Boundary-angle derivative of a disk automorphism ($f_w(e^{i\theta })=e^{i{\varphi }_w(\theta )},{\varphi }_w^{\prime }(\theta )=\frac{1-|w{|}^2}{|e^{i\theta }-w{|}^2}$)

For $|w|<1$, the disk automorphism $f_w(z)=(z-w)/(1-\overline{w}z)$ maps the boundary circle to itself. If $g(\theta )$ is a continuous argument of $w-e^{i\theta }$, direct conjugation gives ${\varphi }_w(\theta )=-\theta +2g(\theta )$ modulo $2\pi$. Differentiating the argument givesand 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 $(2\pi {)}^{-1}{\varphi }_w^{\prime }(\theta )$ for a planar Brownian motion started at $w$. This is the Poisson kernel for the unit disk with its probability normalization.