# Homework Help: Bessel function

1. Sep 27, 2011

### MarkovMarakov

1. The problem statement, all variables and given/known data
How do I integrate $\int_0^1 xJ_0(ax)J_0(bx)dx$ where $J_0$ is the zeroth order Bessel function?

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

3. 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.

2. Sep 27, 2011

### 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}]

3. Sep 27, 2011

### MarkovMarakov

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

4. Sep 27, 2011

### 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 $$\frac{a J_0(b) J_1(a) - b J_0(a) J_1(b)}{a^2-b^2}$$

5. Sep 27, 2011

### MarkovMarakov

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

6. Sep 27, 2011