Complex Analysis/Meromorphic functions and the Riemann sphere

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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.

Exercises[edit]