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.