aimee Ross, a 3rd year mathematics major at University of Massachusetts—Dartmouth, who read the early sections of the manuscript, and provided feedback on the level of difficulty and the clarity of the material. ╬ Francis J. O’Brien, Jr. Newport, Rhode Island October 12, 2008 TABLE OF CONTENTS Section in Gradshteyn and Ryzhik Page MATHEMATICAL AND GRAPHICAL SUMMARY OF SELECTED ELEMENTARY AND SPECIAL FUNCTIONS Exponential Functions 2 Logarithmic Functions 4 Gamma Function 6 Incomplete Gamma Functions 8 Probability Integral or Error Function (Erf) and Imaginary Error Function (Erfi) 10 Exponential-Integral Function 12 Logarithm-Integral Function 14 Euler’s Constant 16 Catalan’s Constant 17 Partial Fractions 18 Miscellaneous (Completing the Square, Finite Binomial Expansions, Double 20 Factorial, Natural Number N, Differentiation Under the Integral Sign) NOTATIONS 22 INTRODUCTION 0.1 Finite Sums 24 0.11 Progressions 25 0.111 Arithmetic progressions 25 0.112 Geometric progressions 26 0.12 Sums of powers of natural numbers 26 INDEFINITE INTEGRALS OF ELEMENTARY FUNCTIONS 2.3 The Exponential Function 28 n 2.31 Forms containing e ax , e ax 30 2.312 31 2.32 The exponential combined with rational functions of x 35 2.7 Logarithms and Inverse and Hyperbolic Functions 61 2.71 The logarithm 62 2.72-2.73 The logarithm and combinations of logarithms and algebraic functions 62 DEFINITE INTEGRALS OF ELEMENTARY FUNCTIONS 3.3–3.4 Exponential Functions 79 3.31 Exponential functions 80 3.310 81 3.311 82 3.32–3.34 Exponentials of more complicated arguments 84 3.321 85 3.322 89 3.323 90 3.324 94 3.326 96 3.327–3.334 Exponentials of exponentials 98 3.327 99 3.328 100 3.331 101 3.35 Combinations of exponentials and rational functions 103 3.351 104 3.353 105 3.36–3.37 Combinations of exponentials and algebraic functions 106 3.361 107 3.362 108 3.363 109 3.371 116 3.38–3.39 Combinations of exponentials and arbitrary powers 120 3.381 121 3.382 125 3.41–3.44 Combinations of rational functions of powers and exponentials 126 3.427 127 3.434 128 3.46–3.48 Combinations of exponentials of more complicated arguments and powers 129 3.461 130 3.462 134 3.464 148 3.471 151 3.473 160 4.2–4.4 Logarithmic Functions 161 4.21 Logarithmic functions 162 4.211 163 4.212 168 4.215 174 4.22 Logarithms of more complicated arguments 177 4.229 178 4.24 Combinations of logarithms and algebraic functions 179 4.241 180 4.26–4.27 Combinations involving powers of the logarithm and other powers 181 4.269 182 4.272 185 4.274 190 4.28 Combinations of rational functions of ln x and powers 192 4.281 193 4.283 194 4.29-4.32 Combinations of logarithmic functions of more complicated arguments 195 and powers 4.326 196 4.33–4.34 Combinations of logarithms and exponentials 197 4.331 198 4.337 199 SPECIAL FUNCTIONS 8.2 The Exponential Integral Function and Functions Generated by It 202 8.21 The exponential integral function Ei(x) 203 8.212 204 8.24 The logarithm integral li(x) 210 8.240 211 8.241 Integral representations 212 8.25 The probability integral, the Fresnel Integrals Φ ( x ), S ( x), C ( x) , the error 214 function erf(x), and the complementary error function erfc(x) 8.250 Definition 215 8.252 Integral representations 216 8.3 Euler’s Integrals of the First and Second Kinds and Functions Generated by Them 221 8.31 The gamma function (Euler’s integral of the second kind): Γ(z) 222 8.313 223 8.33 Functional relations involving the gamma function 224 8.331 225 8.334 232 8.335 238 8.339 Particular Values: For n a natural number 239 8.35 The Incomplete Gamma Function 253 8.350 Definition 254 8.351 255 8.352 Special cases 256 8.353 Integral representations 261 8.356 Functional relations 263 8.359 Relationships with other functions 265 8.36 The Psi function ψ ( x ) 273 8.367 Euler’s constant: Integral representations 274 References 276 Index 278 Index of Formulas 284 LIST OF ILLUSTRATIONS Figure 1. Exponential Function (Growth) 2 Figure 2. Exponential Function (Decay) 4 Figure 3. Natural Logarithm Function 4 Figure 4. Gamma Function 6 Figure 5. Error Function 10 Figure 6. Imaginary Error Function 10 Figure 7. Exponential-integral Function 12 Figure 8. Logarithm-integral Function 14 Figure 9. Euler’s Constant 16 Figure 10. Catalan’s Constant 17 MATHEMATICAL & GRAPHICAL SUMMARY OF SELECTED ELEMENTARY AND SPECIAL FUNCTIONS GAMMA FUNCTION 2 EXPONENTIAL FUNCTIONS e− x x e x x Figure 1. Exponential Function (Growth) Figure 2. Exponential Function (Decay) Notation: e x = exp(x ) = ln −1 (x ) Definitions & Laws of Exponents: a 0 = 1, a ≠ 0 a x = e x ln a x = e ln x 1 ( a) p [( )] q p a = a = a = q p q p q a x+ y = a x a y a − x = e − x ln a 1 1 ( ) =e x n = e ln x n n ln x = [(− a ) ] = [(− 1) ] [a ] 1 (− a ) q = (e ) = e p q q q ax = ex n n ax ln a p p p a ln x n a n a ln x a x− y = xn ay a −x = e−x n n ln a 1 (a ) (− 1) = i (imaginary) 2 x y =a xy (ab )x = a xb x 64 4 4 4 74 4 4 4 8 Let a be e for exp. Limits: lim e x = 1 lim e x = +∞ ex e − ax − e − bx x →0 x → +∞ lim a = +∞ lim =b−a x → +∞ x x →0 x −x lim e =1 lim e − x = 0 x →0 x → +∞ e−x xa e −ax − e −bx lim a = lim x = 0 lim =0 x → +∞ x x → +∞ e x →∞ x 3 Derivatives: a, e, n are assumed constants except where noted du v de v ln u n n dv ln u du dv da x de x ln a de x = = uv = vu v −1 + u v ln u = = nx n −1a x ln a = ex n dx dx dx dx dx dx dx dx da u de u ln a du x x ln a de − x = = a u ln a da = de = a x ln a = −e − x dx dx dx dx dx dx u de ax de = eu du d x 1 = ae ax − x ln a dx dx dx a = de =− x ln a de − ax 644444444 44444444 7 8 dx dx a = −ae − ax chain rule n n dx da x de x ln a = = a x x n ln x ln a n dn dn [n not a constant ] Indefinite Integrals: −x −x ∫ e dx = e ∫ e dx = −e x x e ax − ax e − ax ∫ e dx = ∫ e dx = − ax a a ∫x ± m ± ax n • See Section 2.32 for generalized indefinite exponential integrals of form, e dx, expressed in terms of incomplete gamma functions. Summarized in INCOMPLETE GAMMA FUNCTIONS. • These exponential integrals are used often in the evaluation of definite integrals of elementary and some special functions. References • Bers, Calculus. Ch. 6, pp. 367 ff. & 375 ff ; Ch. 7, pp. 453 ff.; Ch. 8, pp. 547 ff. • Carr, 1970 • Dwight, Table of Integrals, 1961 • Spiegel, Chaps. 7 & 20 • Table of Integrals, Series, and Products. Sects. 0.245, 1.2 (“The Exponential Function”), 2.01 (“The basic integrals”) 4 LOGARITHMIC FUNCTIONS (Base e) 1 ln x x 1 x x 1 1 Figure 3. Natural Logarithm Functions, ln x & ,& ln x x Notation: log x = ln x (natural logarithm, base e) log m x = (log x ) m x dt Definition: ∫ 1 t = log x, x > 0 5 Properties: log ( xy ) = log x + log y log x n = n log x ln e x = x x log = log x − log y log 1 = − log x ln e − x = − x y x ln e ax = ax ⎛ b⎞ ( ) a ln x + b ln y = ln x a y b = a ln⎜ xy a ⎟ ⎜ ⎟ ln e − ax = − ax ⎝ ⎠ ⎛ ⎞ ⎛ xa ⎞ ⎜ x ⎟ a ln x − b ln y = ln⎜ b ⎟ = a ln⎜ b ⎟ ⎜y ⎟ ⎝ ⎠ ⎜ ya ⎟ ⎝ ⎠ Limits: lim log x = +∞ xa log x ex ln(1) = 0 x → +∞ lim = +∞ lim =0 lim = +∞ lim log x = −∞ x → +∞ log x x → +∞ e x x → +∞ log x ln (e ) = 1 x →0 + log x log x lim x = −∞ x e ⎛1⎞ lim a = 0 x →0 + e lim =0 ln⎜ ⎟ = −1 x → +∞ x x →0 + log x ⎝e⎠ Derivatives: d ln x 1 d ln m (a + bx ) mb ln m−1 (a + bx ) d ln u 1 du = (chain rule) = ,x ≠ 0 = dx x dx a + bx dx u dx Indefinite Integrals: dx ∫ x = ln x NOTE: see Section 2.32 for special logarithmic function—the dilogarithm, ln (1 − x ) Li 2 ( x ) = − ∫ dx x ∞ xk Li 2 ( x ) = ∑ 2 , x < 1 k =1 k References • Bers, Calculus. Ch. 6, pp. 357 ff.; Ch. 7, pp. 454 ff.; Ch. 8, pp. 547 ff. • Carr, 1970 • Dwight, Table of Integrals, 1961 • Spiegel, Chaps. 7 & 20 • Table of Integrals, Series, and Products. Sects. 1.5 (“The logarithm”), 2.01 (“The basic integrals”) • Wolfram Functions website, http://functions.wolfram.com/GammaBetaErf/ 6 GAMMA FUNCTION Figure 4. Gamma Function, Γ( x ) Definition: ∞ Γ(z ) = ∫ t z −1e −t dt , real z > 0 [Formula 8.310.1] 0 z −1 ⎛1⎞ 1 Γ(z ) = ∫ ln⎜ ⎟ dt , real z > 0 [Formula 8.312.1] 0 ⎝t ⎠ Integral, Product, and Series Representations: ∞ Γ( x ) = z x ∫t x −1 − zt e dt , real z , x > 0 [Formula 8.312.2] 0 p! p x px Γ( x ) = lim = lim , real x > 0 [Artin, Formula 2.7] ⎛ x⎞ p →∞ p p →∞ p x∏ ( x + k ) x ∏ ⎜1 + ⎟ k =1 k =1 ⎝ k⎠ x ⎛ x⎞ ⎛ 1⎞ exp⎜ ⎟ ∞ ⎜ 1+ ⎟ exp(− γx ) ∞ ⎝k⎠ = 1 ⎝ k ⎠ = lim n x n Γ(x ) = k x ∏ x x∏ x n →∞ x ∏ k + x, real x > 0 [Formula 8.322] k =1 1+ k =1 1+ k =1 k k n −1 ⎛ k⎞ ∏ Γ⎜ z + n ⎟ & Γ(nz ) & Γ(x ) [Section 8.335] 1 k =0 ⎝ ⎠ ⎧ ∞ ⎡ ⎛ x ⎞ x ⎤⎫ log Γ( x ) = − ⎨ln x + γx + ∑ ⎢ln⎜1 + ⎟ − ⎥ ⎬, real x > 0 [Artin, Formula 2.9] ⎩ k =1 ⎣ ⎝ k ⎠ k ⎦⎭ Properties: Γ( x ) = (x − 1)! Γ(1 − x ) lim Γ( x ) = +∞ lim Γ( x ) = +∞ Γ(− x ) = − , x not a natural x →0 + x → +∞ x num. Γ( x ) ≠ 0 Γ( x + 1) = xΓ( x ) lim Γ( x ) = −∞ lim Γ( x ) is indeterminate x →0 − x → −∞ 7 Derivatives: dΓ( x ) ∞ ∞ = Γ ′( x ) = Γ′(1) = ∫ e −t ln tdt d x −1 −t dx ∫ t e dt = dx 0 0 ∞ ∂ ∞ = −γ (Euler' s constant ) ∫ ∂x e ( x −1) ln t e −t dt = ∫ t x −1e −t ln tdt = Γ( x )Ψ ( x ) 0 0 = Ψ (1) (Formula 8.366.1) ∞ d ln Γ(x ) Γ′(x ) Γ (n ) ( x ) = ∫ t x −1e −t (ln t ) dt n • = = Ψ (x ) dx Γ( x ) 0 (digamma or Psi function), Formula 8.330 d ⎛ 1 ⎞ 1 dΓ( x ) Ψ (x ) • ⎜ ⎜ Γ( x ) ⎟ = − ⎟ =− Γ( x ) dx ⎝ ⎠ [Γ(x )] dx 2 Relation to Incomplete Gamma Functions, γ (a, x ) & Γ(a, x ) : Γ(a ) = γ (a, x ) + Γ(a, x ) ∞ x ∞ ∫t = ∫ t e dt + ∫ t a −1e −t dt a −1 −t a −1 −t e dt [Formula 8.356.3] 4 3 1 24 0 4 3 x4 3 1 24 1 24 0 GAMMA 1444 444 2 UPPER3 LOWER COMPLETE INCOMPLETE GAMMA • NOTE: Relation is additive; e.g., γ (a, x ) = Γ(a ) − Γ(a, x ) Trigonometric Functions: • Bers, Ch. 6. • Dwight, Table of Integrals, 1961 • Spiegel, Ch. 5. • Table of Integrals, Series, and Products, Sects. 1.3-1-4, 8.31-8.35 & Index NOTE on Γ(x ) : Emil Artin’s brief 1964 book—The Gamma Function—provides a complete statement for real variables of this special transcendental function called the complete gamma function, and derives the fundamental mathematical properties originated in the classical 18th & 19th century works of Euler, Gauss, Legendre, Riemann, Stirling, Weierstrass, and others. See Artin’s book for other definitions, theorems, and derivations of Γ(x ) not given in this elementary book. Whittaker & Watson is also recommended for derivations. References • Abramowitz & Stegun, Ch. 6. • Artin, 1964 • Bers, Calculus , Chaps. 4 & 6, pp. 402-3 • Carr, 1970 • Moll, http://www.math.tulane.edu/~vhm/Table.html • Table of Integrals, Series, and Products, Sects 1.3-1.4 and Sects. 8.31-8.35 & Index • Whittaker & Watson, 1934 • Wolfram Functions website, http://functions.wolfram.com/GammaBetaErf/ • O’Brien, http://www.docstoc.com/docs/5473276/Pochhammer-Symbol-Selected-Proofs • O’Brien, http://www.docstoc.com/docs/5836783/Selected-Transformations-Identities--and-Special- Values--for-the-Gamma-Function 8 INCOMPLETE GAMMA FUNCTIONS 1. Lower Incomplete Gamma Function Definition: γ (a, x ) = ∫ t a −1e −t dt , real a > 0 [Formula 8.350.1] x 0 NOTE: • See Sect. 8.352 for integer special cases, γ (n, x ) , etc. 1 Integral Representation: γ (a, x ) = x a ∫ t a −1e − xt dt , real a,x > 0 [Sections 3.331 & 8.353] 0 Properties: • γ (a, x ) = Γ(a ) − Γ(a, x ) γ (a + 1, x ) + x a e − x • γ (a, x ) = , a ≠ 0 [Formula 8.356.1] a • γ( 1 , x ) = π Φ(x ) , 2 [ Φ(x ) = erf(x), error function, 8.250.1] ( ) 2 • γ ( , x) = π Φ x 1 2 • γ (a,0) = 0 [Formula 8.350.5] • γ (a, ∞ ) = Γ(a ) dγ (a, x ) dΓ(a, x ) a −1 − x • =− = x e [Form. 8.356.4] dx dx dγ (a, u ) dΓ(a, u ) dγ (a, u ) du • =− = (chain rule) dx dx du dx dγ (a, x ) 1 • = γ (a, x ) ln x + x a ∫ t a −1e − xt ln tdt [Sect. 8.356] da 0 2. Upper Incomplete Gamma Function ∞ Definition: Γ(a, x ) = ∫ t a −1e −t dt [Formula 8.350.2] x NOTE: • See Sect. 8.352 for integer special cases, Γ(n, x ) , etc. ∞ Integral Representation: Γ(a, x ) = x a ∫t a −1 − xt e dt , real a,x > 0 [Sections 3.331 & 8.353] 1 Properties: • Γ(a, x ) = Γ(a ) − γ (a, x ) Γ(a + 1, x ) − x a e − x • Γ(a, x ) = , a ≠ 0 [Formula 8.356.2] a • Γ(1 , x 2 ) = π − π Φ ( x ) [ Φ(x ) = erf(x), error function, 8.250.1] ( x) 2 • Γ( 1 , x ) = π − π Φ 2 9 • Γ(a,0 ) = Γ(a )[Formula 8.350.3] • Γ(a, ∞ ) = 0 [Formula 8.350.4] dΓ(a, x ) dΓ(a, u ) • & (see 1. above) dx dx dΓ(a, x ) ∞ • = Γ(a, x ) ln x + x ∫ t a −1e − xt ln tdt a [Sect. 8.356] da 1 d ⎡ γ (a, x ) ⎤ γ (a, x ) x a 1 = [ln x − Ψ (a )] Γ(a ) Γ(a ) ∫ + t a −1e − xt ln tdt , real a,x > 0 [Section 8.356] da ⎢ Γ(a ) ⎥ ⎣ ⎦ 0 NOTE: • See exponential-integral function, Ei( x) , for relation to Γ(a, x ) Indefinite Integrals: (− 1)1−γ Γ(γ ,−ax n ) = (− 1)1−γ ∞ m +1 ∫ x e dx = ∫t γ −1 −t e dt , γ = n m ax γ γ na na − ax n n ( Γ γ , βx n 1 ) ∞ m +1 ∫ x e dx = − ∫t m − βx γ −1 −t =− γ e dt , γ = n γ [Formula 2.33.10] nβ nβ βx n n e ax n (− 1) z +1 ( a z Γ − z ,− ax n ) (− 1)z +1 a z ∞ 1 m −1 ∫ x m dx = n = n ∫ t z +1 e −t dt , z= n [Formula 2.325.6] − ax n e − βx n β z Γ − z , βx n ( βz ) ∞ e −t m −1 ∫ xm dx = − n =− n ∫n t z +1 dt , z= n [Formula 2.33.19] βx NOTE: • These elementary algebraic-exponential formulas are used extensively in this book to derive the definite integral expressions (see Formulas—3.326.2, 3.462.19, 3.381.8-3.381.10 & others in those sections—for negative exponential forms), to prove existing formulas, and to create new integrals for a useful class of transcendental functions and special functions. ⎛ 1 ⎞ ⎛1 ⎞ See Section 8.359 for special forms, Γ⎜ ± ,± x⎟ &γ ⎜ , ± x⎟ ⎝ 2 ⎠ ⎝2 ⎠ References • Abramowitz & Stegun, Ch. 6 • Boas, Ch. 11 • Moll, http://www.math.tulane.edu/~vhm/Table.html • Table of Integrals, Series, and Products, Use of the Tables (“The factorial gamma”), Sect. 8.35 • Wolfram Functions website, http://functions.wolfram.com/GammaBetaErf/Gamma2/ 10 PROBABILITY INTEGRAL OR ERROR FUNCTION (ERF) AND IMAGINARY ERROR FUNCTION (ERFI) x x Figure 5. Error Function (erf) Figure 6. Imaginary Error Function (erfi) Definition: x Φ (x ) = erf (x ) = 2 π ∫ −t 2 e dt [Formula 8.250.1] 0 ∞ erfc(x ) = 1 − Φ (x ) = 2 ∫ e dt [Complimentary Error Function, Formula 8.250.4] −t 2 π x Integral Representation: x2 e −t Φ (x ) = erf (x ) = 1 π ∫ t dt [Formula 8.251.1] 0 Properties: Φ (− x ) = −Φ ( x ) Φ (0 ) = 0 Sect. 8.359 Φ (± ∞ ) = ±1 11 Relation to Incomplete Gamma Function: ⎛1 ⎞ ⎛1 ⎞ γ ⎜ , x2 ⎟ γ ⎜ , x⎟ Φ(x ) = ⎝ 2 ⎠ Φ ( ) x = ⎝ 2 π ⎠ π Imaginary Error Function, erfi(z): erf(iz ) erfi( z ) = i (iz )2 e −t z iz 2 2 1 ∫e ∫e ∫ −t 2 erfi( z ) = t2 dt = dt = dt π 0 i π 0 i π 0 t Relation to Incomplete Gamma Function: ⎛1 ⎞ γ ⎜ ,− z 2 ⎟ erfi(z ) = ⎝ ⎠ 2 i π Indefinite Integrals: π ∫e −t 2 dt = erf (t ) 2 π ∫e t2 dt = erfi(t ) 2 References • Abramowitz & Stegun, Ch. 7. • Boas, Ch. 11. • Dwight, Table of Integrals, 1961 • Moll, http://www.math.tulane.edu/~vhm/Table.html • Spiegel, Ch. 35 • Table of Integrals, Series, and Products, Use of the Tables (“Exponential and related functions”), and Sect. 8.250. • Wolfram Functions website, http://functions.wolfram.com/GammaBetaErf/Erf2/ 12 EXPONENTIAL-INTEGRAL FUNCTION Ei ( x ) x e−x Figure 7. Exponential-integral Function, Ei(x), − x Definition: Formulas 8.211.1 & 8.211.2 ∞ e −t et x ⎡−ε e −t ∞ −t ⎤ Ei( x ) = − ∫ dt = ∫ dt , x < 0 Ei( x ) = − lim ⎢ ∫ dt + ∫ e dt ⎥, x > 0 (Cauchy Principal Value PV) t t ε → +0 −x −∞ ⎣− x t ε t ⎦ Properties: Ei(+ ∞ ) = +∞ Ei(− ∞ ) = 0 Ei(0 ) is not defined Relation to Logarithm-Integral Function: ( ) Ei( x ) = li e x , x < 0 ( ) Ei(ax ) = li e ax , x < 0, a ≠ 0 13 Relation to Incomplete Gamma Function: ∞ ∞ e −t e −t Ei( x ) = − ∫ dt = −Γ(0,− x ) Ei(− x ) = − ∫ dt = −Γ(0, x ) −x t x t ∞ ∞ e −t e −t − Ei( x ) = ∫ dt = Γ(0,− x ) − Ei(− x ) = ∫ dt = Γ(0, x ) Form. 8.359.1 −x t x t Indefinite Integrals: e±s ∫ s ds = Ei(± s ) References • Abramowitz & Stegun, Ch. 5. • Moll, http://www.math.tulane.edu/~vhm/Table.html • Table of Integrals, Series, and Products, Use of the Tables (“Exponential and related functions”), and Sect. 8.21 & Sects. 3.04-3.05 (“Improper Integrals”; “Principal Values”) • Weisstein, Eric W. "Exponential Integral." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ExponentialIntegral.html • Wolfram Functions website, http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/ 14 LOGARITHM-INTEGRAL FUNCTION Figure 8. Logarithm-integral Function, li(x) Definition: Formulas 8.240.1 & 8.240.2 x ⎡1−ε dt x dt ⎤ li(x ) = ∫ dt , x <1 li( x ) = lim ⎢ ∫ + ∫ ⎥ = Ei(ln x ), x > 1 (Cauchy Principal Value PV) ln t ε →0 0 ⎣ 0 ln t 1+ε ln t ⎦ Properties: li(0 ) = 0 li(+ ∞ ) = +∞ li(1) not defined Relation to Exponential-Integral Function: li( x ) = Ei(ln x ), x < 1 ( ) li x a = Ei(a ln x ), x < 1, a ≠ 0 Relation to Incomplete Gamma Function: ⎛ 1⎞ − li( x ) = Γ⎜ 0, ln ⎟ = Γ(0,− ln x ) [Formula 8.359.2] ⎝ x⎠ 15 Indefinite Integrals: ∫ ln s = li(s ) ds References • Abramowitz & Stegun, Ch. 5. • Moll, http://www.math.tulane.edu/~vhm/Table.html • Table of Integrals, Series, and Products, Use of the Tables (“Exponential and related functions”), Sects. 8.24 & Sects. 3.04-3.05 (“Improper Integrals”; “Principal Values”) • Weisstein, Eric W. "Logarithmic Integral." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LogarithmicIntegral.html 16 EULER’S CONSTANT (C or γ ) − e − x ln(x) x Figure 9. Euler’s Constant Definition: ⎡ n−1 1 ⎤ ⎛ 1 1 1 ⎞ γ = lim ⎢∑ − ln n⎥ = lim⎜1 + + + K + − ln n ⎟ [Formula 8.367.1] n →∞ ⎣ k =1 k ⎦ n →∞ ⎝ 2 3 n ⎠ Integral Representations: ∞ ⎛1⎞ 1 γ = − ∫ exp(− x) ln( x)dx = − ∫ ln ln⎜ ⎟dt [Many other integral representations] 0 0 ⎝t⎠ Properties: C or γ = 0.577 215 664 ... Relation to Complete Gamma Function and Psi (Digamma) Function: Γ ′(1) = Ψ (1) = −γ References • Abramowitz & Stegun, Ch. 23. • Bers, Calculus, pp. 512-3 • Moll, http://www.math.tulane.edu/~vhm/Table.html • Table of Integrals, Series, and Products, Use of the Tables, and Sect. 8.367 & Index • Weisstein, Eric W. "Euler-Mascheroni Constant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Euler-MascheroniConstant.html 17 CATALAN’S CONSTANT (G or K ) arctan( x ) x x Figure 10. Catalan’s Constant Definition: G=∑ ∞ (− 1)m = 1 1 1 1 − 2 + 2 − 2 +K [Formula 0.234.3] m =0 (2m + 1) 2 2 1 3 5 7 Integral Representations: arctan( x ) 1 G=∫ dx [Many other integral representations] 0 x Properties: G or K = 0.915 965 594 ... References • Abramowitz & Stegun, Ch. 23. • Dwight, Table of Integrals, 1961 • Marichev, Oleg; Sondow, Jonathan; and Weisstein, Eric W. "Catalan's Constant." From MathWorld-- A Wolfram Web Resource. http://mathworld.wolfram.com/CatalansConstant.html • Spiegel, Ch. 34 • Table of Integrals, Series, and Products, Use of the Tables, Formula 4.531.1, & Index 18 PARTIAL FRACTIONS Elementary Identities a = a t = t bt 1 t−a m = x+t ⎛ t ⎜1 + x⎞ x(a ± bx ) ax a (a ± bx ) t−a t−a ⎟ ⎝ t⎠ t t t 1 1 a =± m = + a = a x( x ± a ) ax a(x ± a ) x x+a x+a x x+a x−t ⎛x ⎞ t ⎜ − 1⎟ 1 1 1 1 1 a ⎝t ⎠ =± m = − x( x ± a ) ax a(x ± a ) x x−a x−a x x−a x a 1 = 1− = 1 1 1 a+x a+x a = − x a + bx a 1+ t (t − 1) t −1 t = − x a + bx b b a + bx x a 1 1 1 1 = −1 = = + x a a − bx a−x a−x a t (1 − t ) 1− t t = − −1 a − bx b a − bx b x 1 1 1 = − a + bx u (t − u ) ut b a x = 1+ a = 1 t (t − u ) = + x−a x−a a x a + bx x a + bx 1− 1 1 1 = − u (t − u ) a − bx x a b t (u − t ) ut = − t 1 1 = 1− = 1 1 1 x x a − bx a − bx t +1 t +1 1 = − 1+ 2 t x ( x + 1) x x +1 t =1− u = 1 t +u 2 t +u 2 u x = 1+ 1 = 1 u = 1 − 1 1+ x −1 x −1 1− 1 t (t − u ) t −u t t2 x t2 u 1 u = 1 + 1 = −1 = t (u − t ) u −t u−t x 1 1 2 2 u −t u = −1 = t −1 1− x 1− x 1 t2 −1 x 1 a x = − t2 u 1 t x 1 a + bx b b( a + bx) =1+ = = 1− = t −u 2 t −u 2 u x+t x+t x x 1 a 1− 1+ = + t2 t bx − a b b(bx − a ) t 1 a+ x−t x a−x x a 1 = − a = 1− a a − bx = b( a − bx ) − b (a + x ) 2 a+x (a + x )2 a x−a x 1 a = 1− a 1 = − x x a + bx = bx (a + x ) 2 a+x (a + x )2 1 1+ x= a x 1 a x −1 = − 1− x a = 1 (a + bx )2 b(a + bx ) b(a + bx )2 ±n a − bx bx x a 1 ⎛ a⎞ 1− = − (a ± x ) ±n ±n = x ⎜1 + ⎟ a (a − bx ) b(a − bx ) b(a − bx ) 2 2 ⎝ x⎠ ⎛ ⎞ ⎛ ⎞ ⎛ x⎞ ±n ⎜ 1 ⎟ ⎜ 1 ⎟ x2 x x = a ± n ⎜1 + ⎟ 1 1 1⎜ = + = ⎜ ⎟= ⎟ x −a 2 2 2( x − a ) 2( x + a ) ⎝ a⎠ ax + b ax ⎜ b ⎟ b⎜ a ⎟ ⎜1+ ⎟ ⎜1+ x ⎟ ⎝ ax ⎠ ⎝ b ⎠ 19 ⎛ ⎞ ⎛ ⎞ x2 = x − x = ax ⎜ 1 ⎟ = ⎜ ax ⎟ a 2 − x2 2( a− x ) 2( a+ x ) ±n 1 1 1 1 (x − a )±n = x ± n ⎛1 − a ⎞ ⎜ ⎟ ax−b ⎜ 1− b ⎟ b ⎜ −1 ⎟ x = x + x ⎝ x⎠ ⎝ ax ⎠ ⎝ b ⎠ a 2 − x 2 2 a ( a + x ) 2 a( a − x ) ⎛x ⎞ ±n = a ⎜ − 1⎟ ±n x ( ) ( ) = a 1 − b 1 ( x+ a ) ( x+b ) a −b x+ a a −b x+b x = x − x x 2 − a 2 2 a( x − a ) 2 a ( x + a ) = a ( 1 )+ b ( 1 ) ⎝a ⎠ x 1 = 1 1 ⎛ a ⎞ ( x+a ) ( x−b ) a +b x+ a a+b x−b − x 2 − a 2 2 a( x − a ) 2 a ( x + a ) a ± bx = bx⎜ ± 1⎟ ⎝ bx ⎠ x = a ( 1 )+ b ( 1 ) 1 1 1 ( x−a ( x+b ) a +b x−a a+b x+b ) = + a 2 − x 2 2 a ( a + x ) 2 a( a − x ) ⎛ bx ⎞ = a ⎜1 ± ⎟ ⎝ a⎠ NOTE: These identities often simplify integral calculations See Sects. 2.32 & 3.363, below, for examples dx Example: GR Formula 2.313.1, I = ∫ a + be mx . Change of variable: s = e mx , ds = me mx dx = msdx 1 ds I= ∫ m s (a + bs ) 1 t t bt The fraction, , is simplified by above partial fraction, = − , s (a + bs ) x(a + bx ) ax a(a + bx ) 1 1 ⎛1 b ⎞ with t = 1 , so that = ⎜ − ⎟. s (a + bs ) a ⎝ s a + bs ⎠ 1 ⎛1 b ⎞ 1 ⎛ ds b ⎞ Then, I = ma ⎝ ∫ ⎜ s − a + bs ⎟ds = ma ⎜ ∫ s − ∫ a + bs ds ⎟. ⎠ ⎝ ⎠ Now, apply change of variable, t = a + bs, dt = bds, which leads to solution, I =∫ dx a + be mx = 1 ⎛ ds ⎜∫ − ∫ ⎟ = ma ⎝ s dt ⎞ 1 t ⎠ ma [ ( (ln s − ln t ) = 1 ln⎛ s ⎞ = 1 mx − ln a + bemx ⎜ ⎟ ma ⎝ t ⎠ ma )] References • Bers, Calculus, pp. 421 ff. • Table of Integrals, Series, and Products, Sect. 2.1 (Rational Functions). 20 MISCELLANEOUS Completing The Square 2 ⎛ b ⎞ ax + bx + c = a⎜ x + ⎟ − b 2 − 4ac 2 ( ) [The second form is most useful for integrals. ⎝ 2a ⎠ 4a See Sects. 2.32 & 3.323, below, for uses] ⎛ = ⎜ ax + b ⎞ ⎟ − 2 b 2 − 4ac ( ) ⎜ ⎟ ⎝ 2 a⎠ 4a Finite Binomial Expansions n ⎛n⎞ n • (a + b ) = ∑ ⎜ ⎟a n − j b j = ∑ n!