Complex Analysis/Meromorphic functions and the Riemann sphere

Definition (singularity):

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

Definition (pole):

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

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

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

Definition (order):

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

Definition (essential singularity):

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

Definition (meromorphic):

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

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

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

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

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

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

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

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