### Summary

*A Guide to the Evaluation of Integrals*

**Special Integrals of Gradshetyn and Ryzhik: the Proofs** provides self-contained proofs of a variety of entries in the frequently used table of integrals by I.S. Gradshteyn and I.M. Ryzhik. The book gives the most elementary arguments possible and uses *Mathematica*^{®} to verify the formulas. You will discover the beauty, patterns, and unexpected connections behind the formulas.

**Volume II** collects 14 papers from *Revista Scientia* covering elliptic integrals, the Riemann zeta function, the error function, hypergeometric and hyperbolic functions, Bessel-K functions, logarithms and rational functions, polylogarithm functions, the exponential integral, and Whittaker functions. Many entries have a variety of proofs that can be evaluated using a symbolic language or point to the development of a new algorithm.

### Table of Contents

**Complete elliptic integrals **

Introduction

Some examples

An elementary transformation

Some principal value integrals

The hypergeometric connection

Evaluation by series expansions

A small correction to a formula in Gradshteyn and Ryzhik

**
****The Riemann zeta function**

Introduction

A first integral representation

Integrals involving partial sums of ζ(s)

The alternate version

The logarithmic scale

The alternating logarithmic scale

Integrals over the whole line

**
****Some automatic proofs**

Introduction

The class of holonomic functions

A first example: The indefinite form of Wallis’ integral

A differential equation for hypergeometric functions in two variables

An integral involving Chebyshev polynomials

An integral involving a hypergeometric function

An integral involving Gegenbauer polynomials

The product of two Bessel functions

An example involving parabolic cylinder functions

An elementary trigonometric integral

**
****The error function**

Introduction

Elementary integrals

Elementary scaling

A series representation for the error function

An integral of Laplace

Some elementary changes of variables

Some more challenging elementary integrals

Differentiation with respect to a parameter

A family of Laplace transforms

A family involving the complementary error function

A final collection of examples

**
****Hypergeometric functions**

Introduction

Integrals over [0, 1]

A linear scaling

Powers of linear factors

Some quadratic factors

A single factor of higher degree

Integrals over a half-line

An exponential scale

A more challenging example

One last example: A combination of algebraic factors and exponentials

**
****Hyperbolic functions**

Introduction

Some elementary examples

An example that is evaluated in terms of the Hurwitz zeta function

A direct series expansion

An example involving Catalan’s constant

Quotients of hyperbolic functions

An evaluation by residues

An evaluation via differential equations

Squares in denominators

Two integrals giving beta function values

The last two entries of Section 3.525

**
****Bessel-K functions **

Introduction

A first integral representation of modified Bessel functions

A second integral representation of modified Bessel functions

A family with typos

The Mellin transform method

A family of integrals and a recurrence

A hyperexponential example

**
****Combination of logarithms and rational functions **

Introduction

Combinations of logarithms and linear rational functions

Combinations of logarithms and rational functions with denominators that are squares of linear terms

Combinations of logarithms and rational functions with quadratic denominators

An example via recurrences

An elementary example

Some parametric examples

Integrals yielding partial sums of the zeta function

A singular integral

**
****Polylogarithm functions**

Introduction

Some examples from the table by Gradshteyn and Ryzhik

**
****Evaluation by series **

Introduction

A hypergeometric example

An integral involving the binomial theorem

A product of logarithms

Some integrals involving the exponential function

Some combinations of powers and algebraic functions

Some examples related to geometric series

**
****The exponential integral**

Introduction

Some simple changes of variables

Entries obtained by differentiation

Entries with quadratic denominators

Some higher degree denominators

Entries involving absolute values

Some integrals involving the logarithm function

The exponential scale

**
****More logarithmic integrals**

Introduction

Some examples involving rational functions

An entry involving the Poisson kernel for the disk

Some rational integrands with a pole at x = 1

Some singular integrals

Combinations of logarithms and algebraic functions

An example producing a trigonometric answer

**
****Confluent hypergeometric and Whittaker functions **

Introduction

A sample of formulas

**
****Evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets **

Introduction

The method of brackets

Examples of index 0

Examples of index 1

Examples of index 2

The goal is to minimize the index

The evaluation of a Mellin transform

The introduction of a parameter

**
****The list of integrals **

The list

**
**References