A double integral containing the modified Bessel function: asymptotics and computation
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- by N. M. Temme PDF
- Math. Comp. 47 (1986), 683-691 Request permission
Abstract:
A two-dimensional integral containing $\exp ( - u - t){I_0}(2\sqrt {ut} )$ is considered. ${I_0}(z)$ is the modified Bessel function and the integral is taken over the rectangle $0 \leqslant u \leqslant x$, $0 \leqslant t \leqslant y$. The integral is difficult to compute when x and y are large, especially when x and y are almost equal. Computer programs based on existing series expansions are inefficient in this case. A representation in terms of the error function (normal distribution function) is discussed, from which more efficient algorithms can be constructed.References
- Donald E. Amos, Computation of exponential integrals, ACM Trans. Math. Software 6 (1980), no. 3, 365–377. MR 585343, DOI 10.1145/355900.355908
- Walter Gautschi, Recursive computation of certain integrals, J. Assoc. Comput. Mach. 8 (1961), 21–40. MR 119392, DOI 10.1145/321052.321054
- Walter Gautschi, A computational procedure for incomplete gamma functions, Rend. Sem. Mat. Univ. Politec. Torino 37 (1979), no. 1, 1–9 (Italian). MR 547763
- 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, DOI 10.1098/rspa.1953.0137
- C. G. van der Laan and N. M. Temme, Calculation of special functions: the gamma function, the exponential integrals and error-like functions, CWI Tract, vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1984. MR 777869
- Keith R. Lassey, On the computation of certain integrals containing the modified Bessel function $I_{0}(\xi )$, Math. Comp. 39 (1982), no. 160, 625–637. MR 669654, DOI 10.1090/S0025-5718-1982-0669654-6
- Yudell L. Luke, Integrals of Bessel functions, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1962. MR 0141801
- Yudell L. Luke, The special functions and their approximations. Vol. II, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR 0249668
- F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Math. Comp. 47 (1986), 683-691
- MSC: Primary 33A40; Secondary 41A60, 65D30
- DOI: https://doi.org/10.1090/S0025-5718-1986-0856712-X
- MathSciNet review: 856712