= Harmonic measure of a semicircle in a disk
{title2=$F(x,y)=\frac12+\frac1\pi\arctan\frac{2y}{1-x^2-y^2}$}
The bounded <harmonic function> on the open unit disk with boundary values one on its upper semicircle and zero on its lower semicircle is the displayed function. The <Möbius transformation> $w=i(1-z)/(1+z)$ has real part $2y/|1+z|^2$ and imaginary part $(1-|z|^2)/|1+z|^2$. Pulling back the upper-half-plane step-data solution proves the formula by <conformal invariance of harmonicity>. The discontinuity points $z=\pm1$ have no path-independent boundary limit.
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