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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 hyperbolic function ： ウィキペディア英語版 hyperbolic function In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular functions. The basic hyperbolic functions are the hyperbolic sine "sinh" ( or ),〔(1999) ''Collins Concise Dictionary'', 4th edition, HarperCollins, Glasgow, ISBN 0 00 472257 4, p.1386〕 and the hyperbolic cosine "cosh" (),〔''Collins Concise Dictionary'', p.328〕 from which are derived the hyperbolic tangent "tanh" ( or ),〔''Collins Concise Dictionary'', p.1520〕 hyperbolic cosecant "csch" or "cosech" (〔''Collins Concise Dictionary'', p.328〕 or ), hyperbolic secant "sech" ( or ),〔''Collins Concise Dictionary'', p.1340〕 and hyperbolic cotangent "coth" ( or ),〔''Collins Concise Dictionary'', p.329〕〔(tanh )〕 corresponding to the derived trigonometric functions. The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also called "asinh" or sometimes "arcsinh")〔(Some examples of using arcsinh ) found in Google Books.〕 and so on. Just as the points (cos ''t'', sin ''t'') form a circle with a unit radius, the points (cosh ''t'', sinh ''t'') form the right half of the equilateral hyperbola. The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. Hyperbolic functions occur in the solutions of some important linear differential equations, for example the equation defining a catenary, of some cubic equations, and of Laplace's equation in Cartesian coordinates. The latter is important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. When considered defined by a complex variable, the hyperbolic functions are rational functions of exponentials, and are hence meromorphic. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.〔Robert E. Bradley, Lawrence A. D'Antonio, Charles Edward Sandifer. ''Euler at 300: an appreciation.'' Mathematical Association of America, 2007. Page 100.〕 Riccati used ''Sc.'' and ''Cc.'' (''()sinus circulare'') to refer to circular functions and ''Sh.'' and ''Ch.'' (''()sinus hyperbolico'') to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today.〔Georg F. Becker. ''Hyperbolic functions.'' Read Books, 1931. Page xlviii.〕 The abbreviations ''sh'' and ''ch'' are still used in some other languages, like French and Russian. ==Standard analytic expressions== The hyperbolic functions are: * Hyperbolic sine: ::$\backslash sinh\; x\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; \}$. * Hyperbolic cosine: ::$\backslash cosh\; x\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; \}$. * Hyperbolic tangent: ::$\backslash tanh\; x\; =\; \backslash frac\; =\; \backslash frac\; \}\; =$ ::$=\; \backslash frac\; =\; \backslash frac\}$. * Hyperbolic cotangent: ::$\backslash coth\; x\; =\; \backslash frac\; =\; \backslash frac\; \}\; =$ ::$=\; \backslash frac\; =\; \backslash frac\}$ * Hyperbolic secant: ::$\backslash operatorname\backslash ,x\; =\; \backslash left(\backslash cosh\; x\backslash right)^\; =\; \backslash frac\; =\; \backslash frac\}$ * Hyperbolic cosecant: ::$\backslash operatorname\backslash ,x\; =\; \backslash left(\backslash sinh\; x\backslash right)^\; =\; \backslash frac\; =\; \backslash frac\}$ Hyperbolic functions can be introduced via imaginary circular angles: * Hyperbolic sine: ::$\backslash sinh\; x\; =\; -i\; \backslash sin\; (i\; x)$ * Hyperbolic cosine: ::$\backslash cosh\; x\; =\; \backslash cos\; (i\; x)$ * Hyperbolic tangent: ::$\backslash tanh\; x\; =\; -i\; \backslash tan\; (i\; x)$ * Hyperbolic cotangent: ::$\backslash coth\; x\; =\; i\; \backslash cot\; (i\; x)$ * Hyperbolic secant: ::$\backslash operatorname\; x\; =\; \backslash sec\; (i\; x)$ * Hyperbolic cosecant: ::$\backslash operatorname\; x\; =\; i\; \backslash csc\; (i\; x)$ where ''i'' is the imaginary unit with the property that . The complex forms in the definitions above derive from Euler's formula. 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「hyperbolic 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.