Let $a, b, c, d$ be complex numbers with $ad - bc \ne 0$. Then $$ f(z) = \frac{az + b}{cz + d}$$ is called the Mobius transformation.
If $H := \{ z \in \mathbb{C} : \Im(z) > 0 \}$ is the open upper half plane, show that any Mobius transformation from $H$ onto itself can be written with real coefficients $a, b, c, d$ with $ad - bc = 1$.
Can someone please show me how to do this problem?
I already showed that the Cayley transform $C(z) = \frac{z-i}{z+i}$ is a biholomorphic map (analytic, injective, and onto) from $H$ onto the unit disk $B(0,1)$. I also showed that any biholomorphic map from $H$ onto $H$ is a Mobius transformation. How can I go further and use this?
There is also another hint telling me that such transformation maps the real axis into the real axis, but I don't see how that would help me.
Thank you.