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.
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.
regular singularities in the finite part of the complex plane of the linear differential equation
{\displaystyle Lf:={\frac {d^{n}f}{dz^{n}}}+q_{1}{\frac {d^{n-1}f}{dz^{n-1}}}+\cdots +q_{n-1}{\frac {df}{dz}}+q_{n}f}
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
[2] This means the coefficient
has poles of order at most
be a Fuchsian equation of order
and the point at infinity.
be the roots of the indicial polynomial relative to
be the roots of the indicial polynomial relative to
, which is given by the indicial polynomial of
Then the so called Fuchs relation holds: The Fuchs relation can be rewritten as infinite sum.
denote the indicial polynomial relative to
of the Fuchsian equation
defect :
{\displaystyle \operatorname {defect} :\mathbb {C} \cup \{\infty \}\to \mathbb {C} }
gives the trace of a polynomial
denotes the sum of a polynomial's roots counted with multiplicity.
defect ( ξ ) = 0
{\displaystyle \operatorname {defect} (\xi )=0}
for any ordinary point
, due to the fact that the indicial polynomial relative to any ordinary point is
, that is used to obtain the indicial equation relative to
, motivates the changed sign in the definition of
The rewritten Fuchs relation is: