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Bessel function

  1. Sep 27, 2011 #1
    1. The problem statement, all variables and given/known data
    How do I integrate [itex]\int_0^1 xJ_0(ax)J_0(bx)dx[/itex] where [itex]J_0[/itex] is the zeroth order Bessel function?


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


    3. The attempt at a solution
    Surely we must use the fact that [itex]J_0[/itex] 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. jcsd
  3. Sep 27, 2011 #2

    phyzguy

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    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}]
     
  4. Sep 27, 2011 #3
    Thank you @phyzguy. I tried it out but it doesn't seem to be working. What should the inout format be?
     
  5. Sep 27, 2011 #4

    phyzguy

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    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 [tex]\frac{a J_0(b) J_1(a) - b J_0(a) J_1(b)}{a^2-b^2}[/tex]
     
  6. Sep 27, 2011 #5
    @phyzguy: Thanks! :-) How did you figure out the inout format for WA? Do you know how I can get the steps as well?
     
  7. Sep 27, 2011 #6

    phyzguy

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