Rank of an ordinary linear differential equation

in the complex domain,

The number , where

if the coefficients in equation

are series which are convergent for large :

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

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

except for a finite number of directions.

References

[a1]	E. Hille, "Lectures on ordinary differential equations" , Addison-Wesley (1969)