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| In [[mathematics]], an '''abelian integral''', named after the Norwegian mathematician [[Niels Henrik Abel|Niels Abel]], is an integral in the [[complex plane]] of the form
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| :<math>\int_{z_0}^z R\left(x,w\right)dx,</math>
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| where <math>R\left(x,w\right)</math> is an arbitrary [[rational function]] of the two variables <math>x</math> and <math>w</math>. These variables are related by the equation
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| :<math>F\left(x,w\right)=0, \, </math>
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| where <math>F\left(x,w\right)</math> is an irreducible polynomial in <math>w</math>,
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| :<math>F\left(x,w\right)\equiv\phi_n\left(x\right)w^n+\cdots+\phi_1\left(x\right)w+\phi_0\left(x\right), \, </math>
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| whose coefficients <math>\phi_j\left(x\right)</math>, <math>j=0,1,\ldots,n</math> are [[rational function]]s of <math>x</math>. 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 <math>z</math>.
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| Abelian integrals are natural generalizations of [[elliptic integral]]s, which arise when
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| :<math>F\left(x,w\right)=w^2-P\left(x\right), \, </math>
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| where <math>P\left(x\right)</math> is a polynomial of degree 3 or 4. Another special case of an abelian integral is a [[hyperelliptic integral]], where <math>P\left(x\right)</math>, in the formula above, is a polynomial of degree greater than 4.
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| == History ==
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| The theory of abelian integrals originated with the paper by Abel <ref>a</ref> 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 variety]], 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 [[mathematical analysis]].
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| == Modern view ==
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| In [[Riemann surface]] 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 <math>S</math> and on it a [[Differential form|differential 1-form]] <math>\omega</math> that is everywhere [[holomorphic]] on <math>S</math>, and fix a point <math>P_0</math> on <math>S</math>, from which to integrate. We can regard
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| :<math>\int_{P_0}^P \omega</math>
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| as a [[multi-valued function]] <math>f\left(P\right)</math>, or (better) an honest function of the chosen path <math>C</math> drawn on <math>S</math> from <math>P_0</math> to <math>P</math>. Since <math>S</math> will in general be [[multiply connected]], one should specify <math>C</math>, but the value will in fact only depend on the [[homology class]] of <math>C</math>.
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| In the case of <math>S</math> a [[compact Riemann surface]] of [[genus (mathematics)|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 <math>f</math>.
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| Such functions were first introduced to study [[hyperelliptic integral]]s, i.e. for the case where <math>S</math> is a [[hyperelliptic curve]]. This is a natural step in the theory of integration to the case of integrals involving [[algebraic function]]s <math>\sqrt{A}</math>, where <math>A</math> is a [[polynomial]] of degree <math>>4</math>. The first major insights of the theory were given by [[Niels Abel]]; it was later formulated in terms of the [[Jacobian variety]] <math>J\left(S\right)</math>. Choice of <math>P_0</math> gives rise to a standard [[holomorphic]] [[function (mathematics)|mapping]]
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| :<math>S\to J\left(S\right) \, </math>
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| of [[complex manifold]]s. It has the defining property that the holomorphic 1-forms on <math>S\to J\left(S\right)</math>, of which there are ''g'' independent ones if ''g'' is the genus of ''S'', [[pullback (differential geometry)|pull back]] to a basis for the differentials of the first kind on ''S''.
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| == References ==
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| * {{Citation | last1=Appell | first1=Paul | author1-link=Paul Appell | last2=Goursat | first2=Édouard | author2-link=Édouard Goursat| title=Theorie des Fonctions Algebraiques et de Leurs Integrales | publisher=[[Gauthier-Villars]] | location=Paris | year=1895}}.
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| * {{Citation | last1=Bliss | first1=Gilbert A. | author1-link=Gilbert Ames Bliss| title=Algebraic Functions | publisher=[[American Mathematical Society]] | location=Providence | year=1933}}.
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| * {{Citation | last1=Forsyth | first1=Andrew R. | author1-link=Andrew Forsyth| title=Theory of Functions of a Complex Variable | publisher=[[Cambridge University Press]] | location=Providence | year=1893}}.
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| * {{Citation | last1=Griffiths | first1=Phillip | last2=Harris | first2=Joseph | title=Principles of Algebraic Geometry | publisher=[[John Wiley & Sons]] | location=New York | year=1978}}. Lucidly presented modern perspective.
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| * {{Citation | 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}}.
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| <References/>
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| [[Category:Riemann surfaces]]
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| [[Category:Algebraic curves]]
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| [[Category:Abelian varieties]]
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