Residue of $\frac{\tan(z)}{z^3}$ What is the easiest way to calculate the residue of $\dfrac{\tan(z)}{z^3}$ at zero? I could either use the line integral theorem, or expand it out as a series. Is there a right way to do it?

You don't have to calculate the series expansion. The $z^{-1}$-coefficient in the series expansion of $\tan(z)/z^3$ is the $z^2$-coefficient in the series expansion of $\tan(z)$. But this is zero since $\tan$ is an odd function.
