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A linear ordinary differential equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p0729501.png" /> of the form
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$#C+1 = 25 : ~/encyclopedia/old_files/data/P072/P.0702950 Pochhammer equation
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<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/p/p072/p072950/p0729502.png" /></td> </tr></table>
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{{TEX|done}}
  
<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/p/p072/p072950/p0729503.png" /></td> </tr></table>
+
A linear ordinary differential equation of order  $  n $
 +
of the form
  
<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/p/p072/p072950/p0729504.png" /></td> </tr></table>
+
$$
 +
Q ( z) w  ^ {(} n) - \mu Q  ^  \prime  ( z) w ^ {( n - 1 ) } + \dots +
 +
$$
  
<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/p/p072/p072950/p0729505.png" /></td> </tr></table>
+
$$
 +
+
 +
( - 1 )  ^ {n}
 +
\frac{\mu \dots ( \mu + n - 1 ) }{n!}
 +
Q  ^ {(} n) ( z) w +
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p0729506.png" /> is a complex constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p0729507.png" /> are polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p0729508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p0729509.png" />, respectively. The Pochhammer equation was studied by L. Pochhammer [[#References|[1]]] and C. Jordan [[#References|[2]]].
+
$$
 +
- \left [ R ( z) w ^ {( n - 1 ) } - ( \mu +
 +
1 ) R ^ { \prime } ( z) w ^ {( n - 2 ) } + \dots \right . +
 +
$$
 +
 
 +
$$
 +
+ \left .
 +
( - 1 ) ^ {( n - 1 ) }
 +
\frac{( \mu + 1 ) \dots ( \mu + n - 1
 +
) }{( n - 1 ) ! }
 +
R ^ {( n - 1 ) } ( z) w \right ]  = 0 ,
 +
$$
 +
 
 +
where  $  \mu $
 +
is a complex constant and $  Q ( z) , R ( z) $
 +
are polynomials of degree $  \leq  n $
 +
and $  \leq  n - 1 $,  
 +
respectively. The Pochhammer equation was studied by L. Pochhammer [[#References|[1]]] and C. Jordan [[#References|[2]]].
  
 
The Pochhammer equation has been integrated using the [[Euler transformation|Euler transformation]], and its particular integrals have the form
 
The Pochhammer equation has been integrated using the [[Euler transformation|Euler transformation]], and its particular integrals have the form
  
<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/p/p072/p072950/p07295010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
w ( z)  = \int\limits _  \gamma  ( t - z ) ^ {\mu + n - 1 } u ( t) \
 +
d t ,
 +
$$
  
<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/p/p072/p072950/p07295011.png" /></td> </tr></table>
+
$$
 +
u ( t)  =
 +
\frac{1}{Q ( t) }
 +
  \mathop{\rm exp}  \left [ \int\limits ^ { t } 
 +
\frac{
 +
R ( \tau ) }{Q ( \tau ) }
 +
  d \tau \right ] ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p07295012.png" /> is some contour in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p07295013.png" />-plane. Let all roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p07295014.png" /> of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p07295015.png" /> be simple and let the residues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p07295016.png" /> at these points be non-integers. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p07295017.png" /> be a fixed point such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p07295018.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p07295019.png" /> be a simple closed curve with origin and end at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p07295020.png" />, positively oriented and containing only the root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p07295021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p07295022.png" />, inside it. Formula (*) gives the solution of the Pochhammer equation, if with
+
where $  \gamma $
 +
is some contour in the complex $  t $-
 +
plane. Let all roots $  a _ {1} \dots a _ {m} $
 +
of the polynomial $  Q ( z) $
 +
be simple and let the residues of $  R ( z) / Q ( z) $
 +
at these points be non-integers. Let $  a $
 +
be a fixed point such that $  Q ( a) \neq 0 $
 +
and let $  \gamma _ {j} $
 +
be a simple closed curve with origin and end at $  a $,  
 +
positively oriented and containing only the root $  a _ {j} $,  
 +
$  j = 1 \dots m $,  
 +
inside it. Formula (*) gives the solution of the Pochhammer equation, if with
  
<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/p/p072/p072950/p07295023.png" /></td> </tr></table>
+
$$
 +
\gamma  = \gamma _ {j} \gamma _ {k} \gamma _ {j}  ^ {-} 1
 +
\gamma _ {k}  ^ {-} 1 ,\ \
 +
j \neq k ,\  j , k = 1 \dots m ,
 +
$$
  
exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072950/p07295024.png" /> of these solutions are linearly independent. To construct the other solutions other contours are used, including non-closed ones (see [[#References|[3]]], [[#References|[4]]]). The [[Monodromy group|monodromy group]] for the Pochhammer equation has been calculated (see [[#References|[3]]]).
+
exactly $  m $
 +
of these solutions are linearly independent. To construct the other solutions other contours are used, including non-closed ones (see [[#References|[3]]], [[#References|[4]]]). The [[Monodromy group|monodromy group]] for the Pochhammer equation has been calculated (see [[#References|[3]]]).
  
 
Particular cases of the Pochhammer equation are the Tissot equation (see [[#References|[4]]]), i.e. the Pochhammer equation in which
 
Particular cases of the Pochhammer equation are the Tissot equation (see [[#References|[4]]]), i.e. the Pochhammer equation in 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/p/p072/p072950/p07295025.png" /></td> </tr></table>
+
$$
 +
Q ( z)  = \prod _ {i = 1 } ^ { {n }  - 1 } ( z - a _ {j} ) ,\ \
 +
R ( z)  = Q ( z) \left ( 1 +
 +
\sum _ {j = 1 } ^ { {n }  - 1 }
 +
 
 +
\frac{b _ j}{z - a _ {j} }
 +
\right ) ,
 +
$$
  
 
and the [[Papperitz equation|Papperitz equation]].
 
and the [[Papperitz equation|Papperitz equation]].

Revision as of 08:06, 6 June 2020


A linear ordinary differential equation of order $ n $ of the form

$$ Q ( z) w ^ {(} n) - \mu Q ^ \prime ( z) w ^ {( n - 1 ) } + \dots + $$

$$ + ( - 1 ) ^ {n} \frac{\mu \dots ( \mu + n - 1 ) }{n!} Q ^ {(} n) ( z) w + $$

$$ - \left [ R ( z) w ^ {( n - 1 ) } - ( \mu + 1 ) R ^ { \prime } ( z) w ^ {( n - 2 ) } + \dots \right . + $$

$$ + \left . ( - 1 ) ^ {( n - 1 ) } \frac{( \mu + 1 ) \dots ( \mu + n - 1 ) }{( n - 1 ) ! } R ^ {( n - 1 ) } ( z) w \right ] = 0 , $$

where $ \mu $ is a complex constant and $ Q ( z) , R ( z) $ are polynomials of degree $ \leq n $ and $ \leq n - 1 $, respectively. The Pochhammer equation was studied by L. Pochhammer [1] and C. Jordan [2].

The Pochhammer equation has been integrated using the Euler transformation, and its particular integrals have the form

$$ \tag{* } w ( z) = \int\limits _ \gamma ( t - z ) ^ {\mu + n - 1 } u ( t) \ d t , $$

$$ u ( t) = \frac{1}{Q ( t) } \mathop{\rm exp} \left [ \int\limits ^ { t } \frac{ R ( \tau ) }{Q ( \tau ) } d \tau \right ] , $$

where $ \gamma $ is some contour in the complex $ t $- plane. Let all roots $ a _ {1} \dots a _ {m} $ of the polynomial $ Q ( z) $ be simple and let the residues of $ R ( z) / Q ( z) $ at these points be non-integers. Let $ a $ be a fixed point such that $ Q ( a) \neq 0 $ and let $ \gamma _ {j} $ be a simple closed curve with origin and end at $ a $, positively oriented and containing only the root $ a _ {j} $, $ j = 1 \dots m $, inside it. Formula (*) gives the solution of the Pochhammer equation, if with

$$ \gamma = \gamma _ {j} \gamma _ {k} \gamma _ {j} ^ {-} 1 \gamma _ {k} ^ {-} 1 ,\ \ j \neq k ,\ j , k = 1 \dots m , $$

exactly $ m $ of these solutions are linearly independent. To construct the other solutions other contours are used, including non-closed ones (see [3], [4]). The monodromy group for the Pochhammer equation has been calculated (see [3]).

Particular cases of the Pochhammer equation are the Tissot equation (see [4]), i.e. the Pochhammer equation in which

$$ Q ( z) = \prod _ {i = 1 } ^ { {n } - 1 } ( z - a _ {j} ) ,\ \ R ( z) = Q ( z) \left ( 1 + \sum _ {j = 1 } ^ { {n } - 1 } \frac{b _ j}{z - a _ {j} } \right ) , $$

and the Papperitz equation.

References

[1] L. Pochhammer, "Ueber ein Integral mit doppeltem Umlauf" Math. Ann. , 35 (1889) pp. 470–494
[2] C. Jordan, "Cours d'analyse" , 3 , Gauthier-Villars (1915)
[3] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)
[4] E. Kamke, "Handbuch der gewöhnliche Differentialgleichungen" , Chelsea, reprint (1947)
How to Cite This Entry:
Pochhammer equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pochhammer_equation&oldid=48198
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article