Zeros, Singularities, Residues: Singularites
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Example: The Laurent series of $\exp(1/z)$ about the point $z = 0$ is given by MATH. Since $a_{n} \neq 0$ for infinitely many $n < 0$ in the Laurent series of $\exp(1/z)$ about the point $z =0$, the point $z = 0$ is an essential singularity of $\exp(1/z)$.

The properties of the removable 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 a removable singularity at $z_{0}$.
2. MATH for MATH.
3. MATH exists and is finite.
4. $f$ is bounded in a neighborhood of $z_{0}$.
5. MATH.

   
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