If $z=re^{i\theta}$ and $\theta\in (-\frac{\pi}{2},0)$, then argument of $z^3$ lies between ______.
My approach is as follow $z^3=r^3e^{i3\theta}$
$arg(z^3)=3\theta$
Hence $3\theta\in (-\frac{3\pi}{2},0)$ or $3\theta\in (0,\frac{\pi}{2})$ but this may not be correct. I need help
The Argument of a complex number $z$, if defined as $\theta \in \mathbb{R}$ such that $z=re^{i\theta}$, is multivalued because adding a factor of $2\pi$ doesn't change $e^{i\theta}$. However, to make argument a proper function, we restrict the range of the function to a suitable interval of the real line, say $(-\pi,\pi]$. This restricted entity is now a function which we call $Arg$. Other argument functions can be made by adding any integer multiple of $2\pi$ to this range.
Here $Arg(z)\in(-\pi/2,0)$ and therefore, $Arg(z^3) \in (-\pi,0)\cup(\pi/2,\pi]$. You can add integer multiples of $2\pi$ to this to get other answers.
I have provided to contradictory answers, and so they cannot both be correct. The first answer is correct though: $z^3$ has an argument in $\left(-\frac{3\pi}2,0\right)$ and every number that has an argument from that interval is of the form $z^3$, for some $z=re^{i\theta}$, with $\theta\in\left(-\frac\pi2,0\right)$.
