A version of Watson lemma for Laplace integrals and some applications (Q2055118)

Summary: 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. For the entire collection see [Zbl 1465.35005].