 Solution

ID: past-exam-of-the-complex-analysis-conformal-maps-and-geometry-course-of-the-university-of-oxford/2023/3/a/i/solution

The radial harmonic function with value zero at

∣ z ∣ = r

and one at

∣ z ∣ = 1

ω_{A (r, 1)} (z, T) = \frac{lo g ( ∣ z ∣/ r )}{lo g ( 1/ r )} .

Indeed

lo g ∣ z ∣

is harmonic away from zero, and the displayed expression has the required boundary values. Uniqueness of the Dirichlet problem identifies it as the harmonic measure of an annulus.

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