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On numerical evaluation of integrals involving Bessel functions. (English) Zbl 0614.65012
The authors discuss the numerical computation of the integral $$ I=\int\sp{\infty}\sb{0}f(x)J\sb n(rx)\quad dx $$ for fixed integer $n\ge 0$ and a given set of real values of r. In a first approach, they replace the Bessel function $J\sb n(x)$ by a well-known trigonometric integral and then compute I by using a fast Fourier transform procedure. The second method consists of the construction of weights and abscissas for a Gaussian integration formula with $J\sb n(x)$ a weight function over the intervals between two consecutive zeros of $J\sb n(x)$. A comparison shows that the second method, once the weights and abscissas have been obtained, is more efficient. A table of weights and abscissas for a five- point Gauss rule is given and an error analysis is presented. An appropriate application of any of the existing adaptive quadrature procedures is not discussed.
Reviewer: K.S.Kölbig

65D20Computation of special functions, construction of tables
65T40Trigonometric approximation and interpolation (numerical methods)
65D32Quadrature and cubature formulas (numerical methods)
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
42A16Fourier coefficients, special Fourier series, etc.
Full Text: EuDML
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