Zeros, Singularities, Residues: Singularites
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The properties of the essential singularity of a function is described in the following theorem:
Theorem: Suppose that $f$ is analytic in MATH for some $r > 0$. The following are equivalent:

1. $f$ has an essential singularity at $z_{0}$.

2. Casorati-Weierstrass Theorem: For each complex number $w$ with possibly one exception, there exists a sequnence $\{ z_{n} \}$ (that depends on $w$) in the region MATH converging to $z_{0}$ such that $f(z_{n}) \to w$ whenever $\{ z_{n} \}$ converges to $z_{0}$.

3. MATH for MATH with $a_{n} \neq 0$ for infinitely many $n < 0$.

The Casorati Weierstrass theorem tells that in any neighborhood of $z_{0}$ the function $f(z)$ comes arbitrarily close to any specified complex number. One can prove (with the available tools) the above said Casorati Weierstrass theorem which is rephrased below:
Casorati-Weierstrass Theorem: Let $f(z)$ have an essential singularity at $z_{0}$. If a complex number $w$ and $\epsilon > 0$ are given, then for each $\delta > 0$, there exists $z$ such that MATH and MATH.

 

   
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