Zeros, Singularities, Residues: Singularites
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Non-Isolated Singular Point: If $f(z)$ is not analytic at $z_{0}$ and if there is no punctured neighborhood MATH such that $f(z)$ is analytic in $D_{r}(z_{0})$ then the point $z_{0}$ is called a non-isolated singular point of $f(z)$.

Examples: We have seen that in the previous example, the point $z = \frac{1}{n}$ is an isolated singular point of MATH if $n$ is a non-zero integer. Whereas the point $z = 0$ is a non-isolated singular point of MATH. For the principal branch MATH of the logarithm function, all non-positive real numbers are non-isolated singular points.

Classification of Isolated Singularity: Suppose that the function $f(z)$ has an isolated singularity at the point $z = z_{0}$. Based on the behaviour of $f(z)$ as $z$ approaches $z_{0}$, the isolated singular point $z_{0}$ is classified into three kinds, namely, removable singularity, pole and essential singularity as follows:

Definition: Let $f(z)$ have an isolated singularity at $z = z_{0}$. Then,
1. the point $z_{0}$ is a removable singularity if MATH where MATH.
2. the point $z_{0}$ is a pole if MATH.
3. the point $z_{0}$ is an essential singularity if it is neither a pole nor a removable singularity.

   
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