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Math Help - Complex Analysis: Is Sqrt(z) entire?

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    Complex Analysis: Is Sqrt(z) entire?

    This question is part of a bigger one, but I need to determine for what domain \sqrt{z} is entire. I am not sure how to get started. I suspect I am having a really stupid moment and just can't figure out how to make it fit into Cauchy-Riemann, but I can't work it out. Thanks for your help.
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    MHF Contributor Drexel28's Avatar
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    Re: Complex Analysis: Is Sqrt(z) entire?

    Quote Originally Posted by tarheelborn View Post
    This question is part of a bigger one, but I need to determine for what domain \sqrt{z} is entire. I am not sure how to get started. I suspect I am having a really stupid moment and just can't figure out how to make it fit into Cauchy-Riemann, but I can't work it out. Thanks for your help.
    I feel as though, at least from my experience, this is non-standard terminology. I assume what you means is that you want to find some domain D\subseteq\mathbb{C} for which \sqrt{z} is holomorphic on the entire domain? If so, what if you just take the principal branch D=\mathbb{C}-(\mathbb{R}-\mathbb{R}^+)?
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    Re: Complex Analysis: Is Sqrt(z) entire?

    So I'm basically taking the domain to leave off the negative reals?
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    MHF Contributor Drexel28's Avatar
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    Re: Complex Analysis: Is Sqrt(z) entire?

    Quote Originally Posted by tarheelborn View Post
    So I'm basically taking the domain to leave off the negative reals?
    And zero.
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    Re: Complex Analysis: Is Sqrt(z) entire?

    Quote Originally Posted by tarheelborn View Post
    This question is part of a bigger one, but I need to determine for what domain \sqrt{z} is entire. I am not sure how to get started. I suspect I am having a really stupid moment and just can't figure out how to make it fit into Cauchy-Riemann, but I can't work it out. Thanks for your help.
    I have a bit of a different take on this question from that of reply #2.
    If f(z)=z^{\frac{1}{2}}=\sqrt r \cos \left( {\frac{\theta }{2}} \right) + \mathbf{i}\sqrt r \sin \left( {\frac{\theta }{2}} \right)
    where r>0~\&~-\pi<\theta<\pi then using the polar form of the C-R equations show the derivative exist.
    That is a standard example is many basic textbooks.
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    Re: Complex Analysis: Is Sqrt(z) entire?

    Thanks; this is a little more along the lines of the way I was taught to address these problems.
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    MHF Contributor chisigma's Avatar
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    Re: Complex Analysis: Is Sqrt(z) entire?

    Quote Originally Posted by tarheelborn View Post
    This question is part of a bigger one, but I need to determine for what domain \sqrt{z} is entire. I am not sure how to get started. I suspect I am having a really stupid moment and just can't figure out how to make it fit into Cauchy-Riemann, but I can't work it out. Thanks for your help.
    As in...

    Entire Function -- from Wolfram MathWorld

    ... according to the 'standard definition', an f(z) is called entire if it is analytic on the whole complex plane \mathbb{C}. That means that in any case the expression 'for what domain \sqrt{z} is entire' is 'selfcontraductory' and the expression 'for what domain \sqrt{z} is analytic' should be used instead of it...

    Kind regards

    \chi \sigma
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