Let
be defined as a function of in terms of a parameter by
(1)
Then Lagrange's inversion theorem, also called a Lagrange expansion, states that any function of can be expressed as a power series
in
which converges for sufficiently small and has the form
(2)
The theorem can also be stated as follows. Let and where , then
(3)
(4)
Expansions of this form were first considered by Lagrange (1770; 1868, pp. 680-693).
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 Algebra6, 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 Berlin24, 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, France: 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."
C. R. Acad. Sci. Paris53, 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,
England: Longmans, pp. 158-159, 1895.
be defined as a function of 
in terms of a parameter 
by
can be expressed as a power series
in 
which converges for sufficiently small 
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)}+....](/images/equations/LagrangeInversionTheorem/NumberedEquation2.svg)
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)](/images/equations/LagrangeInversionTheorem/NumberedEquation3.svg)
![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)
