The special functions and their approximations. Volume 1 / Yudell L. Luke.

Format:

Book

Author/Creator:

Luke, Yudell L.

Series:

Mathematics in science and engineering ; 53I.

Mathematics in science and engineering ; 53I

Language:

English

Subjects (All):

Functions, Special.

Approximation theory.

Physical Description:

1 online resource (373 p.)

Place of Publication:

New York : Academic Press, 1969.

Language Note:

English

Summary:

The Special Functions and Their Approximations: v. 1

Contents:

Front Cover; The Special Functions and Their Approximations; Copyright Page; Preface; Contents; 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 Г(z); 2.3. Multiplication Formula; 2.4. The Logarithmic Derivative of the Gamma Function; 2.5. Integral Representations for ψ(z) and In Г(z)

2.6. The Beta Function and Related Functions2.7. Contour Integral Representations for Gamma and Beta Functions; 2.8. Bernoulli Polynomials and Numbers; 2.9. The D and δ 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 Solution3.11. 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-Function5.1. 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 Physics6.1. 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

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

1-282-29022-3

9786612290220

0-08-095560-6

OCLC:

428101252