# Meromorphic function

A Meromorphic Function on an open subset of $\mathbb {C}$ is, in the field of Complex Analysis, a function that is holomorphic except at Isolated singularity. These poles must be isolated. The set of all meromorphic functions on a subdomain $U$ of the complex plane has an advantage over the set of holomorphic functions: it forms not only a ring but also a field. In fact, it can be shown that it is the field of fractions of the ring of holomorphic functions.

## Definition

Let $U\subseteq \mathbb {C}$ be open. A meromorphic function $f$ on $U$ is a function with a discrete set of singularities $S\subseteq U$ with
* $f\colon U\setminus S\to \mathbb {C}$ is holomorphic.
* $f$ has a pole at each point $s\in S$.
We say that $f$ is meromorphic on $U$ and write $f\in {\mathcal {M}}(U)$.
Note that a meromorphic function on $U$ is not defined on the entire set $U$, but only on the complement of a discrete subset.

## Remark

In the definition of singularities, the following three types of singularities were mentioned:
* Removable singularities,
* Poles (of order $m$),
* Essential singularities.
Meromorphic functions may only have poles in the set of singularities $S$; they must not have essential singularities.

## Properties

* The sum, difference, and product of two functions $f,g\in {\mathcal {M}}(U)$ are again meromorphic, so ${\mathcal {M}}(U)$ is an algebra over the ring of holomorphic functions on $U$.Let $S$ be the set of poles of $f$ and $T$ the set of poles of $g$. Then $S\cup T$ is a discrete subset of $U$, and $f+g,f-g,fg$ are holomorphic on $U\setminus (S\cup T)$, with removable singularities or poles on $S\cup T$.
* If $U$ is connected, $f,g\in {\mathcal {M}}(U)$, and $g\neq 0$, then $f/g$ is meromorphic on $U$. In this case, ${\mathcal {M}}(U)$ is a field.Let $S$ be the set of poles of $f$ and $T$ the set of poles of $g$. Since $U\setminus T$ is a domain, the zero set $N$ of $g$ is a discrete subset of $U$, by the Identity Theorem. Now $f/g\colon U\setminus (S\cup T\cup N)\to \mathbb {C}$ is holomorphic, on $S\cup T\cup N$ has $f/g$ removable singularities or poles.
* Locally, every meromorphic function is a quotient of two holomorphic functions. That is, if $f\in {\mathcal {M}}(U)$ and $z_{0}\in U$, there exist a neighborhood $V$ of $z_{0}$ and holomorphic functions $p,q\colon V\to \mathbb {C}$ such that $f=p/q$.A deeper result shows that for domains $U$, such a representation is globally always possible. In this case, the field of meromorphic functions is the field of fractions of the ring of holomorphic functions on $U$.

## Equivalent Description as Holomorphic Functions with Values in ${\hat {\mathbb {C} }}$

Another way to describe meromorphic functions on an open set $U\subseteq \mathbb {C}$ is to define them as holomorphic functions with values in the Riemann sphere.

### Definition

Let $U\subseteq \mathbb {C}$. A function $f\colon U\to {\hat {\mathbb {C} }}$ is called "holomorphic" at a point $z_{0}\in U$ with $f(z_{0})\neq \infty$ if there exists a neighborhood $V$ of $z_{0}$ such that $f(V)\subseteq \mathbb {C}$ and $f|_{V}\colon V\to \mathbb {C}$ is holomorphic at $z_{0}$.$f$ is called "holomorphic" at a point $z_{0}$ with $f(z_{0})=\infty$ if ${\frac {1}{f}}$ is holomorphic at $z_{0}$ in the above sense.

### Poles and Points at Infinity

Let $f\colon U\to {\hat {\mathbb {C} }}$ be holomorphic with $f(z_{0})=\infty$. If $f$ is not constant on any neighborhood of $z_{0}$, then by the Identity Theorem, there exists a neighborhood $V$ of $z_{0}$ such that $f|_{V\setminus {z_{0}}}\colon V\setminus {z_{0}}\to \mathbb {C}$. Since $f$ is holomorphic at $z_{0}$, ${\frac {1}{f}}$ has a power series expansion at $z_{0}$, say
${\frac {1}{f(z)}}=\sum _{k=0}^{\infty }a_{k}(z-z_{0})^{k}$
Let $m:=\min\{k:a_{k}\neq 0\}$. Then
${\frac {1}{f(z)}}=(z-z_{0})^{m}\underbrace {\sum _{k=0}^{\infty }a_{k+m}(z-z_{0})^{k}} _{g(z):=}$
Here,$g$ is holomorphic with $g(z_{0})\neq 0$, so $1/g$ is holomorphic at $z_{0}$. It follows that
$f(z)(z-z_{0})^{m}={\frac {1}{g(z)}},\qquad z\in V\setminus \{z_{0}\}$
thus, $f$ in $z_{0}$ has a pole of order $m$.

### Characterization of Meromorphic Functions

Since poles can be described as points at infinity, we have: A meromorphic function $f$ on $U\subseteq \mathbb {C}$ is a holomorphic function $f\colon U\to {\hat {\mathbb {C} }}$ whose points at infinity do not accumulate (equivalently, $f$ is not constantly equal to $\infty$ on any component of $U$).