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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

$I={\int }_{0}^{\infty }f\left(x\right){J}_{n}\left(rx\right)\phantom{\rule{1.em}{0ex}}dx$

for fixed integer $n\ge 0$ and a given set of real values of r. In a first approach, they replace the Bessel function ${J}_{n}\left(x\right)$ 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 ${J}_{n}\left(x\right)$ a weight function over the intervals between two consecutive zeros of ${J}_{n}\left(x\right)$. 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, construction of tables 65T40 Trigonometric approximation and interpolation (numerical methods) 65D32 Quadrature and cubature formulas (numerical methods) 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 42A16 Fourier coefficients, special Fourier series, etc.