In mathematics, an abelian integral, named after the Norwegian mathematician Niels Abel, is an integral in the complex plane of the form
where is an arbitrary rational function of the two variables and . These variables 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, and 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.
Read more about Abelian Integral: History, Modern View
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