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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 Lagrange reversion theorem ： ウィキペディア英語版 Lagrange reversion theorem : ''This page is about Lagrange reversion. For inversion, see Lagrange inversion theorem.'' In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions. Let ''v'' be a function of ''x'' and ''y'' in terms of another function ''f'' such that :$v=x+yf(v)$ Then for any function ''g'', :$g(v)=g(x)+\backslash sum\_^\backslash infty\backslash frac\backslash left(\backslash frac\backslash partial\backslash right)^\backslash left(f(x)^kg\text{'}(x)\backslash right)$ for small ''y''. If ''g'' is the identity :$v=x+\backslash sum\_^\backslash infty\backslash frac\backslash left(\backslash frac\backslash partial\backslash right)^\backslash left(f(x)^k\backslash right)$ In 1770, Joseph Louis Lagrange (1736–1813) published his power series solution of the implicit equation for ''v'' mentioned above. However, his solution used cumbersome series expansions of logarithms.〔Lagrange, Joseph Louis (1770) "Nouvelle méthode pour résoudre les équations littérales par le moyen des séries," ''Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin'', vol. 24, pages 251–326. (Available on-line at: http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=41070 .)〕〔Lagrange, Joseph Louis, ''Oeuvres'', (1869 ), Vol. 2, page 25; Vol. 3, pages 3–73.〕 In 1780, Pierre-Simon Laplace (1749–1827) published a simpler proof of the theorem, which was based on relations between partial derivatives with respect to the variable x and the parameter y.〔Laplace, Pierre Simon de (1777) "Mémoire sur l'usage du calcul aux différences partielles dans la théories des suites," ''Mémoires de l'Académie Royale des Sciences de Paris,'' vol. , pages 99–122.〕〔Laplace, Pierre Simon de, ''Oeuvres'' (1843 ), Vol. 9, pages 313–335.〕〔Laplace's proof is presented in: *Goursat, Édouard, ''A Course in Mathematical Analysis'' (translated by E.R. Hedrick and O. Dunkel) (N.Y.: Dover, 1959 ), Vol. I, pages 404–405.〕 Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration.〔Hermite, Charles (1865) "Sur quelques développements en série de fonctions de plusieurs variables," ''Comptes Rendus de l'Académie des Sciences des Paris'', vol. 60, pages 1–26.〕〔Hermite, Charles, ''Oeuvres'' (1908 ), Vol. 2, pages 319–346.〕〔Hermite's proof is presented in: *Goursat, Édouard, ''A Course in Mathematical Analysis'' (translated by E. R. Hedrick and O. Dunkel) (N.Y.: Dover, 1959 ), Vol. II, Part 1, pages 106–107. *Whittaker, E.T. and G.N. Watson, ''A Course of Modern Analysis'', 4th ed. (England: Cambridge University Press, 1962 ) pages 132–133.〕 Lagrange's reversion theorem is used to obtain numerical solutions to Kepler's equation. ==Simple proof== We start by writing: :$g(v)\; =\; \backslash int\; \backslash delta(y\; f(z)\; -\; z\; +\; x)\; g(z)\; (1-y\; f\text{'}(z))\; \backslash ,\; dz$ Writing the delta-function as an integral we have: :$$ \begin g(v) & = \iint \exp(ik(f(z) - z + x )) g(z) (1-y f'(z)) \, \frac \, dz \\() & =\sum_^\infty \iint \frac g(z) (1-y f'(z)) e^\, \frac \, dz \\() & =\sum_^\infty \left(\frac\right)^n\iint \frac g(z) (1-y f'(z)) e^ \, \frac \, dz \end The integral over ''k'' then gives $\backslash delta(x-z)$ and we have: :$$ \begin g(v) & = \sum_^\infty \left(\frac\right)^n \left(\frac g(x) (1-y f'(x))\right ) \\() & =\sum_^\infty \left(\frac\right)^n \left( \frac - \frac\left\\right\} \right ) \end Rearranging the sum and cancelling then gives the result: :$g(v)=g(x)+\backslash sum\_^\backslash infty\backslash frac\backslash left(\backslash frac\backslash partial\backslash right)^\backslash left(f(x)^kg\text{'}(x)\backslash right)$ 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「Lagrange reversion theorem」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.