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Math Help - Complex Number Proof

  1. #1
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    Complex Number Proof

    For the complex equation z^4 = cosx + isinx, prove that the sum of the four solutions is always zero, no matter what size x is.

    This is what I've done so far.

    Z = (cis(x + 360k))^1^/^4<br />
= cis(x/4 + 90k)

    Can someone please teach me how to do the rest? Thanks in advance
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  2. #2
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    Rewrite it as \displaystyle z^4 = e^{ix}.

    This means that the first fourth root is \displaystyle z = \left(e^{ix}\right)^{\frac{1}{4}} = e^{i\frac{x}{4}} = \cos{\left(\frac{x}{4}\right)} + i\sin{\left(\frac{x}{4}\right)}.

    The other fourth roots are evenly spaced around a circle, so differ by an angle of \displaystyle \frac{\pi}{4}. What are the other solutions? What do you get when you add them together?
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  3. #3
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    A little twist on ProveIt's approach:

    Let w = \cos x + i \sin x, so the equation can be written
    z^4 - w = 0.

    If the roots are r_1, r_2, r_3, r_4, then we must have
    z^4 - w = (z - r_1) (z - r_2) (z - r_3) (z - r_4) .

    If we expand the right-hand side of this equation, what can we say about the coefficient of z^3?
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