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Math Help - Need proof of theorem about log branch-cuts

  1. #1
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    Need proof of theorem about log branch-cuts

    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 p and q polynomials, there exists a holomorphic extension \text{Log}_h(p/q)=\ln|p/q|+i\arg(p/q) to the entire complex plane except along branch-cuts extending from each of the zeros and poles of the expression p/q with \arg(p/q) 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 2n\pi i for a zero of order n and -2m\pi i for a pole of order m.

    I believe I have a proof but not sure it's correct.
    Last edited by shawsend; June 8th 2009 at 07:16 AM. Reason: not sure about proof I have
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  2. #2
    MHF Contributor chisigma's Avatar
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    Setting...

    \lambda (z) = \ln \frac{p(z)}{q(z)} = \ln p(z) - \ln q(z) (1)

    ... is...

    \lambda^{'} (z) = \frac{p^{'}(z)}{p(z)} - \frac{q^{'}(z)}{q(z)} (2)

    From (2) we observe that in all z where is p(z) \ne 0 and q(z) \ne 0 the derivative of \lambda (*) does exist and therefore \lambda (*) is holomorphic...

    Kind regards

    \chi \sigma
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  3. #3
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    Quote Originally Posted by chisigma View Post
    Setting...

    \lambda (z) = \ln \frac{p(z)}{q(z)} = \ln p(z) - \ln q(z) (1)

    ... is...

    \lambda^{'} (z) = \frac{p^{'}(z)}{p(z)} - \frac{q^{'}(z)}{q(z)} (2)

    From (2) we observe that in all z where is p(z) \ne 0 and q(z) \ne 0 the derivative of \lambda (*) does exist and therefore \lambda (*) is holomorphic...

    Kind regards

    \chi \sigma
    Hi Chisigma. I'm extending the principal branch logarithm \text{Log} 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.
    Last edited by shawsend; June 8th 2009 at 05:34 AM.
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