Lucas decomposition and extrapolation methods for the evaluation of infinite integrals involving the product of three Bessel functions of arbitrary order

A method to efficiently evaluate integrals containing the product of three Bessel functions of the first kind and of any non negative real order is presented. The numerical problem is particularly challenging since standard integration techniques are completely unsuccessful in integrating such anomalously oscillating (and possibly slow decaying) functions. The proposed method is based on a decomposition of such a product into a sum of functions which asymptotically approach sinusoidal functions, for which integration schemes based on integration then summation procedures followed by extrapolation methods can be applied. Different extrapolation procedures are compared in order to identify the most efficient extrapolation strategy. Several numerical examples and comparisons with known analytic results are provided to show the robustness and the accuracy of the proposed approach and to help in identifying the most efficient and reliable extrapolation scheme. Particular attention is given to a class of integrals emerging in many fields of physics, in particular in quantum mechanics and particle physics, showing that the proposed method can be even more efficient (sometimes hundred of times faster) than the available analytical formulas.