Numerische Mathematik

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

An approximation for the zeros of Bessel functions

  • Á. Elbert
Article

Summary

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)

65H05 

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References

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    Abramowitz, M., Stegun, I.A. (1978): Handbook of mathematical functions. Dover, New YorkGoogle Scholar
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    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

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