Let $f(z)=\frac{e^{az}}{1+e^z}$ where $0
Can anyone help me find the residues of this function?
So $e^z+1=0 \Rightarrow z=i\pi(1+2k)$ where $k\in \mathbb{Z}$, so these are simple poles (if someone could explain a simple way of showing this that'd be great, other than expansion)
$\lim_{z\rightarrow i\pi(1+2k)}\frac{(z-i\pi(1+2k))e^{az}}{1+e^z}=\lim_{z\rightarrow i\pi(1+2k)}\frac{a(z-i\pi(1+2k))e^{az}+e^{az}}{e^z}=e^a$
So i'm trying to evaluate $\int_{\infty}^\infty f(z)$ you see, so will I need to pick a contour with fixed height otherwise the integral around the contour will be equal to $2\pi i \sum_{n=0}^\infty e^a$