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I need to show the following result:

$$ \int_{-\infty}^\infty \frac{1}{(1+x^2)^{n+1}}dx\, = \frac{1\cdot 3\cdot\ldots\cdot(2n-1)}{2\cdot 4\cdot\ldots\cdot(2n)}\pi $$

With n=1,2,3,...

This function has a pole at i and -i. I've tried a semicircle in the upperhalf of the plain, but the residue then goes to infinity. I've also tried a rectangle in the upperhalf that stays beneath i, but all 3 sides that do not include the integral we're looking for go to zero because of the R in the denominator.

Anyone with tips?

  • 1
    What do you mean by "the residue then goes to infinity"?2012-08-07

4 Answers 4