Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T07:13:33.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Monotonicity properties of the zeros of Bessel functions

Published online by Cambridge University Press:  17 February 2009

Roger C. McCann  and
E. R. Love
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let jν, denote the first positive zero of Jν. It is shown that jν/(ν + α) is a strictly decreasing function of ν for each positive α provided ν is sufficiently large. For each α lowe bounds on ν are given to assure the monotonicity of jν/(ν + α). From this it is shown that jν > ν + j0 for all ν > 0, which is both simpler and an improvement on the well known inequality Jν ≥ (ν (ν + 2))1/2.

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 1 , July 1982 , pp. 67 - 85
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Abramowitz, M. and Stegun, I. A. (eds.), Handbook of mathematical funcrions (Dover, New York, 1970).Google Scholar
[2]Courant, R. and Hilbert, D., Methods of mathematical physics, Vol.1 (Interscience, New York, 1953).Google Scholar
[3]Gray, A., Mathews, G. B. and MacRobert, T. M., A treatise on the theory of Bessel functions (2nd edition, Dover, New York, 1966).Google Scholar
[4]Hethcote, H. W., “Error bounds for asymptotic approximations to zeros of transcendental functions”, SIAM J. Math. Anal. 1 (1970), 147–152.CrossRefGoogle Scholar
[5]Jahnke, E. and Emde, F., Tables of functions (4th edition, Dover, New York, 1945).Google Scholar
[6]Lewis, J. T. and Muldoon, M. E., “Monotonicity and convexity properties of zeros of Bessel functions”, SIAM J. Math. Anal. 8 (1977), 171–178.CrossRefGoogle Scholar
[7]Luke, Y. L., Integrals of Bessel functions (McGraw-Hill, New York, 1962).Google Scholar
[8]McCann, R. C., “Inequalities for the zeros of Bessel functions”, SIAM J. Math. Anal. 8 (1977), 166–170.CrossRefGoogle Scholar
[9]McCann, R. C., “Lower bounds for the zeros of Bessel functions”, Proc. Amer. Math. Soc. 64 (1977), 101–103.CrossRefGoogle Scholar
[10]Mikhlin, S. G., Variational methods of mathematical physics (Macmillan, New York, 1964).Google Scholar
[11] National Bureau of Standards, Tables of spherical Bessel functions, Vol. 2 (Columbia University Press, New York, 1950).Google Scholar
[12]Olver, F. W. J., (ed), Royal society mathematical tables, Vol. 7, Bessel functions, Part III, Zeros and associated values (Cambridge University Press, Cambridge, 1960).Google Scholar
[13]Royden, H. L., Real analysis (Macmillan, New York, 1963).Google Scholar
[14]Tricomi, F. G., “Sulle funzioni di Bessel di ordine e argomento pressoché uguale”, Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. 83 (1949), 3–20.Google Scholar
[15]Watson, G. N., A treatise on the theory of Bessel functions (Cambridge University Press, 2nd edition, Cambridge, 1945).Google Scholar