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

Solvability of Equations 237
Every one of the functions and can be obtained
from the others by quadratures. It follows that none of the integrals
and can be expressed in terms of single-valued by means
of generalized quadratures, compositions, and meromorphic operations.
In the following section we will generalize the example above, finding
all polygons, bounded by arcs of circles, to which the upper semi-plane
can be sent by functions representable by generalized quadratures.
A.10 Mapping of the semi-plane to a
polygon bounded by arcs of circles
A.10.1 Application of the symmetry principle
In the complex plane consider a polygon G bounded by arcs of circles. By
Riemann’s theorem, there exists a function sending the upper semiplane
to polygon the G. This mapping was studied by Riemann, Schwarz,
Christoffel, Klein, and others (cf., for example, [11]). Let us recall some
classical results which will be useful.
Denote by the pre-image of the set of the vertices of the
polygon G under the mapping by H(G) the group of conformal transformations
of the sphere generated by the inversions with respect to the
sides of the polygon, and by L(G) the subgroup of homographic mappings
(the quotient of two linear functions). L(G) is a subgroup of index 2 of
the group H(G). From the Riemann–Schwarz symmetry principle one
obtains the following results.
PROPOSITION.
1) The function can be meromorphically continued along any
curve avoiding the set B.
2) All germs of the multi-valued functions in a non-singular point
are obtained by applying to a given germ the group L(G)
of homographic mappings.
3) The monodromy group of the function is isomorphic to the
group L(G).
4) The singularities of the function are of the following types at
the points If at the vertex of the polygon G that corresponds