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Polar complex differentiation

  1. Jul 17, 2009 #1
    Does there exist anything like a polar complex differentiation? So there exists a gradient equation in polar coordinates something like
    [tex]\nabla{f} = \frac{\partial f}{\partial r} e_r + \frac{1}{r}\;\frac{\partial f}{\partial \theta} e_{\theta}[/tex]

    But this is not for a complex number [tex]f(z)[/tex] where [tex] z=r\,e^{i\theta}[/tex]. Now for cartesian coordinates, there exists a complex gradient formula as
    [tex]\nabla{f}(z) = \frac{\partial f}{\partial x} e_x - i\;\frac{\partial f}{\partial y} e_y[/tex]

    So I would like to know if there exists a formula like [tex]\nabla{f}(z) = \frac{\partial f}{\partial r} e_r -i\; \frac{1}{r}\;\frac{\partial f}{\partial \theta} e_{\theta}[/tex], if [tex] z=r\,e^{i\theta}[/tex].

    I can differentiate by [tex]z[/tex] directly. But I would like to know if anything like this exists.

  2. jcsd
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