How are powers of complex numbers defined? How are powers of complex numbers defined? Suppose I have some number $z\in\mathbb{C}$. It makes sense that there are $n$ unique solutions to $w^n=z$. Where we define this power in terms of complex multiplication.
How is this extended to $z^w$ where $w\in \mathbb{C}$. I assume some sort of limiting process is required from natural to rational to real to complex? Is there a good resource for finding where this is defined?
 A: It's defined using Euler's Formula:
$$e^{i\theta} = \cos\theta + i \sin\theta$$
Or, adding a real part to the exponent:
$$e^{x+iy} = e^xe^{iy} = e^x(\cos y + i \sin y)$$
So if you have a complex number expressed in polar coordinates $z = r(\cos \theta + i \sin \theta)$, then:
$$\log z = \log r + i\theta$$
And you can then calculate $z^w = e^{w \log z}$ just like you can for real numbers.
The tricky part with complex logarithms is that because the trig functions are periodic, the choice of $\theta$ isn't unique.
$$\log z = \log r + i(\theta + 2\pi n), n \in \mathbb{Z}$$
So, do you use an angle in the interval $[0, 2\pi)$, $[-\pi, \pi)$, or some other $2\pi$-wide interval?  This choice, called a “branch cut”, is a matter of context or convenience.
A: Fix a branch of natural log, the most common one being $\log : B = \mathbb{C} \setminus (-\infty, 0] \to \mathbb{C}$ that inverts $\exp : A = (-\infty, \infty) \times (-\pi, \pi) \to B$. Then for $z \in B$ and $w \in \mathbb{C}$, define $z^w = \exp(w \log(z))$. For details see sections 3 and 4 of https://mtaylor.web.unc.edu/notes/complex-analysis-course/
