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On the Points of Inflection of Bessel Functions of Positive Order, II

Published online by Cambridge University Press:  20 November 2018

R. Wong  and
T. Lang
R. Wong
Affiliation:
Department of Applied Mathematics, University of Manitoba, Winnipeg Manitoba R3T2N2.
T. Lang
Affiliation:
Department of Applied Mathematics, University of Manitoba, Winnipeg Manitoba R3T2N2.
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Let j ν, 1, jν,2, … denote the positive zeros of the Bessel function Jν(x), and similarly, let j'v,1 , j'v,2 , … denote the positive zeros of J'v(x), which are the positive critical points of Jv(x). It is well-known that when v is positive, both jν ,k . it and j'ν k are increasing functions of ν; see, e.g., [12, pp. 246 and 248]. Recently, Lorch and Szego [6] have attempted to show that the same is true for the positive zeros j″v,1 , j″v,2 , … of j″v (x), which are the positive inflection points of Jv(x).

Keywords

33A4041A60Inflection pointBessel functionintegral transformuniform asymptotic expansionAiry function

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 3 , 01 June 1991 , pp. 628 - 651
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Abramowitz, M. and Stegun, I., Handbook of mathematical functions, N.B.S. Applied Math. Ser. 55 (Washington, D. C, 1964).Google Scholar
2. Gatteschi, L., Limitazione degli errori nelleformule asintoticheper le funzioni speciali, Rend. Semin. Mat. Univ. e Pol, Torino 16 (1956-1957., 83–97.Google Scholar
3. Hethcote, H.W., Error bounds for asymptotic approximations of zeros of transcendental functions, SIAM J. Math. Anal., 1 (1970), 147–152.Google Scholar
4. Hethcote, H.W., Bounds for zeros of some special functions, Proc. Amer. Math. Soc, 25 (1970), 12–1.A.Google Scholar
5. Lang, T., Asymptotic analysis of the inflection points of the Bessel function Jv(x), M.Sc. Thesis, University of Manitoba, 1989.Google Scholar
6. Lorch, L. and Szego, P., On the points of inflection of Bessel functions of positive order, . ., Can. J. Math., 42 (1990), 933–948.Google Scholar
7. O.I. Marichev, Handbookof integral transforms of higher transcendental functions, theory and algorithmic tables (Ellis Horwood Ltd., West Sussex, England, 1983).Google Scholar
8. Olver, F.W.J., The asymptotic expansion of Bessel functions of large order, Philos. Trans. Roy. Soc. London Ser. A, 247 (1954), 328–368.Google Scholar
9. Olver, F.W.J., Tables of Bessel functions of moderate or large orders, Math. Tables Nat. Phys. Lab., Vol. 6 (Her Majesty's Stationery Office, London, 1962).Google Scholar
10. Olver, F.W.J., Error bounds for first approximations in turning point problems, J. Soc. Indust. Appl. Math., 11 (1963), 748–772.Google Scholar
11. Olver, F.W.J., Error bounds for asymptotic expansions in turning point problems, J. Soc. Indust. Appl. Math., 12 (1964), 200–214.Google Scholar
12. Olver, F.W.J., Asymptotics and special functions (Academic Press, New York, 1974).Google Scholar
13. Watson, G.N., Theory of Bessel functions (Cambridge University Press, Cambridge, 1944).Google Scholar
14. Wong, R., Explicit error terms for asymptotic expansions ofMellin convolutions, J. Math. Anal. Appl., 72 (1979), 740–756.Google Scholar
15. Wong, R., Error bounds for asymptotic expansions of integrals, SIAM Rev., 22 (1980), 401–435.Google Scholar
16. Wong, R., Applications of some recent results in asymptotic expansions, Proc. 12th Winnipeg Conf. on Numerical Methods of Computing, Congress Numer., 37(1983), 145–182.Google Scholar