Complex power

Remarks

The powers of $z$ with exponent $w$ are the numbers $e^{w \operatorname{Log} z} e^{2 \pi i k w}$ with $k \in \mathbb{Z}$. There is one value if $w$ is an integer, there are $q$ values if $w = p/q$ in lowest terms with $q \ge 2$, and there are infinitely many values otherwise. For an integer $w = n$, $z^n$ is the usual product of $n$ factors $z$, or of $-n$ factors $1/z$. For example, $i^i = e^{i \cdot i \pi/2} = e^{-\pi/2}$, a positive real number. The usual rules can fail: with principal values, $(z^a)^b \neq z^{ab}$ for some $z, a, b$. A common convention for $z = 0$ is $0^0 = 1$ and $0^w = 0$ for $w \neq 0$.