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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 Gaussian function ： ウィキペディア英語版 Gaussian function In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form: :$f\backslash left(x\backslash right)\; =\; a\; \backslash exp\; \}\; \backslash right)\}$ for arbitrary real constants , and . It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, is the position of the center of the peak and (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are widely used in statistics where they describe the normal distributions, in signal processing where they serve to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics where they are used to solve heat equations and diffusion equations and to define the Weierstrass transform. ==Properties== Gaussian functions arise by composing the exponential function with a concave quadratic function. The Gaussian functions are thus those functions whose logarithm is a concave quadratic function. The parameter is related to the full width at half maximum (FWHM) of the peak according to : $\backslash mathrm\; =\; 2\; \backslash sqrt\backslash \; c\; \backslash approx\; 2.35482\; c.$〔Using the logarithmic identity $\backslash log\; \backslash left(\; x\; \backslash right)\; =\; -\backslash log\; \backslash left(\; \backslash frac\; \backslash right)$, this expression can be transformed to $\backslash mathrm\; =\; 2\; \backslash sqrt\backslash \; c$.〕 Alternatively, the parameter can be interpreted by saying that the two inflection points of the function occur at and . The ''full width at tenth of maximum'' (FWTM) for a Gaussian could be of interest and is : $\backslash mathrm\; =\; 2\; \backslash sqrt\backslash \; c\; \backslash approx\; 4.29193\; c.$〔Using the logarithmic identity $\backslash log\; \backslash left(\; x\; \backslash right)\; =\; -\backslash log\; \backslash left(\; \backslash frac\; \backslash right)$, this expression can be transformed to $\backslash mathrm\; =\; 2\; \backslash sqrt\backslash \; c$.〕 Gaussian functions are analytic, and their limit as is 0 (for the above case of ). Gaussian functions are among those functions that are elementary but lack elementary antiderivatives; the integral of the Gaussian function is the error function. Nonetheless their improper integrals over the whole real line can be evaluated exactly, using the Gaussian integral :$\backslash int\_^\backslash infty\; e^\backslash ,dx=\backslash sqrt$ and one obtains :$\backslash int\_^\backslash infty\; a\; e^\backslash ,dx=ac\backslash cdot\backslash sqrt.$ This integral is 1 if and only if $a\; =\; \backslash tfrac\backslash left(\backslash frac\backslash right)^2\; \}.$ These Gaussians are plotted in the accompanying figure. Gaussian functions centered at zero minimize the Fourier uncertainty principle. The product of two Gaussian functions is a Gaussian, and the convolution of two Gaussian functions is also a Gaussian, with variance being the sum of the original variances: $c^2\; =\; c\_^2\; +\; c\_^2$. The product of two Gaussian probability density functions, though, is not in general a Gaussian PDF. Taking the Fourier transform (unitary, angular frequency convention) of a Gaussian function with parameters , and yields another Gaussian function, with parameters $\backslash sqrtac$, and $\backslash frac$. So in particular the Gaussian functions with and $c\; =\; \backslash frac\}\backslash exp\backslash left(-\backslash pi\backslash cdot\backslash left(\backslash frac\backslash right)^2\backslash right)\; =\; c\backslash cdot\backslash sum\_\{k\backslash in\backslash mathbb\{Z\}\}\backslash exp(-\backslash pi\backslash cdot(kc)^2).$ 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「Gaussian 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.