# Mathematical Functions and Their Approximations

## 1st Edition

**Authors:**Yudell L. Luke

**eBook ISBN:**9781483262451

**Imprint:**Academic Press

**Published Date:**28th September 1975

**Page Count:**586

## Description

Mathematical Functions and their Approximations is an updated version of the Applied Mathematics Series 55 Handbook based on the 1954 Conference on Mathematical Tables, held at Cambridge, Massachusetts. The aim of the conference is to determine the need for mathematical tables in view of the availability of high speed computing machinery.

This work is composed of 14 chapters that cover the machinery for the expansion of the generalized hypergeometric function and other functions in infinite series of Jacobi and Chebyshev polynomials of the first kind. Numerical coefficients for Chebyshev expansions of the more common functions are tabulated. Other chapters contain polynomial and rational approximations for certain class of G-functions, the coefficients in the early polynomials of these rational approximations, and the Padé approximations for many of the elementary functions and the incomplete gamma functions. The remaining chapters describe the development of analytic approximations and expansions.

This book will prove useful to mathematicians, advance mathematics students, and researchers.

## Table of Contents

Preface

I. The Gamma Function and Related Functions

1.1 Definitions and Elementary Properties

1.2 Power Series and Other Series Expansions

1.3 Asymptotic Expansions

1.4 Rational Approximations for Ѱ (z)

1.5 Inequalities

1.6 Bibliographic and Numerical Data

1.6.1 General References

1.6.2 Description of and References to Tables

1.6.3 Description of and References to Other Approximations and Expansions

II. The Binomial Function

2.1 Power Series

2.2 Expansions in Series of Jacobi and Chebyshev Polynomials

2.3 Expansions in Series of Bessel Functions

2.4 Padé Approximations

2.4.1 (1 + 1/z)-c

2.4.2 The Square Root

2.4.3 Padé Coefficients

2.4.4 The Function e-w

2.5 Inequalities

III. Elementary Functions

3.1 Logarithmic Functions

3.1.1 Power Series

3.1.2 Expansion in Series of Chebyshev Polynomials

3.1.3 Padé Approximations

3.1.4 Inequalities

3.2 Exponential Function

3.2.1 Series Expansions

3.2.2 Expansions in Series of Jacobi and Chebyshev Polynomials and Bessel Functions

3.2.3 Padé Approximations

3.2.4 Inequalities

3.3 Circular and Hyperbolic Functions

3.3.1 Power Series

3.3.2 Expansions in Series of Jacobi and Chebyshev Polynomials and Bessel Functions

3.3.3 Rational and Padé Approximations

3.3.4 Inequalities

3.4 Inverse Circular and Hyperbolic Functions

3.4.1 Power Series

3.4.2 Expansions in Series of Chebyshev Polynomials

3.4.3 Padé Approximations

3.4.4 Inequalities

3.5 Bibliographic and Numerical Data

3.5.1 Description of and References to Tables

3.5.2 Description of and References to Other Approximations and Expansions

IV. Incomplete Gamma Functions

4.1 Definitions and Series Expansions

4.2 Differential Equations and Difference Equations

4.3 Padé Approximations

4.3.1 1F1 (1; v + 1;-z)

4.3.2 z1-vezГ(v,z)

4.3.3 The Error Tn(v,z) for | arg z/k| < π

4.3.4 The Negative Real Axis and the Zeros of Fn(v,z)

4.4 Inequalities

4.4.1 H(v,z)

4.4.2 Г(v,z)

4.5 Notes on the Computation of the Incomplete Gamma Function

4.6 Exponential Integrals

4.6.1 Relation to Incomplete Gamma Function and Other Properties

4.6.2 Expansions in Series of Chebyshev Polynomials

4.6.3 Rational and Padé Approximations

4.7 Cosine and Sine Integrals

4.7.1 Relation to Exponential Integral and Other Properties

4.7.2 Expansions in Series of Chebyshev Polynomials

4.8 Error Functions

4.8.1 Relation to Incomplete Gamma Function and Other Properties

4.8.2 Expansions in Series of Chebyshev Polynomials and Bessel Functions

4.8.3 Padé Approximations

4.8.4 Trapezoidal Rule Approximations

4.8.5 Inequalities

4.9 Fresnel Integrals

4.9.1 Relation to Error Functions and Other Properties

4.9.2 Expansions in Series of Chebyshev Polynomials

4.10 Bibliographic and Numerical Data

4.10.1. References

4.10.2 Description of and References to Tables

4.10.3 Description of and References to Other Approximations and Expansions

V. The Generalized Hypergeometric Function pFq and the G-Function

5.1 Introduction

5.2 The pFq

5.2.1 Power Series

5.2.2 Derivaties and Contiguous Relations

5.2.3 Integral Representations and Integrals Involving the pFq

5.2.4 Evaluation for Special Values of the Variable and Parameters

5.3 The G-Function

5.3.1 Definition and Relation to the pFq

5.3.2 Elementary Properties

5.3.3 Analytic Continuation of Gm,p, np (z)

5.4 The Confluence Principle

5.5 Multiplication Theorems

5.6 Integrals Involving G-Functions

5.7 Differential Equations

5.7.1 The pFq

5.7.2 The G-Function

5.8 Series of G-Functions

5.8.1 Introduction

5.8.2 Notation

5.8.3 Expansion Theorems

5.9 Asymptotic Expansions

5.9.1 Gq,p, nq (z), n = 0,1

5.9.2 Gm,p, nq (z)

5.9.3 pFq(z)

5.10 Expansions in Series of Generalized Jacobi, Generalized Laguerre and Chebyshev Polynomials

5.10.1 Expansions for G-Functions

5.10.2 Expansions for pFq

5.11 Expansions in Series of Bessel Functions

5.12 Polynomial and Rational Approximations

5.13 Recurrence Formulas for Polynomials and Functions Occurring in Approximations to Generalized Hypergeometric Functions

5.13.1 Introduction

5.13.2 Recursion Formulas for Extended Jacobi and Laguerre Functions

5.13.3 Recursion Formulas for the Numerator and Denominator Polynomials in the Rational Approximations for the Generalized Hypergeometric Function

5.13.4 Recursion Formula for Coefficients in the Expansion of the G-Function in Series of Extended Jacobi Polynomials

5.14 Inequalities

VI. The Gaussian Hypergeometric Function 2F1

6.1 Introduction

6.2 Elementary Properties

6.2.1 Derivatives

6.2.2 Contiguous Relations

6.2.3 Integral Representations

6.3 Differential Equations

6.4 Kummer Solutions and Transformation Formulae

6.5 Analytic Continuation

6.6 The Complete Solution and Wronskians

6.7 Quadratic Transformations

6.8 The 2F1 for Special Values of the Argument

6.9 Expansion in Series of Chebyshev Polynomials

6.10 Padé Approximations for 2F1 (1, σ;ρ + 1;-1/z)

6.11 Inequalities

6.12 Bibliographic and Numerical Data

6.12.1 References

6.12.2 Description of and References to Tables

VII. The Confluent Hypergeometric Function

7.1 Introduction

7.2 Integral Representations

7.3 Elementary Relations

7.3.1 Derivatives

7.3.2 Contiguous Relations

7.3.3 Products of Confluent Functions

7.4 Differential Equations

7.5 The Complete Solution and Wronskians

7.6 Asymptotic Expansions

7.7 Expansions in Series of Chebyshev Polynomials

7.8 Expansions in Series of Bessel Functions

7.9 Inequalities

7.10 Other Notations and Related Functions

7.11 Bibliographic and Numerical Data

7.11.1 References

7.11.2 Description of and References to Tables and Other Approximations

VIII. Identification of thd pFq and G-Functions with the Special Functions

8.1 Introduction

8.2 Named Special Functions Expressed as pFq's

8.2.1 Elementary Functions

8.2.2 The Incomplete Gamma Function and Related Functions

8.2.3 The Gaussian Hypergeometric Function

8.2.4 Legendre Functions

8.2.5 Orthogonal Polynomials

8.2.6 Complete Elliptic Integrals

8.2.7 Confluent Hypergeometric Functions, Whittaker Functions and Bessel Functions

8.3 Named Functions Expressed in Terms of the G-Function

8.4 The G-Function Expressed as a Named Function

IX. Bessel Functions and Their Integrals

9.1 Introduction

9.2 Definitions, Connecting Relations and Power Series

9.3 Difference-Differential Formulas

9.4 Products of Bessel Functions

9.5 Asymptotic Expansions for Large Variable

9.6 Integrals of Bessel Functions

9.7 Expansions in Series of Chebyshev Polynomials

9.8 Expansions in Series of Bessel Functions

9.9 Rational Approximations

9.9.1 Introduction

9.9.2 Iv(z), z Small

9.9.3 Kv(z), z Large

9.10 Computation of Bessel Functions by Use of Recurrence Formulas

9.10.1 Introduction

9.10.2 Backward Recurrence Schemata for Generating Iv(z)

9.10.3 Closed Form Expressions

9.10.4 Expressions for Iv(z)

9.10.5 Numerical Examples

9.11 Evaluation of Bessel Functions by Application of Trapezoidal Type Integration Formulas

9.12 Inequalities

9.13 Bibliographic and Numerical Data

9.13.1 References

9.13.2 Description of and References to Tables

9.13.3 Description of and References to Other Approximations and Expansions

X. Lommel Functions, Struve Functions, and Associated Bessel Functions

10.1 Definitions, Connecting Relations and Power Series

10.2 Asymptotic Expansions

10.3 Expansions in Series of Chebyshev Polynomials and Bessel Functions

10.4 Rational Approximations for Hv(z) - Yv(z) and the Errors in These Approximations

10.5 Bibliographic and Numerical Data

10.5.1 References

10.5.2 Description of and References to Tables

XI. Orthogonal Polynomials

11.1 Introduction

11.2 Orthogonal Properties

11.3 Jacobi Polynomials

11.3.1 Expansion Formulae

11.3.2 Difference-Differential Formulae

11.3.3 Integrals

11.3.4 Expansion of xP in Series of Jacobi Polynomials

11.3.5 Convergence Theorems for the Expansion of Arbitrary Functions in Series of Jacobi Polynomials

11.3.6 Evaluation and Estimation of the Coefficients in the Expansion of a Given Function f(x) in Series of Jacobi Polynomials

11.4 The Chebyshev Polynomials Tn(x) and Un(x)

11.5 The Chebyshev Polynomials T

*n(x) and U*n(x)

11.6 Coefficients for Expansion of Integrals of Functions in Series of Chebyshev Polynomials of the First Kind

11.6.1 Introduction

11.6.2 Series of Shifted Chebyshev Polynomials

11.6.3 Series of Chebyshev Polynomials of Even Order

11.6.4 Series of Chebyshev Polynomials of Odd Order

11.7 Orthogonality Properties of Chebyshev Polynomials with Respect to Summation

11.8 A Nesting Procedure for the Computation of Expansions in Series of Functions Where the Functions Satisfy a Linear Finite Difference Equation

XII. Computation by Use of Recurrence Formulas

12.1 Introduction

12.2 Homogeneous Difference Equations

12.3 Inhomogeneous Difference Equations

XIII. Some Aspects of Rational and Polynomial Approximations

13.1 Introduction

13.2 Approximations in Series of Chebyshev Polynomials of the First Kind

13.3 The Padé Table

13.4 Approximation of Functions Defined by a Differential Equation - The τ-Method

13.5 Approximations of Functions Defined by a Series

13.6 Solution of Differential Equations in Series of Chebyshev Polynomials of the First Kind

XIV. Miscellaneous Topics

14.1 Introduction

14.2 Bernoulli Polynomials and Numbers

14.3 D and δ Operators

14.4 Computation and Check of the Tables

14.5 Mathematical Constants

14.6 Late Bibliography

Bibliography

Notation Index

Subject Index

## Details

- No. of pages:
- 586

- Language:
- English

- Copyright:
- © Academic Press 1975

- Published:
- 28th September 1975

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483262451