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Math Help - Complex help

  1. #1
    Senior Member Danneedshelp's Avatar
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    Complex help

    Q: For fixed a\in{\mathbb{C}} find the value of |z^{n}-a| that is maximized when |z|<1.

    A: Well, I would think I need to find the the roots to the equation z^{n}=a firs. After that I am not sure. Since this is a max question, I feel that I need to take the derivative at some point with respect to z.

    Any help would be appreciated.

    Thanks
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  2. #2
    MHF Contributor chisigma's Avatar
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    Setting z= e^{\rho + i\ \theta} and a= e^{\alpha + i\ \beta} is...

    \displaystyle |z^{n} - a|^{2} = (e^{n\ \rho + i\ n\ \theta} - e^{\alpha + i\ \beta})\ (e^{n\ \rho - i\ n\ \theta} - e^{\alpha - i\ \beta}) (1)

    Now what You have to do is to find the values of \rho and \theta [with the condition \rho <0, \implies Lagrange's multipliers ...] so that (1) is maximized...

    Kind regards

    \chi \sigma
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  3. #3
    Senior Member
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    If \; |z|<1 \; then \; f(z)=z^n \; |f(z)|<1

    max |z^n-a| \; is \; the \; same \; as \; max|z-a|

    max|z-a|=|a|+1
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