Exercise 1, § 3

For every finite sequence $s = (z_k)_{k \in I}$ of complex numbers, put

(cf. Chapter VII, § 3, no. 1, Proposition 2).

a) \* If $z_k = r_k (\cos \varphi_k + i \sin \varphi_k)$, where $r_k = |z_k|$ and $\varphi_k$ is the principal radian measure of the amplitude of $z_k$, put

for every $\varphi \in [0, 2\pi[$. Show that the least upper bound of $f(\varphi)$ in the interval $[0, 2\pi[$ is equal to $\sup_{J \subset I} \left| \sum_{k \in J} z_k \right|$. By considering the integral deduce that for all finite sequences $s$ we have $\rho_s \leq \pi$, and that there is no finite sequence $(z_k) = s$ such that $\rho_s = \pi$.

b) Show that, for every $\varepsilon > 0$, there exists a finite sequence $(z_k) = s$ such that $\rho_s > \pi - \varepsilon$ (take the $z_k$ to be the roots of a binomial equation of sufficiently high degree). \*