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Full Description
This book concerns the question of how the solution of a
system of ODE's varies when the differential equation
varies. The goal is to give nonzero asymptotic expansions
for the solution in terms of a parameter expressing how some
coefficients go to infinity. A particular classof families
of equations is considered, where the answer exhibits a new
kind of behavior not seen in most work known until now. The
techniques include Laplace transform and the method of
stationary phase, and a combinatorial technique for
estimating the contributions of terms in an infinite series
expansion for the solution. Addressed primarily to
researchers inalgebraic geometry, ordinary differential
equations and complex analysis, the book will also be of
interest to applied mathematicians working on asymptotics of
singular perturbations and numerical solution of ODE's.
Contents
Ordinary differential equations on a Riemann surface.- Laplace transform, asymptotic expansions, and the method of stationary phase.- Construction of flows.- Moving relative homology chains.- The main lemma.- Finiteness lemmas.- Sizes of cells.- Moving the cycle of integration.- Bounds on multiplicities.- Regularity of individual terms.- Complements and examples.- The Sturm-Liouville problem.
