= Disk potential with two semicircular boundary values
{title2=$u(r,\theta)=\frac{V_1+V_2}{2}+\frac{V_1-V_2}{\pi}\arctan\frac{2ar\sin\theta}{a^2-r^2}$}
On a disk of radius $a$, prescribe $V_1$ on the upper open semicircle and $V_2$ on the lower one. The bounded <harmonic extension> has the formula in the title for $r<a$. The boundary data have only a constant term and odd sine <Fourier coefficients>, giving $u=(V_1+V_2)/2+[2(V_1-V_2)/\pi]\sum_{m\geq0}(r/a)^{2m+1}\sin((2m+1)\theta)/(2m+1)$. The complex series is half of $\log[(1+w)/(1-w)]$, where $w=(r/a)e^{i\theta}$. Its quotient has positive real part, so taking the imaginary part yields the displayed principal <arctangent> without an argument ambiguity. At the two boundary jumps, the radial limit is the mean; the harmonic function has no continuous extension matching both adjacent constant arcs there.
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