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Lagrange inversion theorem
From Wikipedia, the free encyclopedia
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. 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
This formula can for instance be used to find the Taylor series of the Lambert W function (by setting f(w) = w exp(w) and a = b = 0).
The formula is also valid for formal power series and can be generalized in various ways. If 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 and generalized by Bürmann, 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).
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The Lambert W function is the function W(z) that satisfies the implicit equation
We may use the theorem to compute the Taylor series of W(z) at z = 0. We take f(w) = wew and a = b = 0. Recognising that
this gives
There is a special case of the theorem that is used in combinatorics and applies when f(w) = w / φ(w) and 
Take a = 0 to obtain b = f(0) = 0. We have
or
which can be written alternatively as
![[z^n] g(z) = \frac{1}{n} [w^{n-1}] \phi(w)^n,](../../../math/a/6/2/a620741e8de2f4325fa2ee458addaab2.png)
where [wr] is an operator which extracts the coefficient of wr in what follows it.
Consider the set 
of unlabelled binary trees. An element of is either a leaf of size zero, or a root node with two subtrees (planar, i.e. no symmetry between them). The Fundamental theorem of combinatorial enumeration (unlabelled case) applies.
The group acting on the two subtrees is E2, which contains a single permutation consisting of two fixed points. The set satisfies
This yields the functional equation of the OGF B(z) by the number of internal nodes:
Let 
to obtain
Now apply the theorem with φ(w) = (w + 1)2:
![[z^n] B_{\ge 1}(z) = \frac{1}{n} [w^{n-1}] (w+1)^{2n} = \frac{1}{n} {2n \choose n-1} = \frac{1}{n+1} {2n \choose n},](../../../math/b/6/9/b697c3d126fc4ad77d7531867c86bea4.png)
the Catalan numbers.
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 nth derivative of a composite function.
- Lagrange reversion theorem for another theorem sometimes called the inversion theorem