Complex Analysis - Sketch a curve

[kjartan]

Homework Statement

sketch the curve in the z-plane and sketch its image under w=z^2

|z-1|=1

Homework Equations

z=|z|e^(iArgz)
argw=2argz

The Attempt at a Solution

At first I simply sketched the solution for a circle centered at (1,0) in the z-plane and then mapped that to another circle in the w-plane also centered at (1,0). Then I realized that mapping a circle to a circle under the squaring function only works for circles centered at the origin. So, what I have now is sort of ellipse-ish shaped, but I'm not sure how to characterize the solution set in general.

Any help appreciated.

 

[hunt_mat]

What is the equation you get.

 

[kjartan]

I don't have an equation at this point. Basically, I just plotted some points.

If theta is taken with respect to the origin, not the center of this circle, I identified the points on the original circle (before squaring) corresponding to +/- (0, pi/12, pi/8, pi/6, pi/4, pi/3) and then actually just calculated what happened to those points when squared, and plotted them.

That's why I'm asking here, I'm not sure how to characterize this in a better way.

Thanks for taking a look.

 

[kjartan]

Update: after squaring, the image obtained is what is called a "cardioid shape" not "ellipse-ish."

Now that I have the shape, I am working "backwards" to obtain a description.

This confirms the shape:
http://en.wikipedia.org/wiki/Cardioid#Cardioids_in_complex_analysis

The rest is algebraic manipulation, so no further help is necessary.