Open mapping theorem (complex analysis): Difference between revisions

Revision as of 00:28, 29 December 2014

Revision as of 00:56, 29 December 2014

Line 7:

==Proof==

[[Image:OpenMapping1Openmappingtheorem.png|framethumb|right|Blue500px| Black dots represent zeros of ''g''(''z''). Black spikesannuli 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 circledisk is the boundary of the disk ''B'', centered at ''z''₀.]]

Assume ''f'' : ''U'' → '''C''' is a non-constant holomorphic function and ''U'' is a [[Domain (mathematical analysis)|___domain]] of the complex plane. We have to show that every [[Point (geometry)|point]] in ''f''(''U'') is an [[interior point]] of ''f''(''U''), i.e. that every point in ''f''(''U'') has a neighborhood (open disk) which is also in ''f''(''U'').