ABSTRACT

A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool.

chapter 1|3 pages

Origin of Asymptotic Expansions

chapter 2|2 pages

The Symbols ~, o, and 0

chapter 3|2 pages

The Symbols ~, o, and 0 (continued)

chapter 8|3 pages

Operations with Asymptotic Expansions

chapter 10|3 pages

Generalizations of Poincard's Definition

chapter 11|4 pages

Error Analysis; Variational Operator

chapter 1|8 pages

The Gamma Function

chapter 2|1 pages

The Psi Function

chapter 5|1 pages

Incomplete Gamma Functions

chapter 6|2 pages

Orthogonal Polynomials

chapter 7|5 pages

The Classical Orthogonal Polynomials

chapter 8|2 pages

The Airy Integral

chapter 9|5 pages

The Bessel Function Jv(z)

chapter 10|1 pages

The Modified Bessel Function Iv(z)

chapter 11|5 pages

The Zeta Function

chapter 1|1 pages

Integration by Parts

chapter 2|4 pages

Laplace Integrals

chapter 3|2 pages

Watson's Lemma

chapter 4|2 pages

The Riemann–Lebesgue Lemma

chapter 5|1 pages

Fourier Integrals

chapter 6|4 pages

Examples; Cases of Failure

chapter 7|5 pages

Laplace's Method

chapter 10|4 pages

Examples

chapter 11|2 pages

The Method of Stationary Phase

chapter 12|2 pages

Preliminary Lemmas

chapter 3|4 pages

Watson's Lemma

chapter 6|4 pages

Laplace's Method for Contour Integrals

chapter 7|2 pages

Saddle Points

chapter 8|3 pages

Examples

chapter 6|1 pages

Large Values of the Independent Variable

chapter 7|2 pages

Numerically Satisfactory Solutions

chapter 8|3 pages

The Hypergeometric Equation

chapter 9|4 pages

The Hypergeometric Function

chapter 11|1 pages

Generalized Hypergeometric Functions

chapter 12|5 pages

The Associated Legendre Equation

chapter 15|5 pages

Ferrers Functions

chapter 1|3 pages

The Liouville Transformation

chapter 2|4 pages

Error Bounds: Real Variables

chapter 4|3 pages

Convergence of v (F) at a Singularity

chapter 7|3 pages

A Special Extension

chapter 8|3 pages

Zeros

chapter 9|3 pages

Eigenvalue Problems

chapter 10|3 pages

Theorems on Singular Integral Equations

chapter 11|3 pages

Error Bounds: Complex Variables

chapter 13|5 pages

Choice of Progressive Paths

chapter 1|3 pages

Formal Series Solutions

chapter 2|4 pages

Asymptotic Nature of the Formal Series

chapter 3|1 pages

Equations Containing a Parameter

chapter 4|4 pages

Hankel Functions; Stokes' Phenomenon

chapter 5|3 pages

The Function Yv(z)

chapter 6|4 pages

Zeros of Jv (z)

chapter 8|4 pages

Modified Bessel Functions

chapter 9|2 pages

Confluent Hypergeometric Equation

chapter 11|2 pages

Whittaker Functions

chapter 13|4 pages

Error Bounds for Hankel's Expansions

chapter 14|4 pages

Inhomogeneous Equations

chapter 15|5 pages

Struve's Equation

chapter 2|5 pages

Applications

chapter 3|4 pages

Contour Integral for the Remainder Term

chapter 4|2 pages

Stirling's Series for In Г(z)

chapter 5|4 pages

Summation by Parts

chapter 7|5 pages

Further Examples

chapter 8|2 pages

Asymptotic Expansions of Entire Functions

chapter 10|4 pages

Examples

chapter 11|7 pages

Inverse Laplace Transforms; Haar's Method

chapter 1|3 pages

Logarithmic Singularities

chapter 2|4 pages

Generalizations of Laplace's Method

chapter 3|2 pages

Example from Combinatoric Theory

chapter 5|2 pages

Examples

chapter 6|4 pages

More General Kernels

chapter 8|2 pages

Oscillatory Kernels

chapter 9|2 pages

Bleistein's Method

chapter 10|5 pages

Example

chapter 12|6 pages

Anger Functions of Large Order

chapter 13|4 pages

Extension of the Region of Validity

chapter 2|2 pages

Case I: Formal Series Solutions

chapter 3|2 pages

Error Bounds for the Formal Solutions

chapter 7|4 pages

Modified Bessel Functions of Large Order

chapter 10|2 pages

Inhomogeneous Equations

chapter 1|2 pages

Airy Functions of Real Argument

chapter 2|3 pages

Auxiliary Functions for Real Variables

chapter 3|4 pages

The First Approximation

chapter 5|2 pages

Real Zeros of the Airy Functions

chapter 6|3 pages

Zeros of the First Approximation

chapter 7|5 pages

Higher Approximations

chapter 8|3 pages

Airy Functions of Complex Argument

chapter 10|7 pages

Bessel Functions of Large Order

chapter 11|3 pages

More General Form of Differential Equation

chapter 12|6 pages

Inhomogeneous Equations

chapter 2|2 pages

Case III: Formal Series Solutions

chapter 3|3 pages

Error Bounds: Positive ζ

chapter 4|4 pages

Error Bounds: Negative ζ

chapter 5|2 pages

Asymptotic Properties of the Expansions

chapter 6|2 pages

Determination of Phase Shift

chapter 7|2 pages

Zeros

chapter 8|4 pages

Auxiliary Functions for Complex Arguments

chapter 9|3 pages

Error Bounds: Complex u and ζ

chapter 11|1 pages

Behavior of the Coefficients at Infinity

chapter 14|6 pages

Other Types of Transition Points

chapter 1|2 pages

Introduction

chapter 3|1 pages

Differential Equations with a Parameter

chapter 4|4 pages

Connection Formula for Case III

chapter 5|3 pages

Application to Simple Poles

chapter 6|1 pages

Example: The Associated Legendre Equation

chapter 8|3 pages

Two Turning Points

chapter 9|4 pages

Bound States

chapter 10|2 pages

Wave Penetration through a Barrier. I

chapter 12|1 pages

Example: Airy's Equation

chapter 13|2 pages

Choice of Progressive Paths

chapter 15|6 pages

Wave Penetration through a Barrier. II

chapter 1|3 pages

Numerical Use of Asymptotic Approximation

chapter 2|1 pages

Converging Factor

chapter 3|4 pages

Exponential Integral

chapter 4|4 pages

Exponential Integral (continued)

chapter 5|5 pages

Confluent Hypergeometric Function

chapter 6|4 pages

Euler's Transformatio

chapter 7|5 pages

Application to Asymptotic Expansion

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