Analytic branches of $z^{-i}$.

Question

How to describe all the branches of the function $z^{-i}$, analytic in the whole complex plane except the positive real axis?

I consider $z^{-i}=e^{-i \log z}$ and the branch becomes whole complex plane except the negative real axis. Taking into account the hypothesis, we should exclude the whole real line from $\mathbb{C}$.

Actually I don't believe what I am doing. For example, what if we had a rational exponent i.e. consider the function $z^{4/7}$. My solution gives the same branch and this seems to me incorrect. Can anyone help?

Answer 1

Function $\log (z)=\log|z| + i \arg (z)$ has a single-valued analytic (holomorphic) branch in $\mathbb C\setminus \mathbb R_+$ (it differs from main branch of the logarithm, which has usually the branch cut along the negative real axis, but it is still-single valued). Two branches of this function differ there in an integer multiple of $2\pi$.

Hence, indeed $$ z^{-i}=\exp (- i \log (z))=\exp(-i (\log|z| + i \arg (z))= e^{-i \log|z|} e^{\arg(z) } . $$ Thus, in $\mathbb C\setminus \mathbb R_+$ the ratio of two branches of $z^{-i}$ is a constant of the form $e^{2\pi n }$, $n\in \mathbb Z$.