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'':
:$g(v)=g(x)+\sum_^\infty\frac\left(\frac\partial\right)^\left(f(x)^kg'(x)\right).$
If ''g'' is the identity, this becomes
:$v=x+\sum_^\infty\frac\left(\frac\partial\right)^\left(f(x)^k\right)$

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

In 1770, Joseph Louis Lagrange (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. Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration.Hermite's proof is presented in:
*Goursat, Édouard, ''A Course in Mathematical Analysis'' (translated by E. R. Hedrick and O. Dunkel) .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. ambridge, England: Cambridge University Press, 1962pages 132–133.

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

Simple proof

We start by writing:

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

Writing the delta-function as an integral we have:

:
\begin
g(v) & = \iint \exp(ik f(z) - z + x g(z) (1-y f'(z)) \, \frac \, dz \\0pt& =\sum_^\infty \iint \frac g(z) (1-y f'(z)) e^\, \frac \, dz \\0pt& =\sum_^\infty \left(\frac\right)^n\iint \frac g(z) (1-y f'(z)) e^ \, \frac \, dz
\end

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

:
\begin
g(v) & = \sum_^\infty \left(\frac\right)^n \left \frac g(x) (1-y f'(x))\right\\0pt& =\sum_^\infty \left(\frac\right)^n \left
\frac - \frac\left\ \right\end

Rearranging the sum and cancelling then gives the result:

:$g(v)=g(x)+\sum_^\infty\frac\left(\frac\partial\right)^\left(f(x)^kg'(x)\right)$

References

External links

Lagrange Inversion [Reversion] Theorem
on MathWorld
Cornish–Fisher expansion
an application of the theorem

on equation of time contains an application to Kepler's equation.

{{DEFAULTSORT:Lagrange Reversion Theorem
Theorems in analysis
Inverse functions

fr:Théorème d'inversion de Lagrange