Solving Tricky Supergaussian-Rician Integral: Methods & Challenges

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SUMMARY

The discussion focuses on solving the integral of a supergaussian multiplied by a Rician distribution, specifically the integral of the form \(\int_0^{\infty}e^{-ax^4}e^{-bx^2}xI_0(cx)dx\). The author notes that while they have successfully solved a normal Gaussian times a Rician by completing the square, this approach does not extend to higher-order cases. They have experimented with substitution methods, including \(u = x^2\) and the Laplace transform, but have not yet found a definitive solution.

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I am trying to solve the integral of a supergaussian multiplied by a Rician distribution.

Basically, I am trying to solve an integral of the form

[itex] \int_0^{\infty}e^{-ax^4}e^{-bx^2}xI_0(cx)dx[/itex]

I have no particular reason to believe this has a closed form.
However, I have solved a normal gaussian times a Rician; however, that involved completing the square and the integral being a valid Rician, thus summing to 1 and leaving multipliers, which will not generalize to higher order.

I have tried a few methods, including substituting u = x^2 and then Laplace transform.
 
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