On the computation of certain integrals containing the modified Bessel function 𝐼₀(𝜉)

Lassey, Keith R.

doi:10.1090/S0025-5718-1982-0669654-6

On the computation of certain integrals containing the modified Bessel function $I\sb{0}(\xi )$

Author: Keith R. Lassey
Journal: Math. Comp. 39 (1982), 625-637
MSC: Primary 65D20
DOI: https://doi.org/10.1090/S0025-5718-1982-0669654-6
MathSciNet review: 669654
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Abstract: Efficient stratagems are developed for numerically evaluating one- and two-dimensional integrals over x, y with integrand $\exp ( - x - y){I_0}(2\sqrt {pxy} )$ . The integrals are expressed in terms of convergent series, which exhibit the correct limiting behavior, and which can be evaluated recursively. The performances of these stratagems are compared with numerical integration.

[1] Stuart R. Brinkley Jr., Heat transfer between a fluid and a porous solid generating heat, J. Appl. Phys. 18 (1947), 582–585. MR 21210
[2] H. C. Thomas, "Heterogeneous ion exchange in a flowing system," J. Amer. Chem. Soc., v. 66, 1944, pp. 1664-1666.
[3] J. E. Walter, "Rate-dependent chromatographic adsorption," J. Chem. Phys., v. 13, 1945, pp. 332-336.
[4] H. C. Thomas, "Chromatography: A problem in kinetics," Ann. New York Acad. Sci., v. 49, 1948, pp. 161-182.
[5] S. Goldstein, On the mathematics of exchange processes in fixed columns. I. Mathematical solutions and asymptotic expansions, Proc. Roy. Soc. London Ser. A 219 (1953), 151–171. MR 58824, https://doi.org/10.1098/rspa.1953.0137
[6] A. Ogata, Mathematics of Dispersion with a Linear Adsorption Isotherm, Prof. Pap. U. S. Geol. Surv. 411-H, 1964.
[7] J. Hubert & A. Lenda, "A solution of the dispersion-adsorption equation with linear adsorption isotherm," Nucleonika, v. 16, 1971, pp. 271-278.
[8] F. De Smedt & P. J. Wierenga, "Mass transfer in porous media with immobile water," J. Hydrol., v. 41, 1979, pp. 59-67.
[9] S. R. Brinkley & R. F. Brinkley, "Table of the probability of hitting a circular target," MTAC, v. 2, 1947, p. 221.
[10] Yudell L. Luke, Integrals of Bessel functions, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1962. MR 0141801
[11] K. R. Lassey, "The interception and retention of aerosols by vegetation: I--The formulation of a filtration model," Atmos. Environ., v. 16, 1982, pp. 13-24.
[12] K. R. Lassey, "Conceptually simple mathematical models of filtration (and exchange) processes," Ecol. Modelling. (Submitted.)
[13] C. Hastings & J. P. Wong, "Analytical approximations," MTAC, v. 7, 1953, pp. 212-213.
[14] W. J. Cody, Preliminary Report on Software for the Modified Bessel Functions of the First Kind, Argonne National Lab. Report TM-357, 1980.
[15] D. E. Amos, Computation of modified Bessel functions and their ratios, Math. Comp. 28 (1974), 239–251. MR 333287, https://doi.org/10.1090/S0025-5718-1974-0333287-7
[16] Walter Gautschi and Josef Slavik, On the computation of modified Bessel function ratios, Math. Comp. 32 (1978), no. 143, 865–875. MR 470267, https://doi.org/10.1090/S0025-5718-1978-0470267-9
[17] Marietta J. Tretter and G. W. Walster, Further comments on the computation of modified Bessel function ratios, Math. Comp. 35 (1980), no. 151, 937–939. MR 572867, https://doi.org/10.1090/S0025-5718-1980-0572867-3
[18] E. E. Allen, Polynomial approximations to some modified Bessel functions, Math. Tables Aids Comput. 10 (1956), 162–164. MR 80774, https://doi.org/10.1090/S0025-5718-1956-0080774-4
[19] F. W. J. Olver, Handbook of Mathematical Functions (M. Abramowitz and I. A. Stegun, Eds.), Chapter 9, Dover, New York, 1965.
[20] J. M. Blair & C. A. Edwards, Stable Rational Minimax Approximations to the Modified Bessel Functions ${I_0}(X)$ and ${I_1}(X)$ , Atomic Energy of Canada Ltd. Report AECL-4928, 1974.
[21] M. B. Carver & V. J. Jones, An Evaluation of Available Quadrature Algorithms and Selection for the AECL FORTRAN Mathematical Library, Atomic Energy of Canada Ltd. Report AECL-5605, 1977.
[22] J. Oliver, "A doubly-adaptive Clenshaw-Curtis quadrature method," Comput. J., v. 15, 1972, pp. 141-147.
[23] H. O’Hara and Francis J. Smith, The evaluation of definite integrals by interval subdivision, Comput. J. 12 (1969/70), 179–182. MR 242369, https://doi.org/10.1093/comjnl/12.2.179
[24] J. F. Hart et al., "Computer approximations," (Chapter 6.2), SIAM Series in Applied Mathematics (R. F. Drenick, H. Hochstadt and D. Gillette, Eds.), Wiley, New York, 1968.

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