Hi
I need a help in solving the following problem. the problem says: find the image of the unit circle |z|=1, under the mapping w=2z+z^2.
Thanks
Given a point on the unit circle, it will have the form $\displaystyle z=e^{(it)}$ where $\displaystyle 0\leq t \leq 2\pi$. Have: $\displaystyle w=2z+z^2 = 2e^{(it)}+e^{(2it)}$.
This means that the real part is $\displaystyle x=2cos(t)+cos(2t)$ and the imaginary part is $\displaystyle y=2sin(t)+sin(2t)$. These are the parametric equations of a cardioid. You can take several values of t and try to draw it.
Cardioid - Wikipedia, the free encyclopedia