Lagrange reversion theorem

: "This page is about Lagrange reversion. For inversion, see Lagrange inversion theorem."

In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; 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",:$g(v)=g(x)+sum\_\{k=1\}^inftyfrac\{y^k\}\{k!\}left(fracpartial\{partial\; x\}\; ight)^\{k-1\}left(f(x)^kg\text{'}(x)\; ight)$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:$g(v)\; =\; int\; dz\; delta(y\; f(z)\; -\; z\; +\; x)\; g(z)\; (1-y\; f\text{'}(z))$Writing the delta-function as an integral we have:$g(v)\; =\; int\; dz\; int\; frac\{dk\}\{2pi\}\; exp(ik\; [y\; f(z)\; -\; z\; +\; x]\; )\; g(z)\; (1-y\; f\text{'}(z))$:$=sum\_\{n=0\}^infty\; int\; dz\; int\; frac\{dk\}\{2pi\}\; frac\{(ik\; y\; f(z))^n\}\{n!\}\; g(z)\; (1-y\; f\text{'}(z))\; e^\{ik(x-z)\}$:$=sum\_\{n=0\}^infty\; left(frac\{partial\}\{partial\; x\}\; ight)^nint\; dz\; int\; frac\{dk\}\{2pi\}\; frac\{(y\; f(z))^n\}\{n!\}\; g(z)\; (1-y\; f\text{'}(z))\; e^\{ik(x-z)\}$The integral over "k" then gives $delta(x-z)$ and we have:$g(v)\; =sum\_\{n=0\}^infty\; left(frac\{partial\}\{partial\; x\}\; ight)^n\; left\; [\; frac\{(y\; f(x))^n\}\{n!\}\; g(x)\; (1-y\; f\text{'}(x))\; ight]$:$=sum\_\{n=0\}^infty\; left(frac\{partial\}\{partial\; x\}\; ight)^n\; left\; [\; frac\{y^n\; f(x)^n\; g(x)\}\{n!\}\; -\; frac\{y^\{n+1\{(n+1)!\}left\{\; (g(x)\; f(x)^\{n+1\})\text{'}\; -\; g\text{'}(x)\; f(x)^\{n+1\}\; ight\}\; ight]$Rearranging the sum and cancelling then gives the result:$g(v)=g(x)+sum\_\{k=1\}^inftyfrac\{y^k\}\{k!\}left(fracpartial\{partial\; x\}\; ight)^\{k-1\}left(f(x)^kg\text{'}(x)\; ight)$

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] on equation of time contains an application to Kepler's equation