Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 2 / b

Substitute $U=R(r)F(\theta )$ into the axisymmetric Laplace equation and multiply by $r^2/(RF)$. The radial and angular terms must be separate constants:ThusWith $x=\cos \theta$, the angular equation becomesRequiring boundedness on the entire polar interval, and using the supplied characterization of bounded Legendre polynomial solutions, gives $\lambda =n(n+1)$ and $F\propto P_n(\cos \theta )$ for $n=0,1,2,\dots$. Trying $R=r^s$ gives $s(s+1)=n(n+1)$, hence $s=n$ or $s=-n-1$.

The globally angular-regular separated modes and their superposition are thereforeCoefficients and convergence are determined by the radial domain and boundary data. A domain containing the origin excludes negative radial powers for a regular solution. If the angular domain excludes a polar axis of the spherical coordinate system and singular angular behavior is allowed, additional Legendre-function branches are possible; the displayed family is the one regular over the complete sphere, as intended by the hint.

The far field is proportional to $r{P}_1(\cos \theta )$, since $P_1(\cos \theta )=\cos \theta$. After matching the growing terms, write the exterior solution asAn added constant has here been excluded by interpreting the far-field condition as $U-v_{0}r\cos \theta \to 0$. Differentiate and impose the radial Neumann boundary condition:The Orthogonality of Legendre polynomials makes each coefficient vanish separately. Thus $B_n=0$ for $n\ne 1$, and $2B_1/r_0^3=v_0$. ConsequentlyIndeed ${\partial }_{r}U=v_0(1-r_0^3/r^3)\cos \theta$, which vanishes at the boundary, while the correction decays at infinity. If the far-field condition is intended to specify only the leading growing term, an arbitrary additive constant is also allowed; its gradient has no effect on the Neumann boundary condition.