Abelian Integral
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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 Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
, is an
integral In 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 i ...
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 rat ...
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 mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted ...
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 rat ...
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 In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
of z. Abelian integrals are natural generalizations of
elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s, 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 In mathematics, ''differential of the first kind'' is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1 ...
, 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 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, or more precisely in the way an
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
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 a ...
'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 ve ...
, an abelian integral is a function related to the
indefinite integral In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
of a
differential of the first kind In mathematics, ''differential of the first kind'' is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1 ...
. 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 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 derivati ...
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 In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
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 In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space ...
, one should specify C, but the value will in fact only depend on the
homology class Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor *Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chromo ...
of C. In the case of S a
compact Riemann surface 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 versio ...
of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
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. If ...
, such functions are the
elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s. Logically speaking, therefore, an abelian integral should be a function such as f. Such functions were first introduced to study
hyperelliptic integral In mathematics, ''differential of the first kind'' is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1 ...
s, i.e., for the case where S is a
hyperelliptic curve In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dist ...
. This is a natural step in the theory of integration to the case of integrals involving
algebraic function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...
s \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 exa ...
of degree >4. The first major insights of the theory were given by Abel; it was later formulated in terms of the
Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian var ...
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 derivativ ...
: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 com ...
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 In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
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