[Math] Cardioid in complex plane.

Question

I have a question in complex analysis. The question is this:

• sketch the circle $|z-1|=1$. Find (geometrically) the polar equation of the image of this circle under the mapping $z \mapsto z^2$. Sketch image curve.

Thanks.

Answer 1

$$ | z - 1 | = 1 $$ $$ z = e^{i\theta} + 1 $$

$$ f(z)= z^2 $$

                       working in the w plane now.

$$\rho e^{i\phi} = (1 + e^ {i \theta})^2 $$

$$\rho e^{i\phi} = 1 + 2e^{i \theta} + e^{2i \theta} $$

$$\rho e^{i\phi} = e^{i\theta} ( e^{-i\theta} + 2 + e ^{i\theta}) $$

             using an identity for the complex cosine function

$$ \rho e^{i\phi} = e^{i\theta} ( 2 + 2 cos\theta) $$

$$ \rho= ( 2 + 2 cos\theta) , (\phi = \theta )$$

                    Which is a cardioid in the w plane 

Another method:

Parametric equation for the circle:

$x(t) = cos(t) + 1$ and $y(t) = sin(t) $

$f(z) = z^2 $ which takes $(u,v)$ to $(u^2 - v^2, 2uv)$

we get, after playing around a bit:

$ (1 + 2 cos(t) + cos(2t) , 2sin(t) + sin(2t) ) $

The equation of a cardioid.

Only, it should be in the form $ r = 2(1 + cos(t)) $ , same as above. So, a little work perhaps, it's the same cardioid...