Zeros, Singularities, Residues: Singularites
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Singular Point: A point $z_{0}$ is said to be a singular point of a function $f(z)$ if $f(z)$ is not analytic at $z_{0}$ and every neighborhood $N(z_{0})$ of the point $z_{0}$ contains at least one point at which $f(z)$ is analytic.
We say it as the function $f(z)$ has a singularity at $z = z_{0}$.

Examples: The point $z = 1$ is a singular point of MATH. The point $z = n \pi$ where $n \in \QTR{Bbb}{Z}$ is a singular point of $f(z) = 1 / \sin z$. The points $z = \frac{1}{n}$ for $n \in \QTR{Bbb}{Z}$ are singular points of MATH.

Isolated Singular Point: If $f(z)$ is not analytic at $z_{0}$ and $f(z)$ is analytic in the punctured neighborhood MATH for some $r > 0$ then the point $z_{0}$ is called an isolated singular point of $f(z)$.

Examples: The points $z = 1$ and $z =2$ are isolated singular points of MATH. The point $z = 0$ is an isolated singular point of the function $f(z) = \exp(1/z)$. Similarly, the function MATH has an isolated singularity at $z = 0$. The point $z = \frac{1}{n}$ is an isolated singular point of MATH if $n$ is a non-zero integer

 
 
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