Definition 2

Tag 089W, permanent. Cite it as [Bourbaki, Tag 089W].

Functions of a Real Variable, chapter III, § 1. DERIVATIVES OF THE EXPONENTIAL AND CIRCULAR FUNCTIONS

The homomorphism $x + iy \mapsto e^z \mathbf{e}(y/2\pi)$ of $\mathbf{C}$ onto $\mathbf{C}^*$ is called the complex exponential; its value at an arbitrary complex number $z$ is denoted by $e^z$ or $\exp z$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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