Meromorphic function

A Meromorphic Function on an open subset of ${\displaystyle \mathbb {C} }$ 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 ${\displaystyle U}$ 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.

Let ${\displaystyle U\subseteq \mathbb {C} }$ be open. A meromorphic function ${\displaystyle f}$ on ${\displaystyle U}$ is a function with a discrete set of singularities ${\displaystyle S\subseteq U}$ with

1. ${\displaystyle f\colon U\setminus S\to \mathbb {C} }$ is holomorphic.
2. ${\displaystyle f}$ has a pole at each point ${\displaystyle s\in S}$.

We say that ${\displaystyle f}$ is meromorphic on ${\displaystyle U}$ and write ${\displaystyle f\in {\mathcal {M}}(U)}$.

Note that a meromorphic function on ${\displaystyle U}$ is not defined on the entire set ${\displaystyle U}$, but only on the complement of a discrete subset.

In the definition of singularities, the following three types of singularities were mentioned:

• Removable singularities,
• Poles (of order ${\displaystyle m}$),
• Essential singularities.

Meromorphic functions may only have poles in the set of singularities ${\displaystyle S}$; they must not have essential singularities.

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

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

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

Let ${\displaystyle f\colon U\to {\hat {\mathbb {C} }}}$ be holomorphic with ${\displaystyle f(z_{0})=\infty }$. If ${\displaystyle f}$ is not constant on any neighborhood of ${\displaystyle z_{0}}$, then by the Identity Theorem, there exists a neighborhood ${\displaystyle V}$ of ${\displaystyle z_{0}}$ such that ${\displaystyle f|_{V\setminus {z_{0}}}\colon V\setminus {z_{0}}\to \mathbb {C} }$. Since ${\displaystyle f}$ is holomorphic at ${\displaystyle z_{0}}$, ${\displaystyle {\frac {1}{f}}}$ has a power series expansion at ${\displaystyle z_{0}}$, say

${\displaystyle {\frac {1}{f(z)}}=\sum _{k=0}^{\infty }a_{k}(z-z_{0})^{k}}$

Let ${\displaystyle m:=\min\{k:a_{k}\neq 0\}}$. Then

${\displaystyle {\frac {1}{f(z)}}=(z-z_{0})^{m}\underbrace {\sum _{k=0}^{\infty }a_{k+m}(z-z_{0})^{k}} _{g(z):=}}$

Here,${\displaystyle g}$ is holomorphic with ${\displaystyle g(z_{0})\neq 0}$, so ${\displaystyle 1/g}$ is holomorphic at ${\displaystyle z_{0}}$. It follows that

${\displaystyle f(z)(z-z_{0})^{m}={\frac {1}{g(z)}},\qquad z\in V\setminus \{z_{0}\}}$

thus, ${\displaystyle f}$ in ${\displaystyle z_{0}}$ has a pole of order ${\displaystyle m}$.

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

• Complex Analysis

This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

• Source: Meromorphe Funktion - URL:

https://de.wikiversity.org/wiki/Meromorphe_Funktion

• Date: 01/07/2024