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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2006, Volume 46, Number 7, Pages 1195–1210
(Mi zvmmf438)
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This article is cited in 9 scientific papers (total in 9 papers)
Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions
S. L. Skorokhodova, D. V. Khristoforovb a Dorodnicyn Computing Center, Russian Academy of Sciences,
ul. Vavilova 40, Moscow, 119991, Russia
b Faculty of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, 119992, Russia
Abstract:
A method for calculating eigenvalues $\lambda_{mn}(c)$ corresponding to the wave spheroidal functions in the case of a complex parameter c is proposed, and a comprehensive numerical analysis is performed. It is shown that some points $c_s$ are the branch points of the functions $\lambda_{mn}(c)$ with different indexes $n_1$ and $n_2$ so that the value $\lambda_{mn_1}(c_s)$ is a double one: $\lambda_{mn_1}(c_s)=\lambda_{mn_2}(c_s)$. The numerical analysis suggests that, for each fixed $m$, all the branches of the eigenvalues $\lambda_{mn}(c)$ corresponding to the even spheroidal functions form a complete analytic function of the complex argument $c$. Similarly, all the branches of the eigenvalues $\lambda_{mn}(c)$ corresponding to the odd spheroidal functions form a complete analytic function of $c$. To perform highly accurate calculations of the branch points $c_s$ of the double eigenvalues $\lambda_{mn}(c)$, the Padé approximants, the Hermite–Padé quadratic approximants, and the generalized Newton iterative method are used. A large number of branch points are calculated.
Key words:
wave spheroidal functions, computation of eigenvalues, computation of branch points of eigenvalues, Padé approximants, generalized Newton iterative method.
Received: 21.12.2005
Citation:
S. L. Skorokhodov, D. V. Khristoforov, “Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions”, Zh. Vychisl. Mat. Mat. Fiz., 46:7 (2006), 1195–1210; Comput. Math. Math. Phys., 46:7 (2006), 1132–1146
Linking options:
https://www.mathnet.ru/eng/zvmmf438 https://www.mathnet.ru/eng/zvmmf/v46/i7/p1195
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