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Thread: harmonic, boundary values

  1. #1
    Nov 2008

    harmonic, boundary values

    Let R >0, and let h(Re^{i \theta}) be a continuous function on the circle \{ |z|=R \}. Show that the function

    \tilde{h}(z)=\int^{\pi}_{-\pi} \frac{R^2-r^2}{R^2+r^2-2rR\cos(\theta - \phi)}h(Re^{i \phi}) \frac{d \phi}{2 \pi}, |z|<R,

    is harmonic on the disk \{ |z|<R \} and has boundary values h(Re^{i \theta}) on the boundary circle.

    I am not sure how to show that this is harmonic on that disk. In this section, it talks about Poisson Integral Formula and the Poisson Integral. I am not sure if this somehow applies here or how to apply them. Thanks in advance.
    Last edited by zelda2139; Mar 7th 2010 at 02:44 PM.
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