
Re: existence of holomorphic log in simplyconnected region not containing {0}
Posted:
Oct 13, 2012 2:52 AM


On Friday, October 12, 2012 11:48:16 PM UTC7, smn wrote: > On Thursday, October 11, 2012 4:37:37 PM UTC7, (unknown) wrote: > > > Hi, All: > > > > > > > > > > > > I'm looking for a proof of the existence of a holomorphic log in > > > > > > > > > > > > a region R that are simplyconnected but do not wind around the origin. > > > > > > > > > > > > My idea is: > > > > > > > > > > > > logz is defined as the integral Int_Gamma dz/z , for Gamma a simpleclosed > > > > > > > > > > > > curve. The log is then welldefined , since, in simplyconnected regions, > > > > > > > > > > > > the integral is independent of path. In addition, 1/z is holomorphic > > > > > > > > > > > > since z=/0 in R . Then the integral is welldefined and holomorphic, > > > > > > > > > > > > (integral of holomorphic function is holomorphic ) , so the log exists. > > > > > > > > > > > > Is this O.K? > > > > > > > > > > > > > > > > > > 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 d/dz(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

