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An integral about Bessel function

  1. Sep 22, 2011 #1
    Is there somebody who knows the solution (closed form) for the integral
    $$\int^\infty_0\frac{J^3_1(ax)J_0(bx)}{x^2}dx$$
    where $a>0,b>0$ and $J(\cdot)$ the bessel function of the first kind with integer order?

    Reference, or solution from computer programs all are welcome. Thanks!
     
  2. jcsd
  3. Jun 7, 2012 #2
    even if belatedly believe that this property of bessel funtions is useful
    $$\frac{J_{n}(x)}{x}=\frac{J_{n-1}+J_{n+1}}{2n}$$
    with the orthonormality of bessel functions
    $$\int^\infty_0 J_{n}(ax)J_{n}(bx)xdx=\frac{1}{a}\delta(a-b)$$
    $$\int^\infty_0\left(\frac{J_1(ax)}{x}\right)^{3}J_0(bx)xdx =\int^\infty_0\left(\frac{J_{1-1}+J_{1+1}}{2\cdot1}\right)^{3}J_0(bx)xdx=\dots$$
     
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