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 
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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     author = {S. L. Skorokhodov and D. V. Khristoforov},
     title = {Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1195--1210},
     year = {2006},
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     number = {7},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_7_a3/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Flammer K., Tablitsy volnovykh sferoidalnykh funktsii, VTs AN SSSR, M., 1962 | MR

[2] Mors F. M., Feshbakh G., Metody teoreticheskoi fiziki, v. 1, Izd-vo inostr. lit., M., 1958; т. 2, 1960

[3] Komarov I. V., Ponomarev L. I., Slavyanov S. Yu., Sferoidalnye i kulonovskie sferoidalnye funktsii, Nauka, M., 1976 | MR | Zbl

[4] http://mathworld.wolfram.com/SpheroidalWaveFunction.html

[5] Abramovits M., Stigan I. (red.), Spravochnik po spetsialnym funktsiyam, Nauka, M., 1979 | MR

[6] Li L.-W., Kang X.-K.,Leong M.-S., Spheroidal wave functions in electromagnetic theory, Wiley, New York, 2001

[7] Falloon P. E., Abbott P. C., Wang J. B., “Theory and computation of the spheroidal wave functions”, J. Phys. A: Math. General., 36 (2003), 5477–5495 {http://arxiv.org/ftp/math-ph/papers/0212/0212051.pdf} http://internal.physics.uwa.edu.au/falloon/spheroidal/spheroidal.html | DOI | MR | Zbl

[8] Markushevich A. I., Teoriya analiticheskikh funktsii, v. 1, 2, Nauka, M., 1968 | Zbl

[9] Beiker Dzh., Greivs-Morris P., Approksimatsii Pade, Mir, M., 1986 | MR

[10] Suetin S. P., “Approksimatsii Pade i effektivnoe analiticheskoe prodolzhenie stepennogo ryada”, Uspekhi matem. nauk, 57:1 (2002), 45–142 | MR | Zbl

[11] Shafer R. E., “On quadratic approximation”, SIAM J. Numer. Analys., 11:2 (1974), 447–460 | DOI | MR | Zbl

[12] Blanch G., Clemm D. S., “The double points of Mathieu's differential equation”, Math. Comput., 23:15 (1969), 97–108 | MR | Zbl

[13] Abramov A. A., Dyshko A. L., Konyukhova N. B., Levitina T. V., “O chislenno-analiticheskom issledovanii zadach difraktsii ploskoi zvukovoi volny na idealnykh vytyanutykh sferoidakh i trekhosnykh ellipsoidakh”, Zh. vychisl. matem. i matem. fiz., 35:9 (1995), 1374–1400 | MR | Zbl

[14] Levitina T. V., Brandas E.J., “Computational techniques for prolate shperoidal wave functions in signal processing”, J. Comput. Meth. Sci. Eng., 1:2s–3s (2001), 287–313 | Zbl

[15] Abramov A. A., Kurochkin C. B., “Vysokotochnoe vychislenie uglovykh sferoidalnykh funktsii”, Zh. vychisl. matem. i matem. fiz., 46:1 (2006), 12–17 | MR | Zbl

[16] Skorokhodov S. L., “Kvaziavtomodelnost sobstvennykh znachenii sferoidalnykh volnovykh funktsii”, Spectral and Evolution Problems, Proc. XV Crimean Autumn Math. School-Symp. (KROMSH-2004), v. 15, Simferopol, Crimea, Ukraine, 2005, 22 | MR | Zbl