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

The complex numbers 87
property is possessed by a quite wide class of functions. In particular, it
is possessed by all functions we have considered, for instance, by all functions
representable by radicals (§2.11) and by algebraic functions (§2.14)
(these two classes are special cases of a wider class of functions, called
analytic, which possess this property as well).
The proof of this statement lies outside the purposes of this book. We
have therefore to accept this proposition without proof, knowing that the
proof does exist 15 . The reader can thus pass directly to §2.11.
However, a sense of dissatisfaction may arise in the reader about something.
Although we cannot destroy it completely, we prove that the property
formulated above follows from another property, the so called monodromy
property, which turns out to be more evident.
We know that after having decomposed the function into continuous
single-valued branches (in a certain region of the plane), the
function is defined by continuity in the same way along two arbitrary
curves and lying in this region and joining two arbitrary
points and The monodromy property is related to this condition.
Suppose a multi-valued function be such that if one fixes one of
its values at an arbitrary point then the value of the function can
be defined by continuity (possibly in a unique way) along an arbitrary
curve beginning at the point (and not passing through the points at
which is not defined). We say that the function possesses the
monodromy property if it satisfies the following condition.
MONODROMY PROPERTY. Suppose that two continuous curves
and on the plane join two points and passing neither through
the branch points nor through the points of non-uniqueness of the function
Furthermore, suppose that the curve can be transformed,
varying continuously, into the curve in such a way that none of the
curves during the deformation passes through the branch points, and
that the ends of these curves are fixed (see Figure 35; and are branch
points). Thus the value is uniquely defined by continuity along the
curves and (when a value is chosen).
309. Suppose a function possess the monodromy property. On
the plane make the cuts, not intersecting each other, from the branch
points of to infinity. Prove that in this way the function is
decomposed into continuous single-valued branches.
15 Cf., for example, Springer G., (1957), Introduction to Riemann Surfaces, (Addison-
Wesley: Reading, Mass).