Revision as of 10:50, 25 October 2010 edit 195.178.216.56 (talk) added link to Russian version of this page ← Previous edit		Revision as of 19:41, 8 May 2011 edit undo 131.111.185.1 (talk) →Proof: script e and roman e were being used interchangably to refer to the same variable. It's a simple mistake to make: using <math>e</math> when the last time you used ''e''. Next edit →
Line 17: The boundary of ''B'' is a circle and hence a [[compact set]], and \|''g''(''z'')\| is a [[continuous function]], so the [[extreme value theorem]] guarantees the existence of a minimum. Let ''e'' be the minimum of \|''g''(''z'')\| for ''z'' on the boundary of ''B'', a positive number. Denote by <math>D</math> the disk around <math>w_0</math> with [[radius]] ~~<math>~~''e~~</math>~~''. 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 ==

Open mapping theorem (complex analysis): Difference between revisions