Let $f(z)= \frac{e^{az}}{1+e^{z}}$ and $0<a<1$. Find Laurent series expansion centered at $z_0 = -i \pi$.
The biggest problem for me is to find this series centered at $z_0 $ which does not equal $0$. (I can do it in point $z_0 = 0$ and then principal part equal 0).