How regular are regular singularities?
In this paper, the author proposes an abstract look at the works of Fuchs and Frobenius on the solutions of ordinary differential equations at regular singularities. Starting from the study of the solutions of very simple equations like \begin{align*} &x^2y''-3xy'+3y=0\\ &x^3y'''-4x^2y''+9xy'-9=0\\ &x^2y''-3xy'+3y+x^3y=0\\ &x^2y''-3xy'+3y+x^2y=0 \end{align*} the author introduces various tools and various fundamental results necessary to understand the solutions in the neighborhood of the singularity \(x=0\). In particular, he shows that all relevant informations on the structure of the solutions are contained in the initial form of the differential operator. For the entire collection see [Zbl 1478.11004].
scientific article; zbMATH DE number 7615268
scientific article