Abelian integral
From Wikipedia, the free encyclopedia
In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form
where is an arbitrary rational function of the two variables
and
, which are related by the equation
where is an irreducible polynomial in
,
whose coefficients ,
are rational functions of
. 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
.
Abelian integrals are natural generalizations of elliptic integrals, which arise when
where is a polynomial of degree 3 or 4. Another special case of an abelian integral is a hyperelliptic integral, where
, in the formula above, is a polynomial of degree greater than 4.