# Complex Analysis/Meromorphic functions and the Riemann sphere

**Definition (singularity)**:

Let be an open subset of , let and finally . Suppose that is unbounded in every neighbourhood of . Then (and only then) is called a **singularity** of .

**Definition (pole)**:

Let be an open subset of , let and finally . Suppose that is a singularity of , but there exists an such that the function

is bounded in a neighbourhood of (whence by the Riemann removability theorem it may be holomorphically continued into the whole of ). Then (and only then) is called a **pole** of .

**Definition (order)**:

Let be an open subset of , let and finally . If is a pole of , the natural number from the definition of a pole is called the **order** of the pole .

**Definition (essential singularity)**:

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

**Definition (meromorphic)**:

Let be an open subset of , let be discrete and let . We call a **meromorphic** function on if and only if at least one of the elements of is a pole of and all elements of are either poles or singularities of .

**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 is normal if and only if for every sequence in , either the sequence or the sequence contains a subsequence that uniformly converges to a holomorphic function.

**Proof:** Suppose first that is normal. Then there exists a constant such that

- .

Let then be a sequence in . Then either is a bounded sequence, or there exists a subsequence of such that for a certain constant . Since

- , we have ,

we may infer from Montel's theorem that is normal, whence it contains a convergent subsequence.

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