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Math Help - principal branch of square root

  1. #1
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    principal branch of square root

    If \sqrt w denotes the principal branch of square root of w, for w in \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 \theta), for r>1 and - \pi/3< \theta< \pi/3 ?

    Thanks.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: principal branch of square root

    Quote Originally Posted by Veve View Post
    If \sqrt w denotes the principal branch of square root of w, for w in \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 \theta), for r>1 and - \pi/3< \theta< \pi/3 ?

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

    Kind regards

    \chi \sigma
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