500 Integrals

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 ¾ 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 e +1 x 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 [email protected] —or— [email protected].

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 Logarithmic Functions Gamma Function Incomplete Gamma Functions Probability Integral or Error Function (Erf) and Imaginary Error Function (Erfi) Exponential-Integral Function Logarithm-Integral Function Euler’s Constant Catalan’s Constant Partial Fractions Miscellaneous (Completing the Square, Finite Binomial Expansions, Double Factorial, Natural Number N, Differentiation Under the Integral Sign) NOTATIONS

2 4 6 8 10 12 14 16 17 18 20

22

INTRODUCTION 0.1 Finite Sums

24

0.11 Progressions 0.111 Arithmetic progressions 0.112 Geometric progressions 0.12 Sums of powers of natural numbers

25 25 26 26

INDEFINITE INTEGRALS OF ELEMENTARY FUNCTIONS 2.3 The Exponential Function

28 n

2.31 Forms containing e ax , e ax 2.312 2.32 The exponential combined with rational functions of x 2.7 Logarithms and Inverse and Hyperbolic Functions 2.71 The logarithm

30 31 35 61 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 3.31 Exponential functions 3.310 3.311 3.32–3.34 Exponentials of more complicated arguments 3.321 3.322 3.323 3.324 3.326 3.327–3.334 Exponentials of exponentials 3.327 3.328 3.331 3.35 Combinations of exponentials and rational functions 3.351 3.353 3.36–3.37 Combinations of exponentials and algebraic functions 3.361 3.362 3.363 3.371 3.38–3.39 Combinations of exponentials and arbitrary powers 3.381 3.382 3.41–3.44 Combinations of rational functions of powers and exponentials 3.427 3.434 3.46–3.48 Combinations of exponentials of more complicated arguments and powers 3.461 3.462 3.464 3.471 3.473

79 80 81 82 84 85 89 90 94 96 98 99 100 101 103 104 105 106 107 108 109 116 120 121 125 126 127 128 129 130 134 148 151 160

4.2–4.4 Logarithmic Functions

161

4.21 Logarithmic functions 4.211

162 163

4.212 4.215 4.22 Logarithms of more complicated arguments 4.229 4.24 Combinations of logarithms and algebraic functions 4.241 4.26–4.27 Combinations involving powers of the logarithm and other powers 4.269 4.272 4.274 4.28 Combinations of rational functions of ln x and powers 4.281 4.283 4.29-4.32 Combinations of logarithmic functions of more complicated arguments and powers 4.326 4.33–4.34 Combinations of logarithms and exponentials 4.331 4.337

168 174 177 178 179 180 181 182 185 190 192 193 194 195 196 197 198 199

SPECIAL FUNCTIONS 8.2 The Exponential Integral Function and Functions Generated by It

202

8.21 The exponential integral function Ei(x) 8.212 8.24 The logarithm integral li(x) 8.240 8.241 Integral representations 8.25 The probability integral, the Fresnel Integrals Φ ( x ), S ( x), C ( x) , the error function erf(x), and the complementary error function erfc(x) 8.250 Definition 8.252 Integral representations

203 204 210 211 212 214

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) 8.313 8.33 Functional relations involving the gamma function 8.331 8.334 8.335 8.339 Particular Values: For n a natural number 8.35 The Incomplete Gamma Function 8.350 Definition 8.351 8.352 Special cases

215 216

222 223 224 225 232 238 239 253 254 255 256

8.353 Integral representations 8.356 Functional relations 8.359 Relationships with other functions 8.36 The Psi function ψ ( x ) 8.367 Euler’s constant: Integral representations

261 263 265 273 274

References

276

Index Index of Formulas

278 284

LIST OF ILLUSTRATIONS Figure 1. Figure 2. Figure 3. Figure 4. Figure 5. Figure 6. Figure 7. Figure 8. Figure 9. Figure 10.

Exponential Function (Growth) Exponential Function (Decay) Natural Logarithm Function Gamma Function Error Function Imaginary Error Function Exponential-integral Function Logarithm-integral Function Euler’s Constant Catalan’s Constant

2 4 4 6 10 10 12 14 16 17

MATHEMATICAL & GRAPHICAL SUMMARY OF SELECTED ELEMENTARY AND SPECIAL FUNCTIONS

GAMMA FUNCTION

2

EXPONENTIAL FUNCTIONS

e

e− x

x

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

a x+ y = a x a y

a − x = e − x ln a

a x− y =

ax ay

(a )

=a

(ab )x

= a xb x

x y

ax = ex n

n

ln a

a −x = e−x n

p q

n

ln a

xy

(− a ) q

p

(− 1)

1 2

1 q

( a) = [(− a ) ] = [(− 1) ] [a ]

[( )]

a = a

p

= a = q

p

p

1 q

x = e ln x

p

q

p

1 q

p

1 q

( ) =e = (e ) = e

x n = e ln x xn

a

n

a ln x n

n ln x n a ln x

= i (imaginary)

64 4 4 4 74 4 4 4 8 Let a be e for exp.

Limits: lim e x = 1 x →0

lim e x →0

−x

=1

lim e x = +∞

x → +∞

lim e − x = 0

x → +∞

ex lim a = +∞ x → +∞ x e−x xa lim a = lim x = 0 x → +∞ x x → +∞ e

e − ax − e − bx lim =b−a x →0 x e −ax − e −bx lim =0 x →∞ x

3

Derivatives: ¾

a, e, n are assumed constants except where noted

du v de v ln u dv ln u du dv = = uv = vu v −1 + u v ln u dx dx dx dx dx u u ln a da de du = = a u ln a dx dx dx u de du = eu dx dx 6444444447444444448 chain rule

n

n

n

n

n da x de x ln a = = nx n −1a x ln a dx dx x x ln a da de = = a x ln a dx dx 1 d x − x ln a ln a a = de =− x dx dx a n da x de x ln a = = a x x n ln x ln a dn dn [n not a constant ]

de x = ex dx de − x = −e − x dx de ax = ae ax dx de − ax = −ae − ax dx

Indefinite Integrals: x x ∫ e dx = e e ax ax ∫ e dx = a

• •

−x −x ∫ e dx = −e e − ax − ax ∫ e dx = − a

See Section 2.32 for generalized indefinite exponential integrals of form,

∫x

± m ± ax n

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

Figure 3. Natural Logarithm Functions, ln x &

Notation: log x = ln x (natural logarithm, base e) log m x = (log x )

m

x

Definition:

∫ 1

dt = log x, x > 0 t

1 1 ,& x ln x

5

Properties: log ( xy ) = log x + log y x log = log x − log y y

log x n = n log x log

ln e x = x ln e − x = − x

1 = − log x x

ln e ax = ax

⎛ b⎞ a ln x + b ln y = ln x a y b = a ln⎜⎜ xy a ⎟⎟ ⎝ ⎠ ⎞ ⎛ ⎜ x ⎟ ⎛ xa ⎞ a ln x − b ln y = ln⎜⎜ b ⎟⎟ = a ln⎜ b ⎟ ⎝y ⎠ ⎜ ya ⎟ ⎠ ⎝

(

)

ln e − ax = − ax

Limits: lim log x = +∞

x → +∞

lim log x = −∞

x →0 +

xa lim = +∞ x → +∞ log x log x lim a = 0 x → +∞ x

log x =0 x → +∞ e x log x lim x = −∞ x →0 + e lim

ex lim = +∞ x → +∞ log x x

e =0 x →0 + log x lim

ln(1) = 0 ln (e ) = 1

⎛1⎞ ln⎜ ⎟ = −1 ⎝e⎠

Derivatives: d ln x dx

1 = ,x ≠ 0 x

d ln m (a + bx ) mb ln m−1 (a + bx ) = dx a + bx

d ln u 1 du (chain rule) = dx u dx

Indefinite Integrals:

∫

dx = ln x x

NOTE: see Section 2.32 for special logarithmic function—the dilogarithm,

ln (1 − x ) dx x ∞ xk Li 2 ( x ) = ∑ 2 , x < 1 k =1 k Li 2 ( x ) = − ∫

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⎞ Γ(z ) = ∫ ln⎜ ⎟ ⎝t ⎠ 0 1

dt ,

real z > 0

[Formula 8.312.1]

Integral, Product, and Series Representations: Γ( x ) = z

∞ x

∫t

x −1 − zt

e dt ,

real z , x > 0

p! p x

= lim

[Formula 8.312.2]

0

Γ( x ) = lim

p →∞

p

x∏ ( x + k ) k =1

px , p p →∞ ⎛ x⎞ x ∏ ⎜1 + ⎟ k⎠ k =1 ⎝

real x > 0

[Artin, Formula 2.7]

x

⎛ x⎞ ⎛ 1⎞ exp⎜ ⎟ 1+ ⎟ ∞ ∞ ⎜ exp(− γx ) nx k⎠ 1 k⎠ ⎝ ⎝ Γ(x ) = = ∏ = lim ∏ n →∞ x x x x x k =1 k =1 1+ 1+ k k n −1 k 1 ⎛ ⎞ Γ⎜ z + ⎟ & Γ(nz ) & [Section 8.335] ∏ n⎠ Γ( x ) k =0 ⎝

n

k

∏ k + x,

real x > 0

[Formula 8.322]

k =1

∞ ⎧ ⎡ ⎛ x ⎞ x ⎤⎫ log Γ( x ) = − ⎨ln x + γx + ∑ ⎢ln⎜1 + ⎟ − ⎥ ⎬, real x > 0 k ⎠ k ⎦⎭ k =1 ⎣ ⎝ ⎩

[Artin, Formula 2.9]

Properties: Γ( x ) = (x − 1)! Γ( x ) ≠ 0

Γ(− x ) = − num.

Γ(1 − x ) , x

Γ( x + 1) = xΓ( x )

x not a natural

lim Γ( x ) = +∞

x →0 +

lim Γ( x ) = −∞

x →0 −

lim Γ( x ) = +∞

x → +∞

lim Γ( x ) is indeterminate

x → −∞

7

Derivatives: dΓ( x ) d x −1 −t t e dt = = Γ ′( x ) = dx dx ∫0

∞

∞

∞

∂

∫ ∂x e

( x −1) ln t

0

Γ′(1) = ∫ e −t ln tdt 0

= −γ (Euler' s constant ) = Ψ (1) (Formula 8.366.1)

∞

e −t dt = ∫ t x −1e −t ln tdt = Γ( x )Ψ ( x ) 0

∞

Γ (n ) ( x ) = ∫ t x −1e −t (ln t ) dt

•

n

0

d ln Γ(x ) Γ′(x ) = = Ψ (x ) dx Γ( x )

(digamma or Psi function), Formula 8.330

•

1 dΓ( x ) d ⎛ 1 ⎞ Ψ (x ) ⎜⎜ ⎟⎟ = − =− 2 dx ⎝ Γ( x ) ⎠ Γ( x ) [Γ(x )] dx

Relation to Incomplete Gamma Functions, γ (a, x ) & Γ(a, x ) : Γ(a ) ∞

∫t

= γ (a, x )

e dt 0 1424 3

•

+ Γ(a, x ) ∞

= ∫ t e dt + ∫ t a −1e −t dt 0 x 1 424 3 1 424 3 LOWER UPPER 1 4 4 4 2 4 4 4 3 GAMMA

a −1 −t

COMPLETE

x

a −1 −t

[Formula 8.356.3]

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-SpecialValues--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]

•

γ(

•

a , x ) = π Φ(x ) , 2

( )

[ Φ(x ) = erf(x), error function, 8.250.1]

•

γ ( , x) = π Φ x

• •

γ (a,0) = 0 [Formula 8.350.5] γ (a, ∞ ) = Γ(a ) dγ (a, x ) dΓ(a, x ) a −1 − x

• • •

2.

1 2

1 2

=− = x e [Form. 8.356.4] dx dx dγ (a, u ) dΓ(a, u ) dγ (a, u ) du (chain rule) =− = dx dx du dx 1 dγ (a, x ) = γ (a, x ) ln x + x a ∫ t a −1e − xt ln tdt [Sect. 8.356] da 0

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(12 , x 2 ) = π − π Φ ( x ) [ Φ(x ) = erf(x), error function, 8.250.1] Γ( 12 , x ) = π − π Φ

( x)

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 ) a = Γ(a, x ) ln x + x ∫ t a −1e − xt ln tdt da 1

•

[Sect. 8.356]

γ (a, x ) x a d ⎡ γ (a, x ) ⎤ [ ( ) ] = ln x − Ψ a + t a −1e − xt ln tdt , real a,x > 0 [Section 8.356] da ⎢⎣ Γ(a ) ⎥⎦ Γ(a ) Γ(a ) ∫0 1

NOTE: • See exponential-integral function, Ei( x) , for relation to Γ(a, x )

Indefinite Integrals: m ax ∫ x e dx = n

(− 1)1−γ Γ(γ ,−ax n ) = (− 1)1−γ

m − βx ∫ x e dx = − n

(− 1) e ax ∫ x m dx = n

na

(

γ

na

)

Γ γ , βx n 1 =− γ γ nβ nβ z +1

(

)

∞

∫t

a z Γ − z ,− ax n = n

(

)

∫t

γ −1 −t

e dt , γ =

− ax n

γ −1 −t

e dt , γ =

βx n

β z Γ − z , βx n βz e − βx = − = − dx ∫ xm n n n

γ

∞

(− 1)z +1 a z n

m +1 [Formula 2.33.10] n

∞

∫

− ax n

1 t

∞

e −t ∫n t z +1 dt , βx

m +1 n

z=

z +1

e −t dt ,

z=

m −1 n

[Formula 2.325.6]

m −1 [Formula 2.33.19] n

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 ⎞ ,± x⎟ &γ ⎜ , ± x⎟ ⎝ 2 ⎠ ⎝2 ⎠

¾ See Section 8.359 for special forms, Γ⎜ ±

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

Figure 5. Error Function (erf)

x

Figure 6. Imaginary Error Function (erfi)

Definition: Φ (x ) = erf (x ) =

2

x

e π ∫

−t 2

dt [Formula 8.250.1]

0

erfc(x ) = 1 − Φ (x ) =

2

π

∞

−t ∫ e dt [Complimentary Error Function, Formula 8.250.4] 2

x

Integral Representation: Φ (x ) = erf (x ) =

x2

e −t dt [Formula 8.251.1] π ∫0 t

1

Properties: Φ (− x ) = −Φ ( x ) Φ (0 ) = 0 Φ (± ∞ ) = ±1

Sect. 8.359

11

Relation to Incomplete Gamma Function: ⎛1 2 Φ(x ) = ⎝

⎞ ⎠

γ ⎜ , x2 ⎟ Φ

π

⎛1 2 x = ⎝

( )

⎞ ⎠

γ ⎜ , x⎟ π

Imaginary Error Function, erfi(z): erf(iz ) i

erfi( z ) =

z

2

erfi( z ) =

∫e

π

t2

dt =

0

2 i π

iz

∫e 0

−t 2

dt =

1 i π

(iz )2

∫ 0

e −t dt t

Relation to Incomplete Gamma Function: ⎛1 2 erfi(z ) = ⎝ i π

⎞ ⎠

γ ⎜ ,− z 2 ⎟

Indefinite Integrals:

∫e

−t 2

∫e

t2

dt =

dt =

π 2

π 2

erf (t )

erfi(t )

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

Figure 7. Exponential-integral Function, Ei(x), −

e−x x

Definition: Formulas 8.211.1 & 8.211.2 ∞

e −t et Ei( x ) = − ∫ dt = ∫ dt , x < 0 t t −x −∞ x

∞ −t ⎤ ⎡−ε e −t e Ei( x ) = − lim ⎢ ∫ dt + ∫ dt ⎥, x > 0 (Cauchy Principal Value PV) ε → +0 ε t ⎣− x t ⎦

Properties: Ei(+ ∞ ) = +∞

Ei(− ∞ ) = 0

Relation to Logarithm-Integral Function:

( )

Ei( x ) = li e x , x < 0

( )

Ei(ax ) = li e ax , x < 0, a ≠ 0

Ei(0 ) is not defined

13

Relation to Incomplete Gamma Function: ∞

e −t dt = −Γ(0,− x ) t −x

Ei( x ) = − ∫

∞

e −t − Ei( x ) = ∫ dt = Γ(0,− x ) t −x

∞

Ei(− x ) = − ∫ x

∞

− Ei(− x ) = ∫ x

e −t dt = −Γ(0, x ) t e −t dt = Γ(0, x ) Form. 8.359.1 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

li(x ) = ∫

dt , x <1 ln t 0

x ⎡1−ε dt dt ⎤ li( x ) = lim ⎢ ∫ + ∫ ⎥ = Ei(ln x ), x > 1 (Cauchy Principal Value PV) ε →0 ⎣ 0 ln t 1+ε ln t ⎦

Properties: li(0 ) = 0

li(+ ∞ ) = +∞

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 ) x⎠ ⎝

[Formula 8.359.2]

li(1) not defined

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 n →∞ ⎣ k =1 k

⎤

⎛ n →∞ ⎝

1 2

1 3

1 n

⎞ ⎠

γ = lim ⎢∑ − ln n⎥ = lim⎜1 + + + K + − ln n ⎟ ⎦

[Formula 8.367.1]

Integral Representations: ∞

1

0

0

⎛1⎞ ⎝t⎠

γ = − ∫ exp(− x) ln( x)dx = − ∫ ln ln⎜ ⎟dt [Many other integral representations]

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=∑

m =0

(− 1)m

(2m + 1)

2

=

1 1 1 1 − 2 + 2 − 2 +K 2 1 3 5 7

[Formula 0.234.3]

Integral Representations: 1

G=∫ 0

arctan( x ) dx [Many other integral representations] 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

=

x+t

t

⎛ ⎝

x(a ± bx )

x⎞

t ⎜1 +

⎟ t⎠

t

a

a = x−t ⎛x ⎞ t ⎜ − 1⎟ ⎝t ⎠ x a+x x a−x

a

= 1−

=

a+x

a

1+

−1 =

a−x

x−a t t +1 x x −1 x 1− x

= 1+

a x−a 1

= 1−

t +1

x −1

1

=

x+t x a a x

= 1−

x+t

t (u − t )

=

u

1

t (t − u )

x

u

t (u − t )

1

−1

x a + bx

1 1+

x

x

t

bx − a x a − bx

a x−a

= 1−

1− t

m

1

x x+a 1

m

a(x ± a ) a(x ± a )

x

t

a + bx

+

1

x

1

=

1

= = =

= = =

−

x

−

1 t −u 1 u −t 1 b 1

b

− +

−

a − bx

1

1

ut

⎛ a⎞ = x ⎜1 + ⎟ ⎝ x⎠

±n

±n

⎛ x⎞ = a ± n ⎜1 + ⎟ ⎝ a⎠

±n

x x−a

−

t

a + bx x

1

u (t − u )

a − bx x

1

t−a

=

1

t−a 1

= =

=

a + bx b

b a − bx b

=

a + bx

t

−

1

t +u

+

1

2

x a − bx u t +u 2

a b a + bx a − bx

−

+

a

=1−

x x−a

−

a

=

x x+a a

−

x−a

=

a

+

x+a 1

2

x +1

b a

x a + bx b

−

a − bx 1

=

u

1+

t2

t

t2 u −t

t a

t2

b( a + bx)

t −u 2

a b(bx − a ) a

b( a − bx )

2

=

u u−t

2

−1 =

1 u

−1

t2

−

(a + x )

=

x

=

2

b

(a + x )

2

a = a − bx

1 bx 1− a ⎛ ⎞ ⎜ 1 1 ⎜ 1 ⎟⎟ = = b ⎟ ax + b ax ⎜ ⎜1+ ⎟ ax ⎠ ⎝

=1+

t

1

1 = bx a + bx 1+ a

1 x −1 1− x

(a ± x )

1

t

t−a 1

a (a ± bx )

a

x

±n

ax

1

u (t − u ) ut

x ( x + 1)

t 1

1

bt

=

1

a−x

= 1−

x=

x

x

1 x

t

1

1

1−

−1 =

1− x

a

ax

t −1

=

t (1 − t )

t

1

=

t (t − 1)

t (t − u )

1

=

=±

1

1−

1+

1

1

1

=

=

1

= 1+

x

−1

x x

a

1 a

x( x ± a )

1

m

ax

=±

x( x ± a )

1

=

t

=

⎛ ⎞ ⎜ 1 ⎜ 1 ⎟⎟ a ⎟ b⎜ ⎜1+ x ⎟ b ⎠ ⎝

u

1

2

=

1

t2 a+ x−t

t −u

a+x 1 a+x

− −

1−

u

(a + x )2 a

(a + x )2

x 1 a = − (a + bx )2 b(a + bx ) b(a + bx )2 x a 1 = − 2 2 (a − bx ) b(a − bx ) b(a − bx ) x2 x x = + 2 2 x −a 2( x − a ) 2( x + a )

19

(x − a )±n = x ± n ⎛⎜1 − a ⎞⎟ ⎝

±n

x⎠

⎛x ⎞ = a ⎜ − 1⎟ ⎝a ⎠ ⎞ ⎛ a a ± bx = bx⎜ ± 1⎟ ⎝ bx ⎠ ⎛ bx ⎞ = a ⎜1 ± ⎟ a⎠ ⎝

±n

±n

⎛ ⎞ ⎛ ⎞ 1 1 1 = ax ⎜ 1 ⎟ = ⎜ ax1 ⎟ ax−b ⎜ 1− b ⎟ b ⎜ −1 ⎟ ⎝ b ⎠ ⎝ ax ⎠

( ) ( ) x = a ( 1 )+ b ( 1 ) x a x b + − ( )( ) a+b x+a a+b x−b x = a ( 1 )+ b ( 1 ) x−a ( x+b ) a +b x−a a+b x+b x = a 1 − b 1 ( x+ a ) ( x+b ) a −b x+ a a −b x+b

(

)

x2

x x − 2( a− x ) 2( a+ x ) x = x x + a 2 − x 2 2 a ( a + x ) 2 a( a − x ) x = x x − x 2 − a 2 2 a( x − a ) 2 a ( x + a ) 1 = 1 1 − x 2 − a 2 2 a( x − a ) 2 a ( x + a ) 1 1 1 = + a 2 − x 2 2 a ( a + x ) 2 a( a − x ) a 2 − x2

=

NOTE: These identities often simplify integral calculations ¾ See Sects. 2.32 & 3.363, below, for examples ¾ Example: GR Formula 2.313.1, I =

dx

∫ 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 b ⎞ 1 with t = 1 , so that = ⎜ − ⎟. s (a + bs ) a ⎝ s a + bs ⎠ 1 ⎛1 b ⎞ 1 ⎛ ds b ⎞ Then, I = ds ⎟. ⎜ − ⎟ds = ⎜∫ − ∫ ∫ ma ⎝ s a + bs ⎠ ma ⎝ s a + bs ⎠ Now, apply change of variable, t = a + bs, dt = bds, which leads to solution, dx 1 ⎛ ds dt ⎞ 1 (ln s − ln t ) = 1 ln⎛⎜ s ⎞⎟ = 1 mx − ln a + bemx I =∫ = ⎜∫ − ∫ ⎟ = mx a + be ma ⎝ s t ⎠ ma 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

(

b ⎞ b 2 − 4ac ⎛ ax + bx + c = a⎜ x + ⎟ − 2a ⎠ 4a ⎝ 2

2

2

)

(

b ⎞ b 2 − 4ac ⎛ = ⎜⎜ a x + ⎟⎟ − 4a 2 a⎠ ⎝

)

[The second form is most useful for integrals. See Sects. 2.32 & 3.323, below, for uses]

Finite Binomial Expansions n n ⎛n⎞ n! n • (a + b ) = ∑ ⎜⎜ ⎟⎟a n − j b j = ∑ a n− j b j j =0 ⎝ j ⎠ j = 0 j!(n − j )! n(n − 1) n − 2 2 n(n − 1)(n − 2) n −3 3 = a n + na n −1b + a b + a b + K + bn 2! 3! n n ⎛n⎞ n! n j j j j a n − j (bx ) • (a − bx ) = ∑ ⎜⎜ ⎟⎟(− 1) a n − j (bx ) = ∑ (− 1) j!(n − j )! j =0 ⎝ j ⎠ j =0

= a n − na n −1bx +

n(n − 1) n − 2 n(n − 1)(n − 2) n −3 2 3 n a (bx ) − a (bx ) + K + (bx ) , 2! 3!

⎛n⎞

where ⎜⎜ ⎟⎟ is the binomial coefficient, defined as, j

⎝ ⎠

⎛n⎞ n! n(n − 1)(n − 2 )L(n − j + 1) ⎜⎜ ⎟⎟ = , = 1 ⋅ 2 ⋅ 3L j ⎝ j ⎠ j!(n − j )! for factorial n!= n(n − 1)(n − 2)L3 ⋅ 2 ⋅1 = 1 ⋅ 2 ⋅ 3 ⋅ L(n − 2)(n − 1)n and, by definition, 0!= 1!= 1 NOTE: Stirling asymptotic factorial Formula: n!~ n n e − n 2πn

Double Factorial ⎧2 ⋅ 4 ⋅ 6L (n − 2)n for n > 0, even ⎪ n!!= ⎨1 ⋅ 3 ⋅ 5L (n − 2)n for n > 0, odd ⎪1 for n = 0,−1 ⎩

(2n + 1)!!= 1 ⋅ 3 ⋅ 5L(2n − 1)(2n + 1) (2n − 1)!!= 1 ⋅ 3 ⋅ 5L(2n − 3)(2n − 1) (2n )!!= 2 ⋅ 4 ⋅ 6L(2n − 2)(2n)

Note that, by definition, 0!!= −1!!= 1

• See Wolfram Website, •

http://mathworld.wolfram.com/DoubleFactorial.html

O’Brien, http://www.docstoc.com/docs/5606124/Double-Factorials-Selected-Proofs-and-Notes

Natural Number N: meaning differs across fields and textbooks to mean all positive integers (1,2,3,...) either with or without zero included. Here, zero is included in definition, so N = 0,1,K .

21

Differentiation Under The Integral Sign 1)

d dϕ (a ) dψ (a ) ∂f ( x, a ) f ( x, a )dx = f (ϕ (a), a ) dx − f (ψ (a ), a ) + ∫ ∫ da ψ ( a ) da da a ∂ ψ (a)

2)

d da

ϕ (a)

d dx

3)

ϕ (a)

∫ c

ϕ ( x)

∫

ϕ (a)

dϕ (a ) + f ( x, a)dx = f (ϕ (a), a ) da f (t , a)dt = f (ϕ ( x), a )

c

ϕ (a)

∫ c

∂f ( x, a ) dx ∂a

[c

constant ]

dϕ ( x ) [c constant ] dx

4)

d dψ (a ) ∂f ( x, a ) f ( x, a)dx = − f (ψ (a ), a ) dx + ∫ ∫ da ψ ( a ) da ∂a ψ (a)

5)

d dψ ( x ) f (t , a )dt = − f (ψ ( x), a ) [c constant ] ∫ dx ψ ( x ) dx

6)

d ∂f ( x, t ) f ( x, t )dt = ∫ dt ∫ dx a ∂x a

7)

dv( x) du ( x) d f (t )dt = f (v( x) ) − f (u ( x) ) ∫ dx dx dx u ( x )

8)

d dx

c

c

[c

constant ]

c

b

b

[a, b

constant ]

v( x)

ϕ ( x)

∫

f (t )dt = f (ϕ ( x) )

a

dϕ ( x ) dx

[a

constant ]

d dψ ( x ) [a constant ] f (t )dt = − f (ψ ( x) ) ∫ dx ψ ( x ) dx a

9)

NOTEs: • Rule # 1 traditionally is called “Leibnitz’s Rule for Differentiating Integrals” (Form. 0.41). Rules 2-9 are special cases for one and two variable functions with or without constant lower/upper limits of integration. Example—Rule # 6 provides

dΓ( x ) with f ( x, t ) = t x −1e − t , a = 0, b = ∞ limits; dx

¾ Some integrals cannot be differentiated using these rules, such as— ∞

∫ exp(−t

2

)dt or

0

d d γ (a, x ) & Γ(a, x ). For example, no rule seems to apply to da da

dγ (a, x ) d d t a −1 exp(−t )dt = = ∫ da da 0 da x

ϕ (x )

∫ f (t , a )dt . These forms must be transformed to equivalent c

integrals for which Leibnitz’s rules do apply. See Sects. 3.331 & 8.356 for derivations. References •

Bers, Calculus, Vol. 2, pp. 808 ff. & Boas, Ch. 4, pp. 233-236

22

SYMBOL int (x ) (a )n

∑u

= a(a + 1)(a + 2)L(a + n − 3)(a + n − 2)(a + n − 1) =

Γ(a + n ) Γ(1 − a ) n = (− 1) Γ(a ) Γ(1 − a − n )

(Pochhammer symbol) = um + um +1 + K + un .

n

k =m

MEANING The integer part of the real number x

k

n

If n < m, we define ∑ uk = 0 k =m

n

∏ f (k ) k =m

= f (m) f (m + 1)L f (n). n

If n < m, we define ∏ f(k) = 1 k =m

Γ(1 − a ) is cited in Integrals and Series, Γ(1 − a − n ) Vol. 1, Elementary Functions, A. P. Prudnikov, et al., 1986, page 772. ¾ Proof : x =1− a (− 1)n Γ(x ) (by Sect. 8.331, below ) = (− 1)n Γ(1 − a ) , Denominator Γ(1 − a − n ) = Γ(x − n ) = (1 − x )n (a )n

¾ Suggested new Pochhammer relation: (a )n = (− 1)n

n n ( ( − 1) Γ(1 − a ) − 1) Γ(1 − a ) so that, = (a )n . = Γ(1 − a − n ) (− 1)n Γ(1 − a ) (a )n

♦ Pochhammer Symbol derivations O’Brien, http://www.docstoc.com/docs/5473276/Pochhammer-SymbolSelected-Proofs

23

INTRODUCTION

24

25

Finite Arithmetic Progression FORMULA

PROOF OUTLINE (formula references are to G & R 7e) Change Formula 0.111 to read: Last form is suggested addition to Formula 0.111. Last term in sum is n −1 l−a l = a + (n − 1)r ⇒ n = + 1. (a + kr ) = n [2a + (n − 1)r ] = n (a + l ) ∑ r 2 2 k =0 l−a n 1 Substituting + 1 into (a + l ) provides the = (a + l )(l − a + r ) 2 r 2r proposed alternative solution for the finite sum of [l = a + (n − 1)r is the last term] an arithmetic progression, independent of knowledge of n. This new form (not seen in common schoolbooks) is useful when the number of terms n in a finite progression in unknown, such as, r =3 (3336 )(3333) 3 + 6 + 9 + K + 3333 = = 1,853,148, 2(3) giving n = 1,111 terms.

—Formulas 0.111 & 0.112 & 0.121—

26

Finite Geometric Progression FORMULA

PROOF OUTLINE (formula references are to G & R 6e) Last form is suggested addition to Formula 0.112. Last term in sum is aq n−1 = l . lq = aq n is substituted into aq n − a ql − a ⇒ . This new form (derived in q −1 q −1 many schoolbooks) is useful when the number of terms in a finite geometric progression is unknown.

Change Formula 0.112 to read: n

∑ aq k −1 = k =1

(

)

a q n − 1 ql − a = q −1 q −1

[q ≠ 1, l = aq

n −1

is the last term

]

Change Formula 0.121.1 to read: n

∑k = k =1

n(n + 1) 2

New comment refers reader to more general expression for arithmetic progression. [cf. 0.111]

—Formulas 0.111 & 0.112 & 0.121—

27

INDEFINITE INTEGRALS OF ELEMENTARY FUNCTIONS

28

29

30

FORMULA

PROOF OUTLINE (formula references are to G & R 7e)

1 s γ −1e ± s ds γ ∫ na m + 1⎤ ⎡ n ⎢ s = ax , γ = n ⎥ ⎣ ⎦

ds m +1 ⎛ s ⎞n Set s = ax , = nax n−1 , x = ⎜ ⎟ , γ = .A dx n ⎝a⎠ useful transform for elementary integrals of this form. Similar to above n 1 integral, ∫ x m e ± ax dx = γ ∫ s γ −1e ± s ds . na

1

m ± ax ∫ x e dx = n

e ± ax 1 z −1 ± s ∫ x m dx = na z ∫ s e ds n

n

1− m⎤ ⎡ n ⎢⎣ s = ax , z = n ⎥⎦

——Sect. 2.31——

31

FORMULA

( )

x ∫ exp a dx =

( ) [ln a > 0]

Ei a x ln a

PROOF OUTLINE (formula references are to G & R 7e) Change of variable: s = a x = e x ln a , ds = a x ln adx = s ln adx

( )

1 es Ei a x I= ds = ln a ∫ s ln a by definition of exponential-integral function, Ei( x) , Math. Summary. n exp a x xn NOTE: ∫ exp a dx, ∫ dx do not seem easily x solved;math.com, Mathematica not able to solve. But try setting n n s = a x = e x ln a , ds = snx n −1 ln adx which may be reduced by integration by parts. ¾ verified math.com • Change of variable: s = a x = e x ln a , ds = a x ln adx = s ln adx, ln s = ln a x

( )

( )

(− 1) ax ∫ x a dx = (ln a )1−a Γ(1 − a,− x ln a ) [ln a > 0] a

(− 1) ax ∫ x b dx = (ln a )1−b Γ(1 − b,− x ln a ) [ln a > 0] b

( )

2x 2x x ( ) = − dx ln 2 li 2 ∫ x2 x

⎛ ln s ⎞ = x ln a, x = ⎜ ⎟ ⎝ ln a ⎠ to give,

a

a

1 1 s ds (ln a ) ds ds = = . a a 1− a ∫ ∫ ∫ ln a x s ln a (ln s ) (ln a ) (ln s )a o Second change of variable: ds t = ln s, dt = , e t = s , to give: s et 1 I= dt . (ln a )1−a ∫ t a ƒ Now use result of 2.325.6, z +1 z ax n (− 1) a Γ − z,−ax n e = dx ∫ xm n z +1 z ∞ (− 1) a 1 −t m −1 = e dt z = z +1 ∫ n n t − ax n a

I=

(

)

with parameters [m = a, a = n = 1, z = a − 1] , giving, (− 1)a Γ(1 − a,−t ) = (− 1)a Γ(1 − a,− x ln a ) I= (ln a )1−a (ln a )1−a NOTE: derivable directly from 2.325.6 with parameters [a = ln a, m = a, n = 1].

—Sect.2.312—

32

(− 1) Γ(1 − b,− x ln a ) similar ax 2 integral, ∫ b dx = x (ln a )1−b to above, and more general 3rd integral substitutes a = b = 2 into (− 1)b Γ(1 − b,− x ln a ) = ln 2Γ(− 1,− x ln 2) = ax = dx ∫ xb (ln a )1−b b

• •

nd

⎡ ⎡ x e x ln 2 ⎤ 2x ⎤ = ln 2⎢Ei( xln2) − ⎥ = ln 2⎢li 2 − ⎥ x ln 2 ⎦ x ln 2 ⎦ ⎣ ⎣ 2x = ln 2li 2 x − x by Γ(− 1,− x ) , in Sect. 8.352, below. ¾ Verified math.com • First Integral • Change of variable: n n n s = a x = e x ln a , ds = nx n −1 ln a x dx = nx n −1 s ln adx,

( )

( )

∫a

xn

1

dx = −

n(− ln a ) [n ≠ 0, ln a < 0] dx

∫a

=−

1

n(ln a ) [n ≠ 0, ln a > 0]

∫x −

xn

m

1 n

1 n

⎞ ⎛1 Γ⎜ ,− x n ln a ⎟ ⎠ ⎝n

⎞ ⎛1 Γ⎜ , x n ln a ⎟ ⎠ ⎝n

a x dx = n

1 m +1 n

⎞ ⎛ m +1 ,− x n ln a ⎟ Γ⎜ ⎠ ⎝ n

n(ln a ) [n ≠ 0, ln a > 0]

1

1−

1

⎛ ln s ⎞ n n −1 ⎛ ln s ⎞ n ln s = x ln a, x = ⎜ ⎟ ,x = ⎜ ⎟ , to give, ⎝ ln a ⎠ ⎝ ln a ⎠ 1 s 1 1 I= ds = ds = n −1 ∫ ∫ n ln a sx n ln a x n −1 1 1 ds. 1 ∫ 1 1− n(ln a ) n (ln s ) n • Second Change of variable ds t = ln s, dt = , e t = s, to give, s 1 s 1 et I= dt = dt. 1 ∫ 1 1 ∫ 1 1− 1− n(ln a ) n t n n(ln a ) n t n o Evaluate I with Formula 2.325.6, z +1 ax n ( − 1) a z Γ − z ,− ax n e ∫ x m dx = n z +1 z ∞ (− 1) a 1 −t m −1 = e dt z= z +1 ∫ n n t − ax n n

(

)

1 1⎤ ⎡ with parameters ⎢a = n = 1, m = 1 − , z = − ⎥ , to n n⎦ ⎣ give: 1− ( 1 − 1) n ⎛ 1 ⎞ ⎛1 ⎞ Γ⎜ ,− x n ln a ⎟ I= Γ⎜ ,− x n ln a ⎟ = − 1 1 ⎠ ⎠ n(− ln a ) n ⎝ n n(ln a ) n ⎝ n 1

NOTE: As a check on calculations, if n = 1, then

—Sect. 2.312—

33

1 x ln a ax 1 Γ(1,− x ln a ) = = = ∫ a x dx. e (− ln a ) ln a ln a x by Γ(1,− x ) = e , Sect. 8.352, below. • Second and 3rd integrals similar to first NOTE: integrals derivable from formula, Sect. 2.32, below, (− 1)1−γ Γ γ ,−ax n m ax n = x e dx ∫ na γ (− 1)1−γ ∞ t γ −1e −t dt m +1 γ= , = γ ∫ n na − ax n (GR 7 errata submission). NOTE: See Sect. 3.311, below, for definite integrals. I =−

(

∫a

x2

dx =

[ln a > 0]

(

1 π erfi ln a x 2 ln a

erf (ix ) where erfi( x) = i

1 π erf 2 ln a [ln a > 0] dx

∫a

x2

=

(

ln a x

)

)

)

¾ verified math.com • Re-express and change of variable:

I = ∫ a dx = ∫ e x2

x 2 ln a

dx = ∫ e

[

ln a x

]

2

dx;

s = x ln a , ds = ln a dx , 2 1 e s ds . to give, I = ∫ ln a o Since the indefinite integral for imaginary error function, erfi, is, 2 t2 erfi(t ) = ∫ e dt ,

π

then, 1

∫e ln a

I=

s2

(

(

⎛ π⎞ ⎟erfi x ln a ⎜ ln a ⎜⎝ 2 ⎟⎠ 1

ds =

)

)

1 π erfi x ln a . 2 ln a NOTE: See Math. Summary (ERROR FUNCTION) NOTE: See Formula 2.33.2, where by change of variable s = a x, ds = a dx , =

(

)

2

•

•

( )

)2 dx s ==a x 1 π erfi a x . 2 a nd 2 integral similar to first using Formula 2.33.16 1 −s 2 for error function, erf, calculation, e ∫ ds . ln a verified mathematica

( ax ∫ e dx = ∫ e

ax

—Sect. 2.312—

34

NOTE: All solutions in this section can be verified by differentiating under the integral sign. Three examples follow. Ei(a x ) , we transform to a one-variable • First example— To verify the solution, ∫ exp(a x )dx = ln a definite integral and differentiate the integral. One approach: from Math. Summary, ax x φ (x) e Ei a x li e a 1 ⎛⎜ dt ⎞⎟ ( ) = = x < 0 which is of the form, ∫c f (t )dt. Thus, using Rule # 8 ln a ln a ln a ⎜ ∫0 ln t ⎟ ⎝ ⎠

( )

( )

(with Φ(x) = e ) for differentiating integrals [Math. Summary], and the chain rule, dt d ⎡ Ei(a )⎤ 1 d 1 ⎡e d a ⎤ = 1 [e 1 ⎡ 1 d (e )⎤ = ⎥ ⎢ ⎥ = ln a dx ∫ t = ln a ⎢ ⎢ ⎥ ln a(a ) ⎣ dx ⎦ ln a (a ) ln dx ln a ln (e ) dx ax

x

ea

x

ax

ax

x

x

x

x

ax

]

( )

a x ln a = exp a x .

⎣ ⎦ 0 ⎣ ⎦ NOTE: The Principal Value solution for x > 0 is left as an exercise. Hint—use Rule # 9, where 2nd term ∞ ⎞ ⎛ ⎜ in PV definition ⎜ of limiting form, ∫ f (t )dt ⎟⎟ is assumed to differentiate to 0, so that we solve only the 0 ⎠ ⎝ first term, ⎡ 1 ε e −t ⎤ ⎫⎪ d Ei a x d ⎧⎪ = ⎨− lim ⎢ dt ⎥ ⎬ (x > 0). dx ln a dx ⎪⎩ ε →0 ⎢⎣ ln a −∫a x t ⎥⎦ ⎪⎭ a

( )

•

2nd ex.—Verify solution, ∫ ∞

Restating Γ(1 − a,− x ln a ) =

∫t

−a

(− 1) Γ(1 − a,− x ln a ) . ax dx = a x (ln a )1− a a

e −t dt , from Math. Summary, and using Rule # 9 for differentiating

− x ln a

integrals [Math. Summary], a d ⎡ (− 1) ⎢ dx ⎣ (ln a )1− a

a ⎤ ( − 1) ∫ t e dt ⎥⎦ = (ln a )1− a − x ln a ∞

− a −t

x ⎡ ⎤ a − a x ln a d ( ) ( ) ln ln x a e x a = − − − ⎢⎣ ⎥⎦ x a dx

(

)

1 π erfi ln a x [ln a ≠ 0] . From Math Summary 2 ln a (Incomplete Gamma Functions), restate solution as, ln a x ln a x ⎞ 2 1 π 1 π ⎛⎜ 2 t2 ⎟= 1 e dt e t dt , erfi ln a x = ∫ ∫ ⎟ 2 ln a 2 ln a ⎜⎝ π 0 ln a 0 ⎠ so that, by Rule # 8 for differentiating integrals [Math. Summary], and laws of exponents, ln a x 2 d 1 1 ⎡ ( ln a x )2 d ln a x ⎤ t2 x 2 ln a e dt = = ax ⎢e ⎥=e ∫ dx ln a 0 dx ⎦ ln a ⎣

•

3rd ex. —Verify solution, ∫ a x dx = 2

(

)

———— Note: See Section 8.356, below, for derivatives of algebraic-exponential function integrals ¾ Exercise: Verify other solutions in this section.

—Sect. 2.312—

35

36

FORMULA

∫e

± μx

ln p xdx =

1⎡ ln p −1 xe ± μx ⎤ p ± μx xe p ln m ± ⎢ ∫ x dx⎥⎥ μ ⎢⎣ ⎦

∫e 1

μ

± μx

[μ ≠ 0]

ln xdx =

[± ln xe

± μx

] [μ ≠ 0]

m Ei(± μx )

Formula 2.325.4 e ax − e ax ae ax a 2 e ax a 3 dx = − 2 − + Ei(ax ) ∫ x4 3x 3 6x 6x 6

∫x

m

n

e ax dx =

x m +1− n ax n m + 1 − n m − n ax n x e dx e − na ∫ na

PROOF OUTLINE (formula references are to G & R 7e) • Integration by parts on each integral with, u = ln x p , dv = e ± μx dx. NOTE: see def. integrals below in Sects. 4.331, 4.335. ¾ Mathematica could not do calculation— “Mathematica could not find a formula for your integral. Most likely this means that no formula exists.” • Set p = 1 in above integral

1⎡ ln p-1 xe± μx ⎤ p ± μx ln m xe p ± ∫ x dx⎥⎦ μ ⎢⎣ and exponential integral, Ei(± x ) , in Form. 2.33.18, Form. 2.325.7 o Or—integration by parts on each integral with, u = ln x, dv = e ± μx dx, and apply definition of exponential integral, Ei(± x ) ¾ verified mathematica.com Derived from expansion Formula (2.324.2) in G & R 6e, which see below ± μx p ∫ e ln xdx =

• Use Change of Variable & Integration by Parts.

1) Let I = ∫ x m e ax dx; n

set s = ax n ,

ds m +1 , to get, = nax n−1 , γ = dx n

1 s γ −1e s ds. γ ∫ na 2) Do integration by parts on ∫ s γ −1e s ds and I=

re-express in terms of x. Let u = sγ −1 , du = (γ −1)sγ −2 ds; dv = es ds, v = es ; uv − vdu = sγ −1es − (γ −1) sγ −2es ds

∫

∫

Then, 1 I = γ sγ −1es − (γ −1) sγ −2es ds na

[

(ax ) =

n γ −1

γ

na

Sect. 2.32

]

∫

e

axn

( ) (nax )e

m +1 − n axn − n

∫

γ −2

n−1

γ

na

axn

dx

37

=

x m+1−n ax n m + 1 − n m−n ax n e − x e dx na na

∫

¾ If n = 1, Form. 2.321.1 follows as special case. Formula 2.33.4

Same as above for

m ± ax ∫ x e dx = n

x

±

na

∫x =

m +1− n

m ax n

e

n

e ± ax m

m + 1 − n m − n ± ax n x e dx na ∫

1−γ ( − 1) Γ(γ ,− ax n ) dx =

(− 1)1−γ na

γ

na γ

∞

∫t

γ −1 −t

− ax n

e dt γ =

m +1 n

e

(

dx and signs

∞

1−γ ( − 1) =

na

)

d f (t , γ )dt dx ψ ∫( x )

na γ γ

⎡ n ⎢− − ax ⎣

(

)

e (

γ −1 − − ax n

n ) d (− ax )⎤ = x m e ax ⎥

dx

⎦

¾ verified mathematica.com • Improved Formula 2.33.5 for integer γ . • Derived from Form. 8.352 .4 , xm m = 0 m! n −1

( )

γ − k −1 ⎤ ( e ⎡ γ −1 γ − 1)! x n k ⎢ ∑ (− 1) ⎥ (γ − k − 1)! a k +1 ⎥⎦ n ⎢⎣ k =0 m +1 ⎡ ⎤ ⎢γ = n = 1,2,..., n ≠ 0, a ≠ 0⎥ ⎣ ⎦ ax n

m − ax n

assigned. NOTE: algebraic-exponential formulas of the form x ± m exp ± ax n generalize many long-standing formulas with n = 1 and solve many elementary indefinite and definite integrals. • See Sect. 2.31 for derivation; then apply definition of incomplete gamma function. • Γ(•,•) is incomplete gamma function, Formula 8.350.2, given as Γ(α , x ) ƒ Formula omitted in GR7; sent in as erratum ƒ Verify derived integral by differentiating the integral using Rule # 5 in Math. Summary with ψ ( x ) = −ax n :

(− 1)1−γ

Formula 2.33.5 m ax n ∫ x e dx =

∫x

Γ(n, x ) = (n − 1)!e − x ∑

•

[n = 1,2,...]

Restate integral as incomplete gamma function, as previously disclosed (omitted in GR 7):

∫x

m axn

e

•

1−γ ( m +1 −1) Γ(γ ,−axn ) dx = ,γ =

naγ Plug into Form. 8.352.4: n −1

Γ (n , x ) = (n − 1)! e − x ∑

m =0

xm m!

where substitute d is n =

to give:

Sect. 2.32

n

[ n = 1, 2,...] m +1 = γ , x = − ax n n

n

38

γ −1

∫x •

(−1) (γ −1)!e 1−γ

m axn

e dx =

n

ax

∑(− ax )

n k

k =0

γ

k!

na

Now write summation in reverse order and simplify, noting that its first term is positive, (− 1)1−γ (− 1)γ −1 = (− 1)0 :

( )

( )

⎤ ⎡ axn γ −1 axn γ −2 − ⎥ ⎢ ( γ −1)! axn ⎢ (γ −1)! (γ − 2)! ⎥ m axn ∫ x e dx= naγ e ⎢ axn γ −3 ⎥ 1−γ ⎥ ⎢+ −K− (−1) ⎥⎦ ⎢⎣ (γ − 3)! ⎡ xn γ −1 (γ −1) xn γ −2 (γ −1)(γ − 2) xn γ −3 ⎤ n ⎢ − + ⎥ 2 3 eax ⎢ a a a ⎥ = 1−γ ⎥ n ⎢ (γ −1)!(−1) ⎥ ⎢−K− aγ ⎦ ⎣

( )

( )

n e ax ⎡ = ⎢ n ⎣⎢

γ −1

( )

(γ − 1)!

∑ (− 1) (γ − k − 1)! k =0

k

( )

(x )

n γ − k −1

a k +1

⎤ ⎥ ⎦⎥

by inspection, noting relation, (γ − 1)! k=> 0(γ − 1)(γ − 2 )(γ − 3)K (γ − k ), (γ − k − 1)! where the vaule of 1 is understood for k = 0.

NOTE: new version translates more readily into calculations for γ = 1, 2, 3, 4, given in Forms. 2.33.6 – 2.33.9. NOTE: Formula is useful for writing the logexpansions in Section 2.7 which can be transformed from log. function to exp. n function of the form x m e ax . Three (3) examples from Sect. 2.7: 1) Form. 2.711, transformed to exponential form, m m s ∫ ln xdx = ∫ s e ds by change of

•

Sect. 2.32

variable dx s ⎛ ⎞ ⎜ s = ln x, ds = , e = x ⎟, where x ⎝ ⎠ m is integer Applying Form. 2.33.5,

39

⎡ ⎢ ⎢⎣

n

eax n

γ −k−1 ( γ −1)! (xn ) ⎤ (−1) ⎥ ∑ (γ −k −1)! ak+1 ⎥⎦ k =0 γ −1

k

with s = x = ln x, n = 1, n → m, γ = m + 1, a = 1, to give, m s =ln x m m s s (−1)k m! (s)m−k . ln xdx s e ds e = = ∑ ∫ ∫ (m − k )! k =0 m

= x∑ (−1)

k

k =0

m! lnm−k x, (m − k )!

which differs from Form. 2.711 only in expression of factorial term. 2) Form. 2.722, n m m ( n +1)s ∫ x ln xdx = ∫ s e ds , by change of variable dx s ⎛ n+1 (n+1)s ⎞ ⎜ s = ln x, ds = , e = x, x = e ⎟ x ⎝ ⎠ Applying n γ − k −1 ⎤ ( γ − 1)! x n e ax ⎡ γ −1 k ⎢ ∑ (− 1) ⎥ (γ − k − 1)! a k +1 ⎦⎥ n ⎣⎢ k = 0

•

( )

with

s = x = ln x, n = 1, n → m,γ = m + 1, a = n + 1, to give, s =ln x

n m ∫ x ln xdx =

=e

( n+1)s

∫s

m

e (n+1)s ds =

(− 1) m! (s ) k +1 ∑ (m − k )! (n + 1) k =0 m

m−k

k

ln m−k x m! , (m − k )! (n + 1)k +1 k =0 which differs from Form. 2.722 only in expression of factorial term. 3) Formula 2.723.1, m

= x n+1 ∑ (− 1)

∫x

n

k

s = ln x

ln xdx =

∫ se

( n +1)s

ds by

change of variable,

dx s ⎛ n +1 ( n +1)s ⎞ ⎜ s = ln x, ds = , e = x, x = e ⎟ x ⎝ ⎠ •

Sect. 2.32

Applying n γ − k −1 ⎤ ( eax ⎡ γ −1 γ − 1)! xn k ⎢ ∑(− 1) ⎥ (γ − k − 1)! ak +1 ⎥⎦ n ⎢⎣ k =0

( )

40

with

s = x = ln x, n =1, n → m =1,γ = 2, a = n +1, to give, s =ln x

n ∫ x ln xdx =

∫ se

( n +1)s

ds

⎡ s 1 ⎤ = e (n +1)s ⎢ − 2⎥ ⎣ n + 1 (n + 1) ⎦ ⎡ ln x 1 ⎤ . = x n +1 ⎢ − 2⎥ 1 + n ( ) 1 + n ⎣ ⎦ n ax NOTE: Form. 2.321.2, ∫ x e dx, is a special case of

n

e ax na

n

m +1 ⎤ ⎡ ⎢γ = n = 1⎥ ⎦ ⎣

Formula 2.33.7

∫x

m

e

ax n

e ax dx = n

n

m ax n

e dx, with

m = n, n = 1, γ = n + 1 . Substitute values of integer γ in 2.33.5

Formula 2.33.6 m ax ∫ x e dx =

∫x

⎛ xn 1 ⎞ ⎜⎜ − 2 ⎟⎟ a a ⎝ ⎠

m +1 ⎡ ⎤ ⎢ γ = n = 2⎥ ⎣ ⎦

Sect. 2.32

41

Formula 2.33.8

∫x e m

ax n

e ax dx = n

n

⎛ x 2n 2 x n 2 ⎜⎜ − 2 + 3 a a ⎝ a

⎞ ⎟⎟ ⎠

m +1 ⎤ ⎡ ⎢γ = n = 3⎥ ⎣ ⎦

Formula 2.33.9 m ax ∫ x e dx = n

e ax n

n

⎛ x 3n 3x 2 n 6 x n 6 ⎞ ⎜⎜ − 2 + 3 − 4 ⎟⎟ a a a a ⎝ ⎠

m +1 ⎡ ⎤ ⎢γ = n = 4⎥ ⎣ ⎦

Formula 2.33.2 1 π ax 2 ∫ e dx = 2 a erfi a x

[ ]

•

Use change of variable, t = a x & indefinite integral 2 t2 definition of erfi(t ) : erfi (t ) = ∫ e dt.

π

¾ Erfi derived from

Formula 2.325.13

(ax +2bx +c )dx = ∫e 2

⎛ ac − b 2 ⎞ b ⎞ 1 π ⎛ ⎟⎟ erfi ⎜ a x + exp ⎜⎜ ⎟ 2 a a⎠ ⎝ ⎝ a ⎠ [a ≠ 0]

∫x

m ax n

e

dx with γ =

1 and Formula 2

⎛1 ⎞ 8.359.4, γ ⎜ , x 2 ⎟ with x 2 → − z 2 . ⎝2 ⎠ Complete the square on exponent, use change of variable, and definition of imaginary error function, erfi, from above. • Thus, completing the square and changing the variable: b ⎞ 4b 2 − 4ac ⎛ = ax 2 + 2bx + c = a⎜ x + ⎟ − 4a a⎠ ⎝ 2

2

⎛ b ⎞ b 2 − ac b ds ;s = ax + ; = a; ⎜⎜ a x + ⎟⎟ − a a⎠ a dx ⎝ 2

⎛ ac − b 2 ⎞ ⎛ b ⎞ ⎟⎟ ∫ exp⎜⎜ a x + I = exp⎜⎜ ⎟⎟ dx = a⎠ ⎝ ⎝ a ⎠ ⎛ ac − b 2 ⎞ s 2 1 ⎟⎟ ∫ e ds exp⎜⎜ a ⎝ a ⎠ which leads to result. Mathematica solution at http://integrals.wolfram.com/ shows the awesome speed of computer processing, and the frequent need for post-processing simplification:

•

CHECK: If b = c = 0 in Sect. 2.32

∫

e (ax

2

+ 2 bx +c

) dx , the integral

42

is identical to ∫ e ax dx = 2

Formula 2.33.10

∫x

m

=−

e

− βx n

∞

1 nβ

(

Γ γ , βx n dx = − nβ γ

∫t

γ

γ −1 −t

e dt

βx n

)

γ =

m +1 n

[ ]

1 π erfi a x of Form. 2 a

2.33.2. • NOTE: see Form. 2.33.1, GR 7, for a similar negative exponential integral; uses same approach 2 −t 2 involving erf ( x ) = e dt. π ∫ • NOTE: Sect. 3.323, below, has many similar definite integrals • Verify solution by differentiating the integral • See Sect. 2.31 for derivation; then apply definition and properties of incomplete gamma function • Γ (• ,• ) is incomplete gamma function, Formula 8.350 2, p. 890 , given as Γ(α , x ) • NOTE: this elementary integral solves and creates many integrals in Sects. 3.3-3.4, below. • Exercises: o Verify solution by differentiating under the integral sign o Show that the solution, a −1 − x ∫ x e dx = −Γ(a, x ), reproduces the definitions in Math. Summary of Γ(a ), Γ(a, x ), γ (a, x ); e.g., show that, ∞

∫x

e dx = −Γ(a, x )

a −1 − x

0

Formula 2.33.11

∫x

m

(

)

)

x =0

= Γ(a ) by indefinite integral

solution and properties of gamma functions • verified mathematica.com • Improved Formula 2.33.11 for integer γ . Similar in derivation to above integer-expansion, n γ − k −1 ⎤ ( e ax ⎡ γ −1 γ − 1)! x n k m ax n ( ) x e dx 1 = − ⎥ ⎢ ∑ ∫ (γ − k − 1)! a k +1 ⎦⎥ n ⎣⎢ k =0

( )

exp − β x n dx =

(

x =∞

( )

γ − k −1 ⎤ exp − βx n ⎡ γ −1 (γ − 1)! x n − ⎥ ⎢∑ k +1 n ⎦⎥ ⎣⎢ k =0 (γ − k − 1)! β

m +1 ⎤ ⎡ ⎢⎣γ = n = 1,2,..., n ≠ 0, β ≠ 0⎥⎦

m +1 ⎡ ⎤ ⎢⎣γ = n = 1,2,...⎥⎦ • Based on incomplete gamma (special case) expansion, Form. 8.352.4, Γ(n, x ) = (n − 1)! e − x

n −1

xm

∑ m!

[n = 1,2,...]

m=0

m +1 = γ , x = βx n n (a) State integral as incomplete gamma function, Form 2.33.10, (b) translate into incomplete gamma (special case), Formula 8.352 4, noting that all terms in summation expression are positive:

where substituted is n = •

Sect. 2.32

43

∫x

m − βx n

e

dx =

(

)

(γ − 1)!e −βx Γ γ , βx n − = − nβ γ nβ γ

( )

⎡ γ −1 βx n k ⎤ ⎢∑ ⎥ ⎢⎣ k =0 k! ⎥⎦ • Writing summation in reverse order, and simplifying m − βx n ∫ x e dx =

−

(γ − 1)!e

n

( )

− βx n

( )

⎡ β γ −1 x n γ −1 β γ −2 x n γ −2 ⎤ + ⎢ ⎥ (γ − 2)! ⎥ ⎢ (γ − 1)! γ −3 ⎢ ⎥ γ −3 xn 1 ⎥ ⎢+ β +K+ ⎢⎣ (γ − 3)! 0! ⎥⎦

( )

nβ γ

( )

( )

⎡ x n γ −1 (γ − 1) x n γ −2 ⎤ + ⎥ n ⎢ β2 e − βx ⎢ β ⎥ =− γ −3 ⎢ n n (γ − 1)!⎥⎥ ⎢+ (γ − 1)(γ − 2) x + + K β3 β γ ⎦⎥ ⎣⎢

( )

− e − βx = n

n

( )

⎡ γ −1 (γ − 1)! x n γ − k −1 ⎤ ⎢∑ ⎥ by inspection, noting k +1 ⎢⎣ k =0 (γ − k − 1)! β ⎥⎦

relation, (γ − 1)! k=>0(γ − 1)(γ − 2)(γ − 3)K (γ − k ), (γ − k − 1)!

where the vaule of 1 is understood for k = 0. NOTE: new version translates more readily into calculations for γ = 1, 2, 3, 4, Formulas 2.33.12 – 2.33.15. NOTE: new version is easily derived from above solution, m ax n x ∫ e dx, by setting a = − β , to give,

∫x

m − βx n

e

( )

n γ − k −1 ⎤ ( e − βx ⎡ γ −1 γ − 1)! x n k dx = ⎢ ∑ (− 1) ⎥= (γ − k − 1)! (− β )k +1 ⎦⎥ n ⎣⎢ k = 0

( )

n γ − k −1 ⎤ e − βx ⎡ γ −1 (γ − 1)! x n − ⎢∑ k +1 ⎥ n ⎣⎢ k = 0 (γ − k − 1)! (β ) ⎦⎥

where (− 1) cancels and negative sign is taken outside. NOTE: Formula useful for writing the log-expansions in Section 2.7 which can be transformed from log. function to k

n

exponential function of the form x m e − βx ; e.g., see Sect. 2.7, below, for new formula, (ln x )n dx = s n −(m−1)s , by change of variable ∫ x m ∫ e ds dx s ⎛ 1− m − ( m −1)s ⎞ ⎜ s = ln x, ds = , e = x, x = e ⎟. x ⎝ ⎠ Sect. 2.32

44

Formula 2.33.12 n

e − βx nβ

n

∫s e (

n − m −1)s

ds , is solved below from x equivalent expression, (ln x )n dx = s n (1− m )s . ∫ xm ∫ e ds • Substitute values of integer γ into Formula 2.33.11 • Exercises given below, this section. NOTE:

m − βx ∫ x e dx = −

∫

(ln x )n dx =

m +1 ⎤ ⎡ ⎢⎣γ = n = 1⎥⎦

m

Formula 2.33.13

∫x

m

n

e − βx dx = −

e − βx n

n

⎛ xn 1 ⎞ ⎜⎜ + 2 ⎟⎟ β β ⎝ ⎠

m +1 ⎤ ⎡ ⎢γ = n = 2⎥ ⎦ ⎣

Formula 2.33.14 m − βx ∫ x e dx = − n

e − βx n

n

⎛ x2n 2 xn 2 ⎞ ⎜⎜ + 2 + 3 ⎟⎟ β β β ⎠ ⎝

m +1 ⎡ ⎤ ⎢γ = n = 3⎥ ⎣ ⎦

Formula 2.33.15 m − βx ∫ x e dx = − n

e − βx n

n

⎛ x 3n 3 x 2 n 6 x n 6 ⎞ ⎜⎜ + 2 + 3 + 4 ⎟⎟ β β β ⎠ ⎝ β

m +1 ⎡ ⎤ ⎢γ = n = 4⎥ ⎣ ⎦

Formula 2.33.16

∫e

− βx 2

1 π dx = erf 2 β

[

βx

Change of variable, t = β x , and definition of

•

]

erf (x ) = o Set

⎤ 1 ⎡ e ax e ax e ax = − + dx ⎥ dx na ⎢ m −1 m−n ∫ xm ∫ x m − 1 ⎣⎢ x ⎦⎥ n

n

n

2

x

2

2

π

e π ∫

−t ∫ e dt

−t

0

2

= erf (t ) and apply limits, noting that

erf (0 ) = 0. • See Form. 2.33.2 for similar case Use Integration by Parts. n n Let u = e ax , du = nax n−1e ax dx

dv = x −m dx,

v = ∫ x −m dx = −

1 (m − 1)x m−1

⎤ 1 ⎡ e ax e ax − + ∴ uv − ∫ vdu = dx ⎥ na ⎢ m−1 m−n ∫ m − 1 ⎢⎣ x x ⎥⎦ Check : If n = 1, then Formula 2.324.1 gives, n

I=

1 ⎡ e ax e ± ax ⎤ e ± ax = − ± dx na ⎢ ∫ xm ∫ x m−n dx⎥⎥ m − 1 ⎢⎣ x m−1 ⎦ Formula 2.325.6

1 ⎡ e ax e ax e ax ⎤ = − + dx a dx ⎥ ⎢ m − 1 ⎣ x m −1 xm x m−1 ⎦

∫

n

Formula 2.325.5 n

∫

n

n

n

e − ax Same as above for ∫ m dx and signs assigned. x • See Sect. 2.31, above, for derivation; then apply definition of incomplete gamma function Sect. 2.32

45

(

)

( − 1) a z Γ − z ,− ax n e ax ∫ x m dx = n z +1 z ∞ (− 1) a 1 −t m −1 = e dt z = z +1 ∫ n n t − ax n

• Γ(•,•) is incomplete gamma function, Formula 8.350 .2, given as Γ(α , x ) • Verify solution by differentiating under the integral sign

Formula 2.325.8

• •

z +1

n

ax n

e dx = xm n e ax ⎡ z −1 ( z − k − 1)! a k ⎤ − ⎢∑ ⎥ n ⎣ k =0 z! x n ( z −k ) ⎦

∫

+ az

( ) n

Ei ax n z!

m −1 ⎡ ⎤ ⎢n ≠ 0, a ≠ 0, z = n = 1,2,..., ⎥ ⎢ ⎥ ⎣m = 2,3,... ⎦

Improved Formula 2.325.8 for integer z. Derived from Formula 8.352 .3, as given in GR 6 for real x, Γ(− n, x ) =

(− 1)n

n −1 k! ⎤ ⎡ k Γ(0, x ) − e − x ∑ (− 1) k +1 ⎥ ⎢ n! ⎣ x ⎦ k =0

•

Restate integral as incomplete gamma function, as previously disclosed in Form. 2.325.6: n (− 1)z +1 a z Γ − z,−ax n , z = m − 1 e ax = dx ∫ xm n n • Plug into: (− 1)n ⎡Γ(0, x ) − e − x n −1 (− 1)k k! ⎤ , substituting Γ(− n, x ) = ∑

(

n! ⎢⎣

n=z=

)

x k +1 ⎥⎦

k =0

m-1 , x = −ax n , n

and noting that Γ(0,− x ) = − Ei( x ) to give: n

e ax ∫ x m dx = ⎧ (− 1)z +1 a z ⎪ (− 1)z ⎨ n ⎪ z! ⎩

(

)

⎡Γ 0,−ax n − ⎢ z −1 k! ⎢e ax n (− 1)k ⎢ ∑ k =0 − ax n ⎣

z +1 z ( − 1) (− 1) (− 1)a z Ei(ax n ) = −

(

⎤⎫ ⎥⎪ ⎥⎬ k +1 ⎪ ⎥ ⎦⎭

)

n z!

(− 1)z +1 (− 1)z a z e ax n z!

n

(− 1)k k!

z −1

∑ (− 1) (−1)a k =0

k

k +1

xn

( k +1)

( )

k! az a z ax n z −1 + Ei ax n e ∑ ( k +1) k n + 1 n z! n z! k =0 a x • Write summation term in reverse order and simplify, =−

Sect. 2.32

46

(z − 2)! (z − 3)! ⎤ ⎡ (z − 1)! + z −1 n ( z −1) + z − 2 n ( z − 2 ) ⎥ z nz ⎢ a ax n a x a x a x − e ⎢ ⎥ 1 n z! ⎥ ⎢+ K + n ⎥⎦ ⎢⎣ ax z

⎡ ( z − 1)! ( z − 2 )!a ( z − 3)!a 2 ⎤ + + ⎥ n ( z −1) e ax ⎢ z! x nz z ! x z! x n ( z −2 ) ⎥ ⎢ =− n ⎢ ⎥ a z −1 K + + ⎢ ⎥ n z! x ⎣ ⎦ n

a ⎤ ⎡ 1 ⎥ ⎢ zx nz + z ( z − 1)x n ( z −1) + e ⎢ ⎥ =− n ⎢ a2 a z −1 ⎥ +K+ ⎥ ⎢ n ( z −2 ) z! x n ⎦ ⎣ z (z − 1)( z − 2 )x ax n

e ax ⎡ z −1 ( z − k − 1)! a k ⎤ =− ⎢ ⎥ n ⎣ k =0 z! x n ( z −k ) ⎦ by inspection, noting the factorial relation, (z − k − 1)! = 1 . z! z ( z − 1)(z − 2)K( z − k ) NOTE: new version translates more readily into calculations for z = 1,2,3, Formulas 2.325.9 – 2.325.11. NOTE: Formula 2.324.2 may be restated from above integral expansion by letting n

∑

m = n, n = 1, z = n − 1, to give :

Formula 2.325.7 n e ax Ei ax n dx = ∫ x n

( )

Formula 2.325.9

n−2 ( e ax n − k − 2 )! a k ⎤ a n −1 ax ⎡ Ei(ax ) + ∫ x n = −e ⎢⎣∑ (n − 1)! x n − k −1 ⎥⎦ (n − 1)! k =0 which differs only in the start/end summation limits and statement of factorial terms. e ax NOTE: Formula 2.324.2, ∫ n dx, is a special case of x ax m m −1 e ∫ x n dx, z = n , with n = 1, m → n, z = m − 1. n Γ(0,− ax n ) Ei(ax n ) e ax Derived from, ∫ , = dx = − x n n m −1 =0. in Form. 8.359.1, where z = n ¾ See Mathematical Summary and Sect. 8.212, below, for definitions of ± Ei(± x ) in terms of incomplete gamma function, Γ(α , x ). Substitute values of integer z in 2.325.6

Sect. 2.32

47

n

n

e ax aEi(ax n ) − e ax ∫ x m dx = nx n + n m -1 ⎤ ⎡ ⎢ z = n = 1⎥ ⎦ ⎣ Formula 2.325.10 n

n

n

e ax ae ax a 2 Ei(ax n ) − e ax dx = − + ∫ xm 2n 2nx 2 n 2nx n m -1 ⎤ ⎡ ⎢ z = n = 2⎥ ⎦ ⎣

Formula 2.325.11 n

e ax ∫ x m dx = n

n

n

ae ax a 2 e ax a 3 Ei(ax n ) − e ax − − + 6n 3nx 3n 6nx 2 n 6nx n m -1 ⎡ ⎤ ⎢ z = n = 3⎥ ⎣ ⎦ Formula 2.325.12 2 2 e ax e ax ∫ x 2 dx = − x + aπ erfi a x Formula 2.33.19 n β z Γ − z, βx n e − βx ∫ x m dx = − n

[ ]

(

=−

βz

∞

e −t n β∫x n t z +1

Formula 2.33.19

)

z=

m −1 n

Reduction formula 2.325.5 and apply 2.33.2,

∫e •

ax 2

dx.

See Sect. 2.31 for derivation; then apply definition of incomplete gamma function o Set m → − m in

• • • • • •

∫x

m − βx n

e

Γ(•,•) is incomplete gamma function,

Formula 8.350.2, p. 890 , given as Γ(α , x ) Useful formula for solutions in 3.3-3.4, below. Verify solution by differentiating under the integral sign verified mathematica.com Improved Formula 2.33.19 for integer z Derived from Formula 8.352 .3, as given in GR 6 for real x, (− 1)n ⎡Γ(0, x ) − e − x n−1 (− 1)k k! ⎤ Γ(− n, x ) =

∑

n! ⎢⎣

•

k =0

x k +1 ⎥⎦

Restate integral as incomplete gamma function, as previously disclosed in 2.33.17: n e − βx − β z Γ − z, βx n m −1 = ,z= dx ∫ xm n n Plug into:

(

•

dx

Sect. 2.32

)

48

(

)

(

)

exp − βx n ∫ xm dx = k ⎤ exp − βx n ⎡ z−1 k ( z − k − 1)! β ( ) − − + 1 ⎢∑ n( z −k ) ⎥ n z! x ⎣ k =0 ⎦

(−1)z β

z

(

Ei − βx n

)

(− 1)n

n −1 k! ⎤ ⎡ k Γ(0, x ) − e − x ∑ (− 1) k +1 ⎥ , ⎢ n! ⎣ x ⎦ k =0

Γ(− n, x ) =

m-1 , x = βx n , and noting that n Γ(0, x ) = − Ei(− x ) , to give: substituting n = z = n

nz! ⎡ m −1 ⎤ ⎢ z = n = 1,2,..., m = 2,3,...,⎥ ⎢ ⎥ ⎣n ≠ 0, β ≠ 0 ⎦

e − βx ∫ x m dx = z − β z ⎧⎪ (− 1) ⎨ n ⎪ z! ⎩

z −1 ⎡ (− 1)k k! ⎤ ⎫⎪ n − βx n 0 , Γ − β x e ⎢ ⎥⎬ ∑ n k +1 k =0 β x ⎢⎣ ⎥⎦ ⎪⎭

(

)

( )

− (− 1) β z = − Ei − β x n − n z! z

(− 1)(− 1)z β z e − βx n z!

=

(

n

)

z −1

∑β k =0

(− 1)k k! k +1

x n ( k +1)

(− 1)z β z Ei(− βx n ) + (− 1)z β z e − βx

n

z −1

∑

(− 1)k k!

k +1 n ( k +1) n z! n z! x k =0 β • Write summation term in reverse order and simplify, noting that its first term in oscillating series is negative, ⎡ (− 1)z −1 ( z − 1)! (− 1)z −2 ( z − 2)!⎤ + ⎥ (− 1)z β z e −βxn ⎢⎢ β z x nz β z −1 x n ( z −1) ⎥ ⎢ ⎥ n z! 1 ⎢− K − n ⎥ βx ⎣ ⎦

e − βx = n

n

⎡ (− 1)2 z −1 ( z − 1)! (− 1)2 z −2 ( z − 2 )! β ⎤ + ⎢ ⎥ z! x nz z! x n ( z −1) ⎢ ⎥ z −1 ⎢ ⎥ β z ⎢− K − (− 1) ⎥ n z! x ⎣ ⎦

k ⎤ − e − βx ⎡ z −1 k ( z − k − 1)! β ( ) = − 1 ∑ ⎢ n ( z −k ) ⎥ n ⎣ k =0 z! x ⎦ by inspection, noting relation, (z − k − 1)! = 1 . z! z ( z − 1)( z − 2 )K ( z − k ) NOTE: new version translates more readily into calculations for z = 1, 2, 3, Formulas 2.33.20 – 2.33.22. n

n

e ax NOTE: easily derived from above expansion, ∫ m dx, by x setting a → − β , and rearranging, which adds a

(− 1)k multiplier to summation symbol. Formula 2.33.18 n Ei − βx n e − βx = dx ∫ x n

(

)

See 2.325.7 for similar case ¾ Verify solution by differentiating under the integral sign (convert Ei(•) to definite integral. Sect. 2.32

49

Formula 2.33.20

β Ei(− βx e − n n x nx m −1 ⎤ ⎡ z = = 1⎥ ⎢⎣ n ⎦ Formula 2.33.21 n n n e − βx e − βx β e − βx ∫ x m dx = − 2nx 2n + 2nx n + β 2 Ei − βx n 2n m −1 ⎡ ⎤ ⎢⎣ z = n = 2⎥⎦ Formula 2.33.22 n n n e − βx e − βx β e − βx ∫ x m dx = − 3nx 3n + 6nx 2n −

∫

e

− βx n m

dx = −

(

β 2 e − βx

− βx n

n

Substitute values for integer z in 2.33.19

)

)

n

−

β 3 Ei(− βx n )

6nx n 6n m −1 ⎡ ⎤ z = = 3⎥ ⎢⎣ n ⎦ Formula 2.33.23 2 2 e − βx e − βx ∫ x 2 dx = − x − βπ erf

[

β x

]

Integration by parts (Form. 2.325.5), change of variable on second term, and definition of erf ( x ) , Form. 8.250.1,

2

π Formula 2.321.2 n ax ax ⎡ ∫ x e dx = e ⎢⎣

n

n!

∑ (− 1) (n − k )! k =0

k

x ⎤ a k +1 ⎥⎦ n−k

x

∫e

−t 2

dt .

0

• Proposed alternative restatement of 2.321.2. In above expansion, n γ − k −1 ⎤ ( e ax ⎡ γ −1 γ − 1)! x n k m ax n ( ) x e dx = − 1 ⎢ ⎥ ∑ ∫ (γ − k − 1)! a k +1 ⎥⎦ n ⎢⎣ k =0 m +1 ⎡ ⎤ ⎢⎣γ = n = 1,2,...⎥⎦ set n = 1, m → n, γ = n + 1 and substitute. NOTE: proposed formula is essentially the same as suggested in online GR 6 errata, and published in GR 7, http://www.mathtable.com/errata/gr6_errata.pdf, Errata #47, by Drs. Moll and Boros— except we state n! for the equivalent expression, (n − k )! ⎛n⎞ n! k!⎜⎜ ⎟⎟ = k! , k!(n − k )! ⎝k ⎠ n! noting that, = n(n − 1)(n − 2 )K (n − k + 1). (n − k )!

( )

Sect. 2.32

50

Formula 2.324.2

e ax ∫ x n dx = ⎡ n−2 (n − k − 2 )! a k ⎤ + − e ax ⎢∑ n − k −1 ⎥ ( ) n 1 ! x − k = 0 ⎦ ⎣ a n−1 Ei(ax ) (n − 1)!

− βx

⎛x 1 ⎞ dx = − e − βx ⎜⎜ + 2 ⎟⎟ ⎝β β ⎠

[β ≠ 0]

∫x e

2 − βx

⎛x 2x 2 ⎞ dx = − e − βx ⎜⎜ + 2 + 3 ⎟⎟ β ⎠ ⎝ β β 2

[β ≠ 0]

3 3x2 6x 6 ⎞ 3 −βx −βx ⎛ x ⎜ = − + x e dx e ∫ ⎜ β β 2 + β 3 + β 4 ⎟⎟ ⎝ ⎠ [β ≠ 0]

e − βx dx ∫ x = Ei(− βx ) [β ≠ 0] e − βx dx ∫ xn = k ⎡ n−2 ⎤ k (n − k − 2 )! β + e − βx ⎢∑ (− 1) n − k −1 ⎥ (n − 1)! x ⎦ ⎣ k =0 (− 1)n−1 β n−1 Ei(− βx ) [β ≠ 0] (n − 1)! e − βx dx − e − βx ∫ x 2 = x − βEi(− βx ) [β ≠ 0] e − βx dx − e − βx β e − βx 2 Ei(− β x ) ∫ x3 = 2x2 + 2x + β 2 [β ≠ 0] e − βx dx ∫ x4 =

( )

n − e ax ⎡ z −1 ( z − k − 1)! a k ⎤ e ax z Ei ax = + dx a ⎢∑ ⎥ ∫ xm n ⎣ k =0 z! n z! x n( z −k ) ⎦ m −1 ⎤ ⎡ ⎢ z = n = 1,2,..., m = 2,3,...⎥ ⎦ ⎣ n

n

[a ≠ 0]

n n! x n −k ⎤ n − βx − βx ⎡ x e dx e = − ⎢∑ k +1 ⎥ ∫ ⎣ k =0 (n − k )! β ⎦ [β ≠ 0]

∫ xe

• Alternative expression for Formula 2.324.2: Set n = 1, m → n, z = n − 1 in above integral,

n n! x n−k ⎤ −βx ⎡ n − βx = − x e dx e ⎢∑ (n − k )! β k +1 ⎥ is derived from ∫ ⎣ k =0 ⎦ general case, n γ − k −1 ⎤ − e − βx ⎡ γ −1 (γ − 1)! x n m − βx n = x e dx ⎢ ⎥ ∑ ∫ n ⎣⎢ k =0 (γ − k − 1)! β k +1 ⎦⎥

•

( )

m +1 ⎡ ⎤ γ = = 1,2,...⎥ ⎢⎣ n ⎦ by substituting n = 1, m → n, γ = n + 1 •

∫ xe

− βx

dx,∫ x 2 e − βx dx & ∫ x 3 e − βx dx obtained by setting n

= 1, 2, 3, respectively, in n n! x n − k ⎤ n − βx − βx ⎡ = − x e dx e ⎢∑ (n − k )! β k +1 ⎥. ∫ ⎣k =0 ⎦

e − βx dx Ei(− βx n ) = From ∫ with n = 1. x n n

•

k n−2 ⎤ e − βx k (n − k − 2 )! β − βx ⎡ (− 1) + n − k −1 ⎥ ∫ x n dx = −e ⎢⎣∑ (n − 1)! x ⎦ k =0

(− 1)n−1 β z Ei(− βx ) (n − 1)! n

is derived from general case, n n k ⎤ e −βx − e −βx ⎡ z−1 k ( z − k −1)! β ( ) dx = − 1 ⎢∑ ⎥+ ∫ xm n ⎣k =0 z! x n( z −k ) ⎦

(−1) z β z Ei(− βx

)

⎡ m −1 ⎤ z= = 1,2,..., m = 2,3,...⎥ ⎢ n z! n ⎣ ⎦ by substituting n = 1, m → n, z = n − 1 •

n

e − βx dx e − βx dx e − βx dx ∫ x 2 ,∫ x3 & ∫ x 4 : set n = 2, 3, 4, e − βx respectively in ∫ n dx. x

Sect. 2.32

51

Ei(− βx ) − e − βx β e − βx β 2 e − βx + − − β3 3 2 3x 6x 6x 6 [β ≠ 0]

∫x

n

x n+1 (n ≠ −1). Show derivation n +1 using exponential functions. One approach: In integral, I = ∫ x n dx , set x n = e n ln x from definition.

dx

A basic integral equal to

Then, I = ∫ e n ln x dx . Use change of variable,

dx s , e = x , to give, x et ( n +1)s I = ∫e ds leading to I = ∫ dt by a second n +1 change of variable, t = (n + 1)s , which leads to answer, s = ln x, ds =

et e (n+1)s e ( n +1) ln x x n +1 = = = n +1 n +1 n +1 n +1 by back substitution and laws of exponents. NOTE: Other approaches are possible once I = ∫ e (n +1)s ds is obtained. I=

1 ∫ x n dx 1 ∫ x 2 dx & x ∫ x dx

Exercise. Similar to ∫ x n dx .

1

∫x

3

dx

Exercise. Check against solution above for

∫x

e

1− γ e ( − 1) dx =

m +1 γ= n

n

dx .

Computers cannot solve this indefinite integral in closed form. Try it on mathematica.com. We can write, from definition,

x x = (e ln x ) = e x ln x . And the derivative is given by the chain rule, dx x e x ln x d ( x ln x ) x ln x =d = e = (ln x + 1)e x ln x dx dx dx x = x (ln x + 1). Substitute and see result. No elementary forms can be substituted to provide a closed form solution. See GR, Section 1.211, for series expansion of x x by looking at a x and setting a → x. Derived by above formula (with t set to s ) , x

m ax n

1

∫x

na γ

ax n

∫ 0

γ −1

⎛ 1⎞ ⎜ ln ⎟ dt ⎝ t⎠

∫x

m ax n

e

1− γ ( − 1) Γ(γ ,−ax n ) dx =

(− 1)1−γ

naγ

∞

m +1 . na − ax n n Use change of variable on definite integral, =

γ

Sect. 2.32

∫s

γ −1 − s

e ds, γ =

52

(− 1)1−γ na

∞

∫s

γ

γ −1 − s

e ds :

− ax n

t = e − s , dt = −e − s ds = −tds n

t −ax n = e ax , t ∞ = 0, ln t = − s, s = − ln t = ln

1 t

Then,

1−γ ( − 1) I=

na

0

∫ (− ln t )

γ

1−γ ( − 1) =

e ax e ax

na γ Thus,

n

∫ 0

m ax ∫ x e dx = n

(− 1)1−γ na γ

∞

∫s

− ax n

γ −1

n

⎛ 1⎞ ⎜ ln ⎟ ⎝ t⎠

t dt −t

γ −1

dt

(− 1)1−γ Γ(γ ,−ax n ) = na γ

γ −1 − s

e ds

1−γ ( − 1) =

na γ

e ax

n

∫ 0

⎛ 1⎞ ⎜ ln ⎟ ⎝ t⎠

γ −1

dt ,

m +1 . n • Derive similar logarithmic expressions for these 3 algebraic-exponential integrals given earlier: n n e ax e − βx m − βx n ∫ x e dx & ∫ x m dx & x m dx ¾ Verify each solution by diff. under the integral We show two alternative solutions. • Separate the terms, ax e −1 e ax 1 dx = ∫ eax +1 ∫ eax +1dx − ∫ eax +1 dx

γ=

∫

e ax − 1 2 dx = ln e ax + 1 − x ax e +1 a

(

)

ax ax − 2 ⎛ 2 ⎜ = ln ⎜ e + e 2 a ⎝

⎞ ⎟ ⎟ ⎠

• Change of variable: ax s = e , ds = ae ax dx = asdx Then, s ds 1 ds 1 ⎡ ds ds ⎤ I =∫ −∫ = ⎢∫ −∫ (s + 1) as (s + 1) as a ⎣ s + 1 s(s + 1)⎥⎦ Use partial fractions for second integral, 1 1 1 , to give, = − s (s + 1) s s + 1

I= =

1 ⎡ ds 1 ⎞ ⎤ ⎛1 −∫ ⎜ − ⎟ds ⎢ ∫ a ⎣ s + 1 ⎝ s s + 1 ⎠ ⎥⎦ 1 [ln(s + 1) − ln s + ln(s + 1)] = 1 [2 ln(s + 1) − ln s ] a a Sect. 2.32

53

•

At this stage two solutions can be obtained. First solution is obtained by substituting e ax : 1 I = [2 ln (s + 1) − ln s ] a 1 = 2 ln e ax + 1 − ln e ax a 1 2 ln e ax + 1 = 2 ln e ax + 1 − ax = −x a a ¾ This is computer solution at mathematica.com Also, using the log. relation in Math. Summary, ⎞ ⎛ ⎜ x ⎟ a ln x − b ln y = a ln⎜ b ⎟, with a = 2, b = 1, ⎜ ya ⎟ ⎠ ⎝ the second solution is: ⎛ ⎞ 1 2 ⎜ s + 1⎟ I = [2 ln(s + 1) − ln s ] = ln⎜ 1 ⎟ a a ⎜ 2 ⎟ ⎝ s ⎠

[ (

)

[ (

)

]

]

(

)

1 1 ax ax − ⎞ − ⎞ 2 ⎛ 2 2 ⎛ ln⎜⎜ s + s 2 ⎟⎟ = ln⎜⎜ e 2 + e 2 ⎟⎟ a ⎝ ⎠ a ⎝ ⎠ ¾ Cited in Prudnikov, et al., Vol. 1, Formula 1.3.1.7, p. 137 ax ax − ⎞ e ax + 1 2 ⎛ ¾ Exercise: Show ∫ ax dx = ln⎜⎜ e 2 − e 2 ⎟⎟ e −1 a ⎝ ⎠ ¾ One solution solved by separation of terms, change of variable, s = e ax , partial fraction, 1 1 1 , and relation, =± m x(x ± a ) ax a( x ± a )

=

m ax nax − ⎞ e ax − m m + n ⎛⎜ m + n m+n ⎟ ln dx e ne = + ∫ eax + n ⎟ ⎜ na ⎠ ⎝

⎞ ⎛ ⎜ x ⎟ a ln x − b ln y = a ln⎜ b ⎟ . ⎜ ya ⎟ ⎠ ⎝ ¾ Computer provides alternative solution, (m + n ) ax e ax − m mx ∫ e ax + n dx = na ln e + n − n . e ax − 1 ¾ Check against integral solution, ∫ ax dx e +1

(

m ax nax − ⎞ e ax + m m + n ⎛⎜ m + n m+n ⎟ = − dx ln e ne ax ∫ e −n ⎜ ⎟ na ⎝ ⎠

[(

n − m m ax e ax + m 1 ax ∫ eax + n dx = na ln e + n e

)

]

)

with m = n = 1, and by differentiation. Exercises. Similar to above integral, nax m ax − ⎞ e ax − m m + n ⎛⎜ m+ n m+ n ⎟ = + dx ln e ne ∫ eax +n ⎜ ⎟ na ⎝ ⎠ • Each integral has two alternative solutions. We give Sect. 2.32

54

[(

n − m m ax e ax − m 1 ax ∫ eax −n dx = na ln e − n e

)

]

only one solution, based on the log. relations, ⎛ ba ⎞ a b a ln x + b ln y = ln x y = a ln⎜⎜ xy ⎟⎟ ⎠ ⎝

(

)

⎞ ⎛ ⎜ x ⎟ ⎞ ⎟⎟ = a ln⎜ b ⎟ ⎠ ⎜ ya ⎟ ⎠ ⎝ ¾ The computer gives the second solution. Verify solution to this basic integral by earlier exponential integral formula, (− 1)1−γ Γ γ ,−ax n = (− 1)1−γ ∞ t γ −1e −t dt m ax n x e dx = ∫ ∫n na γ na γ −ax ⎛ xa a ln x − b ln y = ln⎜⎜ b ⎝y

∫ e dx = e x

x

(

)

m +1 , with γ = 1, and Γ(n,-x ), Sect. 8.352, below. n Show by Formula 2.33.11, above, ∞ Γ γ , βx n 1 m − βx n x e dx t γ −1e −t dt = − = − γ γ ∫ ∫ nβ nβ β x n

γ=

∫e

−x

dx = −e − x

(

)

m +1 , with γ = 1, and Γ(n, x ), Sect. 8.352, below. n • Use change of variable, s = e ax , ds = ae ax dx = asdx • Then, by partial fraction expansion and rules of logarithms, 1 1 ds 1 ⎛ 1 1 ⎞ = ∫⎜ − I= ∫ ⎟ds a (s + 1) s a ⎝ s s + 1 ⎠

γ=

(

)

dx ln e ax + 1 x = − ∫ 1 + eax a

ax 1 [ln(s ) − ln(s + 1)] = 1 ln⎛⎜⎜ e ax ⎞⎟⎟ a a ⎝1+ e ⎠ 1 ln 1 + e ax = ln e ax − ln 1 + e ax = x − a a • Use change of variable, s = x + b, ds = dx, x = s − b • Then, as ea (s −b ) − ab e I =∫ ds = e ∫ ds = e − ab Ei(as ) s s − ab = e Ei(ax + ab )

=

[ ( ) (

e ax − ab ∫ x + bdx = e Ei(ax + ab)

(

)]

n

(

)

( )

( )

e ax Ei ax n dx = , with n = 1. By Formula 2.325.7, x n Exercise. Check answer on mathematica.com

∫

x +1 − e − ax (ax + a + 1) dx = ∫ eax a2 x x2 x dx = x log 1 − e − + Li 2 e x ∫ e x −1 2

)

•

Uses a special function, the dilogarithm, Li 2 ( x ), defined

Sect. 2.32

55

∞

zk , 2 k =1 k

as an infinite series, Li 2 (z ) = ∑

z < 1. The

indefinite integral is taken as: Li 2 (x ) = − ∫

log(1 − x ) dx. x

x dx , use change of variable, s = e x , and e −1 partial fraction expansion to give, ln s ln s I =∫ ds − ∫ ds . s −1 s The second integral is solved by change of variable, t = ln s, to give, •

In I = ∫

x

t 2 (ln s ) x2 ln s ds = tdt = = = . ∫ s ∫ 2 2 2 The first integral is more involved. The exercise below will show that, after integration by parts, ln s ln s ∫ s − 1ds = ∫ 1 − s ds = ln s ln(1 − s ) + Li2 (s ) , so that, x x2 ( ) ( ) s − s + s − ln ln 1 Li dx = 2 ∫ e x −1 2 x which leads to solution when e is substituted, x x2 x dx = x log 1 − e − + Li 2 e x ∫ e x −1 2 NOTE: Do not confuse notation of Li 2 ( x ) with logarithmintegral function, li( x ). • Exercise: derive selected common dilogarithm relations: 2

(

Sect. 2.32

)

( )

56

ln (1 ± x )dx

∫

= −Li 2 (m x ) =

x

⎛ ⎝

ln xdx

∫

∫ ∫

ln xdx

=

a−x

∫

⎛ x⎞ ⎟ + Li 2 ⎜ m ⎟ a⎠ ⎝ a⎠

ln xdx x−a

ln (a ± x )dx

2

(see above ) ⎡

⎛ ⎝

= ln x ⎢ln (a ± x ) − ln⎜ 1 ±

⎣

x

⎛ 1⎞ 2⎜m x ⎟ ⎝ ⎠

x⎞

= ln x ln⎜ 1 ±

x±a

(ln x )2 − Li

x ⎞⎤

⎛ x⎞ ⎟⎥ −Li 2 ⎜ m ⎟ a ⎠⎦ ⎝ a⎠

⎛ x ⎞ (ln x ) ⎛ a⎞ − Li 2 ⎜ m ⎟ ⎟= 2 ⎝ x⎠ ⎝ a⎠ 2

= ln a ln x − Li 2 ⎜ m ∫ ∫

ln ( x − a )dx x

⎡

⎛ ⎝

= ln x ⎢ln ( x − a ) − ln ⎜ 1 −

ln (a ± bx )dx

⎣

⎡

2

⎟⎥ −Li 2 ⎜

a ⎠⎦

⎛ ⎝

= ln x ⎢ln (a ± bx ) − ln ⎜ 1 ±

⎣

x

⎛ x ⎞ (ln x ) ⎛a⎞ − Li 2 ⎜ ⎟ ⎟= 2 a ⎝ ⎠ ⎝x⎠

x ⎞⎤

bx ⎞ ⎤

⎛ bx ⎞ ⎟⎥ −Li 2 ⎜ m ⎟ a ⎠⎦ ⎝ a⎠

⎛ bx ⎞ (ln bx ) ⎛ a⎞ − Li 2 ⎜ m ⎟= ⎟ 2 ⎝ a⎠ ⎝ bx ⎠ 2

= ln a ln x − Li 2 ⎜ m

∫

ln (bx − a )dx

⎣

x

(ln bx )

2

=

⎡

2

⎛ ⎝

= ln x ⎢ln (bx − a ) − ln⎜1 −

bx ⎞ ⎤

⎛ bx ⎞ ⎟⎥ −Li 2 ⎜ ⎟ a ⎠⎦ ⎝a⎠

⎛a⎞ ⎟ ⎝ bx ⎠

− Li 2 ⎜

⎡ ⎛ ⎡ ⎛ a ⎞⎤ x ⎞⎤ ln ⎢a⎜⎜1 ± ⎟⎟ ⎥ dx ln ⎢ x⎜⎜ ± 1⎟⎟ ⎥ dx ln (a ± x )dx ⎣ ⎝ a ⎠⎦ ⎣ ⎝ x ⎠⎦ =∫ =∫ Ex: ∫ ; first x x x

integral requires two integrations by parts with partial fraction identity on du term; second and third solutions use log. relation, log xy = log x + log y, and definition of Li 2 ( x ).

∫

dx dx dx & & b b a − be mx a + mx a − mx e e

∫

∫x e

3 − βx 2

∫x e

2 − βx

dx

•

NOTE: Li 2 ( x ) has useful results important for definite π2 π2 . See The integrals; e.g., Li 2 (1) = ; Li 2 (− 1) = − 6 12 Wolfram Function Site http://functions.wolfram.com/.

•

Three exercises. See

mx

derivation in the

Mathematical Summary. • Check solution on mathematica.com Derive from Formula 2.33.12, above,

∫x dx

dx

∫ a + be

m − βx n

e

dx

m +1 ⎤ ⎡ ⎢⎣γ = n = 2⎥⎦

Derive from Formula 2.33.12, above,

∫x •

m − βx n

e

dx

m +1 ⎤ ⎡ ⎢⎣γ = n = 3⎥⎦

Try other combinations of m, n for γ =

Sect. 2.32

m +1 = integer. n

57

∞

x

e−t et Ei( x ) = − ∫ dt = ∫ dt = li e x , x < 0 t t −x −∞

( )

In this definition, show equivalence of two integrals by stating the indefinite integral forms, substituting the limits, and applying properties of the exponentialintegral function summarized in Mathematical ex

( ) = ∫ lndtt

Summary. Then show Ei( x ) = li e

x

by change

0

−ε

⎡ e ⎤ e Ei( x ) = − lim ⎢ ∫ dt + ∫ dt ⎥, x > 0 ε → +0 t ε ⎣− x t ⎦ −t

∞

−t

x

dt = Ei(ln x ), x < 1 ln t 0

li( x ) = ∫

x ⎡1−ε dt dt ⎤ li( x ) = lim ⎢ ∫ + ∫ ⎥ = Ei(ln x ), ε →0 ⎣ 0 ln t 1+ε ln t ⎦ x >1

xe x ex dx = ∫ (1 + x )2 1 + x

of variable, s = ln t. Cauchy Principal Value (PV) for definition of exponentialintegral function, x > 0 . • Show the limit calculation by stating the indefinite integral forms, substituting the limits, applying properties of the exponential-integral function in Mathematical Summary, and evaluating the limit, ε → +0. In this definition, show the relation, li(x ) = Ei(ln x ), x < 1, by change of variable, x dt s = ln t on integral, ∫ . ln t 0 Cauchy Principal Value PV for definition of logarithmintegral function, li( x ), x > 1. • Perform same operations as for above exponentialintegral function, Ei( x ) • Change of variable,

s = x + 1, ds = dx

(s − 1)e s −1 ds = e −1 ⎡

es es ⎤ − ds ⎢∫ s ∫ s 2 ds ⎥⎦ s2 ⎣ Integration by parts on first integral with 1 ds u = , du = − 2 ; dv = e s ds, v = e s , s s to produce, s es es ⎤ −1 ⎡ e I = e ⎢ + ∫ 2 ds − ∫ 2 ds ⎥ s s ⎣s ⎦ I =∫

•

⎛ es ⎞ e x +1 ex = e −1 ⎜⎜ ⎟⎟ = e −1 = . x +1 x +1 ⎝ s ⎠ ¾ NOTE: Advanced solution. In first integral above, es ∫ s ds = Ei(s ) , from definition, es ∫ s 2 ds = Γ(− 1,−s ) by Formula 2.325.7, above, (− 1) z +1 a z Γ(− z,−ax n ) (− 1) z +1 a z ∞ 1 −t m −1 e ax dx = e dt , z = = n

∫

xm

n

Then, Γ(− 1,− s ) = Ei(s ) − Sect. 2.32

n

s

∫

− ax n

t z +1

n

e by Γ(− 1,− x ) in Sect. 8.352, s

58

below so that, (s − 1)e s −1 ds = e −1 ⎡ e s ds − e s ds ⎤ I =∫ ⎢∫ ∫ s 2 ⎥⎦ s2 ⎣ s

xe ax e ax dx = ∫ (1 + ax )2 a 2 (1 + ax )

∫e

−

x2 2

dx

⎡ es ⎤ es ex . = e −1 ⎢Ei(s ) − Ei(s ) + ⎥ = e −1 = s s x + 1 ⎣ ⎦ ¾ Cited Dwight, 1961, ITEM 570 xe x dx . Exercise. Similar to above, ∫ (1 + x )2 Check solution on mathematica.com, and compare with above integral for a = 1. ¾ Cited Dwight, 1961, ITEM 570.1 We show that this integral is equal to 2π for limits ± ∞. It is the basis for the Gaussian probability density function. • From Formula 2.33.16, above, 1 1 π − βx ∫ e dx = 2 β erf [ β x]. Set β = 2 and use property for 2

error function, Φ (± ∞ ) = ±1 (see Sect. 8.359, below). Thus, ⎡1 ⎛ x ⎞⎤ + ∞ 1 ⎟⎟⎥ 2π erf ⎜⎜ 2π [erf (+ ∞ ) − erf (− ∞ )] = dx = ⎢ ⎝ 2 ⎠⎦ − ∞ 2 −∞ ⎣2 1 1 2π [1 − (− 1) )] = = 2π (2 ) = 2π . 2 2 +∞

∫e

1 − x2 2

¾ Compare solution with standard approach involving a double integral transformed to polar coordinates. See Bers, Calculus, Vol. 2, p. 928. Also shown in books on probability theory. x2

¾ NOTE: Since the integral ∫ e − 2 dx contains an even function, f ( − x) = f ( x), (see GR, Sect. 3.03), then, +∞

∫e

1 − x2 2

−∞

+∞

dx =2 ∫ e

1 − x2 2

dx .

0

In this approach, first solve the indefinite integral,

∫e

x2 − 2

∫x

dx = −

⎛ 1 x2 Γ⎜⎜ , ⎝2 2 ⎛1⎞ 2⎜ ⎟ ⎝2⎠

m

e

− βx n

⎞ ⎟⎟ ⎠

by Formula 2.33.10,

1 2

(

)

1 Γ γ , βx n =− γ dx = − γ nβ nβ

so that,

Sect. 2.32

∞

∫t

βx n

γ −1 −t

e dt , γ =

m +1 , n

59

+∞ ⎡ ⎛ 1 x2 ⎞ ⎤ 2Γ⎜⎜ , ⎟⎟ ⎥ ⎢ +∞ +∞ 1 1 − x2 − x2 ⎢ ⎝ 2 2 ⎠⎥ 2 2 e dx =2 e dx = 2⎢− ⎥ 2 −∞ 0 ⎥ ⎢ ⎥⎦ ⎢⎣ 0

∫

∫

⎡ ⎛ 1 ⎞ ⎛ 1 ⎞⎤ ⎛1⎞ = 2 ⎢Γ⎜ ,0 ⎟ − Γ⎜ , ∞ ⎟⎥ = 2Γ⎜ ⎟ = 2π ⎝2⎠ ⎣ ⎝ 2 ⎠ ⎝ 2 ⎠⎦

by the properties of incomplete gamma functions, and use ⎛1⎞ of known relation, Γ⎜ ⎟ = π . ⎝2⎠ +∞

NOTE: In the integral

∫

xe

1 − x2 2 dx,

the integrand is

−∞

considered an odd function, f ( − x) = − f ( x), (see GR, Sect. 3.03), so that +∞

∫

1 − x2 xe 2 dx

= 0.

−∞

The integral,

∫

ax

e dx = Ei(ax ) x

xn = 0 for all values of n x → +∞ e x

lim

1 2π

+∞

∫

xe

1 − x2 2 dx

= 0, is the “mean” of the

−∞

Gaussian probability density. See series expansion in Dwight, 1961, ITEM 568.1 NOTE: Dwight book is useful for other series expansions, and contains some information on the gamma function, Γ( x ). • This is an important limit for a fundamental relation of the gamma function, ∞

Γ(x + 1) = ∫ t x e −t dt = xΓ(x ). Using integration by parts 0

with u = t , dv = e −t dt , then, x

∞ t x t = +∞ Γ(x + 1) = ∫ t e dt = − t + x ∫ t x −1e − t dt t 0 = e 0 0 ∞

x −t

⎡ ⎤ tx = − ⎢ lim t − 0⎥ + xΓ( x ) ⎣t → +∞ e ⎦ = xΓ( x ). •

tx = 0 follows from the theorem which t → +∞ e t ⎛∞ ⎞ is derived from l'Hôpital's rule ⎜ case ⎟ by taking ⎝∞ ⎠ repeated derivatives of numerator and denominator (see Bers, Calculus, pp. 547 ff.).

The limit lim

Sect. 2.32

60

xn =0 x → +∞ e x by showing that the numerator eventually becomes n! by repeated differentiation. For example, o Try to prove that lim

dx 2 2dx = = 2 ⋅1 = 2! dx dx dx 3 3dx 2 dx = = 3 ⋅ 2 = 3 ⋅ 2 ⋅1 = 3! dx dx dx n n −1 dx dx dx n−2 =n = n(n − 1) dx dx dx n −3 dx = n(n − 1)(n − 2) dx dx dx = n(n − 1)(n − 2)(n − 3)K3 ⋅ 2 ⋅1 = n! = n(n − 1)(n − 2)(n − 3)K3 ⋅ 2

1 tx = x! lim t = 0. t t → +∞ e t →+∞ e

¾ Apply this result to lim

Sect. 2.32

61

62

FORMULA Sect. 2.711 m

m ∫ ln xdx = x∑ (− 1) k =0

[m ≥ 0]

k

m! ln m−k x (m − k )!

PROOF OUTLINE (formula references are to G & R 7e) • Alternative restatement of 2.711 (2nd formula) • Derived above in Sect. 2.32 under Formula 2.33.5, (γ − 1)! (x n )γ − k −1 ⎤ e ax ⎡ γ −1 k m ax = − x e dx 1 ( ) ⎢ ⎥ ∑ ∫ (γ − k − 1)! a k +1 ⎥⎦ n ⎢⎣ k = 0 n

n

m +1 ⎡ ⎤ ⎢γ = n = 1,2,..., n ≠ 0, a ≠ 0⎥ ⎣ ⎦

NOTE: unlike 2.711, if m = 0, I = x, a trivial result. NOTE on notation: (ln x ) = ln m x ≠ ln x m • Change of variable, b s = ln(a + bx ), ds = dx, e s = a + bx, to a + bx give, 1 es m m! k sm−k , I = ∫ s m e s ds = ∑ (− 1) (m − k )! b k =0 b by above integer-expansion Formula in Sect. 2.32, (γ −1)! (xn )γ −k −1 ⎤ ⎡γ = m + 1 = 1,2,...⎤ eax ⎡ γ −1 k m ax = − ( ) x e dx 1 ⎢ ∑ ⎥ ∫ ⎥ (γ − k −1)! ak +1 ⎦⎥ ⎢⎣ n ⎣⎢ k =0 n ⎦ with parameters, [s = x = ln(a + bx ), m = m, a = n = 1, γ = m + 1] ¾ Reduces to Formula 2.711 for a = 0, b = 1. Change of variable: m

∫ ln (a + bx )dx = m

a + bx m (− 1)k m! ln m−k (a + bx ) ∑ (m − k )! b k =0

n

n

Sect. 2.721 For m = −1 dx ∫ ln x = Ei(ln x ) = li(x )

ds 1 s = , e = e ln x = x. dx x x es ds = ∫ ds = Ei (ln x ) = li ( x ) s s

s = ln x , I =

∫

by integral previously submitted, Form. 2.325.7, n e ax Ei ax n dx = , and Formula 8.240.1. ∫ x n

( )

Sect. 2.721 For m = n = 1

A standard textbook integral, from Formula 2.721.1 with m = n = 1.

x2 x2 ∫ x ln xdx = 2 ln x − 4

—Sect. 2.71 & 2.72-2.73—

63

Sect. 2.722 m! ln m−k x k n m n+1 ( ) ln = − 1 x xdx x ∑ ∫ (m − k )! (n + 1)k +1 k =0 [m ≥ 0] m

• Alternative statement of 2.722 Derived above in Sect. 2.32, under • Formula 2.33.5, (γ − 1)! (x n )γ − k −1 ⎤ e ax ⎡ γ −1 k m ax ( ) 1 x e dx = − ⎥ ⎢ ∑ • ∫ (γ − k − 1)! a k +1 ⎥⎦ n ⎢⎣ k = 0 n

n

m +1 ⎡ ⎤ ⎢γ = n = 1,2,..., n ≠ 0, a ≠ 0⎥ ⎣ ⎦

NOTE: unlike 2.722, if m = 0, I =

x n+1 ,a n +1

trivial result. Sect 2.721 or new Section n n ∫ ln(ln x ) dx = x ln(ln x ) − nli( x)

• Change of variable: dx s = ln x, ds = , e s = eln x = x . Then, x n s I = ∫ x ln s = ∫ e (n ln s )ds by relation log x n = n log x • Integration by parts: n u = n ln s, du = ds s dv = e s ds, v = ∫ e s ds = e s es ds s dx n Substituting s = ln x, ds = , e s = x, n ln s = ln(ln x ) , x n n I = x ln(ln x ) − nEi(s ) = x ln(ln x ) − nEi(ln x ) = uv − ∫ vdu = e s (n ln s ) − n∫

x ln(ln x ) − nli(x ) by 8.240.1. n

¾ Verified Mathematica

—Sect. 2.71 & 2.72-2.73—

64

Sect 2.721 or new section m x n +1 ln (ln x ) − mli( x n +1 ) m n ∫ x ln(ln x ) dx = n +1

For n = −1 m ln (ln x ) m ∫ x dx = ln x ln(ln x ) − m

[

•

Math.com calculations; same approach as for above integral, n n ∫ ln(ln x ) dx = x ln(ln x ) − nli( x)

•

]

For first integral, change of variable: dx s = ln x, ds = , e s = e ln x = x, x n+1 = e (n+1)s x I = ∫x

n +1

(ln s )m ds = ∫ e (n+1)s ln s m ds = ∫ e (n+1)s (m ln s )ds

by relation log x n = n log x • Then, int. by parts with, u = m ln s, dv = e (n+1)s ds. NOTE: for n = 0, m = n, see above integral, n n ∫ ln(ln x ) dx = x ln(ln x ) − nli( x)

•

For n = −1 : change of variable & int. by parts; dx s = ln x, ds = x m x ln(s ) I =∫ ds = ∫ m ln sds. x int. by parts : m u = m ln s,du = ds s dv = ds, v = ∫ ds = s s uv − ∫ vds = s (m ln s ) − m ∫ ds = s dx m m ln x ln (ln x ) − m ∫ = ln x ln (ln x ) − m ln x x

—Sect. 2.71 & 2.72-2.73—

65

•

Section 2.724 x n dx ∫ (ln x )m = ⎡m − 2 (m − k − 2 )! (n + 1)k ⎤ (n + 1)m −1 − x n +1 ⎢ ∑ + li x n +1 m − k −1 ⎥ ⎣ k = 0 (m − 1)! (ln x ) ⎦ (m − 1)!

( )

Change of variable, dx s = ln x, ds = , e s = x, x n +1 = e (n +1)s , to give: x n x xds e(n +1)s I = ∫ m = ∫ m ds s s • In above integral, Sect. 2.32, − e ax ⎡ z −1 ( z − k − 1)! a k ⎤ Ei(ax n ) e ax + az , dx = n

n

n +1

n

⎢∑ ( z −k ) ⎥ z! xn ⎣ k =0 ⎦ set s = x = ln x,n = 1,a = n + 1,z = m − 1

∫

( )

x dx x n +1 ∫ (ln x )2 = − ln x + (n + 1)li x

2 x n dx 1 1 ⎤ (n + 1) n +1 ⎡ n +1 ∫ (ln x )3 = − x ⎢⎣ 2(ln x )2 + 2 ln x ⎥⎦ + 2 li x

( )

x

m

n

n z!

e(n +1)s ds . sm • Alternative statement of solution: ⎤ ⎡m −2 (n + 1)m−k −2 x n dx n +1 x = − ⎢∑ k +1 ⎥ ∫ (ln x )m ⎣ k =0 (m − 1)(m − 2 )K (k + 1)(ln x ) ⎦ and substitute for solution of I = ∫

+

(n + 1)m−1 li(x n+1 ) (m − 1)!

e(n +1)s ds into gamma sm n e ax function via Formula 2.325.8, ∫ m dx, x m m −1 to give, (− 1) (n + 1) Γ[− m + 1,−(n − 1)ln x ] , and using Formula 8.352 5, Γ(− n + 1, x ) , which produces a different ordering of the series terms. NOTE: 2.724.2 follows if m = 1. NOTE: m = 2,3 : substitute into expansion formula ds 1 s Set s = ln x, = , e = e ln x = x. dx x Then, based on converting I = ∫

Formula 2.724.2 Change Formula 2.724.2 to read:

( )

n +1

∫

n

( )

x dx = Ei[(n + 1) ln x ] = li x n+1 ln x

xx n es I =∫ ds = ∫ ds = Ei[(n + 1)s ] s s = Ei[(n + 1) ln x ] = li e (n+1)ln x = li x n+1 , by Formula 2.325 7, n e ax Ei ax n x ∫ x dx = n , Formula 8.240.1, Ei(x ) = li e , and relation e x ln a = a x . NOTE: new version relates exponential- and logarithm-integrals, Ei( x ) & li( x ), Formula 8.211.1.

(

( )

—Sect. 2.71 & 2.72-2.73—

) ( )

( )

66

New Section m n ∫ (a + bx ) ln (c + ln x )dx = ⎡(a + bx )m +1 ln n (c + ln x ) − ⎤ 1 ⎢ ⎥ m +1 ( a + bx ) ln n −1 (c + ln x ) ⎥ ⎢ (m + 1)b ⎢n ∫ dx ⎥ x(c + ln x ) ⎣ ⎦

For m = −1 ln n (c + ln x ) ln(a + bx ) ln n (c + ln x ) = − dx ∫ a + bx b n ln (a + bx ) ln n −1 (c + ln x ) dx b∫ x(c + ln x )

New Section

∫ (a + bx )

m

ln n (c + kx )dx =

⎡(a + bx )m +1 ln n (c + kx ) − ⎤ 1 ⎢ ⎥ n −1 m +1 ⎢ ⎥ ( ) ( ) + + a bx c kx ln (m + 1)b ⎢nk dx ⎥ ∫ c + kx ⎣ ⎦ For m = −1

ln(c + kx ) dx = a + bx ⎡ln (a + bx ) ln n (c + kx ) − ⎤ 1⎢ ⎥ n −1 ( ) ( ) + + ln ln a bx c kx ⎢ ⎥ b nk dx ⎥ ⎢⎣ ∫ c + kx ⎦

∫

n

•

Integration by parts: ⎛1⎞ n⎜ ⎟ ln n −1 (c + ln x ) n x u = ln (c + ln x ), du = ⎝ ⎠ dx c + ln x s = a + bx 1 (a + bx )m+1 m m dv = (a + bx ) dx, v = s ds = b∫ b(m + 1) uv − ∫ vdu =

⎤ ⎡(a + bx )m +1 ln n (c + ln x ) − 1 ⎥ ⎢ m +1 ( a + bx ) ln n −1 (c + ln x ) ⎥ ⎢ (m + 1)b ⎢n ∫ dx ⎥ x(c + ln x ) ⎦ ⎣

• For m = −1 : Integration by parts with, dx u = ln n (c + ln x ), dv = . a + bx NOTE: a difficult (to reduce further) but general integral useful for deriving existing forms by setting a, b, c, m, n as desired. • Integration by parts: nk ln n −1 (c + kx ) u = ln n (c + kx ), du = dx c + kx s = a + bx 1 m m dv = (a + bx ) dx, v = ∫ (a + bx ) dx = s m ds = b∫ (a + bx )m+1 b(m + 1) uv − ∫ vdu =

1 (a + bx )m+1 ln n (c + kx ) (m + 1)b

( nk a + bx ) ln n −1 (c + kx ) − dx b(m + 1) ∫ c + kx • NOTE: reduces to 2.278.1 if a = 0, b = 1, m = m, c = a, k = b, n = 1. • Test: if a = c = m = 0, n = b = k = 1, I = ∫ ln xdx = x ln x − x. m +1

•

For m = −1 :

dx a + bx Reduces to 2.721.2 if c = a = 0, b = k = 1. NOTE: a difficult (to reduce further) but general integral useful for deriving existing forms (such as 2.711, 2,722, 2.733, 2.726) by setting a, b, c, k , m, n as desired.

Integration by parts: u = ln(c + kx), dv = • •

—Sect. 2.71 & 2.72-2.73—

67

Sect. 2.727 or New Section (ln x )n dx = ∫ (a + bx )m

• Integration by parts: n −m u = (ln x ) , dv = (a + bx ) dx • NOTE: 2.727.1 results if n = 1. • NOTE: may be derived from

⎡ (ln x )n + ⎤ − ⎢ ⎥ m −1 1 ⎢ (a + bx ) ⎥ n −1 ⎥ b(m − 1) ⎢ (ln x ) ⎢n ∫ ⎥ dx m −1 ⎢⎣ x(a + bx ) ⎥⎦

if c = 0, k = 1, m = −m. • m =1:

For m = 1

Integration by parts with u = (ln x ) , dv =

∫

(a + bx )m ln n (c + kx )dx =

∫

m +1 n −1 ⎡ ⎤ 1 (a + bx )m +1 ln n (c + kx ) − nk ∫ (a + bx ) ln (c + kx ) dx⎥ (m + 1)b ⎢⎣ c + kx ⎦

n

dx . a + bx

(ln x )n dx =

a + bx (ln x )n ln(a + bx) − n (ln x)n−1 ln(a + bx) dx b b∫ x New Section • First integral: n n −1 n Set a = 0, b = 1 in above integral, (ln x ) dx = 1 ⎡− (ln x ) + n (ln x ) dx ⎤ ∫ ⎢ ⎥ m ∫ xm (ln x )n dx = 1 ⎡− (ln x )n + n (ln x )n−1 dx⎤ x m − 1 ⎣ x m −1 ⎦ ∫ ⎢ ⎥ m m−1 m−1 ∫

(a + bx)

For m = 1, see 2.721.2

(ln x )n dx = −

n−k 1 ⎡ n n! (ln x ) ⎤ ⎢∑ ⎥ ∫ xm x m −1 ⎣ k = 0 (n − k )! (m − 1)k +1 ⎦ ln x 1 ⎡ ln x 1 ⎤ ∫ x m dx = − x m −1 ⎢⎣ m − 1 + (m − 1)2 ⎥⎦

∫

(ln x )2 dx = xm

2 1 ⎡ (ln x ) 2 ln x 2 ⎤ − m −1 ⎢ + + ⎥ 2 x ⎣ m − 1 (m − 1) (m − 1)3 ⎦

∫

(ln x )

3

xm

dx =

⎡ (ln x )3 3(ln x )2 6 ln x ⎤ + + ⎥ ⎢ 2 m − 1 (m − 1) (m − 1)3 ⎥ 1 − m−1 ⎢ ⎥ x ⎢ 6 ⎥ ⎢+ 4 ⎦ ⎣ (m − 1)

For m = 1 and n = −1 , see 2.721.3

b(m − 1) ⎣

(a + bx )

x(a + bx )

• •

For m = 1 : see 2.721.2 with m → n. Expansion formula: ™ Change of variable; dx s = ln x, ds = , e s = x, x1− m = e(1− m )s x n s I = ∫ m xds = ∫ s n e(1− m )s . x ™ Now use indefinite integral previously derived (omitted in GR 7, but submitted as erratum), 1−γ ( m +1 − 1) Γ γ ,−ax n m ax n , γ = γ ∫ x e dx = n na with parameters [a = 1 − m, γ = n + 1] to give:

(

)

(− 1)− n Γ[n + 1,−(1 − m)ln x] (1 − m )n+1 Γ[n + 1, (m − 1)ln x ] =− (m − 1)n+1 ⎡ m!e −(m−1)ln x n [(m − 1)ln x ]k ⎤ = −⎢ ⎥ n +1 ∑ k! k =0 ⎣ (m − 1) ⎦ k ⎡ m! 1 n [(m − 1)ln x ] ⎤ = −⎢ ⎥ by 8.352.2, n +1 m −1 ∑ k! ⎣ (m − 1) x k =0 ⎦ Γ(n + 1,x ). I=

™ Writing summation in reverse order: —Sect. 2.71 & 2.72-2.73—

⎦

68

∫

(ln x)n dx = − xm

n −1 n−2 1 ⎡ ln x n n(ln x ) n(n − 1)(ln x ) n! ⎤ + + +K+ ⎢ m−1 2 3 x ⎣ m − 1 (m − 1) (m − 1) (m − 1)n+1 ⎥⎦

n−k 1 ⎡ n n! (ln x ) ⎤ ⎥ by inspection. ⎢∑ x m −1 ⎣ k = 0 (n − k )! (m − 1)k +1 ⎦ NOTE: this formula may be derived readily by restating n ( ln x ) I = ∫ m dx = ∫ s n e − (m −1)s ds and applying integral x derived above in Sect. 2.32, γ − k −1 ⎤ ⎡ m +1 − e − βx ⎡ γ −1 (γ − 1)! (x n ) ⎤ m − βx x e dx = = 1,2,...⎥ ⎢ ⎥ ⎢γ = ∑ k +1 ∫ n ⎢⎣ k = 0 (γ − k − 1)! β n ⎦ ⎥⎦ ⎣ with parameters n = 1, m → n, γ = n + 1, β x n = e − (m −1)s , s = x = ln x. • For n = 1,2,3 : substitute into expansion formula above • For m = 1 and n = −1 , see 2.721.3. • New reduction formula; same reduction approach as Formula 2.728.1 • Integration by parts: ln n −1 (a + bx ) u = ln n (a + bx ), du = nb dx, a + bx x m +1 dv = x m dx, v = ∫ x m dx = ; m +1

=−

n

n

[

Sect. 2.728

∫x

m

ln n (a + bx )dx =

⎡ x m+1 ln (a + bx )n − ⎤ 1 ⎢ ⎥ m +1 n −1 ( ) x a bx ln + ⎢ m + 1 nb dx ⎥⎥ a + bx ⎣⎢ ∫ ⎦

For n = −1 dx li(a + bx ) ∫ ln(a + bx ) = b For n = 1, m = −1 , see 2.728.2 . Also, see Sect. 2.32, above, for multiple solutions to ln (a + bx ) ∫ x dx (in terms of dilogarithms, defined in Math. Summary) all of which differ from equivalent GR solution! x

∫ ln(a + bx ) dx =

Ei[2ln (a + bx )] − aEi[ln (a + bx )] = b2 2 li (a + bx ) − ali(a + bx ) b2

[

]

]

uv − ∫ vdu

x m +1 n nb x m +1 ln n −1 (a + bx ) = dx ln (a + bx ) − m +1 m +1∫ a + bx • n = −1 : o Change of variable: u = a + bx, du = bdx 1 du Ei(ln u ) li(u ) li(a + bx ) I= ∫ = = = b ln u b b b by Formula 8.240.1. ¾ Verified math.com • Change of variable: s−a s = a + bx, ds = bdx, x = b 1 (s − a ) / b 1 ⎡ s ds ⎤ I= ∫ ds = 2 ⎢ ∫ ds − a ∫ b ln s ln s ⎥⎦ b ⎣ ln s s o ∫ ds : ln s

—Sect. 2.71 & 2.72-2.73—

69

du 1 u = ,e = s ds s s2 e 2u I = ∫ du = ∫ ds =Ei(2u ) = u u 2 Ei(2 ln s ) = Ei[2 ln (a + bx )] = li (a + bx ) by 8.211.1 and relation e x ln a = a x . ds : o a∫ ln s du 1 u = ,e = s u = ln s, ds s eu s I = a ∫ ds = a ∫ du = aEi(u ) = u u aEi(ln s ) = aEi[ln(a + bx )] = ali(a + bx ) u = ln s,

[

by 8.211.1 and relation e x ln a = a x . Ei[2ln (a + bx )] − aEi[ln (a + bx )] xdx • ∫ = ln(a + bx ) b2 NOTE: if x a = 0, b = 1, I = ∫ dx = li x 2 by 2.724 2. ln x ¾ verified math.com • Change of variable: dx s = a + ln x, ds = x s a + ln x a e =e = xe ⇒ x = e s e −a

( )

New Section ln m (a + ln x ) ∫ x n dx = m ln m −1 (a + ln x ) ln m (a + ln x ) dx − + (n − 1)x n −1 n − 1 ∫ x n (a + ln x )

For n = 1 ln m (a + ln x ) dx = ∫ x (a + ln x )ln m (a + ln x ) − m∫

ln m −1 (a + ln x ) dx x

For m = 1 and n = 1 See integral below, ln (a + ln x ) ∫ x dx = (a + ln x ) ln(a + ln x ) − ln x

]

x1−n =

e (1−n )s e (1−n )a

x(ln s ) ds m I =∫ = ∫ x1− n (ln s ) ds = n x k =1− n 1 1 m m (1− n )s ( ) e ln s ds = ka ∫ e ks (ln s ) ds. (1− n )a ∫ e e m • Integration by parts on ∫ e ks (ln s ) ds : m

m(ln s ) u = (ln s ) , du = ds s 1 dv = e ks ds, v = ∫ e ks ds = e ks k m m −1 ks e (ln s ) m e ks (ln s ) ds uv − vdu = − ∫ k k s Substituting m −1

m

—Sect. 2.71 & 2.72-2.73—

70

dx , k = 1 − n, e(1− n )s = x1− n e(1− n )a , x 1− n (1− n )a ⎡x e ⎤ ln m (a + ln x ) ⎢ ⎥ 1− n ⎢ ⎥= ⎢ m x1− n e(1− n )a ln m −1 (a + ln x ) ⎥ dx ⎥ ⎢− 1 − n ∫ x(a + ln x ) ⎣ ⎦

s = a + ln x, ds =

I=

1

e

(1− n )a

ln m (a + ln x ) m ln m −1 (a + ln x ) − + dx. (n − 1)x n −1 n − 1 ∫ x n (a + ln x ) NOTE: easier approach: int. by parts on dx u = ln m (a + ln x ), dv = n x •

For n = 1 :

ln m (a + ln x ) dx = ∫ x

m −1 (a + ln x ) ln m (a + ln x ) − m∫ ln (a + ln x ) dx

x

derived by change of variable with dx s = a + ln x, ds = , to give, x

I = ∫ (ln s ) ds, and integration by parts with : m

u = (ln s ) , dv = ds. ¾ Mathematica provided solution in terms of hypergeometric function. m

New Section

∫x

n

ln m (a + ln x )dx =

⎡ x n+1 ln m (a + ln x ) ⎤ 1 ⎢ ⎥ n m −1 n + 1 ⎢− m x ln (a + ln x ) dx ⎥ ∫ a + ln x ⎢⎣ ⎥⎦

For m = 1

∫ x ln(a + ln x )dx = n

n +1 ⎫⎪ 1 ⎪⎧ x ln(a + ln x ) − ⎨ −(n +1)a ⎬ n + 1 ⎪⎩e Ei[(n + 1)(a + ln x )]⎪⎭

• •

m Similar to ∫ ln (a +n ln x )dx above: make n → −n. x

For m =1: o Change of variable: dx s = a + ln x, ds = x s a + ln x a e =e = xe ⇒ x = e s e − a e (n +1)s k = n +1 e ks = ka e (n +1)a e 1 I = ∫ x n +1 ln s = ka ∫ e ks ln sds e o Integration by parts: x n +1 =

For m = 1and n = −1 ln (a + ln x ) ∫ x dx = (a + ln x ) ln(a + ln x ) − ln x

—Sect. 2.71 & 2.72-2.73—

71

ds s 1 dv = e ks ds, v = e ks k ks ln e ns 1 e ks uv − vdu = ds = − ∫ k k s ln e ks ns Ei(ks ) − . k k Substituting dx s = a + ln x, ds = , k = n + 1, e (n +1)s = x n +1e (n +1)a , x n +1 ( n +1)a ⎡x e ln (a + ln x ) ⎤ −⎥ 1 ⎢ n +1 ⎥ I = (n +1)a ⎢ e ⎢ 1 ⎥ ⎢⎣ n + 1 Ei(n + 1)(a + ln x ) ⎥⎦ 1 = x n +1 ln (a + ln x ) − e −(n +1)a li x n +1e (n +1)a n +1 by Formula 8.211.1, Ei( x ) = li e x , and relation, e x ln a = a x . • For m = 1 and n = −1 : math.com calculation; (change of variable, s = a + ln x and int. by parts on u = ln s,dv = ds ) u = ln s, du =

[

(

( )

New Section

∫ x ln(a − ln x )dx = n

n +1 ⎫⎪ 1 ⎧⎪ x ln(a − ln x ) − ⎨ −(n +1)a ⎬ n + 1 ⎪⎩e Ei[(n + 1)(ln x − a )]⎪⎭

For n = −1 ln(a − ln x ) ∫ x dx = (ln x − a )ln(a − ln x ) − ln x New Section dx ∫ (a − ln x )n = x 1 dx − n −1 ∫ n − 1 (a − ln x )n −1 (n − 1)(a − ln x )

For n = 1 dx a ∫ a − ln x = −e Ei(ln x − a )

• •

Similar to ∫ x n ln(a + ln x )dx , above n = −1 :

Similar to

∫

ln (a + ln x ) dx , above x

•

Change of variable: dx s = a − ln x, ds = − x 1 ln ea s a − ln x a x e =e =e e = x −s −a a −s x = e e , e = xe −s x a e = − ds e ∫ s n ds sn Integration by parts:

I = −∫ •

—Sect. 2.71 & 2.72-2.73—

)]

72

dx

∫ (a − ln x ) n−2

n

(− 1)n x∑ (− 1)k k =0

(− 1)n

u = e − s , du = −e − s ds

=

(n − k − 2)! + (n − 1)!(a − ln x )n−k −1

a

e Ei(ln x − a ) (n − 1)!

dx x a ∫ (a − ln x )2 = a − ln x + e Ei(ln x − a )

s 1− n 1 dv = s ds, v = ∫ s ds = =− 1− n (n − 1)s n−1 −n

−n

uv − ∫ vdu = e −s − e −s ds − − (n − 1)s n−1 ∫ − (n − 1)s n−1 =− −

s = a − ln x e −s e −s ds − = (n − 1)s n−1 ∫ (n − 1)s n−1

xe − a

(n − 1)(a − ln x )n−1

−

1 − xe − a dx. n − 1 ∫ x(a − ln x )n −1

⎤ ⎡ xe − a − ⎥ ⎢ n −1 a ⎢ (n − 1)(a − ln x ) ⎥ I = −e −a ⎥ ⎢ 1 − xe dx ⎥ ⎢− n −1 ∫ ⎥⎦ ⎢⎣ n − 1 x(a − ln x ) x 1 dx = − n −1 ∫ n − 1 (a − ln x )n −1 (n − 1)(a − ln x ) • For n = 1 : Change of variable; 1 ln dx s ea a − ln x a =e e x = ⇒ s = a − ln x, ds = − , e = e x x x = e a e −s . e −s ds = −e a Ei(− s ) = − e a Ei(ln x − a ) s by 8.211.1 and relation, e x ln a = a x . • Integer-expansion formula: From above, e− s I = −e a ∫ n ds s Apply integral formula, above, Sect. 2.32, n n z k ⎤ ( e−βx −1) e−βx ⎡ z−1 k ( z − k − 1)! β ( ) = − dx 1 ∑ ⎢ ⎥ ( ) z k − m ∫x n z! xn ⎦ ⎣ k =0 I = −e a ∫

+ (−1) β z z

(

Ei − βxn n z!

)

⎡ m-1 ⎤ ⎢⎣ z = n = 1,2,..., m = 2,3,...⎥⎦

with parameters, [s = x = a − ln x, n = 1, m = n, β = 1, z = n − 1] o Alternative expression of expansion formula:

—Sect. 2.71 & 2.72-2.73—

73

dx

∫ (a − ln x )

n

=

k ( − 1) (− 1) x∑ + k +1 k = 0 (n − 1)(n − 2 )K (k + 1)(a − ln x ) (− 1)n ea Ei(ln x − a ) (n − 1)! n

n−2

e−βx based on expansion formula for ∫ m dx previously x submitted as Form. 2.325.19, which produces a different ordering of the series terms. n = 2 : In above integral, • dx x 1 dx ∫ (a − ln x )n = (n − 1)(a − ln x )n−1 − n − 1∫ (a − ln x )n−1 , set n = 2, and note, n

New Section dx

∫ x (ln x ) n

−

m

= 1

(m − 1)x (ln x ) n −1

m −1

−

n −1 dx ∫ n m − 1 x (ln x )m −1

For m = 1 dx ⎛ 1 ⎞ ∫ x n ln x = li⎜⎝ x n−1 ⎟⎠

dx s = a −ln x a e − s a ∫ a − ln x = − e ∫ s ds = −e Ei(− s ) = − e a Ei(ln x − a ) NOTE: Formula 4.212.4 results for this integral with limits from 0 to 1. • First integral: o Change of variable: dx s = ln x, ds = , e s = x, x1− n = e(1− n )s to give, x e − (n −1)s e(1− n )s x I = ∫ n m ds = ∫ m ds = ∫ m ds s s x s o Integration by parts: u = e (1−n )s , du = (1 − n)e (1−n )s 1 dv = s −m ds, v = ∫ s −m ds = − (m − 1)s m−1 o

For n = 1 dx 1 ∫ x(ln x )m = − (m − 1)(ln x )m −1 For n = m = 1, see 2.721 .3

uv − ∫ vdu = −

e (1−n)s − (n − 1) e (1−n )s ds − (m − 1)s m−1 − (m − 1) ∫ s m−1

( n − 1) x1−n dx =− − (m − 1)(ln x)m−1 (m − 1) ∫ (ln x)m−1 x (n − 1) 1 dx =− − ∫ m−1 n−1 n (m − 1) x (ln x)m−1 (m − 1)x (ln x) x1−n

NOTE: setting m → −m, n → −n in 2.721.1 produces a poor reduction formula since powers of ln x are increasing. Setting n → − n in 2.724 produces desired results. •

For m = 1:

—Sect. 2.71 & 2.72-2.73—

74

dx

∫ x (ln x )

m

n

=

dx s , e = x, x1− n = e(1− n )s , to give x 1− n x e (1−n )s I =∫ ds = ∫ ds =Ei[(1 − n ) ln x ] = li e (1−n )ln x s s ⎡ 1 ⎤ = li x1−n = li ⎢ n−1 ⎥ ⎣x ⎦ s = ln x, ds =

k ⎤ 1 ⎡m − 2 k (m − k − 2 )! (n − 1) + − n −1 ⎢ ∑ (− 1) m − k −1 ⎥ (m − 1)! (ln x ) ⎦ x ⎣ k =0

m −1 ( n − 1) ⎛ 1 (− 1) li⎜ (m − 1)! ⎝ x [n > 1] m −1

n −1

⎞ ⎟ ⎠

[

[ ]

o NOTE: it is unwise to set n = 1 in this formula to solve Formula 2.721.3, dx n=1 ∫ x ln x = li(1). • For n = 1 : dx 1 ∫ x(ln x )m dx = − (m − 1)(ln x )m −1 by change of variable, s = ln x. • Integer-expansion formula: e − (n−1)s Define from above, I = ∫ m ds s − βx n e and use integral, ∫ m dx, Section 2.32, above, x − βx n − βxn k ⎤ ⎡ z−1 e e k ( z − k − 1)! β (−1) ( z −k ) ⎥ + ∫ xm dx = − n ⎢⎣∑ n z! k =0 x ⎦

(−1)z β z Ei(− βx

n

n z!

)

⎡ m −1 ⎤ ⎢⎣z = n = 1,2,...,m = 2,3,...⎥⎦

with parameters [x = s = ln x, n = 1, β = n − 1, z = m − 1]. • Alternative form for expansion: dx ∫ x n (ln x )m = −

⎤ 1 ⎡m−2 (− 1)k (n − 1)m−k −2 ⎥+ n −1 ⎢ ∑ x ⎣ k =0 (m − 1)(m − 2)K (k + 1)(ln x )k +1 ⎦

(− 1)m−1 (n − 1) (m − 1)!

m −1

⎛ 1 ⎞ li⎜ n −1 ⎟ ⎝x ⎠

based on first submission for expansion of n e − βx ∫ x m dx , Formula 2.325.19, which produces a different ordering of the series terms. ¾ EXAMPLES: dx s = ln x 1 o ∫ = − 2 ln x x(ln x ) —Sect. 2.71 & 2.72-2.73—

]

75

dx s = ln x ⎛1⎞ ∫ x 2 ln x = Ei(− ln x ) = li⎜⎝ x ⎟⎠ ; or by expansion formula above ( n = 2, m = 1) 1 dx s = ln x ⎛1⎞ = − − li⎜ ⎟ ; or by o ∫ 2 2 x ln x x (ln x ) ⎝x⎠ expansion formula above ( n = m = 2). o

•

New Section

dx

∫ (a + ln x ) −

n

=

x

(n − 1)(a + ln x )

n −1

+

1 dx ∫ n − 1 (a + ln x )n −1

For n = 1 dx −a ∫ a + ln x = e Ei(a + ln x ) =

n

( )

n e ax − e ax ⎡ z −1 (z − k − 1)! a k ⎤ z Ei ax dx a + = ∑ ⎥ ⎢ ∫ xm n ⎣k =0 z! x n( z − k ) ⎦ n z! m −1 ⎤ ⎡ ⎢ z = n = 1,2,..., m = 2,3,...⎥ ⎦ ⎣ n

dx

∫ (a + ln x )

First two integrals: Similar to above integrals, dx dx ∫ (a − ln x )n & ∫ a − ln x . • Integer-expansion formula Change of variable: dx s = a + ln x, ds = , e s = e a eln x = xe a . x s e ds I = e− a ∫ n . s Apply above integral, Sect. 2.32, n

(n − k − 2)! e −a + Ei(a + ln x ) n − k −1 (n − 1)! k = 0 (n − 1)!(a + ln x ) with parameters

n−2

− x∑

[x = s = a + ln x, n = 1, m = n, a = 1, z = n − 1]

dx

∫ (a + ln x )

2

=−

x + e −a Ei(a + ln x ) a + ln x

o Alternative form for expansion: n−2 1 dx = − x + ∑ k +1 ∫ (a + ln x )n k = 0 ( n −1)( n − 2 ) K ( k +1)( a + ln x ) Ei(a + ln x ) e− a (n − 1)! n

e ax based on first submission for expansion of ∫ m dx , x Form. 2.325.8, which produces a different ordering of the series terms. • n = 2 : Similar to above integral, dx x a ∫ (a − ln x )2 = (a − ln x ) + e Ei(ln x − a ) , by setting n = 2 in, n−2 (n − k − 2)! + e−a Ei(a + ln x ) dx x = − ∑ ∫ (a + ln x)n k =0 (n − 1)!(a + ln x)n−k −1 (n − 1)!

o NOTE: Formula 4.212.3 results for this integral with limits 0 to 1.

—Sect. 2.71 & 2.72-2.73—

76

New Section xn ∫ (a + ln x )m dx =

⎡m − 2 (m − k − 2 )! (n + 1)k ⎤ − x n +1 ⎢ ∑ + m − k −1 ⎥ ⎣ k =0 (m − 1)! (a + ln x ) ⎦ e −(n +1) a

(n + 1) (m − 1)!

m −1

Ei[(n + 1)(a + ln x )]

For m =1: xn −( n +1) a ∫ a + ln x dx = e Ei[(n + 1)(a + ln x )] xn ( n +1) a ∫ a − ln x dx = −e Ei[(n + 1)(ln x − a )] ln x

∫ (a + ln x )

2

dx =

ax + (1 − a )e −a Ei(a + ln x ) a + ln x ln x ∫ (a − ln x )2 dx = ax + (a + 1)e a Ei(ln x − a ) a − ln x n

e ax ∫ x m dx m ax n ∫ x e dx

• First integral Change of variable: dx s s = a + ln x, ds = , e = xe a , x n+1 = e (n+1)s e −(n+1)a , x to get : e (n+1)s ds sm o Solve I with expansion formula in Sect. 2.32, above: e ax ⎡ z −1 (z − k − 1)! a k ⎤ Ei(ax n ) + az dx = −

I = e −(n+1)a ∫

n

∫

e ax xm

n

⎢∑ n ⎣k =0

z!

⎥ x n( z −k ) ⎦

n z!

m −1 ⎡ ⎤ ⎢n ≠ 0, a ≠ 0, z = n = 1,2,..., m = 2,3,...⎥, ⎣ ⎦

with parameters [a = n + 1, n = 1, m = m, z = m − 1] • Second & 3rd integrals: substitute m = 1 into expansion formula & s = a − ln x into 3rd. ¾ Verified Mathematica math.com calculation. See derivation of definite integral, Sect. 4.212, below. math.com calculation See derivation of definite integral, Sect. 4.212, below. Express exponential integrals as equivalent integrals in logarithms. NOTE: see Sect. 4.215, below, for similar definite integrals.

—Sect. 2.71 & 2.72-2.73—

275

REFERENCES Abramowitz, Milton and Irene Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington, DC: U.S.Government Printing Office, 1964 (Reprinted Dover, New York, 1972). Artin, Emil. The Gamma Function. New York, NY: Holt, Rinehart and Winston, 1964. Bers, Lipman. Calculus. 2 volumes. NY: Holt, 1969. Boas, Mary L. Mathematical Methods in the Physical Sciences, 3 ed. NY: Wiley, 2006. Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, 2004. Carr, G.S. Formulas and Theorems in Pure Mathematics, 2 ed., New York: Chelsea, 1970 Dwight, H.B. Table of Integrals and Other Mathematical Data, 4th ed. New York: Macmillan, 1961. Gautschi, W. “The incomplete gamma functions since Tricomi.” In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei, Roma, 1998, pp. 203-237. Gradshteyn, I.S. and I.M. Ryzhik (6th Edition). Table of Integrals, Series, and Products. Alan Jeffrey and Daniel Zwillinger, Editors. NY: Academic Press, 2000. _______ (7th Edition). Table of Integrals, Series, and Products. Alan Jeffrey and Daniel Zwillinger, Editors. NY: Academic Press, 2007. Moll, Victor H. The integrals in Gradshteyn and Rhyzik. Part 2: Elementary logarithmic integrals. Scientia, Series A: Mathematical Sciences, Vol. 14 (2007), 7-15 _______. The integrals in Gradshteyn and Ryzhik. Part 3: Combinations of Logarithms and Exponentials. arXiv:0705.0175v1 [math.CA]. 1 May 2007 _______.The integrals in Gradshteyn and Ryzhik. Part 4: The Gamma function arXiv:0705.0179v1 [math.CA]. 1 May 2007. _______. The integrals in Gradshteyn and Ryzhik. Part 13: Evaluation using The Error Function. http://www.math.tulane.edu/~vhm/web_html/erfweb.pdf, 4 October 2006

_______. The integrals in Gradshteyn and Ryzhik. Part 14: The Exponential-Integral Function. http://www.math.tulane.edu/~vhm/web_html/expintweb.pdf, 6 Semptember 2006.

_______. The integrals in Gradshteyn and Ryzhik. Part 15: The Incomplete Gamma Function http://www.math.tulane.edu/~vhm/web_html/erfweb.pdf, 4 September 2006

276

Moll, Victor H., Jason Rosenberg, Armin Straub, Pat Whitworth. The Integrals in Gradshteyn and Ryzhik. Part 8: Combinations of powers, exponentials and logarithms. arXiv:0707.2123v1 [math.CA]. 14 Jul 2007. O’Brien, F.J. 100 Proposed Formulas Submitted for Publication in I.S. Gradshteyn and I.M. Ryzhik Table of Integrals, Series, and Products, 7th Edition. Alan Jeffrey and Daniel Zwillinger, Editors. NY: Academic Press, 2005. Newport, RI: Naval Undersea Warfare Center, Division, Newport. November 1, 2005. _______. Selected Transformations, Identities, and Special Values for the Gamma Function. Submittted to Mathworld, http://functions.wolfram.com/GammaBetaErf/Gamma, June, 2006. _______. 400 Proposed Formulas Submitted for Publication in I.S. Gradshteyn and I.M. Ryzhik Table of Integrals, Series, and Products, 8th Edition. Alan Jeffrey and Daniel Zwillinger, Editors. NY: Academic Press, 2007. Newport, RI: Naval Undersea Warfare Center, Division, Newport. May 9, 2007. _______. Summary of Four Generalized Exponential Models (GEM) For Continuous Probability Distributions, Jan. 18, 2008. arXiv:0801.2941v2 [math.GM] Prudnikov, A. P., Yu. A. Brychkov, and O. I. Marichev. Integrals and Series. Vol. 1: Elementary Functions. Gordon & Breach, New York, 1986. _______. Integrals and Series. Vol. 2: Special Functions. Gordon & Breach, New York, 1986. _______. Integrals and Series. Vol. 3: More Special Functions. Gordon & Breach, New York, 1990. Sokolnikoff, I. S. Advanced Calculus. New York: McGraw Hill Book Co., 1939. Spiegel, M. R. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1968. [Reprinted Spiegel, Murray R., John Liu, and Seymour Lipschutz. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1999.] Whittaker, E. T. and G. N. Watson. A Course of Modern Analysis. Cambridge University Press; 4th edition, 1934. Wolfram Research, Inc. The Wolfram Functions Site. http://functions.wolfram.com/, 1998-2008. _______. Wolfram Mathworld. http://mathworld.wolfram.com/, 2008. Zwillinger, Daniel. Handbook of Integration. Boston: Jones & Bartlett Publishers, Inc., 1992.

277

INDEX OF SYMBOLS, FUNCTIONS, AND CONCEPTS This index shows the occurrence of symbols, functions, and concepts used in the expressions within the text. The numbers refer to pages on which the symbols, functions, or concepts appear.

(a )n see Pochhammer symbol C or γ see Euler' s constant G or K see Catalan' s constant ε see epsilon ∉ not an element of a set Ei see exponential - integral function erf see error functions erfi see error functions exp or e see exponential function i imaginary number γ and Γ see gamma functions

γ ∗ see gamma functions int see integer function

li see logarithm - integral function Li 2 see dilogarithm lim see limit ln & log see logarithm n! see factorial Re real variable Φ see error functions π mathematical constant 3.14159265... Π see product Σ see summation Ψ see Psi (Digamma) function ⎛n⎞ ⎜⎜ ⎟⎟ see binomial coefficient ⎝ j⎠ ! and !! see factorials

278

A algebraic functions and exponentials 9, 106-119 and logarithmic functions 62-77, 178-180 alternating series see oscillating series arctan see arctangent arctangent 17 arithmetic progression 25-26

B binomial coefficient 20, 37-39,42-43, 45-46, 48-49, 50-51, 62, 65, 67-68, 73-76, 198

C Catalan’s constant (G or K ) 17, 82, 180 Cauchy principal value (PV) see Principal Value complement or reflection formula see gamma functions complementary error function see error functions complete gamma function see gamma functions complex (imaginary) number i 2, 11, 33, 40, 47, 265-266 completing the square 20, 41, 89, 91 computer solution see mathematical software constant of integration Preface, 128 cos cosine 229-234, 250 cosec see csc csc cosecant 227-231

D definite integrals 4-21, 81-275 derivatives see differentiation differentiation 3, 5, 6, 8-9, 21, 102, 112-114, 264 of integrals 21, 34, 37, 42, 45, 81, 102, 114-115, 128, 264 of special functions 6, 8, 9, 21, 198, 263-264 digamma function see Psi function dilogarithm Li 2 ( x ) 5, 55-56 double factorials see factorial, double double integral see integral

279

E elementary functions 1-201 graphs 1-18 epsilon 12, 14 error functions 8-11, 33, 40-41, 44, 47, 49, 58-59, 85, 148-150, 214-229, 265-267 and gamma functions 11, 215-216 complementary 10, 215, 265-266 imaginary 10, 11, 33, 40-41, 47, 55, 269-272 proof of limits 58, 266-270 Euler’s constant (C or γ ) 6-7, 16, 101-102, 127, 178, 193, 198, 274-275 Euler’s integral see gamma functions even function 59, 86, 122, 132, 267 exponential function 2-3 exponential-integral function 9, 12-14, 31-34, 36, 45-51, 57, 62-63, 65, 69, 71-77, 82, 99, 104, 152, 154, 163-164, 168-169, 170-171, 173, 193, 196, 198-201, 204-209, 211-213, 257-259, 265, 275 exponentials 2-3 and algebraic functions 9, 105-119 and complicated arguments 84-97, 129-159 and exponential integrals 209 and gamma functions 31-34, 37-60 and logarithmic functions 34, 36, 51, 77, 176 and powers 120-125, 129-159 and rational functions 35-60, 103-105, 126-128 of exponentials 98-102

F factorial (!) 6, 20, 37-51, 62-63, 65, 67-68, 72-76, 87-88, 130-132, 146,165, 182-184, 239-240, 252, 256-257 factorial, double (!!) 20, 86, 87, 116-119, 130-133, 160, 182-189, 239-240, 243, 251, 268-272 finite Binomial expansion 20, 198 finite sums 24-26 Frullani Integral 81, 128

G gamma functions 6-9 and complement or reflection formula 227-237 and exponentials 31-34 and logarithms 198 and multiplication theorem 227, 234, 238 and product theorem 227, 238 and trigonometric functions 225-238 complete (Γ(x)) 6-7, 225-254 doubling formula 229, 235, 238

280

(

)

incomplete γ ∗ ( x ) 255 integer arguments 86, 239-252 lower incomplete (γ(x, y)) 7-8, 56-60, 254-272 upper incomplete (Γ(x, y)) 7-9, 34, 36-60, 254-272 Gaussian probability density function 58-59, 86 geometric progression 26 graphs of elementary and special functions 1-18, 267

H hypergeometric function 70

I identities 7, 18-19 algebraic 18-19 trigonometric 7, 227-238 imaginary error function see error functions imaginary number i see complex (imaginary) number improper integrals 13, 15 incomplete gamma function see gamma functions incorrect listing 212 indefinite integrals 4, 9, 11, 13-14, 27-77 integer function 22, 119, 244, 245-248 integral, double 59, 81, 128 inverse trigonometric functions see trigonometric functions

L Leibnitz’s Rule for differentiating integrals see differentiation L’Hôpital Rule 60 limit lim 2, 5, 12, 14, 16, 57, 60, 151, 159, 159, 255, 266-267 logarithm-integral function 12, 14-15, 31-32, 34, 62-66, 68-69, 71, 73-75, 82, 163-167, 137, 211, 212213 logarithms 4-5 and algebraic functions 62-78, 179-184 and dilogarithms 5, 55-56 and exponentials 77, 180, 195-201, 212 and gamma functions 198 and powers 62-78, 181-189 and rational functions 193-196 of more complicated arguments 177-178, 195-201

M Math.com see quickmath.com mathematical software Preface, also see Mathematica, quickmath.com Mathematica Preface, 41, throughout text

281

mathematical summary 1-21 Meijer’s G Function 264 multiplication theorem see gamma functions

N natural logarithm see logarithms natural numbers 6, 21, 118-119, 189, 225-226, 236-237, 242, 244-248 notations 22, also see Symbols

O odd function 59 oscillating series 48, 56

P partial fractions 18-19, 53-58, 111-113, 115 Pochhammer symbol 22, 225-226, 241, 242-252 polar coordinates 59 powers and exponential functions 120-125, 129-159 and logarithmic functions 62-78, 181-191 principal value (PV) 12-15, 34, 57 probability integral Φ ( x ) see error function product Π 22, 225, 238-252 product theorem see gamma functions progressions 25-26 Psi function 6, 16

Q quickmath.com Preface, throughout text

R rational functions and exponentials 103-105, 126-128 and logarithmic functions 192-193 references 3, 5, 7, 9 11, 13-17, 19-22, 58, 112, 298, 225-232, 238, 255, 276 reflection or complement formula see gamma functions

S sec secant 234 series 5, 48, 55-57, 60, 65, 73, 75, 76

282

sin sine 174, 232-238, 251 special functions 1-21, 202-275 graphs 10, 12, 14, 16-17, 267 Stirling asymptotic formula 20 summation Σ 18, 20, 22, 25-26, 37-40, 46, 48-51, 53-56, 57,59, 62-63, 65, 67-68, 73-76, 198, 256-259

T transforms 18-19, 38, 43, 113, 238, 248, 264 transcendental functions 7, 9, 151 trigonometric functions 7, 227-238, also see arctangent, cosecant, cosine, secant, sine

283

INDEX OF FORMULAS This index lists the formulas which are solved in the text. The formulas are identified by the section/subsection and page numbers where they appear in the text. Similar expressions are stated in combined algebraic form. The Mathematical and Graphical Summary (page 1 and following) contains other formulas not listed below. FORMULA

int (x ) (a )n k =m

NOTATIONS (22)

— —

n

∑u

SECTION —

k

n

∏ f (k )

—

k =m n −1

∑ (a + kr )

0.111

FINITE SUMS (25-26)

k =0 n

∑ aq

0.112

k −1

k =1

0.121.1

n

∑k k =1

∫x

e ± ax dx

2.31

e ± ax ∫ x m dx x ∫ exp a dx

2.31

m

n

n

( )

2.312 2.312 2.312 2.312

dx x

2.312 2.312

ax ∫ x a dx ax ∫ x b dx 2x ∫ x 2 dx x2 ∫ a dx

∫a

2

x ∫ a dx n

2.312

dx

2.312

∫a

x

THE EXPONENTIAL FUNCTION (29-34)

n

∫ x a dx μ ∫ e ln xdx μ ∫ e ln xdx m

± x ± x

xn

p

2.312 2.32

THE EXPONENTIAL COMBINED WITH RATIONAL FUNCTIONS OF x (35-60)

284

e ax ∫ x 4 dx m ± ax n ∫ x e dx

2.32

∫ ln xdx ∫ x ln xdx ∫ x ln xdx

2.32

m

n

m

2.32

2.32

n

2.32

erfi( x )

2.32

∫e ( ∫e

ax 2

2.32

dx

ax 2 +2 bx + c

)dx

Γ(a ), γ (a, x ), Γ(a, x )

2.32 2.32

ln x n ∫ x m dx erf ( x )

2.32

∫x ∫x

dx

2.32

e dx

2.32

dx

2.32

∫e

m − βx n

e

a −1 − x

− βx 2 n

e ax ∫ x m dx n e ax ∫ x dx Ei(ax n ) n − βx n e ∫ x m dx n e − βx ∫ x dx 2 e − βx ∫ x 2 dx n ax ∫ x e dx

2.32

2.32 2.32 2.32 2.32 2.32 2.32 2.32

e ax ∫ x n dx n − βx ∫ x e dx

2.32

∫ xe dx β ∫ x e dx β ∫ x e dx − βx

2.32

2 − x

2.32

3 − x

2.32

e − βx dx ∫ x

2.32

2.32

285

e − βx dx ∫ xn e − βx dx ∫ x2 e − βx dx ∫ x3 e − βx dx ∫ x4 n ∫ x dx

2.32

1 ∫ x n dx 1 ∫ x 2 dx & x ∫ x dx

2.32

(− 1)1−γ na γ

2.32 2.32 2.32 2.32

e ax

n

∫ 0

1

∫x

3

⎛ 1⎞ ⎜ ln ⎟ ⎝ t⎠

dx

2.32 2.32 2.32

γ −1

dt

e −1 e ax + 1 & dx ∫ eax +1 ∫ eax −1 dx

2.32

e ax − m ∫ eax +n dx

2.32

e ax + m ∫ eax +n dx

2.32

e ax − m ∫ eax −n dx

2.32

∫ e dx ∫ e dx x

2.32

−x

2.32

ax

2.32

dx

∫ 1+ e

ax

e ax ∫ x + bdx x +1 ∫ e ax dx x ∫ e x − 1 dx ln (a ± bx )dx ∫ x ln ( x )dx ∫ a±x

2.32 2.32 2.32 2.32 2.32

286

∫

2.32

dx

b e mx dx ∫ b a − mx e dx ∫ a − bemx a+

2.32

2.32

∫ x e dx β ∫ x e dx 3 − βx 2

2.32

2 − x

2.32

Ei( x ) li( x )

2.32 2.32

xe x ∫ (1 + x )2 dx

2.32

xe ax ∫ (1 + ax )2 dx

2.32

∫e

−

x2 2

2.32

dx

Ei(ax )

2.32 2.32

xn lim =0 x → +∞ e x m ∫ ln xdx

∫ ln(a + bx )

2.71

m

dx

2.71

dx ∫ ln x ∫ x ln xdx

2.72-2.73

∫ x ln xdx ∫ ln(ln x ) dx ∫ x ln(ln x ) dx

2.72-2.73

n

2.72-2.73

m

n

m

n

2.72-2.73 2.72-2.73

ln (ln x ) ∫ x dx xn ∫ (ln x )m dx

2.72-2.73

x n dx ∫ ln x 2 x n dx ∫ ln x 3 m n ∫ (a + bx ) ln(c + ln x ) dx

2.72-2.73

m

2.72-2.73

2.72-2.73 2.72-2.73

LOGARITHMS (61-77)

287

ln(c + ln x ) ∫ a + bx dx m n ∫ (a + bx ) ln(c + kx ) dx n

ln(c + kx ) ∫ a + bx dx n

m

2.72-2.73

dx

m

2.72-2.73

ln x n dx ∫ a + bx ln x n ∫ x m dx ln x n ∫ x m dx ln x ∫ x m dx ln x 2 ∫ x m dx ln x 3 ∫ x m dx

∫x

2.72-2.73

2.72-2.73

ln x n

∫ (a + bx )

2.72-2.73

2.72-2.73 2.72-2.73 2.72-2.73 2.72-2.73 2.72-2.73

ln(a + bx ) dx n

ln (a ± bx )dx x dx ∫ ln(a + bx ) x ∫ ln(a + bx ) dx

∫

x n dx ∫ ln x m ln (a + ln x ) dx ∫ xn m ln (a + ln x ) dx ∫ x

2.72-2.73 2.72-2.73 2.72-2.73 2.72-2.73 2.72-2.73 2.72-2.73 2.72-2.73

∫ x ln(a + ln x ) dx ∫ x ln(a + ln x )dx ln (a + ln x ) dx

2.72-2.73

∫ x ln(a − ln x )dx ln (a − ln x ) dx

2.72-2.73

m

n

n

∫

2.72-2.73

x

n

∫

2.72-2.73

x dx

∫ (a − ln x )

2.72-2.73 2.72-2.73

n

288

∫ a − ln x

dx

2.72-2.73

dx ln x m dx ∫ x n ln x dx ∫ (a + ln x )n dx ∫ a + ln x xn

2.72-2.73

∫x

n

∫ (a + ln x )

m

2.72-2.73 2.72-2.73 2.72-2.73 2.72-2.73

dx

xn ∫ a − ln x dx ln x ∫ (a ± ln x )2 dx n

2.72-2.73 2.72-2.73 2.72-2.73

e ax ∫ x m dx m ax n ∫ x e dx

2.72-2.73 3.310

v

−ρ x ∫ e dx u

∞

e − x − e −a x ∫0 x dx

3.310

∞

3.310

∫

f ( ax )− f ( bx )

x

dx

0

∞

e − ax − e − bx dx ∫0 x p

p

0

xe x −∫ dx 2x − ∞1 + e u

∫

ex

n

ln a

dx

x

−∞

v

x ∫ a dx n

3.310 3.311

3.311 3.311

u

1

x ∫ a dx n

3.311

0

1

3.311

1

∫a

x

n

dx

0

∞

1

∫a 0

dx xn

3.311

EXPONENTIAL FUNCTIONS (81-83)

289

3.311

1

x ∫ a dx 0

3.311

1

x ∫ a dx n

0

3.321

u

−x ∫ e dx 2

0

2

π

3.321

u

−x ∫ e dx 2

0

3.321

v

−q x ∫ e dx 2 2

u

∞

3.321

−q x ∫ e dx 2 2

u

∞

3.321

−q x ∫ e dx 2 2

−∞

3.321

u

n −q x ∫ x e dx 2 2

0

⎛ x2 ⎞ exp ∫u ⎜⎜⎝ − 4β − γx ⎟⎟⎠dx

3.322

∞

v

(

)

3.323

(

)

3.323

(

)

3.323

2 ∫ exp − ax − bx dx 0

∞

2 ∫ exp − ax − bx dx

u u

2 ∫ exp − ax − bx dx 0

v

− ∫e

(ax

2

(ax

2

(ax

2

(ax

2

(ax

2

(ax

2

+ kbx + c

)

+ 2 bx + c

)

+ kbx + c

)

3.323

dx

u v

− ∫e

3.323

dx

u

∞

− ∫e

3.323

dx

0

∞

− ∫e

+ bx + c

)

+ bx + c

)

3.323

dx

0

∞

− ∫e

3.323

dx

u

u

− ∫e 0

+ kbx + c

)

3.323

dx

EXPONENTIALS OF MORE COMPLICATED ARGUMENTS (84-97)

290 ∞

∫ (cx

(ax

)

+ kgx e −

2

+ jbx

2

)

3.323

dx

−∞ ∞

∫ (cx

)

(ax

+ bx

)

( px

+ kqx

2

2

+ gx e −

2

+ bx + c e −

3.323

dx

0

∞

∫ (ax

)

2

)

dx

3.323

−∞ ∞

∫ (ax

2

)

+ kbx e −

( px

2

+ qx + c

)

3.323

dx

−∞

∞

(∫ ax + kbx+ c)e (

2 − px + jqx+r

2

)

3.323

dx

−∞

⎛ a+b⎞ ∫0 exp⎜⎝ − x 2 ⎟⎠dx

3.324

(

)

3.326

exp − β x n ∫u x m dx

3.326

∞

3.326

u

v

m n ∫ x exp − βx dx u v

(

)

(

)

m n ∫ x exp − βx dx 0

m ∫ x exp(− ρx − b )dx

3.326

∞

3.326

v

u

(

)

m + n +1 n +1 ∫ x exp − βx dx 0

∞

− β ( x −b) dx ∫ ( x − a )e n

3.326

0

∞

∫ ( x − a )e

− β ( x −b) n

3.326

dx

u

∞

− β ( x −b) dx ∫ ( x + a )e n

3.326

0

∞

− β ( x −b) dx ∫ ( x + a )e n

3.326

u

u

− β ( x −b) dx ∫ ( x + a )e n

3.326

0

v

(

)

nx ∫ exp − ae dx u

3.327

EXPONENTIALS OF EXPONENTIALS (99-102)

291 ∞

− (x + e − x ) dx ∫ xe

3.327

∞

3.327

0

− (x − e − x ) dx ∫ xe 0

(

v

3.328

)

x μx ∫ exp − e e dx u

(

v

)

−x μx ∫ exp − e e dx

3.328

u

∞

(

)

3.331

x ∫ x exp x − e dx

−∞

(

v

)

−x ∫ exp − βe − μx dx

3.331

u v

(

)

3.331

(

)

3.331

x ∫ exp − βe − μx dx u v

x ∫ exp − βe + μx dx u

∞

∫ exp(− βe

x

)

3.331

+ μx dx

0

(

v

)

−x ∫ exp − βe + μx dx

3.331

u

∞

β μ ∫ x μ −1e − βx dx

3.331

1

β

1

μ −1 − βx ∫ x e dx

μ

3.331

0

∞

β μ ∫ x μ −1e −βx dx

3.331

d γ (μ , β ) dβ d Γ(μ , β ) dβ

3.331

e − ρx ∫u x n+1 dx

3.351

0

v

∞

3.331

e−x −∫ dx x −u

3.351

u

3.351

− μx ∫ xe dx 0

u

2 − μx ∫ x e dx 0

3.351

COMBINATIONS OF EXPONENTIALS AND RATIONAL FUNCTIONS (103-105)

292

3.351

u

3 − μx ∫ x e dx 0

3.351

v

n − μx ∫ x e dx u v

3.353

xe x

∫ (1 + x )

2

dx

2

dx

u v

xe − x

∫ (1 − x )

3.353

u v

∫ u

∞

∫ u v

∫ u

∞

∫ u

u

∫ 0

∞

∫ u

e − qx x e − qx x

3.361

dx 3.361

dx

e − ρx dx ax ± b

3.362

e − μx dx x+β

3.362

e − μx dx x+β

3.362

e − μx dx x−u

3.362

∞

− μx ∫ x x − u e dx

3.363

u v

− px ∫ x a + bxe dx

3.363

u v

∫ x(a + bx )

m

e − px dx

3.363

u

∞

∫ u

x x−u

e − μx dx

3.363

∞

3.363

v

x − px ∫u (a + bx )m e dx

3.363

∞

ae − px ∫0 a + x 2 dx

3.363

∞

3.363

x − px ∫0 a + xe dx

∫ u

2

x − u − qx e dx x

COMBINATIONS OF EXPONENTIALS AND ALGEBRAIC FUNCTIONS (106-118)

293 ∞

n−

1 2

∫x

n+

1 2

∞

1

∫x

3.371

e − μx dx

0

∞

3.371

e − μx dx

0

3.371

−n

− μx ∫ x 2 e dx 0

∞

∫x

n−

p q

n+

p q

3.371

e − μx dx

0

∞

∫x

3.371

e − μx dx

0

∞

∫x

p −n q

3.371

e − μx dx

0

3.381

u

m − βx ∫ x e dx n

COMBINATIONS OF EXPONENTIALS AND ARBITRARY POWERS (120-125)

0

∞

3.381

m − βx ∫ x e dx n

u

∞

3.381

m − βx ∫ x e dx n

0

3.381

u

ρ −1 − x ∫ x e dx 0

∞

e−x ∫u xν dx

3.381

v

3.381

ν −1 − μx ∫ x e dx u

∫

∞

(

)

x 2 m exp − β x 2 n dx

−∞

∞

3.381

n±b

3.381

n±b

3.381

n±b

3.381

m ± a − βx ∫x e 0

u

m ± a − βx ∫x e 0

∞

m ± a − βx ∫x e u

∞

−ν − μx ∫ (1 + x ) e dx

3.382

1 ⎤ x ⎡1 ∫−∞⎢⎣ x − e x − 1⎥⎦e dx

3.427

0

0

COMBINATIONS OF RATIONAL FUNCTIONS OF POWERS AND EXPONENTIALS (126-128)

294 ∞

e − vx − e − μx ∫0 x ρ +1 dx

3.434

exp(− ax ) − exp(− bx )dx ∫0 x

3.434

t

t

∞

∞

∫x

n −1 − ρx 2

e

3.461

dx

0

∞

∫x

2 n −1 − ρx 2

e

3.461

dx

0

∞

2 ( n −1) − ρx ∫ x e dx 2

3.461

0

∞

e−ρ x ∫0 x 2n dx

3.461

∞

3.461

2 2

− ρx ∫ e dx 2

−∞

∞

3.461

2

2 n − ρx dx ∫x e

−∞ v

− qx

∫e

u

2

dx x2

v

3.461

m± a − βx ∫ x e dx n ±b

3.462

u

∞

3.462

− βx ∫ e dx n±a

0

u

3.462

− βx ∫ e dx n±a

0

∞

3.462

− βx ∫ e dx n±a

u

∞

∫ (x ± b )e

−βx n ± b

3.462

dx

0 v

− βx ∫ (x ± b )e

n ±b

3.462

dx

u

n ±b

3.462

∞

n ±b

3.462

∞

n ±b

3.462

e − βx ∫u x m±a dx v

e − βx ∫0 x m±a dx e − βx ∫u x m±a dx

COMBINATIONS OF EXPONENTIALS OF MORE COMPLICATED ARGUMENTS AND POWERS (129-160)

295 n ±b

e − βx ∫0 x m±a dx

3.462

v

m − ρx ∫ (ax ± b ) e dx

3.462

v

3.462

u

u

∫ (b − ax )

m

e − ρx dx

u

∞

m − ρx ∫ (b − ax ) e dx

3.462

u

m − ρx ∫ (b − ax ) e dx

3.462

∞

m − ρx ∫ (b − ax ) e dx

3.462

v

e − ρx ∫u (ax ± b )n dx

3.462

v

e − ρx ∫u (b − ax )n dx

3.462

∞

e − ρx ∫0 (b − ax )n dx

3.462

u

e − ρx ∫0 (b − ax )n dx

3.462

∞

3.462

0

0

u

e − ρx ∫u (b − ax )n dx v

e

⎛ x−a ⎞ −β ⎜ ⎟ ⎝ b ⎠

n±q

∫ ( ) u

x−a b

m± p

3.462

dx

∞

−β ( x± a ) dx ∫ (x − a )e

3.462

∞

m − px ∫ (ax ± b ) e dx

3.462

∞

3.462

0

0

m − px ∫ (ax ± b ) e dx u

3.462

u

∫ (ax ± b )

m

e − px dx

0

∞

e − px ∫0 (ax ± b )n dx

3.462

296 ∞

e − px ∫u (ax ± b )n dx

3.462

e − px ∫0 (ax ± b )n dx

3.462

u

∞

∫ (x + a )e

− β ( x− a )

∞

⎛ x−a⎞ ∫0 ⎜⎝ b ⎟⎠

m± p

⎛ x−a⎞ ∫0 ⎜⎝ b ⎟⎠

m± p

∞

⎛ x−a⎞ ∫u ⎜⎝ b ⎟⎠

m± p

∞

j

3.462

dx

0

u

e

e

e

⎛ x−a ⎞ −β ⎜ ⎟ ⎝ b ⎠

n± q

⎛ x−a ⎞ −β ⎜ ⎟ ⎝ b ⎠

n± q

⎛ x−a ⎞ −β ⎜ ⎟ ⎝ b ⎠

n± q

3.462

dx 3.462

dx 3.462

dx

⎛x−a⎞ ⎛x−a⎞ ∫0 ⎜⎝ b ⎟⎠ exp β ⎜⎝ b ⎟⎠ dx

3.462

∞

e − βx ∫0 x m dx

3.462

k

n

u

e − βx ∫0 x m dx n

3.462

∞

n

3.462

e − βx ∫u x m dx

∫ (e

∞

− μx 2

− e − βx

2

u

) dxx

3.464

2

⎛ − μx 2 − βx 2 ⎞ dx −e ⎟ 2 ∫ ⎜e 0⎝ ⎠x

3.464

u

∫ exp⎜⎝ − x

b ⎞ dx n ⎟ ⎠

3.471

u

⎛ b ⎞ ∫0 exp⎜⎝ − x n ⎟⎠dx

3.471

u

⎛ b⎞ ∫0 exp⎜⎝ − x ⎟⎠dx

3.471

u

⎛ b ⎞ ∫0 exp⎜⎝ − x 2 ⎟⎠dx

3.471

⎛ β ⎞ dx ∫0 exp⎜⎝ − x n ⎟⎠ x n+1

3.471

v

⎛

u

u

v

[

∫ exp − β (ax + b ) u

−n

]dx

3.471

297

]dx

3.471

∫ exp − β (ax + b ) dx −1

]

3.471

∫ exp[− β (ax + b )

−2

]dx

3.471

⎤ ⎥ dx ⎦

3.471

u

[

−n

[

∫ exp − β (ax + b ) 0

u

0

u

0

⎡ β (ax + b ) ∫0 exp⎢⎣− ax + b u

v

[

−n

]dx

3.471

[

−n

]dx

3.471

−1

]dx

3.471

−2

]dx

3.471

∫ exp − β (b − ax ) u

u

∫ exp − β (b − ax ) 0

∫ exp[− β (b − ax ) u

0

[

u

∫ exp − β (b − ax ) 0

u

⎡ β ⎤ ∫0 exp⎢⎣− b − ax ⎥⎦dx

3.471

b ⎤ ⎡ ∫0 exp⎢⎣− a + x ⎥⎦ dx

3.471

u

[

(

)

⎛ b exp⎜ − n ⎝ x ∫0 x m

⎞ ⎟ ⎠ dx

u

n+ p ∫ exp − β x

−1

]dx

3.471

0

u

v

−

β xn

3.471

3.471

e ∫u x m dx ∞

−

β n

3.471

e x ∫0 x m dx

∞

−

β xn

3.471

e ∫u x m dx u

∫x 0

m

⎛ b ⎞ exp⎜ − n ⎟dx ⎝ x ⎠

3.471

298 v

∫ u

(a − x )μ −1 exp⎛ − β ⎞dx ⎜ ⎝

x μ +1

∞

( ∫ exp(− x )x n

m +1 2 )n −1

⎟ x⎠

dx

3.471

3.473

0

∞

(

)

3.473

n (1 2− m )n −1 dx ∫ exp(− βx )x

3.473

v

dx u ± ln x

4.211

∞

4.211

v

4.211

( m −1 2 )n −1 n dx ∫ exp − βx x 0

∞

0

∫

dx ∫e ln x

∫ ln xdx u v

dx ∫u x ln x

4.211

v

x n dx ∫u ln x

4.211

u

x n dx ∫0 ln x

4.211

v

xdx ∫u ln x

4.211

v

4.211

∫ x ln xdx u e

∫ ln x dx

4.211

1

∫ ln(ln x )dx

4.211

(ln x ) p dx

4.211

v

u v

∫ u

x

ln xdx ∫1 x

4.211

e

(ln x ) p dx

4.211

ln xdx

4.212

e

∫ 1 1

x

∫ a + ln x 0

LOGARITHMIC FUNCTIONS (161-178)

299 1

4.212

v

4.212

ln xdx ∫0 a − ln x dx ∫u (a ± ln x )2 v

4.212

dx

∫ (a ± ln x )

n

u v

4.212

v

4.212

v

4.212

ln xdx ∫u (a + ln x )n ln xdx ∫u (a − ln x )n ln xdx ∫u (a ± ln x )2

⎛ 1⎞ ∫0 ⎜⎝ ln x ⎟⎠ 1

1

∫⎛

μ −1

4.215

dx

1⎞ ⎜ ln ⎟ ⎝ x⎠

0

4.215

dx

μ

n

4.215

−1

⎛ 1 ⎞2 ∫0 ⎜⎝ ln x ⎟⎠ dx 1

1

∫ 0

4.215

dx ⎛ 1⎞ ⎜ ln ⎟ ⎝ x⎠

⎛ 1⎞ ∫u ⎜⎝ ln x ⎟⎠ v

n +1 2

μ −1

4.215

dx

⎛ 1⎞ ln ⎟ γ ∫⎜ nβ 0 ⎝ t ⎠ 1

1

βz n

1

∫⎛ 0

γ −1

4.215

dt 4.215

dt

1⎞ ⎜ ln ⎟ ⎝ t⎠

z +1

ln (ln x ) ∫1 x 2 dx

∞

∞

1 ln x dx ∫ 4 1 x ( x + 1) n

⎛ 1 ⎞ 2 p −1 ∫0 ⎜⎝ ln x ⎟⎠ x dx 1

4.229

LOGARITHMS OF MORE COMPLICATED ARGUMENTS (178)

4.241

COMBINATIONS OF LOGARITHMS AND ALGEBRAIC FUNCTIONS (179-180)

4.269

COMBINATIONS INVOLVING POWERS OF THE LOGARITHM AND OTHER POWERS (181-191)

300 1

∫ 0

x p −1

4.269

dx n

⎛ 1 ⎞2 ⎜ ln ⎟ ⎝ x⎠

n p ∫ (ln x ) x dx

4.272

1

4.272

1

0

p ∫ (ln x )x dx 0

1

−n

⎛ 1 ⎞ 2 ν −1 ∫0 ⎜⎝ ln x ⎟⎠ x dx 1

⎛ 1⎞ ∫0 ⎜⎝ ln x ⎟⎠ 1

∞

∫ 1

∞

∫ 1

∞

∫ 1 v

∫

n+

1 2

4.272

4.272

x

ν −1

dx

(ln x ) p ± a ± n−1 dx

4.272

(ln x ) p + a −1 dx

4.272

(ln x ) p −1 dx

4.272

(ln x ) p −1 dx

4.272

x2

x2

x2

x

u

2

∞

dx ∫1 (ln x ) p x 2

4.272

∞

4.272

dx

∫ (ln x ) 1 1 e

∫ 0

∞

p +1

q

x2 4.274

x

x[− (1 + ln x )]

n 2

dx

1 ⎤ dx x

⎡ 1

∫ ⎢⎣ x − 1 − x ln x ⎥⎦ 1

∞

4.281

∫ x (ln p + ln x )

dx

4.281

u

x n + a −1 ∫0 ln x dx

4.281

u

x n −1 ∫0 ln x dx

4.281

∞

4.283

2

1

⎤ dx ⎡ 1 1 ∫1 ⎢⎣ x ln x − ln x(1 + ln x )⎥⎦ x

COMBINATIONS OF RATIONAL FUNCTIONS OF ln x AND POWERS (193-194)

301

1

⎡1

1

⎤

∫ ⎢⎣ x + ln(1 − x )⎥⎦ dx

4.283

0

μ −1 ∫ ln(a + ln x )x dx

4.326

COMBINATIONS OF LOGARITHMIC FUNCTIONS OF MORE COMPLICATED ARGUMENTS AND POWERS (196)

∞

− μx ∫ e (ln x ) dx

4.331

COMBINATIONS OF LOGARITHMS AND EXPONENTIALS (197-201)

v

4.331

1

0

n

0

−μ x ∫ e ln xdx u v

μx ∫ e ln xdx

4.331

u

−μ x ∫ e ln(a ± bx )dx

4.337

∞

−μ x ∫ e ln(a + bx )dx

4.337

∞

ln(a ± bx )dx

4.337

−μ x ∫ e ln(a ± bx )dx

4.337

v

u

0

∫e

−μ x

u

∞

0

± Γ(0,± x )

(

)

8.212

Ei ± x n Ei(± x ) ⎛ 1 ⎞ Ei⎜ ± n ⎟ ⎝ x ⎠ ⎛ 1⎞ Ei⎜ ± ⎟ ⎝ x⎠ Ei( x ± y ) Ei[− ( x + y )]

8.212

⎛ 1 ⎞ ⎟⎟ Ei⎜⎜ ⎝x± y⎠ ⎛ x⎞ Ei⎜⎜ ± ⎟⎟ ⎝ y⎠

8.212

Ei a x li(x ) − Γ(0,− ln x )

8.212

ln x

8.241

( )

et ∫ t dt −∞

THE EXPONENTIAL INTEGRAL FUNCTION Ei(x) (203-209)

8.212 8.212 8.212 8.212 8.212

8.212

8.240

THE LOGARITHM INTEGRAL li(x) (211)

8.240 INTEGRAL REPRESENTATIONS (212-213)

302

( )

li a x li( xy ) erfc( x )

Φ ( xy )

8.241 8.241 8.250

8.252

⎛ x⎞ Φ⎜⎜ ⎟⎟ ⎝ y⎠ ⎛ y ⎞ Φ⎜ ⎟ ⎝ 2x ⎠ ⎛x± y⎞ Φ⎜ ⎟ ⎝ a ⎠ Φ(x ± y ) ⎛x± y⎞ Φ⎜ ⎟ ⎝ 2 ⎠ Φ (a + bx )

8.252

Φ ax

8.252

( )

8.252 8.252 8.252 8.252 8.252

⎛1− z ⎞ ⎟ ⎝ v ⎠

8.313

⎛ z −1⎞ ⎟ ⎝ v ⎠

8.313

Γ⎜

Γ⎜

Γ(x ± a ) Γ(x ± n ) Γ(n − x ) (n − x )Γ(n − x ) Γ(1 − x ) Γ(− x ) ⎛ x⎞ Γ⎜ ± ⎟ ⎝ 2⎠ ⎛ x⎞ Γ⎜1 − ⎟ ⎝ 2⎠ ⎛1− x ⎞ Γ⎜ ⎟ ⎝ 2 ⎠ Γ(m )Γ(1 − m )

⎛ x ± 1⎞ Γ⎜ ⎟ ⎝ 2 ⎠ Γ(x − 1)

THE PROBABILITY INTEGRAL, THE FRESNEL INTEGRALS Φ ( x ), S ( x), C ( x) , THE ERROR FUNCTION ERF(x), AND THE COMPLEMENTARY ERROR FUNCTION ERFC( x ) (214-220)

8.331 8.331 8.331 8.331 8.331 8.331 8.331 8.331 8.331 8.331 8.331

THE GAMMA FUNCTION (EULER’S INTEGRAL OF THE SECOND KIND): Γ(z) (221)

FUNCTIONAL RELATIONS INVOLVING THE GAMMA FUNCTION (222-238)

303

⎞ ⎛1 Γ⎜ − x ⎟ ⎠ ⎝2 Γ(x )Γ(− x ) Γ(1 + x )Γ(1 − x ) Γ(1 + x )Γ(− x ) Γ(− x )Γ( x − 1) 1⎞ ⎛ Γ( x )Γ⎜ x + ⎟ 2⎠ ⎝ 1⎞ ⎛ Γ⎜ x − ⎟ 2⎠ ⎝ 1⎞ ⎛1 ⎞ ⎛ Γ ⎜ x − ⎟ Γ⎜ − x ⎟ 2⎠ ⎝2 ⎠ ⎝ 1⎞ ⎛ 1⎞ ⎛ Γ⎜ x + ⎟Γ⎜ x − ⎟ 2⎠ ⎝ 2⎠ ⎝ Γ(2 x ) ⎛ x⎞ ⎛ x ⎞ Γ⎜1 − ⎟Γ⎜ − 1⎟ ⎝ 2⎠ ⎝2 ⎠ Γ(n ± x )Γ( x − n ) Γ(1 + x )Γ( x − 1) Γ(n − x )Γ( x + n ) Γ(x )Γ(1 − x ) Γ(x )Γ(n − x ) sin πx 2 n −1 n 1 definitions Γ(x ) & Γ( x) n −1 ⎛ k⎞ ∏ Γ⎜ z + ⎟ k =0 ⎝ n⎠ Γ(2 z ), Γ(3 z ), Γ(nz ) Γ(nz + b ) n −1 ⎛k⎞ Γ⎜ ⎟ ∏ k =1 ⎝ n ⎠ n −1 ⎛ k ⎞⎛ k ⎞ Γ⎜ ⎟⎜1 − ⎟ ∏ n⎠ k =1 ⎝ n ⎠⎝ Γ(n ± k ) Γ(n + 1) ⎛ n⎞ Γ⎜ ± ⎟ ⎝ 2⎠

8.331 8.334 8.334 8.334 8.334 8.334 8.334 8.334 8.334 8.334 8.334 8.334 8.334 8.334 8.334 8.334 8.335 8.335 8.335 8.335 8.335 8.335 8.335 8.335 8.339 8.339 8.339

PARTICULAR VALUES: FOR N A NATURAL NUMBER (239-252)

304

⎛n ⎞ + 1⎟ ⎝2 ⎠ p⎞ ⎛ Γ⎜ n ± ⎟ q⎠ ⎝ 1⎞ ⎛ ⎛ 3⎞ Γ⎜ n ± ⎟ ,…, Γ⎜ n ± ⎟ 2⎠ ⎝ 4⎠ ⎝ ⎛p ⎞ Γ⎜ − n ⎟ ⎠ ⎝q ⎛1 ⎞ ⎛3 ⎞ Γ⎜ − n ⎟ ,…, Γ⎜ − n ⎟ ⎝2 ⎠ ⎝4 ⎠ ⎛ p⎞ Γ⎜ ± ⎟ ⎝ q⎠ 1⎞ ⎛1 ⎞ ⎛ Γ⎜ n ± ⎟Γ⎜ − n ⎟ 2⎠ ⎝2 ⎠ ⎝ ⎛n⎞ ⎛ n⎞ Γ ⎜ ⎟Γ ⎜ − ⎟ ⎝2⎠ ⎝ 2⎠ Γ(n + x ) Γ( x − n ) Γ(z ,0 ) Γ⎜

Γ(a, ∞ ) γ (a,0) γ (a, ∞ )

γ (− n, x ) γ (n ± k , x ) *

Γ(n ± k , x ) Γ(− n + k , x ) Γ(0, x ), Γ(1, x )Γ(2, x ), Γ(3, x ), Γ(4, x )

Γ(1,− x ), Γ(2,− x ), Γ(3,− x ), Γ(4,− x ) Γ(− 1, x ), Γ(− 2, x ), Γ(− 3, x ), Γ(− 4, x )

8.339 8.339 8.339 8.339 8.339 8.339 8.339 8.339 8.339 8.350

8.350 8.350 8.350 8.351 8.352 8.352 8.352 8.352 8.352 8.352

γ (1, x ), γ (2, x ), γ (3, x ), γ (4, x )

8.352

γ (1,− x ), γ (2,− x ), γ (3,− x ), γ (4,− x )

γ (α , x ) γ (α , x ± y ) γ (α , xy ) ⎛

x⎞ y⎠

γ ⎜⎜ α , ⎟⎟ ⎝

SPECIAL CASES (256-260)

8.352

Γ(0,− x ), Γ(− 1,− x ), Γ(− 2,− x ), Γ(− 3,− x ), Γ(− 4,− x )

THE INCOMPLETE GAMMA FUNCTION DEFINITION (253-255)

8.352 8.353 8.353 8.353

INTEGRAL REPRESENTATIONS (261-262)

305

Γ(α , x )

8.353

⎛ x⎞ Γ⎜⎜ α , ⎟⎟ ⎝ y⎠ Γ(α , x ± y ) Γ(a ± k , x ) γ (a ± k , x ) dΓ(a, x ) da dγ (a, x ) da d erf ( x ) dx d [Ei(x )] & d [li(x )] dx dx dγ (a, x ) dx dΓ( x ) dx d ⎛ 1 ⎞ ⎟ ⎜ dx ⎜⎝ Γ( x ) ⎟⎠

8.356 8.356 8.356 8.356 8.356 8.356 8.356

8.356

(

)

8.356

(

)

8.356

(

)

8.356

d Γ γ , βu n dγ nβ γ

d γ ν , βu n dν Γ(ν ) ⎞ ⎞ ⎛1 ⎛1 γ ⎜ ,± x k ⎟, Γ⎜ ,± x k ⎟ ⎠ ⎠ ⎝2 ⎝2 2⎤ ⎡1 γ ⎢ , ax ⎥ ⎦ ⎣2 ⎞ ⎛ 1 Γ⎜ ± , ± x ⎟ ⎠ ⎝ 2 ⎞ ⎛ 1 Γ⎜ ± , ± x 2 ⎟ ⎠ ⎝ 2 ⎞ ⎛1 γ ⎜ ,±x ⎟ ⎠ ⎝2

( )

FUNCTIONAL RELATIONS (263-264)

8.356

)

d Γ γ , βu n nβ γ du nβ γ 1

8.356

(

d γ ν , βu n nβ ν du nβ ν 1

8.353

8.356

x

d exp( xy )dy ` dx ∫0

8.353

8.359 8.359 8.359 8.359 8.359

RELATIONSHIPS WITH OTHER FUNCTIONS (265-272)

306

⎞ ⎛1 ⎠ ⎝2 1 ⎡ ⎛1 2⎞ ⎛ 1 2 ⎞⎤ ⎢γ ⎜ 2 , x ⎟ + Γ⎜ 2 , x ⎟⎥ π ⎣ ⎝ ⎠⎦ ⎠ ⎝

γ ⎜ ,± x 2 ⎟

erf (± ∞ ) ⎞ ⎛ n Γ⎜ ± , ± x ⎟ ⎠ ⎝ 2 ⎛n ⎞ Γ⎜ , x ⎟ ⎝k ⎠ ⎛n ⎞ γ ⎜ , x⎟ ⎝k ⎠

8.359 8.359 8.359

ln (ln t ) dt t2 1

∞

8.367

1 ⎤ dt t

⎡ 1

∫ ⎢⎣ t − 1 − t ln t ⎥⎦ 1

∞

⎤ dt ⎡ 1 1 −∫⎢ − t ln t ln t (1 + ln t )⎥⎦ t 1 ⎣ 1 ⎡1 1 ⎤ ∫0 ⎢⎣ t + ln(1 − t )⎥⎦ dt ∞

(

8.359 8.359

−∫ ∞

8.359

)

8.367 8.367 8.367

− ∫ t exp t − e t dt

8.367

0

⎡ et et ⎤ − ∫ ⎢ t e t − 1⎥⎦ dt − ∞⎣

8.367

∞

8.367

−∞

− (t + e − t ) ∫ te dt + Ei(−1) 0

∞

−t Ei(1) − ∫ te −(t −e )dt

0

8.367

THE PSI FUNCTION. EULER’S CONSTANT: INTEGRAL REPRESENTATIONS (273-275)

About The Author

Dr. Francis (Frank) J. O’Brien, Jr. received his Ph.D. from Columbia. He is a Senior Scientist with the Department of Navy and works in the Undersea Warfare Combat Systems Department at the Naval Undersea Warfare Center, Newport, Rhode Island. His areas of expertise are statistical signal processing and logistics modeling. O’Brien has contributed extensively to Gradshteyn and Ryzhik’s Table of Integrals, Series and Products. He holds numerous U.S. Patents involving engineering mathematics.