Residue, singularity and range where function is holomorphic? Find residue, singularity (type) and (maximal) range where the given function is holomorphic?
$f(z)=\frac{z}{sin z}$
Function has singularity in $z_0=0$ (removable). But how do I find residue and range?
I tried to find residue with this formula but can't finish.
$Res(f;0)=\frac{1}{(m-1)!}\lim_{z \to 0} \frac{d^{m-1}}{dz^{m-1}}(z-z_0)f(z) =\lim_{z \to 0} \frac{z^2}{sin z}$ 

That is not the only singularity, there are others at the real multiples of $\pi$. To find the residue is the same for all the poles, LHopitals rule holds for complex limits as well, so use that.
For example, to compute the limit you found, $$\lim_{z\to 0} \frac{z^2}{\sin(z)}=\lim_{z\to 0} \frac{2z}{\cos(z)}=0$$
