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In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. ==Theorem statement== Suppose ''z'' is defined as a function of ''w'' by an equation of the form : where ''f'' is analytic at a point ''a'' and ''f'' '(''a'') ≠ 0. Then it is possible to ''invert'' or ''solve'' the equation for ''w'': : on a neighbourhood of ''f(a)'', where ''g'' is analytic at the point ''f''(''a''). This is also called reversion of series. The series expansion of ''g'' is given by : The formula is also valid for formal power series and can be generalized in various ways. It can be formulated for functions of several variables, it can be extended to provide a ready formula for ''F''(''g''(''z'')) for any analytic function ''F'', and it can be generalized to the case ''f'' '(''a'') = 0, where the inverse ''g'' is a multivalued function. The theorem was proved by Lagrange〔 (Note: Although Lagrange submitted this article in 1768, it was not published until 1770.)〕 and generalized by Hans Heinrich Bürmann,〔Bürmann, Hans Heinrich, “Essai de calcul fonctionnaire aux constantes ad-libitum,” submitted in 1796 to the Institut National de France. For a summary of this article, see: 〕〔Bürmann, Hans Heinrich, "Formules du développement, de retour et d'integration," submitted to the Institut National de France. Bürmann's manuscript survives in the archives of the École Nationale des Ponts et Chaussées (School of Bridges and Roads ) in Paris. (See ms. 1715.)〕〔A report on Bürmann's theorem by Joseph-Louis Lagrange and Adrien-Marie Legendre appears in: ("Rapport sur deux mémoires d'analyse du professeur Burmann," ) ''Mémoires de l'Institut National des Sciences et Arts: Sciences Mathématiques et Physiques'', vol. 2, pages 13–17 (1799).〕 both in the late 18th century. There is a straightforward derivation〔 *E. T. Whittaker and G. N. Watson. ''A Course of Modern Analysis''. Cambridge University Press; 4th edition (January 2, 1927), pp. 129–130〕 using complex analysis and contour integration; the complex formal power series version is clearly a consequence of knowing the formula for polynomials, so the theory of analytic functions may be applied. Actually, the machinery from analytic function theory enters only in a formal way in this proof, in that what is really needed is just some property of the formal residue, and a more direct formal proof is available. 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 inversion theorem」の詳細全文を読む スポンサード リンク
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