Definition:Abelian Integral

Definition
An Abelian integral is a complex Riemann integral of the form


 * $\ds \int_{z_0}^z \map R {x, w} \rd x$

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

These variables are related by the equation:
 * $\map F {x, w} = 0$

where $\map F {x, w}$ is an irreducible polynomial in $w$:


 * $\map F {x, w} \equiv \map {\phi_n} x w^n + \cdots + \map {\phi_1} x w + \map {\phi_0} x$

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