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

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The integral of interest, ∫_0^∞ J_1(x)²/(x) dx, equals 1/2 and is referenced in various sources, including Sakurai's work on scattering theory. Attempts to prove this integral using recurrence relations and series representations of Bessel functions have not been successful. The integral is relevant in diffraction theory, particularly in the context of the airy disc. A request for a proof sketch has been made, indicating a need for clarity on the computation methods involved. Understanding this integral is crucial for applications in physics and engineering.
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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