Lagrange reversion theorem

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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

<math>z=x+yf(z)</math>

Then for any function g,

<math>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)</math>

for small y. If g is the identity

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

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 equationfr:Théorème d'inversion de Lagrange