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Math Help - Harmonic functions and C-R equations

  1. #1
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    Harmonic functions and C-R equations

    Please refer to the attached image.

    Could I get some hints on how to evaluate these question.

    The question asking to find where f(re^{i \theta}) is differentiable doesn't seem to involved,
    however would I use C-R equations, or would it just be for wherever r \neq 0. Although that is given in the domain, so I'm assuming they want us to use C-R equations.
    If i were to use C-R equations, then I would have to convert to cartesian coordinates, correct?
    How would I do that.

    As for the other question,
    could I get any hints on what theorems may be relevant. I am not sure how to approach this.


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  2. #2
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    Re: Harmonic functions and C-R equations

    Yes, use the Cauchy-Riemann Equations. You don't need to convert back to Cartesians, the Cauchy-Riemann Equations in polar form are \displaystyle \begin{align*} \frac{\partial u}{\partial r} = \frac{1}{r} \, \frac{\partial v}{\partial \theta} \end{align*} and \displaystyle \begin{align*} \frac{\partial v}{\partial r} = -\frac{1}{r}\,\frac{\partial u}{\partial \theta} \end{align*}.
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