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.