Lagrange reversion theorem

This page is about Lagrange reversion. For inversion, see Lagrange inversion theorem.

The Lagrange reversion theorem gives series expansions of certain implicitly defined functions; indeed, of compositions with such functions.

Let z be a function of x and y in terms of another function f such that

$z=x+yf(z)$

Then for any function g,

$g(z)=g(x)+\sum _{k=1}^{\infty }{\frac {y^{k}}{k!}}\left({\frac {\partial }{\partial x}}\right)^{k-1}\left(f(x)^{k}g'(x)\right)$

for small y. If g is the identity

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

External links

those which call it "inversion theorem":

• Lagrange Inversion Theorem is Mathworld's name
• Cornish-Fisher expansion, an application

those which call it "reversion theorem":

• Article on Equation of time contains an application to Kepler's equation