# Complex Analysis/Meromorphic functions and the Riemann sphere

Definition (singularity):

Let ${\displaystyle U}$ be an open subset of ${\displaystyle \mathbb {C} }$, let ${\displaystyle z_{0}\in U}$ and finally ${\displaystyle f\in H(U\setminus \{z_{0}\})}$. Suppose that ${\displaystyle f}$ is unbounded in every neighbourhood of ${\displaystyle z_{0}}$. Then (and only then) ${\displaystyle z_{0}}$ is called a singularity of ${\displaystyle f}$.

Definition (pole):

Let ${\displaystyle U}$ be an open subset of ${\displaystyle \mathbb {C} }$, let ${\displaystyle z_{0}\in U}$ and finally ${\displaystyle f\in H(U\setminus \{z_{0}\})}$. Suppose that ${\displaystyle z_{0}}$ is a singularity of ${\displaystyle f}$, but there exists an ${\displaystyle n\in \mathbb {N} }$ such that the function

${\displaystyle U\ni z\mapsto (z-z_{0})^{n}f(z)}$

is bounded in a neighbourhood of ${\displaystyle z_{0}}$ (whence by the Riemann removability theorem it may be holomorphically continued into the whole of ${\displaystyle U}$). Then (and only then) ${\displaystyle z_{0}}$ is called a pole of ${\displaystyle f}$.

Definition (order):

Let ${\displaystyle U}$ be an open subset of ${\displaystyle \mathbb {C} }$, let ${\displaystyle z_{0}\in U}$ and finally ${\displaystyle f\in H(U\setminus \{z_{0}\})}$. If ${\displaystyle z_{0}}$ is a pole of ${\displaystyle f}$, the natural number ${\displaystyle n}$ from the definition of a pole is called the order of the pole ${\displaystyle z_{0}}$.

Definition (essential singularity):

An essential singularity is a singularity which is not a pole.

Definition (meromorphic):

Let ${\displaystyle U}$ be an open subset of ${\displaystyle \mathbb {C} }$, let ${\displaystyle E\subset U}$ be discrete and let ${\displaystyle f\in H(U\setminus E)}$. We call ${\displaystyle f}$ a meromorphic function on ${\displaystyle U}$ if and only if at least one of the elements of ${\displaystyle E}$ is a pole of ${\displaystyle f}$ and all elements of ${\displaystyle E}$ are either poles or singularities of ${\displaystyle f}$.

Theorem (existence and uniqueness of the Laurent expansion in a punctuated ball):

Theorem (existence and uniqueness of the Laurent decomposition in a punctuated ball):

Theorem (existence and uniqueness of the Laurent decomposition in an open set):

Theorem (Marty's theorem):

A family of functions ${\displaystyle {\mathcal {F}}:\mathbb {C} \to S^{2}}$ is normal if and only if for every sequence ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ in ${\displaystyle {\mathcal {F}}}$, either the sequence ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ or the sequence ${\displaystyle (1/f_{n})_{n\in \mathbb {N} }}$ contains a subsequence that uniformly converges to a holomorphic function.

Proof: Suppose first that ${\displaystyle {\mathcal {F}}}$ is normal. Then there exists a constant ${\displaystyle M>0}$ such that

${\displaystyle \forall f\in {\mathcal {F}}:\forall z\in \mathbb {C} :{\frac {|f'(z)|}{1+|f(z)|^{2}}}.

Let then ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ be a sequence in ${\displaystyle {\mathcal {F}}}$. Then either ${\displaystyle (|f_{n}'(z)|)_{n\in \mathbb {N} }}$ is a bounded sequence, or there exists a subsequence ${\displaystyle (f_{n_{k}})_{k\in \mathbb {N} }}$ of ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ such that ${\displaystyle \forall k\in \mathbb {N} :|f_{n_{k}}(z)|>C}$ for a certain constant ${\displaystyle C>0}$. Since

${\displaystyle {\frac {d}{dz}}\left({\frac {1}{f(z)}}\right)=-{\frac {f'(z)}{f(z)^{2}}}}$, we have ${\displaystyle \left|{\frac {d}{dz}}\left({\frac {1}{f(z)}}\right)\right|\leq \left({\frac {1}{C}}+1\right)M}$,

we may infer from Montel's theorem that ${\displaystyle (1/f_{n_{k}})_{k\in \mathbb {N} }}$ is normal, whence it contains a convergent subsequence.

The opposite direction follows immediately from Montel's theorem and the symmetry of the spherical derivative. ${\displaystyle \Box }$