Zeroes And Poles

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 $(\frac{1}{f} (z) = 0)$ (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:

$z \to z_{0} lim (z - z_{0}) \cdot f (z) \neq = 0$

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 $or d_{p} (n u m) - or d_{p} (d e n o m)$ . Generally, order of zero of a zero function is termed as infinity.

Similarly, take $f (x) = \frac{1}{x}; g (x) = \frac{1}{x ^{2}}$ , 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 $or d_{p} (f)$ , extend the definition to meromorphic functions $f = \frac{g}{h}$ where $g, h$ are holomorphic, then $or d_{p} (f) = or d_{p} (g) - or d_{p} (h)$ .

$or d_{p} (f) \in Z \cup {\infty}$ , where $or d_{p} (f) = \infty ⟺ f = 0$ .
$or d_{p} (f . g) = or d_{p} (f) + or d_{p} (g)$
$or d_{p} (f + g) \leq min {or d_{p} (f), or d_{p} (g)}$

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 $f : G \to C$ is called holomorphic if, at every point $z \in G$ , the complex derivative

$f^{'} (z) = h \to 0 lim \frac{f ( z + h ) - f ( z )}{h}$

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

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

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