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Math Help - Harmonic Functions

  1. #1
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    Harmonic Functions

    Show by example that a harmonic function need not have an analytic completion in a multiply connected domain. [HINT: Consider ln(|z|), z a complex number]

    well I considered u=ln(x^2+y^2) where z=x+iy

    and I figured out it was harmonic (second partial derivitives are 0) but I do not know where to go from there.
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  2. #2
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    what is the definition of analytic completion?
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  3. #3
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    Quote Originally Posted by xxp9 View Post
    what is the definition of analytic completion?
    Analytic completion for a function u is when there exists a harmonic function v in a simply connected domain such that u+iv is analytic.
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  4. #4
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    So let u=ln|z|, u is hormonic in the domain of punctured plane R^2 - {0}.
    Since an analytic function is determined in any open sub-set of the plane the analytic completion is unique( up to a constant), if it exists.
    So the only possible analytic completion for u would be f=lnz=u + i argz
    While f can only be defined on a plane where a half line is cut.
    Last edited by xxp9; March 8th 2011 at 05:26 PM.
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