Miller P.D. Applied Asymptotic Analysis

American Mathematical Society, 2006. — 467 pages. — (Graduate Studies in Mathematics).
ISBN: 0821840789 ISBN13: 9780821840788.This book is a survey of asymptotic methods set in the current applied research context of wave propagation. It stresses rigorous analysis in addition to formal manipulations. Asymptotic expansions developed in the text are justified rigorously, and students are shown how to obtain solid error estimates for asymptotic formulae.The book relates examples and exercises to subjects of current research interest, such as the problem of locating the zeros of Taylor polynomials of entire nonvanishing functions and the problem of counting integer lattice points in subsets of the plane with various geometrical properties of the boundary. The book is intended for a beginning graduate course on asymptotic analysis in applied mathematics and is aimed at students of pure and applied mathematics as well as science and engineering. The basic prerequisite is a background in differential equations, linear algebra, advanced calculus, and complex variables at the level of introductory undergraduate courses on these subjects.Themes of asymptotic analysis
The nature of asymptotic approximations
Fundamental techniques for integrals
Laplace's method for asymptotic expansions of integrals
The method of steepest descents for asymptotic expansions of integrals
The method of stationary phase for asymptotic analysis of oscillatory integrals
Asymptotic behavior of solutions of linear second-order differential equations in the complex plane
Introduction to asymptotics of solutions of ordinary differential equations with respect to parameters
Asymptotics of linear boundary-value problems
Asymptotics of oscillatory phenomena
Weakly nonlinear waves
Appendix. Fundamental inequalities