Lagrange reversion theorem

{{Short description|Gives power series for certain implict functions}}

{{for|reversion of series|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, for small enough y:

:$g(v)=g(x)+\backslash sum\_\{k=1\}^\backslash infty\backslash frac\{y^k\}\{k!\}\backslash left(\backslash frac\backslash partial\{\backslash partial\; x\}\backslash right)^\{k-1\}\backslash left(f(x)^kg\text{'}(x)\backslash right).$

If g is the identity, this becomes

:$v=x+\backslash sum\_\{k=1\}^\backslash infty\backslash frac\{y^k\}\{k!\}\backslash left(\backslash frac\backslash partial\{\backslash partial\; x\}\backslash right)^\{k-1\}\backslash left(f(x)^k\backslash right)$

In which case the equation can be derived using perturbation theory.

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.Lagrange, Joseph Louis (1770) "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: [https://archive.today/20120630182344/http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=41070] .)Lagrange, Joseph Louis, Oeuvres, [Paris, 1869], Vol. 2, page 25; Vol. 3, pages 3–73. 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.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.Laplace, Pierre Simon de, Oeuvres [Paris, 1843], Vol. 9, pages 313–335.Laplace's proof is presented in:

• Goursat, Édouard, A Course in Mathematical Analysis (translated by E.R. Hedrick and O. Dunkel) [N.Y., N.Y.: Dover, 1959], Vol. I, pages 404–405. Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration.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.Hermite, Charles, Oeuvres [Paris, 1908], Vol. 2, pages 319–346.Hermite's proof is presented in:
• Goursat, Édouard, 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.
• E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. [Cambridge, England: Cambridge University Press, 1962] pages 132–133.

Lagrange's reversion theorem is used to obtain numerical solutions to Kepler's equation.