Asymptotic expansion

Summary

An asymptotic expansion of a function $f$ near a point is a formal series $\sum a_n \varphi_n$ in a scale of functions that shrink one after another. Every partial sum must approximate $f$ with an error smaller than its last term. The series need not converge. Asymptotic expansions describe special functions, integrals and solutions of differential equations for large or small parameters. They are closely tied to the theory of divergent series and to Borel summation.