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CR equations and differentiability

  1. Sep 17, 2007 #1
    1. The problem statement, all variables and given/known data
    Where is f(z) differentiable? Analytic?
    [tex]f(z) = x^{2} + i y^{2}[/tex]

    2. Relevant equations

    Cauchy-Riemann Equations

    3. The attempt at a solution

    I calculated the partial derivatives,

    [tex]u_{x} = 2x[/tex]
    [tex]v_{y} = 2y[/tex]
    [tex]u_{y} = 0[/tex]
    [tex]v_{x} = 0[/tex]

    Then said that for the CR equations to hold,

    [tex]u_{x}=v_{y}[/tex] therefore [tex]y=x[/tex]
    and
    [tex]u_{y}=-v_{x}[/tex] therefore [tex]0=0[/tex]

    Then becuase the partial derivatives are continuous for all [tex]x,y[/tex], [tex]f(z)[/tex] is differentiable along [tex]y=x[/tex]

    [tex]f(z)[/tex] is nowhere analytic because an arbitrarily small open disk centered at any point on the line [tex]y=x[/tex] will always contain points which are not differentiable.

    Is that sufficient to show differentiability? Or am I misapplying the cauchy-riemann conditions?
     
  2. jcsd
  3. Sep 18, 2007 #2
    I suppose what I'm really asking is wether or not I need to look at the limits depending on the direction of approach, or if this is sufficient as is? Help?
     
  4. Sep 18, 2007 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I think that is good enough. You could explicitly show the derivative limit is independent of direction along x=y, but why? That's what CR are for.
     
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