= Bounded exterior disk Green function
{title2=$G(x,y)=\frac1{2\pi}\log\frac{|x-y|}{|y|\,|x-y/|y|^2|}$}
For the exterior of the unit disc in two dimensions, with the sign convention $\Delta_xG=+\delta(x-y)$, this is the <Dirichlet Green function> that vanishes on the circle and is bounded at infinity. The <method of images> places an opposite logarithmic source at the inverse point $y/|y|^2$. The constant $-(2\pi)^{-1}\log|y|$ enforces the zero boundary value. The <inversion in a circle> identity $|x-y|=|y|\,|x-y/|y|^2|$ for $|x|=1$ verifies that condition directly.
The outward normal of the exterior domain is $-x$ on the unit circle. Its <normal derivative> gives the positive exterior <Poisson kernel>
$$
\partial_nG(x,y)=\frac{|y|^2-1}{2\pi|x-y|^2},\qquad |x|=1<|y|.
$$
For bounded <harmonic functions> this kernel integrates boundary data to the interior value. Boundedness at infinity is essential to uniqueness: without it, a multiple of $\log|y|$ can be added without changing the circle's <Dirichlet boundary condition>.
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