Asymptotics and Special Functions
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Asymptotics and Special Functions provides a comprehensive introduction to two important topics in classical analysis: asymptotics and special functions. The integrals of a real variable and contour integrals are discussed, along with the Liouville-Green approximation and connection formulas for solutions of differential equations. Differential equations with regular singularities are also considered, with emphasis on hypergeometric and Legendre functions. Comprised of 14 chapters, this volume begins with an introduction to the basic concepts and definitions of asymptotic analysis and special functions, followed by a discussion on asymptotic theories of definite integrals containing a parameter. Contour integrals as well as integrals of a real variable are described. Subsequent chapters deal with the analytic theory of ordinary differential equations; differential equations with regular and irregular singularities; sums and sequences; and connection formulas for solutions of differential equations. The book concludes with an evaluation of methods used in estimating (as opposed to bounding) errors in asymptotic approximations and expansions. This monograph is intended for graduate mathematicians, physicists, and engineers.
Table of Contents
Preface
1 Introduction to Asymptotic Analysis
1 Origin of Asymptotic Expansions
2 The Symbols ~, o, and Ο
3 The Symbols ~, o, and Ο (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 Expansions
10 Generalizations of Poincaré'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 References
3 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 References
4 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
9 The Hypergeometric Function
10 Other Solutions of the Hypergeometric Equation
11 Generalized Hypergeometric Functions
12 The Associated Legendre Equation
13 Legendre Functions of General Degree and Order
14 Legendre Functions of Integer Degree and Order
15 Ferrers Functions
Historical Notes and Additional References
6 The Liouville-Green Approximation
1 The Liouville Transformation
2 Error Bounds: Real Variables
3 Asymptotic Properties with Respect to the Independent Variable
4 Convergence of V (F) at a Singularity
5 Asymptotic Properties with Respect to Parameters
6 Example: Parabolic Cylinder Functions of Large Order
7 A Special Extension
8 Zeros
9 Eigenvalue Problems
10 Theorems on Singular Integral Equations
11 Error Bounds: Complex Variables
12 Asymptotic Properties for Complex Variables
13 Choice of Progressive Paths
Historical Notes and Additional References
7 Differential Equations with Irregular Singularities; Bessel and Confluent Hypergeometric Functions
1 Formal Series Solutions
2 Asymptotic Nature of the Formal Series
3 Equations Containing a Parameter
4 Hankel Functions; Stokes' Phenomenon
5 The Function Yv(z)
6 Zeros of Jv(z)
7 Zeros of Yv(z) and Other Cylinder Functions
8 Modified Bessel Functions
9 Confluent Hypergeometric Equation
10 Asymptotic Solutions of the Confluent Hypergeometric Equation
11 Whittaker Functions
12 Error Bounds for the Asymptotic Solutions in the General Case
13 Error Bounds for Hankel's Expansions
14 Inhomogeneous Equations
15 Struve's Equation
Historical Notes and Additional References
8 Sums and Sequences
1 The Euler-Maclaurin Formula and Bernoulli's Polynomials
2 Applications
3 Contour Integral for the Remainder Term
4 Stirling's Series for In Γ(z)
5 Summation by Parts
6 Barnes' Integral for the Hypergeometric Function
7 Further Examples
8 Asymptotic Expansions of Entire Functions
9 Coefficients in a Power-Series Expansion; Method of Darboux
10 Examples
11 Inverse Laplace Transforms; Haar's Method
Historical Notes and Additional References
9 Integrals: Further Methods
1 Logarithmic Singularities
2 Generalizations of Laplace's Method
3 Example from Combinatoric Theory
4 Generalizations of Laplace's Method (Continued)
5 Examples
6 More General Kernels
7 Nicholson's Integral for J2v(z) + Y2v(z)
8 Oscillatory Kernels
9 Bleistein's Method
10 Example
11 The Method of Chester, Friedman, and Ursell
12 Anger Functions of Large Order
13 Extension of the Region of Validity
Historical Notes and Additional References
10 Differential Equations with a Parameter: Expansions in Elementary Functions
1 Classification and Preliminary Transformations
2 Case I: Formal Series Solutions
3 Error Bounds for the Formal Solutions
4 Behavior of the Coefficients at a Singularity
5 Behavior of the Coefficients at a Singularity (Continued)
6 Asymptotic Properties with Respect to the Parameter
7 Modified Bessel Functions of Large Order
8 Extensions of the Regions of Validity for the Expansions of the Modified Bessel Functions
9 More General Forms of Differential Equation
10 Inhomogeneous Equations
11 Example: An Inhomogeneous Form of the Modified Bessel Equation
Historical Notes and Additional References
11 Differential Equations with a Parameter: Turning Points
1 Airy Functions of Real Argument
2 Auxiliary Functions for Real Variables
3 The First Approximation
4 Asymptotic Properties of the Approximation; Whittaker Functions with m Large
5 Real Zeros of the Airy Functions
6 Zeros of the First Approximation
7 Higher Approximations
8 Airy Functions of Complex Argument
9 Asymptotic Approximations for Complex Variables
10 Bessel Functions of Large Order
11 More General Form of Differential Equation
12 Inhomogeneous Equations
Historical Notes and Additional References
12 Differential Equations with a Parameter: Simple Poles and Other Transition Points
1 Bessel Functions and Modified Bessel Functions of Real Order and Argument
2 Case III: Formal Series Solutions
3 Error Bounds: Positive ζ
4 Error Bounds: Negative ζ
5 Asymptotic Properties of the Expansions
6 Determination of Phase Shift
7 Zeros
8 Auxiliary Functions for Complex Arguments
9 Error Bounds: Complex u and ζ
10 Asymptotic Properties for Complex Variables
11 Behavior of the Coefficients at Infinity
12 Legendre Functions of Large Degree: Real Arguments
13 Legendre Functions of Large Degree: Complex Arguments
14 Other Types of Transition Points
Historical Notes and Additional References
13 Connection Formulas for Solutions of Differential Equations
1 Introduction
2 Connection Formulas at a Singularity
3 Differential Equations with a Parameter
4 Connection Formula for Case III
5 Application to Simple Poles
6 Example: The Associated Legendre Equation
7 The Gans-Jeffreys Formulas: Real-Variable Method
8 Two Turning Points
9 Bound States
10 Wave Penetration Through a Barrier. I
11 Fundamental Connection Formula for a Simple Turning Point in the Complex Plane
12 Example: Airy's Equation
13 Choice of Progressive Paths
14 The Gans-Jeffreys Formulas: Complex-Variable Method
15 Wave Penetration through a Barrier. II
Historical Notes and Additional References
14 Estimation of Remainder Terms
1 Numerical Use of Asymptotic Approximations
2 Converging Factors
3 Exponential Integral
4 Exponential Integral (Continued)
5 Confluent Hypergeometric Function
6 Euler's Transformation
7 Application to Asymptotic Expansions
Historical Notes and Additional References
Answers to Exercises
References
Index of Symbols
General Index
Product details
- No. of pages: 588
- Language: English
- Copyright: © Academic Press 1974
- Published: January 28, 1974
- Imprint: Academic Press
- eBook ISBN: 9781483267449
About the Author
F. W. J. Olver
About the Editor
Werner Rheinbolt
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