Asymptotics of a class of integrals

Abstract

Consider an integral $I(s):=\int_0^T e^{-s(x^2-icx)}dx$, where $c>0$ and $T>0$ are arbitrary positive constants. It is proved that $I(s)\sim \frac{i}{sc}$ as $s\to +\infty$. The asymptotic behavior of the integral $J(s):=\int_0^Te^{s(x^2+icx)}dx$ is also derived. One has $J(s)\sim \frac{e^{sT^2+iscT}}{s(2T+ic)}$ as $s\to +\infty$.