Table of Contents
Introduction xi
Chapter 1 Complete elliptic integrals 1
1.1 Introduction 1
1.2 Some examples 2
1.3 An elementary transformation 3
1.4 Some principal value integrals 6
1.5 The hypergeometric connection 7
1.6 Evaluation by series expansions 7
1.7 A small correction to a formula in Gradshteyn and Ryzhik 10
Chapter 2 The Riemann zeta function 15
2.1 Introduction 15
2.2 A first integral representation 16
2.3 Integrals involving partial sums of ζ(s) 18
2.4 The alternate version 20
2.5 The logarithmic scale 21
2.6 The alternating logarithmic scale 23
2.7 Integrals over the whole line 24
Chapter 3 Some automatic proofs 25
3.1 Introduction 25
3.2 The class of holonomic functions 25
3.3 A first example: The indefinite form of Wallis' integral 28
3.4 A differential equation for hypergeometric functions in two variables 29
3.5 An integral involving Chebyshev polynomials 30
3.6 An integral involving a hypergeometric function 33
3.7 An integral involving Gegenbauer polynomials 36
3.8 The product of two Bessel functions 37
3.9 An example involving parabolic cylinder functions 39
3.10 An elementary trigonometric integral 42
Chapter 4 The error function 45
4.1 Introduction 45
4.2 Elementary integrals 45
4.3 Elementary scaling 46
4.4 A series representation for the error function 48
4.5 An integral of Laplace 49
4.6 Some elementary changes of variables 51
4.7 Some more challenging elementary integrals 53
4.8 Differentiation with respect to a parameter 54
4.9 A family of Laplace transforms 55
4.10 A family involving the complementary error function 57
4.11 A final collection of examples 60
Chapter 5 Hyper geometric functions 63
5.1 Introduction 63
5.2 Integrals over [0, 1] 63
5.3 A linear scaling 64
5.4 Powers of linear factors 65
5.5 Some quadratic factors 67
5.6 A single factor of higher degree 68
5.7 Integrals over a half-line 70
5.8 An exponential scale 72
5.9 A more challenging example 72
5.10 One last example: A combination of algebraic factors and exponentials 73
Chapter 6 Hyperbolic functions 75
6.1 Introduction 75
6.2 Some elementary examples 75
6.3 Air example that is evaluated in terms of the Hurwitz zeta function 76
6.4 A direct series expansion 79
6.5 An example involving Catalan's constant 80
6.6 Quotients of hyperbolic functions 80
6.7 An evaluation by residues 84
6.8 An evaluation via differential equations 86
6.9 Squares in denominators 87
6.10 Two integrals giving beta function values 89
6.11 The last two entries of Section 3.525 91
Chapter 7 Bessel-K functions 95
7.1 Introduction 95
7.2 A first integral representation of modified Bessel functions 97
7.3 A second integral representation of modified Bessel functions 101
7.4 A family with typos 104
7.5 The Merlin transform method 105
7.6 A family of integrals and a recurrence 108
7.7 A hyperexponential example 109
Chapter 8 Combination of logarithms and rational functions 117
8.1 Introduction 117
8.2 Combinations of logarithms and linear rational functions 118
8.3 Combinations of logarithms and rational functions with denominators that are squares of linear terms 120
8.4 Combinations of logarithms and rational functions with quadratic denominators 121
8.5 An example via recurrences 123
8.6 An elementary example 124
8.7 Some parametric examples 127
8.8 Integrals yielding partial sums of the zeta function 130
8.9 A singular integral 133
Chapter 9 Polylogarithm functions 135
9.1 Introduction 135
9.2 Some examples from the table by Gradshteyn and Ryzhik 136
Chapter 10 Evaluation by series 141
10.1 Introduction 141
10.2 A hypergeometric example 141
10.3 An integral involving the binomial theorem 142
10.4 A product of logarithms 143
10.5 Some integrals involving the exponential function 145
10.6 Some combinations of powers and algebraic functions 146
10.7 Some examples related to geometric series 150
Chapter 11 The exponential integral 153
11.1 Introduction 153
11.2 Some simple changes of variables 153
11.3 Entries obtained by differentiation 155
11.4 Entries with quadratic denominators 156
11.5 Some higher degree denominators 158
11.6 Entries involving absolute values 160
11.7 Some integrals involving the logarithm function 161
11.8 The exponential scale 163
Chapter 12 More logarithmic integrals 165
12.1 Introduction 165
12.2 Some examples involving rational functions 165
12.3 An entry involving the Poisson kernel for the disk 167
12.4 Some rational integrands with a pole at x = 1 171
12.5 Some singular integrals 173
12.6 Combinations of logarithms and algebraic functions 176
12.7 An example producing a trigonometric answer 179
Chapter 13 Confluent hypergeometric and Whittaker functions 181
13.1 Introduction 181
13.2 A sample of formulas 184
Chapter 14 Evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets 195
14.1 Introduction 195
14.2 The method of brackets 196
14.3 Examples of index 0 197
14.4 Examples of index 1 199
14.5 Examples of index 2 205
14.6 The goal is to minimize the index 207
14.7 The evaluation of a Mellin transform 212
14.8 The introduction of a parameter 213
Chapter 15 The list of integrals 215
15.1 The list 215
References 255
Index 259
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