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# 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

The Bessel equation

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

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 $4$.

## Mathematics Subject Classification

34A25*no label found*

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