I'm working on computing a contour integral (Actually it's a part of a inverse Laplace transform problem, see my question) $$ \oint_{\Gamma}g(s)ds $$ $$ g(s)=e^{-\tau s\sqrt{\frac{s+q}{s+p}}}e^{ts} $$ where $\tau$, $t$, $p$, $q$ are all positive real, and $q>p$. The integral contour $\Gamma$ can be any contour enclosing the branch cut of $g(s)$, which is the real line segment $(-q,-p)$.
I have tried to evaluate this integral by choosing a bog-bone contour with two key-holes as shown in the picture.
But it turns out that the contour integrals around the keyhole doesn't zero as the radius of the keyhole $\epsilon\rightarrow 0$ (the integrand seems to behave like $1/\epsilon$).
So my question is how to evaluate the integral around the branch point? Or if it's too difficult should I choose another integral contour?