The Special Functions and Their Approximations

Name: The Special Functions and Their Approximations | v. 1
Brand: Academic Press
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v. 1

Luke(Autor*in)

Luke(Herausgeber*in)

Academic Press

Erschienen am 14. Mai 2014

373 Seiten

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Front Cover
The Special Functions and Their Approximations
Copyright Page
Contents
Preface
Contents of Volume II
Introduction
Chapter 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
Chapter II. The Gamma Function and Related Functions
2.1. Definitions and Elementary Properties
2.2. Analytic Continuation of ?(z)
2.3. Multiplication Formula
2.4. The Logarithmic Derivative of the Gamma Function
2.5. Integral Representations for ?(z) and In ?(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 d Operators
2.10. Power Series and Other Expansions
2.11. Asymptotic Expansions
Chapter III. Hypergeometric Functions
3.1. Elementary Hypergeometric Series
3.2. A Generalization of the 2F1
3.3. Convergence of the pFq Series
3.4. Elementary Relations
3.5. The Confluence Principle
3.6. Integral Representations
3.7. Differential Equations for the 2F1
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 n+1Fp for Special Values of the Argument
Chapter 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
Chapter V. The Generalized Hypergeometric Function and the G-Function
5.1. The pFq Differential Equation
5.2. The G-Function
5.3. Analytic Continuation of Gmp,np(z)
5.4. Elementary Properties of the G-Function
5.5. Multiplication Theorems
5.6. Integrals Involving G-Functions
5.7. Asymptotic Expansion of Gqp,1q(z) and Gqp,0q(z) for Large z
5.8. Differential Equation for Gmp,nq(z)
5.9. Series of G-Functions
5.10. Asymptotic Expansions of Gmp,nq(z)
5.11. Asymptotic Expansions of pFq(z) for Large z
Chapter VI. Identification of the pFq and G-Functions with the Special Functions of Mathematical Physics
6.1. Introduction
6.2. Named Special Functions Expressed as pFq's
6.3. The pFq Expressed as a Named Function
6.4. Named Functions Expressed in Terms of the G-Function
6.5. The G-Function Expressed as a Named Function
Chapter VII. Asymptotic Expansions of pFq for Large Parameters
7.1. Introduction
7.2. The 2F1
7.3. Some Generalizations of the 2F1 Formulas
7.4. Extended Jacobi Polynomials
Chapter VIII. Orthogonal Polynomials
8.1. Orthogonal Properties
8.2. Jacobi Polynomials
8.3. Expansion of Functions in Series of Jacobi Polynomials
8.4. Evaluation and Estimation of the Coefficients in the Expansion of a Given Function f(x) in Series of Jacobi Polynomials
8.5. Chebyshev Polynomials
8.6. Differential and Integral Properties of Expansions in Series of Chebyshev Polynomials of the First Kind
8.7. A Nesting Procedure for the Computation of Expansions in Series of Functions Where the Functions Satisfy Linear Finite Difference Equations
Bibliography
Notation Index
Subject Index
Mathematics in Science and Engineering

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