Geodesic
Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 7, pp. 1195-1210 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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S. L. Skorokhodov; D. V. Khristoforov. Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 7, pp. 1195-1210. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_7_a3/

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