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Complex variables transformation/mapping of w=z^3

  1. Jul 16, 2008 #1
    Hi all! My basic problem is that I can't figure out how to do a transformation of z^3 from the w to the z plane.

    The problem statement, all variables and given/known data
    w=z^3. Region R in the w plane, R = {-1 [tex]\leq[/tex] u [tex]\leq[/tex] 0, 0[tex]\leq[/tex]v[tex]\leq[/tex]1 }.

    Region Q is the mapping of region R onto the z plane. Sketch region Q.

    The attempt at a solution
    I tried converting to polar coordinates, so w=r^3 * exp[3i*theta]
    Therefore: u=r^3 * cos(3*theta) and v=r^3 * sin(3*theta).
    Then I set those equations equal to the boundary conditions of R. i.e.
    r^3 * cos(3*theta) = -1
    r^3 * cos(3*theta) = 0
    r^3 * sin(3*theta) = 0
    r^3 * sin(3*theta) = 1
    But I have no idea how to plot any of those functions. Then I tried to do it in rectangular coordinates, and I got: w= x^3 + 3iyx^2 - 3xy^2 - iy^3
    Therefore: u= x^3 - 3xy^2 and v= 3yx^2 - y^3
    I had the same problem there in that I couldn't sketch either of those functions - and I don't think they're right anyway. Am I just approaching this problem incorrectly altogether? I could really use any help you could offer.
  2. jcsd
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