# Article

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Keywords:
integrals with Bessel functions; fast Fourier transform; Gaussian integration formula; five-point Gauss rule; error analysis; numerical quadrature
Summary:
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.
References:
[1] M. Ambrožová: Using fast Fourier transform for evaluation of the integral of an oscillating function on the infinite integral. (Czech.) RNDr. thesis. Matematicko-fyzikální fakulta Univerzity Karlovy, Praha 1979.
[2] N. S. Bahvalov: Numerical Methods. (Russian.) Nauka, Moskva 1973. MR 0362811
[3] I. S. Berezin I. P. Židkov: Methods of Computation. (Russian.) Vol. 1. Nauka, Moskva 1966.
[4] V. Bezvoda K. Segeth: A contribution to the theory of electromagnetic induction of a line source. Studia Geod. et Geoph. 20 (1976), 366-377. DOI 10.1007/BF01617648
[5] V. Bezvoda K. Segeth: Mathematical Modeling in Electromagnetic Prospecting Methods. Univerzita Karlova, Praha 1982.
[6] P. I. Davis P. Rabinowitz: Methods of Numerical Integration. Academic Press, New York 1975. MR 0448814
[7] R. H. Farzan: Propagation of electromagnetic waves in stratified media with local inhomogeneity. Mathematical Models in Physics and Chemistry and Numerical Methods of Their Realization. Teubner Texte zur Mathematik 61. Teubner, Leipzig 1984, 237-247.
[8] L. N. G. Filon: On a quadrature formula for trigonometric integrals. Proc. Roy. Soc. Edinburgh 49 (1928), 38-47.
[9] I. S. Gradštein I. M. Ryžik: Tables of Integrals, Sums, Series, and Products. (Russian.) 5th ed. Nauka, Moskva 1971.
[10] S.-Å. Gustafson G. Dahlqaist: On the computation of slowly convergent Fourier integrals. Methoden und Verfahren der mathematischen Physik. Band 6. Bibliographisches Institut, Mannheim 1972, 93-112. MR 0359377
[11] T. Kaneko B. Liu: Accumulation of round-off error in fast Fourier transforms. J. Assoc. Comput. Mach. 17 (1970), 637-654. DOI 10.1145/321607.321613 | MR 0275710
[12] V. I. Krylov: Approximate Computation of Integrals. (Russian.) Nauka, Moskva 1967. MR 0218015
[13] I. M. Longman: Note on a method for computing infinite integrals of oscillatory functions. Proc. Camb. Phil. Soc. 52 (1956), 764-768. DOI 10.1017/S030500410003187X | MR 0082193 | Zbl 0072.33803
[14] R. Piessens E. de Doncker-Kapenga C. W. Überhuber D. K. Kahaner: QUADPACK. A Subroutine Package for Automatic Integration. Springer Series in Computational Mathematics 1. Springer-Verlag, Berlin 1983. MR 0712135
[15] A. Ralston: A First Course in Numerical Analysis. McGraw-Hill, New York 1965. MR 0191070 | Zbl 0139.31603
[16] K. Segeth: Roundoff errors in the fast computation of discrete convolutions. Apl. Mat. 26 (1981), 241-262. MR 0623505 | Zbl 0474.65025
[17] H. J. Stetter: Numerical approximation of Fourier transforms. Numer. Math. 8 (1966), 235-249. DOI 10.1007/BF02162560 | MR 0198716 | Zbl 0163.39503

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