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Mathematics of Computation

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Roots of two transcendental equations determining the frequency spectra of standing spherical electromagnetic waves


Authors: Robert L. Pexton and Arno D. Steiger
Journal: Math. Comp. 31 (1977), 1000-1002
MSC: Primary 65A05
DOI: https://doi.org/10.1090/S0025-5718-1977-0443286-5
MathSciNet review: 0443286
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Abstract: Roots of the transcendental equations \[ \frac {{{j_l}(\lambda )}}{{{y_l}(\lambda )}} = \frac {{{j_l}(\alpha \lambda )\frac {{{i_{l - 1}}(\alpha \lambda \sqrt {|\varepsilon |} )}}{{{i_l}(\alpha \lambda \sqrt {|\varepsilon |} }} - \frac {1}{{\sqrt {|\varepsilon |} }}{j_{l - 1}}(\alpha \lambda )}}{{{y_l}(\alpha \lambda )\frac {{{i_{l - 1}}(\alpha \lambda \sqrt {|\varepsilon |} )}}{{{i_l}(\alpha \lambda \sqrt {|\varepsilon |} }} - \frac {1}{{\sqrt {|\varepsilon |} }}{y_{l - 1}}(\alpha \lambda )}}\] and \[ \frac {{\eta {j_{l - 1}}(\eta ) - l{j_l}(\eta )}}{{\eta {y_{l - 1}}(\eta ) - l{y_l}(\eta )}} = \frac {{\frac {{|\varepsilon |}}{{1 + |\varepsilon |}}\alpha \eta {j_{l - 1}}(\alpha \eta ) - l{j_l}(\alpha \eta ) + \frac {{\sqrt {|\varepsilon |} }}{{1 + |\varepsilon |}}\alpha \eta {j_l}(\alpha \eta )\frac {{{i_{l - 1}}(\alpha \eta \sqrt {|\varepsilon |)} }}{{{i_l}(\alpha \eta \sqrt {|\varepsilon |)} }}}}{{\frac {{|\varepsilon |}}{{1 + |\varepsilon |}}\alpha \eta {y_{l - 1}}(\alpha \eta ) - l{y_l}(\alpha \eta ) + \frac {{\sqrt {|\varepsilon |} }}{{1 + |\varepsilon |}}\alpha \eta {y_l}(\alpha \eta )\frac {{{i_{l - 1}}(\alpha \eta \sqrt {|\varepsilon |)} }}{{{i_l}(\alpha \eta \sqrt {|\varepsilon |)} }}}}\] for the spherical Bessel functions of the first and second kind, ${j_l}(x)$ and ${y_l}(x)$, and for the modified spherical Bessel functions of the first kind, ${i_l}(x)$, have been computed. The ranges for the parameters $\sqrt {|\varepsilon |}$ and $\alpha$, the order l and the root index n are: \[ \sqrt {|\varepsilon |} = 1.0,10.0,100.0,500.0;\quad \alpha = 0.1(0.1)0.7;\quad l = 1(1)15;\quad n = 1(1)30.\]


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Keywords: Roots of transcendental equations, spherical Bessel functions, modified spherical Bessel functions, electromagnetic cavity resonators
Article copyright: © Copyright 1977 American Mathematical Society