Abelian integral

Encyclopedia

In mathematics

, an

, is an integral in the complex plane

of the form

where is an arbitrary rational function

of the two variables and . These variables are related by the equation

where is an irreducible polynomial in ,

whose coefficients , are rational function

s of . The value of an abelian integral depends not only on the integration limits but also on the path along which the integral is taken, and it is thus a multivalued function

of .

Abelian integrals are natural generalizations of elliptic integral

s, which arise when

where is a polynomial of degree 3 or 4. Another special case of an abelian integral is a hyperelliptic integral, where , in the formula above, is a polynomial of degree greater than 4.

, or more precisely in the way an algebraic curve

can be mapped into abelian varieties. The Abelian Integral was later connected to the prominent mathematician David Hilbert

's 16th Problem and continues to be considered one of the foremost challenges to contemporary analysis (mathematics).

theory, 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 and on it a differential 1-form

that is everywhere holomorphic on , and fix a point on , from which to integrate. We can regard

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

In the case of a compact Riemann surface

of genus

1, i.e. an elliptic curve

, such functions are the elliptic integral

s. Logically speaking, therefore, an abelian integral should be a function such as .

Such functions were first introduced to study hyperelliptic integrals, i.e. for the case where is a hyperelliptic curve. This is a natural step in the theory of integration to the case of integrals involving algebraic function

s , where is a polynomial

of degree . The first major insights of the theory were given by Niels Abel; it was later formulated in terms of the Jacobian variety

. Choice of gives rise to a standard holomorphic mapping

of complex manifold

s. It has the defining property that the holomorphic 1-forms on , of which there are

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an

**abelian integral**, named after the Norwegian mathematician Niels AbelNiels Henrik Abel

Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...

, is an integral in the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

of the form

where is an arbitrary rational function

Rational function

In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

of the two variables and . These variables are related by the equation

where is an irreducible polynomial in ,

whose coefficients , are rational function

Rational function

In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

s of . The value of an abelian integral depends not only on the integration limits but also on the path along which the integral is taken, and it is thus a multivalued function

Multivalued function

In mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...

of .

Abelian integrals are natural generalizations of elliptic integral

Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...

s, which arise when

where is a polynomial of degree 3 or 4. Another special case of an abelian integral is a hyperelliptic integral, where , in the formula above, is a polynomial of degree greater than 4.

## History

The theory of abelian integrals originated with the paper by Abel published in 1841. This paper was written during his stay in Paris in 1826 and presented to 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 varietyAbelian variety

In mathematics, particularly in algebraic geometry, complex analysis and 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 functions...

, or more precisely in the way an algebraic curve

Algebraic curve

In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...

can be mapped into abelian varieties. The Abelian Integral was later connected to the prominent mathematician David Hilbert

David Hilbert

David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

's 16th Problem and continues to be considered one of the foremost challenges to contemporary analysis (mathematics).

## Modern view

In Riemann surfaceRiemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

theory, an abelian integral is a function related to the indefinite integral of a differential of the first kind

Differential of the first kind

In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces and algebraic curves , for everywhere-regular differential 1-forms...

. Suppose we are given a Riemann surface and on it a differential 1-form

Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

that is everywhere holomorphic on , and fix a point on , from which to integrate. We can regard

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

In the case of a compact Riemann surface

Compact Riemann surface

In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact and the open .A compact Riemann surface C that is a...

of genus

Genus (mathematics)

In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...

1, i.e. an elliptic curve

Elliptic curve

In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

, such functions are the elliptic integral

Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...

s. Logically speaking, therefore, an abelian integral should be a function such as .

Such functions were first introduced to study hyperelliptic integrals, i.e. for the case where is a hyperelliptic curve. This is a natural step in the theory of integration to the case of integrals involving algebraic function

Algebraic function

In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...

s , where is a polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

of degree . The first major insights of the theory were given by Niels Abel; it was later formulated in terms of the Jacobian variety

Jacobian variety

In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...

. Choice of gives rise to a standard holomorphic mapping

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

of complex manifold

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

s. It has the defining property that the holomorphic 1-forms on , 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*.