Zeros, Singularities, Residues: Singularites
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Examples: The function MATH has a removable singularity at $z = 0$. The function MATH has a pole at $z = 1$. The function $f(z) = \exp(1/z)$ has an essential singularity at $z = 0$.

Note 1: Suppose that $f(z)$ has a removable singularity at $z = z_{0}$. Since $z_{0}$ is an isolated singularity, there exists a punctured neighborhood MATH such that $f(z)$ is analytic in $D_{r}(z_{0})$. Set $g(z) = f(z)$ for $z \in D_{r}(z_{0})$ and MATH. Then, $g(z)$ will become analytic in the neighborhood MATH and in particular it is analytic at $z_{0}$. Thus, if $f(z)$ has a removable singularity at $z = z_{0}$, then we can redefine the function $f(z)$ suitably at the point $z_{0}$ so that $f(z)$ is analytic at $z = z_{0}$. That is why, it is called as a removable singularity and it is actually not considered as a singular point of $f(z)$.

Note 2: A point $z_{0}$ is a pole of $f(z)$ if and only if $z_{0}$ is a zero of $1/f(z)$. We say that the point $z_{0}$ is a pole of order $m$ of $f(z)$ if and only if it is a zero of order $m$ of $1/f(z)$. Therefore, if $f(z)$ has a pole of order $m$ at $z_{0}$ then it can be written as MATH where $g(z)$ is analytic at $z = z_{0}$. A pole of order $1$ is called the simple pole of $f(z)$.

   
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