For each complex number $w$, we denote by $\operatorname{Re}(w)$ the real part of $w$. Show that, for all $z \in \stackrel{\circ}{\mathbb{D}}$:
$$\ln|1-z| = -\operatorname{Re}\left(\sum_{n=1}^{\infty} \frac{z^n}{n}\right)$$
To do this, one may write $z = re^{i\theta}$ with $0 \leq r < 1$ and $\theta \in \mathbb{R}$, then study the function:
$$\begin{aligned} F : [0,1[ &\rightarrow \mathbb{R} \\ \rho &\mapsto \ln\left|1 - \rho e^{i\theta}\right| \end{aligned}$$