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Thread: Complex Analysis Question

  1. #1
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    Cool Complex Analysis Question

    I'm having trouble with this showing question.


    Any help would be much appreciated. thanks!
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  2. #2
    MHF Contributor chisigma's Avatar
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    Setting $\displaystyle z= x + i\cdot y$ and $\displaystyle f(z)= u(x,y) + i\cdot v(x,y)$, $\displaystyle f(*)$ is analytic in a region $\displaystyle A$ of the complex plane if and only if in that region is...

    $\displaystyle \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$

    $\displaystyle \frac{\partial u}{\partial y}=- \frac{\partial v}{\partial x}$ (1)

    But $\displaystyle f(*)$ is a real function in $\displaystyle A$ so that is $\displaystyle v(z)=0$ and therefore for the (1) is $\displaystyle \frac{\partial u}{\partial x} = \frac{\partial u}{\partial y}=0$...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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  3. #3
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    how did you arise to f(*) is a real function in A so that v(z) = 0 ?

    thanks for the reply!
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