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    Scenarios where the Cauchy-Riemann equations aren't true?

    z and z* are the same single complex variable, one the conjugate of the other. I'm wondering what would happen if we gave the plane of the single complex coordinate z a metric? Essentially, break up z into it's component parts x and y and give the x-y plane a non-Euclidean metric. If we...
  2. T

    Scenarios where the Cauchy-Riemann equations aren't true?

    Are there any scenarios where the Cauchy-Riemann equations aren't true? And if so, would there really be any difference between C^1 and R^2 in those cases? The function: f(z, z*) = z* z doesn't solve the Cauchy-Riemann equations yet I would think is quite useful. Couldn't we add a metric...
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