An
integral
of an
algebraic function,
i.e. an integral of the form
where
is some rational function in variables
that are related by an algebraic equation
with coefficients
that are polynomials in
,
.
To equation
(2)
there corresponds a compact
Riemann surface
which is an
-sheeted
covering of the Riemann sphere. On this Riemann surface
,
and consequently also
,
can be considered as single-valued functions of the points on
.
The integral
(1)
is then given as the integral
of the
Abelian differential
on
taken along some rectifiable path
.
In general, specifying only the initial and end points
and
of the path
does not completely determine the value of the Abelian integral
(1),
or,
which is the same, the integral
(1)
turns out to be a
multi-valued function of the initial and end points of the path
.
The behaviour of an Abelian integral on
depends, first of all, on the topological structure of
,
in particular on the topological invariant called the genus
of the surface
(cf.
Genus of a surface).
The genus
is connected with the number of sheets
and with the number of branch points
(counted with multiplicities) by the relation
.
For
the variables
and
can be rationally expressed in some parameter
,
and the Abelian integral becomes the integral of a rational function in
.
This will happen, for example, in the elementary cases
and
.
If
,
any Abelian integral can be expressed in the form
of a linear combination of elementary functions and
canonical Abelian integrals
of the three kinds. The integral
is called an Abelian integral of the first kind if
is an Abelian differential of the first kind. In
other words, Abelian integrals of the first kind are characterized
by the fact that for a fixed initial point
of the path
they are a function of the upper bound
that is an everywhere finite, usually multi-valued, function on
.
Such a characterization may be used, for example, to
construct analogues of Abelian integrals of the first kind
on non-compact Riemann surfaces. Any Abelian integral of the first kind
can be represented in the form of a linear combination of
linearly-independent
normal Abelian integrals of the first kind
of differentials
that constitute a canonical basis for the Abelian
differentials of the first kind. If the surface
is cut along the cycles
of a canonical basis for the homology, a simply-connected domain
is obtained. The integrals
are single-valued functions of the upper bound
for all paths
with fixed initial point
and fixed end point
.
The multi-valuedness of the integrals
along an arbitrary path
joining
with
is now completely described by the fact that it differs from the integrals
only by an integral linear combination of the
-periods
and the
-periods
of a basis of the differentials of the first kind. These constitute the
period matrix
of dimension
,
which satisfies the bilinear Riemann relations (cf.
Abelian differential).
An integral
where
is an Abelian differential of the second kind is said to be an
Abelian integral of the second kind.
Considered as a function of the upper bound, it has no singularities anywhere on
except for poles. An Abelian integral of a normalized Abelian
differential of the second kind is known as a
normal Abelian integral of the second kind.
An
Abelian integral of the third kind
is an arbitrary Abelian integral. It usually has logarithmic singularities on
;
however, such singularities can only occur in pairs. An Abelian
integral of a normal Abelian differential of the third kind is called a
normal Abelian integral of the third kind.
Any Abelian integral can be represented as a linear combination of
normal Abelian integrals of the first, second and third kinds.
Unlike Abelian integrals of the first and second kinds, Abelian
integrals of the third kind usually also have the so-called
polar periods,
beside the
-
and
-periods
(which are called
cyclic periods).
Polar periods are taken along cycles which are homologous to
zero, but encircle the logarithmic singularities of the Abelian
integral. They are caused by the poles of the Abelian differential
with non-zero residues.
A number of relations depending on the topological and conformal structure of
exist for arbitrary Abelian integrals on the same Riemann surface
.
Thus, if
is a normal Abelian differential of the third kind with simple poles in
and
,
then the following
theorem on the permutation of the parameters and the bounds
of an Abelian integral of the third kind holds for all points
:
 |
Relations which connect Abelian integrals with rational functions on
are known as
Abelian theorems.
In terms of divisors, for example, the Abelian theorem for Abelian
integrals of the first kind has the following form: A divisor
on
is the divisor of a meromorphic function if and only if there exists a chain
with
and
for all Abelian differentials of the first kind on
.
There also exist variants of Abelian theorems for
Abelian integrals of the second and third kinds
[4].
Abelian integrals and, in particular, Abelian theorems, are
the basis of the transcendental construction of the
Jacobi variety
of a Riemann surface. The question of the inversion of an Abelian
integral as a function of its upper bound also leads to the concepts of an
Abelian function;
an
elliptic function;
and theta-functions (cf.
Theta-function;
Jacobi inversion problem).
Historically, the theory of Abelian integrals followed from
the consideration of a surface of genus
.
If one writes the corresponding equation in the form
where
is a polynomial in
of the third or fourth degree, then one obtains elliptic integrals (cf.
Elliptic integral)
as the respective Abelian integrals. They first appeared at the
end of the
17th century
and the beginning of the
18th century
as the result of the rectification of curves of the second
order in the studies of
Jacob
and
Johann Bernoulli
and of
G. Fagnano.
L. Euler
tackled the addition theorem of elliptic integrals, which
is a special case of a theorem of
N.H. Abel
(
1752).
Abel and
C.G.J. Jacobi
(
1827)
stated the problem
of inversion of elliptic integrals and obtained the solution. The beginnings
of the theory of elliptic functions were thus established. However,
some facts concerning this theory had been established by
C.F. Gauss
early in the
18th century.
Abel and Jacobi dealt with
the much more difficult case of inversion of Abelian integrals in the case
.
During the very first stages of development
stress was laid on hyper-elliptic integrals, where
with
a polynomial of the fifth or sixth degree without multiple roots. Here
and the difficulty of the inversion problem can already be noticed.
The principal advances in the theory of inversion of Abelian
integrals are due to
B. Riemann
(
1851),
who introduced the
concept of Riemann surfaces and formulated and gave
proofs of a large number of important results.
Multi-dimensional generalizations of the theory of Abelian integrals
form the subject matter of algebraic geometry and the theory of complex manifolds.