(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that if z_{0}is an isolated singularity of f(z) that is not removable, then z_{0}is an essential singularity for e^{f(z)}.

2. Relevant equations

z_{0}is a pole of f(z) of order N iff f(z) = g(z)/(z-z_{0})^N, where g is analytic at z_{0}and g(z_{0}) is not 0, iff 1/f(z) is analytic at z_{0}and has a zero of order N, iff |f(z)| → ∞ as z → z_{0}.

Casorati-Weierstrass theorem

3. The attempt at a solution1. The problem statement, all variables and given/known data

I know what to do if z is an essential singularity. But I'm having trouble if z is a pole. It seems that what I must do is find a sequence of points going to z_{0}such that f(z) goes to infinity, and also a sequence of points going to z_{0}such that f(z) approaches some complex number. I don't know how to do that.

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# Essential Singularity

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