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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 $I=\int^{\infty}_{0}f(x)J_ n(rx)\quad dx$ for fixed integer $$n\geq 0$$ and a given set of real values of r. In a first approach, they replace the Bessel function $$J_ n(x)$$ 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 $$J_ n(x)$$ a weight function over the intervals between two consecutive zeros of $$J_ n(x)$$. 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

MSC:
 65D20 Computation of special functions and constants, construction of tables 65T40 Numerical methods for trigonometric approximation and interpolation 65D32 Numerical quadrature and cubature formulas 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series