Basically, what is the best method to calculate residues, specifically, something like this: \begin{equation*} f(z)=\frac{1+z}{1-\cos(z)}. \end{equation*}
For simple poles, I can just use L'Hopital by using the fact that the Larent series will only have one negative power term in it (the ^-1 term). But if there are more poles, like a double pole or an essential singularity, how do I calculate the residue then? Thank you
Calculate the order of the pole: 2 in this case. Write the Laurent series of $f$: $$f(z) = \sum_{n=-2}^\infty a_n z^n.$$ Now, $$1 + z = f(z)(1 - \cos z) = (a_{-2}z^{-2} + a_{-1}z^{-1} + a_0 + \cdots)(\frac{z^2}2 + \cdots)$$ Write the first terms of the product and... can you continue?