Lagrange inversion theorem: Difference between revisions

Revision as of 22:12, 31 July 2025

Revision as of 23:06, 31 July 2025

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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}