Harmonic measure of a semicircle in a disk ($F(x,y)=\frac{1}{2}+\frac{1}{\pi }\arctan \frac{2y}{1-x^2-y^2}$)

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 $z=\pm 1$ have no path-independent boundary limit.