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Thread: harmonic, boundary values

  1. #1
    Nov 2008

    harmonic, boundary values

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

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

    is harmonic on the disk $\displaystyle \{ |z|<R \}$ and has boundary values $\displaystyle h(Re^{i \theta})$ 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 01:44 PM.
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