$$f(z) = \frac{az + b}{cz + d}, \quad \text{ for } a, b, c, d \in \mathbb{C}$$
We have that $f \in H(\mathbb{C} - \{ z = -\frac{d}{c} \} )$.
My notes say that for any point $z_0$ in the plane, except $-\frac{d}{c}$, $f$ is holomorphic in the open disc centred at $z_0$ with radius $\mid z_0 + \frac{d}{c} \mid$.
Why is that?