# Fuchsian singularity

Suppose that $D$ is an open subset of $\mathbb{C}$ and that the $n$ functions $c_{k}\colon D\to\mathbb{C},\quad k=0,\ldots,n-1$ are meromorphic. Consider the ordinary differential equation

 ${d^{n}w\over dz^{n}}+\sum_{k=0}^{n-1}c_{k}(z){d^{k}w\over dz^{k}}=0$

A point $p\in D$ is said to be a regular singular point or a Fuchsian singular point of this equation if at least one of the functions $c_{k}$ has a pole at $p$ and, for every value of $k$ between $0$ and $n$, either $c_{k}$ is regular 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

 ${dv_{i}\over dz}={1\over z}\sum_{j=1}^{n}b_{ij}(z)v_{j}(z)$

in which the coefficient functions $b_{ij}$ are analytic at $z$. This fact helps explain the restiction on the orders of the poles of the $c_{k}$’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 point of a differential equation which is not a regular singular point is known as an irregular singular point.

Examples

 $w^{\prime\prime}+{1\over z}w^{\prime}+{z^{2}-1\over z^{2}}w=0$

has a Fuchsian singularity at $z=0$ since the coefficient of $w^{\prime}$ has a pole of order $1$ and the coefficient of $w$ has a pole of order $2$.

On the other hand, the Hamburger equation

 $w^{\prime\prime}+{2\over z}w^{\prime}+{z^{2}-1\over z^{4}}w=0$

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