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

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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.[1]

Indicial equation

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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  .[1]

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.[2] 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  .[3]

Example: Regular singularity

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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

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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

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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

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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.[1]

Fundamental system at regular singularity

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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  .[4]

General result

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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  .[5]

  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

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  1. ^ a b c Tenenbaum, Morris; Pollard, Harry (1963). Ordinary Differential Equations. New York, USA: Dover Publications. pp. Lesson 40. ISBN 9780486649405.
  2. ^ Ince, Edward Lindsay (1956). Ordinary Differential Equations. New York, USA: Dover Publications. pp. 160. ISBN 9780486158211.
  3. ^ Ince, Edward Lindsay (1956). Ordinary Differential Equations. New York, USA: Dover Publications. pp. 370. ISBN 9780486158211.
  4. ^ Ince, Edward Lindsay (1956). Ordinary Differential Equations. New York, USA: Dover Publications. pp. Section 16.3. ISBN 9780486158211.
  5. ^ Kauers, Manuel; Paule, Peter (2011). The Concrete Tetrahedron. Vienna, Austria: Springer-Verlag. pp. Theorem 7.3. ISBN 9783709104453.
  • Ince, Edward Lindsay (1956). Ordinary Differential Equations. New York, USA: Dover Publications. ISBN 9780486158211.
  • Poole, Edgar Girard Croker (1936). Introduction to the theory of linear differential equations. New York: Clarendon Press.
  • Tenenbaum, Morris; Pollard, Harry (1963). Ordinary Differential Equations. New York, USA: Dover Publications. pp. Lecture 40. ISBN 9780486649405.
  • Horn, Jakob (1905). Gewöhnliche Differentialgleichungen beliebiger Ordnung. Leipzig, Germany: G. J. Göschensche Verlagshandlung.
  • Schlesinger, Ludwig Lindsay (1897). Handbuch der Theorie der linearen Differentialgleichungen (2. Band, 1. Teil). Leipzig, Germany: B. G.Teubner. pp. 241 ff.
  • Lay, Wolfgang (2024). Higher Special Functions. Stuttgart, Germany: Cambridge University Press. pp. 114–156. ISBN 9781009128414.