Bessel Function, Orthogonality and More

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Discussion Overview

The discussion revolves around the evaluation of an integral involving Bessel functions, specifically the relationship between the integral of the product of two Bessel functions and the square of a Bessel function at a root. Participants explore methods to demonstrate this relationship and discuss orthogonality properties of Bessel functions.

Discussion Character

  • Technical explanation, Mathematical reasoning, Homework-related

Main Points Raised

  • One participant seeks to prove the integral of the product of Bessel functions, specifically that 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).
  • Another participant suggests expanding J0 in a power series and integrating, proposing that this could lead to a demonstration of the equality.
  • A third participant expresses uncertainty about handling the power series expansion, noting the complexity of dealing with two infinite sums and indicating difficulty in progressing with the problem.
  • One participant references a specific equation from an external document, questioning its relevance to the problem at hand.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the best approach to demonstrate the relationship. There are differing opinions on the utility of power series expansions and the relevance of external resources.

Contextual Notes

Participants have not resolved the mathematical steps necessary to show the relationship, and there are assumptions regarding the properties of Bessel functions that remain unexamined.

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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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