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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 Nonelementary integral ： ウィキペディア英語版 Nonelementary integral In mathematics, a nonelementary antiderivative is an antiderivative for which it can be shown that there exists no formula in terms of ''elementary functions'' (i.e. involving polynomials, and the standard functions sin, cos, exp, and so on).〔Weisstein, Eric W. ("Elementary Function." ) From MathWorld--A Wolfram Web Resource. Accessed 7 Dec 2012.〕 A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. This theorem also provides a basis for determining (with difficulty) which expressions are integrable. It can be shown that, if one is given a function of any complexity, the probability that it will have an elementary antiderivative is very low. Some examples of such functions are: *$\backslash sqrt$〔 *$\backslash ln(\backslash ln\; x)\; \backslash ,$ *$\backslash frac$〔(Impossibility theorems for elementary integration; ) Brian Conrad. Clay Mathematics Institute: 2005 Academy Colloquium Series. Accessed 14 Jul 2014.〕 (see Logarithmic integral) *$e^\; \backslash ,$ *$e^\}\; \backslash ,$〔 (see Normal distribution) The evaluation of nonelementary antiderivatives can often be done using Taylor series. This is because Taylor series can ''always'' be integrated as one would an ordinary polynomial (using the fact that any Taylor series is uniformly convergent within its radius of convergence), even if there is no elementary antiderivative of the function that generated the Taylor series. However, in some cases it is not possible to rely on Taylor series. For example, if the function is not ''infinitely differentiable'', one cannot generate a Taylor series. Even if a Taylor series can be generated, there is a good possibility that it will diverge and ''not'' represent the function one is attempting to antidifferentiate. Many functions which are infinitely differentiable have higher order derivatives that are unmanageable by hand. In these cases, it is not possible to evaluate indefinite integrals, but definite integrals can be evaluated numerically, for instance by Simpson's rule. There are yet other cases where definite integrals can be evaluated exactly without numerical methods, but indefinite integrals cannot, for lack of an elementary antiderivative. The integrals for many of these functions can be written down if one allows so-called "special" (nonelementary) functions. For example, the first example's integral is expressible using incomplete elliptic integrals of the first kind, the second and third use the logarithmic integral, the fourth the exponential integral, and the fifth the error function. Still, there exist functions, such as $x^x$ and $\backslash sin(\backslash sin(x))$ for which no notation currently exists to describe their integrals (other than the use of the integrals themselves). ==See also== * List of integrals * Derivatives * Risch algorithm * Symbolic integration * Algebraic and transcendental functions 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「Nonelementary 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.