Lagrange reversion theorem

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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 + y f (v)

Then for any function g,

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

for small y. If g is the identity

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

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 [3-5]. 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.

[edit] Simple proof

We start by writing

$g(v) = \int dz \delta(y f(z) - z + x) g(z) (1-y f'(z))$

Writing the delta-function as an integral we have

$g(v) = \int dz \int \frac{dk}{2\pi} \exp(ik[y f(z) - z + x]) g(z) (1-y f'(z))$

$=\sum_{n=0}^\infty \int dz \int \frac{dk}{2\pi} \frac{(ik y f(z))^n}{n!} g(z) (1-y f'(z)) e^{ik(x-z)}$

$=\sum_{n=0}^\infty \left(\frac{\partial}{\partial x}\right)^n\int dz \int \frac{dk}{2\pi} \frac{(y f(z))^n}{n!} g(z) (1-y f'(z)) e^{ik(x-z)}$

The integral over k then gives $δ(x - z)$ and we have

$g(v) =\sum_{n=0}^\infty \left(\frac{\partial}{\partial x}\right)^n \left[ \frac{(y f(x))^n}{n!} g(x) (1-y f'(x))\right]$

$=\sum_{n=0}^\infty \left(\frac{\partial}{\partial x}\right)^n \left[ \frac{y^n f(x)^n g(x)}{n!} - \frac{y^{n+1}}{(n+1)!}\left\{ (g(x) f(x)^{n+1})' - g'(x) f(x)^{n+1}\right\} \right]$

Rearranging the sum and cancelling then gives the result

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

[edit] 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.