# Residue at a non isolated essential singularity

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.

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.