In complex analysis, functions are studied that are differentiable (in the complex space) at almost all points. Zeroes and poles are very important concept.

**Zeros** are termed as points at which a function $f,f(z)=0$.

A **Pole** $(f1 (z)=0)$ (also called an *isolated singularity*) is a point where the limit of a complex function inflates dramatically with polynomial growth.

More specifically, a point $z_{0}$ is a pole of a complex valued function $f$ if the function value $f(z)$ tends to infinity as $z$ gets closer to $z_{0}$. If the limit is finite, then $z_{0}$ is not a pole.

$z_{0}$ is a pole of order $n$ if:

$z→z_{0}lim (z−z_{0})⋅f(z)=0$

What is done here is $f(z)$ is multiplied by $(z−z_{0})_{n}$ and then taking limit as $z$ approaches $z_{0}$. If the result is not equal to $0$, then $z_{0}$ is a pole.

### Order of Zeros

let’s take an example of $f(x)=x$ and $g(x)=x_{2}$. It has a zero at $x=0$ and it’s order is $1$ for $f$ and $2$ for $g$, since $g$ has a repeated root. We say order of zeroes for $f$ is 1 and for $g$, it’s 2. for rational functions (can be written as fraction), order of zero of whole function is $ord_{p}(num)−ord_{p}(denom)$. Generally, order of zero of a zero function is termed as infinity.

Similarly, take $f(x)=x1 ;g(x)=x_{2}1 $, order of poles for $f$ at $x=0$ is 1, and $g$ is 2.

Denote order of zero a holomorphic function $f$ at point $p$ by $ord_{p}(f)$, extend the definition to meromorphic functions $f=hg $ where $g,h$ are holomorphic, then $ord_{p}(f)=ord_{p}(g)−ord_{p}(h)$.

- $ord_{p}(f)∈Z∪{∞}$, where $ord_{p}(f)=∞⟺f=0$.
- $ord_{p}(f.g)=ord_{p}(f)+ord_{p}(g)$
- $ord_{p}(f+g)≤min{ord_{p}(f),ord_{p}(g)}$

## Meromorphic Functions

**Holomorphic function** (also known as Analytic function) usually refer to functions that are infinitely differentiable.

Formal Definition:

Let G be an open set in $C$. A function $f:G→C$ is called holomorphic if, at every point $z∈G$, the complex derivative

$f_{′}(z)=h→0lim hf(z+h)−f(z) $

exists as a complex number where $C$ = complex realm.

Meromorphic Function is the ratio of two analytic function which are analytic except for poles.