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 <math>v</math> be a function of <math>x</math> and <math>y</math> in terms of another function <math>f</math> such that

<math>v=x+yf(v)</math>

Then for any function <math>g</math>, for small enough <math>y</math>:

<math>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).</math>

In particular, if <math>g</math> is the identity function <math>g(x)=x</math>, this reduces to

<math>v=x+\sum_{k=1}^\infty\frac{y^k}{k!}\left(\frac\partial{\partial x}\right)^{k-1}\left(f(x)^k\right),</math>

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 <math>v</math> mentioned above. However, his solution used cumbersome series expansions of logarithms.<ref>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] .)</ref><ref>Lagrange, Joseph Louis, Oeuvres, [Paris, 1869], Vol. 2, page 25; Vol. 3, pages 3–73.</ref> 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 <math>x</math> and the parameter <math>y</math>.<ref>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.</ref><ref>Laplace, Pierre Simon de, Oeuvres [Paris, 1843], Vol. 9, pages 313–335.</ref><ref>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.</ref> Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration.<ref>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.</ref><ref>Hermite, Charles, Oeuvres [Paris, 1908], Vol. 2, pages 319–346.</ref><ref>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.</ref>

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

Simple proof

We start by writing

<math> g(v) = \int \delta(y f(z) - z + x) g(z) (1-y f'(z)) \, dz.</math>

Writing the delta-function as an integral, we have:

<math>

\begin{align} g(v) & = \iint \exp(\mathrm{i}k[y f(z) - z + x]) g(z) (1-y f'(z)) \, \frac{dk}{2\pi} \, dz \\[10pt] & =\sum_{n=0}^\infty \iint \frac{(\mathrm{i}k y f(z))^n}{n!} g(z) (1-y f'(z)) \mathrm{e}^{\mathrm{i}k(x-z)}\, \frac{dk}{2\pi} \, dz \\[10pt] & =\sum_{n=0}^\infty \left(\frac{\partial}{\partial x}\right)^n\iint \frac{(y f(z))^n}{n!} g(z) (1-y f'(z)) \mathrm{e}^{\mathrm{i}k(x-z)} \, \frac{dk}{2\pi} \, dz \end{align} </math>

The integral over <math>k</math> then gives <math>\delta(x-z)</math> and we have:

<math>

\begin{align} 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] \\[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{align} </math>

Rearranging the sum and cancelling then gives the result:

<math>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)</math>

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.