Complex Analysis - Sketch a curve

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Homework Help Overview

The problem involves sketching a curve in the z-plane defined by the equation |z-1|=1 and determining its image under the transformation w=z^2. The subject area is complex analysis, focusing on the mapping of geometric shapes in the complex plane.

Discussion Character

  • Exploratory, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to sketch the curve and its image, initially considering it as a circle but later questioning the nature of the mapping due to the center of the circle not being at the origin. They explore specific points on the circle and their transformations.

Discussion Status

The discussion is ongoing, with participants exploring the characteristics of the curve and its image. Some guidance has been offered regarding the shape of the image, with the original poster updating their understanding to identify it as a cardioid shape.

Contextual Notes

The original poster notes that they do not have a formal equation at this point and are working from plotted points. There is an indication of algebraic manipulation needed to further describe the shape, but no resolution has been reached.

kjartan
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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.
 
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What is the equation you get.
 
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.
 
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.
 
Last edited:

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