Finding the solutions of a complex number

[EmmaK]

Homework Statement

Find the 3 solutions of e^i$$\pi$$/3z³=1/(1+i)

Homework Equations

e^i$$\theta$$=cos($$\theta$$)+isin($$\theta$$)

The Attempt at a Solution

i have put i/(1+i) into polar form,1/$$\sqrt{2}$$ e^{i$$\stackrel{\pi}{4}$$}

So i get z³ = $$\stackrel{1}{\sqrt{2}}$$e^i-$$\pi/12$$

Then i got stuck... z³=r³e^i$$\theta$$

So shouldn't r³=1/$$\sqrt{2}$$ and $$\theta$$=-$$\pi$$/36 ..but that's only 1 solution?

 

[latentcorpse]

well.
if $z^3=\frac{1}{\sqrt{2}} e^{-\frac{i \pi}{12}}$
then $z=(\frac{1}{\sqrt{2}})^{\frac{1}{3}} e^{-\frac{i \pi}{36}}$

 

[EmmaK]

but that is only one solution and it asks for 3

 

[Count Iblis]

You can add any multiple of 2 pi i to theta.

 

[EmmaK]

ahh, of course. thank you!

 

[RoyalCat]

ahh, of course. thank you!

Note that for unique solutions, you need to add $$n\cdot 2\pi i$$ to the exponent of the complex number describing $$z^3$$

Otherwise you're just describing the same number over and over!

 

[EmmaK]

haha, oh yea.
where do you get n2/pi i from?

 

[RoyalCat]

haha, oh yea.
where do you get n2/pi i from?

That's how much you need to add to the angle of the exponent, $$r e^{i\theta}$$ so that you get the same value.

$$re^{i\theta}=re^{i(\theta+2\pi)}$$

You can easily see this using Euler's identity since the sine and cosine both have a period of $$2\pi$$ radians.

When you take the cube root, you use De-Moivre and divide the angle by 3. Note that you get different angles depending on whether you add $$2\pi$$ once, twice, or three times to the original exponent's angle.

 

[latentcorpse]

yes i should have mentioned that. sorry.

 

[EmmaK]

so the final answer is $$\stackrel{n2\pi}{3}$$? i think i just misread your post as $$\stackrel{2n}{\pi}$$ or something :)

 

[latentcorpse]

well, is $(\frac{2n \pi}{3})^3=\frac{1}{\sqrt{2}}e^{-\frac{1 \pi}{12}}$?

go to my first line of working in post 2, the other two solutions will correspond to $e^{-\frac{25 i \pi}{12}}$ and $e^{-\frac{49 i \pi}{12}}$.
then of course, you have to do the division by 3 etc as before to get to the final answer.