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Use the method of Fourier analysis to calculate the following integral:

$$ \int_{0}^{\infty} \frac{\cos x}{1+4x^2} \operatorname{d} x .$$

Could someone help about this question? what skills should I use? Should I change the $\cos$ function to $\exp$?

Vobo
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mnmn1993
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1 Answers1

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One can see $\int\limits_0^{\infty}\frac{\cos(x)}{1+4x^2}dx=\frac{1}{2}\int\limits_{-\infty}^{\infty}\frac{\cos(x)}{1+4x^2}dx$ as the Fourier trasform of $f(x)=\frac{\sqrt{2\pi}}{2(1+4x^2)}$ valuated in 1, which gives $\frac{\pi}{4\sqrt{e}}$.

Matteo
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  • Sorry,how can i get pi/2sqrt(e)?? – mnmn1993 Nov 30 '14 at 14:08
  • I edited. The Fourier transform $\widehat{f}(k)=\int_{-\infty}^{\infty}\frac{e^{-ikx}}{2(1+4x^2)}=\frac{1}{4}\pi e^{-\frac{|k|}{2}}$. This integral can be performed using the Residue Theorem. – Matteo Nov 30 '14 at 17:01