Integrating \int xJ_0(ax)J_0(bx)dx w/ Bessel Functions

[MarkovMarakov]

Homework Statement

How do I integrate \int_0^1 xJ_0(ax)J_0(bx)dx where J_0 is the zeroth order Bessel function?

Homework Equations

See above.
Also, the zeroth order Bessel equation is (xy')'+xy=0

The Attempt at a Solution

Surely we must use the fact that J_0 is a Bessel function, since we can't integrate any old function in the given integral. But I don't know how.

Thanks for any help.

 

[phyzguy]

If you're like me, you look it up, either online, or using a tool like Mathematica. Wolfram Alpha is a good online source, and it gave the following answer:

http://www.wolframalpha.com/input/?i=Integrate[x+BesselJ[0%2C+a+x]+BesselJ[0%2C+b+x]%2C+{x%2C+0%2C+1}]

 

[MarkovMarakov]

Thank you @phyzguy. I tried it out but it doesn't seem to be working. What should the inout format be?

 

[phyzguy]

The input should be:

Integrate[x BesselJ[0, a x] BesselJ[0, b x], {x, 0, 1}]

The output is:

(a BesselJ[0, b] BesselJ[1, a] -
b BesselJ[0, a] BesselJ[1, b])/(a^2 - b^2)

which is \frac{a J_0(b) J_1(a) - b J_0(a) J_1(b)}{a^2-b^2}

 

[MarkovMarakov]

@phyzguy: Thanks! :-) How did you figure out the inout format for WA? Do you know how I can get the steps as well?

 

[phyzguy]

The input format is just the same as Mathematica, so if you learn that program you will have it. As for the steps, I think if you start with the integral formulation of the Bessel functions:
http://en.wikipedia.org/wiki/Bessel_function#Bessel.27s_integrals
and then integrate by parts you will get there.