= Poisson integral on the unit disk
{c}
{title2=$P_r[u](t)$}
= Disk Poisson integral
{synonym}
The disk <Poisson integral> is the <harmonic function> with boundary data $u$, given by
$$
P_r[u](t)=\int_{\mathbb T}\frac{1-r^2}{|e^{2\pi is}-re^{2\pi it}|^2}u(s)\,ds.
$$
It is the real part of the <Schwarz integral on the unit disk> for real data. The nonnegative kernel has integral one; smooth boundary data extend smoothly to the circle. This disk formula should be distinguished from the half-plane formula.
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