= Exterior harmonic extension of disk boundary data
{title2=$u(r,\theta)=a_0+\sum_{n\geq1}(R/r)^n[a_n\cos(n\theta)+b_n\sin(n\theta)]$}
For bounded piecewise continuous <Dirichlet boundary data> $f$ on a circle of radius $R$, write its <Fourier series> as $a_0+\sum_{n\geq1}[a_n\cos(n\theta)+b_n\sin(n\theta)]$. The displayed series gives the bounded exterior <harmonic extension> for $r>R$. The radial <inversion in a circle> $r\mapsto R^2/r$ converts it into the ordinary interior <Poisson kernel> solution. Values at discontinuities are understood through the usual boundary trace, with midpoint radial limits at jumps; changing the assigned values at finitely many boundary points does not change the extension.
The limit at infinity is forced to equal the angular mean $a_0=(2\pi)^{-1}\int_{-\pi}^{\pi}f(\theta)d\theta$. Indeed the angular mean of any exterior <harmonic function> solves $(r\overline u')'=0$, and boundedness excludes its logarithmic term. Equivalently, inversion makes infinity an isolated removable singularity of a bounded planar <harmonic function>. Uniqueness follows from interior uniqueness after inversion, including the <logarithmic cutoff proof of harmonic uniqueness with finitely many boundary exceptions> when needed.
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