Introduction to Asymptotics and Special Functions

Introduction to Asymptotics and Special Functions is a comprehensive introduction to two important topics in classical analysis: asymptotics and special functions. The integrals of a real variable are discussed, along with contour integrals and differential equations with regular and irregular singularities. The Liouville-Green approximation is also considered. Comprised of seven chapters, this volume begins with an overview of the basic concepts and definitions of asymptotic analysis and special functions, followed by a discussion on integrals of a real variable. Contour integrals are then examined, paying particular attention to Laplace integrals with a complex parameter and Bessel functions of large argument and order. Subsequent chapters focus on differential equations having regular and irregular singularities, with emphasis on Legendre functions as well as Bessel and confluent hypergeometric functions. A chapter devoted to the Liouville-Green approximation tackles asymptotic properties with respect to parameters and to the independent variable, eigenvalue problems, and theorems on singular integral equations. This monograph is intended for students needing only an introductory course to asymptotics and special functions.

Preface

Preface to Asymptotics and Special Functions

1 Introduction to Asymptotic Analysis

1 Origin of Asymptotic Expansions

2 The Symbols ~, o, and O

3 The Symbols ~, o, and O (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

Answers to Exercises

References

Index of Symbol

General Index