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

254 Appendix by Khovanskii
given by the mapping of the complex line on the complex plane
such that the image of the line is contained in A, i.e.,
for every Let be the pre-image of the point under this mapping,
i.e., What can we say about the multi-valued analytic
function, on the complex line, generated by the germ of at the
point obtained as the result of the composition of the germs of the
rational function at the point and of the germ of the function
at the point It is clear that the analytic properties of this function
depend essentially on the continuation of the germ along the singular
curve A.
Nothing like this may happen under the composition of functions of
a single variable. Indeed, the set of singularities of an of one
variable consists of isolated points. If the image of the complex space
under an analytic mapping is entirely contained in the set of the singular
points of a function then the function is a constant. It is evident that
if the function is constant, after having defined on its set of singular
points, the function turns out to to be constant, too.
In the one-dimensional case, for our purpose it suffices to study the
values of an analytic multi-valued function only in the complement of
its singular points. In the multi-dimensional case we have to study the
possibility of continuing those germs of functions which meet along their
set of singularities (if, of course, the germ of the function is defined in an
arbitrary point of the set of singularities). It happens that the germs of
multi-valued functions sometimes are automatically continued along their
set of singularities [24]: this thus allows us to pass all difficulties.
An important role is played by the following definition:
DEFINITION. A germ of an analytic function at the point of the
space is called an if the following condition is satisfied. For
every connected complex analytic manifold M, every analytic mapping
and every pre-image c of the point there exists a
thin subset such that for every curve beginning at
the point and having no intersection with the set A, except,
at most, at the initial point, i.e., for the germ can be
analytically continued along the curve
PROPOSITION. If the set of singular points of an is an
analytic set then every germ of this function is an
This proposition follows directly from the results set out in [24].
It is evident that every is the germ of an For the