Numerical calculation of singular integrals related to Hankel transform (Q2639589)

The singular integral \(S=\int^{\infty}_{0}f(x)e^{-x}J_ 0(\omega x)dx\) is calculated numerically \((J_ i\) is the Bessel function of order i, \(i=0,1)\) by using an integral expression for \(J_ 0\). If f(x) is bounded and analytic in some complex domain, the double integral obtained in this way is calculated for \(| \omega | \leq 1.5\) by Gauss-Laguerre and Gauss-Chebyshev \(formulae;\) \(| \omega | >1.5\) by Gauss-Laguerre formulae, changes of variables, and Gauss-Legendre formulae. The bound 1.5 is searched by trial. Further the singular integral \(S'=\int^{\infty}_{0}f(x)e^{-x}J_ 1(\omega x)dx\) is derived from S. It is stated that the FORTRAN subroutines run very fast and give a relative precision better than \(5\times 10^{-6}\) (for all \(\omega\)).