# Thread: equation involving complex numbers

1. ## equation involving complex numbers

hi,
'Find all the solutions of z bar
= z^3 in cartesian form ?', thanks in advance for any help

2. Originally Posted by speckmagoo
hi,
'Find all the solutions of z bar
= z^3 in cartesian form ?', thanks in advance for any help
Hi

You can solve by setting $\displaystyle z=x+iy$ and expand $\displaystyle z^3 = (x+iy)^3$
or more simply by setting $\displaystyle z = r\:e^{i\theta}$ and solve $\displaystyle r\:e^{-i\theta} = r^3\:e^{3i\theta}$

3. hi thanks for ur assistance!!, id really appreciate if you could explain a little further as im not quite understanding how to go about using the approach u suggested, cheers

4. Let $\displaystyle z = r\:e^{i\theta}$

$\displaystyle r\:e^{-i\theta} = r^3\:e^{3i\theta}$

leads to $\displaystyle r = r^3$ and $\displaystyle 3\theta = -\theta +2k\pi$

$\displaystyle r = r^3$ is solved by $\displaystyle r \r^2-1)=0$ which leads to $\displaystyle r=0$ or $\displaystyle r=1$

$\displaystyle 3\theta = -\theta +2k\pi$ leads to $\displaystyle \theta = k\:\frac{\pi}{2}$

The solutions are therefore $\displaystyle z=0, z=1, z=i, z=-1, z=-i$

5. hi,
thanks for ur assistance again, it now makes more sense, thanku