Numerical quadrature for Bessel transformations

In this paper we explore higher order numerical quadratures for the integration of systems containing Bessel functions. Two families of methods are presented. One is based on a truncation of the asymptotic series and the other is extending an approach in the work of Levin. The decay of the error drastically improves as frequency grows. The effectiveness and accuracy of the quadrature is tested for large arguments of Bessel functions.