Fuchsian singularity

Suppose that D is an open subset of and that the n functions ck:D,k=0,,n-1 are meromorphic. Consider the ordinary differential equationMathworldPlanetmath


A point pD is said to be a regular singular pointMathworldPlanetmath or a Fuchsian singular point of this equation if at least one of the functions ck has a pole at p and, for every value of k between 0 and n, either ck is regularPlanetmathPlanetmathPlanetmath at p or has a pole of order not greater than n-k.

If p is a Fuchsian singular point, then the differential equation may be rewritten as a system of first order equations


in which the coefficient functions bij are analytic at z. This fact helps explain the restiction on the orders of the poles of the ck’s.

If an equation has a Fuchsian singularity, then the solution can be expressed as a Frobenius series in a neighborhood of this point.

A singular pointMathworldPlanetmathPlanetmathPlanetmathPlanetmath of a differential equation which is not a regular singular point is known as an irregular singular point.


The Bessel equation


has a Fuchsian singularity at z=0 since the coefficient of w has a pole of order 1 and the coefficient of w has a pole of order 2.

On the other hand, the Hamburger equation


has an irregular singularity at z=0 since the coefficient of w has a pole of order 4.

Title Fuchsian singularity
Canonical name FuchsianSingularity
Date of creation 2013-03-22 14:47:26
Last modified on 2013-03-22 14:47:26
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 14
Author rspuzio (6075)
Entry type Definition
Classification msc 34A25
Synonym Fuchsian singular point
Synonym regular singular point
Synonym regular singularity
Related topic FrobeniusMethod
Defines irregular singular point
Defines irregular singularity
Defines Hamburger equation