# An integral about Bessel function

1. Sep 22, 2011

### zluo

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. Jun 7, 2012

### <MIT|s|Mag>

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
$$\int^\infty_0 J_{n}(ax)J_{n}(bx)xdx=\frac{1}{a}\delta(a-b)$$
$$\int^\infty_0\left(\frac{J_1(ax)}{x}\right)^{3}J_0(bx)xdx =\int^\infty_0\left(\frac{J_{1-1}+J_{1+1}}{2\cdot1}\right)^{3}J_0(bx)xdx=\dots$$