设 
定义为 
的函数，以参数 
表示，由
 |
(1)
|
那么拉格朗日反演定理，也称为拉格朗日展开，指出任何 
的函数都可以表示为 
的幂级数，对于足够小的 
收敛，并具有以下形式
![F(z)=F(w)+alpha/1phi(w)F^'(w)+(alpha^2)/(1·2)partial/(partialw){[phi(w)]^2F^'(w)}
+...+(alpha^(n+1))/((n+1)!)(partial^n)/(partialw^n){[phi(w)]^(n+1)F^'(w)}+....](/images/equations/LagrangeInversionTheorem/NumberedEquation2.svg) |
(2)
|
该定理也可以如下表述。设 
和 
其中 
, 那么
![x=x_0+sum_(k=1)^infty((y-y_0)^k)/(k!){(d^(k-1))/(dx^(k-1))[(x-x_0)/(f(x)-y_0)]^k}_(x=x_0)](/images/equations/LagrangeInversionTheorem/NumberedEquation3.svg) |
(3)
|
![g(x)=g(x_0)+sum_(k=1)^infty((y-y_0)^k)/(k!){(d^(k-1))/(dx^(k-1))[g^'(x)((x-x_0)/(f(x)-y_0))^k]}_(x=x_0).](/images/equations/LagrangeInversionTheorem/NumberedEquation4.svg) |
(4)
|
这种形式的展开式最早由拉格朗日 (1770; 1868, pp. 680-693) 考虑。
拉格朗日反演定理
另请参阅
比尔曼定理, 麦克劳林级数, 舒尔-雅博廷斯基定理, 泰勒级数, 特谢拉定理使用 Wolfram|Alpha 探索
参考文献
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972.Goursat, E. A Course in Mathematical Analysis, Vol. 2: Functions of a Complex Variable & Differential Equations. New York: Dover, pp. 106 and 120, 1959.Henrici, P. "An Algebraic Proof of the Lagrange-Burmann Formula." J. Math. Anal. Appl. 8, 218-224, 1964.Henrici, P. "The Lagrange-Bürmann Theorem." §1.9 in Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros. New York: Wiley, pp. 55-65, 1988.Joni, S. A. "Lagrange Inversion in Higher Dimensions and Umbral Operators." J. Linear Multi-Linear Algebra 6, 111-121, 1978.Lagrange, J.-L. "Nouvelle méthode pour résoudre les problèmes indéterminés en nombres entiers." Mém. de l'Acad. Roy. des Sci. et Belles-Lettres de Berlin 24, 1770. Reprinted in Oeuvres de Lagrange, tome 2, section deuxième: Mémoires extraits des recueils de l'Academie royale des sciences et Belles-Lettres de Berlin. Paris: Gauthier-Villars, pp. 655-726, 1868.Moulton, F. R. An Introduction to Celestial Mechanics, 2nd rev. ed. New York: Dover, p. 161, 1970.Popoff, M. "Sur le reste de la série de Lagrange." Comptes Rendus Herbdom. Séances de l'Acad. Sci. 53, 795-798, 1861.Roman, S. "The Lagrange Inversion Formula." §5.2. in The Umbral Calculus. New York: Academic Press, pp. 138-140, 1984.Whittaker, E. T. and Watson, G. N. "Lagrange's Theorem." §7.32 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 132-133, 1990.Williamson, B. "Remainder in Lagrange's Series." §119 in An Elementary Treatise on the Differential Calculus, Containing the Theory of Plane Curves, with Numerous Examples, 9th ed. London: Longmans, pp. 158-159, 1895.在 Wolfram|Alpha 中被引用
拉格朗日反演定理请引用本文为
韦斯坦因，埃里克·W. "拉格朗日反演定理。" 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/LagrangeInversionTheorem.html