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We need some sort of analytic expression for the integral:

$$\int^\infty _{-\infty} \frac{\sqrt{\alpha^2 + 1}}{(\mathrm{i}\alpha)^{\frac{3}{4}}}\mathrm{e}^{\mathrm{i}\alpha \chi} \mathrm{d}\alpha$$

where $\chi$ is a real number. Any thoughts?

EDIT (thanks Math1000 for the feedback): For context, it's an integral that arises in fluid dynamics as part of an inverted fourier transform.

Other terms that arise do not have a square root so we can formulate this as a gamma function. However here such a trivial rotation doesn't seem to give us anything useful. Plugging the integral into MAPLE gives a rather complicated looking result in terms of hypergeometric functions, but experience has taught us that usually a simpler solution should exist.

Hopefully this complies a little better with the etiquette.

QMRush
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  • What are your thoughts? – Math1000 Jul 10 '15 at 13:26
  • We can do similar integrations without the square root by rotating the contour and using the definition for a gamma function, but here the square root makes that rather difficult! – QMRush Jul 10 '15 at 13:28
  • My point is, it's against site etiquette to ask "How to compute _____" without any of your own context or insight. So you may want to add some of that to the question. – Math1000 Jul 10 '15 at 13:29
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    No problem, I've added a little more to the question, hopefully it's a little better now! :) – QMRush Jul 10 '15 at 13:35
  • I am not sure that this is defined - am having a hard time seeing how the denominator can be single-valued (surely $i$ has four fourth roots). – Marconius Jul 10 '15 at 23:03

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