Fuchsian equation in nLab

Idea

A Fuchsian (differential) equation is a linear homogeneous ordinary differential equation with analytic coefficients in the complex domain whose singular points, including at infinity, are all regular singular points. In other words, it is the 1-dimensional case of the theory of meromorphic differential equations with only regular singular points. Hilbert's 21st problem is concerned with finding a Fuchsian equation with prescribed points of singularities and prescribed monodromies. The corresponding connection is also called Fuchsian.

References

• M.V. Fedoryuk, Fuchsian equation, Springer online Enc. of Math.

• J.L. Fuchs, J. Reine Angew. Math. 66, 121–160 (1866); 68, 354–385 (1868)

• A. Haefliger, Local theory of meromorphic connections in dimension one (Fuchs theory), pages 129-149 of “Algebraic D-modules”, A. Borel, ed.

• P. Deligne, Équations différentielles à points singuliers réguliers, Lect. Notes in Math. 163, Springer-Verlag (1970)

• M. Yoshida, Fuchsian differential equations, Aspects of Math. Eil, Vieweg, Braunschweig 1987.

• Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, Masaaki Yoshida, From Gauss to Painlevé, A modern theory of special functions, 184 pp.