Complex variables transformation/mapping of w=z^3

[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.

Homework Statement
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.

 

[mighty2000]

Hi there, it looks like you're on the right track with your approach to converting to polar coordinates. However, it seems like you may have made a mistake in the equations for u and v. Let's go through it together step by step.

First, we start with the equation w=z^3. In polar coordinates, this becomes w=r^3 * exp[3i*theta]. Now, we want to find the equations for u and v in terms of r and theta. We can do this by equating the real and imaginary parts of the equation w=r^3 * exp[3i*theta].

For the real part, we have u=r^3 * cos(3*theta). This is correct.

For the imaginary part, we have v=r^3 * sin(3*theta). This is also correct.

Now, we have to set these equations equal to the boundary conditions of R. In R, we have -1≤u≤0 and 0≤v≤1. This means that -1≤r^3 * cos(3*theta)≤0 and 0≤r^3 * sin(3*theta)≤1.

Solving for r, we get -1/r^3≤cos(3*theta)≤0 and 0≤sin(3*theta)≤1/r^3. Now, we can use the inverse trigonometric functions to solve for theta in terms of r. This gives us the following equations:

-1/r^3≤cos(3*theta)≤0
arccos(-1/r^3)≤3*theta≤arccos(0)
-arccos(-1/r^3)/3≤theta≤0

0≤sin(3*theta)≤1/r^3
0≤3*theta≤arcsin(1/r^3)
0≤theta≤arcsin(1/(3*r^3))

Now, we have the equations for theta in terms of r. To sketch region Q, we can plot these equations in the polar coordinate system, with r as the radius and theta as the angle. This will give us a polar plot of region Q.

I hope this helps! Let me know if you have any further questions or if you need any clarification on the steps. Good