Geodesic
On numerical evaluation of integrals involving Bessel functions
Applications of Mathematics, Tome 31 (1986) no. 5, pp. 396-410
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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.
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.
DOI : 10.21136/AM.1986.104216
Classification : 33C10, 42A16, 65D20, 65D30, 65D32, 65T40
Keywords: integrals with Bessel functions; fast Fourier transform; Gaussian integration formula; five-point Gauss rule; error analysis; numerical quadrature 
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Bezvoda, Václav; Farzan, Ruszlán; Segeth, Karel; Takó, Galina. On numerical evaluation of integrals involving Bessel functions. Applications of Mathematics, Tome 31 (1986) no. 5, pp. 396-410. doi: 10.21136/AM.1986.104216

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