Numerical methods for integrals involving product of Bessel functions of the first kind (1st order) I am looking for the best (in terms of low computation times) numerical methods for calculating the following integrals:
$$\int_0^{\infty}\,f(k)\,J_1(ak)\,J_1(bk)\,dk$$
with for instance
$$f(k)=\frac{\exp(-ak^2)}{k}\quad \text{or}\quad f(k)=\frac{k\exp(-ak^2)}{k^2+\alpha^2}$$
Thanks.
 A: The integral does not have the solution in closed form.
 But for the first case one can use the result from A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev Integrals and Series: Special Functions, Vol. 2.:
$$\int_0^{\infty }e^{-k^2 p} J_{\mu }(b k) J_{\nu }(c k) \ k^{\alpha-1} \ \mathrm dk= \\
\frac{\left(b^{\mu } c^{\nu } p^{-\frac{1}{2} (\alpha +\mu +\nu )}\right) }{2^{\mu +\nu +1} \Gamma (\nu +1)}\sum _{n=0}^{\infty } \frac{\Gamma \left(n+\frac{1}{2} (\alpha +\mu +\nu )\right) }{n! \  \Gamma (n+\mu +1)}\left(-\frac{b^2}{4 p}\right)^n\,_2F_1\left(-n,-n-\mu ;\nu +1;\left(\frac{c}{b}\right)^2\right)$$
which is valid while $\Re(p), \Re(\alpha+\mu+\nu)>0$
So setting $\mu=\nu=1, \ p=b=a,  c=b, \alpha=0$  one will obtain the result.
$$\int_0^{\infty }\frac{e^{-k^2 a} J_{1 }(a k) J_{1}(b k)}{k} \ \mathrm dk= 
\frac{a}{8} \sum _{n=0}^{\infty } \frac{(-1)^n}{(n+1) n!}\left(\frac{a}{4}\right)^n \, _2F_1\left(-n,-(n+1);2;\left(\frac{b}{a}\right)^2\right)$$
So after that you can set some restrictions (if you'd like to) on the hypergeometric function depending on its' argument in order to find an efficient computational solution (so that you would be able to truncate the series).
