In this paper the authors write down the following relation (p. 5, the last equation before the 3rd section): $$-\frac{1}{2}\ln\left(\frac{1-|z|^2}{|z-\zeta|^2}\right)=\lim\limits_{r\rightarrow \zeta}\left[L(z,r) - L(0,r)\right],\quad (*)$$ where $L(z_1,z_2)$ is the Poincaré distance between points $z_1$ and $z_2$ on hyperbolic unit disk $D$.
I know that geodesic length between two points $z_1$ and $z_2$ can be expressed as follows: consider the Möbius map of arc $(z_1,z_2)$ to diameter $(0,u)$, where $$u = \left|\frac{z_2-z_1}{1-\bar{z}_2z_1}\right|,$$ then the geodesic length is $$L(z_1,z_2)=\ln\frac{1+u}{1-u}.$$ Fine. However, I cannot rederive the expression $(*)$. Can anyone clarify this derivation?
You may do it directly for the unit disk. It is convenient, though, to make a rotation so that $\zeta=1$. If I am not mistaking it also looks as if Douady included a factor of one half in the Poincaré distance. Let $r=1-\epsilon$ and expand in $\epsilon\to 0^+$. Then $$ u = \frac{|1-\epsilon-z|}{|1-(1-\epsilon)z|} = \left|1-\epsilon \frac{1+z}{1-z} + ...\right| = 1-{\epsilon} \frac{1-|z|^2}{|1-z|^2}+ ...$$ so that $$\frac12 \ln\frac{1+u}{1-u} = -\frac12 \ln(\epsilon) -\frac12 \ln \frac{1-|z|^2}{|1-z|^2} + ... $$ The first part cancels against the $L(0,1-\epsilon)$ term.