Why is it that functions with nonisolated singularities at a point do not have Laurent series at that point?

complex-analysis laurent-series

It's not true. Any function that is analytic on an annulus has a Laurent series about the centre of the annulus that converges on the annulus. Thus $\csc(1/z)$ has Laurent series about $0$ for each of the annuli $1/((n+1)\pi) < |z| < 1/(n \pi)$.

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