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

Solvability of Equations 223
function at the same point We say that the function defined
by the germ is representable as the sum of the functions and
This sum is, however, not defined in a unique. For example, one
easily sees that there are exactly two functions representable as the sum
namely, and
The closure of a class of multi-valued functions with respect to the
addition is a class which contains, together with any two functions, all
functions representable by their sum. One can say the same for all the
operations on the multi-valued functions that we shall encounter in this
chapter.
EXAMPLE 2. Elementary functions. Basic elementary functions are
those functions which one learns at school and which are usually represented
on the keyboard of calculators. Their list is the following: the
constant function, the identity function (associating with every value
of the argument the value itself), the roots the exponential
the logarithm ln the trigonometrical functions:
The allowed operations are: the arithmetic
operations, the composition.
Elementary functions are expressed by formulae, for instance:
From the beginning of the study of analysis we learn that the integration
of elementary functions is very far from being an easy task. Liouville
proved, in fact, that the indefinite integrals of elementary functions are
not, in general, elementary functions.
EXAMPLE 3. Functions representable by quadratures. The basic functions
in this class are the basic elementary functions. The allowed operations
are the arithmetic operations, the composition and the integration.
A class is said to be closed with respect to integration if it also contains
together with every function a function such that
For example, the function
is representable by quadratures. But, as Liouville had proved, this function
is not elementary.
Examples 2 and 3 can be modified. We shall say that a class of
functions is closed with respect to the solutions of the algebraic equations