I have recently encountered a truely terrible integral which I need to compute. I am not sure it's doable but before throwing the whole project in the bin I thought I would ask here. At the moment, a step I require is evaluating the integral of $f(z)$ along a closed coutour $C$ containing zero where $f(z)$ is something like

$f(z)=e^{(\frac{a}{z}+\frac{b}{z-c})}g(z)$

$c$ is located outside the contour $g(z)$ is holomorphic inside the disk enclosed by $C$ which has a very long but finite taylor expansion. The reason no traditional tricks work (using the coordinate change $z=1/w$, looking at the series and trying to collect together all the $\frac{1}{z}$ terms) is of course this $\exp((\frac{b}{z-c}))$ term, for which the series expansion has an infinite number of terms, so trying to sum up all those containing $1/z$ is doomed from the start... Has anybody ever encountered something similar? I tried reading about Bessel functions, but they didn't quite fix the problem.