Let $\gamma$ be a closed curve in the right half plane that has index $N$ with respect to the point $1$. Find $$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $$
This is a problem from an old qualifying exam and I have tried to compute the reside of the integrand at the point $1$ where it has an essential singularity, but could not make any progress. I found the same question posted before at How to compute the complex integral $\int_{\gamma} e^{\frac{1}{z^2 - 1}}\sin{\pi z} \, \mathrm dz$? but I feel that it was not satisfactorily answered even there. Can someone please help?