Numerical Techniques for Finding ν-Zeros of Hankel Functions

Cochran, James Alan; Hoffspiegel, Judith N.

doi:10.2307/2004488

Cited by 4 publications

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“…To be specific, the eigenvalues in the exponential potential are expressed via zeros of the modified Bessel function of the second kind (known also as the Macdonald function) K iν (z) of imaginary order. The computation of the K iν (z) zeros with respect to order, known as ν-zeros, is not readily implemented in modern software even though algorithms for this were proposed decades ago [8,10]. Moreover, available asymptotic expansions for large zeros reported in the literature [4,9,20] provide rather inaccurate estimations not applicable for direct calculations.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of K_iν(z) with Respect to Order

Krynytskyi¹,

Rovenchak²

2021

SIGMA

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The paper presents the derivation of the asymptotic behavior of ν-zeros of the modified Bessel function of imaginary order K iν (z). This derivation is based on the quasiclassical treatment of the exponential potential on the positive half axis. The asymptotic expression for the ν-zeros (zeros with respect to order) contains the Lambert W function, which is readily available in most computer algebra systems and numerical software packages. The use of this function provides much higher accuracy of the estimation comparing to known relations containing the logarithm, which is just the leading term of W (x) at large x. Our result ensures accuracies sufficient for practical applications.

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Section: Introductionmentioning

confidence: 99%

Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of K_iν(z) with Respect to Order

Krynytskyi¹,

Rovenchak²

2021

SIGMA

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“…For the location of these, the Newton method has been used. Let us mention, however, other existing techniques like, for instance, the auxiliary tables for the evaluation of the ν-zeros of K ν (z), for z real or pure imaginary, proposed by Cochran and Hoffspiegel [4], the finite element approximation method, applied by Leung and Ghaderpanah [22] to a very precise determination of the zeros of K n (z), and a procedure, applied by Segura [28] to the computation of zeros of Bessel and other special functions, that uses fixed point iterations and does not require the evaluation of the functions. A double precision FORTRAN code was used to compute the expression…”

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confidence: 99%

Zeros of the Macdonald function of complex order

Ferreira

Sesma

2008

Journal of Computational and Applied Mathematics

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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

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Complex zeros of the modified Bessel function 𝐾_{𝑛}(𝑍)

Parnes¹

1972

Math. Comp.

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Abstract.The complex zeros of Kn(Z) are computed for integer orders n = 2(1)10, to 9D figures, using an iterative interpolation scheme. Introduction.The investigation of wave propagation and scattering in elastic media is often performed by means of integral transform methods. The analysis of such problems in cylindrical coordinates often leads to waves whose transformed potential functions are expressed in terms of modified Bessel functions. In particular, the potentials for outgoing radiating waves which decay with increasing distance from the source are expressed in terms of modified Bessel functions of the second kind, Kn(Z). In the course of a recent study using the Laplace transform, it was necessary to determine complex zeros of K£Z) in order to locate poles of the solution required for the inversion. It is believed that the tabulated zeros given below will permit the evaluation of several significant scattering problems.Several methods for the evaluation of zeros of Bessel functions, notably by means of asymptotic expansions, have been given by Olver [1] and Luke [2]. The methods developed by Olver, however, are not entirely applicable in the present case, since the convergence only improves with large orders of n. On the other hand, the rational approximations given by Luke have been proved, under appropriate restrictions of the parameters, to converge in the first quadrant; convergence for |arg Z\ < it has recently been established by Fields [3]. It should be observed that Cochran and Hoffspiegel [4] have extended the work of Olver for determining the positive real or purely imaginary zeros of Hankel functions with respect to noninteger orders when the variable is held fixed. However, their methods are inapplicable, since the zeros of the modified Bessel functions are seen to be always complex. A method for the evaluation of complex zeros of cylinder functions was given by Döring [5] based on MacMahon and Olver expansions. Tabulated values were given for some zeros of Hankel functions Hn(Z) which can be related to those of Kn(Z).The zeros presented below to 9D significant figures were calculated according to a method described in the following section.

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Numerical Techniques for Finding ν-Zeros of Hankel Functions

Cited by 4 publications

References 0 publications

Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of K_iν(z) with Respect to Order

Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of K_iν(z) with Respect to Order

Zeros of the Macdonald function of complex order

Complex zeros of the modified Bessel function 𝐾_{𝑛}(𝑍)

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Numerical Techniques for Finding ν-Zeros of Hankel Functions

Cited by 4 publications

References 0 publications

Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kiν(z) with Respect to Order

Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of Kiν(z) with Respect to Order

Zeros of the Macdonald function of complex order

Complex zeros of the modified Bessel function 𝐾_{𝑛}(𝑍)

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Product

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Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of K_iν(z) with Respect to Order

Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of K_iν(z) with Respect to Order