= Harmonic Fourier expansions in planar concentric domains
A single-valued <harmonic function> in a planar <annulus> has radial <Fourier modes> $1,\log r$ and $r^n,r^{-n}$ multiplying $\cos(n\theta),\sin(n\theta)$ for integers $n\ge1$. Substitute $R(r)\Theta(\theta)$ in <Laplace's equation>: periodicity gives $\Theta''+n^2\Theta=0$, while $r^2R''+rR'-n^2R=0$. Regularity at the origin excludes $\log r,r^{-n}$; boundedness at infinity excludes $\log r,r^n$. Coefficients must give locally convergent <series>, with the boundary regularity required by the particular <Dirichlet problem>. Mere finiteness at each finite exterior point does not imply boundedness at infinity.
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