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][4][5]Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration.[6][7][8]
Lagrange's reversion theorem is used to obtain numerical solutions to
Kepler's equation.
Simple proof
We start by writing:
Writing the delta-function as an integral we have:
The integral over k then gives and we have:
Rearranging the sum and cancelling then gives the result:
References
^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:
[1] .)
^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.
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.
^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.
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.