Abelian integral

${\displaystyle {\int }_{z_0}^{z}R(x,w)dx,}$

where ${\displaystyle R(x,w)}$ is an arbitrary rational function of the two variables ${\displaystyle x}$ and ${\displaystyle w}$, which are related by the equation

${\displaystyle F(x,w)=0,}$

where ${\displaystyle F(x,w)}$ is an irreducible polynomial in ${\displaystyle w}$,

${\displaystyle F(x,w)\equiv {\phi }_n(x)w^n+\cdots +{\phi }_1(x)w+{\phi }_0\left(x\right),}$

whose coefficients ${\displaystyle {\phi }_j(x)}$, ${\displaystyle j=0,1,\dots ,n}$ are rational functions of ${\displaystyle x}$. 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 ${\displaystyle z}$.

Abelian integrals are natural generalizations of elliptic integrals, which arise when

${\displaystyle F(x,w)=w^2-P(x),}$

where ${\displaystyle P\left(x\right)}$ is a polynomial of degree 3 or 4. Another special case of an abelian integral is a hyperelliptic integral, where ${\displaystyle P(x)}$, 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. Abelian integrals were later connected to the prominent mathematician David Hilbert's 16th problem, and they continue to be considered one of the foremost challenges in contemporary mathematics.