The function $f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}$ represents a rotation around some complex number $c$. Find $c$.
Hello, I am having some trouble trying to do this problem. I thought of the given equation as a "circle" equation, like $(x-a)^2+(y-b)^2=r^2$, but in complex form. Can anyone help me?
Hint: if you will rotate the complex plane with nonzero angle, there will be exactly one point $p $ that will be fixed, $ f(p) = p $.
Another way is as follows: a rotation of angle $ \theta $ about the origin is just multiplication by $ e^{i \theta} $. About a different center $ c $ $$ f(z) = e^{i \theta} ( z - c) + c \\ $$ so you can check a different way.