In mathematics, the Fuchs relation is a relation between the starting exponents of formal series solutions of certain linear differential equations, so called Fuchsian equations. It is named after Lazarus Immanuel Fuchs.
Definition Fuchsian equation
A linear differential equation in which every singular point, including the point at infinity, is a regular singularity is called Fuchsian equation or equation of Fuchsian type.[1] For Fuchsian equations a formal fundamental system exists at any point, due to the Fuchsian theory.
Coefficients of a Fuchsian equation
Let be the regular singularities in the finite part of the complex plane of the linear differential equation
with meromorphic functions . For linear differential equations the singularities are exactly the singular points of the coefficients. is a Fuchsian equation if and only if the coefficients are rational functions of the form
with the polynomial and certain polynomials for , such that .[2] This means the coefficient has poles of order at most , for .
Fuchs relation
Let be a Fuchsian equation of order with the singularities and the point at infinity. Let be the roots of the indicial polynomial relative to , for . Let be the roots of the indicial polynomial relative to , which is given by the indicial polynomial of transformed by at . Then the so called Fuchs relation holds:
- .[3]
The Fuchs relation can be rewritten as infinite sum. Let denote the indicial polynomial relative to of the Fuchsian equation . Define as
where gives the trace of a polynomial , i. e., denotes the sum of a polynomial's roots counted with multiplicity.
This means that for any ordinary point , due to the fact that the indicial polynomial relative to any ordinary point is . The transformation , that is used to obtain the indicial equation relative to , motivates the changed sign in the definition of for . The rewritten Fuchs relation is:
References
- Ince, Edward Lindsay (1956). Ordinary Differential Equations. New York, USA: Dover Publications. ISBN 9780486158211.
- 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 (1897). Handbuch der Theorie der linearen Differentialgleichungen (2. Band, 1. Teil). Leipzig, Germany: B. G.Teubner. pp. 241 ff.
- ↑ Ince, Edward Lindsay (1956). Ordinary Differential Equations. New York, USA: Dover Publications. p. 370. ISBN 9780486158211.
- ↑ Horn, Jakob (1905). Gewöhnliche Differentialgleichungen beliebiger Ordnung. Leipzig, Germany: G. J. Göschensche Verlagshandlung. p. 169.
- ↑ Ince, Edward Lindsay (1956). Ordinary Differential Equations. New York, USA: Dover Publications. p. 371. ISBN 9780486158211.
- ↑ Landl, Elisabeth (2018). The Fuchs Relation (Bachelor Thesis). Linz, Austria. chapter 3.