Numerische Mathematik

, Volume 59, Issue 1, pp 647–657 | Cite as

An approximation for the zeros of Bessel functions

  • Á. Elbert


LetCvk be thekth positive zero of the cylinder functionCv(x)=cosαJv(x)−sinαYv(x), whereJv(x),Yv(x) are the Bessel functions of first kind and second kind, resp., andv>0, 0≦α<π. Definejvk byjvk=Cvk with\(\kappa = k - \frac{\alpha }{\pi }\). Using the notation 1/K=ε, we derive the first two terms of the asymptotic expansion ofjvk in terms of the powers of ε at the expense of solving a transcendental equation. Numerical examples are given to show the accuracy of this approximation.

Mathematics Subject Classification (1991)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramowitz, M., Stegun, I.A. (1978): Handbook of mathematical functions. Dover, New YorkGoogle Scholar
  2. 2.
    Elbert, Á., Laforgia, A. (1985): On the convexity of the zeros of Bessel functions. SIAM J. Math. Anal. Appl.16, 614–619Google Scholar
  3. 3.
    Elbert, Á., Laforgia, A. (1984): An asymptotic relation for the zeros of Bessel functions. J. Math. Anal. Appl.98, 502–511Google Scholar
  4. 4.
    Elbert, Á., Laforgia, A., Lorch, L.: Additional monotonicity properties of the zeros of Bessel functions. Analysis (to appear)Google Scholar
  5. 5.
    Watson, G.N. (1944): A treatise on the theory of Bessel functions, 2nd ed. Cambridge Univ. Press, London New YorkGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Á. Elbert
    • 1
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

Personalised recommendations