 Solution

The domains are conformally equivalent. The Möbius transformationmaps the upper half of the unit disc onto the first quadrant: its diameter $(-1,1)$ maps to the positive real axis and its upper semicircle maps to the positive imaginary axis. Squaring maps the first quadrant bijectively and conformally onto the upper half-plane. Thereforeis the required equivalence, as recorded by the conformal equivalence between the upper half-disc and upper half-plane.