I am studying Complex Analysis and I need to solve a question, that is, I want to compute the residue on $z_0=0$ of this funcion $f(z)=\frac{\exp(4z)-1}{\sin^2(z)} $.
Ok, I know that $z_0=0$ is a pole of second order, so $Res[f,0] = \displaystyle \lim_{z \to 0} \frac{1}{1!} \frac{d}{dz} \frac{z^2 (\exp(4z)-1)}{\sin^2 z}$.
Ok, but this limit is very hard..
How To solve that?
The pole is actually first order since the numerator goes like $4z$ as $z\to 0.$ So you can just find $$\lim_{z\to 0}z\frac{e^{4z}-1}{\sin^2(z)}.$$