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TucanHolmes (talk | contribs) Add disambiguation hatnote for similarly named Lagrange reversion theorem |
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{{Short description|Formula for inverting a Taylor series}}
{{for|the formal power series expansion of certain implicitly defined functions|Lagrange reversion theorem}}
In [[mathematical analysis]], the '''Lagrange inversion theorem''', also known as the '''Lagrange–Bürmann formula''', gives the [[Taylor series]] expansion of the [[inverse function]] of an [[analytic function]]. Lagrange inversion is a special case of the [[inverse function theorem]].
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===Lagrange–Bürmann formula===
There is a special case of Lagrange inversion theorem that is used in [[combinatorics]] and applies when <math>f(w)=w/\phi(w)</math> for some analytic <math>\phi(w)</math> with <math>\phi(0)\ne 0.</math> Take <math>a=0</math> to obtain <math>f(a)=f(0)=0.</math> Then for the inverse <math>g(z)</math> (satisfying <math>f(g(z))\equiv z</math>), we have
:<math>\begin{align}
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*[[Faà di Bruno's formula]] gives coefficients of the composition of two formal power series in terms of the coefficients of those two series. Equivalently, it is a formula for the ''n''th derivative of a composite function.
*[[Lagrange reversion theorem]] for another theorem sometimes called the inversion theorem
*
==References==
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