How can I use the asymptotic approximation for integrand functions with simple poles?
Let's consider the integral $$\int_{\infty e^ {\pi 3/4}}^{\infty e^{\pi/4}} \frac {f(s)}{s-s_0}e^{-\Omega{(s^2-2s+1)}}ds$$ let $s_0=\rho_0 e^{j\pi /4}$.
The sadle point is in $s=1.$
I have to include the residue so $$I=-2\pi jf(s_0)e^{-(s_0+1)^2}+\int_{-\infty}^{\infty}\frac{f(s+1)}{s-s_0}e^{-\Omega s^2}$$
But what can I do now? Can I use the Q-integral? How?
Many thanks