Z (conjugate) not analytic?

  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


    Edit: Is there another approach? Because the CR Equations is something we learned later on.
  2. jcsd
  3. Dick

    Dick 25,913
    Science Advisor
    Homework Helper

    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. I think I get it now.


    (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. Dick

    Dick 25,913
    Science Advisor
    Homework Helper

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