Numerical calculation of singular integrals related to Hankel transform

Abstract The singular integral S = ∫ 0 ∞ f ( x ) e − x J 0 ( ωx ) dx , related to the Hankel transform of order 0, is calculated numerically by using an integral expression for the Bessel function of order zero, J 0 . With the assumptions that the function f ( x ) is bounded and is analytic in some complex domain, the double integral obtained in this way is calculated by a combination of changes of variables and Gauss methods using Laguerre, Chebyshev and Legendre polynomials. The singular integral S ′ = ∫ 0 ∞ f ( x ) e − x J 1 ( ωx ) dx is derived from S . The subroutines written in FORTRAN run very fast on a personal computer and give a relative precision better than 5 × 10 −6 .