1st Edition

Asymptotic Analysis and Perturbation Theory

By William Paulsen Copyright 2014
    550 Pages 104 B/W Illustrations
    by Chapman & Hall

    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
    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




    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.