In mathematics, an abelian integral, named after the Norwegian mathematician Niels Abel, is an integral in the complex plane of the form
Contents
where
where
whose coefficients
Abelian integrals are natural generalizations of elliptic integrals, which arise when
where
History
The theory of abelian integrals originated with the paper by Abel published in 1841. This paper was written during his stay in Paris in 1826 and presented to 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 Riemann surface theory, 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
as a multi-valued function
In the case of
Such functions were first introduced to study hyperelliptic integrals, i.e. for the case where
of complex manifolds. It has the defining property that the holomorphic 1-forms on