Setting...
(1)
... is...
(2)
From (2) we observe that in all z where is and the derivative of does exist and therefore is holomorphic...
Kind regards
I'm going to need the following theorem for a problem in here, and I was wondering if anyone could help me prove it?
Theorem:
For and polynomials, there exists a holomorphic extension to the entire complex plane except along branch-cuts extending from each of the zeros and poles of the expression with a continuous and analytic function in this domain. Additionally, the difference across each branch-cut is determined by the order of the branch point such that traveling in the positive sense around the branch point, this difference is for a zero of order n and for a pole of order m.
I believe I have a proof but not sure it's correct.
Hi Chisigma. I'm extending the principal branch logarithm to the entire complex plane. At each zero and pole, we have to define a branch-cut over which the function is not analytic so it's more than just at the zeros of p and q the function is non-analytic. Looks to me anyway. It's these branch-cuts that I wish to determine how the function value changes across, and I believe that is only dependent on the order of the zeros and poles of p/q.