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January 8th, 2011, 03:25 AM   #1
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Existance of analytic function at C - {0}

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.
vitalik.t is offline  
January 9th, 2011, 10:08 AM   #2
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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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