Computation of modified Bessel functions and their ratios
Author:
D. E. Amos
Journal:
Math. Comp. 28 (1974), 239251
MSC:
Primary 33A40; Secondary 65D20
MathSciNet review:
0333287
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: An efficient algorithm for calculating ratios , is presented. This algorithm in conjunction with the recursion relation for gives an alternative to other recursive methods for when approximations for loworder Bessel functions are available. Sharp bounds on and are also established in addition to some monotonicity properties of and .
 [1]
Milton
Abramowitz and Irene
A. Stegun, Handbook of mathematical functions with formulas,
graphs, and mathematical tables, National Bureau of Standards Applied
Mathematics Series, vol. 55, For sale by the Superintendent of
Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
(29 #4914)
 [2]
D.
E. Amos, Bounds on iterated coerror functions
and their ratios, Math. Comp. 27 (1973), 413–427. MR 0331723
(48 #10055), http://dx.doi.org/10.1090/S00255718197303317232
 [3]
D. E. Amos, Evaluation of Some Cumulative Distribution Functions by Numerical Quadrature, Symposium Proceedings, Sixth Annual Symposium of Interface: Computer Science and Statistics, University of California, Berkeley, October 1617, 1972.
 [4]
C.
W. Clenshaw, Chebyshev series for mathematical functions,
National Physical Laboratory Mathematical Tables, Vol. 5. Department of
Scientific and Industrial Research, Her Majesty’s Stationery Office,
London, 1962. MR
0142793 (26 #362)
 [5]
C.
W. Clenshaw and Susan
M. Picken, Chebyshev series for Bessel functions of fractional
order, National Physical Laboratory Mathematical Tables, Vol. 8, Her
Majesty’s Stationery Office, London, 1966. MR 0203095
(34 #2948)
 [6]
Walter
Gautschi, Computational aspects of threeterm recurrence
relations, SIAM Rev. 9 (1967), 24–82. MR 0213062
(35 #3927)
 [7]
Yudell
L. Luke, On generating Bessel functions by use of the backward
recurrence formula, Aerospace Research Laboratories, Air Force Systems
Command, United States Air Force, WrightPatterson Air Force Base, Ohio,
1972. ARL 720030. MR 0326022
(48 #4368)
 [8]
Yudell
L. Luke, Inequalities for generalized hypergeometric
functions, J. Approximation Theory 5 (1972),
41–65. Collection of articles dedicated to J. L. Walsh on his 75th
birthday, I. MR
0350082 (50 #2575)
 [9]
Yudell
L. Luke, The special functions and their approximations. Vol.
II, Mathematics in Science and Engineering, Vol. 53, Academic Press,
New YorkLondon, 1969. MR 0249668
(40 #2909)
 [10]
F.
W. J. Olver, Numerical solution of secondorder linear difference
equations, J. Res. Nat. Bur. Standards Sect. B 71B
(1967), 111–129. MR 0221789
(36 #4841)
 [11]
F.
W. J. Olver, Tables for Bessel functions of moderate or large
orders, National Physical Laboratory Mathematical Tables, Vol. 6.
Department of Scientific and Industrial Research, Her Majesty’s
Stationery Office, London, 1962. MR 0147676
(26 #5190)
 [12]
F.
W. J. Olver, The asymptotic expansion of Bessel functions of large
order, Philos. Trans. Roy. Soc. London. Ser. A. 247
(1954), 328–368. MR 0067250
(16,696a)
 [1]
 M. Abramowitz & I. A. Stegun (Editors), Handbook of Mathematical Functions, With Formulas, Graphs and Mathematical Tables, Nat. Bur. Standards Appl. Math. Ser., 55, Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 29 #4914. MR 0167642 (29:4914)
 [2]
 D. E. Amos, "Bounds on iterated coerror functions and their ratios," Math. Comp., v. 27, 1973, pp. 413427. MR 0331723 (48:10055)
 [3]
 D. E. Amos, Evaluation of Some Cumulative Distribution Functions by Numerical Quadrature, Symposium Proceedings, Sixth Annual Symposium of Interface: Computer Science and Statistics, University of California, Berkeley, October 1617, 1972.
 [4]
 C. W. Clenshaw, Chebyshev Series for Mathematical Functions, National Physical Laboratory Mathematical Tables, vol. 5, Department of Scientific and Industrial Research, Her Majesty's Stationery Office, London, 1962. MR 26 #362. MR 0142793 (26:362)
 [5]
 C. W. Clenshaw & S. M. Picken, Chebyshev Series for Bessel Functions of Fractional Order, National Physical Laboratory Mathematical Tables, vol. 8, Her Majesty's Stationery Office, London, 1966. MR 34 #2948. MR 0203095 (34:2948)
 [6]
 W. Gautschi, "Computational aspects of threeterm recurrence relations," SIAM Rev., v. 9, 1967, pp. 2482. MR 35 #3927. MR 0213062 (35:3927)
 [7]
 Y. L. Luke, On Generating Bessel Functions by Use of the Backward Recurrence Formula, Aerospace Research Laboratories Report ARL 720030, WrightPatterson Air Force Base, Ohio. MR 0326022 (48:4368)
 [8]
 Y. L. Luke, "Inequalities for generalized hypergeometric functions," J. Approximation Theory, v. 5, 1972, pp. 4165. MR 0350082 (50:2575)
 [9]
 Y. L. Luke, The Special Functions and Their Approximations. Vol. II, Math. in Sci. and Engineering, vol. 53, Academic Press, New York, 1969. MR 40 #2909. MR 0249668 (40:2909)
 [10]
 F. W. J. Olver, "Numerical solution of second order linear difference equations," J. Res. Nat. Bur. Standards Sect. B, v. 71B, 1967, pp. 111129. MR 36 #4841. MR 0221789 (36:4841)
 [11]
 F. W. J. Olver, Tables for Bessel Functions of Moderate or Large Orders, National Physical Laboratory Mathematical Tables, vol. 6, Department of Scientific and Industrial Research, Her Majesty's Stationery Office, London, 1962. MR 26 #5190. MR 0147676 (26:5190)
 [12]
 F. W. J. Olver, "The asymptotic expansion of Bessel functions of large order," Philos. Trans. Roy Soc. London Ser. A, v. 247. 1954, pp. 328368. MR 16, 696. MR 0067250 (16:696a)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
33A40,
65D20
Retrieve articles in all journals
with MSC:
33A40,
65D20
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197403332877
PII:
S 00255718(1974)03332877
Keywords:
Modified Bessel functions,
recursive computation,
ratios of Bessel functions,
bounds on Bessel functions
Article copyright:
© Copyright 1974
American Mathematical Society
