The local asymptotic behavior of solutions of second order linear differential equations in a nutshell. (English) Zbl 0795.34003

Previous results by H. Gingold [Asymptotic Anal. 1, 317-350 (1988; Zbl 0672.34051); Math. Anal. Appl. 135, 309-325 (1988; Zbl 0667.34044)] are applied to derive the general solution of the differential equation \[ {d\over {dx}} \left( {{1-x^ 2} \over {\sigma^ 2- x^ 2}} {{dy} \over {dx}} \right)+ \left\{ {1\over {\sigma^ 2- x^ 2}} \left( {{m(\sigma^ 2+ x^ 2)} \over {\sigma(\sigma^ 2- x^ 2)}}- m^ 2\right)+ \varepsilon\right\} y=0, \tag{1} \] where \(\sigma\), \(m\) and \(\varepsilon\) are parameters. This equation possesses a variety of different types of singularities. The authors also give asymptotic formulas for solutions of (1) and their derivatives in the following cases: (i) \(x\to 1\) with \(\sigma\neq 1\), (ii) \(x\to 1\) with \(\sigma=1\) and (iii) \(x\to \sigma\) with \(\sigma\neq 1\).


34M99 Ordinary differential equations in the complex domain
34A30 Linear ordinary differential equations and systems
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34D05 Asymptotic properties of solutions to ordinary differential equations