Equivalence of Definitions of Complex Exponential Function

Proof
From Power Series over Factorial: Complex Case, it follows that the power series $\displaystyle \sum_{n \mathop = 0}^\infty \dfrac {z^n} {n!}$ is absolutely convergent.

Hence, the definition of $\exp z$ as a sum of a series is valid.

It remains to demonstrate the logical equivalence of all the definitions.