Mapping unit circle from one complex plane to another

[sr3056]

I want to show that if the complex variables ζ and z and related via the relation

z = (2/ζ) + ζ

then the unit circle mod(ζ) = 1 in the ζ plane maps to an ellipse in the z-plane.

Then if I write z as x + iy, what is the equation for this ellipse in terms of x and y?

Any help would be much appreciated.

Thanks!

 

[Simon Bridge]

Welcome to PF;

Then if I write z as x + iy, what is the equation for this ellipse in terms of x and y?

... you mean: "how should I go about finding the equation of the ellipse?" Nobody is going to spoon-feed you the actual answer here - but we can help you find it for yourself.

You can help us do that by attempting the problem.

Start out by writing out the relations you know:

1. z = (2/ζ) + ζ
2. |ζ| = 1

3. ζ = γ + iλ
4. z = x + iy

5. ... any other relations that must hold true?

Presumably you can expand 1 and 2 in terms of 3?
Presumably you can look up the general equation of an ellipse?

Now where do you get stuck?

 

[sr3056]

I think I've got it now..

Let ζ = u+iv so u²+v²=1 because |ζ| = 1

2/ζ + ζ = 2 / (u+iv) + (u+iv) = 2(u−iv) / (u²+v²) + (u+iv) = 3u−iv

∴ x+iy = 3u−iv and so u=x/3, v=−y

From u²+v² = 1 this yields (x/3)²+y² = 1, an ellipse

Thanks for your help

 

[Simon Bridge]

No worries :-)
Sometimes the trick is starting without knowing whe re you are going.

 

[CellCoree]

Hello,

Thank you for sharing your findings. Your work on mapping the unit circle from one complex plane to another is very interesting.

To show that the unit circle mod(ζ) = 1 in the ζ plane maps to an ellipse in the z-plane, we can use the definition of the unit circle, which is a circle with radius 1 centered at the origin. In the ζ plane, this can be represented as |ζ| = 1.

Now, let's substitute z = (2/ζ) + ζ into this equation. We get:

|(2/ζ) + ζ| = 1

Simplifying, we get:

|2 + ζ^2| = |ζ|

Using the definition of complex modulus, we can write this as:

√(4 + ζ^4) = √(ζ^2)

Squaring both sides, we get:

4 + ζ^4 = ζ^2

Rearranging, we get:

ζ^4 - ζ^2 + 4 = 0

This equation represents an ellipse in the ζ plane. Now, if we express z = x + iy, we can rewrite this equation in terms of x and y:

(x^2 + y^2)^2 - (x^2 - y^2) + 4 = 0

This is the equation of the ellipse in the z-plane that maps to the unit circle in the ζ plane. I hope this helps. Keep up the good work!