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Thread: Complex Factorisation

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    4

    Complex Factorisation

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

    $\displaystyle
    z^2 + i = (z + \frac{1}{\sqrt{2}}(1-i))(z - \frac{1}{\sqrt{2}}(1-i))
    $
    $\displaystyle
    z^2 - i = (z + \frac{1}{\sqrt{2}}(1+i))(z - \frac{1}{\sqrt{2}}(1+i))
    $

    Any help would be much appreciated!

    edit: $\displaystyle i$ is the imaginary unit, as always.

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

    $\displaystyle x^4 + 1$

    in order to evalute this:

    $\displaystyle
    \int_{-\infty}^{+\infty} \frac{e^{isx}}{x^4+1} dx
    $

    using contour integration.
    I have the solution, I just can't replicate those factorisations.
    Last edited by QuestionMark; Sep 8th 2008 at 06:28 AM.
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  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    $\displaystyle z^2=-i$

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

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

    $\displaystyle z^4=-1$

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