Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Can anyone help me with this question:

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

share|cite|improve this question
up vote 7 down vote accepted

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 $\infty$)

share|cite|improve this answer
Thank you for the kind help. – jigar Jun 9 '13 at 15:08

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.