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Homework Help: Show where the functions is anlaytic and differentiable

  1. Apr 7, 2012 #1
    1. The problem statement, all variables and given/known data

    z→x3+ i(1 - y)3: Show where the functions is analytic and differentiable.

    2. Relevant equations

    3. The attempt at a solution
    For a function to be analytic cauchy-riemann equations must hold.. so
    ux = vy and uy = -vx

    Now f(z) = x3 + i(1 - y)3 is already in the form u(x,y) + iv(x,y) with u(x,y) = x3 and v(x,y) = (1 - y)3 So:

    ux = 3x2 ; vy = -3(1 - y)2 and
    uy = 0 ; -vx = 0

    Does this then mean cauchy-riemann does not hold anywhere but 0 so f is not differentiable on a neighbourhood of any point in C which means it is nowhere analytic.
  2. jcsd
  3. Apr 7, 2012 #2


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    Staff Emeritus
    Science Advisor
    Gold Member

    From what you wrote down it doesn't look like the Cauchy Riemann equations hold at x=y=0
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