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Topic: Can a branch point of finite order be turned into a pole?
Replies: 2   Last Post: Apr 15, 2009 4:26 PM

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Posts: 49
Registered: 6/3/05
Can a branch point of finite order be turned into a pole?
Posted: Apr 15, 2009 1:07 AM
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Consider a function F : C->C, where C is the field of complex numbers.
If F is holomorphic near every point except for 0, and F has a branch
point of order N at 0. Then I can show that G(z)=F(z^m) is holomorphic
in C\{0}.

Now my question is can we also show that if G has a singularity at 0,
it must be either removable or a pole. I can show this via Big
Picard's theorem when F is an algebraic function. But can we show this
in general? If so what's the trick? Thank you.

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