Bessel Function, Orthogonality and More

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SUMMARY

The integral of the product of Bessel functions, specifically Integral[x*J0(a*x)*J0(a*x), from 0 to 1], equals 1/2 * J1(a)^2, where 'a' is a root of J0(x). The discussion highlights the orthogonality of Bessel functions, noting that when two different roots are used, the integral evaluates to zero. To prove the relationship, participants suggest expanding J0 in a power series and integrating, while also considering the right side's power series expansion for comparison.

PREREQUISITES
  • Understanding of Bessel functions, specifically J0 and J1.
  • Knowledge of power series expansions and their applications.
  • Familiarity with integral calculus, particularly definite integrals.
  • Basic concepts of orthogonality in mathematical functions.
NEXT STEPS
  • Study the properties of Bessel functions, focusing on J0 and J1.
  • Learn about power series expansions and their convergence.
  • Research orthogonality conditions for special functions.
  • Examine integral calculus techniques for evaluating complex integrals.
USEFUL FOR

Mathematicians, physicists, and engineering students interested in advanced calculus, specifically those working with Bessel functions and their applications in solving differential equations.

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Hello,
I'm trying to show that

Integral[x*J0(a*x)*J0(a*x), from 0 to 1] = 1/2 * J1(a)^2

Here, (both) a's are the same and they are a root of J0(x). I.e., J0(a) = 0.

I have found and can do the case where you have two different roots, a and b, and the integral evaluates to zero (orthogonality). How do I go about showing this relationship? I can't find details anywhere.
 
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Try expanding J0 in a power series, collect terms in like powers, and integrate. Then you can also expand the right side in a power series and show the two are equal.
 
Hi,
Sorry for my ignorance, but if expanding into a power series don't we have two infinite sums multiplied together? I attempted it but wasn't able to get anywhere nicely (maybe it's beyond me)

I was thinking something more along the lines of this:
http://physics.ucsc.edu/~peter/116C/bess_orthog.pdf
but I don't see the proper modifications that will give me my identity.

Any further hints would be amazing!
 
Why isn't equation 15 of the link you sent exactly what you are looking for?
 

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