 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2/2/b/solution

The Fourier orthogonality relation on this angular interval is

\int_{0}^{2 π α} sin \frac{n θ}{2 α} sin \frac{m θ}{2 α} d θ = π α δ_{nm} .

Projecting the outer boundary condition onto each angular eigenfunction gives

A_{n} = \frac{R ^{- n / (2 α)}}{π α} \int_{0}^{2 π α} φ (θ) sin \frac{n θ}{2 α} d θ .

The solution is unique in the class of bounded harmonic functions with the prescribed boundary trace. For continuous boundary data, the difference of two solutions is zero on the boundary and bounded at the vertex, so the maximum principle for harmonic functions makes it zero throughout the sector. Equivalently all its Fourier sine series coefficients vanish. The usual convergence interpretation applies if the data are only piecewise smooth.

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