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I thought I was fairly well-versed with using the residue theorem to evaluate improper integrals, but one problem has been giving me grief.

How does one compute the integral $$\int_{-\infty}^{\infty}\frac{e^{\alpha+ix}}{(\alpha+ix)^\beta} dx$$

for real numbers $\alpha>1$, $\beta>0$?

Note that $\beta$ is not necessarily an integer, which restricts the contours that can be used (ie. one needs to use contours on which a branch of logarithm can be defined).

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