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''in the complex domain,
 
''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/r0775001.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
\sum _ { j= } 0 ^ { n }
 +
P _ {j} ( z) w ^ {( n - j ) }  = 0 ,\ \
 +
P _ {0} ( z)  = 1
 +
$$
  
 
''
 
''
  
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077500/r0775002.png" />, where
+
The number $  r = k + 1 $,  
 +
where
  
<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/r0775003.png" /></td> </tr></table>
+
$$
 +
= \max _ {1 \leq  j \leq  n } 
 +
\frac{n _ {j} }{j}
 +
,
 +
$$
  
 
if the coefficients in equation
 
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" />:
+
are series which are convergent for large $  | z | $:
  
<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/r0775005.png" /></td> </tr></table>
+
$$
 +
P _ {j} ( z)  = \
 +
\sum _ { m = - \infty } ^ { {n _ j } } p _ {jm} z  ^ {m} ,\ \
 +
j = 1 \dots n .
 +
$$
  
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
+
The concept of rank is used only when $  z = \infty $
 +
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 $  z = \infty $.  
 +
If this point is a [[Regular singular point|regular singular point]], then $  r = 0 $;  
 +
if it is an [[Irregular singular point|irregular singular point]], then r > 0 $.  
 +
The number $  k $
 +
is called the subrank. The rank of the equation is an integer or a rational number. If the subrank is rational with denominator $  q \geq  2 $,  
 +
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.
+
by the change of variable $  z = \zeta  ^ {q} $
 +
is an integer. The rank of the equation is invariant with respect to a change of variable of the form $  z = \zeta \phi ( \zeta ) $,  
 +
with $  \phi $
 +
a holomorphic function at the point $  \zeta = \infty $
 +
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
 
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
+
with a singular point at infinity. Let $  Q ( z) $
 +
be a polynomial of degree $  p $,  
 +
let
  
<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/r07750018.png" /></td> </tr></table>
+
$$
 +
\Psi ( \zeta )  = \
 +
\sum _ { m= } 0 ^  \infty 
 +
\psi _ {m} \zeta  ^ {-} m
 +
$$
  
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
+
be a formal series, and let $  s \geq  1 $
 +
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>
+
$$ \tag{2 }
 +
= e ^ {Q ( z  ^ {1/s} ) } z  ^  \rho  \Psi ( z  ^ {1/s} )
 +
$$
  
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
+
is a normal (subnormal, respectively) series of order $  p / s $
 +
if $  s = 1 $(
 +
$  s \geq  2 $,  
 +
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]]]).
+
which is represented by a normal (subnormal) series, convergent in a neighbourhood of $  z = \infty $,  
 +
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
+
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 r $
 +
of equation
  
 
is an integer, then
 
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" />.
+
has at least one formal solution of the form (2) of order r $.  
 +
The substitution $  w ( z) = e ^ {Q ( z) } u ( z) $
 +
does not alter the rank of the equation. If the subrank is $  k = p / q $,  
 +
where $  p , q $
 +
are mutually prime integers and $  q \geq  2 $,  
 +
then the equation has no less than $  q $
 +
formal solutions of the form (2) of order r $.
  
 
A Hamburger equation is an equation
 
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]]]).
+
with rational coefficients which has exactly two singular points: a regular one $  z = 0 $
 +
and an irregular one $  z = \infty $.  
 +
For a Hamburger equation one can obtain sufficient conditions for it to have normal solutions; when $  n = 2 $
 +
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
 
One also introduces the concept of rank in the case when equation
Line 47: Line 105:
 
has a finite singular point (see [[#References|[2]]], [[#References|[3]]]).
 
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,
+
In the case of a linear system of $  n $
 +
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>
+
$$ \tag{3 }
 +
w  ^  \prime  = z  ^ {r} A ( z) w ,
 +
$$
  
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]]]).
+
where r \geq  - 1 $
 +
is an integer and the matrix-function $  A ( z) $
 +
is holomorphic at $  z = \infty $
 +
and $  A ( \infty ) \neq 0 $,  
 +
the number r + 1 $
 +
is called the rank of the system (3) or the rank of the singular point $  z = \infty $,  
 +
the number r $
 +
is its subrank (see [[#References|[4]]]–[[#References|[6]]]). If $  r = - 1 $,  
 +
then the point $  z = \infty $
 +
is a regular singular point; in contrast to a scalar equation , the point $  z = \infty $
 +
can be a regular singular point if r > - 1 $(
 +
see [[#References|[4]]]).
  
 
====References====
 
====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>
 
<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====
 
====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
+
For the word rank sometimes the term grade is used. The following can be proved, see [[#References|[a1]]]. For every r $
 +
a solution $  w( z) $
 +
exists for which
  
<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/r07750059.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {| z | \rightarrow \infty }  | z |  ^ {-} r  \mathop{\rm log}  | w( z) |  > 0 ,
 +
$$
  
 
except for a finite number of directions.
 
except for a finite number of directions.

Revision as of 08:09, 6 June 2020


in the complex domain,

$$ \tag{1 } \sum _ { j= } 0 ^ { n } P _ {j} ( z) w ^ {( n - j ) } = 0 ,\ \ P _ {0} ( z) = 1 $$

The number $ r = k + 1 $, where

$$ k = \max _ {1 \leq j \leq n } \frac{n _ {j} }{j} , $$

if the coefficients in equation

are series which are convergent for large $ | z | $:

$$ P _ {j} ( z) = \ \sum _ { m = - \infty } ^ { {n _ j } } p _ {jm} z ^ {m} ,\ \ j = 1 \dots n . $$

The concept of rank is used only when $ z = \infty $ is a singular point of the differential equation . The rank of the differential equation is also called the rank of the singular point $ z = \infty $. If this point is a regular singular point, then $ r = 0 $; if it is an irregular singular point, then $ r > 0 $. The number $ k $ is called the subrank. The rank of the equation is an integer or a rational number. If the subrank is rational with denominator $ q \geq 2 $, then the subrank of the equation obtained from

by the change of variable $ z = \zeta ^ {q} $ is an integer. The rank of the equation is invariant with respect to a change of variable of the form $ z = \zeta \phi ( \zeta ) $, with $ \phi $ a holomorphic function at the point $ \zeta = \infty $ 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 $ Q ( z) $ be a polynomial of degree $ p $, let

$$ \Psi ( \zeta ) = \ \sum _ { m= } 0 ^ \infty \psi _ {m} \zeta ^ {-} m $$

be a formal series, and let $ s \geq 1 $ be an integer. The series

$$ \tag{2 } w = e ^ {Q ( z ^ {1/s} ) } z ^ \rho \Psi ( z ^ {1/s} ) $$

is a normal (subnormal, respectively) series of order $ p / s $ if $ s = 1 $( $ s \geq 2 $, respectively). A solution to equation

which is represented by a normal (subnormal) series, convergent in a neighbourhood of $ z = \infty $, 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 $ r $ of equation

is an integer, then

has at least one formal solution of the form (2) of order $ r $. The substitution $ w ( z) = e ^ {Q ( z) } u ( z) $ does not alter the rank of the equation. If the subrank is $ k = p / q $, where $ p , q $ are mutually prime integers and $ q \geq 2 $, then the equation has no less than $ q $ formal solutions of the form (2) of order $ r $.

A Hamburger equation is an equation

with rational coefficients which has exactly two singular points: a regular one $ z = 0 $ and an irregular one $ z = \infty $. For a Hamburger equation one can obtain sufficient conditions for it to have normal solutions; when $ n = 2 $ 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 $ n $ ordinary differential equations in the complex domain,

$$ \tag{3 } w ^ \prime = z ^ {r} A ( z) w , $$

where $ r \geq - 1 $ is an integer and the matrix-function $ A ( z) $ is holomorphic at $ z = \infty $ and $ A ( \infty ) \neq 0 $, the number $ r + 1 $ is called the rank of the system (3) or the rank of the singular point $ z = \infty $, the number $ r $ is its subrank (see [4][6]). If $ r = - 1 $, then the point $ z = \infty $ is a regular singular point; in contrast to a scalar equation , the point $ z = \infty $ can be a regular singular point if $ r > - 1 $( 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 $ r $ a solution $ w( z) $ exists for which

$$ \lim\limits _ {| z | \rightarrow \infty } | z | ^ {-} r \mathop{\rm log} | w( z) | > 0 , $$

except for a finite number of directions.

References

[a1] E. Hille, "Lectures on ordinary differential equations" , Addison-Wesley (1969)
How to Cite This Entry:
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
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article