= Holomorphic derivative of a quadrant Dirichlet solution
{title2=$u_z$}
For the bounded <Dirichlet Poisson integral in a quadrant>, the <Wirtinger derivative> is
$$
u_z(z)=\frac{2z}{\pi i}\int_0^\infty r\left[\frac{g_2(r)}{(r^2-z^2)^2}+\frac{g_1(r)}{(r^2+z^2)^2}\right]dr.
$$
This follows by differentiating the <Schwarz integral formula> composed with $z^2$, remembering that for real boundary data $u_z$ is half the derivative of a <holomorphic function> whose <real part> is $u$. For complex boundary data, apply the construction to the real and imaginary parts separately and use linearity. If the data and their <derivatives> decay sufficiently, <integration by parts> gives
$$
u_z(z)=\frac z{\pi i}\left[\int_0^\infty\frac{g_2'(r)}{r^2-z^2}\,dr+\int_0^\infty\frac{g_1'(r)}{r^2+z^2}\,dr\right].
$$
The endpoint terms cancel exactly when $g_1(0)=g_2(0)$. Interior denominators do not vanish because $\operatorname{Im}z^2>0$.
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