Book:Yudell L. Luke/The Special Functions And Their Approximations, Volume I

Yudell L. Luke: Higher Transcendental Functions, Volume II

Published $1969$, Academic Press.

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Contents

Preface

Contents of Volume II

Introduction

I. Asymptotic Expansions

1.1. The order symbols $O$ and $o$

1.2. Definition of an Asymptotic Expansion

1.3. Elementary Properties of Asymptotic Series

1.4. Watson's Lemma

II. The Gamma function and Related Functions

2.1. Definitions and Elementary Properties

$(1)$

2.2. Analytic Continuation of $\Gamma(z)$

2.3. Multiplication Formula

2.4. The Logarithmic Derivative of the Gamma Function

2.5. Integral Representations for $\psi(z)$ and $\ln \Gamma(z)$

2.6. The Beta Function and Related Functions

2.7. Contour Integral Representations for Gamma and Beta Functions

2.8. Bernoulli Polynomials and Numbers

2.9. The $D$ and $\delta$ Operators

2.10. Power Series and Other Expansions

2.11. Asymptotic Expansions

III. Hypergeometric Functions

3.1. Elementary Hypergeometric Series

3.2. A Generalization of the ${}_2F_1$

3.3. Convergence of the ${}_pF_q$ series

3.4. Elementary Relations

3.5. The Confluence Principle

3.6. Integral Representations

3.7. Differential Equations for the ${}_2F_1$

3.8. Kummer's Solutions

3.9. Analytic Continuation

3.10. The Complete Solution

3.11. Kummer Type Relations for the Logarithmic Solutions

3.12. Quadratic Transformations

3.13. The ${}_{p+1}F_q$ for Special Values of the Argument

IV. Confluent Hypergeometric Functions

4.1. Introduction

4.2. Integral Representations

4.3. Elementary Relations for the Confluent Functions

4.4. Confluent Differential Equation

4.5. The Complete Solution

4.6. Kummer Type Relations for the Logarithmic Solutions

4.7. Asymptotic Expansions for Large $z$

4.8. Asymptotic Behavior for Large Parameters and Variable

4.9. Other Notations and Related Functions

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

5.1. The ${}_pF_q$ Differential Equation

5.2. The $G$ Function

5.3. Analytic Continuation of $G$

5.4. Multiplcation Theorems

5.6. Integrals Involving $G$ Functions