= Explicit disk map of a slit half disk
{title2=$u^2=\dfrac{4(z^2+1)}{z^2+16},\quad F(z)=\dfrac{u+u^{-1}+i}{u+u^{-1}-i}$}
For the upper half disk $|z|<2$ slit along $z=iy$, $0<y\le1$, squaring maps onto $|w|<4$ minus $[-1,4)$. After scaling the radius-four disc, the <disk automorphism> $v=4(w+1)/(w+16)$ sends this slit to $[0,1)$. Its <square root> with argument in $(0,\pi)$ maps onto the upper half <unit disc>. There $u+u^{-1}$ maps bijectively onto the <lower half-plane>: its <imaginary part> is $\operatorname{Im}u(1-|u|^{-2})<0$, and the two roots of its inverse equation are reciprocal, with exactly one inside the unit <circle>. The final <Möbius map> gives the displayed conformal disk <bijection>.
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