Abelian integral

In mathematics, an abelian integral in Riemann surface theory is a function related to the indefinite integral of a differential of the first kind. Suppose we are given a Riemann surface "S" and on it a differential 1-form ω that is everywhere holomorphic on "S", and fix a point "P" on "S" from which to integrate. We can regard

:$int\_P^Q\; omega$

as a multi-valued function "f"("Q"), or (better) an honest function of the chosen path "C" drawn on "S" from "P" to "Q". Since "S" will in general be multiply-connected, one should specify "C", but the value will in fact only depend on the homology class of "C" "modulo" cycles on "S".

In the case of "S" a compact Riemann surface of genus 1, i.e. an elliptic curve, such functions are the elliptic integrals. Logically speaking, therefore, an abelian integral should be a function such as "f".

Such functions were first introduced to study hyperelliptic integrals, i.e. for the case where "S" is a hyperelliptic curve. This is a natural step in the theory of integration to the case of integrals involving algebraic functions √"A", where "A" is a polynomial of degree > 4. The first major insights of the theory were given by Niels Abel; it was later formulated in terms of the Jacobian variety "J"("S"). Choice of "P" gives rise to a standard holomorphic mapping

:"S" → "J"("S")

of complex manifolds. It has the defining property that the holomorphic 1-forms on "J"("S"), of which there are "g" independent ones if "g" is the genus of "S", pull back to a basis for the differentials of the first kind on "S".