It's related to the Riemann mapping theorem which asserts that there is a unique analytic one-one function maps any simply connected region (which is not the whole plane) to a unit open disc. However, I'm not sure how to find the function explicitly.
Such as in this case, suppose $G=\{z:|z|<1,\Re(z)>0\}$,find an analytic function maps $G$ onto $B(0,1)$ in a one-one fashion.
Best Answer
The mapping $z \mapsto \dfrac{1+z}{1-z}$ maps the full unit disc conformally onto the right half plane. You can check that the interval $(-1,1)$ maps onto the real axis, so it takes the upper semi-disc onto a quadrant. (You can verify that the image is the first quadrant.)
Then continue with the mapping $z \mapsto z^2$, taking the first quadrant conformally onto the upper half plane, and finish off with a Möbius map taking the upper half-plane back to the unit disc.