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Math Help - Determine is function is Analytic

  1. #1
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    Determine is function is Analytic

    I have a complex function

    f(z) =ln|z| + jArg(z)

    how do I show it is analytic?
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  2. #2
    MHF Contributor chisigma's Avatar
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    The function...

    f(z) = \ln |z| + j \angle z (1)

    ... is the principal branch of the function \varphi (z) = \ln z. It is analytic in tha whole complex plane with the exception of z=0 where it has a branch point...

    Kind regards

    \chi \sigma
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  3. #3
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    Quote Originally Posted by chisigma View Post
    The function...

    f(z) = \ln |z| + j \angle z (1)

    ... is the principal branch of the function \varphi (z) = \ln z. It is analytic in tha whole complex plane with the exception of z=0 where it has a branch point...

    Kind regards

    \chi \sigma
    Can we use Cauchy-Riemann equation to prove it is analytic except  z=0 ?
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  4. #4
    MHF Contributor chisigma's Avatar
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    Setting z = x + j y and f(z)= \ln z we have ...

    f(z) = u(x,y) + j\cdot v(x,y) = \frac{1}{2}\cdot \ln (x^{2} + y^{2}) + j\cdot \tan^{-1} (\frac{y}{x}) (1)

    ... so that is...

    \frac{\partial u}{\partial x} = \frac{x}{x^{2} + y^{2}} = \frac{\partial v}{\partial y}

    \frac{\partial u}{\partial y} = \frac{y}{x^{2} + y^{2}} = - \frac{\partial v}{\partial x} (2)

    The Cauchy-Riemann condition are then satisfied in the whole complex plane with the only exception of the point x=y=0...

    Kind regards

    \chi \sigma
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