View source for Rank of an ordinary linear differential equation

''in the complex domain,

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''

The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r0775002.png" />, where

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if the coefficients in equation

are series which are convergent for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r0775004.png" />:

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The concept of rank is used only when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r0775006.png" /> is a [[Singular point|singular point]] of the differential equation . The rank of the differential equation is also called the rank of the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r0775007.png" />. If this point is a [[Regular singular point|regular singular point]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r0775008.png" />; if it is an [[Irregular singular point|irregular singular point]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r0775009.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750010.png" /> is called the subrank. The rank of the equation is an integer or a rational number. If the subrank is rational with denominator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750011.png" />, then the subrank of the equation obtained from

by the change of variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750012.png" /> is an integer. The rank of the equation is invariant with respect to a change of variable of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750013.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750014.png" /> a holomorphic function at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750015.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750016.png" /> be a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750017.png" />, let

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be a formal series, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750019.png" /> be an integer. The series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

is a normal (subnormal, respectively) series of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750025.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750026.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750027.png" />, respectively). A solution to equation

which is represented by a normal (subnormal) series, convergent in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750028.png" />, is called a normal (subnormal) solution of the same order (see [[#References|[2]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750033.png" /> of equation

is an integer, then

has at least one formal solution of the form (2) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750034.png" />. The substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750035.png" /> does not alter the rank of the equation. If the subrank is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750037.png" /> are mutually prime integers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750038.png" />, then the equation has no less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750039.png" /> formal solutions of the form (2) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750040.png" />.

A Hamburger equation is an equation

with rational coefficients which has exactly two singular points: a regular one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750041.png" /> and an irregular one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750042.png" />. For a Hamburger equation one can obtain sufficient conditions for it to have normal solutions; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750043.png" /> there are necessary and sufficient conditions for the existence of normal and subnormal solutions (see [[#References|[2]]]).

One also introduces the concept of rank in the case when equation

has a finite singular point (see [[#References|[2]]], [[#References|[3]]]).

In the case of a linear system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750044.png" /> ordinary differential equations in the complex domain,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750046.png" /> is an integer and the matrix-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750047.png" /> is holomorphic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750049.png" />, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750050.png" /> is called the rank of the system (3) or the rank of the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750051.png" />, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750052.png" /> is its subrank (see [[#References|[4]]]–[[#References|[6]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750053.png" />, then the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750054.png" /> is a regular singular point; in contrast to a scalar equation , the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750055.png" /> can be a regular singular point if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750056.png" /> (see [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,   "Sur les intégrales irregulières des équations linéaires"  ''Acta Math.'' , '''8'''  (1866)  pp. 295–344</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.Ya. Latysheva,   N.I. Tereshchenko,   G.S. Orel,   "Normally regular solutions and their applications" , Kiev  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.A. Coddington,   N. Levinson,   "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Kamke,   "Handbuch der gewöhnliche Differentialgleichungen" , Chelsea, reprint  (1947)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  W. Wasov,   "Asymptotic expansions for ordinary differential equations" , Interscience  (1965)</TD></TR></table>



====Comments====
For the word rank sometimes the term grade is used. The following can be proved, see [[#References|[a1]]]. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750057.png" /> a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r07750058.png" /> exists for which

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except for a finite number of directions.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hille,   "Lectures on ordinary differential equations" , Addison-Wesley  (1969)</TD></TR></table>