Difference between revisions of "Riemann differential equation"
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| + | $#C+1 = 9 : ~/encyclopedia/old_files/data/R081/R.0801870 Riemann differential equation | ||
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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 | + | 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 $, |
| + | $ b $ | ||
| + | and $ c $ | ||
| + | having characteristic exponents $ \alpha , \alpha ^ \prime $, | ||
| + | $ \beta , \beta ^ \prime $, | ||
| + | $ \gamma , \gamma ^ \prime $ | ||
| + | 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 $- | ||
| + | function | ||
| + | |||
| + | $$ | ||
| + | w = P \left \{ | ||
| + | |||
| + | Riemann differential equations belong to the class of Fuchsian equations (cf. [[Fuchsian equation|Fuchsian equation]]) with three singular points. A particular case of Riemann differential equations is the [[Hypergeometric equation|hypergeometric equation]] (the singular points are $ 0, 1, \infty $); | ||
| + | therefore, a Riemann differential equation itself is sometimes known as a generalized hypergeometric equation. A Riemann differential equation can be reduced to a [[Pochhammer equation|Pochhammer equation]], and its solution can thus be written in the form of an integral over a special contour in the complex plane. | ||
For references see [[Papperitz equation|Papperitz equation]]. | For references see [[Papperitz equation|Papperitz equation]]. | ||
Revision as of 08:11, 6 June 2020
A linear homogeneous ordinary differential equation of the second order in the complex plane with three given regular singular points (cf. Regular singular point) $ a $,
$ b $
and $ c $
having characteristic exponents $ \alpha , \alpha ^ \prime $,
$ \beta , \beta ^ \prime $,
$ \gamma , \gamma ^ \prime $
at these points. The general form of such an equation was first given by E. Papperitz, because of which it is also known as a Papperitz equation. Solutions of a Riemann differential equation are written in the form of the so-called Riemann $ P $-
function
$$ w = P \left \{
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 $ 0, 1, \infty $); 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.
Riemann differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_differential_equation&oldid=48545