 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 $n\ge 1$. Substitute $R(r)\Theta (\theta )$ in Laplace's equation: periodicity gives ${\Theta }^{\prime \prime }+n^2\Theta =0$, while $r^2{R}^{\prime \prime }+r{R}^{\prime }-n^{2}R=0$. 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.