Feuillebois, F. Numerical calculation of singular integrals related to Hankel transform. (English) Zbl 0718.65010 Comput. Math. Appl. 21, No. 2-3, 87-94 (1991). 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 \(\omega\)). Reviewer: W.Moldenhauer (Erfurt) Cited in 1 Review MSC: 65D20 Computation of special functions and constants, construction of tables 65D32 Numerical quadrature and cubature formulas 65R10 Numerical methods for integral transforms 33E30 Other functions coming from differential, difference and integral equations Keywords:Hankel transform; Gauss-Laguerre formulae; Gauss-Chebyshev formulae; singular integral; Bessel function; Gauss-Legendre formulae; FORTRAN subroutines PDFBibTeX XMLCite \textit{F. Feuillebois}, Comput. Math. Appl. 21, No. 2--3, 87--94 (1991; Zbl 0718.65010) Full Text: DOI Digital Library of Mathematical Functions: §10.77(ix) Integrals of Bessel Functions ‣ §10.77 Software ‣ Computation ‣ Chapter 10 Bessel Functions References: [1] Abramowitz, M.; Stegun, I. A., (Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1964), N.B.S.) · Zbl 0171.38503 [2] Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series and Products (1965), Academic Press · Zbl 0918.65002 [3] Mineur, H., Techniques de calcul numerique (1966), Dunod: Dunod Paris · Zbl 0145.40102 [4] Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T., Numerical Recipes. The Art of Scientific Computing (1986), Cambridge Univ. Press · Zbl 0587.65003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.