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A linear homogeneous ordinary differential equation of the second order in the complex plane with three given regular singular points (cf. [[Regular singular point|Regular singular point]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081870/r0818701.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081870/r0818702.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081870/r0818703.png" /> having characteristic exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081870/r0818704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081870/r0818705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081870/r0818706.png" /> at these points. The general form of such an equation was first given by E. Papperitz, because of which it is also known as a [[Papperitz equation|Papperitz equation]]. Solutions of a Riemann differential equation are written in the form of the so-called Riemann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081870/r0818708.png" />-function
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A linear homogeneous ordinary differential equation of the second order in the complex plane with three given regular singular points (cf. [[Regular singular point|Regular singular point]])  $  a $,
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Riemann differential equations belong to the class of Fuchsian equations (cf. [[Fuchsian equation|Fuchsian equation]]) with three singular points. A particular case of Riemann differential equations is the [[Hypergeometric equation|hypergeometric equation]] (the singular points are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081870/r08187010.png" />); therefore, a Riemann differential equation itself is sometimes known as a generalized hypergeometric equation. A Riemann differential equation can be reduced to a [[Pochhammer equation|Pochhammer equation]], and its solution can thus be written in the form of an integral over a special contour in the complex plane.
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$  b $
 
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and  $  c $
 
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having characteristic exponents  $  \alpha , \alpha  ^  \prime  $,
 
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$  \beta , \beta  ^  \prime  $,
 
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$  \gamma , \gamma  ^  \prime  $
 
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at these points. The general form of such an equation was first given by E. Papperitz, because of which it is also known as a [[Papperitz equation|Papperitz equation]]. Solutions of a Riemann differential equation are written in the form of the so-called Riemann  $  P $-
 
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function
 
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$$
 
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w  =  P \left \{
 
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Riemann differential equations belong to the class of Fuchsian equations (cf. [[Fuchsian equation|Fuchsian equation]]) with three singular points. A particular case of Riemann differential equations is the [[Hypergeometric equation|hypergeometric equation]] (the singular points are $  0, 1, \infty $);  
 
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therefore, a Riemann differential equation itself is sometimes known as a generalized hypergeometric equation. A Riemann differential equation can be reduced to a [[Pochhammer equation|Pochhammer equation]], and its solution can thus be written in the form of an integral over a special contour in the complex plane.
 
  
 
For references see [[Papperitz equation|Papperitz equation]].
 
For references see [[Papperitz equation|Papperitz equation]].

Revision as of 14:53, 7 June 2020

A linear homogeneous ordinary differential equation of the second order in the complex plane with three given regular singular points (cf. Regular singular point) , and having characteristic exponents , , at these points. The general form of such an equation was first given by E. Papperitz, because of which it is also known as a Papperitz equation. Solutions of a Riemann differential equation are written in the form of the so-called Riemann -function

Riemann differential equations belong to the class of Fuchsian equations (cf. Fuchsian equation) with three singular points. A particular case of Riemann differential equations is the hypergeometric equation (the singular points are ); therefore, a Riemann differential equation itself is sometimes known as a generalized hypergeometric equation. A Riemann differential equation can be reduced to a Pochhammer equation, and its solution can thus be written in the form of an integral over a special contour in the complex plane.

For references see Papperitz equation.

How to Cite This Entry:
Riemann differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_differential_equation&oldid=49402
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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