Definition:Abelian Integral

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An Abelian integral is a complex Riemann integral of the form

$\displaystyle \int_{z_0}^z R \left({x, w}\right) \mathrm d x$

where $R \left({x, w}\right)$ is an arbitrary rational function of the two variables $x$ and $w$.

These variables are related by the equation:

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

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

$F \left ({x, w}\right) \equiv \phi_n \left({x}\right) w^n + \cdots + \phi_1 \left({x}\right) w + \phi_0 \left({x}\right)$

whose coefficients $\phi_j \left({x}\right), j = 0, 1, \ldots, n$ are rational functions of $x$.

Source of Name

This entry was named for Niels Henrik Abel.

Historical Note

The foundations for the theory of Abelian integrals were laid in Abel's paper Mémoire sur une Propriété Générale d'une Classe Très-Étendue de Fonctions Transcendantes.