Remarks on the Zeros of Cross-Product Bessel Functions

Cochran, James Alan

doi:10.1137/0112049

Cited by 28 publications

(22 citation statements)

References 5 publications

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“…However, unlike the widely developed theory of zeros of Bessel functions , , and corresponding cylinder functions (see, e.g., the surveys [7,11,19] and references therein), significantly less inequalities and asymptotic results are known for zeros of , . Among known ones, the inequality of McCann [14, (10)] reads as , ≥ √︂ 2 0, + 2 2 , (1.1) and the following approximation of , for a fixed of McMahon [15, (24)] (see also [5,Theorem,p. 583]) states that…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Recalling that any zero of , is simple (see, e.g., [5]), the rate of change of = ( ) with respect to can be found from the total derivative Finally, recalling that ( ) = ( ) = 0, > 1, and that 2 ( ) + 2 ( ) decreases with respect to > 0 [19, p. 446], we have 2 ( ) 2 ( ) = 2 ( ) + 2 ( ) 2 ( ) + 2 ( ) > 1, and hence (1.3) follows. The proof of (1.3) in the case ( ) = ( ) = 0 can be handled in much the same way as above; see also [20].…”

Section: Pleijel's Constant For Annulimentioning

confidence: 99%

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Asymptotic relation for zeros of cross-product of Bessel functions and applications

Bobkov

2019

Journal of Mathematical Analysis and Applications

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Let , be the -th positive zero of the cross-product of Bessel functions ( ) ( )− ( ) ( ), where ≥ 0 and > 1. We derive an initial value problem for a first order differential equation whose solution ( ) characterizes the limit behavior of , in the following sense: lim →∞ , = ( ), ≥ 0.Moreover, we show thatWe use ( ) to obtain an explicit expression of the Pleijel constant for planar annuli and compute some of its values.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Pleijel's Constant For Annulimentioning

confidence: 99%

Asymptotic relation for zeros of cross-product of Bessel functions and applications

Bobkov

2019

Journal of Mathematical Analysis and Applications

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“…For given waveguide dimensions (a, Tt, r2), dielectric medium, and operating frequency, the admissible angular propagation constants (it is certainly inappropriate to talk in terms of linear propagation constants) appear as the zeros with respect to 11 of one or the other of the cross-product combinations of Bessel functions (6) or (7). It is known [Cochran, 1964] that for positive h both of these sets of 11-zeros are discrete and countably infinite, in general, and only a finite number of the zeros are real, the. remainder being pure imaginary.…”

Section: General Remarksmentioning

confidence: 99%

Mode Propagation in Continuously Curved Waveguides

Cochran¹,

Pecina²

1966

Radio Science

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This paper presents the results of precise numerical computations and extensive analytical investigations of the angular propagation constants for the various electromagnetic modes which may exist in waveguide bends of rectangular cross section. Although the values of the several independent parameters have been chosen to be commensurate with microwave applications, the results are displayed and discussed in a manner which should be convenient for both the analysis and the design of such components in analogous problems in other and diverse practical situations. Moreover, special consideration is given to the relationship of this work to various efforts in the study of radio wave propagation. Two important approximations, one for “high” frequencies and the other for gradual bends, are also considered as a part of the present inquiry. These approximations, especially the latter, turn out to be remarkably good, and rather accurate values of the propagation constants can be obtained with their use for wide ranges of parameter values. The following are typical of the conclusions which can be drawn from this research: (1) cutoff frequencies for modes existing in continuously curved waveguides are curvature dependent; (2) there exist waveguide bends such that circularly polarized waves are undistorted by transmission along the curved section; (3) the angular propagation constants are generally monotonically increasing functions of the operating frequency f (for fixed mean radius) and of the mean radius R (for fixed frequency). Moreover, they vary linearly with f for frequencies far above cutoff and with R for small curvatures.

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“…The paper is organised as follows. In Section 2 we present a detailed study of the mode structure in the bend, which is closely related to the work of Cochran [15,16,17]. In comparison to the work of others, ours is directed more towards the application of the mode structure to scattering calculations.…”

Section: Introductionmentioning

confidence: 98%

The bends on a quantum waveguide and cross-products of Bessel functions

Horvat¹,

Prosen²

2007

J. Phys. A: Math. Theor.

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A detailed analysis of the wave-mode structure in a bend and its incorporation into a stable algorithm for calculation of the scattering matrix of the bend is presented. The calculations are based on the modal approach. The stability and precision of the algorithm is numerically and analytically analysed. The algorithm enables precise numerical calculations of scattering across the bend. The reflection is a purely quantum phenomenon and is discussed in more detail over a larger energy interval. The behaviour of the reflection is explained partially by a one-dimensional scattering model and heuristic calculations of the scattering matrix for narrow bends. In the same spirit we explain the numerical results for the Wigner-Smith delay time in the bend.

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Remarks on the Zeros of Cross-Product Bessel Functions

Cited by 28 publications

References 5 publications

Asymptotic relation for zeros of cross-product of Bessel functions and applications

Asymptotic relation for zeros of cross-product of Bessel functions and applications

Mode Propagation in Continuously Curved Waveguides

The bends on a quantum waveguide and cross-products of Bessel functions

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