500 Integrals of Elementary and Special Functions

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					     500 INTEGRALS
           OF
ELEMENTARY AND SPECIAL
       FUNCTIONS
                ⊕
        Francis J. O’Brien, Jr.




       GAMMA FUNCTION
500 Integrals of Elementary and
       Special Functions
Library of Congress Cataloging-in-Publication Data

O’Brien, Francis Joseph, Jr.
      500 Integrals of Elementary and Special Functions
          p. cm.
      Includes bibliographical references and index.
      ISBN: 1-4392-1981-8
      ISBN-13: 978-1439219812
      1. Mathematics. 2. Differential and integral calculus.
      3. Proofs and derivations.
      Library of Congress Catalog Card Number:




Copyright © 2008 by Francis Joseph O’Brien, Jr. All rights reserved. No part of this publication
may be reproduced, stored in retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise, without the written permission of
the author, except as permitted under Sections 107 or 108 of the 1976 United States Copyright
Act.

                            Printed in the United States of America
500 Integrals of Elementary
   and Special Functions
          Francis J. O’Brien, Jr.
      Naval Undersea Warfare Center
       Naval Sea Systems Command
         Newport, Rhode Island
For my daughter, Miss Lily-Rae O’Brien

                  ⊗
In gratitude to The Department of Defense
                                       PREFACE
                                                ▲
                                                ●
                                                ▼
        This book is a listing of solved formulas for about 500 integrals, sums, series, and
products. The intended audience is students and practitioners in the fields of mathematics, pure
and applied science, and engineering. The fundamental purpose of the book is to present a
modest listing of formulas with worked out solutions structured on the widely used desk
reference, Gradshteyn & Ryzhik’s Table of Integrals, Series, and Products (2007).
        We provide “new” formulas in a limited number of areas—exponential and logarithmic
functions and selected special functions with emphasis on the gamma and related functions. The
level of difficulty of the material in this book ranges from easy to moderately difficult, and
covers selected topics in first year calculus through advanced calculus. The focus is on
understanding how to evaluate the unsolved indefinite and definite integrals covered in the text.
Throughout the computer is used as an answer- checker rather than the primary evaluator or
prover.
        In this age of computer technology, one may wonder why books such as Table of
Integrals, Series, and Products are even necessary. One of the many sophisticated computer
programs used in the scientific, engineering and mathematical fields is Mathematica
(http://www.wolfram.com/). This computer algebra system can display solutions to derivatives
& integrals in a matter of seconds. It has been asserted that Mathematica can solve
approximately 80% of the formulas in Table of Integrals, Series, and Products. Yet the
Gradshteyn & Ryzhik book is still widely purchased, and is now available in the 7th edition.
        Computer calculators such as Mathematica can solve the majority of integrals in
Gradshteyn & Ryzhik and this book. In addition, comprehensive mathematics websites such as
The Wolfram Functions Site at http://functions.wolfram.com/ provide thousands of formulas, yet
very little exists in the way of proofs, derivations, and justifications. The only significant effort I
am aware of is a series of journal articles by Prof. V.H. Moll of Tulane University. He is in the
process of deriving and verifying the formulas in Gradshteyn & Ryzhik. See the website,
http://www.math.tulane.edu/~vhm/.
        Recent papers that amplify the derivations given in this book may be found at the
docstoc.com website. They derive relations for the Pochhammer Symbol, double factorials, and
gamma          transformations,       identities       and        special     values.              See
http://www.docstoc.com/profile/waabu.
        Given the significant decline in enrollments in mathematics, science and engineering in
the United States, it is important to supplement major mathematical reference works such as the
Gradshteyn & Ryzhik handbook. Other writers may be motivated to assist in this effort.
                                   Structure of the Book
        As mentioned, 500 Integrals is based on the architecture of Gradshteyn & Ryzhik’s Table
of Integrals, Series, and Products, 7th Edition (GR for short). In essence, the book contains
previously published formulas in GR as well as candidate formulas for the next edition. The
material is presented in the order of appearance in GR using their section nomenclature and
formula numbering system. This format is used instead of conventional chapter and sub-chapter
headings.
        The topics covered reflect the author’s own interests in theoretical fundamentals and
potential applications in education, science and engineering. The first section, Mathematical and
Graphical Summary of Selected Elementary and Special Functions, contains an abbreviated
tutorial on all the notations, definitions and properties of the functions used in this book. This
material is based on various standard sources including GR (Sections 0. and 1.) and new results.
Key references are provided for additional background reading and study. The Math. Summary
should be consulted for guidance in the solution of the formulas presented. The Notations
section also contains useful information. The first major section of results consists of Indefinite
Integrals of selected Elementary Functions (Sections 2.3 & 2.7 in GR). This is followed by
Definite Integrals of Elementary Functions (Sections 3.3–3.4 & 4.2.–4.4 in GR), and lastly,
selected Special Functions (Sections 8.2 & 8.3 in GR).
        A number of integrals are left as exercises. In addition, certain related integrals are
solved “from scratch” so that readers can search for a simplified solution based on earlier
material (or their own creation) while other calculations are stated as “similar to above” without
solution, inviting readers to supply details.
        Speed of identification of a formula is a major concern for many users of mathematical
reference handbooks. At the rear of the book is an Index of Formulas which lists all of the
formulas presented in the book, arranged by section and page numbers corresponding to the
Table of Contents. An Index of Symbols, Functions and Concepts may also be found in the rear
of the book.
        In the author’s opinion the most useful formulas presented in this volume are those
involving three-parameter algebraic-exponential functions expressed in terms of gamma
functions (see Mathematical Summary [Incomplete Gamma Function—Indefinite Integrals], and
Sections 2.32, 3.326, 3.381, 3.462). I view them as “reduction” formulas which help solve and
create a number of useful integrals.
        Some formulas developed here have been used in the derivation of new elementary
probability models; see “Summary of Four Generalized Exponential Models (GEM) For
Continuous Probability Distributions,” Jan. 18, 2008, arXiv:0801.2941v2 [math.GM].

       Notes on the entries in the book—

           •  All integrals in this book omit the constant of integration
           •  In most cases, derived solutions may be verified by ordinary differentiation or by
              differentiation under the integral sign using Leibnitz’s rules provided in
              Mathematical Summary
             o Computer verification is suggested when this is possible
           • Variables are considered to be real quantities, unless otherwise indicated
                                         1
          •   A square root x or x 2 or π π is taken to be positive unless otherwise
              specified
          •   Logarithms are natural logarithms, denoted ln x (base e), unless otherwise
              indicated
          •   Errata for the 6th and 7th editions of Gradshteyn and Rhyzik can be obtained
              online at http://www.mathtable.com/gr/
          •   “GR” or “G & R 7e” refer to Gradshteyn and Rhyzik’s Table of Integrals, Series
              and Products, 7th edition, unless otherwise indicated
          •   math.com refers to the public Internet website, http://www.quickmath.com/,
              which calculates indefinite and definite one-dimensional integrals, and performs
              other services including derivatives, partial fraction expansion, graphical plotting,
              matrix inversion, etc.
                  o math.com displays the solutions for the lower incomplete gamma function
                       as Γ(a ) − Γ(a, x ) vice γ (a, x )
          •    The      Mathematica         Integrator       calculates      indefinite     integrals at
              http://integrals.wolfram.com/index.jsp
              o Sometimes math.com unable but Mathematica able to do an indefinite integral
              o Sometimes an indefinite integral can be calculated while the definite one
                  cannot
              o Sometimes computer provides a solution at higher level of complexity than
                                                   dγ (a, x )
                  needed; e.g., the derivative,                (see Section 8.356)
                                                      da
              o Sometimes computer returns a “different” solution compared to paper and
                  pencil answer
                         Computer always provides only a single solution when multiple
                                                              ln (a + bx )dx       e ax − 1
                         answers are possible; e.g., ∫                        or ∫ ax dx (see Section
                                                                     x             e +1
                         2.32)
          •    “verified on math.com”, “verified on mathematica” means author’s answer
              confirmed by computer when possible
              o Post-solution simplification is often required for final form
              o Logarithmic indefinite integrals: computer calculation usually returns an
                  incomplete gamma function solution vice desired reduction formula by
                  integration by parts/change of variable or combination solution
          •   “not verifiable on math.com” & “not verifiable on mathematica” means computer
              unable to calculate answer directly from the input, providing evidence that the
              human calculator is still useful in this electronic age.

                                              ———


        I would appreciate communications regarding misprints and any helpful suggestions for
improvement of presentation of the material. I can be reached by e-mail at either
francis.j.obrien@navy.mil —or— francis28@cox.net.
                                    Acknowledgements
       The author would like to acknowledge and thank all those who have funded, assisted, and
encouraged him over the years. These include the Office of Naval Research, the Base
Commander, my research colleagues in the USW Combat Systems Department, and the
attorneys and paralegals in Office of Patent Counsel of the Naval Undersea Warfare Center,
Newport, Rhode Island.
       Professor Alan Jeffrey of the University of Newcastle Upon Tyne (England), Editor of
Table of Integrals, Series and Products, has been most helpful in his encouragement. I also thank
his co-editor, Dr. Daniel Zwillinger of Rensselaer Polytechnic Institute, for correspondence
regarding errata and related matters.
       I also want to acknowledge the assistance of Aimee Ross, a 3rd year mathematics major at
University of Massachusetts—Dartmouth, who read the early sections of the manuscript, and
provided feedback on the level of difficulty and the clarity of the material.

                                               ╬

Francis J. O’Brien, Jr.
Newport, Rhode Island
October 12, 2008
                           TABLE OF CONTENTS
                       Section in Gradshteyn and Ryzhik                                 Page
MATHEMATICAL AND GRAPHICAL SUMMARY OF SELECTED ELEMENTARY
AND SPECIAL FUNCTIONS

    Exponential Functions                                                              2
    Logarithmic Functions                                                              4
    Gamma Function                                                                     6
    Incomplete Gamma Functions                                                         8
    Probability Integral or Error Function (Erf) and Imaginary Error Function (Erfi)   10
    Exponential-Integral Function                                                      12
    Logarithm-Integral Function                                                        14
    Euler’s Constant                                                                   16
    Catalan’s Constant                                                                 17
    Partial Fractions                                                                  18
    Miscellaneous (Completing the Square, Finite Binomial Expansions, Double           20
       Factorial, Natural Number N, Differentiation Under the Integral Sign)


NOTATIONS                                                                               22

                                     INTRODUCTION
0.1 Finite Sums                                                                        24

0.11 Progressions                                                                      25
       0.111 Arithmetic progressions                                                   25
       0.112 Geometric progressions                                                    26
0.12 Sums of powers of natural numbers                                                 26


                         INDEFINITE INTEGRALS OF
                          ELEMENTARY FUNCTIONS

2.3 The Exponential Function                                                           28
                                 n
  2.31 Forms containing e ax , e ax                                                    30
      2.312                                                                            31
  2.32 The exponential combined with rational functions of x                           35

2.7 Logarithms and Inverse and Hyperbolic Functions                                    61

  2.71 The logarithm                                                                   62
  2.72-2.73 The logarithm and combinations of logarithms and algebraic functions    62


                          DEFINITE INTEGRALS OF
                         ELEMENTARY FUNCTIONS

3.3–3.4 Exponential Functions                                                       79

  3.31 Exponential functions                                                        80
      3.310                                                                         81
      3.311                                                                         82
  3.32–3.34 Exponentials of more complicated arguments                              84
      3.321                                                                         85
      3.322                                                                         89
      3.323                                                                         90
      3.324                                                                         94
      3.326                                                                         96
  3.327–3.334 Exponentials of exponentials                                          98
      3.327                                                                         99
      3.328                                                                         100
      3.331                                                                         101
  3.35 Combinations of exponentials and rational functions                          103
      3.351                                                                         104
      3.353                                                                         105
  3.36–3.37 Combinations of exponentials and algebraic functions                    106
      3.361                                                                         107
      3.362                                                                         108
      3.363                                                                         109
      3.371                                                                         116
  3.38–3.39 Combinations of exponentials and arbitrary powers                       120
      3.381                                                                         121
      3.382                                                                         125
  3.41–3.44 Combinations of rational functions of powers and exponentials           126
      3.427                                                                         127
      3.434                                                                         128
  3.46–3.48 Combinations of exponentials of more complicated arguments and powers   129
      3.461                                                                         130
      3.462                                                                         134
      3.464                                                                         148
      3.471                                                                         151
      3.473                                                                         160


4.2–4.4 Logarithmic Functions                                                       161

  4.21 Logarithmic functions                                                        162
      4.211                                                                         163
       4.212                                                                                  168
       4.215                                                                                  174
   4.22 Logarithms of more complicated arguments                                              177
        4.229                                                                                 178
   4.24 Combinations of logarithms and algebraic functions                                    179
        4.241                                                                                 180
   4.26–4.27 Combinations involving powers of the logarithm and other powers                  181
        4.269                                                                                 182
        4.272                                                                                 185
        4.274                                                                                 190
   4.28 Combinations of rational functions of ln x and powers                                 192
        4.281                                                                                 193
        4.283                                                                                 194
   4.29-4.32 Combinations of logarithmic functions of more complicated arguments              195
               and powers
       4.326                                                                                  196
   4.33–4.34 Combinations of logarithms and exponentials                                      197
        4.331                                                                                 198
        4.337                                                                                 199

                                  SPECIAL FUNCTIONS
8.2 The Exponential Integral Function and Functions Generated by It                           202

   8.21 The exponential integral function Ei(x)                                               203
       8.212                                                                                  204
   8.24 The logarithm integral li(x)                                                          210
       8.240                                                                                  211
       8.241 Integral representations                                                         212
   8.25 The probability integral, the Fresnel Integrals Φ ( x ), S ( x), C ( x) , the error   214
        function erf(x), and the complementary error function erfc(x)
       8.250 Definition                                                                       215
       8.252 Integral representations                                                         216

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

   8.31 The gamma function (Euler’s integral of the second kind): Γ(z)                        222
       8.313                                                                                  223
   8.33 Functional relations involving the gamma function                                     224
       8.331                                                                                  225
       8.334                                                                                  232
       8.335                                                                                  238
       8.339 Particular Values: For n a natural number                                        239
   8.35 The Incomplete Gamma Function                                                         253
       8.350 Definition                                                                       254
       8.351                                                                                  255
       8.352 Special cases                                                                    256
       8.353 Integral representations                     261
       8.356 Functional relations                         263
       8.359 Relationships with other functions           265
   8.36 The Psi function ψ ( x )                          273
       8.367 Euler’s constant: Integral representations   274
References                                                276

Index                                                     278
Index of Formulas                                         284


                           LIST OF ILLUSTRATIONS

Figure 1.    Exponential Function (Growth)       2
Figure 2.    Exponential Function (Decay)        4
Figure 3.    Natural Logarithm Function          4
Figure 4.    Gamma Function                      6
Figure 5.    Error Function                      10
Figure 6.    Imaginary Error Function            10
Figure 7.    Exponential-integral Function       12
Figure 8.    Logarithm-integral Function         14
Figure 9.    Euler’s Constant                    16
Figure 10.   Catalan’s Constant                  17
MATHEMATICAL & GRAPHICAL SUMMARY
            OF SELECTED
 ELEMENTARY AND SPECIAL FUNCTIONS



            GAMMA FUNCTION
                                                                                                                                                                      2



                                        EXPONENTIAL FUNCTIONS

                                                                                                                                e− x
                                x
                            e




                                                                                   x                                                                                  x




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



Notation: e x = exp(x ) = ln −1 (x )

Definitions & Laws of Exponents:

       a 0 = 1, a ≠ 0      a x = e x ln a                                                                                       x = e ln x
                                                                                       1


                                                                                             ( a)
                                                               p

                                                                         [( )]
                                                                                       q                          p
                                                             a = a                         = a =
                                                               q               p            q       p     q

       a x+ y = a x a y    a − x = e − x ln a                                                   1                 1                       ( ) =e
                                                                                                                                x n = e ln x
                                                                                                                                                 n      n ln x


                                                                             = [(− a ) ] = [(− 1) ] [a ]
                                                                                                                          1


                                                             (− a ) q                                                                    = (e ) = e
                                                                         p                      q                 q       q
                           ax = ex
                                n           n
                     ax                          ln a                                      p                  p       p              a
                                                                                                                                             ln x n
                                                                                                                                                    a
                                                                                                                                                           n a ln x
       a x− y =                                                                                                                 xn
                     ay    a −x = e−x
                                    n             n
                                                      ln a           1



       (a )                                                  (− 1)           = i (imaginary)
                                                                     2
          x y
                =a    xy


       (ab )x   = a xb x
       64 4 4 4 74 4 4 4 8
          Let a be e for exp.

Limits:

       lim e x = 1                      lim e x = +∞                                      ex                                       e − ax − e − bx
       x →0                             x → +∞                                      lim a = +∞                                lim                  =b−a
                                                                                   x → +∞ x                                   x →0        x
                −x
       lim e         =1                 lim e − x = 0
       x →0                             x → +∞
                                                                                          e−x        xa                            e −ax − e −bx
                                                                                    lim a = lim x = 0                         lim                  =0
                                                                                   x → +∞ x   x → +∞ e                        x →∞        x
                                                                                                                                        3


Derivatives:
           a, e, n are assumed constants except where noted

    du v de v ln u
                                                                     n         n
                            dv ln u           du            dv   da x     de x ln a                               de x
         =            = uv          = vu v −1    + u v ln u            =               = nx n −1a x ln a                 = ex
                                                                                                   n


     dx       dx              dx              dx            dx    dx        dx                                     dx
    da u
           de  u ln a
                                 du                                 x       x ln a                                de − x
         =            = a u ln a                                 da
                                                                      =
                                                                         de
                                                                                     = a x ln a                           = −e − x
     dx       dx                 dx                               dx       dx                                      dx
       u                                                                                                          de ax
    de
         = eu
               du
                                                                 d x
                                                                    1                                                     = ae ax
                                                                              − x ln a                             dx
    dx         dx                                                  a = de              =− x
                                                                                            ln a
                                                                                                                  de − ax
    644444444 44444444         7                         8        dx         dx              a                             = −ae − ax
                        chain rule                                   n         n                                    dx
                                                                 da x    de x ln a
                                                                      =            = a x x n ln x ln a
                                                                                        n


                                                                  dn       dn
                                                                  [n not a constant ]

Indefinite Integrals:
                          −x       −x
∫ e dx = e             ∫ e dx = −e
   x       x


          e ax            − ax    e − ax
∫ e dx =               ∫ e dx = −
   ax

            a                       a


                                                                                               ∫x
                                                                                                    ± m ± ax n
     •     See Section 2.32 for generalized indefinite exponential integrals of form,                  e         dx, expressed in
            terms of incomplete gamma functions. Summarized in INCOMPLETE GAMMA FUNCTIONS.
•        These exponential integrals are used often in the evaluation of definite integrals of elementary and some
         special functions.


References
• Bers, Calculus. Ch. 6, pp. 367 ff. & 375 ff ; Ch. 7, pp. 453 ff.; Ch. 8, pp. 547 ff.
• Carr, 1970
• Dwight, Table of Integrals, 1961
• Spiegel, Chaps. 7 & 20
• Table of Integrals, Series, and Products. Sects. 0.245, 1.2 (“The Exponential Function”), 2.01 (“The basic
   integrals”)
                                                                                            4




                                LOGARITHMIC FUNCTIONS
                                        (Base e)


                                                        1
                                                       ln x




                                                                                        x




                                     1
                                     x




                                                                                 x




                                                                             1      1
                            Figure 3. Natural Logarithm Functions, ln x &        ,&
                                                                            ln x    x

Notation:

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



                     x
                         dt
Definition:          ∫
                     1
                         t
                            = log x, x > 0
                                                                                                                         5


Properties:

                    log ( xy ) = log x + log y                       log x n = n log x        ln e x = x
                         x
                    log = log x − log y                              log
                                                                           1
                                                                             = − log x        ln e − x = − x
                         y                                                 x                  ln e ax = ax
                                                        ⎛ b⎞
                                        (        )
                    a ln x + b ln y = ln x a y b = a ln⎜ xy a ⎟
                                                        ⎜     ⎟
                                                                                              ln e − ax = − ax
                                                        ⎝     ⎠
                                                      ⎛     ⎞
                                        ⎛ xa ⎞        ⎜ x ⎟
                    a ln x − b ln y = ln⎜ b ⎟ = a ln⎜ b ⎟
                                        ⎜y ⎟
                                        ⎝ ⎠           ⎜ ya ⎟
                                                      ⎝     ⎠
Limits:

           lim log x = +∞                  xa                    log x                     ex              ln(1) = 0
           x → +∞                   lim         = +∞       lim         =0          lim          = +∞
           lim log x = −∞
                                   x → +∞ log x            x → +∞ e x              x → +∞ log x            ln (e ) = 1
           x →0 +                                                log x
                                          log x            lim x = −∞
                                                                                          x
                                                                                           e                 ⎛1⎞
                                    lim a = 0              x →0 + e                lim          =0         ln⎜ ⎟ = −1
                                   x → +∞ x                                        x →0 + log x              ⎝e⎠

Derivatives:

       d ln x        1                      d ln m (a + bx ) mb ln m−1 (a + bx )         d ln u 1 du
                                                                                               =      (chain rule)
                    = ,x ≠ 0                                =
         dx          x                             dx             a + bx                  dx     u dx

Indefinite Integrals:

    dx
∫    x
       = ln x


NOTE: see Section 2.32 for special logarithmic function—the dilogarithm,
                   ln (1 − x )
 Li 2 ( x ) = − ∫             dx
                        x
               ∞
                   xk
 Li 2 ( x ) = ∑ 2 , x < 1
              k =1 k




References
• Bers, Calculus. Ch. 6, pp. 357 ff.; Ch. 7, pp. 454 ff.; Ch. 8, pp. 547 ff.
• Carr, 1970
• Dwight, Table of Integrals, 1961
• Spiegel, Chaps. 7 & 20
• Table of Integrals, Series, and Products. Sects. 1.5 (“The logarithm”), 2.01 (“The basic integrals”)
• Wolfram Functions website, http://functions.wolfram.com/GammaBetaErf/
                                                                                                                                                6



                                                           GAMMA FUNCTION




                                                           Figure 4. Gamma Function, Γ( x )

Definition:
          ∞
Γ(z ) = ∫ t z −1e −t dt ,                  real z > 0            [Formula 8.310.1]
          0
                             z −1
            ⎛1⎞
          1
Γ(z ) = ∫ ln⎜ ⎟                     dt ,    real z > 0           [Formula 8.312.1]
        0   ⎝t ⎠

Integral, Product, and Series Representations:
                  ∞
Γ( x ) = z    x
                  ∫t
                       x −1 − zt
                          e dt ,              real z , x > 0      [Formula 8.312.2]
                  0

                          p! p x                            px
Γ( x ) = lim                                 = lim                ,          real x > 0          [Artin, Formula 2.7]
                                                            ⎛ x⎞
          p →∞           p                      p →∞     p
                      x∏ ( x + k )                   x ∏ ⎜1 + ⎟
                        k =1                           k =1 ⎝  k⎠
                                                                         x
                           ⎛ x⎞         ⎛ 1⎞
                        exp⎜ ⎟      ∞ ⎜
                                         1+ ⎟
        exp(− γx )  ∞
                           ⎝k⎠ = 1      ⎝ k ⎠ = lim n
                                                      x                                    n
Γ(x ) =
                                                                                                   k
           x
                   ∏ x x∏                   x   n →∞ x
                                                                                          ∏ k + x,        real x > 0       [Formula 8.322]
                   k =1
                         1+        k =1
                                         1+                                               k =1
                            k               k
n −1
      ⎛   k⎞
∏ Γ⎜ z + n ⎟ & Γ(nz ) & Γ(x ) [Section 8.335]
                              1
k =0 ⎝      ⎠
               ⎧             ∞
                                 ⎡ ⎛ x ⎞ x ⎤⎫
log Γ( x ) = − ⎨ln x + γx + ∑ ⎢ln⎜1 + ⎟ − ⎥ ⎬, real x > 0                                          [Artin, Formula 2.9]
               ⎩            k =1 ⎣ ⎝ k ⎠ k ⎦⎭

Properties:

Γ( x ) = (x − 1)!                                 Γ(1 − x )                               lim Γ( x ) = +∞         lim Γ( x ) = +∞
                                Γ(− x ) = −                 ,   x not a natural           x →0 +                  x → +∞
                                                     x
                                num.
Γ( x ) ≠ 0                      Γ( x + 1) = xΓ( x )                                       lim Γ( x ) = −∞         lim Γ( x ) is indeterminate
                                                                                          x →0 −                  x → −∞
                                                                                                                               7


Derivatives:

    dΓ( x )
                                          ∞                                            ∞
            = Γ ′( x ) =                                                       Γ′(1) = ∫ e −t ln tdt
                         d x −1 −t
                         dx ∫
                              t e dt =
     dx                     0                                                           0
    ∞
        ∂
                                         ∞
                                                                                    = −γ (Euler' s constant )
    ∫ ∂x e
                ( x −1) ln t
                               e −t dt = ∫ t x −1e −t ln tdt = Γ( x )Ψ ( x )
    0                                    0
                                                                                    = Ψ (1) (Formula 8.366.1)
                       ∞                                                            d ln Γ(x ) Γ′(x )
    Γ (n ) ( x ) = ∫ t x −1e −t (ln t ) dt
                                              n                                •              =        = Ψ (x )
                                                                                       dx       Γ( x )
                       0
                                                                               (digamma or Psi function), Formula 8.330
                                                                                    d ⎛ 1 ⎞              1 dΓ( x )    Ψ (x )
                                                                               •       ⎜
                                                                                       ⎜ Γ( x ) ⎟ = −
                                                                                                ⎟                  =−
                                                                                                                      Γ( x )
                                                                                    dx ⎝        ⎠     [Γ(x )] dx
                                                                                                             2



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

                   Γ(a )                          = γ (a, x )     + Γ(a, x )
                        ∞                           x              ∞

                        ∫t                = ∫ t e dt + ∫ t a −1e −t dt
                               a −1 −t                  a −1 −t
                           e dt                                                 [Formula 8.356.3]
                          4 3
                        1 24
                        0
                                              4 3 x4 3
                                            1 24 1 24
                                            0

                                     GAMMA 1444 444  2 UPPER3
                                               LOWER
            COMPLETE
                                                    INCOMPLETE GAMMA
            •        NOTE: Relation is additive; e.g., γ (a, x ) = Γ(a ) − Γ(a, x )

Trigonometric Functions:
•   Bers, Ch. 6.
•   Dwight, Table of Integrals, 1961
•   Spiegel, Ch. 5.
•   Table of Integrals, Series, and Products, Sects. 1.3-1-4, 8.31-8.35 & Index

NOTE on Γ(x ) : Emil Artin’s brief 1964 book—The Gamma Function—provides a complete statement for
real variables of this special transcendental function called the complete gamma function, and derives the
fundamental mathematical properties originated in the classical 18th & 19th century works of Euler, Gauss,
Legendre, Riemann, Stirling, Weierstrass, and others. See Artin’s book for other definitions, theorems, and
derivations of Γ(x ) not given in this elementary book. Whittaker & Watson is also recommended for
derivations.

References
• Abramowitz & Stegun, Ch. 6.
• Artin, 1964
• Bers, Calculus , Chaps. 4 & 6, pp. 402-3
• Carr, 1970
• Moll, http://www.math.tulane.edu/~vhm/Table.html
• Table of Integrals, Series, and Products, Sects 1.3-1.4 and Sects. 8.31-8.35 & Index
• Whittaker & Watson, 1934
• Wolfram Functions website, http://functions.wolfram.com/GammaBetaErf/
• O’Brien, http://www.docstoc.com/docs/5473276/Pochhammer-Symbol-Selected-Proofs
• O’Brien, http://www.docstoc.com/docs/5836783/Selected-Transformations-Identities--and-Special-
   Values--for-the-Gamma-Function
                                                                                                                     8



                            INCOMPLETE GAMMA FUNCTIONS
1.   Lower Incomplete Gamma Function

Definition: γ (a, x ) = ∫ t a −1e −t dt , real a > 0 [Formula 8.350.1]
                               x

                               0

NOTE:
     •   See Sect. 8.352 for integer special cases, γ (n, x ) , etc.
                                                        1
Integral Representation: γ (a, x ) = x a ∫ t a −1e − xt dt , real a,x > 0 [Sections 3.331 & 8.353]
                                                        0

Properties:

     •   γ (a, x ) = Γ(a ) − Γ(a, x )
                     γ (a + 1, x ) + x a e − x
     •   γ (a, x ) =                           , a ≠ 0 [Formula 8.356.1]
                               a
     •   γ(   1
                  , x ) = π Φ(x ) ,
                    2
                                      [ Φ(x ) = erf(x), error function, 8.250.1]
                             ( )
              2

     •   γ ( , x) = π Φ x
              1
              2

     •   γ (a,0) = 0 [Formula 8.350.5]
     •   γ (a, ∞ ) = Γ(a )
         dγ (a, x )     dΓ(a, x ) a −1 − x
     •              =−               = x e [Form. 8.356.4]
            dx              dx
         dγ (a, u )      dΓ(a, u ) dγ (a, u ) du
     •              =−              =                      (chain rule)
            dx              dx             du dx
         dγ (a, x )
                                           1
     •              = γ (a, x ) ln x + x a ∫ t a −1e − xt ln tdt [Sect. 8.356]
            da                             0

2.   Upper Incomplete Gamma Function
                               ∞
Definition: Γ(a, x ) = ∫ t a −1e −t dt       [Formula 8.350.2]
                               x

NOTE:
     •   See Sect. 8.352 for integer special cases, Γ(n, x ) , etc.

                                                            ∞
Integral Representation:                 Γ(a, x ) = x   a
                                                            ∫t
                                                                 a −1 − xt
                                                                    e dt , real a,x > 0   [Sections 3.331 & 8.353]
                                                            1

Properties:

     •   Γ(a, x ) = Γ(a ) − γ (a, x )
                      Γ(a + 1, x ) − x a e − x
     •   Γ(a, x ) =                            , a ≠ 0 [Formula 8.356.2]
                               a
     •   Γ(1 , x 2 ) = π − π Φ ( x )             [ Φ(x ) = erf(x), error function, 8.250.1]
                                      ( x)
           2

     •   Γ( 1 , x ) = π − π Φ
            2
                                                                                                                                                        9

    •           Γ(a,0 ) = Γ(a )[Formula 8.350.3]
    •           Γ(a, ∞ ) = 0 [Formula 8.350.4]
                dΓ(a, x ) dΓ(a, u )
    •                     &         (see 1. above)
                      dx           dx
                    dΓ(a, x )
                                                   ∞
    •                         = Γ(a, x ) ln x + x ∫ t a −1e − xt ln tdt
                                                 a
                                                                                              [Sect. 8.356]
                      da                           1

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


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

Indefinite Integrals:

                          (− 1)1−γ Γ(γ ,−ax n ) = (− 1)1−γ                   ∞
                                                                                                                   m +1
∫ x e dx =                                                                   ∫t
                                                                                    γ −1 −t
                                                                                       e dt , γ =
                n
   m ax
                                             γ                       γ
                                        na                    na           − ax n
                                                                                                                    n
                                (
                              Γ γ , βx n    1    )            ∞
                                                                                                 m +1
∫ x e dx = −                                                  ∫t
   m − βx                                                            γ −1 −t
                                         =− γ                            e dt , γ =
                    n

                                     γ
                                                                                                      [Formula 2.33.10]
                                nβ         nβ                βx n
                                                                                                  n
 e ax
        n
           (− 1)             z +1
                                             (
                                    a z Γ − z ,− ax n    )            (− 1)z +1 a z       ∞
                                                                                                     1                       m −1
∫ x m dx =                              n
                                                      =
                                                                            n             ∫      t   z +1
                                                                                                            e −t dt ,   z=
                                                                                                                              n
                                                                                                                                    [Formula 2.325.6]
                                                                                        − ax n

 e − βx
            n
               β z Γ − z , βx n     (
                                   βz                )              ∞
                                                                        e −t                          m −1
∫ xm    dx = −
                      n
                                =−
                                   n                                 ∫n t z +1 dt ,       z=
                                                                                                       n
                                                                                                           [Formula 2.33.19]
                                                                    βx

NOTE:
    •           These elementary algebraic-exponential formulas are used extensively in this book to derive the definite integral
                expressions (see Formulas—3.326.2, 3.462.19, 3.381.8-3.381.10 & others in those sections—for negative
                exponential forms), to prove existing formulas, and to create new integrals for a useful class of transcendental
                functions and special functions.
                                                                                        ⎛ 1     ⎞    ⎛1     ⎞
                        See Section 8.359 for special forms, Γ⎜ ±                           ,± x⎟ &γ ⎜ , ± x⎟
                                                                                        ⎝ 2     ⎠    ⎝2     ⎠


References
• Abramowitz & Stegun, Ch. 6
• Boas, Ch. 11
• Moll, http://www.math.tulane.edu/~vhm/Table.html
• Table of Integrals, Series, and Products, Use of the Tables (“The factorial gamma”), Sect. 8.35
• Wolfram Functions website, http://functions.wolfram.com/GammaBetaErf/Gamma2/
                                                                                                             10


PROBABILITY INTEGRAL OR ERROR FUNCTION (ERF)
                      AND
        IMAGINARY ERROR FUNCTION (ERFI)




                                                                x
                                                                                                         x




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


Definition:
                          x
Φ (x ) = erf (x ) =
                      2
                      π ∫
                                  −t 2
                          e              dt [Formula 8.250.1]
                          0
                                    ∞
erfc(x ) = 1 − Φ (x ) =
                              2
                                    ∫ e dt [Complimentary Error Function, Formula 8.250.4]
                                       −t  2


                              π     x


Integral Representation:
                          x2
                          e −t
Φ (x ) = erf (x ) =
                      1
                      π ∫ t
                               dt [Formula 8.251.1]
                        0

Properties:

Φ (− x ) = −Φ ( x )
Φ (0 ) = 0                               Sect. 8.359

Φ (± ∞ ) = ±1
                                                                                                           11


Relation to Incomplete Gamma Function:

        ⎛1                      ⎞                          ⎛1                 ⎞
              γ ⎜ , x2 ⎟                                       γ ⎜ , x⎟
Φ(x ) = ⎝
          2                     ⎠            Φ   ( )   x = ⎝
                                                             2
                                                                      π
                                                                              ⎠
                        π


Imaginary Error Function, erfi(z):

                   erf(iz )
erfi( z ) =
                      i
                                                                                  (iz )2
                                                                                           e −t
                            z                     iz
                    2                        2                            1
                           ∫e                     ∫e                                ∫
                                                        −t 2
erfi( z ) =                     t2
                                     dt =                      dt =                             dt
                     π      0               i π   0                   i π           0         t

Relation to Incomplete Gamma Function:

           ⎛1                         ⎞
                 γ ⎜ ,− z 2 ⎟
erfi(z ) = ⎝                          ⎠
             2
             i π

Indefinite Integrals:

                       π
∫e
     −t 2
            dt =            erf (t )
                     2
                    π
∫e
     t2
          dt =           erfi(t )
                   2


References
• Abramowitz & Stegun, Ch. 7.
• Boas, Ch. 11.
• Dwight, Table of Integrals, 1961
• Moll, http://www.math.tulane.edu/~vhm/Table.html
• Spiegel, Ch. 35
• Table of Integrals, Series, and Products, Use of the Tables (“Exponential and related functions”), and
   Sect. 8.250.
•  Wolfram Functions website, http://functions.wolfram.com/GammaBetaErf/Erf2/
                                                                                                                     12




                       EXPONENTIAL-INTEGRAL FUNCTION

                                                           Ei ( x )


                                                                                                 x




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

Definition:

Formulas 8.211.1 & 8.211.2

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

Properties:

                               Ei(+ ∞ ) = +∞     Ei(− ∞ ) = 0      Ei(0 ) is not defined



Relation to Logarithm-Integral Function:

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


             ( )
Ei(ax ) = li e ax , x < 0, a ≠ 0
                                                                                                           13

Relation to Incomplete Gamma Function:

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

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



Indefinite Integrals:

 e±s
∫ s ds = Ei(± s )

References
• Abramowitz & Stegun, Ch. 5.
• Moll, http://www.math.tulane.edu/~vhm/Table.html
• Table of Integrals, Series, and Products, Use of the Tables (“Exponential and related functions”), and
   Sect. 8.21 & Sects. 3.04-3.05 (“Improper Integrals”; “Principal Values”)
•  Weisstein, Eric W. "Exponential Integral." From MathWorld--A Wolfram Web Resource.
    http://mathworld.wolfram.com/ExponentialIntegral.html
•   Wolfram Functions website, http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/
                                                                                                                     14




                        LOGARITHM-INTEGRAL FUNCTION




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

Definition:

Formulas 8.240.1 & 8.240.2

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

Properties:

                             li(0 ) = 0         li(+ ∞ ) = +∞     li(1) not defined


Relation to Exponential-Integral Function:

li( x ) = Ei(ln x ), x < 1


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


Relation to Incomplete Gamma Function:

             ⎛      1⎞
− li( x ) = Γ⎜ 0, ln ⎟ = Γ(0,− ln x )     [Formula 8.359.2]
             ⎝      x⎠
                                                                                                          15

Indefinite Integrals:


∫ ln s = li(s )
  ds



References
• Abramowitz & Stegun, Ch. 5.
• Moll, http://www.math.tulane.edu/~vhm/Table.html
• Table of Integrals, Series, and Products, Use of the Tables (“Exponential and related functions”), Sects.
   8.24 & Sects. 3.04-3.05 (“Improper Integrals”; “Principal Values”)
•  Weisstein, Eric W. "Logarithmic Integral." From MathWorld--A Wolfram Web Resource.
    http://mathworld.wolfram.com/LogarithmicIntegral.html
                                                                                                 16

                                             EULER’S CONSTANT
                                                             (C or γ )
                                         − e − x ln(x)




                                                                                    x

                                                   Figure 9. Euler’s Constant



Definition:

         ⎡ n−1 1        ⎤        ⎛       1    1          1       ⎞
γ = lim ⎢∑ − ln n⎥ = lim⎜1 + + + K + − ln n ⎟                        [Formula 8.367.1]
    n →∞
         ⎣ k =1 k       ⎦   n →∞
                                 ⎝       2    3          n       ⎠

Integral Representations:
      ∞
                                             ⎛1⎞
                                     1
γ = − ∫ exp(− x) ln( x)dx = − ∫ ln ln⎜ ⎟dt [Many other integral representations]
      0                              0       ⎝t⎠

Properties:

C or γ = 0.577 215 664 ...


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

Γ ′(1) = Ψ (1) = −γ

References
   • Abramowitz & Stegun, Ch. 23.
   • Bers, Calculus, pp. 512-3
   • Moll, http://www.math.tulane.edu/~vhm/Table.html
   • Table of Integrals, Series, and Products, Use of the Tables, and Sect. 8.367 & Index
   •   Weisstein, Eric W. "Euler-Mascheroni Constant." From MathWorld--A Wolfram Web Resource.
          http://mathworld.wolfram.com/Euler-MascheroniConstant.html
                                                                                                  17



                                     CATALAN’S CONSTANT
                                                     (G or K )
                                      arctan( x )
                                          x




                                                                             x


                                             Figure 10. Catalan’s Constant


Definition:


G=∑
     ∞
            (− 1)m        =
                               1 1     1   1
                                 − 2 + 2 − 2 +K      [Formula 0.234.3]
    m =0   (2m + 1)   2        2
                              1 3     5   7

Integral Representations:

         arctan( x )
     1
G=∫                 dx [Many other integral representations]
     0
             x

Properties:

G or K = 0.915 965 594 ...

References
   • Abramowitz & Stegun, Ch. 23.
   • Dwight, Table of Integrals, 1961
   • Marichev, Oleg; Sondow, Jonathan; and Weisstein, Eric W. "Catalan's Constant." From MathWorld--
       A Wolfram Web Resource. http://mathworld.wolfram.com/CatalansConstant.html
   • Spiegel, Ch. 34
   • Table of Integrals, Series, and Products, Use of the Tables, Formula 4.531.1, & Index
                                                                                                                                                                                                18

                                                                PARTIAL FRACTIONS
                                                                         Elementary Identities
    a
         =
                      a                                             t
                                                                                 =
                                                                                         t                       bt                       1               t−a
                                                                                                 m                                                =
x+t           ⎛
             t ⎜1 +
                          x⎞                                x(a ± bx )               ax                  a (a ± bx )              t−a                     t−a
                           ⎟
              ⎝           t⎠                                     t                           t                   t                 1                              1                    a
                                                                             =±                      m                                                =                   +
    a
    =
          a                                                 x( x ± a )                   ax                  a(x ± a )           x x+a                        x+a              x x+a
x−t     ⎛x ⎞
      t ⎜ − 1⎟                                                   1                       1                       1                 1                           1                 a
        ⎝t ⎠                                                                 =±                      m                                                =                   −
                                                            x( x ± a )                   ax                  a(x ± a )           x x−a                        x−a              x x−a
    x                     a                    1
         = 1−                      =                            1                1                   1
a+x                   a+x                          a                     =                       −                                        x                   a + bx                    a
                                       1+                   t (t − 1)        t −1                    t                                                =                    −
                                                   x                                                                               a + bx                         b            b a + bx
    x             a                            1                1                1                   1
         =                    −1 =                                       =                       +                                        x                       a                    a − bx
a−x           a−x                      a                    t (1 − t )       1− t                    t                                                =                       −
                                               −1                                                                                  a − bx                 b a − bx                          b
                                           x                    1                        1                       1
                                                                         =                                   −                     a + bx
                                                                                 u (t − u ) ut
                                                                                                                                                                  b                        a
    x
         = 1+
                          a
                                   =
                                               1            t (t − u )                                                                                =                    +
x−a                   x−a                          a                                                                                      x                   a + bx           x a + bx
                                       1−                       1                1                           1
                                                                         =               −
                                                                                                 u (t − u )                        a − bx
                                                   x                                                                                                              a                         b
                                                            t (u − t )           ut                                                                   =                       −
    t                 1                1
        = 1−                   =                                1                1                   1                                    x               x a − bx                     a − bx
t +1              t +1                     1                                 =        −
                                   1+                                                                                                     2
                                           t                x ( x + 1)           x               x +1                                 t
                                                                                                                                                  =1−
                                                                                                                                                                  u
                                                                                                                                                                          =
                                                                                                                                                                                   1
                                                                                                                                 t +u
                                                                                                                                  2
                                                                                                                                                           t +u
                                                                                                                                                              2
                                                                                                                                                                                       u
    x
         = 1+
                       1
                               =
                                           1                    u
                                                                         =
                                                                                     1
                                                                                                 −
                                                                                                     1                                                                        1+
x −1                  x −1
                                   1−
                                               1            t (t − u )           t −u                    t                                                                             t2
                                               x                                                                                      t2                  u                        1
                                                                u
                                                                         =
                                                                                     1
                                                                                                 +
                                                                                                         1                                        =               −1 =
                                                            t (u − t )                                                           u −t                 u−t
    x             1                        1                                                                                                  2               2
                                                                                 u −t                                                                                          u
         =                −1 =                                                                           t                                                                         −1
1− x         1− x                  1                                                                                                                                          t2
                                           −1                  x             1                           a
                                       x                                 =        −                                                   t2                          u                1
    t                     x                1                a + bx           b           b( a + bx)                                               =1+                     =
         = 1−                  =                                                                                                 t −u
                                                                                                                                  2
                                                                                                                                                           t −u
                                                                                                                                                              2
                                                                                                                                                                                       u
x+t                   x+t                      x               x             1                           a                                                                    1−
                                   1+                                    =        +                                                                                           t2
                                               t            bx − a           b           b(bx − a )
                                                                                                                                          t                   1           a+ x−t
x            a−x                                               x                         a                       1                                    =               −
a
    = 1−
              a                                             a − bx
                                                                         =
                                                                             b( a − bx )
                                                                                                             −
                                                                                                                 b
                                                                                                                                 (a + x )         2
                                                                                                                                                          a+x             (a + x )2
a            x−a                                                                                                                          x                   1                a
    = 1−                                                       a       1                                                                              =               −
x                 x
                                                            a + bx
                                                                   =
                                                                        bx                                                       (a + x )         2
                                                                                                                                                          a+x             (a + x )2
           1                                                         1+
x=                                                                       a                                                           x          1          a
           x −1                                                                                                                             =          −
        1−
             x
                                                              a
                                                                   =
                                                                      1                                                          (a + bx )2 b(a + bx ) b(a + bx )2
                                                   ±n       a − bx     bx                                                            x          a           1
                      ⎛ a⎞                                         1−                                                                       =           −
(a ± x )     ±n           ±n
                  = x ⎜1 + ⎟                                            a                                                        (a − bx ) b(a − bx ) b(a − bx )
                                                                                                                                          2           2
                      ⎝ x⎠
                                                                      ⎛      ⎞                                         ⎛     ⎞
                          ⎛ x⎞
                                                       ±n
                                                                      ⎜ 1 ⎟                                            ⎜ 1 ⎟        x2         x          x
                  = a ± n ⎜1 + ⎟                              1     1                                                 1⎜                 =           +
                                                                  = ⎜        ⎟=                                              ⎟   x −a
                                                                                                                                  2    2
                                                                                                                                           2( x − a ) 2( x + a )
                          ⎝ a⎠                              ax + b ax ⎜   b ⎟                                         b⎜  a ⎟
                                                                      ⎜1+    ⎟                                         ⎜1+ x ⎟
                                                                      ⎝   ax ⎠                                         ⎝  b ⎠
                                                                                                                      19



                                             ⎛      ⎞   ⎛    ⎞              x2
                                                                                     =
                                                                                           x
                                                                                                  −
                                                                                                        x
                                        = ax ⎜ 1 ⎟ = ⎜ ax ⎟               a 2 − x2    2( a− x ) 2( a+ x )
                              ±n     1    1           1 1
(x − a )±n = x ± n ⎛1 − a ⎞
                   ⎜      ⎟        ax−b      ⎜ 1− b ⎟ b ⎜ −1 ⎟                x =         x      +     x
                  ⎝    x⎠                    ⎝ ax ⎠     ⎝ b ⎠
                                                                          a 2 − x 2 2 a ( a + x ) 2 a( a − x )
                ⎛x ⎞
                ±n
          = a ⎜ − 1⎟
                              ±n          x
                                                       ( ) ( )
                                                   = a 1 − b 1
                                   ( x+ a ) ( x+b ) a −b x+ a a −b x+b
                                                                              x =         x      −     x
                                                                          x 2 − a 2 2 a( x − a ) 2 a ( x + a )
                                                  = a ( 1 )+ b ( 1 )
                ⎝a ⎠                     x                                    1 =         1            1
            ⎛ a    ⎞               ( x+a ) ( x−b ) a +b x+ a a+b x−b                             −
                                                                          x 2 − a 2 2 a( x − a ) 2 a ( x + a )
a ± bx = bx⎜ ± 1⎟
            ⎝ bx ⎠                       x        = a ( 1 )+ b ( 1 )          1             1              1
                                   ( x−a ( x+b ) a +b x−a a+b x+b
                                        )                                           =              +
                                                                          a 2 − x 2 2 a ( a + x ) 2 a( a − x )
          ⎛ bx ⎞
      = a ⎜1 ± ⎟
          ⎝     a⎠

  NOTE: These identities often simplify integral calculations

          See Sects. 2.32 & 3.363, below, for examples
                                                      dx
          Example: GR Formula 2.313.1, I =       ∫ a + be   mx
                                                                 .

            Change of variable:
            s = e mx , ds = me mx dx = msdx
                           1         ds
                      I= ∫
                           m s (a + bs )
                                 1                                                      t       t      bt
            The fraction,               , is simplified by above partial fraction,            =   −          ,
                            s (a + bs )                                             x(a + bx ) ax a(a + bx )
                                          1        1 ⎛1       b ⎞
            with t = 1 , so that                 = ⎜ −             ⎟.
                                     s (a + bs ) a ⎝ s a + bs ⎠
                          1 ⎛1            b ⎞         1 ⎛ ds           b       ⎞
            Then, I =
                         ma ⎝ ∫ ⎜ s − a + bs ⎟ds = ma ⎜ ∫ s − ∫ a + bs ds ⎟.
                                               ⎠         ⎝                     ⎠
            Now, apply change of variable, t = a + bs, dt = bds, which leads to solution,

            I =∫
                       dx
                  a + be   mx
                               =
                                   1 ⎛ ds
                                       ⎜∫ − ∫ ⎟ =
                                  ma ⎝ s
                                                  dt ⎞ 1
                                                  t ⎠ ma
                                                                                           [          (
                                                             (ln s − ln t ) = 1 ln⎛ s ⎞ = 1 mx − ln a + bemx
                                                                                   ⎜ ⎟
                                                                             ma ⎝ t ⎠ ma
                                                                                                                 )]
  References
     • Bers, Calculus, pp. 421 ff.
     • Table of Integrals, Series, and Products, Sect. 2.1 (Rational Functions).
                                                                                                                    20


                                                   MISCELLANEOUS

Completing The Square

   2           ⎛    b ⎞
ax + bx + c = a⎜ x + ⎟ −
                         b 2 − 4ac
                                2
                                       (             )             [The second form is most useful for integrals.
               ⎝    2a ⎠     4a                                    See Sects. 2.32 & 3.323, below, for uses]
                     ⎛
                   = ⎜ ax +
                             b ⎞
                               ⎟ −
                                           2
                                   b 2 − 4ac   (          )
                     ⎜         ⎟
                     ⎝      2 a⎠       4a



Finite Binomial Expansions
                  n
                      ⎛n⎞            n
• (a + b ) = ∑ ⎜ ⎟a n − j b j = ∑
                                              n!
                                               
				
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Description: 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 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.
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