# Integral involving exponential of fractional power

Can anything be said about the Fourier integral

$\int_{-\infty}^{\infty} \exp\left[ika - (\gamma + ik)^{2/3}\right]dk$

where $a > 0$ and $\gamma > 0$?

Can it be related to some special function? It appears in the physics application described in this MO question.

• Closed form for the case $k=1,\gamma=0$ is not found by Maple 13. – Gerald Edgar Apr 29 '10 at 12:26
• He did say gamma > 0. I'm almost positive it converges to a real value when alpha = gamma = 1, as the imaginary part cancels itself out (limit of the imaginary component's integral from -a to a as a goes to infinity is zero, as it's an odd function). I wouldn't be surprised if this was the case for all alpha and gamma. I'll do some more experiments. – Gabriel Benamy Apr 29 '10 at 13:58

## 1 Answer

upon a change of variables, $\gamma+ik\mapsto ik'$ it takes the form of the generating function of a socalled stable distribution (with stability parameter $\alpha=2/3$ http://en.wikipedia.org/wiki/Stable_distribution