Evaluating infinite integrals involving Bessel functions of arbitrary order

The evaluation of intergrals of the form I_n = ∫ 0 ∞ f{hook}(x)J_n(x)dx is considered. In the past, the method of dividing an oscillatory integral at its zeros, forming a sequence of partial sums, and using extrapolation to accelerate convergence has been found to be the most efficient technique available where the oscillation is due to a trigonometric function or a Bessel function of order n = 0, 1. Here, we compare various extrapolation techniques as well as choices of endpoints in dividing the integral, and establish the most efficient method for evaluating infinite integrals involving Bessel functions of any order n, not just zero or one. We also outline a simple but very effective technique for calculating Bessel function zeros.