Conformal automorphism of the upper half-plane ($\varphi (z)=(az+b)/(cz+d)$)

A conformal bijection of the complex upper half-plane to itself is a real Möbius transformation with $ad-bc>0$. Three distinct boundary values determine it uniquely. Its real derivative is positive away from its pole, and $\mathrm{Im}\varphi (z)=(ad-bc)\mathrm{Im}z/|cz+d{|}^2$.