= Dirichlet Green function inside the unit disk
{title2=$G_D(z,\zeta)=\frac1{2\pi}\log\left|\frac{z-\zeta}{1-z\bar\zeta}\right|$}
With the convention $\Delta_zG_D=\delta(z-\zeta)$, the displayed <Dirichlet Green function> vanishes on the <unit circle>. Indeed $|z-\zeta|=|1-z\bar\zeta|$ when $|z|=1$. For $\zeta\ne0$, the second logarithm corresponds to an opposite image source at $\zeta/|\zeta|^2$, outside the <unit disc>; it is a <harmonic function> throughout the interior. At $\zeta=0$ the formula becomes $(2\pi)^{-1}\log|z|$. Its outward <normal derivative> in the boundary variable is $(2\pi)^{-1}(1-|\zeta|^2)/|z-\zeta|^2$, the <Poisson kernel> divided by $2\pi$. <Green's second identity> therefore gives the <Poisson integral on the unit disk>.
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