Laurent Series of $f(z) = \frac{z}{\sinh(z)}$ in the region $ 4 < |z| <5 $

Question

Determine all coefficients, belonging to $ z^n $ with $ n<5 $, of the Laurent series of the function $f(z)=\frac{z}{\sinh(z)}$ in the region $4 < |z| <5 $.

Could someone help me to find the solution to this problem?

Answer 1

$\sinh(z)$ has simple zeroes at $z\in \pi i \mathbb{Z}$, hence $f(z)=\frac{z}{\sinh z}$ is regular at $z=0$ but has two simple poles at $z=\pm \pi i$, having residues $\mp \pi i$. So we have that: $$ g(z) = f(z)-\frac{2\pi^2}{\pi^2+z^2} $$ is a holomorphic function over $|z|<2\pi$. By computing its Taylor series we have: $$ g(z) = -1+\left(\frac{2}{\pi^2}-\frac{1}{6}\right)z^2+\left(\frac{7}{360}-\frac{2}{\pi ^4}\right)z^4 +O(z^6)$$ so the Laurent expansion of $f(z)$ over $4<|z|<5$ is given by: $$ f(z) = 2\sum_{m\geq 1}(-1)^{m+1}\frac{\pi^{2m}}{z^{2m}}+(-1)+\left(\frac{2}{\pi^2}-\frac{1}{6}\right)z^2+\left(\frac{7}{360}-\frac{2}{\pi ^4}\right)z^4 +O(z^6).$$