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Math Help - Using polar form of Cauchy-Riemann equations

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    Using polar form of Cauchy-Riemann equations

    How would I go about using the Cauchy-Riemann equations to show that f(z) = z^n is an entire function for any n using polar coordinates?
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    Re: Using polar form of Cauchy-Riemann equations

    Well for starters, what is f(z) in polar form? What is the polar form of the Cauchy-Riemann equations?
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    Re: Using polar form of Cauchy-Riemann equations

    f(z) = r^ne^{i\theta n}

    Cauchy-Riemann is u_r = \frac{1}{r}v_{\theta} and v_r = \frac{-1}{r}u_{\theta}

    I split f(z) up like so:

     u(r,\theta) = r^n \cos(\theta n)

    v(r,\theta) = r^n \sin(\theta n)

    I'm stuck on what to do next... am I allowed to differentiate as I would over R?
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    Re: Using polar form of Cauchy-Riemann equations

    Quote Originally Posted by jgv115 View Post
    f(z) = r^ne^{i\theta n}

    Cauchy-Riemann is u_r = \frac{1}{r}v_{\theta} and v_r = \frac{-1}{r}u_{\theta}

    I split f(z) up like so:

     u(r,\theta) = r^n \cos(\theta n)

    v(r,\theta) = r^n \sin(\theta n)

    I'm stuck on what to do next... am I allowed to differentiate as I would over R?
    Just take your derivatives.

    $u_r(r,\theta)=n r^{n-1}\cos(n\theta)$

    $u_\theta(r,\theta) = -n r^n \sin(n\theta)$

    etc. and then check that the polar CR eqs are satisfied.
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