# Lagrange reversion 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

${\displaystyle v=x+yf(v)}$

Then for any function g, for small enough y:

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

If g is the identity, this becomes

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

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:

${\displaystyle g(v)=\int \delta (yf(z)-z+x)g(z)(1-yf'(z))\,dz}$

Writing the delta-function as an integral we have:

{\displaystyle {\begin{aligned}g(v)&=\iint \exp(ik[yf(z)-z+x])g(z)(1-yf'(z))\,{\frac {dk}{2\pi }}\,dz\\[10pt]&=\sum _{n=0}^{\infty }\iint {\frac {(ikyf(z))^{n}}{n!}}g(z)(1-yf'(z))e^{ik(x-z)}\,{\frac {dk}{2\pi }}\,dz\\[10pt]&=\sum _{n=0}^{\infty }\left({\frac {\partial }{\partial x}}\right)^{n}\iint {\frac {(yf(z))^{n}}{n!}}g(z)(1-yf'(z))e^{ik(x-z)}\,{\frac {dk}{2\pi }}\,dz\end{aligned}}}

The integral over k then gives ${\displaystyle \delta (x-z)}$ and we have:

{\displaystyle {\begin{aligned}g(v)&=\sum _{n=0}^{\infty }\left({\frac {\partial }{\partial x}}\right)^{n}\left[{\frac {(yf(x))^{n}}{n!}}g(x)(1-yf'(x))\right]\\[10pt]&=\sum _{n=0}^{\infty }\left({\frac {\partial }{\partial x}}\right)^{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}\right\}\right]\end{aligned}}}

Rearranging the sum and cancelling then gives the result:

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

## References

1. ^ 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: 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, 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.
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:
• 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.
• Whittaker, E.T. and G.N. Watson, A Course of Modern Analysis, 4th ed. [Cambridge, England: Cambridge University Press, 1962] pages 132–133.