Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 4 / 5G (Methods)

Write $\mu =\cos \theta$. Separation of the axisymmetric Laplace equation in spherical coordinates, with regularity at the poles, gives the Legendre polynomial expansionThe discarded interior terms are singular at the origin; the discarded exterior terms fail the prescribed decay at infinity. Continuity at $r=1$ gives $a_{\ell }=b_{\ell }=c_{\ell }$.

Take the radial derivative jump to mean the exterior derivative minus the interior derivative. ThenSince $1-{\mu }^2=\frac{2}{3}(P_0(\mu )-P_2(\mu ))$, comparison with the prescribed jump gives $c_0=-2A/3$, $c_2=2A/15$, and all other coefficients zero. ThusThese expressions satisfy the Laplace equation, continuity, regularity and decay, and their jump is $A{\sin }^2\theta$. If the opposite convention for the word jump is chosen, both expressions change sign. Although only the axisymmetric expansion is needed, no additional nonaxisymmetric solution can be added: zero jump and continuity glue such an addition into an everywhere regular harmonic function decaying at infinity, which vanishes by the maximum principle for harmonic functions.