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Table of Integrals, Series, and Products, 8th Edition

Book 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.

  • 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

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Preface to the Eighth Edition
  6. Acknowledgments
  7. The Order of Presentation of the Formulas
  8. Use of the Tables*
    1. Bernoulli and Euler Polynomials and Numbers
    2. Elliptic Functions and Elliptic Integrals
    3. The Jacobi Zeta Function and Theta Functions
    4. The Factorial (Gamma) Function
    5. Exponential and Related Integrals
    6. Hermite and Chebyshev Orthogonal Polynomials
    7. Bessel Functions
    8. Parabolic Cylinder Functions and Whittaker Functions
    9. Mathieu Functions
  9. Index of Special Functions
  10. Notation
  11. Note on the Bibliographic References
  12. 0. Introduction
    1. 0.1 Finite sums
    2. 0.2 Numerical series and infinite products
    3. 0.3 Functional series
    4. 0.4 Certain formulas from differential calculus
  13. 1. Elementary Functions
    1. 1.1 Power of Binomials
    2. 1.2 The Exponential Function
    3. 1.3–1.4 Trigonometric and Hyperbolic Functions
    4. 1.5 The Logarithm
    5. 1.6 The Inverse Trigonometric and Hyperbolic Functions
  14. 2. Indefinite Integrals of Elementary Functions
    1. 2.0 Introduction
    2. 2.1 Rational Functions
    3. 2.2 Algebraic functions
    4. 2.3 The Exponential Function
    5. 2.4 Hyperbolic Functions
    6. 2.5–2.6 Trigonometric Functions
    7. 2.7 Logarithms and Inverse-Hyperbolic Functions
    8. 2.8 Inverse Trigonometric Functions
  15. 3-4. Definite Integrals of Elementary Functions
    1. 3.0 Introduction
    2. 3.1-3.2 Power and Algebraic Functions
    3. 3.3–3.4 Exponential Functions
    4. 3.5 Hyperbolic Functions
    5. 3.6–4.1 Trigonometric Functions
  16. 4. Definite Integrals of Elementary Functions
    1. 4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers
    2. 4.13 Combinations of trigonometric and hyperbolic functions and exponentials
    3. 4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and powers
    4. 4.2–4.4 Logarithmic Functions
    5. 4.5 Inverse Trigonometric Functions
    6. 4.6 Multiple Integrals
  17. 5. Indefinite Integrals of Special Functions
    1. 5.1 Elliptic Integrals and Functions
    2. 5.13 Jacobian elliptic functions
    3. 5.14 Weierstrass elliptic functions
    4. 5.2 The Exponential Integral function
    5. 5.3 The Sine Integral and the Cosine Integral
    6. 5.4 The Probability Integral and Fresnel Integrals
    7. 5.5 Bessel Functions
    8. 5.6 Orthogonal Polynomials
    9. 5.7 Hypergeometric Functions
  18. 6-7. Definite Integrals of Special Functions
    1. 6.1 Elliptic Integrals and Functions
    2. 6.2–6.3 The Exponential Integral Function and Functions Generated by It
    3. 6.22–6.23 The exponential integral function
    4. 6.24–6.26 The sine integral and cosine integral functions
    5. 6.27 The hyperbolic sine integral and hyperbolic cosine integral functions
    6. 6.28–6.31 The probability integral
    7. 6.32 Fresnel integrals
    8. 6.4 The Gamma Function and Functions Generated by It
    9. 6.46-6.47 The Function ψ(x)
    10. 6.5-6.7 Bessel Functions
    11. 6.8 Functions Generated by Bessel Functions
    12. 6.9 Mathieu Functions
  19. 7. Definite Integrals of Special Functions
    1. 7.1-7.2 Associated Legendre Functions
    2. 7.3–7.4 Orthogonal Polynomials
    3. 7.325* Complete System of Orthogonal Step Functions
    4. 7.33 Combinations of the polynomials Cnv(x) and Bessel functions. Integration of Gegenbauer functions with respect to the index
    5. 7.34 Combinations of Chebyshev polynomials and powers
    6. 7.35 Combinations of Chebyshev polynomials and elementary functions
    7. 7.36 Combinations of Chebyshev polynomials and Bessel functions
    8. 7.37−7.38 Hermite polynomials
    9. 7.39 Jacobi polynomials
    10. 7.41–7.42 Laguerre polynomials
    11. 7.5 Hypergeometric Functions
    12. 7.53 Hypergeometric and trigonometric functions
    13. 7.54 Combinations of hypergeometric and Bessel functions
    14. 7.6 Confluent Hypergeometric Functions
    15. 7.68 Combinations of confluent hypergeometric functions and other special functions
    16. 7.69 Integration of confluent hypergeometric functions with respect to the index
    17. 7.7 Parabolic Cylinder Functions
    18. 7.8 Meijer's and MacRobert's Functions (G and E)
  20. 8. Special Functions
    1. 8.1 Elliptic Integrals and Functions
    2. 8.2 The Exponential Integral Function and Functions Generated by It
    3. 8.3 Euler's Integrals of the First and Second Kinds and Functions Generated by Them
    4. 8.4–8.5 Bessel Functions and Functions Associated with Them
  21. 9. Special Functions
    1. 9.1 Hypergeometric Functions
    2. 9.2 Confluent Hypergeometric Functions
    3. 9.3 Meijer's G-Function
    4. 9.4 MacRobert's E-Function
    5. 9.5 Riemann's Zeta Functions ζ(z, q), and ζ(z), and the Functions Φ(z, s, v) and ξ(s)
    6. 9.6 Bernoulli Numbers and Polynomials, Euler Numbers, the Functions v(x), v(x, α), μ(x, β), μ(x, β, α), λ(, α), μ(x, β), μ(x, β, α), λ(x, y) and Euler Polynomials
    7. 9.7 Constants
  22. 10. Vector Field Theory
    1. 10.1–10.8 Vectors, Vector Operators, and Integral Theorems
  23. 11. Integral Inequalities
    1. 11.11 Mean Value Theorems
    2. 11.21 Differentiation of Definite Integral Containing a Parameter
    3. 11.31 Integral Inequalities
    4. 11.41 Convexity and Jensen's Inequality
    5. 11.51 Fourier Series and Related Inequalities
  24. 12. Fourier, Laplace, and Mellin Transforms
    1. 12.1–12.4 Integral Transforms
  25. Bibliographic References
  26. Supplementary References
  27. Index of Functions and Constants
  28. Index of Concepts