= Harmonic matching across a circle with a derivative jump
{title2=$K_n=-j_n/(2n)$}
For a single-valued <harmonic function> tending to zero both at the origin and at infinity, match an angular <Fourier series> across the unit circle. A nonzero mode with continuous interface value $K_n\cos(n\theta)$ has interior field $K_nr^n\cos(n\theta)$ and exterior field $K_nr^{-n}\cos(n\theta)$. Its exterior-minus-interior radial <derivative> is $-2nK_n\cos(n\theta)$. Thus a prescribed jump coefficient $j_n$ determines $K_n=-j_n/(2n)$. Sine modes obey the same relation. The constant mode must have zero jump under these decay conditions.
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