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

The complex numbers 89
means will be clarified during the exposition of the proof’s arguments.
Suppose the conditions required by the monodromy property be satisfied.
Let and be the continuous images of the curves and
under the mapping with as the initial point. We have
to prove that the curves and end at the same point.
Suppose first that all the curves obtained deforming into pass
neither through the branch points, nor through the non-uniqueness points
of the function. Let C be one of these curves. Thus there exists a unique
image of the curve C under the mapping beginning at the point
If the function is ‘sufficiently good’ 16 then during the
continuous deformation of the curve C from to the curve is
continuously deformed from to The end point of the curve is
continuously displaced as well. But the curve C ends at the point thus
the final point of the curve must coincide with one of the images of
If the function takes at every (in particular at only a finite
number of values (and we consider only functions of this type), then the
final point of the curve cannot jump from an image of the point to
another, because this should destroy the continuity of the deformation.
Hence the final points of the curves and, in particular, of and of
do coincide.
Consider now what happens when the curve C passes through a nonuniqueness
point (when it is not a branch point) of the function
Consider only the particular case in which the curve varies only in a
neighbourhood of one non-uniqueness point, say (Figure 36). If at the
point one has chosen a value then one defines by continuity
the value of at the point A. Afterwards the values of at the
point E are uniquely defined by continuity along the curves ADE and
ABE, because otherwise, making a turn along the curve EDABE, the
value of the function should change and the point should be a
branch point of the function Afterwards, as we had uniquely defined
the value of at the point E along the two curves, we also define by
continuity along the curve the value of at the point
So, the ‘obscure’ point in our exposition remains the claim that all
functions which we shall consider are ‘sufficiently good’.
The reader either accepts this proposition or will apply himself to the
16 The monodromy property is usually proved for arbitrary analytic functions, cf.,
for example, G. Springer, (1957), Introduction to Riemann Surfaces, (Addison-Wesley:
Reading, Mass.).