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 point 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 
The generalized series at  is defined by

 

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

 

where  denotes the falling factorial notation.

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  in . 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  of ,  must be a root of the indicial polynomial at , i. e.,  needs to solve the indicial equation .

If  is an ordinary point, the resulting indicial equation is given by . If  is a regular singularity, then  and if  is an irregular singularity,  holds. This is illustrated by the later examples. The indicial equation relative to  is defined by the indicial equation of , where  denotes the differential operator  transformed by which is a linear differential operator in , at .

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 
If  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.

Fundamental system at regular singularity 
If  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 .

General result 
One can show that a linear differential equation of order  always has  linearly independent solutions of the form

 

where  and , and the formal power series .

 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  at .

References 

Differential equations