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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 Elliptic function ： ウィキペディア英語版 Elliptic function In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. Such a doubly periodic function cannot be holomorphic, as it would then be a bounded entire function, and by Liouville's theorem every such function must be constant. In fact, an elliptic function must have at least two poles (counting multiplicity) in a fundamental parallelogram, as it is easy to show using the periodicity that a contour integral around its boundary must vanish, implying that the residues of any simple poles must cancel. Historically, elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, and their theory was improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. Jacobi's elliptic functions have found numerous applications in physics, and were used by Jacobi to prove some results in elementary number theory. A more complete study of elliptic functions was later undertaken by Karl Weierstrass, who found a simple elliptic function in terms of which all the others could be expressed. Besides their practical use in the evaluation of integrals and the explicit solution of certain differential equations, they have deep connections with elliptic curves and modular forms. ==Definition== Formally, an elliptic function is a function $f$ meromorphic on $\backslash mathbb$ for which there exist two non-zero complex numbers $\backslash omega\_1$ and $\backslash omega\_2$ with $\backslash frac\backslash notin\backslash mathbb$, such that $f(z)=f(z+\backslash omega\_1)$ and $f(z)=f(z+\backslash omega\_2)$ for all $z\backslash in\backslash mathbb$. Denoting the "lattice of periods" by $\backslash Lambda=\backslash left\backslash \backslash right\backslash \}$, it follows that $f(z)=f(z+\backslash omega)$ for all $\backslash omega\backslash in\backslash Lambda$. There are two families of 'canonical' elliptic functions: those of Jacobi and those of Weierstrass. Although Jacobi's elliptic functions are older and more directly relevant to applications, modern authors mostly follow Karl Weierstrass when presenting the elementary theory, because his functions are simpler, and any elliptic function can be expressed in terms of them. 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「Elliptic function」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.