Conformal map from a parabolic domain to the unit disc
= Conformal map from a parabolic domain to the unit disc
{title2=$C(z)=\tan^2(\pi\sqrt z/4)$}
The map $C(z)=\tan^2(\pi\sqrt z/4)$ is a <conformal map> from $\{x+iy:y^2<4(1-x)\}$ onto the <unit disc>. A <principal square root> first maps the slit parabolic region to a vertical strip, and a scaled <tangent> followed by squaring maps the strip to a slit disc. Evenness in the square-root variable gives a <holomorphic> extension across the removed axis, filling the target slit.