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What is the answer(s) of this equation? ($w$ is a complex number.)

$$\frac{w + \bar{w}}{w - \bar{w}} = -w^2$$

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...and $\omega$ is supposed to be what? – J. M. Jul 23 '11 at 17:30
Are the omega's ($\omega$) and w's supposed to be different things? – Zev Chonoles♦ Jul 23 '11 at 17:30
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Are $\omega$ and $w$ supposed to be different variables? If so, I'd consider that remarkably unfortunate notation. – Ilmari Karonen Jul 23 '11 at 17:30
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An equation is a statement of fact (that the left hand side is equal to the right hand side), as as such, is not at question. Questions have answers. Statements don't have answers. What exactly is it that you want done with this equation you wrote? – Willie Wong♦ Jul 23 '11 at 17:33
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Whatever happened to "leave a comment if you think this question can be improved" (my paraphrase) ??? – The Chaz 2.0 Jul 23 '11 at 20:22
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3 Answers

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Hint: write $w=a+bi$ with $a, b$ real. Plug this in and separate into real and imaginary parts. Can you do that?

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Thank you. I tried to solve it directly, not using that method, and it doesn't work well. – Sqrt4 Jul 23 '11 at 17:40
@Sqrt4: Whenever I see a conjugate bar I think of this approach. Your left hand side simplifies nicely this way. – Ross Millikan Jul 23 '11 at 23:03
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HINT $\ $ Conjugating the equation yields $\ \bar w^{\:2}\: =\: - w^{\:2}\ $ so $\ (\bar w/w)^2 = -1\ $ so $\:\bar w = \pm\: i\: w\ \ldots$

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up vote 1 down vote

Note that $w +\bar{w}$ = 2 Re $w$, while $w -\bar{w}$ = 2i Im $w$. Substituting, we derive the equation Re $w$ = -i$w^2$ Im $w$. Thus, $w^2$ must be pure imaginary. Work from there.

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There is no mistake. If you follow my hint you will find 4 roots: $\: w = (\pm 1\pm i)/\sqrt{2}\:.$ – Gone Jul 23 '11 at 22:33
Does conj(1)^2 = - 1^2? conj(w)^2 = conj(w^2) – Stephen J. Herschkorn Jul 23 '11 at 22:34
The equation $\:\bar w^2 = - w^2\:$ is a consequence of taking the conjugate of the given equation. – Gone Jul 23 '11 at 22:40
While your contention that conj(w)^2 = -w^2 remains incorrect, I do see my error, viz., w - conj(w) = 2i Im w. IS there a way to remove my post?) – Stephen J. Herschkorn Jul 23 '11 at 22:46
What I wrote is correct. You have yet to clarify why you think otherwise. If you don't see the "delete" choice at the bottom of your post then click on "flag" and ask for a moderator to delete it for you. – Gone Jul 23 '11 at 22:52
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