1933/01/01 by George D. Birkhoff, W. J. Trjitzinsky · 160 citations
Mathematics · Medicine · #Applied mathematics #Calculus (dental) #Differential Equations and Numerical Methods #Mathematical analysis #Mathematical and Theoretical Epidemiology and Ecology Models #Mathematics #Nonlinear Differential Equations Analysis #Singular solution
paper · pdf · doi:10.1007/bf02398269
published in Acta Mathematica 60(0), 1-89 (Mittag-Leffler Institute)
openalex publication_date 1933/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
Definition 6.An operator L, (y) (or equation Ln (y)--o) which is proper in G will be called completely proper in G if, in G, proper fundamental sets of solutions exist which are connected by proper periodic functions.Definition 7. A set Q, (x) , . . .q,, (x) has a point of division in G if this set can be separated into two groups Analytic Theory of Singular Difference Equations.q, (x),... Q,,(x); Q~+~ (~),... Qn(x) (I ~ 1"< n) so that for x in G Q'~ (x) => ~ Q'~+.(x) (~= I',... F; ~= I, ... n--F).Definition 8. Let G denote a subregion of F, the lower boundaries of G and F being coincident.An operator L,,(y) (or equation L~(y)= o) with coefficie~ts of the same kind, in G, as in (I) and (2) will be said to be Q-factorable in G if the set of Q's, Q, (x), ... Q, (x), belonging to L~ (y), has a point of divi,~ion in G.Definition 9. Let F be a curve extending to infinity and lying in I. Let RF be the portion of F between F and the right boundary of F. The curve 9 and the region RF will be termed proper for the set Q, (x), Q, (~) if this set is proper along F and ah'o in RF.2.Let us write