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

92 Chapter 2
when we already have the schemes (with the same cuts)
of the Riemann surfaces of the functions and In correspondence
with every pair of branches and we take a sheet on which we
consider defined the branch If in the schemes of the
Riemann surfaces of the functions and at the point arrows
indicate, respectively, the passage from the branch to the branch
and the passage from the branch to the branch then
in the scheme of the Riemann surface of the function we indicate by
an arrow over the point the passage from the branch to the
branch
314. Draw the schemes of the Riemann surfaces of the following functions:
a) b) c) d)
The formal method described above for building the scheme of the
Riemann surface of the function does not always give
the correct result, because it may happen that some of the branches
coincide.
For simplicity we shall suppose that the cuts do not pass through the
non-uniqueness points of the function In this case, traversing a
cut, by virtue of the uniqueness, we shall move from one set of sheets,
corresponding to the same branch of the function to a new set of
sheets, all corresponding to a different branch. As a consequence, if we
join the sheets corresponding to the same branches of the function
i.e., if we substitute every set of sheets with a single sheet, then the
passages between the sheets so obtained are uniquely defined by any turn
round an arbitrary branch point
315. Find all values of if: a) b)
c)
316. Draw the schemes of the Riemann surfaces using the formal
method and the correct schemes for the following functions:
a) b) c)
We obtain finally that to draw the scheme of the Riemann surface of
the function starting from the schemes of the Riemann
surfaces of the functions and (with the same cuts), it suffices to
build the scheme using the formal method described above and afterwards
identify the sheets corresponding to equal values.
It is easy to see that one can apply this procedure to build the schemes