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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
of the form :\int_^z R(x,w) \, dx, where R(x,w) is an arbitrary
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
of the two variables x and w, which are related by the equation :F(x,w)=0, where F(x,w) is an irreducible polynomial in w, :F(x,w)\equiv\varphi_n(x)w^n+\cdots+\varphi_1(x)w +\varphi_0\left(x\right), whose coefficients \varphi_j(x), j=0,1,\ldots,n are
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s of x. The value of an abelian integral depends not only on the integration limits, but also on the path along which the integral is taken; it is thus a multivalued function of z. Abelian integrals are natural generalizations of elliptic integrals, which arise when :F(x,w)=w^2-P(x), \, where P\left(x\right) is a polynomial of degree 3 or 4. Another special case of an abelian integral is a hyperelliptic integral, where P(x), in the formula above, is a polynomial of degree greater than 4.


History

The theory of abelian integrals originated with a paper by Abel published in 1841. This paper was written during his stay in Paris in 1826 and presented to
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...
in October of the same year. This theory, later fully developed by others,. was one of the crowning achievements of nineteenth century mathematics and has had a major impact on the development of modern mathematics. In more abstract and geometric language, it is contained in the concept of
abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functi ...
, or more precisely in the way an algebraic curve can be mapped into abelian varieties. Abelian integrals were later connected to the prominent mathematician
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
's 16th Problem, and they continue to be considered one of the foremost challenges in contemporary mathematics.


Modern view

In the theory of
Riemann surfaces In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
, an abelian integral 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 In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
\omega that is everywhere holomorphic on S, and fix a point P_0 on S, from which to integrate. We can regard :\int_^P \omega as a multi-valued function f\left(P\right), or (better) an honest function of the chosen path C drawn on S from P_0 to P. 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. In the case of S a compact Riemann surface of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
1, i.e. an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
, 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 \sqrt, where A is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
of degree >4. The first major insights of the theory were given by Abel; it was later formulated in terms of the Jacobian variety J\left(S\right). Choice of P_0 gives rise to a standard
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
:S\to J(S) of
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a ...
s. It has the defining property that the holomorphic 1-forms on S\to 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''.


Notes


References

* * * * * *{{Cite book , last1=Neumann , first1=Carl , author1-link=Carl Neumann , title=Vorlesungen über Riemann's Theorie der Abel'schen Integrale , publisher= B. G. Teubner , edition=2nd , location=Leipzig , year=1884 Riemann surfaces Algebraic curves Abelian varieties