= Dirichlet Green function in a quadrant
{c}
{title2=$G_Q(\mathbf r,\mathbf r_0)$}
For the positive Laplacian convention $\Delta G=\delta$, the <method of images> in the first quadrant reflects an interior pole $(a,b)$ across each axis with negative sign and across both with positive sign:
$$
G_Q(x,y;a,b)=\frac1{2\pi}\left[\log|(x,y)-(a,b)|-\log|(x,y)-(-a,b)|-\log|(x,y)-(a,-b)|+\log|(x,y)-(-a,-b)|\right].
$$
The <Dirichlet Green function> vanishes on both axes. Its outward normal derivative on the horizontal axis is
$$
\partial_nG_Q(t,0;a,b)=\frac{4abt}{\pi[(t-a)^2+b^2][(t+a)^2+b^2]},
$$
and the vertical-axis kernel follows by exchange of coordinates. <Green second identity> converts these derivatives into the quadrant <Poisson integral> for boundary data.
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