Integration of Bessel function products (J_1(x)^2/xdx)

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SUMMARY

The integral of the product of the Bessel function J_1(x) squared divided by x, represented as ∫_0^∞ J_1(x)^2 (dx/x) = 1/2, is a well-established result in scattering theory, particularly in the context of the Eikonal approximation. Despite attempts to prove this integral using recurrence relations and series representations of Bessel functions, a definitive proof remains elusive for some. This integral is also relevant in diffraction theory, specifically concerning the airy disc phenomenon.

PREREQUISITES
  • Understanding of Bessel functions, specifically J_1(x)
  • Familiarity with integral calculus and improper integrals
  • Knowledge of scattering theory and the Eikonal approximation
  • Experience with mathematical proofs and series representations
NEXT STEPS
  • Study the properties and applications of Bessel functions in mathematical physics
  • Learn about recurrence relations for Bessel functions
  • Explore the Eikonal approximation in scattering theory
  • Investigate the role of Bessel functions in diffraction theory and the airy disc
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Physicists, mathematicians, and engineers interested in scattering theory, diffraction phenomena, and the mathematical properties of Bessel functions.

euphoricrhino
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Hello,
While reading Sakurai (scattering theory/Eikonal approximation section), I encountered a referenced integral
##
\int_0^\infty J_1(x)^2\frac{dx}{x}=1/2
##

I also see this integral from a few places (wolfram, DLMF, etc), so I tried to prove this from various angles (recurrence relations, series representation of J_n, etc) but have not succeeded.

Can anyone provide a sketch of the proof?
Thanks!
 
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