Lagrange reversion theorem

This page is about Lagrange reversion. For inversion, see Lagrange inversion 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 z be a function of x and y in terms of another function f such that

z = x + y f (z)

Then for any function g,

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

for small y. If g is the identity

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

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