Asymptotic Methods in Analysis

Abstract

An asymptotic formula or asymptotic form for a function f(x) is the name usually given to an approximate formula f(x) ≈ g(x) in some domain of values of x, where g(x) is ‘simpler’ then f(x). For example, if f(x) is an integral, then g(x) must either be given in terms of the values of the integrand and its derivatives at a finite number of points, or in terms of some simpler integral. If f(x) is a solution of an ordinary differential equation, then g(x) must either be expressed in quadratures or be the solution of a ‘simpler’ differential equation. This list can be extended—there is an unwritten heirarchy of asymptotic formulae. Of course all these definitions are very blurred. ‘“What is asymptotics?” This question is about as difficult to answer as the question “What is mathematics?”’