## 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 $$\sigma\neq 1$$.

### MSC:

 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

### Citations:

Zbl 0672.34051; Zbl 0667.34044