500 Integrals

500 INTEGRALS

OF

ELEMENTARY AND SPECIAL

FUNCTIONS

⊕

Francis J. O’Brien, Jr.

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.

500 Integrals of Elementary

and Special Functions

Francis J. O’Brien, Jr. Naval Undersea Warfare Center

Naval Sea Systems Command Newport, Rhode Island

⊗

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.

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

• A square root x or _x2 or π π _{is taken to be positive unless otherwise}

specified

• Logarithms are natural logarithms, denoted lnx (base e), unless otherwise indicated

• Errata for the 6th _{and 7}th _{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 needed; e.g., the derivative,

( )

da x a dγ ,

(see Section 8.356)

o Sometimes computer returns a “different” solution compared to paper and pencil answer

¾ Computer always provides only a single solution when multiple answers are possible; e.g.,

∫

(

+

)

x dx bx a ln or dx e e ax ax

∫

+−1 1 (see Section 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

Acknowledgements

The author would like to acknowledge and thank all those who have funded, assisted, and encouraged him over the years. These include the Office of Naval Research, the Base Commander, my research colleagues in the USW Combat Systems Department, and the attorneys and paralegals in Office of Patent Counsel of the Naval Undersea Warfare Center, Newport, Rhode Island.

Professor Alan Jeffrey of the University of Newcastle Upon Tyne (England), Editor of

Table of Integrals, Series and Products, has been most helpful in his encouragement. I also thank

his co-editor, Dr. Daniel Zwillinger of Rensselaer Polytechnic Institute, for correspondence regarding errata and related matters.

I also want to acknowledge the assistance of Aimee Ross, a 3rd _{year mathematics major at}

University of Massachusetts—Dartmouth, who read the early sections of the manuscript, and provided feedback on the level of difficulty and the clarity of the material.

╬ Francis J. O’Brien, Jr.

Newport, Rhode Island October 12, 2008

TABLE OF CONTENTS

Section in Gradshteyn and Ryzhik

Page

MATHEMATICAL AND GRAPHICAL SUMMARY OF SELECTED ELEMENTARY AND SPECIAL FUNCTIONS

Exponential Functions 2

Logarithmic Functions 4

Gamma Function 6

Incomplete Gamma Functions 8

Probability Integral or Error Function (Erf) and Imaginary Error Function (Erfi) 10

Exponential-Integral Function 12

Logarithm-Integral Function 14

Euler’s Constant 16

Catalan’s Constant 17

Partial Fractions 18

Miscellaneous (Completing the Square, Finite Binomial Expansions, Double Factorial, Natural Number N, Differentiation Under the Integral Sign)

20 NOTATIONS 22

INTRODUCTION

0.1 Finite Sums 24 0.11 Progressions 25 0.111 Arithmetic progressions 25 0.112 Geometric progressions 26

0.12 Sums of powers of natural numbers 26

INDEFINITE INTEGRALS OF

ELEMENTARY FUNCTIONS

2.3 The Exponential Function 28

2.31 Forms containing _{ax ax}n

e

e , 30

2.312 31

2.32 The exponential combined with rational functions of x 35

2.7 Logarithms and Inverse and Hyperbolic Functions 61

DEFINITE INTEGRALS OF

ELEMENTARY FUNCTIONS

3.3–3.4 Exponential Functions 79 3.31 Exponential functions 80 3.310 81 3.311 82

3.32–3.34 Exponentials of more complicated arguments 84

3.321 85 3.322 89 3.323 90 3.324 94 3.326 96 3.327–3.334 Exponentials of exponentials 98 3.327 99 3.328 100 3.331 101 3.35 Combinations of exponentials and rational functions 103

3.351 104

3.353 105 3.36–3.37 Combinations of exponentials and algebraic functions 106

3.361 107

3.362 108 3.363 109 3.371 116 3.38–3.39 Combinations of exponentials and arbitrary powers 120

3.381 121 3.382 125 3.41–3.44 Combinations of rational functions of powers and exponentials 126

3.427 127

3.434 128 3.46–3.48 Combinations of exponentials of more complicated arguments and powers 129

3.461 130 3.462 134 3.464 148 3.471 151 3.473 160 4.2–4.4 Logarithmic Functions 161 4.21 Logarithmic functions 162 4.211 163

4.215 174

4.22 Logarithms of more complicated arguments 177

4.229 178

4.24 Combinations of logarithms and algebraic functions 179

4.241 180

4.26–4.27 Combinations involving powers of the logarithm and other powers 181

4.269 182

4.272 185 4.274 190 4.28 Combinations of rational functions of ln x and powers 192

4.281 193

4.283 194 4.29-4.32 Combinations of logarithmic functions of more complicated arguments

and powers

195

4.326 196

4.33–4.34 Combinations of logarithms and exponentials 197

4.331 198

4.337 199

SPECIAL FUNCTIONS

8.2 The Exponential Integral Function and Functions Generated by It 202

8.21 The exponential integral function Ei(x) 203

8.212 204

8.24 The logarithm integral li(x) 210

8.240 211

8.241 Integral representations 212

8.25 The probability integral, the Fresnel Integrals Φ

( )

x ,S(x),C(x), the error

function erf(x), and the complementary error function erfc(x)

214

8.250 Definition 215

8.252 Integral representations 216

8.3 Euler’s Integrals of the First and Second Kinds and Functions Generated by Them 221

8.31 The gamma function (Euler’s integral of the second kind): Γ(z) 222 8.313 223

8.33 Functional relations involving the gamma function 224

8.331 225 8.334 232 8.335 238

8.339 Particular Values: For n a natural number 239

8.35 The Incomplete Gamma Function 253

8.350 Definition 254

8.351 255

8.356 Functional relations 263

8.359 Relationships with other functions 265

8.36 The Psi function ψ

( )

x 273

8.367 Euler’s constant: Integral representations 274

References 276 Index Index of Formulas 278 284

LIST OF ILLUSTRATIONS

Figure 1. Exponential Function (Growth) 2 Figure 2. Exponential Function (Decay) 4 Figure 3. Natural Logarithm Function 4

Figure 4. Gamma Function 6

Figure 5. Error Function 10

Figure 6. Imaginary Error Function 10 Figure 7. Exponential-integral Function 12 Figure 8. Logarithm-integral Function 14

Figure 9. Euler’s Constant 16

Figure 10. Catalan’s Constant 17

MATHEMATICAL & GRAPHICAL SUMMARY

OF SELECTED

ELEMENTARY AND SPECIAL FUNCTIONS

EXPONENTIAL FUNCTIONS

x e

x e −

Figure 1. Exponential Function (Growth) Figure 2. Exponential Function (Decay)

Notation: ex =exp

( )

x =ln−1

( )

x

Definitions & Laws of Exponents:

( )

( )

x x x xy y x y x y x y x y x b a ab a a a a a a a a a a = = = = ≠ = − + 0 , 1 0 a x x a x x a x x a x x n n n n e a e a e a e a ln ln ln ln − − − − = = = =

_{[ ]}

_{( )}

_{( )}

( )

[

( )

]

[ ]

( )

[ ]

( )

1

(

imaginary

)

1 2 1 1 1 1 1 i a a a a a a a q q q q p p p q p p q q p p q p = − − = − = − = = =

( )

( )

x n n x n x n n x n x a a a e e x e e x e x ln ln ln ln ln = = = = = 4 4 4 4 8 4 4 4 4 7 6 exp. for be Let a e Limits: 1 lim 1 lim 0 0 = = − → → x x x x e e 0 lim lim = +∞ = − +∞ → +∞ → x x x x e e 0 lim lim lim = = +∞ = +∞ → − +∞ → +∞ → x a x a x x a x x e x x e x e 0 lim lim 0 = − − = − − − ∞ → − − → x e e a b x e e bx ax x bx ax x x x

Derivatives:

¾ a ,,e n are assumed constants except where noted

4 4 4 4 4 4 4 4 8 4 4 4 4 4 4 4 4 7 6 ln ln ln ln 1 ln rule chain dx du e dx de dx du a a dx de dx da dx dv u u dx du vu dx u dv u dx de dx du u u u a u u v v v u v v = = = + = = = −

[

not aconstant

]

ln ln ln 1 ln ln ln ln ln 1 ln n a x x a dn de dn da a a dx de dx a d a a dx de dx da a a nx dx de dx da n x a x x x a x x x a x x x n a x x n n n n n n = = − = = = = = = − − ax ax ax ax x x x x ae dx de ae dx de e dx de e dx de − − − − − = = − = = Indefinite Integrals: a e dx e a e dx e e dx e e dx e ax ax ax ax x x x x − − − − − = ∫ = ∫ − = ∫ = ∫

• See Section 2.32 for generalized indefinite exponential integrals of form,

∫

± ±

,

dx e x m axn

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”)

LOGARITHMIC FUNCTIONS

(Base e)

Figure 3. Natural Logarithm Functions, lnx & ,

ln 1 x & x 1 Notation: x x ln

log = (natural logarithm, base e)

(

)

m m x x log log = Definition: log , 0 1 > =

∫

x x t dt x x x ln 1 x 1 x

Properties:

( )

(

)

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = + − = + = a b b a a b b a y x a y x y b x a xy a y x y b x a y x y x y x xy ln ln ln ln ln ln ln ln log log log log log log x x x n xn log 1 log log log − = = ax e ax e x e x e ax ax x x − = = − = = − − ln ln ln ln Limits: −∞ = +∞ = + → +∞ → x x x x log lim log lim 0 0 log lim log lim = +∞ = +∞ → +∞ → a x a x x x x x −∞ = = + → +∞ → x x x x e x e x log lim 0 log lim 0 0 log lim log lim 0 = +∞ = + → +∞ → x e x e x x x x

( )

1 1 ln 1 ln 0 ) 1 ln( − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = e e Derivatives: 0 , 1 ln ≠ = x x dx x d

(

)

(

)

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

(

)

rule chain 1 ln dx du u dx u d = Indefinite Integrals:

∫

= x x dx ln

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

( )

(

)

( )

, 1 Li 1 ln Li 1 2 2 2 < = − − =

∑

∫

∞ = x k x x dx x x x k k References • Bers, Calculus. Ch. 6, pp. 357 ff.; Ch. 7, pp. 454 ff.; Ch. 8, pp. 547 ff. • Carr, 1970

• Dwight, Table of Integrals, 1961 • Spiegel, Chaps. 7 & 20

• Table of Integrals, Series, and Products. Sects. 1.5 (“The logarithm”), 2.01 (“The basic integrals”) • Wolfram Functions website, http://functions.wolfram.com/GammaBetaErf/

GAMMA FUNCTION

Figure 4. Gamma Function, Γ

( )

x Definition:

( )

, 0 1 dt e t z z −t ∞ −

∫

= Γ real z >0

[

Formula 8.310.1

]

( )

ln 1 , 1 0 1 dt t z z

∫

⎟ − ⎠ ⎞ ⎜ ⎝ ⎛ = Γ real z>0

[

Formula 8.312.1

]

Integral, Product, and Series Representations:

( )

₌ ∞

_∫

− − Γ 0 1 dt e t z x x x zt , real z,x>0

[

Formula 8.312.2

]

( )

(

)

1 , real 0 lim ! lim 1 1 > ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = + = Γ

∏

∏

= ∞ → = ∞ → x k x x p k x x p p x _p k x p p k x p

[

Artin, Formula 2.7

]

( )

( )

lim , real 0 1 1 1 1 1 exp exp 1 1 1 > + = + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = Γ

∏

∏

∏

= ∞ = ∞ = →∞ x x k k x n k x k x k x k x x x x n k x k k n x γ

[

Formula 8.322

]

( )

_{( )}

x nz n k z n k Γ Γ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + Γ

∏

− = 1 & & 1 0

[

Section 8.335

]

( )

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

∑

∞ = x k x k x x x x k γ

[

Artin,Formula 2.9

]

Properties:

( ) (

= −1

)

! Γ x x

( )

(

1

)

_, x x x =−Γ − − Γ x not a natural num.

( )

=+∞ Γ + → x x 0 lim Γ

( )

=+∞ +∞ → x xlim

( )

≠0

Derivatives:

( )

_{( )}

( ) _{e dt t e tdt}

_{( ) ( )}

_{x x} e x dt e t dx d x dx x d t x t t x t x Ψ Γ = = ∂ ∂ = = Γ′ = Γ

∫

∫

∫

∞ − − ∞ − − ∞ − − 0 1 0 ln 1 0 1 ln

( )

(

)

( )

1 (Formula8.366.1) constant s Euler' ln 1 0 Ψ = − = = Γ′

∫

∞ − γ tdt e t ( )

_{( )}

_{( )}

∫

∞ − − = Γ 0 1 lnt dt e t x x t n n •

( )

( )

( )

( )

x x x dx x d Ψ = Γ Γ′ = Γ ln

(digamma or Psi function), Formula 8.330

•

_{( )}

( )

[

]

( )

( )

( )

x x dx x d x x dx d Γ Ψ − = Γ Γ − = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ Γ 2 1 1

Relation to Incomplete Gamma Functions, γ

( )

a,x &Γ

( )

a,x :

( )

( )

( )

4 4 4 3 4 4 4 2 14243 14243 1 43 42 1 GAMMA INCOMPLETE GAMMA COMPLETE UPPER 1 LOWER 0 1 0 1 , , dt e t dt e t dt e t x a x a a t x a t x a t a − ∞ − − − − ∞ −

∫

∫

∫

= + Γ + = Γ γ

[

Formula 8.356.3

]

• NOTE: Relation is additive; e.g., γ

( ) ( ) ( )

a,x =Γ a −Γa,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 onΓ

( )

x : Emil Artin’s brief 1964 book—The Gamma Function—provides a complete statement for

real variables of this special transcendental function called the complete gamma function, and derives the fundamental mathematical properties originated in the classical 18th & 19th century works of Euler, Gauss, Legendre, Riemann, Stirling, Weierstrass, and others. See Artin’s book for other definitions, theorems, and derivations of Γ

( )

x not given in this elementary book. Whittaker & Watson is also recommended for derivations.

References

• Abramowitz & Stegun, Ch. 6. • Artin, 1964

• Bers, Calculus , Chaps. 4 & 6, pp. 402-3 • Carr, 1970

• Moll,http://www.math.tulane.edu/~vhm/Table.html

• Table of Integrals, Series, and Products, Sects 1.3-1.4 and Sects. 8.31-8.35 & Index • Whittaker & Watson, 1934

• Wolfram Functions website, http://functions.wolfram.com/GammaBetaErf/

• O’Brien, http://www.docstoc.com/docs/5473276/Pochhammer-Symbol-Selected-Proofs

• O’Brien, http://www.docstoc.com/docs/5836783/Selected-Transformations-Identities--and-Special-Values--for-the-Gamma-Function

INCOMPLETE GAMMA FUNCTIONS

1.

Lower Incomplete Gamma Function

Definition:

( )

a x t e tdt x a− − ∫ = 0 1 , γ , real a > 0

[

Formula 8.350.1

]

NOTE:

• See Sect. 8.352 for integer special cases,γ

( )

n,x , etc.

Integral Representation:

( )

=

∫

− − > 1 0 1 0 real , ,x x t e dt a,x a a a xt γ

[

Sections 3.331 & 8.353

]

Properties: • γ

( ) ( ) ( )

a,x =Γa −Γ a,x •

( )

, =

(

+1,

)

+ , ≠0 − a a e x x a x a x a γ γ

[

Formula 8.356.1

]

• γ

( )

1_2,x2 = πΦ

( )

x _,

[

Φ x

( )

=_{erf(x), error function, 8.250.1}

]

• γ

( )

1₂,x = πΦ

( )

x • γ

( )

a,0 =0

[

Formula 8.350.5

]

• γ

(

a,∞

) ( )

=Γ a •

( )

( )

a x e x dx x a d dx x a dγ , ₌₋ Γ , ₌ −1 −

_[

Form. 8.356.4

]

•

( )

,

( )

,

( )

,

(

chain rule

)

dx du du u a d dx u a d dx u a dγ ₌₋ Γ ₌ γ •

( ) ( )

_{= +}

∫

1 − − 0 1 ln ln , , tdt e t x x x a da x a dγ γ a a xt

[

Sect. 8.356

]

2.

Upper Incomplete Gamma Function Definition:

( )

a x t e tdt x a − ∞ − ∫ = Γ 1 ,

[

Formula 8.350.2

]

NOTE:

• See Sect. 8.352 for integer special cases, Γ

( )

n,x , etc.

Integral Representation:

( )

∫

∞ − − _> = Γ 1 1 0 real , ,x x t e dt a,x a a a xt

[

Sections 3.331 & 8.353

]

Properties: • Γ

( ) ( ) ( )

a,x =Γa −γ a,x •

( )

,

(

1,

)

, a e x x a x a x a − − + Γ = Γ a≠0

[

Formula 8.356.2

]

• Γ

( )

x2 = π − πΦ

( )

x 2

1_,

[

Φ x

( )

=_{erf(x), error function, 8.250.1}

]

• Γ 0

( ) ( )

a, =Γ a

[

Formula 8.350.3

]

• Γ a

(

,∞

)

=0

[

Formula 8.350.4

]

•

( )

( )

dx u a d dx x a d , & , Γ Γ (see

1

. above) • Γ

( )

_=Γ

( )

₊

∫

∞ − − 1 1 ln ln , , tdt e t x x x a da x a d _{a a xt}

[

Sect. 8.356

]

( )

( )

⎥_⎦=

[

−Ψ

( )

]

Γ

( )

( )

+Γ

( )

∫

> ⎤ ⎢ ⎣ ⎡ Γ − − 1 0 1 0 real , ln , ln , a,x tdt e t a x a x a a x a x a da d a xt a γ γ

[

Section 8.356

]

NOTE:

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

( )

a,x Indefinite Integrals:

( )

(

)

( )

n m dt e t na na ax dx e x t ax n ax m n n 1 , 1 , 1 1 1 1 ₊ = − = − Γ − = ∞ − − − − −

∫

∫

γ γ γ γ γ γ γ

(

)

n m dt e t n n x dx e x t x n x m n n 1 , 1 , ₌₋ 1 ₌ + Γ − = ∞ − − −

∫

∫

γ β β β γ β γ γ γ β

_[

Formula 2.33.10

]

( )

(

)

( )

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

∫

∫

[

Formula 2.325.6

]

(

)

n m z dt t e n n x z dx x e n n x z t z n z m x 1 , , 1 − = − = − Γ − =

∫

∫

∞ + − − β β _{β β β}

[

Formula 2.33.19

]

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.

¾ See Section 8.359 for special forms, _⎟ ⎠ ⎞ ⎜ ⎝ ⎛ _± ⎟ ⎠ ⎞ ⎜ ⎝ ⎛_{± ±} Γ x , x 2 1 & , 2 1 _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/

PROBABILITY INTEGRAL OR ERROR FUNCTION (ERF)

AND

IMAGINARY ERROR FUNCTION (ERFI)

Figure 5. Error Function (erf) Figure 6. Imaginary Error Function (erfi)

Definition:

( )

₌

( )

₌

_∫

− Φ x t dt e x x 0 2 2 erf π

[

Formula 8.250.1

]

( )

_{= −Φ}

( )

₌ ∞

_∫

− x t dt e x x 1 2 2 erfc

π

[

Complimentary Error Function, Formula 8.250.4

]

Integral Representation:

( )

=

( )

=

_∫

− Φ 2 0 1 erf x t dt t e x x π

[

Formula 8.251.1

]

Properties:

( )

−x =−Φ

( )

x Φ

( )

0 =0 Φ

( )

±∞ =±1 Φ x x Sect. 8.359

Relation to Incomplete Gamma Function:

( )

π γ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = Φ 2 , 2 1 x x

( )

π γ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = Φ x x , 2 1

Imaginary Error Function, erfi(z):

( )

∫

∫

∫

₌ − ₌ − = = 2 2 2 0 0 0 1 2 2 ) erfi( ) erf( ) erfi( iz t iz t z t dt t e i dt e i dt e z i iz z π π π

Relation to Incomplete Gamma Function:

( )

π γ i z z ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 2 , 2 1 erfi Indefinite Integrals:

( )

( )

t dt e t dt e t t erfi 2 erf 2 2 2 π π

∫

∫

= = −

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.

EXPONENTIAL-INTEGRAL FUNCTION

)

(

Ei x

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

x e−x − Definition: Formulas 8.211.1 & 8.211.2

( )

, 0 Ei =−

∫

=

∫

< ∞ − ∞ − − x dt t e dt t e x x t x t

( )

lim , 0 Ei 0 ⎥_⎦ > ⎤ ⎢ ⎣ ⎡ + − =

∫

∫

∞ − − − − + → _t dt x e dt t e x t x t ε ε

ε (Cauchy Principal Value PV)

Properties:

( )

+∞ =+∞

Ei Ei

( )

−∞ =0 Ei

( )

0 is not defined

Relation to Logarithm-Integral Function:

( )

( )

( )

li

( )

, 0, 0 Ei 0 , li Ei ≠ < = < = a x e ax x e x ax x x

Relation to Incomplete Gamma Function:

( )

dt

(

x

)

t e x x t − Γ − = − =

∫

∞ − − , 0 Ei

( )

dt

( )

x t e x x t , 0 Ei − =−

∫

=−Γ ∞ −

( )

dt

(

x

)

t e x x t − Γ = = −

∫

∞ − − , 0 Ei

( )

dt

( )

x t e x x t , 0 Ei− = =Γ −

∫

∞ − Form. 8.359.1 Indefinite Integrals:

( )

s ds s e s ± =

∫

± Ei 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

LOGARITHM-INTEGRAL FUNCTION

Figure 8. Logarithm-integral Function, li(x)

Definition: Formulas 8.240.1 & 8.240.2

( )

, 1 ln li 0 < =

∫

x t dt x x

( )

Ei

( )

ln , 1 ln ln lim li 1 1 0 0 ⎥= > ⎦ ⎤ ⎢ ⎣ ⎡ + =

∫

∫

+ − → _t x x dt t dt x x ε ε

ε (Cauchy Principal Value PV)

Properties:

( )

0 0

li = li

( )

+∞ =+∞ li

( )

1 not defined

Relation to Exponential-Integral Function:

( )

( )

( )

Ei

(

ln

)

, 1, 0 li 1 , ln Ei li ≠ < = < = a x x a x x x x a

Relation to Incomplete Gamma Function:

( )

(

x

)

x x 0,ln1 0, ln li ⎟=Γ − ⎠ ⎞ ⎜ ⎝ ⎛ Γ = −

[

Formula 8.359.2

]

Indefinite Integrals:

( )

s s ds li ln =

∫

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.

EULER’S CONSTANT

(

C

or

γ

)

Figure 9. Euler’s Constant

Definition: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ _{+ + + + −} = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ₋ = ∞ → − = ∞ →

∑

_k n n _n n n k n ln 1 3 1 2 1 1 lim ln 1 lim 1 1 K γ

[

Formula 8.367.1

]

Integral Representations: dt t dx x x

∫

∫

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − − = ∞ 1 0 0 1 ln ln ) ln( ) exp(

γ

[

Many other integral representations

]

Properties: ... 664 215 0.577 or γ = C

Relation to Complete Gamma Function and Psi (Digamma) Function:

( )

=Ψ

( )

=−γ

Γ′1 1

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 x ) ln(x e−x −

CATALAN’S CONSTANT

(

G or

K

)

Figure 10. Catalan’s Constant

Definition:

( )

(

+

)

= − + − +K − =

∑

∞ =0 2 2 2 2 2 7 1 5 1 3 1 1 1 1 2 1 m m m G

[

Formula 0.234.3

]

Integral Representations:

( )

dx x x G=

∫

1 0 arctan

[

Many other integral representations

]

Properties: ... 594 965 0.915 or K = G 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

x

( )

x x

PARTIAL FRACTIONS

Elementary Identities

(

)

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

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ + = ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ + = + − = − + = + − − = − − + = − + − = + + − = − − − = − + − = + − − = − − − = − + − = − − − = − ± ± = ± ± ± = ± ± = ± x b a b ax b ax b ax a bx bx a a ut u a bx bx a a b bx a b a bx a x a bx b a b a bx x bx a b a b bx a x t t u t u t u t u t u t t u x x x x u t u t u t ut u t u t t t t t t t t t t a x a ax a x x a x a t ax t a x x t bx a a bt ax t bx a x t 1 1 1 1 1 1 1 1 1 1 1 1 1 ) ( ) ( 1 ) ( 1 1 1 1 1 1 1 1 ) 1 ( 1 1 1 ) ( 1 1 ) ( 1 1 1 1 ) 1 ( 1 1 1 1 ) 1 ( 1 1 1 1 m m m

(

)

(

)

(

)

(

)

(

)

(

) (

)

(

)

(

)

(

)

(

) (

x a

)

x a x x a x x bx a b bx a b a bx a x bx a b a bx a b bx a x x a a x a x a x x a t x a x a x a t t u u t u u t t t u t u u t u t t u u t u u t t bx a b bx a x a x bx a bx a x a bx a b x bx a b bx a bx a b a bx a x bx a b a b bx a bx a x a x x a a x a x x a x x a a x a x x a t a t a t + + − = − − − − = − + − + = + + − + = + + − + − + = + − = − + = − − = − − = − + = + − = + − − − = − + + + = + − − − = − + − + = + − − − = − + + + = + − − = − 2 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(

)

n n n n n a x a x a x a x ± ± ± ± ± ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − 1 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ± = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ _± = ± a bx a bx a bx bx a 1 1 ( )

( )

( ) ( )

( )

( )

( ) ( )

( )

( )

( ) ( )

a b x b b a x b a a b x a x x b x b a b a x b a a b x a x x b x b a b a x b a a b x a x x b ax b axb ax b ax + + + − + = + − − + + + + = − + + − − + − = + + − = − = − _⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ 1 1 1 1 1 1 1 1 1 1 1 1 1 _{( ) ( )} ( ) ( ) ( ) ( ) ( ) ( ) (a x) a(a x) a x a a x a a x a a x a x a x a x a x a x x x a a x x a a x x a x x a x x a x x a x − + + = − + − − = − + − − = − − + + = − + − − = − 2 1 2 1 1 2 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

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

∫

+ = _mx be a dx I . Change of variable: msdx dx me ds e s= mx, = mx =

(

)

∫

₊ = bs a s ds m I 1 The fraction,

₍

₎

bs a s + 1

, is simplified by above partial fraction,

₍

₎

₍

₎

bx a a bt ax t bx a x t + − = + , with t=1, so that

₍

₎

bs a s + 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = bs a b s a 1 1 . Then, 1 1 1 ⎟. ⎠ ⎞ ⎜ ⎝ ⎛ + − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − =

_∫

_∫

_∫

ds bs a b s ds ma ds bs a b s ma I

Now, apply change of variable, t =a+bs,dt =bds, which leads to solution,

(

)

[

(

mx

)

]

mx mx a be ma t s ma t s ma t dt s ds ma be a dx I ⎟= − + ⎠ ⎞ ⎜ ⎝ ⎛ = − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ₋ = + =

_∫

1

_∫

_∫

1 ln ln 1 ln 1 ln References • Bers, Calculus, pp. 421 ff.

MISCELLANEOUS

Completing The Square

(

)

(

)

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

+

[

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

]

Finite Binomial Expansions

(

)

₍

₎

(

)

( )

( )

_{( ) ( ) ( )}

( )

( )

( )

, ! 3 ) 2 )( 1 ( ! 2 ) 1 ( ! ! ! 1 1 ! 3 ) 2 )( 1 ( ! 2 ) 1 ( ! ! ! 3 3 2 2 1 0 0 3 3 2 2 1 0 0 n n n n n n j j j n j n j j j n j n n n n n n n j j j n n j j j n n bx bx a n n n bx a n n bx na a bx a j n j n bx a j n bx a b b a n n n b a n n b na a b a j n j n b a j n b a + + − − − − + − = − − = − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − • + + − − + − + + = − = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = + • − − − = − = − − − − = − = −

∑

∑

∑

∑

K K where _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ j n

is the binomial coefficient, defined as,

(

)(

) (

)

j j n n n n j n j n j n L L 3 2 1 1 2 1 )! ( ! ! ⋅ ⋅ + − − − = − = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ,

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!~nne−n 2πn

Double Factorial

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

(

)

(

)

( )

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

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

Differentiation Under The Integral Sign

(

) ( ) (

) ( )

( )

(

) ( )

( )

[

]

(

) ( )

[

]

(

) ( )

( )

[

]

(

) ( )

[

]

[

]

(

)

(

)

(

) ( )

[

]

(

( )

) ( )

[

constant

]

) ( ) 9 constant ) ( ) ( ) 8 ) ( ) ( ) ( ) ( ) ( ) 7 constant , ) , ( ) , ( ) 6 constant ), ( ) , ( ) 5 constant , ), ( ) , ( ) 4 constant ), ( ) , ( ) 3 constant , ), ( ) , ( ) 2 , ), ( ), ( ) , ( ) 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( a dx x d x f dt t f dx d a dx x d x f dt t f dx d dx x du x u f dx x dv x v f dt t f dx d b a dt x t x f dt t x f dx d c dx x d a x f dt a t f dx d c dx a a x f da a d a a f dx a x f da d c dx x d a x f dt a t f dx d c dx a a x f da a d a a f dx a x f da d dx a a x f da a d a a f da a d a a f dx a x f da d a x x a x v x u b a b a c x c a c a x c a c a c a a a a

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

− = = − = ∂ ∂ = − = ∂ ∂ + − = = ∂ ∂ + = ∂ ∂ + − = ψ ϕ ψ ψ ψ ϕ ϕ ϕ ϕ ψ ϕ ψ ψ ψ ϕ ϕ ψ ψ ψ ψ ϕ ϕ ϕ ϕ ψ ψ ϕ ϕ 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

( )

dx x dΓ

with f

( )

x,t =tx−1e−t,a=0,b=∞ limits; ¾ Some integrals cannot be differentiated using these rules, such as—

( )

, &

( )

, . or ) exp( 0 2 x a da d x a da d dt t Γ −

∫

∞

γ For example, no rule seems to apply to

( )

( )

_{( )}

∫

∫

− = = − x c x a dt a t f da d dt t t da d da x a dγ ϕ , ) exp( , 0 1

. These forms must be transformed to equivalent integrals for which Leibnitz’s rules do apply. See Sects. 3.331 & 8.356 for derivations.

References

SYMBOL MEANING

( )

x

int The integer part of the real number x

( )

a n

(

)(

) (

)(

)(

)

(

)

( )

( )

(

(

a

)

n

)

a a n a n a n a n a a a a n − − Γ − Γ − = Γ + Γ = − + − + − + + + = 1 1 1 1 2 3 2 1 _L (Pochhammer symbol)

∑

= n m k k u 0 define we , If . 1 = < + + + =

∑

= + n m k k n m m u m n u u u K

∏

= n m k k f( ) 1 define we , If ). ( ) 1 ( ) ( = < + =

∏

= n m k f(k) m n n f m f m f _L

¾

Suggested new Pochhammer relation:

( )

a _n =

( )

₍

(

)

₎

n a a n − − Γ − Γ − 1 1

1 is cited in Integrals and Series,

Vol. 1, Elementary Functions, A. P. Prudnikov, et al., 1986, page 772.

¾ Proof :

(

)

(

) ( ) ( )

₍

_{) (}

) ( ) (

_{( )}

)

( ) (

)

(

)

( ) (

( ) (

)

)

( )

( )

. 1 1 1 1 1 1 1 that, so , 1 1 below 8.331, Sect. by 1 1 1 r Denominato 1 n n n n n n n n n a x a a a a n a a a a x x n x n a = − Γ − − Γ − = − − Γ − Γ − − Γ − = − Γ − = − Γ = − − Γ =−

♦ Pochhammer Symbol derivations

O’Brien,

http://www.docstoc.com/docs/5473276/Pochhammer-Symbol-

—Formulas 0.111 & 0.112 & 0.121— Finite Arithmetic Progression

FORMULA PROOF OUTLINE

(formula references are to G & R 7e) Change Formula 0.111 to read:

(

)

[

(

)

]

(

)

(

)(

)

(

1

)

is thelast term] [ 2 1 2 1 2 2 1 0 r n a l r a l l a r l a n r n a n kr a n k − + = + − + = + = − + = +

∑

− =

Last form is suggested addition to Formula 0.111. Last term in sum is

(

−1

)

⇒ = − +1. + = r a l n r n a l Substituting − +1 r a l into n

(

a+l

)

2 provides the proposed alternative solution for the finite sum of 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,

(

)(

)

, 148 , 853 , 1 ) 3 ( 2 3333 3336 3333 9 6 3 3 = = + + + + _K r= giving n = 1,111 terms.

—Formulas 0.111 & 0.112 & 0.121— Finite Geometric Progression

FORMULA PROOF OUTLINE

(formula references are to G & R 6e) Change Formula 0.112 to read:

(

)

[

1, is thelast term

]

1 1 1 1 1 1 − = − = ≠ − − = − − =

∑

n n k n k aq l q q a ql q q a aq

Last form is suggested addition to Formula 0.112.

Last term in sum is aqn−1 =l. lq=aqn is substituted into . 1 1 − − ⇒ − − q a ql q a aqn

This new form (derived in many schoolbooks) is useful when the

number of terms in a finite geometric progression is unknown.

Change Formula 0.121.1 to read:

∑

= + = n k n n k 1 0.111] [cf. 2 ) 1 (

New comment refers reader to more general expression for arithmetic progression.

INDEFINITE INTEGRALS OF

ELEMENTARY FUNCTIONS

——Sect. 2.31——

FORMULA PROOF OUTLINE

(formula references are to G & R 7e)

ds e s na dx e xm ±axn

∫

− ±s

∫

= 1 γ 1 γ ⎥⎦ ⎤ ⎢⎣ ⎡ _{= =} + n m ax s n,γ 1 Set n m a s x nax dx ds ax s n, n , n, 1 1 1 = + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = = − _γ . A useful transform for elementary integrals of this form. ds e s na dx x e z s z m axn ± − ±

∫

∫

= 1 1 ⎥⎦ ⎤ ⎢⎣ ⎡ − = = n m z ax s n, 1 Similar to above integral, s e ds na dx e xm ±axn

∫

− ±s

∫

= 1 γ 1 γ .

—Sect.2.312—

FORMULA PROOF OUTLINE

(formula references are to G & R 7e)

( )

( )

a a dx a x x ln Ei exp =

∫

[

lna>0

]

Change of variable:

( )

a a ds s e a I adx s adx a ds e a s x s x a x x ln Ei ln 1 ln ln , ln

∫

= = = = = =

by definition of exponential-integral function,Ei x , ( ) Math. Summary. NOTE:

( )

( )

dx x a dx a n n x x

∫

∫

exp , exp do not seem easily solved;math.com, Mathematica not able to solve. But try setting adx snx ds e a s xn xn a n ln , 1 ln = − =

= which may be reduced

by integration by parts. ¾ verified math.com

( )

( )

a

(

a x a

)

dx x a a a a x ln , 1 ln 1 1 Γ − − − = ₋

∫

[

lna>0

]

( )

( )

a

(

b x a

)

dx x a b b b x ln , 1 ln 1 1 Γ − − − = ₋

∫

[

lna>0

]

(

)

( )

x dx x x x x 2 2 li 2 ln 2 2 = −

∫

• Change of variable: give, to ln ln , ln ln ln , ln ln , ln a a x x a x x a s x a x a s adx s adx a ds e a s ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = = = = = =

( )

( ) ( )

∫

( )

∫

∫

= = − = . ln ln 1 ln ln ln ln 1 1 a a a a a s ds a s ds a a s ds x s a I

o Second change of variable:

s e s ds dt s t =ln , = , t = , to give:

( )

−

∫

= dt t e a I _a t a 1 ln 1 .

ƒ Now use result of 2.325.6,

( )

(

)

( )

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

∫

∫

with parameters

[

m=a,a=n=1,z =a−1

]

, giving,

( )

( )

a

(

a t

) ( )

( )

a

(

a x a

)

I _a a a a ln , 1 ln 1 , 1 ln 1 1 1 Γ − − − = − − Γ − = _{− −}

NOTE: derivable directly from 2.325.6 with parameters