Harmonic matching across a circle with a derivative jump ($K_n=-j_n/(2n)$)

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 $K_n{r}^n\cos (n\theta )$ and exterior field $K_n{r}^{-n}\cos (n\theta )$. Its exterior-minus-interior radial derivative is $-2n{K}_n\cos (n\theta )$. 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.