New backward recurrences for Bessel functions

Author:
Henry C. Thacher

Journal:
Math. Comp. **33** (1979), 744-764

MSC:
Primary 65D20; Secondary 33A40

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521289-1

MathSciNet review:
521289

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The recurrences for the coefficients of appropriate power series may be used with the Miller algorithm to evaluate , , and the modulus and phase of . The first converges slightly faster than the power series or the classical recurrence, but requires more arithmetic; the last three give both better ultimate precision and faster convergence than the corresponding asymptotic series. The analysis also leads to a formal continued fraction for the convergence of which increases with . The procedures were tested numerically both for integer and fractional values of *v*, and for real and complex *x*.

**[1]**J. C. P. MILLER,*British Association for the Advancement of Science Mathematical Tables*. Vol. X.*Bessel Functions*. Part II,*Functions of Positive Integer Order*, Cambridge University Press, Cambridge, 1952. MR**0050973 (14:410d)****[2]**W. GAUTSCHI, "Computational aspects of three-term recurrence relations,"*SIAM Rev.*, v. 9, 1967, pp. 24-82. MR**0213062 (35:3927)****[3]**W. GAUTSCHI, "Zur Numerik rekurrenter Relationen,"*Computing*, v. 9, 1972, pp. 107-126. MR**0312714 (47:1270)****[4]**W. GAUTSCHI, "Computational methods in special functions-a survey,"*Theory and Applications of Special Functions*, R. Askey, (ed.), Academic Press, New York, 1975, pp. 1-98. MR**0391476 (52:12297)****[5]**H. C. THACHER, JR., "Series solutions to differential equations by backward recurrence,"*Information Processing*71, North-Holland, Amsterdam, 1972, pp. 1287-1291. MR**0461923 (57:1905)****[6]**F. W. J. OLVER, "9. Bessel functions of integer order,"*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*, M. Abramowitz and I. A. Stegun, (eds.), Nat. Bur. Standards Appl. Math. Series 55, U. S. Government Printing Office, Washington, D. C., 1964, pp. 355-433. MR**0167642 (29:4914)****[7]**H. A. ANTOSIEWICZ, "10. Bessel functions of fractional order,"*Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables*, M. Abramowitz and I. A. Stegun, (eds.), Nat. Bur. Standards Appl. Math. Series 55, U. S. Government Printing Office, Washington, D. C., 1964, pp. 435-478. MR**0167642 (29:4914)****[8]**F. W. J. OLVER,*Asymptotics and Special Functions*, Academic Press, New York, 1974. MR**0435697 (55:8655)****[9]**S. HITOTUMATU, "On the numerical computation of Bessel functions through continued fractions,"*Comment. Math. Univ. St. Paul*, v. 16, 1967/68, pp. 89-113. MR**0233492 (38:1813)****[10]**E. T. WHITTAKER & G. N. WATSON,*A Course of Modern Analysis*, 4th ed., Cambridge University Press, Cambridge, 1927. MR**1424469 (97k:01072)****[11]**B. DÖRING, "Complex zeros of cylinder functions,"*Math. Comp.*, v. 20, 1966, pp. 215-222. MR**0192632 (33:857)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65D20,
33A40

Retrieve articles in all journals with MSC: 65D20, 33A40

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521289-1

Keywords:
Bessel functions,
modified Bessel functions,
Airy function,
Hankel function,
Kelvin function,
power series,
continued fraction,
differential equation,
recurrence,
Miller algorithm

Article copyright:
© Copyright 1979
American Mathematical Society