Open mapping theorem (complex analysis)

In complex analysis, the open mapping theorem states that if

is a domain of the complex plane

is a non-constant holomorphic function, then

is an open map (i.e. it sends open subsets of

to open subsets of

The open mapping theorem points to the sharp difference between holomorphy and real-differentiability.

On the real line, for example, the differentiable function

is not an open map, as the image of the open interval

is the half-open interval

The theorem for example implies that a non-constant holomorphic function cannot map an open disk onto a portion of any line embedded in the complex plane.

Images of holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1.

is a non-constant holomorphic function and

is a domain of the complex plane.

We have to show that every point in

is an interior point of

has a neighborhood (open disk) which is also in

is open, we can find

is fully contained in

is a root of the function.

is non-constant and holomorphic.

are isolated by the identity theorem, and by further decreasing the radius of the disk

has only a single root in

(although this single root may have multiplicity greater than 1).

is a circle and hence a compact set, on which

is a positive continuous function, so the extreme value theorem guarantees the existence of a positive minimum

the open disk around

By Rouché's theorem, the function

will have the same number of roots (counted with multiplicity) in

The image of the ball

is a subset of the image of

Black dots represent zeros of . Black annuli represent poles. The boundary of the open set is given by the dashed line. Note that all poles are exterior to the open set. The smaller red disk is , centered at .