In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions.

such that

:

is the identity function, this reduces to

in which case the equation can be derived using perturbation theory.

In 1770, Joseph Louis Lagrange (1736–1813) published his power series mentioned above. However, his solution used cumbersome series expansions of logarithms. In 1780, Pierre-Simon Laplace (1749–1827) published a simpler proof of the theorem, which was based on relations between partial derivatives. Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration.

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:

and we have:

Rearranging the sum and cancelling then gives the result:

References

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.

fr:Théorème d'inversion de Lagrange