Zeros, Singularities, Residues: Singularites
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The properties of the pole 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 pole of order $m$ at $z_{0}$.
2. MATH where $g(z)$ is analytic at $z_{0}$ and $g(z_{0}) \neq 0$.
3. $f$ is bounded in a neighborhood of $z_{0}$.
4. MATH.

The properties of the pole 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 pole of order $m$ at $z_{0}$.
2. MATH where $g(z)$ is analytic at $z_{0}$ and $g(z_{0}) \neq 0$.
3. MATH has a zero of order $m$ at $z_{0}$.
4. The function MATH has a removable singularity at $z_{0}$.
5. MATH for MATH. Further, $a_{m} \neq 0$.
6. MATH and MATH for $k > m$.
7. MATH for $k < m$ and in particular MATH.

 
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