eulers

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August 7th 2009, 09:53 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
August 7th 2009, 10:12 PM
red_dog

$i=\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}$

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

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

$k=1\Rightarrow z=\cos\frac{5\pi}{4}+i\sin\frac{5\pi}{4}=-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i$
August 7th 2009, 10:26 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 $z_1, z_2$ .
August 7th 2009, 11:46 PM
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.