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Roots of two transcendental equations as functions of a continuous real parameter


Authors: Robert L. Pexton and Arno D. Steiger
Journal: Math. Comp. 32 (1978), 511-518
MSC: Primary 65H10
DOI: https://doi.org/10.1090/S0025-5718-1978-0488704-2
MathSciNet review: 0488704
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Abstract: The roots, $ \lambda $ and $ \eta $, of the transcendental equations $ {j_l}(\alpha \lambda ){y_l}(\lambda ) = {j_l}(\lambda ){y_l}(\alpha \lambda )$ and

$\displaystyle [x{j_l}(x)]_{x = \alpha \eta }^\prime [x{y_l}(x)]_{x = \eta }^\prime = [x{j_l}(x)]_{x = \eta }^\prime [x{y_l}(x)]_{x = \alpha \eta }^\prime $

where $ l = 1,2, \ldots $ are considered as functions of the continuous real parameter $ \alpha $. The symbols $ {j_l}$ and $ {y_l}$ denote the spherical Bessel functions of the first and second kind. The two transcendental equations are invariant under the transformations $ \lambda \to - \lambda $ and $ \eta \to - \eta $, respectively. Therefore, only positive roots are discussed. All the $ \lambda $-roots increase monotonically as $ \alpha $ increases in the open interval (0, 1). For each order l, the smallest $ \eta $-root decreases monotonically as $ \alpha $ increases in (0, 1), tending towards $ \sqrt {l(l + 1)} $ as $ \alpha $ approaches unity. For $ \alpha \in (0,1)$ all the other $ \eta $-roots have a minimum value equal to $ \sqrt {l(l + 1)} /\alpha \,$.

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

  • [1] ROBERT L. PEXTON & ARNO D. STEIGER, "Roots of two transcendental equations involving spherical Bessel functions," Math. Comp., v. 31, 1977, pp. 752-753. MR 0438662 (55:11570)
  • [2] M. ABRAMOWITZ & I. A. STEGUN, Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series No. 55, U. S. Government Printing Office, Washington, D. C., 1965.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1978-0488704-2
Keywords: Roots of transcendental equations, spherical Bessel functions, electromagnetic cavity resonators
Article copyright: © Copyright 1978 American Mathematical Society

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