Fuchsian theory

The Fuchsian theory of linear differential equations, which is named after Lazarus Immanuel Fuchs, provides a characterization of various types of singularities and the relations among them.

At any ordinary point of a homogeneous linear differential equation of order ${\displaystyle n}$ there exists a fundamental system of ${\displaystyle n}$ linearly independent power series solutions. A non-ordinary point is called a singularity. At a singularity the maximal number of linearly independent power series solutions may be less than the order of the differential equation.