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

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  • #1
euphoricrhino
22
6
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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1. What is the purpose of integrating Bessel function products?

The integration of Bessel function products is used to solve various mathematical problems, particularly in physics and engineering. It allows for the calculation of complex integrals involving Bessel functions, which are commonly used in the analysis of wave phenomena.

2. How is the integration of Bessel function products performed?

The integration of Bessel function products is typically done using various techniques, such as substitution, integration by parts, or series expansion. The specific method used depends on the complexity of the integral and the desired level of accuracy.

3. What are the applications of integrating Bessel function products?

The integration of Bessel function products has many applications in physics and engineering. It is commonly used in the analysis of wave propagation, electromagnetic fields, and heat transfer. It is also used in solving differential equations and in the study of special functions.

4. Are there any special properties or identities associated with integrating Bessel function products?

Yes, there are several special properties and identities associated with integrating Bessel function products. These include the orthogonality of Bessel functions, the Bessel differential equation, and various recurrence relations and generating functions.

5. Are there any challenges or limitations to integrating Bessel function products?

Integrating Bessel function products can be challenging due to the complexity of the integrals involved. In some cases, the integrals may not have closed-form solutions and require numerical methods for evaluation. Additionally, the convergence of the integrals may be slow, leading to potential accuracy issues.

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