= Dirichlet Poisson integral in a quadrant
{c}
The <conformal map> $w=z^2$ sends $x,y>0$ to the <complex upper half-plane>. For bounded continuous boundary values $g_2$ on the horizontal axis and $g_1$ on the vertical axis with equal corner values, set $f(s)=g_2(\sqrt s)$ for $s>0$ and $f(s)=g_1(\sqrt{-s})$ for $s<0$. The bounded <harmonic function> with those <Dirichlet boundary conditions> is
$$
u(z)=\frac1\pi\int_{\mathbb R}\frac{\operatorname{Im}(z^2)f(s)}{|s-z^2|^2}\,ds.
$$
Equivalently, if $z^2=a+ic$ with $c>0$, this is $\frac{2c}{\pi}\int_0^\infty r\left[g_2(r)/((r^2-a)^2+c^2)+g_1(r)/((r^2+a)^2+c^2)\right]dr$. The <Poisson kernel> is a positive <approximate identity>, so the prescribed edge values are attained. Boundedness is essential for uniqueness on this unbounded domain.
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