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 {\displaystyle v} be a function of x {\displaystyle x} and y {\displaystyle y} in terms of another function f {\displaystyle f} such that v = x + y f ( v ) {\displaystyle v=x+yf(v)} Then for any function g {\displaystyle g} , for small enough y {\displaystyle y} : g ( v ) = g ( x ) + ∑ k = 1 ∞ y k k !

Source: Wikipedia — Lagrange reversion theorem (CC BY-SA 4.0)

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 {\displaystyle v} be a function of x {\displaystyle x} and y {\displaystyle y} in terms of another function f {\displaystyle f} such that v = x + y f ( v ) {\displaystyle v=x+yf(v)} Then for any function g {\displaystyle g} , for small enough y {\displaystyle y} : g ( v ) = g ( x ) + ∑ k = 1 ∞ y k k !

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Source: Wikipedia "Lagrange reversion theorem" · CC BY-SA 4.0

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