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

226 Appendix by Khovanskii
THEOREM 1. (Liouville) A function is elementary (a generalized elementary
function) if and only if it is an elementary (generalized elementary)
element over the field of rational functions A function is
representable by quadratures (representable by generalized quadratures) if
and only if it is representable by quadratures (representable by generalized
quadratures) over the field of complex numbers
For example, it follows from Theorem 1 that the basic elementary
function is representable by quadratures over the field
Indeed, this becomes clear from the equation
To prove, for example, the part of Theorem 1 which concerns functions
representable by quadratures it suffices to verify first that there exist
analogous representations for all the basic elementary functions, and, furthermore,
that the class of functions representable by quadratures over
the field is closed with respect to the composition.
Liouville constructed a nice theory about the solvability of equations.
Let us show two examples of his results.
THEOREM 2 (Liouville). The indefinite integral of the algebraic
function of one complex variable is representable by generalized elementary
functions if and only if it is representable in the form
where the are algebraic functions.
A priori the integral of an algebraic function could be given by a very
complicated formula. It could have the form
Theorem 2 says that this does not happen. Either the integral of an
algebraic function can be written in a simple way, or in general it is not
a generalized elementary function.
THEOREM 3 (LIOUVILLE). The differential linear equation
where and are rational functions, is solvable by generalized
quadratures if and only if its solution can be written in the form