The complex exponential by its power series

Let $\iota_{\mathbb R}\colon\mathbb N\to\mathbb R$ be the canonical-natural map of The canonical natural $\iota(n) = n \cdot 1_F$ of a field, and let $j\colon\mathbb R\to\mathbb C$ be the real embedding $j(x)=x+i0$ from The complex numbers as $\mathbb R^2$, with their arithmetic, real embedding, and imaginary unit. For every $n\in\mathbb N$, the factorial $n!$ is nonzero by The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$, so $\iota_{\mathbb R}(n!)>0$ by Canonical naturals are positive and strictly increasing and consequently $j(\iota_{\mathbb R}(n!))\ne0$ in the complex field The complex numbers form a field, and every nonzero $x+iy$ has inverse $(x-iy)/(x^2+y^2)$.

For $z\in\mathbb C$, define

$$\exp z:=\sum_{n=0}^{\infty}\frac{z^n}{j(\iota_{\mathbb R}(n!))}.$$

whenever this complex series converges. Inside a complex expression we abbreviate the embedded denominator $j(\iota_{\mathbb R}(n!))$ by $n!$, so the same definition may be written $\exp z=\sum z^n/n!$ without identifying a natural number with a complex number. Powers and series are those of Integer powers in the complex field and Complex series, absolute convergence, complex power series, and radius of convergence. The convergence for every $z\in\mathbb C$ is discharged by The complex exponential series converges absolutely for every complex argument ↗.