Number of roots of $x^a-1=0$ with $a \in \mathbb{C}$

Solution 1:

As pointed out in the comments, taking the logarithm of both sides of $$ x^a = 1, $$ and taking into account the multi-valued nature of the logarithm, gives $$ a \log x = 2 \pi k i $$ for some integer $k$, or $\log x = 2\pi k i/a$ (assuming $a\neq 0$). Then exponentiating both sides of this equation gives $$ x = \exp\left(\frac{2\pi k i}{a}\right) = \cos\left(\frac{2\pi k}{a}\right) + i\sin\left(\frac{2\pi k}{a}\right). $$ If $a$ is rational and equal to $p/q$ in lowest terms, then this takes on exactly $p$ different values; otherwise it takes on infinitely many different values, dense on the unit circle.

Solution 2:

The ONLY entire functions that has finitely many zeros are functions that are/(can be written in) the form $$P(z) e^{g(z)}$$ where $$g$$ is entire.

This might answer your question partly, I suspect.