Lagrange inversion theorem

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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.

Contents

Theorem statement

Suppose the dependence between the variables w and z is implicitly defined by an equation of the form

where f is analytic at a point a and f '(a) ≠ 0. Then it is possible to invert or solve the equation for w:

where g is analytic at the point b = f(a). This is also called reversion of series.

The series expansion of g is given by

The formula is also valid for formal power series and can be generalized in various ways. It can be formulated for functions of several variables, it can be extended to provide a ready formula for F(g(z)) for any analytic function F, and it can be generalized to the case f '(a) = 0, where the inverse g is a multivalued function.

The theorem was proved by Lagrange[1] and generalized by Hans Heinrich Bürmann,[2][3][4] both in the late 18th century. There is a straightforward derivation using complex analysis and contour integration; the complex formal power series version is clearly a consequence of knowing the formula for polynomials, so the theory of analytic functions may be applied. Actually, the machinery from analytic function theory enters only in a formal way in this proof, in that what is really needed is just some property of the formal residue, and a more direct formal proof is available.

Applications

Lagrange–Bürmann formula

There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when f(w) = w / φ(w) and \phi(0)\ne 0. Take a = 0 to obtain b = f(0) = 0. We have

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