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Numerical calculation of singular integrals related to Hankel transform. (English) Zbl 0718.65010

The singular integral S= 0 f(x)e -x J 0 (ωx)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

|ω|1·5 by Gauss-Laguerre and Gauss-Chebyshev formulae;

|ω|>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 ' = 0 f(x)e -x J 1 (ωx)dx is derived from S. It is stated that the FORTRAN subroutines run very fast and give a relative precision better than 5×10 -6 (for all ω).

65D20Computation of special functions, construction of tables
65D32Quadrature and cubature formulas (numerical methods)
65R10Integral transforms (numerical methods)
33E30Functions coming from differential, difference and integral equations