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What type of singularity is that at $z=\pi k+\pi/2$ for any integer $k$ for the function $\phi(z)=e^{\tan z}$? I can see that it is not removable, but I am not sure how to narrow down further. Thank you.

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    That is one way. Here is another: If $f(z)$ has a pole at $w$ then $1/f(z)$ is bounded in a neighbourhood of $w$. (If you haven't seen this one before, ask yourself why?)2012-02-20

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When a meromorphic function $f$ has a pole at $a$, the composition $e^f$ has an essential singularity at $a$. One way to see this is: neither $e^f$ nor $1/e^f=e^{-f}$ are bounded in any neighborhood of $a$. (Pointed out by Harald Hanche-Olsen).