Numerical techniques for finding -zeros of Hankel functions

Authors:
James Alan Cochran and Judith N. Hoffspiegel

Journal:
Math. Comp. **24** (1970), 413-422

MSC:
Primary 65.25

DOI:
https://doi.org/10.1090/S0025-5718-1970-0272157-6

MathSciNet review:
0272157

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with numerical procedures for the evaluation of the zeros, with respect to order, of Hankel functions and their derivatives in cases when the arguments of these functions are held fixed. Using Olver's asymptotic expansions, two auxiliary tables have been computed, one appropriate for real and the other for purely imaginary argument. These tables, included herein, permit the calculation of rather accurate approximations to the desired -zeros for wide ranges of argument and index. Moreover, from the given tabular entries, the errors attendant with any approximate -zero so determined can be easily estimated.

**[1]**James Alan Cochran,*The zeros of Hankel functions as functions of their order*, Numer. Math.**7**(1965), 238–250. MR**0178170**, https://doi.org/10.1007/BF01436080**[2]**F. W. J. Olver,*The asymptotic expansion of Bessel functions of large order*, Philos. Trans. Roy. Soc. London. Ser. A.**247**(1954), 328–368. MR**0067250**, https://doi.org/10.1098/rsta.1954.0021**[3]**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****[4]**G. N. Watson,*A treatise on the theory of Bessel functions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR**1349110****[5]**L. A. Berry,*Computation of Hankel Functions*, Nat. Bur. Standards Tech. Note #216, U.S. Nat. Bur. Standards, Washington, D.C., June 12, 1964.**[6]**G. Pólya, "On the zeros of certain trigonometric integrals,"*J. London Math. Soc.*, v. 1, 1926, pp. 98-99.**[7]**-, "Über trigonometrische Integrale mit nur reellen Nullstellen,"*J. Reine Angew. Math.*, v. 158, 1927, pp. 6-18.**[8]**James Alan Cochran,*Remarks on the zeros of cross-product Bessel functions*, J. Soc. Indust. Appl. Math.**12**(1964), 580–587. MR**0178169****[9]**W. Streifer, "Creeping wave propagation constants for impedance boundary conditions,"*IEEE Trans. Antennas and Propagation*, v. 12, 1964, pp. 764-766.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0272157-6

Keywords:
Special functions,
cylindrical functions,
Bessel functions,
Hankel functions,
rootfinding,
numerical approximation,
function evaluation,
asymptotic expansions,
electromagnetic scattering,
creeping-wave propagation

Article copyright:
© Copyright 1970
American Mathematical Society