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Asymptotics and Special Functions.

Author: Frank Olver
Publisher: Natick : Chapman and Hall/CRC, 1997.
Series: AKP classics.
Edition/Format:   eBook : Document : English : 2nd edView all editions and formats
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Olver, Frank.
Asymptotics and Special Functions.
Natick : Chapman and Hall/CRC, ©1997
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Frank Olver
ISBN: 9781439864548 1439864543
OCLC Number: 1035519046
Notes: 9 The Hypergeometric Function.
Description: 1 online resource (591 pages).
Contents: Cover; Title Page; Copyright Page; Dedication; Table of Contents; Preface to A K Peters Edition; Preface; 1: Introduction to Asymptotic Analysis; 1 Origin of Asymptotic Expansions; 2 The Symbols ~, o, and 0; 3 The Symbols ~, o, and 0 (continued); 4 Integration and Differentiation of Asymptotic and Order Relations; 5 Asymptotic Solution of Transcendental Equations: Real Variables; 6 Asymptotic Solution of Transcendental Equations: Complex Variables; 7 Definition and Fundamental Properties of Asymptotic Expansions; 8 Operations with Asymptotic Expansions. 9 Functions Having Prescribed Asymptotic Expansions10 Generalizations of Poincard's Definition; 11 Error Analysis; Variational Operator; Historical Notes and Additional References; 2: Introduction to Special Functions; 1 The Gamma Function; 2 The Psi Function; 3 Exponential, Logarithmic, Sine, and Cosine Integrals; 4 Error Functions, Dawson's Integral, and Fresnel Integrals; 5 Incomplete Gamma Functions; 6 Orthogonal Polynomials; 7 The Classical Orthogonal Polynomials; 8 The Airy Integral; 9 The Bessel Function Jv(z); 10 The Modified Bessel Function Iv(z); 11 The Zeta Function. Historical Notes and Additional References3: Integrals of a Real Variable; 1 Integration by Parts; 2 Laplace Integrals; 3 Watson's Lemma; 4 The Riemann-Lebesgue Lemma; 5 Fourier Integrals; 6 Examples; Cases of Failure; 7 Laplace's Method; 8 Asymptotic Expansions by Laplace's Method; Gamma Function of Large Argument; 9 Error Bounds for Watson's Lemma and Laplace's Method; 10 Examples; 11 The Method of Stationary Phase; 12 Preliminary Lemmas; 13 Asymptotic Nature of the Stationary Phase Approximation; 14 Asymptotic Expansions by the Method of Stationary Phase. Historical Notes and Additional References4: Contour Integrals; 1 Laplace Integrals with a Complex Parameter; 2 Incomplete Gamma Functions of Complex Argument; 3 Watson's Lemma; 4 Airy Integral of Complex Argument; Compound Asymptotic Expansions; 5 Ratio of Two Gamma Functions; Watson's Lemma for Loop Integrals; 6 Laplace's Method for Contour Integrals; 7 Saddle Points; 8 Examples; 9 Bessel Functions of Large Argument and Order; 10 Error Bounds for Laplace's Method; the Method of Steepest Descents; Historical Notes and Additional References. 5: Differential Equations with Regular Singularities Hypergeometric and Legendre Functions; 1 Existence Theorems for Linear Differential Equations: Real Variables; 2 Equations Containing a Real or Complex Parameter; 3 Existence Theorems for Linear Differential Equations: Complex Variables; 4 Classification of Singularities; Nature of the Solutions in the Neighborhood of a Regular Singularity; 5 Second Solution When the Exponents Differ by an Integer or Zero; 6 Large Values of the Independent Variable; 7 Numerically Satisfactory Solutions; 8 The Hypergeometric Equation.
Series Title: AKP classics.

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The book under review is a very good reference on this material, giving a detailed collection of various asymptotic results, with a special focus on special functions. ... The book is a classic, and Read more...

 
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