<s>[INST] Calculation of a Residue Does anyone know of a good way to calculate the residue at zero of the following function? I was able to calculate it with the higher order pole formula for residues and then used Mathematica to find the limit.  Really just looking for a nice trick without having to get too dirty.
$$ f(z)=\frac{1}{z^2(e^{-z}-1)}  $$
Thank you for any help.
By the way, the answer is $\frac{-1}{12}$
 [/INST] $$\begin{eqnarray}
\frac{1}{z^2(e^{-z}-1)} 
&=& \frac{1}{z^2\left(1-z+\frac{z^2}{2}-\frac{z^3}{6} +\ldots - 1\right)}  \\
&=& -\frac{1}{z^3\left(1-\left(\frac{z}{2}-\frac{z^2}{6}+\ldots\right)\right)} \\
&=& -\frac{1}{z^3}\left(1+\left(\frac{z}{2}-\frac{z^2}{6}+\ldots\right)+\left(\frac{z}{2}-\frac{z^2}{6}+\ldots\right)^2\right) \\
&=& -\frac{1}{z^3}\left(1+\frac{z}{2}+\frac{z^2}{12}+\ldots\right) \\
&=& -\frac{1}{z^3} - \frac{1}{2z^2} - \frac{1}{12 z} + \ldots 
\end{eqnarray}$$
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