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arXiv · 1505.06905

On an extension of Watson's lemma due to Ursell

R. B. Paris
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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 $|\arg\,x|<\pi/2$.

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R. B. Paris. 2015-05-26. On an extension of Watson's lemma due to Ursell. https://arxiv.org/abs/1505.06905

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