harmonic, boundary values

**zelda2139** · March 6th 2010, 12:28 PM

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.

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