Complex equation

First of all you'll have to excuse any miss-spelling an incorrect expressions, I am a swede and am not to good regarding the math grammar in english..

Now for the problem at hand:

Solve the equation and answer in polar form r(cosv + isinv)

$\displaystyle z^4 = -i $


Thanx!

Hello DenOnde

Welcome to Math Help Forum!

Your English is excellent. It's certainly much better than my Swedish!

Quote:

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Originally Posted by DenOnde

First of all you'll have to excuse any miss-spelling an incorrect expressions, I am a swede and am not to good regarding the math grammar in english..

Now for the problem at hand:

Solve the equation and answer in polar form r(cosv + isinv)

$\displaystyle z^4 = -i $

Thanx!

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If we write

$\displaystyle z = r(\cos\theta + i\sin\theta)$

then, using De Moivre's Theorem:

$\displaystyle z^4 = r^4(\cos\theta + i\sin\theta)^4$

$\displaystyle =r^4(\cos4\theta+i\sin4\theta)$

We now express $\displaystyle - i$ in polar form. On the Argand diagram it is represented by the point $\displaystyle (0,-1)$, so has modulus $\displaystyle 1$ and argument $\displaystyle -\pi/2$. So:

$\displaystyle -i=1\Big(\cos(-\pi/2)+i\sin(-\pi/2)\Big)$

So:

$\displaystyle z^4=-i$

$\displaystyle \Rightarrow r^4(\cos4\theta+i\sin4\theta)=\cos(-\pi/2)+i\sin(-\pi/2)$

$\displaystyle \Rightarrow \left\{\begin{array}{l}
r^4\cos4\theta=-\cos(-\pi/2)\\
r^4\sin4\theta = -\sin(-\pi/2)
\end{array}\right .$

$\displaystyle \Rightarrow \left\{\begin{array}{l}
r=1\\
4\theta=-\pi/2+2n\pi, n \in \mathbb{Z}
\end{array}\right .$

So the values of $\displaystyle \theta$ are given by:

$\displaystyle 4\theta = -\pi/2, -\pi/2\pm2\pi,-\pi/2\pm4\pi, ...$

and its principle values (those between $\displaystyle -\pi$ and $\displaystyle \pi$) are given by:

$\displaystyle \theta = -5\pi/8, -\pi/8, 3\pi/8, 7\pi/8$

Grandad

My hat off to you sir :)

Excellent explaination!