# Find all complex solutions

• Apr 26th 2009, 09:47 AM
posix_memalign
Find all complex solutions
How can I find all complex solutions of the following?

z^4 = -1 + sqrt(3*i)
• Apr 26th 2009, 12:45 PM
running-gag
Quote:

Originally Posted by posix_memalign
How can I find all complex solutions of the following?

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

Hi

Isn't it $z^4 = -1 + \sqrt{3}\:i$

One way is to go to exponential form
$z^4 = 2\left(-\frac12 + \frac{\sqrt{3}}{2}\:i\right) = 2\:e^{i\frac{2\pi}{3}}$

Then $z = r\:e^{i\theta}\implies z^4 = r^4\:e^{4i\theta}$

$r^4 = 2$ and $4\theta = \frac{2\pi}{3} + 2k\pi$

$r = 2^{\frac14}$ and $\theta = \frac{\pi}{6} + k\frac{\pi}{2}$