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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 algebraic function ： ウィキペディア英語版 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 can be expressed using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power: :$f(x)=1/x,\; f(x)=\backslash sqrt,\; f(x)=\backslash frac-\backslash sqrt\; x^\}$ are typical examples. However, some algebraic functions cannot be expressed by such finite expressions (as proven by Galois and Niels Abel), as it is for example the case of the function defined by : $f(x)^5+f(x)^4+x=0$. In more precise terms, an algebraic function of degree ''n'' in one variable ''x'' is a function $y\; =\; f(x)$ that satisfies a polynomial equation : $a\_n(x)y^n+a\_(x)y^+\backslash cdots+a\_0(x)=0$ where the coefficients ''a''''i''(''x'') are polynomial functions of ''x'', with coefficients belonging to a set ''S''. Quite often, $S=\backslash mathbb\; Q$, and one then talks about "function algebraic over $\backslash mathbb\; Q$", and the evaluation at a given rational value of such an algebraic function gives an algebraic number. A function which is not algebraic is called a transcendental function, as it is for example the case of $\backslash exp(x),\; \backslash tan(x),\; \backslash ln(x),\; \backslash Gamma(x)$. A composition of transcendental functions can give an algebraic function: $f(x)=\backslash cos\; (\backslash arcsin(x))\; =\; \backslash sqrt$. As an equation of degree ''n'' has ''n'' roots, a polynomial equation does not implicitly define a single function, but ''n'' functions, sometimes also called branches. Consider for example the equation of the unit circle: $y^2+x^2=1.\backslash ,$ This determines ''y'', except only up to an overall sign; accordingly, it has two branches: $y=\backslash pm\; \backslash sqrt.\backslash ,$ An algebraic function in ''m'' variables is similarly defined as a function ''y'' which solves a polynomial equation in ''m'' + 1 variables: :$p(y,x\_1,x\_2,\backslash dots,x\_m)=0.\backslash ,$ It is normally assumed that ''p'' should be an irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem. Formally, an algebraic function in ''m'' variables over the field ''K'' is an element of the algebraic closure of the field of rational functions ''K''(''x''1,...,''x''''m''). == Algebraic functions in one variable == 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「algebraic 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.