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Math Help - Complex Factorisation

  1. #1
    Newbie
    Joined
    Aug 2008
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    4

    Complex Factorisation

    Hi everyone, I was wondering if anyone can show me how these factorisations were carried out?

    <br />
z^2 + i = (z + \frac{1}{\sqrt{2}}(1-i))(z - \frac{1}{\sqrt{2}}(1-i))<br />
    <br />
z^2 - i = (z + \frac{1}{\sqrt{2}}(1+i))(z - \frac{1}{\sqrt{2}}(1+i))<br />

    Any help would be much appreciated!

    edit: i is the imaginary unit, as always.

    2nd edit: For anyone interested they arose while I was factorising this:

    x^4 + 1

    in order to evalute this:

    <br />
\int_{-\infty}^{+\infty} \frac{e^{isx}}{x^4+1} dx<br />

    using contour integration.
    I have the solution, I just can't replicate those factorisations.
    Last edited by QuestionMark; September 8th 2008 at 07:28 AM.
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  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    z^2=-i

    z=\sqrt{-i}=e^{i/2(-\pi/2+2k\pi)};\; k=0,1

    same dif with the other one except the argument of i is \pi/2 as well as:

    z^4=-1

    z=(-1)^{1/4}=e^{i/4(\pi+2k\pi)};\; k=0,1,2,3
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