Abelian integral

In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form
where formula_2 is an arbitrary rational function of the two variables formula_3 and formula_4, which are related by the equation
where formula_6 is an irreducible polynomial in formula_4,
whose coefficients formula_9, formula_10 are rational functions of formula_3. The value of an abelian integral depends not only on the integration limits, but also on the path along which the integral is taken; it is thus a multivalued function of formula_12.
Abelian integrals are natural generalizations of elliptic integrals, which arise when
where formula_14 is a polynomial of degree 3 or 4. Another special case of an abelian integral is a hyperelliptic integral, where formula_15, in the formula above, is a polynomial of degree greater than 4.
History.
The theory of abelian integrals originated with a paper by Abel published in 1841. This paper was written during his stay in Paris in 1826 and presented to Augustin-Louis Cauchy in October of the same year. This theory, later fully developed by others, was one of the crowning achievements of nineteenth century mathematics and has had a major impact on the development of modern mathematics. In more abstract and geometric language, it is contained in the concept of abelian variety, or more precisely in the way an algebraic curve can be mapped into abelian varieties. The Abelian Integral was later connected to the prominent mathematician David Hilbert's 16th Problem and continues to be considered one of the foremost challenges to contemporary mathematical analysis.
Modern view.
In the theory of Riemann surfaces, an abelian integral is a function related to the indefinite integral of a differential of the first kind. Suppose we are given a Riemann surface formula_16 and on it a differential 1-form formula_17 that is everywhere holomorphic on formula_16, and fix a point formula_19 on formula_16, from which to integrate. We can regard
as a multi-valued function formula_22, or (better) an honest function of the chosen path formula_23 drawn on formula_16 from formula_19 to formula_26. Since formula_16 will in general be multiply connected, one should specify formula_23, but the value will in fact only depend on the homology class of formula_23. 
In the case of formula_16 a compact Riemann surface of genus 1, i.e. an elliptic curve, such functions are the elliptic integrals. Logically speaking, therefore, an abelian integral should be a function such as formula_31.
Such functions were first introduced to study hyperelliptic integrals, i.e., for the case where formula_16 is a hyperelliptic curve. This is a natural step in the theory of integration to the case of integrals involving algebraic functions formula_33, where formula_34 is a polynomial of degree formula_35. The first major insights of the theory were given by Abel; it was later formulated in terms of the Jacobian variety formula_36. Choice of formula_19 gives rise to a standard holomorphic function
of complex manifolds. It has the defining property that the holomorphic 1-forms on formula_38, of which there are "g" independent ones if "g" is the genus of "S", pull back to a basis for the differentials of the first kind on "S".