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How to check whether the following functions can be holomorphically extended to $0$: $z\cot(z)$, $z/e^{z}-1$, $z^{2}\sin(1/z)$.

I just know if that a function is bounded on the neighbourhood of $0$ then it can be, but I do not how to check whether it is bounded or not.

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One possible way to do it would be to work out the Laurent series expansion. If it has no negative power terms, then the function can be holomorphically extended to 0; if a negative power term appears, it can't be. To your second question - one way to check whether a function is bounded at a neighborhood of 0: Does a limit exist as $z \to 0$? If it does, the function must be bounded near 0. – Yoni Rozenshein Oct 8 '12 at 15:48

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