How can I find all complex solutions of the following?
z^4 = -1 + sqrt(3*i)
Hi
Isn't it $\displaystyle z^4 = -1 + \sqrt{3}\:i$
One way is to go to exponential form
$\displaystyle z^4 = 2\left(-\frac12 + \frac{\sqrt{3}}{2}\:i\right) = 2\:e^{i\frac{2\pi}{3}}$
Then $\displaystyle z = r\:e^{i\theta}\implies z^4 = r^4\:e^{4i\theta}$
$\displaystyle r^4 = 2$ and $\displaystyle 4\theta = \frac{2\pi}{3} + 2k\pi$
$\displaystyle r = 2^{\frac14}$ and $\displaystyle \theta = \frac{\pi}{6} + k\frac{\pi}{2}$