# Complex variables transformation/mapping of w=z^3

1. Jul 16, 2008

### lelam

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 $$\leq$$ u $$\leq$$ 0, 0$$\leq$$v$$\leq$$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.