Integration of the product of sine and the first Bessel function

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SUMMARY

The integral of the product of sine and the first Bessel function, represented as \int_0^∞sin(ka)J0(kp)dk, evaluates to (a^2 - p^2)^{1/2} for p < a and equals zero for p > a. The discussion highlights attempts to solve the integral by reversing the order of integration and using series expansions, but these methods did not yield a solution. The key to solving this integral lies in recognizing the properties of the Bessel function and the sine function in relation to the Dirac delta function.

PREREQUISITES
  • Understanding of integral calculus, specifically improper integrals.
  • Familiarity with Bessel functions, particularly the first Bessel function J0.
  • Knowledge of Fourier transforms and their properties.
  • Experience with series expansions and convergence of infinite series.
NEXT STEPS
  • Study the properties of Bessel functions, focusing on J0 and its applications.
  • Learn about the Dirac delta function and its role in integral transforms.
  • Explore techniques for changing the order of integration in double integrals.
  • Investigate Fourier sine transforms and their relationship to Bessel functions.
USEFUL FOR

Mathematicians, physicists, and engineering students who are working with integral transforms, particularly those involving Bessel functions and Fourier analysis.

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Homework Statement



I'm supposed to prove that:

\int_0^∞[STRIKE][/STRIKE]sin(ka)J0(kp)dk = (a2 - p2)1/2 if p < a
and = 0 if p > a

J0 being the first Bessel function.

Homework Equations





The Attempt at a Solution



I've tried to inverse the order of integration and then make the integral form of the delta Dirac function appear, but I'm not sure how to do it, and so far my attemps have failed.

I also tried to put both sine and the bessel function as their series form, then transfom the infinite series into a limit of a finite series so I can interchange the sum and the integral, but it doesn't really leads me anywhere.

If anyone could give me some advise on how to resolve this, I would be grateful.
 
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