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I'm trying to evaluate the integral: $\int_0^\infty e^{-(ct)^\alpha} \cos(x t) dt$, where $\alpha$ and $c$ are parameters.

This integral arises from trying to solve for the probability density for a symmetric $\alpha$-stable probability distribution.

Can this integral be expressed in terms of some special functions, or at least does it have a ready-made numerical solution method?

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  • $\begingroup$ I do not think these can be calculated explicitly. $\endgroup$ – Will M. Nov 15 '18 at 23:42
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    $\begingroup$ For $\alpha = 0, 1, 2$, you can use Fourier Transform tables and Fourier Transform theorems. For other values of $\alpha$, I have no idea. $\endgroup$ – Andy Walls Nov 16 '18 at 0:00
  • $\begingroup$ Why not using the imaginary description to define the sum of two exponents and calculate the two resulting integrals? $\endgroup$ – Moti Nov 16 '18 at 2:08
  • $\begingroup$ I believe that if $\alpha$ is an even natural number you can employ Feynman’s Trick. For non natural numbers, I’m not sure it has a closed from solution. I can post the Feynman trick method if you would like? $\endgroup$ – user150203 Nov 24 '18 at 9:01

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