# 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) , 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. N.Kh. Rozov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Riemann_differential_equation&oldid=15928