Given the integral: $I(s)=\int_{-a}^a \exp\{s \cos(t)\}dt$ is it possible to find an expansion of $I(s)$ using the Watson's lemma?
Thanks in advance.
Given the integral: $I(s)=\int_{-a}^a \exp\{s \cos(t)\}dt$ is it possible to find an expansion of $I(s)$ using the Watson's lemma?
Thanks in advance.
Watson's lemma applies to integrals of the form $ \int_0^Te^{-st}g(t)\,dt, $ where $g$ satisfies certain conditions, among them being differentiable in neighbourhood of $t=0$. You could try to put $I(s)$ into that form using symmetry and the change of variables $\cos t=1-x$: $ I(s)=2\,e^s\int_0^{1-\cos a}e^{-sx}\,(2\,x-x^2)^{-1/2}\,dx, $ but $(2\,x-x^2)^{-1/2}$ is not differentiable at $x=0$.
The asymptotic behaviour of $I(s)$ can be found using Lagrange's method. A reference is Applied Asymptotic Analysis by Peter D. Miller, chapter 3.
It turns out that Watson's lemma can be applied to integrals of the form $ \int_0^Te^{-st}\,t^\sigma g(t)\,dt,\quad \sigma>-1. $ It can then be applied to $ I(s)=2\,e^s\int_0^{1-\cos a}e^{-sx}\,x^{-1/2}(2-x)^{-1/2}\,dx. $