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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Abelian integral ： ウィキペディア英語版 Abelian integral In mathematics, an abelian integral, named after the Norwegian mathematician Niels Abel, is an integral in the complex plane of the form :$\backslash int\_^z\; R\backslash left(x,w\backslash right)dx,$ where $R\backslash left(x,w\backslash right)$ is an arbitrary rational function of the two variables $x$ and $w$. These variables are related by the equation :$F\backslash left(x,w\backslash right)=0,\; \backslash ,$ where $F\backslash left(x,w\backslash right)$ is an irreducible polynomial in $w$, :$F\backslash left(x,w\backslash right)\backslash equiv\backslash phi\_n\backslash left(x\backslash right)w^n+\backslash cdots+\backslash phi\_1\backslash left(x\backslash right)w+\backslash phi\_0\backslash left(x\backslash right),\; \backslash ,$ whose coefficients $\backslash phi\_j\backslash left(x\backslash right)$, $j=0,1,\backslash ldots,n$ are rational functions 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, and it is thus a multivalued function of $z$. Abelian integrals are natural generalizations of elliptic integrals, which arise when :$F\backslash left(x,w\backslash right)=w^2-P\backslash left(x\backslash right),\; \backslash ,$ where $P\backslash left(x\backslash right)$ is a polynomial of degree 3 or 4. Another special case of an abelian integral is a hyperelliptic integral, where $P\backslash left(x\backslash right)$, in the formula above, is a polynomial of degree greater than 4. == History == The theory of abelian integrals originated with the paper by Abel 〔a〕 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Abelian integral」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.