# Thread: sqrt of a complex number

1. ## sqrt of a complex number

hello

let's say I have

z^2 = exp (1.5 * pi *i)

how do I find the values of z which satisfy this equation

thanks

2. Originally Posted by parallel
hello

let's say I have

z^2 = exp (1.5 * pi *i)

how do I find the values of z which satisfy this equation

thanks
Well, $\left ( e^a \right ) ^b = e^{ab}$, even when a and/or b is complex. Does this help?

-Dan

3. Originally Posted by topsquark
Well, $\left ( e^a \right ) ^b = e^{ab}$, even when a and/or b is complex. Does this help?

-Dan
That is not correct when $a,b\in \mathbb{C}$.

4. Originally Posted by parallel
hello

let's say I have

z^2 = exp (1.5 * pi *i)

how do I find the values of z which satisfy this equation

thanks
If $z\not = 0$ then $\sqrt{z} = \sqrt{|z|} e^{i\arg(z)/2}$.

Now given $e^{1.5 \pi i}$. And $\arg e^{1.5 \pi i} = -\frac{\pi}{2}$. Thus, its square root is given by $\sqrt{\left| e^{1.5\pi i} \right|} e^{-i\pi/4} = e^{-i\pi/4} = \frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2}$. The other root is the negative of this root.

5. Originally Posted by ThePerfectHacker
If $z\not = 0$ then $\sqrt{z} = \sqrt{|z|} e^{i\arg(z)/2}$.

Now given $e^{1.5 \pi i}$. And $\arg e^{1.5 \pi i} = -\frac{\pi}{2}$. Thus, its square root is given by $\sqrt{\left| e^{1.5\pi i} \right|} e^{-i\pi/4} = e^{-i\pi/4} = \frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2}$. The other root is the negative of this root.
thank you very much for your help