# Thread: principal branch of square root

1. ## principal branch of square root

If \sqrt w denotes the principal branch of square root of w, for w in $\displaystyle \mathbb{C}$\{0}, then what is the maximal open set in which f(z)=\sqrt{z^3 -1} defines an analytic function and compute f^{-1}(3).

The maximal open set in which f is analytic is z=r exp(i$\displaystyle \theta$), for r>1 and -$\displaystyle \pi$/3<$\displaystyle \theta$<$\displaystyle \pi$/3 ?

Thanks.

2. ## Re: principal branch of square root

Originally Posted by Veve
If \sqrt w denotes the principal branch of square root of w, for w in $\displaystyle \mathbb{C}$\{0}, then what is the maximal open set in which f(z)=\sqrt{z^3 -1} defines an analytic function and compute f^{-1}(3).

The maximal open set in which f is analytic is z=r exp(i$\displaystyle \theta$), for r>1 and -$\displaystyle \pi$/3<$\displaystyle \theta$<$\displaystyle \pi$/3 ?

Thanks.
The function $\displaystyle f(z)= \sqrt{z^{3}-1}$ has three brantch points in $\displaystyle z=e^{i\ \frac{2 k \pi}{3}}$ and in all the remaining part of $\displaystyle \mathbb{C}$ is analytic...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$