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
A version of Watson lemma for Laplace integrals and some applications cover
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Abstract

Let f:R+Cf:\mathbb{R}_+\to \mathbb{C} be a bounded measurable function. Suppose that f(t)0f(t)\to 0 at logarithmic (or kk-logarithmic) rate as t0+t\to 0+. We consider the Laplace integral of the function ff, i.e.,

In=0f(t)entdtI_n=\int^\infty_0 f(t)e^{-nt}\,dt

and obtain its asymptotics for n+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.