1 - INTRODUCTION TO ASYMPTOTIC ANALYSIS

Abstract

An asymptotic expansion of a function f(x) is in practice an expression of that function in terms of a series, the partial sums of which do not necessarily converge but are such that taking any initial partial sum provides an asymptotic formula for f. As a rule, asymptotic and order relations can be integrated, subject to obvious restrictions on the convergence of the integrals involved. In case the asymptotic expansion does not converge, for any particular value of the argument, there will be a particular partial sum that provides the best approximation and adding additional terms will decrease the accuracy. Asymptotic expansions typically arise in the approximation of certain integrals such as Laplace's method, saddle-point method, method of steepest descent, or in the approximation of probability distributions. The famous Feynman graphs in quantum field theory are another example of asymptotic expansions that often do not converge.