Abel's theorem in problems and solutions - School of Mathematics

Solvability of Equations 255
the notions of monodromy group and of monodromy pair are
thus well defined.
In the sequel we will need the notion of a holonomic system of linear
differential equations. A system of N linear differential equations
for the unknown function whose coefficients are analytic functions
of complex variables is said to be holonomic if the
space of its solutions has a finite dimension.
THEOREM ON THE CLOSURE OF THE CLASS OF The class
of the in is closed with respect to the following operations:
1) differentiation, i.e., if is an at a point then
for every the germs of the partial derivatives
are also at the point
2) integration, i.e., if where are
at a point then also is an at the point
3) composition with the of variables, i.e., if
are at a point and is an at the point
in the space then is an
at the point as well;
4) solutions of algebraic equations, i.e., if are
at a point the germ is not zero and the germ satisfies
the equation then the germ is also
an at the point
5) solutions of holonomic systems of linear differential equations,
i.e., if the germ of a function at a point satisfies the
holonomic system of N linear differential equations
all of whose coefficients are at the point then
is also an at the point