# A version of Watson lemma for Laplace integrals and some applications

### Stanislas Kupin

Université de Bordeaux I, Talence, France### Sergey Naboko

St. Petersburg University, Russian Federation

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## 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.,

$I_n=\int^\infty_0 f(t)e^{-nt}\,dt$

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.