Integral involving exponential of fractional power - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:14:23Z http://mathoverflow.net/feeds/question/22940 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22940/integral-involving-exponential-of-fractional-power Integral involving exponential of fractional power Gregory Putzel 2010-04-29T05:06:16Z 2010-12-04T19:48:54Z <p>Can anything be said about the Fourier integral</p> <p>$\int_{-\infty}^{\infty} \exp\left[ika - (\gamma + ik)^{2/3}\right]dk$</p> <p>where $a > 0$ and $\gamma > 0$?</p> <p>Can it be related to some special function? It appears in the physics application described <A HREF="http://mathoverflow.net/questions/22281/can-i-relate-the-l1-norm-of-a-function-to-its-fourier-expansion" rel="nofollow">in this MO question</A>.</p> http://mathoverflow.net/questions/22940/integral-involving-exponential-of-fractional-power/48298#48298 Answer by Carlo Beenakker for Integral involving exponential of fractional power Carlo Beenakker 2010-12-04T19:48:54Z 2010-12-04T19:48:54Z <p>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 $\alpha=2/3$ <a href="http://en.wikipedia.org/wiki/Stable_distribution" rel="nofollow">http://en.wikipedia.org/wiki/Stable_distribution</a></p>