= Boundary-angle derivative of a disk automorphism
{title2=$f_w(e^{i\theta})=e^{i\phi_w(\theta)},\qquad\phi_w'(\theta)=\frac{1-|w|^2}{|e^{i\theta}-w|^2}$}
For $|w|<1$, the <disk automorphism> $f_w(z)=(z-w)/(1-\overline wz)$ maps the boundary circle to itself. If $g(\theta)$ is a continuous argument of $w-e^{i\theta}$, direct conjugation gives $\phi_w(\theta)=-\theta+2g(\theta)$ modulo $2\pi$. Differentiating the argument gives
$$
g'(\theta)=\frac{1-\operatorname{Re}(we^{-i\theta})}{|w-e^{i\theta}|^2},
$$
and substitution yields the displayed positive derivative. Its integral over one circuit is $2\pi$, so the boundary map is orientation preserving of degree one. Transforming a uniform exit angle at the centre by its inverse gives the <harmonic measure> density $(2\pi)^{-1}\phi_w'(\theta)$ for a <planar Brownian motion> started at $w$. This is the <Poisson kernel for the unit disk> with its probability normalization.
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