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Adding local short description: "Theorem on holomorphic functions", overriding Wikidata description "Theorem that holomorphic functions on complex domains are open maps" |
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{{Short description|Theorem on holomorphic functions}}
In [[complex analysis]], the '''open mapping theorem''' states that if
The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the [[real line]], for example, the differentiable function
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.
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==Proof==
[[Image:Openmappingtheorem.png|thumb|right|
Assume
Consider an arbitrary
We know that
The boundary of
Denote by
The image of the ball
== Applications ==
*[[Maximum modulus principle]]
*[[Rouché's theorem]]
*[[Schwarz lemma]]
== See also ==
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