# eulers

• Aug 7th 2009, 10:53 PM
acosta0809
eulers
The question is

http://www.webassign.net/www22/latex...d58f9d7ef1.gif

im not sure how to do this, i know that i=sqrt(-1) but im not sure how to compute this...

thank you
• Aug 7th 2009, 11:12 PM
red_dog
$i=\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}$

$z=\sqrt{i}=\cos\frac{\frac{\pi}{2}+2k\pi}{2}+i\sin \frac{\frac{\pi}{2}+2k\pi}{2}, \ k=0,1$

$k=0\Rightarrow z=\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}=\frac{\sqrt {2}}{2}+\frac{\sqrt{2}}{2}i$

$k=1\Rightarrow z=\cos\frac{5\pi}{4}+i\sin\frac{5\pi}{4}=-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i$
• Aug 7th 2009, 11:26 PM
Failure
Quote:

Originally Posted by acosta0809
The question is

http://www.webassign.net/www22/latex...d58f9d7ef1.gif

im not sure how to do this, i know that i=sqrt(-1) but im not sure how to compute this...

thank you

You don't get a unique answer to this:

a) $\sqrt{\mathrm{i}}=\left(\mathrm{e}^{\mathrm{i}\lef t(\frac{\pi}{2}+n\cdot 2\pi\right)}\right)^{1/2}=$(Thinking) $\mathrm{e}^{\mathrm{i}\left(\frac{\pi}{4}+n\cdot\p i\right)}$ where $n\in \mathbb{Z}$
So, basically you have the two answers $\sqrt{\mathrm{i}}=$(Thinking) $\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$ and $\sqrt{\mathrm{i}}=$ (Thinking) $\mathrm{e}^{\mathrm{i}\frac{5\pi}{4}}$
Therefore, wirting an equality sign $\sqrt{\mathrm{i}}=\ldots$ seems rather problematic. Rather, we would want to say that the equation $z^2=\mathrm{i}$ has two solutions, $z_1=\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$ and $z_2=\mathrm{e}^{\mathrm{i}\frac{5\pi}{4}}$.

b) Just use $r\cdot\mathrm{e}^{\mathrm{i\varphi}}=r\cdot\cos(\v arphi)+\mathrm{i}r\cdot \sin(\varphi)$ for the above two solutions $z_1, z_2$.
• Aug 8th 2009, 12:46 AM
alunw
Complex square root without trig
Quote:

Originally Posted by Failure
You don't get a unique answer to this:

a) $\sqrt{\mathrm{i}}=\left(\mathrm{e}^{\mathrm{i}\lef t(\frac{\pi}{2}+n\cdot 2\pi\right)}\right)^{1/2}=$(Thinking) $\mathrm{e}^{\mathrm{i}\left(\frac{\pi}{4}+n\cdot\p i\right)}$ where $n\in \mathbb{Z}$
So, basically you have the two answers $\sqrt{\mathrm{i}}=$(Thinking) $\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$ and $\sqrt{\mathrm{i}}=$ (Thinking) $\mathrm{e}^{\mathrm{i}\frac{5\pi}{4}}$
Therefore, writing an equality sign $\sqrt{\mathrm{i}}=\ldots$ seems rather problematic. Rather, we would want to say that the equation $z^2=\mathrm{i}$ has two solutions, $z_1=\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$ and $z_2=\mathrm{e}^{\mathrm{i}\frac{5\pi}{4}}$.

b) Just use $r\cdot\mathrm{e}^{\mathrm{i\varphi}}=r\cdot\cos(\v arphi)+\mathrm{i}r\cdot \sin(\varphi)$ for the above two solutions $z_1, z_2$.

a)Almost every number has two distinct square roots. But $\surd 2$ conventionally denotes the positive solution to $z^2=2$. So the usual convention for square roots of complex numbers is to say that $\surd x$ is the square root with a positive real part. If x is real and negative then $\surd x$ is the root with positive imaginary part. This means there is a branch cut along the negative real axis.

b) In fact you can find find square roots without resorting to trigonometry. The following pseudo-code, computes the square root (x,y) of the complex number (u,v)
Code:

if (u > 0) then   x = sqrt((u+sqrt(u*u+v*v))/2)   y = v/(2*x) else if (u < 0) then   y = sign(v)*sqrt((-u+sqrt(u*u+v*v))/2)   x = v/(2*y) else   x = sqrt(abs(v)/2)   if (v > 0)     y = x   else     y = -x
The point of testing the sign of u is to minimise round-off error.