An integral of Bessel functions

[S_klogW]

Homework Statement

My teacher gave us a problem as an open question:
To calculate an integral involving Bessel Functions.

Homework Equations

The Attempt at a Solution

I've tried to convert this integral to one in which the Bessel function is in the numerator but failed. Does anyone know how to manipulate this sort of integrals in which the Bessel function is in the denominator?

 

[Mute]

Homework Statement

My teacher gave us a problem as an open question:
To calculate an integral involving Bessel Functions.

Homework Equations

View attachment 53473

The Attempt at a Solution

I've tried to convert this integral to one in which the Bessel function is in the numerator but failed. Does anyone know how to manipulate this sort of integrals in which the Bessel function is in the denominator?

I have to admit, at first I didn't expect this integral to even have a closed form solution! So, I cheated an checked wolframalpha to find that it did indeed have one, and then I scoured the wikipedia page on Bessel functions looking for an identity which I thought might help. I found one that did the trick, but I'm not sure how to point you in the direction of it without giving too much away or sending you on a wild goose chase. =S I'm trying to think about how I would have searched for a helpful identity if I didn't already know the answer.

I'm not sure there is a good way to do the search, but here are some suggestions:

-The integrand involves a 1/x factor, so I would probably look at identities which had a 1/x factor somewhere in them.

-Since it's an integral, I would probably also focus on the identities which have a derivative somewhere in them. (As you may be able to massage one of the identities into the form $d(\mbox{something})/dx = 1/x/J_\nu^2(x)$).

-Sum or integral forms for the bessel functions are not going to be useful because you'd have a (sqaured!) series or integral in the denominator.

Here's the wikipedia page for Bessel functions

Hopefully the suggestions I gave you will help narrow down your search for a useful identity. I'm sorry I can't think of a better way to solve the problem! Also, note that some of the identities hold for more than just $J_\nu(x)$, so read the text to see if a non- J, Y, H, I or K letter can stand for one or more kinds of Bessel functions!

 

[S_klogW]

Well, in fact this morning my teacher revealed the solution. Just think about the Wronski Determinant of the Bessel equations! This can help us get rid of the 1/x factor!