My Math Forum Existance of analytic function at C - {0}

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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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