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Topic: existence of holomorphic log in simply-connected region not
containing {0}

Replies: 4   Last Post: Oct 13, 2012 12:20 PM

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Stuart M Newberger

Posts: 457
Registered: 1/25/05
Re: existence of holomorphic log in simply-connected region not
containing {0}

Posted: Oct 13, 2012 2:48 AM
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On Thursday, October 11, 2012 4:37:37 PM UTC-7, (unknown) wrote:
> Hi, All:
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> I'm looking for a proof of the existence of a holomorphic log in
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> a region R that are simply-connected but do not wind around the origin.
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> My idea is:
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> logz is defined as the integral Int_Gamma dz/z , for Gamma a simple-closed
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> curve. The log is then well-defined , since, in simply-connected regions,
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> the integral is independent of path. In addition, 1/z is holomorphic
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> since z=/0 in R . Then the integral is well-defined and holomorphic,
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> (integral of holomorphic function is holomorphic ) , so the log exists.
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> Is this O.K?
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> that


As George said ,in your region where where the Cauchy theorem holds the function 1/z z|=0 has a primitive f(z) ,that means a function with f'(z)=1/z in the region. If a is in the region, then by adding a constant to f, arrange that exp(f(a)=a Now show that d/dz( (1/z)exp(f(z))) =0 in the region so the indicated function is a constant which is 1 since that is its value at a.
Regards,smn



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