Finding the residue of a mod of a function...

Question

What is the residue of $$f(z)=\frac{1}{|z+c||z-c|}$$ at $z=c$ and $z=-c$. I know to find the residue without mod in the denominator, but I have no idea of finding the residue with a mod in the denominator.

Answer 1

The notion of residue, only makes sense if the function can be expanded in Laurent series of variable $z$: $$ f(z) = \sum_{-\infty}^{\infty} f_n z^{n} $$ Due to the abs symbol $\vert \ldots \vert$, you will have to write the function in terms of both $z, \bar{z}$, residue is not defined.