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How to classify the singularities of $sin(\frac{1}{sin\pi z})$. I think it should be essential singularities but I do not know how to prove it. To prove that it is not a pole nor a removable singularity I need to look at its limit at n, but how to do that?

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Given $n \in \mathbb{Z}$, the limit of your expression as $z \to n$ along the real line (this is a real variable limit) clearly doesn't exist (even in the extended sense) and so each $n \in \mathbb{Z}$ is an essential singularity.

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