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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 Integral of the secant function ： ウィキペディア英語版 Integral of the secant function __NOTOC__ The integral of the secant function of trigonometry was the subject of one of the "outstanding open problems of the mid-seventeenth century", solved in 1668 by James Gregory.〔V. Frederick Rickey and Philip M. Tuchinsky, (An Application of Geography to Mathematics: History of the Integral of the Secant ) in Mathematics Magazine, volume 53, number 3, May 1980, pages 162–166.〕 In 1599, Edward Wright evaluated the integral by numerical methods – what today we would call Riemann sums.〔Edward Wright, ''Certaine Errors in Navigation, Arising either of the ordinaire erroneous making or vsing of the sea Chart, Compasse, Crosse staffe, and Tables of declination of the Sunne, and fixed Starres detected and corrected'', Valentine Simms, London, 1599.〕 He wanted the solution for the purposes of cartography – specifically for constructing an accurate Mercator projection.〔 In the 1640s, Henry Bond, a teacher of navigation, surveying, and other mathematical topics, compared Wright's numerically computed table of values of the integral of the secant with a table of logarithms of the tangent function, and consequently conjectured〔 that : $\backslash int\_0^\backslash theta\; \backslash sec\backslash zeta\backslash ,d\backslash zeta\; =\; \backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\; +\; \backslash frac\backslash right)\backslash right|.$ That conjecture became widely known, and in 1665, Isaac Newton was aware of it.〔H. W. Turnbull, editor, ''The Correspondence of Isaac Newton'', Cambridge University Press, 1959–1960, volume 1, pages 13–16 and volume 2, pages 99–100.〕〔D. T. Whiteside, editor, ''The Mathematical Papers of Isaac Newton'', Cambridge University Press, 1967, volume 1, pages 466–467 and 473–475.〕 The problem was solved by Isaac Barrow. His proof of the result was the earliest use of partial fractions in integration.〔 Adapted to modern notation, Barrow's proof began as follows: : $\backslash int\; \backslash sec\; \backslash theta\; \backslash ,\; d\backslash theta\; =\; \backslash int\; \backslash frac\; =\; \backslash int\; \backslash frac\; =\; \backslash int\; \backslash frac$ Substituting $u$ for $\backslash sin\backslash theta$ reduces the integral to : $$ \begin \int \frac & = \int\frac = \dfrac12\int \left(\frac + \frac\right)\,du \\() & = \frac12 \ln \left|1 + u\right| - \frac12 \ln \left|1 - u\right| + C = \frac12 \ln\left|\frac\right| + C \end Therefore : $$ \int \sec \theta \, d\theta = \left\ \dfrac12 \ln \left|\dfrac\right| + C \\() \ln\left|\sec\theta + \tan\theta\right| + C \\() \ln\left| \tan\left(\dfrac + \dfrac\right) \right| + C \end\right\}\text The second of these follows by first multiplying top and bottom of the interior fraction by $(1+\backslash sin\backslash theta)$. This gives $\backslash cos^2\backslash theta$ in the denominator and the result follows by moving the factor of 1/2 into the logarithm as a square root. The third form follows by replacing $\backslash sin\backslash theta$ by $-\backslash cos(\backslash theta+\backslash pi/2)$ and expanding using the identities for $\backslash cos2x$. It may also be obtained directly by means of the following substitutions: : $$ \begin \sec\theta=\frac\right)} =\frac + \dfrac\right) \cos\left(\dfrac + \dfrac\right)} =\frac + \dfrac\right)} + \dfrac\right)}. \end The conventional solution for the Mercator projection ordinate may be written without the modulus signs since the latitude (φ) lies between −π/2 and π/2: :$$ y= \ln \tan\!\left(\dfrac + \dfrac\right). The integral can also be derived by using the tangent half-angle substitution. ==Hyperbolic forms== Let :$$ \begin \psi &=\ln(\sec\theta+\tan\theta),\\ ^\psi &=\sec\theta+\tan\theta,\\ \sinh\psi &=\frac12(^\psi-^)=\tan\theta,\\ \cosh\psi &=\sqrt=\sec\theta,\\ \tanh\psi &=\sin\theta. \end Therefore :$$ \begin \int \sec \theta \, d\theta& =\psi =\tanh^\! \left(\sin\theta\right) =\sinh^\! \left(\tan\theta\right) =\cosh^\! \left(\sec\theta\right). \end 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「Integral of the secant 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.