Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 334 / 2 / a / ii

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- Solution ii

Solution

The boundary data are

U sgn (sin θ)

, whose Fourier series is

sgn (sin θ) = \frac{4}{π} \sum_{n \geq 1 n odd} \frac{s i n ( n θ )}{n} .

Regular harmonic modes in a disk are

(r / R)^{n} sin (n θ)

and

(r / R)^{n} cos (n θ)

. Matching the odd boundary data gives

u_{z} (r, θ) = \frac{4 U}{π} n \geq 1 n odd \sum \frac{1}{n} (\frac{r}{R})^{n} sin (n θ) .

Every term is regular at

r = 0

, and the series approaches the prescribed values at every boundary point away from the two jump discontinuities.

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