Open mapping theorem (complex analysis): Difference between revisions

Revision as of 14:39, 13 October 2011

Revision as of 18:37, 23 October 2011

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==Proof==

[[Image: OpenMapping1.png |frame|right|Blue dots represent zeros of ''g''(''z''). Black spikes represent poles. The boundary of the open set ''U'' is given by the dashed line. Note that all poles are exterior to the open set. The smaller red circle is the set ''B'' constructed in the proof.]]

Assume ''f'':''U'' &rarr;→ '''C''' is a non-constant holomorphic function and <math>U</math> is a connected [[open subset]] of the complex plane. We have to show that every [[Point (geometry) | point]] in <math>f(U)</math> is an [[interior point]] of <math>f(U)</math>, i.e. that every point in <math>f(U)</math> is contained in a disk which is contained in <math>f(U)</math>.

Consider an arbitrary <math>w_0</math> in <math>f(U)</math>. Then there exists a point <math>z_0</math> in ''U'' such that <math>w_0 = f(z_0)</math>. Since ''U'' is open, we can find <math>d>0</math> such that the closed disk <math>B</math> around <math>z_0</math> with radius ''d'' is fully contained in ''U''. Since ''U'' is connected and ''f'' is not constant on ''U'', we then know that ''f'' is not constant on ''B'' because of [[Analytic continuation]]. Consider the function <math>g(z)=f(z)-w_0</math>. Note that <math>z_0</math> is a [[root of a function| root]] of the function.

We know that ''g''(''z'') is not constant and holomorphic. The reciprocal of any holomorphic ''g''(''z'') is [[meromorphic]] and has isolated poles. Thus the roots of holomorphic non-constant functions are isolated. In particular, the roots of ''g'' are isolated and by further decreasing the radius of the image disk ''d'', we can assure that ''g''(''z'') has only a single root in ''B''.

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Denote by <math>D</math> the disk around <math>w_0</math> with [[radius]] ''e''. By [[Rouché's theorem]], the function <math>g(z)=f(z)-w_0</math> will have the same number of roots in ''B'' as <math>f(z)-w</math> for any <math>w</math> within a distance <math>e</math> of <math>w_0</math>. Thus, for every <math>w</math> in <math>D</math>, there exists one (and only one) <math>z_1</math> in <math>B</math> so that <math>f(z_1) = w</math>. This means that the disk ''D'' is contained in <math>f(B)</math>.

The image of the ball ''B'', <math>f(B)</math> is a subset of the image of ''U'', <math>f(U)</math>. Thus <math>w_0</math> is an [[interior point]] of the image of an open set by a holomorphic function <math>f(U)</math>. Since <math>w_0</math> was arbitrary in <math>f(U)</math> we know that <math>f(U)</math> is open. Since ''U'' was arbitrary, the function <math>f</math> is open.

== Applications ==