1st Edition

# Asymptotic Analysis and Perturbation Theory

550 Pages 104 B/W Illustrations
by Chapman & Hall

550 Pages
by Chapman & Hall

Also available as eBook on:

Beneficial to both beginning students and researchers, Asymptotic Analysis and Perturbation Theory immediately introduces asymptotic notation and then applies this tool to familiar problems, including limits, inverse functions, and integrals. Suitable for those who have completed the standard calculus sequence, the book assumes no prior knowledge of differential equations. It explains the exact solution of only the simplest differential equations, such as first-order linear and separable equations.

With varying levels of problems in each section, this self-contained text makes the difficult subject of asymptotics easy to comprehend. Along the way, it explores the properties of some important functions in applied mathematics. Although the book emphasizes problem solving, some proofs are scattered throughout to give readers a justification for the methods used.

Introduction to Asymptotics
Basic Definitions
Limits via Asymptotics
Asymptotic Series
Inverse Functions
Dominant Balance

Asymptotics of Integrals
Integrating Taylor Series
Repeated Integration by Parts
Laplace's Method
Review of Complex Numbers
Method of Stationary Phase
Method of Steepest Descents

Speeding Up Convergence
Shanks Transformation
Richardson Extrapolation
Euler Summation
Borel Summation
Continued Fractions

Differential Equations
Classification of Differential Equations
First Order Equations
Taylor Series Solutions
Frobenius Method

Asymptotic Series Solutions for Differential Equations
Behavior for Irregular Singular Points
Full Asymptotic Expansion
Local Analysis of Inhomogeneous Equations
Local Analysis for Nonlinear Equations

Difference Equations
Classification of Difference Equations
First Order Linear Equations
Analysis of Linear Difference Equations
The Euler-Maclaurin Formula
Taylor-Like and Frobenius-Like Series Expansions

Perturbation Theory
Introduction to Perturbation Theory
Regular Perturbation for Differential Equations
Singular Perturbation for Differential Equations
Asymptotic Matching

WKBJ Theory
The Exponential Approximation
Region of Validity
Turning Points

Multiple-Scale Analysis
The Multiple-Scale Procedure
Two-Variable Expansion Method

Appendix: Guide to the Special Functions