View source for Riemann differential equation

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

<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/r081/r081870/r0818709.png" /></td> </tr></table>

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.

For references see [[Papperitz equation|Papperitz equation]].