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Z (conjugate) not analytic?

  1. Oct 24, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that f(z) = ¯z is not differentiable for any z ∈ C.

    2. Relevant equations



    3. The attempt at a solution

    Is it because the Cauchy-Reimann Equations don't hold?

    Z (conjugate) = x-iy
    u(x,y)=x
    v(x,y=-iy

    du/dx=1≠dv/dy=-1
    du/dy=0≠-dv/dx=0



    Edit: Is there another approach? Because the CR Equations is something we learned later on.
     
  2. jcsd
  3. Oct 24, 2011 #2

    Dick

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    Sure. Use the definition of f'(z)=lim h->0 (f(z+h)-f(z))/h. Show the limit is different if you pick h to be real from the limit if you pick h to be imaginary. That's really what the content of the CR equations is.
     
    Last edited: Oct 24, 2011
  4. Oct 24, 2011 #3
    I think I get it now.

    (f(z+h)-f(z))/h

    (conjugate((z+h)-z))/h = h(conjugate)/h

    If h=Δx, the ratio equals 1
    If h=Δiy, the ratio equals -1.

    Since the two approaches don't agree for any z, z(conj) is not analytic anywhere. Correct?
     
  5. Oct 24, 2011 #4

    Dick

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    Yep, that's it. That's how you derive CR.
     
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