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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 error function ： ウィキペディア英語版 error function In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:〔Andrews, Larry C.; (''Special functions of mathematics for engineers'' )〕〔Greene, William H.; ''Econometric Analysis'' (fifth edition), Prentice-Hall, 1993, p. 926, fn. 11〕 :$\backslash operatorname(x)\; =\; \backslash frac\backslash int\_0^x\; e^\backslash ,\backslash mathrm\; dt.$ The complementary error function, denoted ''erfc'', is defined as :$\backslash begin$ \operatorname(x) & = 1-\operatorname(x) \\ & = \frac \int_x^ e^\,\mathrm dt \\ & = e^ \operatorname(x), \end which also defines ''erfcx'', the scaled complementary error function (which can be used instead of erfc to avoid arithmetic underflow〔). Another form of $\backslash operatorname(x)$ is known as Craig's formula:〔(John W. Craig, ''A new, simple and exact result for calculating the probability of error for two-dimensional signal constellaions'', Proc. 1991 IEEE Military Commun. Conf., vol. 2, pp. 571–575. )〕 :$\backslash begin$ \operatorname(x) & = \frac \int_0^ \exp \left( - \frac \right) d\theta. \end The imaginary error function, denoted ''erfi'', is defined as :$\backslash begin$ \operatorname(x) & = -i\operatorname(ix) \\ & = \frac \int_0^x e^\,\mathrm dt \\ & = \frac D(x), \end where ''D''(''x'') is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow〔). Despite the name "imaginary error function", $\backslash operatorname(x)$ is real when ''x'' is real. When the error function is evaluated for arbitrary complex arguments ''z'', the resulting complex error function is usually discussed in scaled form as the Faddeeva function: :$w(z)\; =\; e^\backslash operatorname(-iz)\; =\; \backslash operatorname(-iz).$ ==The name "error function"== The error function is used in measurement theory (using probability and statistics), and its use in other branches of mathematics is typically unrelated to the characterization of measurement errors. The error function is related to the cumulative distribution $\backslash Phi$, the integral of the standard normal distribution, by〔 :$\backslash Phi\; (x)\; =\; \backslash frac+\; \backslash frac\; \backslash operatorname\; \backslash left(x/\; \backslash sqrt\backslash right)\; =\; \backslash frac\; \backslash operatorname\; \backslash left(-x/\; \backslash sqrt\backslash right).$ The error function, evaluated at $\backslash frac\{\backslash sigma\; \backslash sqrt\{2\}\}$ for positive ''x'' values, gives the probability that a measurement, under the influence of normally distributed errors with standard deviation $\backslash sigma$, has a distance less than x from the mean value.〔Van Zeghbroeck, Bart; ''Principles of Semiconductor Devices'', University of Colorado, 2011. ()〕 This function is used in statistics to predict behavior of any sample with respect to the population mean. This usage is similar to the Q-function, which in fact can be written in terms of the error function. 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「error 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.