 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2/3/b/solution

The Dirichlet Green function is the solution, with the source point held fixed, of

Δ_{r} G (r; r^{'}) = δ (r - r^{'}) in S, G (r; r^{'}) = 0 on C .

Its singularity is the logarithmic one found in (a); subtracting that fundamental solution leaves a harmonic function. An unbounded region also requires a condition at infinity, here boundedness.

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