Show z^2 maps the circle |z-1|=1 to a cardioid

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In summary, when we square a complex number, we transform the circle |z-1|=1 to a cardioid shape by doubling its argument and multiplying its magnitude. The circle represents equidistant points from z=1 and the cardioid is a mathematical curve used in polar coordinates. This mapping can be generalized to other complex functions and has practical applications in engineering and physics. Real-life examples include the shape of a heart, satellite orbits, hurricanes, and rugs.
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Homework Statement


show z^2 maps the circle |z-1|=1 to a cardioid 2(1+cos(theta))e^(i*theta)


Homework Equations





The Attempt at a Solution


I tried using the restriction |z-1| = 1 and simply got that r^2=2x.

Then I tried to apply that and got

z^2=2xe^(2i*theta)

I've been trying for a good 15 minutes.
 
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  • #2
NVM, take z = 1+cos +isin
 

1. How does z^2 map the circle |z-1|=1 to a cardioid?

When we take the complex number z and square it, we essentially multiply its magnitude and double its argument. This results in the circle |z-1|=1 being mapped to the cardioid shape, which can be seen by plotting various points on both shapes and observing the transformation.

2. What is the significance of the circle |z-1|=1 and the cardioid shape?

The circle |z-1|=1 represents all the points in the complex plane that are equidistant from the point z=1, also known as the center of the circle. The cardioid shape, on the other hand, is a mathematical curve that is often used in the study of polar coordinates. It is formed by the parametric equation r=1+cos(theta), where r represents the distance from the origin and theta represents the angle.

3. Can the mapping of z^2 to a cardioid be generalized to other complex functions?

Yes, the mapping of z^2 to a cardioid is just one example of how a complex function can transform a shape in the complex plane. Other functions, such as z^3 or z^4, can result in different mappings to various shapes. It all depends on the properties of the specific function being used.

4. How is this mapping useful in practical applications?

The mapping of z^2 to a cardioid has many practical applications, particularly in engineering and physics. It can be used to model and analyze various physical phenomena, such as fluid dynamics and electromagnetic fields. It can also help in solving complex equations and predicting the behavior of systems.

5. Are there any real-life examples of the mapping of z^2 to a cardioid?

Yes, there are many real-life examples of this mapping. One notable example is the shape of a heart, which closely resembles a cardioid. The shape of a satellite's orbit around a planet is also a cardioid, with the planet being the center of the circle and the satellite tracing out the cardioid shape as it moves. Other examples include the shape of a hurricane and the pattern of a cardioid-shaped rug.

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