A problem about integral of modified bessel function

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SUMMARY

The discussion focuses on calculating the probability density function (p.d.f.) of a random variable by integrating a product of two modified Bessel functions, specifically using the inverse Laplace transform, denoted as ℒ-1. The integral in question is ℒ-1(abs2 Kn(√(as)) Kn(√(bs))). Participants emphasize the need for specific properties of Bessel functions to solve this integral effectively.

PREREQUISITES
  • Understanding of inverse Laplace transforms
  • Familiarity with modified Bessel functions
  • Knowledge of probability density functions (p.d.f.)
  • Basic calculus and integral techniques
NEXT STEPS
  • Research properties of modified Bessel functions
  • Study techniques for performing inverse Laplace transforms
  • Explore applications of Bessel functions in probability theory
  • Learn about numerical integration methods for complex functions
USEFUL FOR

Mathematicians, statisticians, and researchers working with probability density functions and integral transforms, particularly those interested in the applications of modified Bessel functions.

jianingli
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To calculate a p.d.f. of a r.v., I need to integral a product of two bessel function as

\mathcal{L}^{-1} \left( abs^2 K_n( \sqrt{as}) K_n( \sqrt{bs} ) \right)

where \mathcal{L}^{-1} is the inverse Laplace transform.

I think some properties about the bessel function can solve this integral, but I cannot find it. So, please help me.

Thank you very much.
 
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I am looking for the inverse laplace transform of modified Bessel functions, could you please help me?.
 

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