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Homework Help: Check if the complex function is differentiable

  1. Jan 26, 2014 #1
    The question is to check where the following complex function is differentiable.

    [tex]w=z \left| z\right|[/tex]

    [tex]w=\sqrt{x^2+y^2} (x+i y)[/tex]

    [tex]u = x\sqrt{x^2+y^2}[/tex]
    [tex]v = y\sqrt{x^2+y^2}[/tex]
    Using the Cauchy Riemann equations

    [tex]\frac{\partial }{\partial x}u=\frac{\partial }{\partial y}v[/tex]
    [tex]\frac{\partial }{\partial y}u=-\frac{\partial }{\partial x}v[/tex]

    my results:

    [tex]\frac{x y}{\sqrt{x^2+y^2}}=0[/tex]

    solutions says that it's differentiable at (0,0). But doesn't it blow at (0,0)?
  2. jcsd
  3. Jan 26, 2014 #2
    If you just plug in ##y=0## and ##x=0## you will get an indeterminate form which is meaningless. If you evaluate the limits, I think that you get all expressions equal to ##0##, but double check that.
  4. Jan 26, 2014 #3


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    Division by zero is not allowed in complex analysis, so your final equations are not defined at x=y=0. They are not equal.
  5. Jan 27, 2014 #4
    That is true, but this function is differentiable at ##z=0##. If you evaluate the two limits along the real and imaginary axes (with ##h\in\mathbb{R}##)



    So the function is differentiable at ##0##. I don't remember enough from my complex analysis course (which had a number of students who had not taken real analysis, so it was a bit less rigorous than some courses) to reconcile this. My recollection is that the limits of the Cauchy-Riemann equations could be evaluated, but a quick look online showed that my recollection was incorrect. Perhaps, since the partial derivatives are undefined at 0 the Cauchy-Riemann equations are not applicable?
  6. Jan 27, 2014 #5


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    I would use the definition of the derivative as a difference quotient to show it's differentiable at z=0.
  7. Jan 27, 2014 #6
    Yes, but ##w(0)=0##, so I left it out.
  8. Jan 27, 2014 #7


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    Sure, I'm just saying there is no need work along any particular axes. The complex derivative f'(0) is defined by the limit |h|->0 of (f(0+h)-f(0))/h where h is complex. You can also conclude that that is zero.
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