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1st Edition

Asymptotic Analysis and Perturbation Theory




ISBN 9781466515116
Published July 18, 2013 by Chapman and Hall/CRC
550 Pages - 104 B/W Illustrations

 
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Book Description

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.

Table of Contents

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
Padé Approximants

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
Strained Coordinates Method (Poincaré-Lindstedt)
The Multiple-Scale Procedure
Two-Variable Expansion Method

Appendix: Guide to the Special Functions

Answers to Odd-Numbered Problems

Bibliography

Index

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Author(s)

Biography

William Paulsen is a professor of mathematics at Arkansas State University, where he teaches asymptotics to undergraduate and graduate students. He is the author of Abstract Algebra: An Interactive Approach (CRC Press, 2009) and has published over 15 papers in applied mathematics, one of which proves that Penrose tiles can be three-colored, thus resolving a 30-year-old open problem posed by John H. Conway. Dr. Paulsen has also programmed several new games and puzzles in Javascript and C++, including Duelling Dimensions, which was syndicated through Knight Features. He received a Ph.D. in mathematics from Washington University in St. Louis.

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