Holomorphic derivative of a quadrant Dirichlet solution ($u_z$)

For the bounded Dirichlet Poisson integral in a quadrant, the Wirtinger derivative isThis 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 givesThe endpoint terms cancel exactly when $g_1(0)=g_2(0)$. Interior denominators do not vanish because $\mathrm{Im}{z}^2>0$.