Zeros, Singularities, Residues: Zeros
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Zero of order $m$: A point $z_{0}$ is called a zero of order (or multiplicity ) $m$ for the function $f$ if $f(z_{0}) = 0$, $f^{(k)}(z_{0}) = 0$ for $k =1$, $2$, $\cdots$, $(m-1)$ and MATH.

A zero of order $1$ is called a simple zero .

Examples: The function $f(z) = z^{m}$ where $m \in \QTR{Bbb}{N}$ has a zero of order $m$ at the point $z = 0$. The function MATH has a simple zero at $z=1$ and a zero of order $5$ at $z =2$. The function $\sin z$ has a simple zero at the points $z = n \pi$ where $n \in \QTR{Bbb}{Z}$. The function MATH has a zero of order $2$ at the points $z = 0$ and ; and it has a simple zero at the points $z = n \pi$ where $n \in \QTR{Bbb}{Z}$ with $n \neq 0, \; 1$.

If a function $f(z)$ is analytic at $z_{0}$ and if $f(z)$ has a zero of order $m$ at $z_{0}$ then the Taylor series of $f$ about the point $z = z_{0}$ takes the form

MATH

 

  
   
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