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

v=x+yf(v)

Then for any function g, for small enough y:

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

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:

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

Writing the delta-function as an integral we have:

{\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 $\delta (x-z)$ and we have:

{\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:

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

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: .)
Lagrange, Joseph Louis, Oeuvres, [Paris, 1869], Vol. 2, page 25; Vol. 3, pages 3–73.
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.
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.

External links

Lagrange Inversion [Reversion] Theorem on MathWorld
Cornish–Fisher expansion, an application of the theorem
Article on equation of time contains an application to Kepler's equation.

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[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: .)

[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] 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.

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

[10] 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.

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