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

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 \(\omega\)).

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
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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
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