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Zbl 0718.65010
Feuillebois, F.
Numerical calculation of singular integrals related to Hankel transform. (English)
[J] Comput. Math. Appl. 21, No.2-3, 87-94 (1991). ISSN 0898-1221

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

MSC 2000:: *65D20 Computation of special functions
65D32 Quadrature formulas (numerical methods)
65R10 Integral transforms (numerical methods)
33E30 Functions coming from diff., difference and integral equations

Keywords: Hankel transform; Gauss-Laguerre formulae; Gauss-Chebyshev formulae; singular integral; Bessel function; Gauss-Legendre formulae; FORTRAN subroutines

Cited in: Zbl 0754.65126

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