Consider the integral ##\displaystyle \int_{-\infty}^\infty \frac{e^{-|x|}}{1+x^2}dx ##. I should be able to use contour integration to solve it because it vanishes faster than ## \frac 1 x ## in the limit ## x \to \infty ## in the upper half plane. It has two poles at i and -i. If I use a semicircle in the upper half of the plane, I should only consider the residue at i which is:(adsbygoogle = window.adsbygoogle || []).push({});

## \displaystyle \lim_{z\to i} \frac{e^{-|z|}}{i+z}=\frac{e^{-|i|}}{2i}=\frac{1}{2ie} ##

So the integral is equal to ## \frac{2\pi i}{2ie}=\frac \pi e \approx 1.15 ##. But wolframalpha.com gives an expression involving Ci(x) and Si(x) that approximates to 1.24. What is wrong here?

Thanks

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# A Why can't I use contour integration for this integral?

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