Fuchsian equation

Summary

A Fuchsian equation is a linear differential equation with rational coefficients whose singular points are all regular singular points. This includes the point $\infty$ of the Riemann sphere. Its solutions are multivalued functions of controlled growth, described near each singular point by Fuchs' theorem. The equation is largely determined by its singular points, its exponents and its monodromy. The hypergeometric equation, the Legendre equation and the Riemann equation are Fuchsian. See the definition.