Originally Posted by

**Random Variable** I want to calculate the residue of $\displaystyle f(z) = \frac{z}{(1+z^{2}) \sinh \pi z} $ at $\displaystyle z = i $.

$\displaystyle z = i $ is a pole of order 2 since $\displaystyle \sinh( \pi i) = 0 $

So the standard approach is $\displaystyle Res[f,i] = \lim_{z \to i} \frac{d}{dz} (z-i)^{2} \frac{z}{(1+z^{2})\sinh \pi z} = \lim_{z \to i} \frac{d}{dz} (z-i) \frac{z}{(z+i)\sinh \pi z} $

After differentiating and applying L'Hospital's rule once, the limit is still indeterminate and very messy. Is there a better approach than repeated applications of L'Hospitals rule?