Integral of spherical bessel function (first kind), first order

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SUMMARY

The integral of the first order spherical Bessel function of the first kind, j_1(kr), from 0 to infinity is a common challenge in mathematical physics. The integral can be evaluated using techniques from Fourier analysis and properties of Bessel functions. Users have successfully utilized computational tools like WolframAlpha to derive results for such integrals. This discussion highlights the importance of understanding the properties of spherical Bessel functions in the context of convolution operations in three-dimensional space.

PREREQUISITES
  • Understanding of spherical Bessel functions, specifically j_1(kr).
  • Familiarity with integral calculus and improper integrals.
  • Knowledge of Fourier analysis techniques.
  • Experience with computational tools like WolframAlpha for mathematical evaluations.
NEXT STEPS
  • Research the properties and applications of spherical Bessel functions in physics.
  • Learn about convolution operations in the context of three-dimensional space.
  • Explore advanced integral techniques for evaluating improper integrals.
  • Investigate the use of WolframAlpha for solving complex integrals and mathematical problems.
USEFUL FOR

Mathematicians, physicists, and engineers working with spherical Bessel functions, as well as anyone involved in convolution operations in three-dimensional space.

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Hello,
I am trying to solve the following integral (limits from 0 to inf).

∫j_1(kr) dr

where j_1 is the first order SPHERICAL Bessel function of the first kind, of argument (k*r). Unfortunately, I cannot find it in the tables, nor manage to solve it... Can anybody help?

Thanks a lot! Any help will be very appreciated.

P.S: I came across this integral trying to apply analytically a direct filtering operation (convolution) in space domain to a point source (in a 3D space).
 
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Alright! Thanks a lot!
 

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