Let $D$ be the first open quadrant ($x,y>0$). Find an elementary function that maps $D$ onto the interior of the complement $int(\mathbb{C}-D)$.
Edit: I thought of using De-Moivre's formula, $(\cos(\pi/2)+i\sin(\pi/2) ) ^3 =\cos(3\pi/2)+i\sin(3\pi/2) $. So, the function I am looking for will most likely be something involving $z^3$. But, how can I then "rotate" the complex numbers so that the resulting map will be onto the interior of the complement?
Thank you
You are on the right track. $D$ is the set of all complex numbers with an argument in the range $(0, \pi/2)$. The function $z \mapsto z^3$ maps that to the set $\{ 0 < \arg(z) < 3\pi/2)$.
This can be mapped onto the target domain $int(\Bbb C \setminus D) = \{ \pi/2 < \arg(z) < 2\pi$ with a multiplication with a complex number having the argument $\pi/2$.
So one possible choice for the desired function is $f(z) = iz^3$.