A linear ordinary differential equation of order of the form
The Pochhammer equation has been integrated using the Euler transformation, and its particular integrals have the form
where is some contour in the complex -plane. Let all roots of the polynomial be simple and let the residues of at these points be non-integers. Let be a fixed point such that and let be a simple closed curve with origin and end at , positively oriented and containing only the root , , inside it. Formula (*) gives the solution of the Pochhammer equation, if with
exactly of these solutions are linearly independent. To construct the other solutions other contours are used, including non-closed ones (see , ). The monodromy group for the Pochhammer equation has been calculated (see ).
Particular cases of the Pochhammer equation are the Tissot equation (see ), i.e. the Pochhammer equation in which
and the Papperitz equation.
|||L. Pochhammer, "Ueber ein Integral mit doppeltem Umlauf" Math. Ann. , 35 (1889) pp. 470–494|
|||C. Jordan, "Cours d'analyse" , 3 , Gauthier-Villars (1915)|
|||E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)|
|||E. Kamke, "Handbuch der gewöhnliche Differentialgleichungen" , Chelsea, reprint (1947)|
Pochhammer equation. M.V. Fedoryuk (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Pochhammer_equation&oldid=14095