Fuchsian singularity

Suppose that $D$ is an open subset of $ℂ$ and that the $n$ functions $c_k:D\to ℂ,k=0,\mathrm{\dots },n-1$ are meromorphic. Consider the ordinary differential equation

$$\frac{d^{n}w}{d{z}^n}+\sum _{k=0}^{n-1}{c}_k(z)\frac{d^{k}w}{d{z}^k}=0$$

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

$$\frac{d{v}_i}{dz}=\frac{1}{z}\sum _{j=1}^n{b}_{ij}(z)v_j(z)$$

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

The Bessel equation

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

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

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

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