- Lagrange reversion theorem
: "This page is about Lagrange reversion. For inversion, see
Lagrange inversion theorem ."In
mathematics , the Lagrange reversion theorem gives series orformal power series expansions of certainimplicitly defined function s; indeed, of compositions with such functions.Let "v" be a function of "x" and "y" in terms of another function "f" such that:v=x+yf(v)Then for any function "g",:g(v)=g(x)+sum_{k=1}^inftyfrac{y^k}{k!}left(fracpartial{partial x} ight)^{k-1}left(f(x)^kg'(x) ight)for small "y". If "g" is the identity:v=x+sum_{k=1}^inftyfrac{y^k}{k!}left(fracpartial{partial x} ight)^{k-1}left(f(x)^k ight)
In 1770,
Joseph Louis Lagrange (1736-1813) published his power series solution of the implicit equation for "v" mentioned above. However, his solution used cumbersome series expansions of logarithms [1,2] . In 1780,Pierre-Simon Laplace (1749-1827) published a simpler proof of the theorem, which was based on relations between partial derivatives with respect to the variable x and the parameter y [3-5] .Charles Hermite (1822-1901) presented the most straightforward proof of the theorem by using contour integration [6-8] .Lagrange's reversion theorem is used to obtain numerical solutions to
Kepler's equation .imple proof
We start by writing:g(v) = int dz delta(y f(z) - z + x) g(z) (1-y f'(z))Writing the delta-function as an integral we have:g(v) = int dz int frac{dk}{2pi} exp(ik [y f(z) - z + x] ) g(z) (1-y f'(z)) :sum_{n=0}^infty int dz int frac{dk}{2pi} frac{(ik y f(z))^n}{n!} g(z) (1-y f'(z)) e^{ik(x-z)} :sum_{n=0}^infty left(frac{partial}{partial x} ight)^nint dz int frac{dk}{2pi} frac{(y f(z))^n}{n!} g(z) (1-y f'(z)) e^{ik(x-z)} The integral over "k" then gives delta(x-z) and we have:g(v) =sum_{n=0}^infty left(frac{partial}{partial x} ight)^n left [ frac{(y f(x))^n}{n!} g(x) (1-y f'(x)) ight] :sum_{n=0}^infty left(frac{partial}{partial x} ight)^n left [ frac{y^n f(x)^n g(x)}{n!} - frac{y^{n+1{(n+1)!}left{ (g(x) f(x)^{n+1})' - g'(x) f(x)^{n+1} ight} ight] Rearranging the sum and cancelling then gives the result:g(v)=g(x)+sum_{k=1}^inftyfrac{y^k}{k!}left(fracpartial{partial x} ight)^{k-1}left(f(x)^kg'(x) ight)
References
[1] Lagrange, Joseph Louis (1768) "Nouvelle méthode pour résoudre les équations littérales par le moyen des séries," "Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin", vol. 24, pages 251-326. (Available on-line at: http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=41070 .)
[2] Lagrange, Joseph Louis, "Oeuvres", [Paris, 1869] , Vol. 2, page 25; Vol. 3, pages 3-73.
[3] Laplace, Pierre Simon de (1777) "Mémoire sur l'usage du calcul aux différences partielles dans la théories des suites," "Mémoires de l'Académie Royale des Sciences de Paris," vol. , pages 99-122.
[4] Laplace, Pierre Simon de, "Oeuvres" [Paris, 1843] , Vol. 9, pages 313-335.
[5] Laplace's proof is presented in:
Goursat, Edouard, "A Course in Mathematical Analysis" (translated by E.R. Hedrick and O. Dunkel) [N.Y., N.Y.: Dover, 1959] , Vol. I, pages 404-405. [6] Hermite, Charles (1865) "Sur quelques développements en série de fonctions de plusieurs variables," "Comptes Rendus de l'Académie des Sciences des Paris", vol. 60, pages 1-26.
[7] Hermite, Charles, "Oeuvres" [Paris, 1908] , Vol. 2, pages 319-346.
[8] Hermite's proof is presented in:
(i) Goursat, Edouard, "A Course in Mathematical Analysis" (translated by E. R. Hedrick and O. Dunkel) [N.Y., N.Y.: Dover, 1959] , Vol. II, Part 1, pages 106-107. (ii) Whittaker, E.T. and G.N. Watson, "A Course of Modern Analysis", 4th ed. [Cambridge, England: Cambridge University Press, 1962] pages 132-133.
External links
* [http://mathworld.wolfram.com/LagrangeInversionTheorem.html Lagrange Inversion Reversion Theorem] on
MathWorld
* [http://www.quantlet.com/mdstat/scripts/xfg/html/xfghtmlnode8.html Cornish-Fisher expansion] , an application of the theorem
* [http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.html Article] onequation of time contains an application toKepler's equation
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