What contour would one use to integrate the following equation?
$\int_{0}^{\infty}\frac{x^a}{(x^2+1)^2}dx$ where $-1 < a <3 $ and $x^a= e^{aln(x)}$
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Using integration by parts and the integral in this answer, $$ \begin{align} \int_0^\infty\frac{x^a}{(1+x^2)^2}\,\mathrm{d}x &=\frac12\int_0^\infty\frac{x^{\frac{a-1}2}}{(1+x)^2}\,\mathrm{d}x\\ &=-\frac12\int_0^\infty x^{\frac{a-1}2}\,\mathrm{d}\frac1{1+x}\\ &=\frac{a-1}4\int_0^\infty\frac{x^{\frac{a-3}2}}{1+x}\,\mathrm{d}x\\ &=\pi\,\frac{a-1}4\,\csc\left(\pi\frac{a-1}2\right)\\[3pt] &=\pi\,\frac{1-a}4\,\sec\left(\pi\frac a2\right) \end{align} $$ So we can use the contour used in that answer.
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The question asked about the appropriate contour – jim Apr 07 '16 at 08:29
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@jim: I have referenced another answer. – robjohn Apr 07 '16 at 08:35
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Thanks for the suggestion! – urpi Apr 08 '16 at 06:33