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Fuchsian singularity (Definition)

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

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

$\displaystyle {d v_i \over dz} = {1 \over z} \sum_{j=1}^{n} b_{ij}(z) v_m (z)$
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

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

$\displaystyle w'' + {2 \over z} w' + {z^2 - 1 \over z^4} w = 0$
has an irregular singularity at $ z=0$ since the coefficient of $ w$ has a pole of order $ 4$.



"Fuchsian singularity" is owned by rspuzio.
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See Also: Frobenius method

Other names:  Fuchsian singular point, regular singular point, regular singularity
Also defines:  irregular singular point, irregular singularity
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Cross-references: Bessel equation, singular point, neighborhood, series, solution, analytic, coefficient, first order, differential equation, order, regular, pole, equation, point, ordinary differential equation, meromorphic, functions, open subset
There are 2 references to this entry.

This is version 8 of Fuchsian singularity, born on 2004-11-03, modified 2006-11-01.
Object id is 6442, canonical name is FuchsianSingularity.
Accessed 5016 times total.

Classification:
AMS MSC: 34A25 (Ordinary differential equations :: General theory :: Analytical theory: series, transformations, transforms, operational calculus, etc.)
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