- Abelian integral
In

mathematics , an**abelian integral**inRiemann surface theory is a function related to theindefinite integral of adifferential of the first kind . Suppose we are given a Riemann surface "S" and on it a differential 1-form ω that is everywhereholomorphic 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) anhonest function of the chosen path "C" drawn on "S" from "P" to "Q". Since "S" will in general bemultiply-connected , one should specify "C", but the value will in fact only depend on thehomology class of "C" "modulo " cycles on "S".In the case of "S" a

compact Riemann surface of genus 1, i.e. anelliptic curve , such functions are theelliptic integral s. Logically speaking, therefore, an**abelian integral**should be a function such as "f".Such functions were first introduced to study

hyperelliptic integral s, i.e. for the case where "S" is ahyperelliptic curve . This is a natural step in the theory of integration to the case of integrals involvingalgebraic function s √"A", where "A" is apolynomial of degree > 4. The first major insights of the theory were given byNiels Abel ; it was later formulated in terms of theJacobian variety "J"("S"). Choice of "P" gives rise to a standardholomorphic mapping:"S" → "J"("S")

of

complex manifold s. 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".

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