21. PROOF OF THE EXISTENCE OF LINEAR DIFFERENTIAL EQUATIONS HAVING A PRESCRIBED MONODROMIC GROUP.
In the theory of linear differential equations with one independent variable
, I wish to indicate an important problem, one which very likely Riemann himself may have had in mind. This problem is as follows: To show that there always exists a linear differential equation of the Fuchsian class, with given singular points and monodromic group. The problem requires the production of
functions of the variable
, regular throughout the complex
plane except at the given singular points; at these points the functions may become infinite of only finite order, and when
describes circuits about these points the functions shall undergo the prescribed linear substitutions. The existence of such differential equations has been shown to be probable by counting the constants, but the rigorous proof has been obtained up to this time only in the particular case where the fundamental equations of the given substitutions have roots all of absolute magnitude unity. L. Schlesinger has given this proof,[49] based upon Poincaré's theory of the Fuchsian
-functions. The theory of linear differential equations would evidently have a more finished appearance if the problem here sketched could be disposed of by some perfectly general method.
[49] Handbuch der Theorie der linearen Differentialgleichungen, vol. 2, part 2, No. 366.