How to derive $(iw)^{n}$ into the format $A+ib$ ($A$ is the real part, $B$ is the imaginary part), where $i=\sqrt{-1}$, $w<0$, $n$ is a fractional number such as $\frac{1}{2}, \frac{2}{3}, ...$?
Note that $\sqrt{xy}$ sometimes not equal to $\sqrt{x}\cdot\sqrt{y}$.
You are essentially looking at $$ (iw)^n = |w|^n(-i)^n $$ we know that $$ -i = \cos\left(\frac{3\pi}{2}\right) + i\sin\left(\frac{3\pi}{2}\right) = \mathrm{e}^{i\frac{3\pi}{2}} $$ so we find $$ |w|^n\mathrm{e}^{n\cdot i\frac{3\pi}{2}} $$ or $$ |w|^n\left(\cos\left(\frac{3\pi}{2}n\right) + i\sin\left(\frac{3\pi}{2}n\right)\right) $$