The zeros of Hankel functions as functions of their order

Cochran, James Alan

doi:10.1007/bf01436080

Cited by 31 publications

(23 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Second differences are included in the tables for purposes of interpolation, and certain limiting cases which extend the domain of applicability are also considered. In a sense, then, the work reported herein complements that already appearing in [1].…”

supporting

confidence: 77%

“…In numerous scattering and diffraction problems where circular or spherical boundaries are present, the zeros of Bessel functions (or combinations thereof) are of interest. Previously we have studied, from a theoretical point-of-view [1], the roots of where these Hankel functions and their derivatives are to be considered as functions of their order v with fixed argument w. We now focus attention on procedures which can be used to obtain numerical values for the v-zeros of these functions. Particular consideration is given to (1.1) and (1.2) for the situation wherein the argument u> is either positive real or purely imaginary.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Numerical techniques for finding 𝜈-zeros of Hankel functions

Cochran¹,

Hoffspiegel²

1970

Math. Comp.

Self Cite

View full text Add to dashboard Cite

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 v-zeros for wide ranges of argument and index. Moreover, from the given tabular entries, the errors attendant with any approximate v-zero so determined can be easily estimated.

show abstract

supporting

confidence: 77%

mentioning

confidence: 99%

Numerical techniques for finding 𝜈-zeros of Hankel functions

Cochran¹,

Hoffspiegel²

1970

Math. Comp.

Self Cite

View full text Add to dashboard Cite

show abstract

“…(2) allows one to obtain the zeros of the Macdonald function from those of the Hankel function, and vice versa. Previous research, by other authors [3,11,19,20,25,27], on the -zeros of the Hankel function shows that, for large values of |z|, they occur at z. Asymptotic expansions of H (1) valid in the transition region should then be used to obtain approximate expressions of its z-zeros for | |?1. From Eqs.…”

Section: Zeros Of K (Z) For | |?1mentioning

confidence: 98%

Zeros of the Macdonald function of complex order

Ferreira

Sesma

2008

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

The z-zeros of the modified Bessel function of the third kind K_{nu}(z), also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order nu. Approximate expressions for the zeros, applicable in the cases of very small or very large |nu|, are given. The behaviour of the zeros for varying |nu| or arg(nu), obtained numerically, is illustrated by means of some graphics.Comment: 14 pages, 3 figures, 1 table. Added a table illustrating the numerical procedur

show abstract

“…Introduction. This paper is devoted to an investigation of the validity of an expansion involving the orthogonal functions H(,1,)(kr), where ul, u2,... are the zeros of the function (1) g(U) [t(u )'(/a)+ iZH(u )(ka).…”

mentioning

confidence: 99%

On an Expansion Problem Occurring in the Theory of Diffraction

Νaylor¹,

Barclay²

1975

SIAM J. Math. Anal.

View full text Add to dashboard Cite

This paper considers an eigenfunction expansion arising in certain problems in diffraction theory involving an impedance-type boundary condition. The expansion in question is usually convergent only in part of the domain of interest and involves an infinite set of orthogonal functions which do not form a complete set, in the sense that it is not in general possible to expand a given function in terms of them no matter how well-behaved the function may be. A theorem is established which asserts the validity ot" the expansion whenever it converges. Also a more general expansion formula is obtained which is valid for all sufficiently general functions and which reduces to the basic expansion whenever the latter converges.

show abstract

The zeros of Hankel functions as functions of their order

Cited by 31 publications

References 4 publications

Numerical techniques for finding 𝜈-zeros of Hankel functions

Numerical techniques for finding 𝜈-zeros of Hankel functions

Zeros of the Macdonald function of complex order

On an Expansion Problem Occurring in the Theory of Diffraction

Contact Info

Product

Resources

About