Lagrange inversion theorem

The Lagrange inversion theorem, also known as the Lagrange-Bürmann formula, is a result in mathematical analysis that gives the Taylor series expansion of the inverse function of an analytic function. If z is given as an analytic function f of w near a point a where the derivative of f is not zero, the theorem lets one solve for w as a power series in z and states that this series has a non-zero radius of convergence, so the inverse is analytic near f(a). The statement is also called reversion of series, and it is a special case of the inverse function theorem. If the analyticity assertions are dropped, the formula still holds for formal power series. The theorem was proved by Lagrange and generalized by Hans Heinrich Bürmann, both in the late 18th century, and it is used in combinatorics, for example in counting binary trees, and in the asymptotic approximation of integrals.