I'm looking for the series for $\frac{cot(\pi z)}{z^2}$ using the residue theory, where the function denotes a circle about the origin with a radius of $k+\frac{1}{2}$
I found that the residues of $\frac{cot(\pi z)}{z^2}$ are:
$R(f;0)= \frac{-\pi}{3}$
$R(f;n)=\frac{1}{\pi n^2}$, where $n$ are positive integers.
I know that the residue theory states that the integral of the function f(z) is $2 \pi i$ $\sum Residues$ however I am unsure how to implement the fact that the radius is $k+\frac{1}{2}$
Because from my residue result I would assume the answer would be:
$2\pi i[\frac{-\pi}{3}+\frac{1}{\pi}\sum_{n=1}^k{\frac{1}{n^2}}]$
However the answer is
$2\pi i[\frac{-\pi}{3}+\frac{2}{\pi}\sum_{n=1}^k{\frac{1}{n^2}}]$
I'm not sure where the 2 before the sum cam from.
Thank you!