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Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros

Author: Javier Segura
Journal: Math. Comp. 70 (2001), 1205-1220
MSC (2000): Primary 33C10
Published electronically: June 12, 2000
MathSciNet review: 1710198
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Bounds for the distance $\vert c_{\nu ,s}-c_{\nu \pm 1 , s^{\prime}}\vert$ between adjacent zeros of cylinder functions are given; $s$ and $s^{\prime}$ are such that $\nexists c_{\nu,s^{\prime\prime}}\in ]c_{\nu ,s},c_{\nu\pm 1,s^{\prime}}[$; $c_{\nu ,k}$ stands for the $k$th positive zero of the cylinder (Bessel) function $\mathcal{C}_{\nu}(x)=\cos\alpha J_{\nu}(x) - \sin\alpha Y_{\nu}(x)$, $\alpha \in [0,\pi[$, $\nu \in {\mathbb R}$.

These bounds, together with the application of modified (global) Newton methods based on the monotonic functions $f_{\nu}(x)=x^{2\nu -1}\mathcal{C}_{\nu}(x)/\mathcal{C}_{\nu -1}(x)$ and $g_{\nu}(x)=-x^{-(2\nu +1)}\mathcal{C}_{\nu}(x)/\mathcal{C}_{\nu +1}(x)$, give rise to forward ( $c_{\nu ,k} \rightarrow c_{\nu ,k+1}$) and backward ( $c_{\nu ,k+1} \rightarrow c_{\nu ,k}$) iterative relations between consecutive zeros of cylinder functions.

The problem of finding all the positive real zeros of Bessel functions $\mathcal{C}_{\nu}(x)$ for any real $\alpha$ and $\nu$ inside an interval $[x_{1},x_{2}]$, $x_{1}>0$, is solved in a simple way.

References [Enhancements On Off] (What's this?)

  • 1. M. Abramowitz and I.A. Stegun (editors), Handbook of Mathematical Functions, Dover Publications Inc. (1972). MR 94b:00012
  • 2. C. Belingeri, P.E. Ricci, ``On asymptotic formulas for the first zero of the Bessel function $J_{\nu}(x)$'', J. Inf. Optimization. Sci. 17 (1996) 267-274. MR 98c:33004
  • 3. Ll. G. Chambers, ``An upper bound for the first zero of the Bessel function $J_{\nu}$'', Math. Comp. 38 (1982) 589-591. MR 83h:33011
  • 4. Á. Elbert, A. Laforgia, ``On the convexity of the zeros of Bessel functions'', SIAM J. Math. Anal. 16 (1985) 614-619. MR 86g:33008
  • 5. Á. Elbert, ``An approximation for the zeros of Bessel functions", Numer. Math. 59 (1991) 647-657. MR 92h:33008
  • 6. Á Elbert, A. Laforgia, ``An upper bound for the zeros of the cylinder function $\mathcal{C}_{\nu} (x)$''. Math. Inequal. Appl. 1 (1998) 105-111. MR 99c:33005
  • 7. E.K. Ifantis, P.D. Siafarikas, ``A differential inequality for the positive zeros of Bessel functions'', J. Comput. Appl. Math. 44 (1992) 115-120. MR 94a:33003
  • 8. K.S. Kölbig, CERNLIB-Short Writeups, subroutine DBZEJY(C345), library MATHLIB
  • 9. P. Kravanja, O. Ragos, M. N. Vrahatis, F.A. Zafiropoulos. ``ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument''. Comput. Phys. Commun. 113 (1998) 220-238.
  • 10. N.N. Lebedev, Special Functions and their Applications, Dover Publications Inc. (1972). MR 50:2568
  • 11. Y. L. Luke, Mathematical functions and their approximations, Academic Press, New York, 1975. MR 58:19039
  • 12. L. Lorch, R. Uberti, `` ``Best possible'' upper bounds for the first positive zeros of Bessel functions - the finite part'', J. Comput. Appl. Math. 75 (1996) 249-258. MR 98b:33010
  • 13. M. E. Muldoon, ``Convexity properties of special functions and their zeros'' in Recent Progress in Inequalities, pp. 309-323, Kluwer Academics Publishers, Dordrecht-Boston-London (1998). CMP 98:11
  • 14. F.W.J. Olver (editor), Royal Society Mathematical Tables, Bessel functions, Part III. Zeros and Associated Values, vol. 7, Cambridge University Press, London-New York, 1960. MR 22:10202
  • 15. R. Piessens, ``Rational approximations for the zeros of Bessel functions'', J. Comput. Phys. 42 (1981) 403-405.
  • 16. R. Piessens, ``Chebyshev series approximations for the zeros of Bessel functions'', J. Comput. Phys. 53 (1984) 188-192. MR 85h:65044
  • 17. R. Piessens, ``A series expansion for the first positive zero of the Bessel functionsii, Math. Comp. 42 (1984) 195-197. MR 84m:33014
  • 18. R. Piessens, ``On the computation of zeros and turning points of Bessel functionsii, Bull. Greek Math. Soc. 31 (1990) 117-122. MR 92e:65019
  • 19. W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in Fortran. Cambridge University Press (1992). MR 93i:65001a
  • 20. J. Segura, ``A global Newton method for the zeros of cylinder functions", Num. Algorithms 18 (1998) 259-276. MR 99h:65092
  • 21. J. Segura $\&$ A. Gil, ``ELF and GNOME: Two tiny codes to evaluate the real zeros of the Bessel functions of the first kind for real orders", Comput. Phys. Commun. 117 (1999) 250. CMP 99:10
  • 22. N.M. Temme, ``An algorithm with ALGOL 60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives". J. Comp. Phys. 32 (1979) 270-279.
  • 23. N.M. Temme, Special Functions: an introduction to the Classical Functions of Mathematical Physics. John Wiley $\&$ Sons (1996). MR 97e:33002
  • 24. I.J. Thompson and A.R. Barnett, ``Coulomb and Bessel functions of complex arguments and order". J. Comput. Phys. 64 (1986) 490-509. MR 87h:33012
  • 25. M.N. Vrahatis, O. Ragos, T. Skiniotis, F.A. Zafiropoulos, T.N. Grapsa. ``RFSFNS: A portable package for the numerical determination of the number and the calculation of roots of Bessel functions. Comput. Phys. Commun., 92 (1995) 252-266.
  • 26. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, London (1944). MR 6:64a

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Additional Information

Javier Segura
Affiliation: Instituto de Bioingeniería, Universidad Miguel Hernández, Edificio La Galia, 03202-Elche, Alicante, Spain

Keywords: Bessel functions, cylinder functions, adjacent and consecutive zeros, global Newton method
Received by editor(s): January 7, 1999
Received by editor(s) in revised form: June 28, 1999
Published electronically: June 12, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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