Geodesic
The structure of a fundamental system of solutions of a singularly perturbed equation with a regular singular point
Izvestiya. Mathematics, Tome 21 (1983) no. 2, pp. 415-424 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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S. A. Lomov; A. S. Yudina. The structure of a fundamental system of solutions of a singularly perturbed equation with a regular singular point. Izvestiya. Mathematics, Tome 21 (1983) no. 2, pp. 415-424. http://geodesic.mathdoc.fr/item/IM2_1983_21_2_a10/

[1] Rabinovich Yu. A., Khapaev M. M., “Lineinye uravneniya s malym parametrom pri starshei proizvodnoi v okrestnosti regulyarno-osoboi tochki”, Dokl. AN SSSR, 129:2 (1959), 268–271 | MR | Zbl

[2] Yudina A. S., “Regulyarizovannye asimptoticheskie resheniya uravnenii s regulyarnoi osoboi tochkoi”, Tr. MEI, 499 (1980), 147–151 | MR | Zbl

[3] Lomov S. A., “Metod vozmuschenii dlya singulyarnykh zadach”, Izv. AN SSSR. Ser. matem., 36:3 (1972), 635–651 | MR | Zbl

[4] Golubev V. V., Lektsii po analiticheskoi teorii differentsialnykh uravnenii, GITTL, M., 1950

[5] Lomov S. A., Vvedenie v obschuyu teoriyu singulyarnykh vozmuschenii, Nauka, M., 1981 | MR