 Explicit disk map of a slit half disk

For the upper half disk $|z|<2$ slit along $z=iy$, $0<y\le 1$, 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 $\mathrm{Im}u(1-|u{|}^{-2})<0$, 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.