(i) Let $C$ be an anticlockwise contour defined by a square with vertices at $z=x+i y$ where

$$|x|=|y|=\left(2 N+\frac{1}{2}\right) \pi$$

for large integer $N$. Let

$$I=\oint_{C} \frac{\pi \cot z}{(z+\pi a)^{4}} d z$$

Assuming that $I \rightarrow 0$ as $N \rightarrow \infty$, prove that, if $a$ is not an integer, then

$$\sum_{n=-\infty}^{\infty} \frac{1}{(n+a)^{4}}=\frac{\pi^{4}}{3 \sin ^{2}(\pi a)}\left(\frac{3}{\sin ^{2}(\pi a)}-2\right) .$$

(ii) Deduce the value of

$$\sum_{n=-\infty}^{\infty} \frac{1}{\left(n+\frac{1}{2}\right)^{4}}$$

(iii) Briefly justify the assumption that $I \rightarrow 0$ as $N \rightarrow \infty$.

[Hint: For part (iii) it is sufficient to consider, at most, one vertical side of the square and one horizontal side and to use a symmetry argument for the remaining sides.]