A finite-valued function $f(r, \theta, \phi)$, where $r, \theta, \phi$ are spherical polar coordinates, satisfies Laplace's equation in the regions $r<1$ and $r>1$, and $f \rightarrow 0$ as $r \rightarrow \infty$. At $r=1, f$ is continuous and its derivative with respect to $r$ is discontinuous by $A \sin ^{2} \theta$, where $A$ is a constant. Write down the general axisymmetric solution for $f$ in the two regions and use the boundary conditions to find $f$.

$$\left[\text { Hint }: \quad P_{2}(\cos \theta)=\frac{1}{2}\left(3 \cos ^{2} \theta-1\right) \cdot\right]$$