翻訳と辞書 Words near each other ・ Characteristic vector ・ Characteristic velocity ・ Characteristic X-ray ・ Characteristica universalis ・ Characteristically simple group ・ Characteristics of common wasps and bees ・ Characteristics of dyslexia ・ Characteristics of Harold Pinter's work ・ Characteristics of New York City mayoral elections ・ Characterization ・ Characterization (disambiguation) ・ Characterization (materials science) ・ Characterization (mathematics) ・ Characterization test ・ Characterizations of the category of topological spaces ・ Characterizations of the exponential function ・ Characterology ・ Characters (John Abercrombie album) ・ Characters (Stevie Wonder album) ・ Characters and Caricaturas ・ Characters and Observations ・ Characters and organizations in the Year Zero alternate reality game ・ Characters and races of the Dark Crystal ・ Characters and time combination password ・ Characters from Ile-Rien ・ Characters from the docks of The Wire ・ Characters in As You Like It ・ Characters in Dale Brown novels ・ Characters in Devo music videos ・ Characters in Dragonriders of Pern Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Characterizations of the exponential function ： ウィキペディア英語版 Characterizations of the exponential function In mathematics, the exponential function can be characterized in many ways. The following characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent of and equivalent to each other. As a special case of these considerations, we will see that the three most common definitions given for the mathematical constant ''e'' are also equivalent to each other. == Characterizations == The six most common definitions of the exponential function exp(''x'') = ''e''''x'' for real ''x'' are: :1. Define ''e''''x'' by the limit ::: $e^x\; =\; \backslash lim\_\; \backslash left(1+\backslash frac\backslash right)^n.$ :2. Define ''e''''x'' as the value of the infinite series ::: $e^x\; =\; \backslash sum\_^\backslash infty\; =\; 1\; +\; x\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots$ ::(Here ''n''! denotes the factorial of ''n''. One proof that ''e'' is irrational uses this representation.) :3. Define ''e''''x'' to be the unique number ''y'' > 0 such that ::: $\backslash int\_^\; \backslash frac\; =\; x.$ ::This is as the inverse of the natural logarithm function, which is defined by this integral. :4. Define ''e''''x'' to be the unique solution to the initial value problem :::$y\text{'}=y,\backslash quad\; y(0)=1.$ ::(Here, ''y''′ denotes the derivative of ''y''.) :5. The exponential function ''f''(''x'') = ''e''''x'' is the unique Lebesgue-measurable function with ''f''(1) = ''e'' that satisfies :::$f(x+y)\; =\; f(x)\; f(y)\backslash textx\backslash texty\; \backslash ,$ ::(Hewitt and Stromberg, 1965, exercise 18.46). Alternatively, it is the unique anywhere-continuous function with these properties (Rudin, 1976, chapter 8, exercise 6). The term "anywhere-continuous" means that there exists at least a single point $x$ at which $f(x)$ is continuous. As shown below, if $f(x+y)\; =\; f(x)\; f(y)$ for all $x$ and $y$ and $f(x)$ is continuous at ''any'' single point $x$ then $f(x)$ is necessarily continuous ''everywhere''. ::(As a counterexample, if one does ''not'' assume continuity or measurability, it is possible to prove the existence of an everywhere-discontinuous, non-measurable function with this property by using a Hamel basis for the real numbers over the rationals, as described in Hewitt and Stromberg.) ::Because ''f''(''x'') = ''e''''x'' is guaranteed for rational ''x'' by the above properties (see below), one could also use monotonicity or other properties to enforce the choice of ''e''''x'' for irrational ''x'', but such alternatives appear to be uncommon. ::One could also replace the conditions that $f(1)=e$ and that $f$ be Lebesgue-measurable or anywhere-continuous with the single condition that $f\text{'}(0)=1$. This condition, along with the condition $f(x+y)\; =\; f(x)\; f(y)$ easily implies both conditions in characterization 4. Indeed, one gets the initial condition $f(0)\; =\; 1$ by dividing both sides of the equation :::$f(0)\; =\; f(0+0)\; =\; f(0)\; f(0)$ ::by $f(0)$, and the condition that $f\text{'}(x)=f(x)$ follows from the condition that $f\text{'}(0)=1$ and the definition of the derivative as follows: :::$$ \begin f'(x) & = \lim_\frac \\ & = \lim_\frac \\ & = \lim_f(x)\frac \\ & = f(x)\lim_\frac \\ & = f(x)\lim_\frac \\ & = f(x)f'(0) = f(x). \end :6. Let ''e'' be the unique real number satisfying ::: $\backslash lim\_\; \backslash frac=1.$ This number can be shown to exist and be unique. This definition is particularly suited to computing the derivative of the exponential function. Then define ''e''''x'' to be the exponential function with this base. 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「Characterizations of the exponential 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.