A version of Watson lemma for Laplace integrals and some applications

Abstract

Let $f:{\mathbb{R}}_{+}\to \mathbb{C}$ be a bounded measurable function. Suppose that $f(t)\to 0$ at logarithmic (or $k$-logarithmic) rate as $t\to 0+$. We consider the Laplace integral of the function $f$, i.e.,

and obtain its asymptotics for $n\to +\infty$, which is a version of the classical Watson’s lemma for the integral. Actually, the result is proved for a larger class of “slowly oscillating” functions satisfying some mild regularity conditions.