How are powers of complex numbers defined?

Question

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?

• Тyma Gaidash: Here is a way using complex components. There is a formula at the end.
• user170231: $w^{a+bi}=e^{(a+bi) \ln(w)},ドル where $\ln(w) = \ln|w| + i\arg(w)$ (taking the principal branch of the logarithm)
• John Douma: You can either use the binomial theorem, simplifying powers of $i$ or convert the complex numbers to polar coordinates which is really convenient for multiplication problems.
• fleablood: Any analysis book. For $n \in \mathbb N$ defining it as $b\cdot b\cdot...\cdot b$ is just as acceptable for complex numbers as it is for real numbers. But for $b^x; x\not \in \mathbb Z$ we need a different definition for it just as we needed a different definition for the reals.
• suckling pig: Very carefully. No seriously, you use the complex exponential and logarithmic functions.

Answer 1

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ドル\pi$-wide interval? This choice, called a "branch cut", is a matter of context or convenience.

Answer 2

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/