Zeros, Singularities, Residues: Singularites
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denote the Laurent series expansion of $f$ about the point $z_{0}$ which is valid in some punctured region MATH for some $r > 0$. Then,

1. $z_{0}$ is a removable singularity if and only if $a_{n} = 0$ for all MATH.

2. $z_{0}$ is a pole of order $m \ge 1$ if and only if $a_{m} \neq 0$ and $a_{n} = 0$ for all $n < - m$.

3. $z_{0}$ is an essential singularity if and only if $a_{n} \neq 0$ for infinitely many $n < 0$.

Example: The Laurent series of MATH about the point $z = 0$ is given by MATH. Since $a_{n} =0$ for all $n < 0$ in the Laurent series of MATH, the function MATH has a removable singularity at $z = 0$.

Example: The Laurent series of MATH about the point $z = 0$ is given by
MATH for MATH. Since $a_{n} = 0$ for all $n < -4$ in the Laurent series of MATH about the point $z = 0$, the function MATH has a pole of order $4$ at $z = 0$.

   
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