Remark

The mapping $x + iy \to e^{x}e(y)$ is a strict morphism of the additive group $\mathbf{C}$ onto the multiplicative group $\mathbf{C}^*$, and its restriction to a suitable neighbourhood of 0 is a local isomorphism of $\mathbf{C}$ with $\mathbf{C}^*$. Consequently (Chapter VII, § 2, no. 3) every strict morphism of $\mathbf{C}$ onto $\mathbf{C}^*$ is of the form $x + iy \to e^{\alpha x + \beta y}e(\gamma x + \delta y)$, where $\alpha, \beta, \gamma, \delta$ are any real numbers such that $\alpha \delta - \beta \gamma \neq 0$. We shall see later that there is just one of these homomorphisms, denoted by $z \to e^z$, such that

and the restriction of this homomorphism to the real axis is the same as $e^x$ (whence the notation).