|
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
$$
w_k(z,\varepsilon)=z^{r_k(\varepsilon)/\varepsilon}
\exp\biggl\{\frac1{\varepsilon}\int_0^z\lambda_k(\tau)\,d\tau\biggr\}
\sum_{i=0}^\infty\varepsilon^iw^k_i(z),\quad k=1,2.
$$
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.
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”, Math. USSR-Izv., 21:2 (1983), 415–424
Linking options:
https://www.mathnet.ru/eng/im1698https://doi.org/10.1070/IM1983v021n02ABEH001798 https://www.mathnet.ru/eng/im/v46/i5/p1124
|
|