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

The boundary data are $U\mathrm{sgn}(\sin \theta )$, whose Fourier series isRegular harmonic modes in a disk are $(r/R{)}^n\sin (n\theta )$ and $(r/R{)}^n\cos (n\theta )$. Matching the odd boundary data givesEvery term is regular at $r=0$, and the series approaches the prescribed values at every boundary point away from the two jump discontinuities.