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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Roots of two transcendental equations determining the frequency spectra of standing spherical electromagnetic waves
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by Robert L. Pexton and Arno D. Steiger PDF
Math. Comp. 31 (1977), 1000-1002 Request permission

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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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Math. Comp. 31 (1977), 1000-1002
  • MSC: Primary 65A05
  • DOI: https://doi.org/10.1090/S0025-5718-1977-0443286-5
  • MathSciNet review: 0443286