“Fuchsian theory”的意思、由来-开放百科全书

The Fuchsian theory of linear differential equations, which is named after Lazarus Immanuel Fuchs, provides a characterization of various types of singularities and the relations among them.

At any ordinary point of a homogeneous linear differential equation of order

there exists a fundamental system of

linearly independent power series solutions. A non-ordinary points is called a singularity. At a singularity the maximal number of linearly independent power series solutions may be less than the order of the differential equation.

Generalized series solutions

which is known as Frobenius series, due to the connection with the Frobenius series method. Frobenius series solutions are formal solutions of differential equations. The formal derivative of

, with

, is defined such that

. Let

denote a Frobenius series relative to

, then

Indicial equation

Let

be a Frobenius series relative to

. Let

be a linear differential operator of order

with one valued coefficient functions

. Let all coefficients

be expandable as Laurent series with finite principle part at

. Then there exists a smallest

such that

is a power series for all

. Hence,

is a Frobenius series of the form

, with a certain power series

. The indicial polynomial is defined by

which is a polynomial in

, i.e.,

equals the coefficient of

with lowest degree in

. For each formal Frobenius series solution

must be a root of the indicial polynomial at

, i. e.,

needs to solve the indicial equation

.^[1]

is an ordinary point, the resulting indicial equation is given by

. If

is a regular singularity, then

and if

is an irregular singularity,

holds.^[2] This is illustrated by the later examples. The indicial equation relative to

is defined by the inidical equation of

, where

denotes the differential operator

transformed by

which is a linear differential operator in

, at

.^[3]

Example: Regular singularity

The differential operator of order

, has a regular singularity at

. Consider a Frobenius series solution relative to

with

This implies that the degree of the indicial polynomial relative to

is equal to the order of the differential equation,

Example: Irregular singularity

The differential operator of order

, has an irregular singularity at

. Let

be a Frobenius series solution relative to

Certainly, at least one coefficient of the lower derivatives pushes the exponent of

down. Inevitably, the coefficient of a lower derivative is of smallest exponent. The degree of the indicial polynomial relative to

is less than the order of the differential equation,

Formal fundamental systems

We have given a homogeneous linear differential equation

of order

with coefficients that are expandable as Laurent series with finite principle part. The goal is to obtain a fundamental set of formal Frobenius series solutions relative to any point

. This can be done by the Frobenius series method, which says: The starting exponents are given by the solutions of the indicial equation and the coefficients describe a polynomial recursion. W.l.o.g., assume

Fundamental system at ordinary point

is an ordinary point, a fundamental system is formed by the

linearly independent formal Frobenius series solutions

, where

denotes a formal power series in

with

, for

. Due to the reason that the starting exponents are integers, the Frobenius series are power series.^[1]

Fundamental system at regular singularity

is a regular singularity, one has to pay attention to roots of the indicial polynomial that differ by integers. In this case the recursive calculation of the Frobenius series' coefficients stops for some roots and the Frobenius series method does not give an

-dimensional solution space. The following can be shown independent of the distance between roots of the indicial polynomial: Let

be a

-fold root of the indicial polynomial relative to

. Then the part of the fundamental system corresponding to

is given by the

linearly independent formal solutions

where

denotes a formal power series in

with

, for

. One obtains a fundamental set of

linearly independent formal solutions, because the indicial polynomial relative to a regular singularity is of degree

.^[4]

General result

One can show that a linear differential equation of order

always has

linearly independent solutions of the form

is an irregular singularity if and only if there is a solution with

. Hence, a differential equation is of Fuchsian type if and only if for all

there exists a fundamental system of Frobenius series solutions with

References

1. ^¹²{{Cite book|title=Ordinary Differential Equations|last1=Tenenbaum|first1=Morris|last2=Pollard|first2=Harry|publisher=Dover Publications|year=1963|isbn=9780486649405|location=New York, USA|pages=Lesson 40}}
2. ^{{Cite book|title=Ordinary Differential Equations|last=Ince|first=Edward Lindsay|publisher=Dover Publications|year=1956|isbn=9780486158211|location=New York, USA|pages=160}}
3. ^{{Cite book|title=Ordinary Differential Equations|last=Ince|first=Edward Lindsay|publisher=Dover Publications|year=1956|isbn=9780486158211|location=New York, USA|pages=370}}
4. ^{{Cite book|title=Ordinary Differential Equations|last=Ince|first=Edward Lindsay|publisher=Dover Publications|year=1956|isbn=9780486158211|location=New York, USA|pages=Section 16.3}}
5. ^{{Cite book|title=The Concrete Tetrahedron|last1=Kauers|first1=Manuel|last2=Paule|first2=Peter|publisher=Springer-Verlag|year=2011|isbn=9783709104453|location=Vienna, Austria|pages=Theorem 7.3}}