# Abelian integral

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" &rarr; "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".

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Abelian — Abelian, in mathematics, is used in many different definitions, named after Norwegian mathematician Niels Henrik Abel:In group theory:*Abelian group, a group in which the binary operation is commutative **Category of abelian groups Ab has abelian …   Wikipedia

• Abelian variety — In mathematics, particularly in algebraic geometry, complex analysis and number theory, an Abelian variety is a projective algebraic variety that is at the same time an algebraic group, i.e., has a group law that can be defined by regular… …   Wikipedia

• Abelian von Neumann algebra — In functional analysis, an Abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute. The prototypical example of an abelian von Neumann algebra is the algebra L^infty(X,mu) for μ a σ… …   Wikipedia

• Abelsches Integral — Das abelsche Integrale ist ein Integral mit einem Integranden, der eine bestimmte Form aufweist. Benannt sind diese Integralausdrücke nach dem Mathematiker Niels Henrik Abel; sie werden besonders in der Funktionentheorie oder in der algebraischen …   Deutsch Wikipedia

• Direct integral — In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was… …   Wikipedia

• Timeline of abelian varieties — This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.Early history* c. 1000 Al Karaji writes on congruent numbers [ [http://www.cms.math.ca/Events/summer05/abs/pdf/hm.pdf PDF] ] eventeenth… …   Wikipedia

• List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

• Schottky problem — In mathematics, the Schottky problem is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.Geometric formulationMore precisely, one should consider algebraic curves C of a… …   Wikipedia

• List of algebraic geometry topics — This is a list of algebraic geometry topics, by Wikipedia page. Contents 1 Classical topics in projective geometry 2 Algebraic curves 3 Algebraic surfaces 4 …   Wikipedia

• Évariste Galois — Infobox Scientist name = Évariste Galois box width = caption = Galois age fifteen, drawn by a classmate. birth date = birth date|1811|10|25 birth place = Bourg la Reine, France death date = death date and age|1832|5|31|1811|10|25 death place =… …   Wikipedia