Residues at a finite point

Question

What is the residue of cotz at z=nπ ,where n is integer ? I have calculated the residue of cotz at z=0 and it is equal to 1 via expansion of cotz ....but how can I find the residue at nπ with the help of power series expansion of cotz ?

Answer 1

If $\cot \, z=\sum a_kz^{k}$ near $0$ then $\cot \, z=\sum a_k(z-n\pi)^{k}$ neat $n\pi$. To see this simply replace $z$ by $z-n\pi$ and note that $\cot (z-n\pi)=cot \, z$. Hence the residue at $n\pi$ is same as the residue at $0$.

Answer 2

Since the residue of $\cot(\pi z)$ at $0$ is $1$ and that function is periodic with periode $1$, its residue at each integer multiple of $\pi$ is also $1$.