Is exponentiation in $\mathbb{C}$ somehow not unique?

Question

Referring to precisely the comments on the original question posted here, one user says,

I suppose you are talking about exponentiation for complex numbers, rather than for reals. Also, isn't it the case that $a^b$ is not unique for any $b$ that is not an integer (assuming we're working with complex numbers)?

And it seems like the resolution was that this is true for complex $a$ and $b$, but I didn't think this was correct. I thought exponentiation defined for a base $a \in \mathbb{C}$ and exponent $b \in \mathbb{C}$, that $a^b$ must be unique. Am I misinterpreting the discussion or is there something that I'm missing?

Answer 1

$a^b$ is defined as $\exp(b \log (a))$ for any branch of the logarithm. The complex logarithm is multi-valued: if $L$ is one value, then $L+2\pi i n$ is a value for any $n$. Unless $b$ is an integer, this makes $a^b$ multivalued as well.

To single out a particular value, one sometimes takes the principal branch where $-\pi < \text{Im}(\log(a)) \le \pi$ (and this is almost always done when $a$ is a positive real).

Answer 2

Because $a^b$ is easy to write, it is tempting to assume that it must be easy to define in some sensible and useful way. However, there is no reason why this must be so. $\frac{1}{0}$ is also easy to write but not easy to define in a useful way. Division has a fairly simple restriction, the restrictions on exponentiation are more complicated. This link in one of the answers to the question that you quote describes the situation quire well. https://en.m.wikipedia.org/wiki/Exponentiation#Powers_of_complex_numbers