Expressing an lntegral in Standard, Nonstandard Functions

Question

Let $\alpha \in (0,2]$. For stable symmetric distributions with characteristic function of the form $e^{-\vert t \vert ^{\alpha}}$, their PDFs are given by the Fourier transform $$f(x)=\frac{1}{2\pi}\int _{-\infty} ^ {\infty}e^{-\vert t \vert ^{\alpha}-itx}dt,$$ where $i=\sqrt{-1}$.

Question: How can $f(x)$ or$$F(x)=\frac{1}{2\pi}\int _{-\infty} ^ xe^{-\vert t \vert ^{\alpha}-itx}dt,$$

be expressed in terms of standard (or non-standard) mathematical functions? I know only of a few choices of $\alpha$ for which $f$ can be simplified.

score 1 · Accepted Answer · edited Apr 13 '17 at 12:20

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The Fourier transform of $e^{-|x|^k}$ is given by a Fox-Write function, that is a generalized hypergeometric function. To call them standard or not is just a matter of taste, in my humble opinion, but they have remarkable properties: see, for instance, a recent question of mine and Noam Elkies' excellent answer.

edited Apr 13 '17 at 12:20

Community

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answered Aug 22 '16 at 01:59

Jack D'Aurizio

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(Question updated) Mathematica gives a simpler looking expression (which I am skeptical of). – The Substitute Aug 22 '16 at 02:42
Probably because you did not type the imaginary unit through esc+ii+esc so a relevant piece was parsed as a constant. – Jack D'Aurizio Aug 22 '16 at 02:45
Thank you. This is my first time using Mathematica. I fixed the $i$, but the result is the same according to Mathematica. – The Substitute Aug 22 '16 at 02:51
How can t appear in the integral? Put a space between i, t and x. – Jack D'Aurizio Aug 22 '16 at 02:53

Expressing an lntegral in Standard, Nonstandard Functions

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