# Fuchsian singularity

Suppose that $D$ is an open subset of $\u2102$ and that the $n$ functions ${c}_{k}:D\to \u2102,k=0,\mathrm{\dots},n-1$ 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 |