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

  1. May 6, 2013 #1
    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 within the C^1 dimension that would give us conditions where the Cauchy-Riemann equations wouldn't work?
     
  2. jcsd
  3. May 7, 2013 #2

    lavinia

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    An arbitrary continuously differentiable function from the plane into itself will no satisfy the Cauchy Riemann equations.

    One requirement is that the Jacobian is complex linear: that is, it is a rotation followed by a change of scale (scalar multiplication)

    The coordinate functions of an analytic function can be shown to harmonic functions.
     
    Last edited: May 7, 2013
  4. May 10, 2013 #3

    Bacle2

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    I'm not sure I get your notation; is your f a function of 2 complex variables? Also,
    what do you mean by a difference between C^1 and R^2 ? If you refer to differentiability,
    any function differentiable in one is also (Real) differentiable in the other, since C^1 and R^2
    are diffeomorphic as manifolds.

    For

    functions of a single complex variable, f(z)=z^ , with z^ conjugation, does not satisfy C-R
    anywhere. A complex analytic function must be expressed without any terms containing z^.
    This means, if your function f is expressed in terms of (x,y) , then, after the coordinate change
    x= (z+z^)/2 and y=(z-z^)/2i , your function is analytic if , after cleaning up , the function
    can be expressed in terms of z alone.
     
  5. May 10, 2013 #4
    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 instead had 2 complex variables for our space (C^2), then would we possibly have 3 different metrics? One for between z0 and z1, and then one for each of the x-y planes of z0 and z1 respectively? Or could we somehow wrap all three metrics into just one?
     
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