The Complex Exponential Function

In the setting of Complex Analysis: Standing Notation, let sums of series in $\mathbb{C}$ be as in Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii), and for $z\in\mathbb{C}$ and $k\in\mathbb{N}$ let the terms $z^{k-1}/(k-1)!$ be as in The Complex Exponential Series Converges Absolutely, and Its Sum Is a Function on ℂ (i).

(i) The exponential function. The complex exponential function is the function of The Complex Exponential Series Converges Absolutely, and Its Sum Is a Function on ℂ (iii),

$$\exp:\mathbb{C}\to\mathbb{C},\qquad \exp z=\sum_{k=1}^{\infty}\frac{z^{k-1}}{(k-1)!}.$$