Theorem 1   There is a conformal map from $\Delta$ the unit disk, to $UHP$ the upper half plane.

Proof. Define $f \colon \hat{{\mathbb C}} \to \hat{{\mathbb C}}$ (where $\hat{{\mathbb C}}$ denotes the Riemann Sphere) to be $f(z) = \displaystyle{\frac{z-i}{z+i}}$ Notice that $f^{-1}(w)= \displaystyle{i \frac{1+w}{1-w}}$ and that $f$ (and therefore $f^{-1}$ is a Mobius transformation.

Notice that $f(0)=-1$ $f(1)=\displaystyle{\frac{1-i}{1+i}} = -i$ and $f(-1) = \displaystyle{\frac{-1-i}{-1+i}} = i$ By the Mobius Circle Transformation Theorem, $f$ takes the real axis to the unit circle. Since $f(i)=0$ $f$ maps $UHP$ to $\Delta$ and $f^{-1}:\Delta \to UHP$