On an extension of Watson's lemma due to Ursell

In 1991, Ursell gave a strong form of Watson’s lemma for the Laplace integral $${\int }_0^{\mathrm{\infty }}{e}^{-xt}f(t)dt(x\to +\mathrm{\infty })$$ in which the amplitude function $f(t)$ is regular at the origin and possesses a Maclaurin expansion valid in $0\le t\le R$. He showed that if the asymptotic series for the integral as $x\to +\mathrm{\infty }$ 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 $\mathrm{\mid }\arg x\mathrm{\mid }<\pi \mathrm{/}2$.