Residue at a non isolated essential singularity

Question

For the function $\csc (1/z)$, $z=0$ is a non isolated essential singularity. Does the concept of "residue" extend to the singularity at $0$ for the given function $csc (1/z)$? If so how do I find the residue at $0$ for $\csc (1/z)$?

The function cosec $(1/z)$ does not have a Laurent Series at the non isolated essential singularity $0$ because in a deleted neighborhood of $0$, the function is not analytic. However the function is analytic in the annulus $\{z\in\mathbb C|1<|z|<2\}$ in which it has a Laurent expansion. Can this expansion be used to find residue at 0 ? Will the coefficient of $1/z$ in that expansion give the residue? Or is there any other method of calculating the residue? Or is it true that residue is defined only for isolated singularities? Please clarify.

Answer 1

The concept of residue is defined only for isolated singularities. If an analytic function $f$ has such a singularity at a point $a$, then the residue of $f$ at $a$ is the coefficient of $(z-a)^{-1}$ of the Laurent series whose sum is $f(z)$ near $a$. Here, as you have noticed, $0$ is a non isolated singularity and therefore there is no such Laurent series.