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How to check if function is differentiable at a point

  1. Jan 25, 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^2}{\sqrt{x^2+y^2}}=\frac{y^2}{\sqrt{x^2+y^2}}[/tex]
    [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)?
     
    Last edited: Jan 25, 2014
  2. jcsd
  3. Jan 26, 2014 #2

    mfb

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    2016 Award

    Staff: Mentor

    What do you mean with "blow"?
    The limit of those expressions for x,y -> 0 is well-defined and zero.
     
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