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.