Defining equation

determining equation, characteristic equation

An equation associated with a regular singular point $ z = a $ of an ordinary linear differential equation

$$ \tag{1 } p _ {0} ( z) w ^ {(} n) + \dots + p _ {n} ( z) w = 0. $$

Let

$$ p _ {j} ( z) = ( z- a) ^ {n-} j q _ {j} ( z), $$

where the functions $ q _ {j} ( z) $ are holomorphic at the point $ z= a $ and $ q _ {0} ( a) \neq 0 $. The defining equation takes the form:

$$ \tag{2 } \lambda \dots ( \lambda - n+ 1) q _ {0} ( a) + \dots + \lambda q _ {n-} 1 ( a) + q _ {n} ( a) = 0. $$

If the roots $ \lambda _ {j} $, $ 1 \leq j \leq n $, of equation (2) are such that all differences $ \lambda _ {j} - \lambda _ {k} $, where $ j \neq k $, are not integers, then equation (1) has a fundamental system of solutions of the form

$$ \tag{3 } w _ {j} ( z) = ( z- a) ^ {\lambda _ {j} } \phi _ {j} ( z),\ 1 \leq j \leq n , $$

where the functions $ \phi _ {j} ( z) $ are holomorphic at $ z= a $. Otherwise the coefficients $ \phi _ {j} ( z) $ can be polynomials in $ \mathop{\rm ln} ( z- a) $ with coefficients holomorphic at $ z= a $.

The defining equation for a system of $ n $ equations

$$ \tag{4 } ( z- a) w ^ \prime = A( z) w, $$

corresponding to the regular singular point $ z= a $, takes the form

$$ \mathop{\rm det} \| \lambda I - A( a) \| = 0, $$

where $ A( z) $ is a matrix-function of order $ n \times n $, holomorphic at $ z= a $ and $ A( a) \neq 0 $. If all differences $ \lambda _ {j} - \lambda _ {k} $, where $ j \neq k $, are not integers, where the $ \lambda _ {j} $ are the eigen values of $ A $, then the system (4) has a fundamental system of solutions of the form (3), where $ \phi _ {j} ( z) $ are vector-functions holomorphic at $ z= a $; otherwise, the vector-functions $ \phi _ {j} ( z) $ can be polynomials in $ \mathop{\rm ln} ( z- a) $ with coefficients which are vector-functions holomorphic at $ z= a $.

In another sense, the term "determining equation" is used in research on transformation groups admitted by ordinary partial differential equations (see [3]).

References

[1]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955)
[2]	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971)
[3]	L.V. [L.V. Ovsyannikov] Ovsiannikov, "Group analysis of differential equations" , Acad. Press (1982) (Translated from Russian)

Comments

A defining equation is more commonly called an indicial equation.