How can I find all complex solutions of the following?

z^4 = -1 + sqrt(3*i)

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- Apr 26th 2009, 08:47 AMposix_memalignFind all complex solutions
How can I find all complex solutions of the following?

z^4 = -1 + sqrt(3*i) - Apr 26th 2009, 11:45 AMrunning-gag
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 $\displaystyle \theta = \frac{\pi}{6} + k\frac{\pi}{2}$