 Odd Legendre extension for an equatorial Dirichlet plane

Suppose the exterior of a sphere also has zero Dirichlet boundary conditions on the part of its equatorial plane outside the sphere. The plane separates the upper and lower domains. Extend the sphere data of each hemisphere oddly across the equator, and apply the axisymmetric harmonic extension from a sphere separately to these two odd extensions. Only odd-degree Legendre polynomials occur, and every such polynomial vanishes at $u=\cos \theta =0$. Thus the resulting harmonic function vanishes on the plane. For upper data $f(u)$ on $0\le u\le 1$, the exterior coefficients are $(2\ell +1){\int }_0^{1}f(u)P_{\ell }(u)du$ for odd $\ell$. If $f(0)=0$, the plane and sphere data agree at their shared edge. The same construction with the lower data gives the lower solution; their normal derivatives need not match through a conducting sheet.