= Solution
Write $\mu=\cos\theta$. Separation of the axisymmetric <Laplace equation> in <spherical coordinates>, with regularity at the poles, gives the <Legendre polynomial> expansion
$$
f_-(r,\theta)=\sum_{\ell=0}^\infty a_\ell r^\ell P_\ell(\mu),\qquad
f_+(r,\theta)=\sum_{\ell=0}^\infty b_\ell r^{-\ell-1}P_\ell(\mu).
$$
The 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. Then
$$
\partial_r f_+-\partial_r f_-=-\sum_{\ell\geq0}(2\ell+1)c_\ell P_\ell(\mu).
$$
Since $1-\mu^2=\frac23(P_0(\mu)-P_2(\mu))$, comparison with the prescribed jump gives $c_0=-2A/3$, $c_2=2A/15$, and all other coefficients zero. Thus
$$
\boxed{f_-(r,\theta)=-\frac{2A}{3}+\frac{2A}{15}r^2P_2(\cos\theta),\qquad
f_+(r,\theta)=-\frac{2A}{3r}+\frac{2A}{15r^3}P_2(\cos\theta).}
$$
These 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>.
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