Lagrange Expansion -- from Wolfram MathWorld

Calculus and Analysis > Series > Series Expansions >

Lagrange Expansion

Let and where , then

$x=x_0+sum_(k=1)^infty((y-y_0)^k)/(k!){(d^(k-1))/(dx^(k-1))[(x-x_0)/(f(x)-y_0)]^k}_(x=x_0)$

$g(x)=g(x_0)+sum_(k=1)^infty((y-y_0)^k)/(k!){(d^(k-1))/(dx^(k-1))[g^'(x)((x-x_0)/(f(x)-y_0))^k]}_(x=x_0).$

Expansions of this form were first considered by Lagrange (1770; 1868, pp. 680-693).

REFERENCES:

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, p. 106, 1959.

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.

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, p. 132, 1990.

CITE THIS AS:

Weisstein, Eric W. "Lagrange Expansion." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LagrangeExpansion.html