$\csch z$ has principal part $1/z$ at $0$, and $\csch z-1/z$ is odd and holomorphic on $\abs{z}<\pi$.Suggest a slogan
The function \(\csch z\coloneqq\frac{1}{\sinh z}\) has a simple pole at \(0\), and for \(0<\abs{z}<\pi\) \[ \csch z=\frac1z-\frac{z}{6}+\frac{7z^{3}}{360}-\frac{31z^{5}}{15120}+\cdots. \]
What is the series expansion of $\csch(z)$?