Integer powers in the complex field

Fix $z\in \mathbb{C}$. Apply The recursion theorem to the set $\mathbb{C}$, the initial value $1$, and the function $w\mapsto wz$. This defines the natural powers uniquely by $z^0=1,z^{n+1}=z^{n}z(n\in \mathbb{N})\mathrm{.}$

Let $\mathrm{ȷ}\text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{Z}$ be the embedding of The naturals embed in the integers. For an integer $r\ge 0$, that lemma gives a unique $n\in \mathbb{N}$ with $r=\mathrm{ȷ}(n)$; define $z^r:=z^n$. If $r<0$ and $z\mathrm{\ne }0$, there is a unique $n\in \mathbb{N}$ with $n\ge 1$ and $-r=\mathrm{ȷ}(n)$; define $z^r:=(z^n{)}^{-1}\mathrm{.}$ The inverse exists because $\mathbb{C}$ is a field by $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$. Thus nonnegative integer powers are defined for every $z$, while negative integer powers are defined exactly when $z\mathrm{\ne }0$; in particular $0^0=1$ and no negative power of $0$ is defined. The integer and its unique natural representative are never conflated.