# 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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