 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/ib/paper-3/2g/ii/solution

With the usual exterior-boundary interpretation that the harmonic function remains bounded as

r \to \infty

, positive radial powers and

lo g r

are excluded. Hence

ϕ (r, θ) = A_{0} + n = 1 \sum \infty r^{- n} (C_{n} cos n θ + D_{n} sin n θ) .

The series must converge locally for

r > 1

, with any required finite boundary values at

r = 1

r \mapsto 1/ r

If “finite” means only finite at each finite point, without a condition at infinity, growing solutions such as

r cos θ

and

lo g r

are allowed. Under that literal reading the general exterior expansion is

A_{0} + B_{0} lo g r + \sum_{n \geq 1} [(A_{n} r^{n} + C_{n} r^{- n}) cos n θ + (B_{n} r^{n} + D_{n} r^{- n}) sin n θ],

with local convergence. Boundedness at infinity is the condition selecting the decaying exterior expansion.

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