= Mixed Dirichlet-Neumann Green function in a quadrant
{title2=$G=\dfrac1{4\pi}\log\dfrac{[(x-\xi)^2+(y-\eta)^2][(x+\xi)^2+(y-\eta)^2]}{[(x-\xi)^2+(y+\eta)^2][(x+\xi)^2+(y+\eta)^2]}$}
For the quadrant $x,y>0$, the displayed <Green function> satisfies $\Delta G=\delta_{(\xi,\eta)}$, zero value on $y=0$ and zero outward normal <derivative> on $x=0$. The <method of images> uses the logarithmic <fundamental solution of the Laplace equation> with coefficient $1/(2\pi)$: keep the source sign when reflecting across the Neumann side and reverse it when reflecting across the Dirichlet side. The four image signs are therefore $+,+,-,-$ for sources $(\xi,\eta),(-\xi,\eta),(\xi,-\eta),(-\xi,-\eta)$.
On $y=0$, equal numerator and denominator give zero. Evenness in $x$ gives the Neumann condition at $x=0$. The outward normal <derivative> on the Dirichlet side is the positive kernel
$$
\partial_nG(x,0,\xi,\eta)=\frac\eta\pi\left(\frac1{(x-\xi)^2+\eta^2}+\frac1{(x+\xi)^2+\eta^2}\right).
$$
<Green's second identity> consequently gives $u(P)=\int_DGf+\int_{y=0}g\partial_nG$ for $\Delta u=f$ with these mixed boundary data, assuming any remote-boundary contribution vanishes. The sign convention is $\Delta G=+\delta$; changing to $-\Delta G=\delta$ changes both the Green function and the representation signs.
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