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)
| |||||
g(v)=g(x)+\sum | \left( | ||||
k=1 |
\partial{\partial | |
x}\right) |
k-1\left(f(x)kg'(x)\right).
| |||||
v=x+\sum | \left( | ||||
k=1 |
\partial{\partial | |
x}\right) |
k-1\left(f(x)k\right)
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 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.
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{align} g(v)&=\iint\exp(ik[yf(z)-z+x])g(z)(1-yf'(z))
dk | |
2\pi |
dz\\[10pt] &
infty | |
=\sum | |
n=0 |
\iint
(ikyf(z))n | |
n! |
g(z)(1-yf'(z))eik(x-z)
dk | |
2\pi |
dz\\[10pt] &
infty | ||
=\sum | \left( | |
n=0 |
\partial | |
\partialx |
\right)n\iint
(yf(z))n | |
n! |
g(z)(1-yf'(z))eik(x-z)
dk | |
2\pi |
dz \end{align}
The integral over k then gives
\delta(x-z)
\begin{align} g(v)&=
infty | ||
\sum | \left( | |
n=0 |
\partial | |
\partialx |
\right)n\left[
(yf(x))n | |
n! |
g(x)(1-yf'(x))\right]\\[10pt] &
infty | ||
=\sum | \left( | |
n=0 |
\partial | |
\partialx |
\right)n\left[
ynf(x)ng(x) | |
n! |
-
yn+1 | |
(n+1)! |
\left\{(g(x)f(x)n+1)'-g'(x)f(x)n+1\right\}\right] \end{align}
Rearranging the sum and cancelling then gives the result:
| |||||
g(v)=g(x)+\sum | \left( | ||||
k=1 |
\partial{\partial | |
x}\right) |
k-1\left(f(x)kg'(x)\right)