Rank of an ordinary linear differential equation
in the complex domain,
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The number , where
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if the coefficients in equation
are series which are convergent for large :
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The concept of rank is used only when is a singular point of the differential equation . The rank of the differential equation is also called the rank of the singular point
. If this point is a regular singular point, then
; if it is an irregular singular point, then
. The number
is called the subrank. The rank of the equation is an integer or a rational number. If the subrank is rational with denominator
, then the subrank of the equation obtained from
by the change of variable is an integer. The rank of the equation is invariant with respect to a change of variable of the form
, with
a holomorphic function at the point
that is non-zero at this point.
The concept of the rank of an equation is used in investigating the structure of the solutions to equation
with a singular point at infinity. Let be a polynomial of degree
, let
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be a formal series, and let be an integer. The series
![]() | (2) |
is a normal (subnormal, respectively) series of order if
(
, respectively). A solution to equation
which is represented by a normal (subnormal) series, convergent in a neighbourhood of , is called a normal (subnormal) solution of the same order (see [2], [3]).
The order of a normal (subnormal) solution does not exceed the rank of the equation; this is true also for formal solutions of the form (2). If the rank of equation
is an integer, then
has at least one formal solution of the form (2) of order . The substitution
does not alter the rank of the equation. If the subrank is
, where
are mutually prime integers and
, then the equation has no less than
formal solutions of the form (2) of order
.
A Hamburger equation is an equation
with rational coefficients which has exactly two singular points: a regular one and an irregular one
. For a Hamburger equation one can obtain sufficient conditions for it to have normal solutions; when
there are necessary and sufficient conditions for the existence of normal and subnormal solutions (see [2]).
One also introduces the concept of rank in the case when equation
has a finite singular point (see [2], [3]).
In the case of a linear system of ordinary differential equations in the complex domain,
![]() | (3) |
where is an integer and the matrix-function
is holomorphic at
and
, the number
is called the rank of the system (3) or the rank of the singular point
, the number
is its subrank (see [4]–[6]). If
, then the point
is a regular singular point; in contrast to a scalar equation , the point
can be a regular singular point if
(see [4]).
References
[1] | H. Poincaré, "Sur les intégrales irregulières des équations linéaires" Acta Math. , 8 (1866) pp. 295–344 |
[2] | E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) |
[3] | K.Ya. Latysheva, N.I. Tereshchenko, G.S. Orel, "Normally regular solutions and their applications" , Kiev (1974) (In Russian) |
[4] | E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17 |
[5] | E. Kamke, "Handbuch der gewöhnliche Differentialgleichungen" , Chelsea, reprint (1947) |
[6] | W. Wasov, "Asymptotic expansions for ordinary differential equations" , Interscience (1965) |
Comments
For the word rank sometimes the term grade is used. The following can be proved, see [a1]. For every a solution
exists for which
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except for a finite number of directions.
References
[a1] | E. Hille, "Lectures on ordinary differential equations" , Addison-Wesley (1969) |
Rank of an ordinary linear differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_of_an_ordinary_linear_differential_equation&oldid=48434