A fast numerical algorithm for related bessel function integrations

A new fast numerical algorithm for the double-order double-argument Hankel transformation is proposed in this article. Due to the fact of rapid oscillation and slow attenuation of the kernel function, it is difficult to compute numerically. The New Algorithm break up the infinite integral interval into two subintervals. For the infinite subinterval, the integral can be calculated by Fast Fourier Transformation (FFT) efficiently; For the other finite integral, there has two choice, the one is ordinary quadrature algorithm which is simple but no efficient enough, the other is to express the integral as a summation series of the Bessel function integral, and then the calculus can be done with more efficient manner.