# Table of Integrals, Series, and Products

## 8th Edition

**Editor:**Daniel Zwillinger

**Hardcover ISBN:**9780123849335

**eBook ISBN:**9780123849342

**Imprint:**Academic Press

**Published Date:**18th September 2014

**Page Count:**1184

## Description

The eighth edition of the classic Gradshteyn and Ryzhik is an updated completely revised edition of what is acknowledged universally by mathematical and applied science users as the key reference work concerning the integrals and special functions. The book is valued by users of previous editions of the work both for its comprehensive coverage of integrals and special functions, and also for its accuracy and valuable updates. Since the first edition, published in 1965, the mathematical content of this book has significantly increased due to the addition of new material, though the size of the book has remained almost unchanged. The new 8th edition contains entirely new results and amendments to the auxiliary conditions that accompany integrals and wherever possible most entries contain valuable references to their source.

## Key Features

- Over 10, 000 mathematical entries
- Most up to date listing of integrals, series and products (special functions)
- Provides accuracy and efficiency in industry work
- 25% of new material not including changes to the restrictions on results that revise the range of validity of results, which lend to approximately
**35% of new updates**

## Readership

Engineers, pure and applied mathematicians, scientists, physicists, and graduate students

## Table of Contents

- Preface to the Eighth Edition
- Acknowledgments
- The Order of Presentation of the Formulas
- Use of the Tables*
- Bernoulli and Euler Polynomials and Numbers
- Elliptic Functions and Elliptic Integrals
- The Jacobi Zeta Function and Theta Functions
- The Factorial (Gamma) Function
- Exponential and Related Integrals
- Hermite and Chebyshev Orthogonal Polynomials
- Bessel Functions
- Parabolic Cylinder Functions and Whittaker Functions
- Mathieu Functions

- Index of Special Functions
- Notation
- Note on the Bibliographic References
- 0. Introduction
- 0.1 Finite sums
- 0.2 Numerical series and infinite products
- 0.3 Functional series
- 0.4 Certain formulas from differential calculus

- 1. Elementary Functions
- 1.1 Power of Binomials
- 1.2 The Exponential Function
- 1.3–1.4 Trigonometric and Hyperbolic Functions
- 1.5 The Logarithm
- 1.6 The Inverse Trigonometric and Hyperbolic Functions

- 2. Indefinite Integrals of Elementary Functions
- 2.0 Introduction
- 2.1 Rational Functions
- 2.2 Algebraic functions
- 2.3 The Exponential Function
- 2.4 Hyperbolic Functions
- 2.5–2.6 Trigonometric Functions
- 2.7 Logarithms and Inverse-Hyperbolic Functions
- 2.8 Inverse Trigonometric Functions

- 3-4. Definite Integrals of Elementary Functions
- 3.0 Introduction
- 3.1-3.2 Power and Algebraic Functions
- 3.3–3.4 Exponential Functions
- 3.5 Hyperbolic Functions
- 3.6–4.1 Trigonometric Functions

- 4. Definite Integrals of Elementary Functions
- 4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers
- 4.13 Combinations of trigonometric and hyperbolic functions and exponentials
- 4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and powers
- 4.2–4.4 Logarithmic Functions
- 4.5 Inverse Trigonometric Functions
- 4.6 Multiple Integrals

- 5. Indefinite Integrals of Special Functions
- 5.1 Elliptic Integrals and Functions
- 5.13 Jacobian elliptic functions
- 5.14 Weierstrass elliptic functions
- 5.2 The Exponential Integral function
- 5.3 The Sine Integral and the Cosine Integral
- 5.4 The Probability Integral and Fresnel Integrals
- 5.5 Bessel Functions
- 5.6 Orthogonal Polynomials
- 5.7 Hypergeometric Functions

- 6-7. Definite Integrals of Special Functions
- 6.1 Elliptic Integrals and Functions
- 6.2–6.3 The Exponential Integral Function and Functions Generated by It
- 6.22–6.23 The exponential integral function
- 6.24–6.26 The sine integral and cosine integral functions
- 6.27 The hyperbolic sine integral and hyperbolic cosine integral functions
- 6.28–6.31 The probability integral
- 6.32 Fresnel integrals
- 6.4 The Gamma Function and Functions Generated by It
- 6.46-6.47 The Function
*ψ*(*x*) - 6.5-6.7 Bessel Functions
- 6.8 Functions Generated by Bessel Functions
- 6.9 Mathieu Functions

- 7. Definite Integrals of Special Functions
- 7.1-7.2 Associated Legendre Functions
- 7.3–7.4 Orthogonal Polynomials
- 7.325* Complete System of Orthogonal Step Functions
- 7.33 Combinations of the polynomials Cnv(x)
*and Bessel functions. Integration of Gegenbauer functions with respect to the index* - 7.34 Combinations of Chebyshev polynomials and powers
- 7.35 Combinations of Chebyshev polynomials and elementary functions
- 7.36 Combinations of Chebyshev polynomials and Bessel functions
- 7.37−7.38 Hermite polynomials
- 7.39 Jacobi polynomials
- 7.41–7.42 Laguerre polynomials
- 7.5 Hypergeometric Functions
- 7.53 Hypergeometric and trigonometric functions
- 7.54 Combinations of hypergeometric and Bessel functions
- 7.6 Confluent Hypergeometric Functions
- 7.68 Combinations of confluent hypergeometric functions and other special functions
- 7.69 Integration of confluent hypergeometric functions with respect to the index
- 7.7 Parabolic Cylinder Functions
- 7.8 Meijer's and MacRobert's Functions (
*G*and*E*)

- 8. Special Functions
- 8.1 Elliptic Integrals and Functions
- 8.2 The Exponential Integral Function and Functions Generated by It
- 8.3 Euler's Integrals of the First and Second Kinds and Functions Generated by Them
- 8.4–8.5 Bessel Functions and Functions Associated with Them

*9. Special Functions*- 9.1 Hypergeometric Functions
- 9.2 Confluent Hypergeometric Functions
- 9.3 Meijer's
*G*-Function - 9.4 MacRobert's
*E*-Function *9.5 Riemann's Zeta Functions**ζ*(*z*,*q*), and*ζ*(*z*), and the Functions Φ(*z*,*s*,*v*) and*ξ*(*s*)- 9.6 Bernoulli Numbers and Polynomials, Euler Numbers, the Functions v(
*x*), v(*x*, α), μ(x, β), μ(x, β, α), λ(*x*,*y*) and Euler Polynomials - 9.7 Constants

- 10. Vector Field Theory
- 10.1–10.8 Vectors, Vector Operators, and Integral Theorems

- 11. Integral Inequalities
- 11.11 Mean Value Theorems
- 11.21 Differentiation of Definite Integral Containing a Parameter
- 11.31 Integral Inequalities
- 11.41 Convexity and Jensen's Inequality
- 11.51 Fourier Series and Related Inequalities

- 12. Fourier, Laplace, and Mellin Transforms
- 12.1–12.4 Integral Transforms

- Bibliographic References
- Supplementary References
- Index of Functions and Constants
- Index of Concepts

## Details

- No. of pages:
- 1184

- Language:
- English

- Copyright:
- © Academic Press 2014

- Published:
- 18th September 2014

- Imprint:
- Academic Press

- Hardcover ISBN:
- 9780123849335

- eBook ISBN:
- 9780123849342

## About the Editor

### Daniel Zwillinger

Dr. Daniel Zwillinger is a Senior Principal Systems Engineer for the Raytheon Company. He was a systems requirements “book boss” for the Cobra Judy Replacement (CJR) ship and was a requirements and test lead for tracking on the Ungraded Early Warning Radars (UEWR). He has improved the Zumwalt destroyer’s software accreditation process and he was test lead on an Active Electronically Scanned Array (AESA) radar. Dan is a subject matter expert (SME) in Design for Six Sigma (DFSS) and is a DFSS SME in Test Optimization, Critical Chain Program Management, and Voice of the Customer. He is currently leading a project creating Trust in Autonomous Systems. At Raytheon, he twice won the President’s award for best Six Sigma project of the year: on converting planning packages to work packages for the Patriot missile, and for revising Raytheon’s timecard system. He has managed the Six Sigma white belt training program. Prior to Raytheon, Dan worked at Sandia Labs, JPL, Exxon, MITRE, IDA, BBN, and The Mathworks (where he developed an early version of their Statistics Toolbox).

For ten years, Zwillinger was owner and president of Aztec Corporation. As a small business, Aztec won several Small Business Innovation Research (SBIR) contracts. The company also created several software packages for publishing companies. Prior to Aztec, Zwillinger was a college professor at Rensselaer Polytechnic Institute in the department of mathematics.

Dan has written several books on mathematics on the topics of differential equations, integration, statistics, and general mathematics. He is editor-in-chief of the Chemical Rubber Company’s (CRC’s) “Standard Mathematical Tables and Formulae”, and is on the editorial board for CRC’s “Handbook of Chemistry and Physics”. Zwillinger holds a bachelor's degree in mathematics from the Massachusetts Institute of Technology (MIT). He earned his doctorate in applied mathematics from the California Institute of Technology (Caltech). Zwillinger is a certified Raytheon Six Sigma Expert and an ASQ certified Six Sigma Black Belt. He also holds a pilot’s license.

### Affiliations and Expertise

Rensselaer Polytechnic Institute, Troy, NY, USA

## Reviews

"...if you use this book frequently it’s definitely worth getting the new edition…" **--MAA.org, November 2014**

"The integrals are very useful, but this book includes many other features that will be helpful to the reader, especially graduate students. The sections on Hermite and Legendre polynomials are especially helpful for students of Electricity and Magnetism, Quantum Mechanics, and Mathematical physics (they won't have to hunt in several books to find what they need)." **--Barry Simon, California Institute of Technology**

"This book is to the CRC Mathematical Tables as the unabridged Oxford English Dictionary is to Webster's Collegiate. Besides being big, it's easy to find things in, because of the way the integrals are organized into classes...It really helped me through grad school." **--Phil Hobbs, Amazon Review**