Movable singular point
A singular point of the solution
of a differential equation
(
is an analytic function), where
is considered as a function of the complex variable
, which is such that solutions to the same equation with initial data close to the original data have singular points close to
but not coincident with it. The classical example of a movable singular point arises when considering the equation
![]() |
where and
are holomorphic functions in a certain region of the space
. If the surface
is irreducible and is projected along the
-axis on a region
, then all points in the region
are movable singular points; for the solution with initial condition
, where
![]() |
the point is an algebraic branch point.
References
[1] | V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) |
Comments
For equations of the form
![]() |
where is rational in
and
and analytic in
, it is known which equations have only non-movable singularities, cf. Painlevé equation and [a1].
References
[a1] | E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) |
Movable singular point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Movable_singular_point&oldid=15470