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

S. A. Lomov ; A. S. Yudina

Izvestiya. Mathematics , Tome 21 (1983) no. 2, pp. 415-424

Résumé

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.