Largest domain on which $z^{i}$ is analytic.

Question

Can anyone help me with this question:

What is the largest domain $D$ on which the function $f(z)=z^{i}$ is analytic?

Answer 1

Rewriting as $f(z)=\exp(i\ln(z))$ suggests that $D$ contains at least the slit plane $\mathbb C\setminus(-\infty,0]$. Since the principle value of $\ln z$ jumps by $\pm2\pi i$ at the slit, we see that $f(z)$ jumps by a factor of $ e^{\mp2\pi}$, hence we cannot make $D$ any larger. (Of course there is not "the" largest domain, but only "a" largest domain depending on how we slit from $0$ to $\infty$)