Residue Theorem

**Deadstar** · April 18th 2009, 11:56 AM

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

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

Lets call the integral above I.

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

Then when i find these residues i sub them into $\frac{z^2}{4z^3} = \frac{1}{4z}$ to get the residues but i cant figure out what z should be. It SHOULD be $e^{\frac{\pi i}{4}}$ and $e^{\frac{3 \pi i}{4}}$ but i have no idea how to get those. I got $\sqrt{i}$ and $i \sqrt{i}$ . Are those equivalent?!?

**Opalg** · April 18th 2009, 12:24 PM

If $z^4+1=0$ then $z^4=-1$ . So you are looking for fourth roots of –1. How do you find them? (Answer: De Moivre's theorem. Write –1 as $e^{i\pi}$ and raise it to the power 1/4.)

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