= Two-boundary annular harmonic interpolation
{title2=$R_n(r)=\frac{A_n\sinh(n\log(b/r))+B_n\sinh(n\log(r/a))}{\sinh(n\log(b/a))}$}
For a <harmonic function> on the planar <annulus> $a<r<b$, the radial amplitude of a sine or cosine <Fourier mode> of positive integer index $n$ is a linear combination of $r^n$ and $r^{-n}$. If its values at $a,b$ are $A_n,B_n$, respectively, the displayed hyperbolic-sine formula gives that combination. It follows either by solving the two boundary equations or by putting $s=\log(r/a)$, which changes the radial equation to $R_{ss}-n^2R=0$. The zero angular mode is instead affine in $\log r$:
$$
R_0(r)=A_0\frac{\log(b/r)}{\log(b/a)}+B_0\frac{\log(r/a)}{\log(b/a)}.
$$
Superposing the modes gives the <harmonic function> with suitable prescribed <Dirichlet boundary conditions>. For a fixed interior radius, high angular modes from either boundary are exponentially suppressed. Piecewise smooth boundary data are attained at continuity points, while jumps have the usual mean-value Fourier limit.
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