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Thread: 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 $\displaystyle \mathbb{C}$\{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.
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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 $\displaystyle \mathbb{C}$\{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

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