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January 8th, 2011, 03:25 AM  #1 
Newbie Joined: Jan 2011 Posts: 2 Thanks: 0  Existance of analytic function at C  {0}
Problem: Is there any analytic function that is bounded and non constant at domain of C  {0}? If the answer is positive, I need an example of the function. If the answer is negative, I need a proof. 
January 9th, 2011, 10:08 AM  #2 
Newbie Joined: Dec 2010 Posts: 18 Thanks: 0  Re: Existance of analytic function at C  {0}
Assume f is a bounded analytic function on C{0}. Due to a theorem of Riemann on removable singularities, you can extend f analytically onto C where it is still bounded, hence constant due to Liouvilles Theorem. So the answer to your question is "No"!


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analytic, existance, function 
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