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Izv. Akad. Nauk SSSR Ser. Mat., 1982, Volume 46, Issue 5, Pages 1124–1133 (Mi izv1698)  

This article is cited in 1 scientific paper (total in 1 paper)

The structure of a fundamental system of solutions of a singularly perturbed equation with a regular singular point

S. A. Lomov, A. S. Yudina


Abstract: The method of regularization is applied to obtain a fundamental system of solutions of a singularly perturbed equation with a regular singular point
$$ \varepsilon^2z^2w"+\varepsilon zp(z)w'+g(z)w =0. $$
The solutions are of the form
$$ w_k(z,\varepsilon)=z^{r_k(\varepsilon)/\varepsilon} \exp\{\frac1{\varepsilon}\int_0^z\lambda_k(\tau) d\tau\} \sum_{i=0}^\infty\varepsilon^iw^k_i(z),\quad k=1,2. $$
The series are asymptotically convergent as $\varepsilon\to0$ uniformly in $z$ in some bounded domain. Here the $r_k(\varepsilon)$ are the roots of the indicial equations, the $\lambda_k(z)$ are the roots of the characteristic equation and the functions $w_i^k(z)$ are the solutions of certain recurrent linear differential equations of the first order. The results are applied to an asymptotic expansion of Bessel functions $I_\nu(\nu z)$ as $\nu\to\infty$.
Bibliography: 5 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1983, 21:2, 415–424

Bibliographic databases:

UDC: 517.9
MSC: Primary 34A20, 34B30, 34E15; Secondary 33A40, 34D15
Received: 01.07.1981

Citation: S. A. Lomov, A. S. Yudina, “The structure of a fundamental system of solutions of a singularly perturbed equation with a regular singular point”, Izv. Akad. Nauk SSSR Ser. Mat., 46:5 (1982), 1124–1133; Math. USSR-Izv., 21:2 (1983), 415–424

Citation in format AMSBIB
\Bibitem{LomYud82}
\by S.~A.~Lomov, A.~S.~Yudina
\paper The structure of a~fundamental system of solutions of a~singularly perturbed equation with a~regular singular point
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1982
\vol 46
\issue 5
\pages 1124--1133
\mathnet{http://mi.mathnet.ru/izv1698}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=675534}
\zmath{https://zbmath.org/?q=an:0536.34034|0513.34064}
\transl
\jour Math. USSR-Izv.
\yr 1983
\vol 21
\issue 2
\pages 415--424
\crossref{https://doi.org/10.1070/IM1983v021n02ABEH001798}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. S. Yudina, “The singularly perturbed Bessel equation in complex domains”, Izv. Math., 73:3 (2009), 627–653  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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