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Dirichlet problem on an annulus for one Fourier mode

The harmonic function on

a < r < b

with boundary values

u (a, θ) = 0

and

u (b, θ) = cos (n θ)

is

u (r, θ) = \frac{b ^{n} ( r ^{2 n} - a ^{2 n} )}{r ^{n} ( b ^{2 n} - a ^{2 n} )} cos (n θ) .

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