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Cauchy Riemann Problem

  1. Sep 14, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that when f(z)=x^3+i(1-y)^3, it is legitimate to write:

    f'(z)=u_x+iv_x=3x^2
    only when z=i

    2. Relevant equations

    Cauchy riemann equations:

    u_x=v_y , u_y=-v_x
    f'(z)=u_x+i*v_y
    3. The attempt at a solution
    u=x^3
    v=(1-y)^3
    u_x=3*x^2
    v_y=-3*(1-y)^2
    x^2=-(1-y)^2 =

    u_y=0
    -v_x=0

    f'(z)=3*x^2+i(0)= 3*x^2

    I don't understand why z=i => z=o+i*1? is relevant to show that f'(z)=3*x^2
     
  2. jcsd
  3. Sep 14, 2008 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    f'(z) only has a complex derivative if it satisfies the Cauchy-Riemann equations. You have correctly found that means x^2=-(1-y)^2. How many values of x and y satisfy that?
     
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