Definition:Abelian Integral

Definition
An Abelian integral is a complex 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 $x$.