On an extension of Watson's lemma due to Ursell

TL;DR

This paper extends Ursell's version of Watson's lemma to cases where the amplitude function has a branch point at zero and considers complex variables, broadening the lemma's applicability in asymptotic analysis.

Contribution

It generalizes Ursell's Watson's lemma to include amplitude functions with branch points at zero and complex variables, enhancing its scope.

Findings

Extended Watson's lemma to branch point cases

Included complex variable scenarios with argument less than pi/2

Provided bounds on the remainder term in the asymptotic expansion

Abstract

In 1991, Ursell gave a strong form of Watson's lemma for the Laplace integral \[\int_0^\infty e^{-xt}f(t)\,dt\qquad (x\rightarrow+\infty) \] in which the amplitude function $f(t)$ is regular at the origin and possesses a Maclaurin expansion valid in $0\leq t\leq R$. He showed that if the asymptotic series for the integral as $x\rightarrow+\infty$ is truncated after $rx$ terms, where $0<r<R$, then the resulting remainder is exponentially small of order $O(e^{-rx})$. In this note we extend this result to include situations when $f(t)$ has a branch point at $t=0$ and when $x$ is a complex variable satisfying $|\arg\,x|<\pi/2$.