In which points is the following function complex differentiable?

Question

Let $a,b,c,d \in \Bbb C$ s.t. $cd \neq 0$. We define $$f(z)=\frac{az+b}{cz+d}$$ I am looking at the points, where $f$ is complex differentiable. I though to use the Cauchy-Riemann equations, but I don't know how to use it here...

Many thanks for some help!

Answer 1

$f$ is the fraction of two complex differentiable maps. So $f$ is complex differentiable where the denominator doesn't vanish, i.e. for $z \in \mathbb C\setminus\{-\frac{d}{c}\}$.