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Thread: 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...

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

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

    Kind regards

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

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

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

    Kind regards

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

    $\displaystyle 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...

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

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

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

    Kind regards

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