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Thread: Residue Theorem

  1. #1
    Super Member Deadstar's Avatar
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    Residue Theorem

    I cant figure out where the singularities are on this Q!

    Q: Use a large semicircular contour and the residue theorem to evaluate:
    $\displaystyle \int_{-\infty}^{\infty} \frac{x^2}{x^4 + 1} dx$.

    Lets call the integral above I.

    Then I = $\displaystyle 2 \pi i \sum$ residues of $\displaystyle \frac{z^2}{z^4 + 1}$.

    Then when i find these residues i sub them into $\displaystyle \frac{z^2}{4z^3} = \frac{1}{4z}$ to get the residues but i cant figure out what z should be. It SHOULD be $\displaystyle e^{\frac{\pi i}{4}}$ and $\displaystyle e^{\frac{3 \pi i}{4}}$ but i have no idea how to get those. I got $\displaystyle \sqrt{i}$ and $\displaystyle i \sqrt{i}$. Are those equivalent?!?
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    If $\displaystyle z^4+1=0$ then $\displaystyle z^4=-1$. So you are looking for fourth roots of 1. How do you find them? (Answer: De Moivre's theorem. Write 1 as $\displaystyle e^{i\pi}$ and raise it to the power 1/4.)
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