= Solution
The boundary data are $U\operatorname{sgn}(\sin\theta)$, whose <Fourier series> is
$$
\operatorname{sgn}(\sin\theta)
=\frac4\pi\sum_{\substack{n\geq1\\n\ {\rm odd}}}
\frac{\sin(n\theta)}n.
$$
Regular harmonic modes in a disk are $(r/R)^n\sin(n\theta)$ and $(r/R)^n\cos(n\theta)$. Matching the odd boundary data gives
$$
\boxed{
u_z(r,\theta)
=\frac{4U}{\pi}
\sum_{\substack{n\geq1\\n\ {\rm odd}}}
\frac1n\left(\frac rR\right)^n\sin(n\theta)}.
$$
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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