We denote $\mathbb{U}$ the multiplicative group of complex numbers of modulus 1. Show that there exist a function $\beta \in \mathcal{E}$ and a constant $C \in ]0,1[$ such that, for all $\zeta \in \mathbb{U}$, $$\mathrm{e}^{\zeta} - 1 = \zeta(1 + \zeta \beta(\zeta)) \quad \text{and} \quad |\beta(\zeta)| \leqslant C.$$