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integrals with Bessel functions; fast Fourier transform; Gaussian integration formula; five-point Gauss rule; error analysis; numerical quadrature
The paper is concerned with the efficient evaluation of the integral $\int^\infty_0 f(x)J_n(rx)dx$, where $J_n$ is the Bessel function of index $n$ and $n$ is a nonnegative integer, for a given sequence of values of a real parameter $r$. Two procedures are proposed and compared. One of them consists in a direct generalization of a procedure for the evaluation of of a similar integral with the weight function exp $(irx), which employs the fast Fourier transform. The other approach is based on the construction of a special Gaussian quadrature formula where $J_n$ appears as a weight. The results of the comparison show that the application of the Gaussian formula is much more efficient.
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