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Lagrange Inversion Theorem

Let z be defined as a function of w in terms of a parameter alpha by

z=w+alphaphi(z).
(1)

Then Lagrange's inversion theorem, also called a Lagrange expansion, states that any function of z can be expressed as a power series in alpha which converges for sufficiently small alpha and has the form

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)}+....
(2)

The theorem can also be stated as follows. Let y=f(x) and y_0=f(x_0) where f^'(x_0)!=0, 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)
(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).
(4)

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

See also

Bürmann's Theorem, Maclaurin Series, Schur-Jabotinsky Theorem, Taylor Series, Teixeira's Theorem

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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, 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.

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Lagrange Inversion Theorem

Cite this as:

Weisstein, Eric W. "Lagrange Inversion Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LagrangeInversionTheorem.html

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