Explicit Series Formulae for the Evaluation of Integrals by the Method of Steepest Descents

Abstract

Explicit series formulae are given for use in the method of steepest descents, where it is required to find the contribution to an integral of the form ∫e vw ( z ) φ ( z ) dz from a well‐isolated saddle point z 0 . Here v is a large positive parameter, and w ( N ) ( z 0) ≠ 0 while lower‐order derivatives at z 0 are equal to zero. Examples where the formulae are helpful are presented.