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.