Lagrange inversion 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.
Statement.
Suppose is defined as a function of by an equation of the form
where is analytic at a point and formula_2 Then it is possible to "invert" or "solve" the equation for , expressing it in the form formula_3 given by a power series
where
The theorem further states that this series has a non-zero radius of convergence, i.e., formula_6 represents an analytic function of in a neighbourhood of formula_7 This is also called reversion of series.
If the assertions about analyticity are omitted, 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 for any analytic function ; and it can be generalized to the case formula_8 where the inverse is a multivalued function.
The theorem was proved by Lagrange and generalized by Hans Heinrich 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 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 some property of the formal residue, and a more direct formal proof is available.
If is a formal power series, then the above formula does not give the coefficients of the compositional inverse series directly in terms for the coefficients of the series . If one can express the functions and in formal power series as
with and , then an explicit form of inverse coefficients can be given in term of Bell polynomials:
where
is the rising factorial. 
When , the last formula can be interpreted in terms of the faces of associahedra 
where formula_13 for each face formula_14 of the associahedron formula_15
Example.
For instance, the algebraic equation of degree 
can be solved for by means of the Lagrange inversion formula for the function , resulting in a formal series solution
By convergence tests, this series is in fact convergent for formula_18 which is also the largest disk in which a local inverse to can be defined.
Sketch of the proof.
For simplicity suppose formula_19. We can then compute
If we expand the integrand using the geometric series we get
where in the last step we used the fact that formula_22 has one simple zero.
Finally we can integrate over formula_23 taking into account formula_24
Upon a redefiniton of the summation index we get the stated formula.
Applications.
Lagrange–Bürmann formula.
There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when formula_26 for some analytic formula_27 with formula_28 Take formula_29 to obtain formula_30 Then for the inverse formula_6 (satisfying formula_32), we have
which can be written alternatively as
where formula_35 is an operator which extracts the coefficient of formula_36 in the Taylor series of a function of .
A generalization of the formula is known as the Lagrange–Bürmann formula:
where is an arbitrary analytic function.
Sometimes, the derivative can be quite complicated. A simpler version of the formula replaces with to get 
which involves instead of .
Lambert "W" function.
The Lambert function is the function formula_39 that is implicitly defined by the equation
We may use the theorem to compute the Taylor series of formula_39 at formula_42 We take formula_43 and formula_44 Recognizing that
this gives
The radius of convergence of this series is formula_47 (giving the principal branch of the Lambert function).
A series that converges for larger (though not for all ) can also be derived by series inversion. The function formula_48 satisfies the equation
Then formula_50 can be expanded into a power series and inverted. This gives a series for formula_51
formula_53 can be computed by substituting formula_54 for in the above series. For example, substituting for gives the value of formula_55
Binary trees.
Consider the set formula_56 of unlabelled binary trees. An element of formula_56 is either a leaf of size zero, or a root node with two subtrees. Denote by formula_58 the number of binary trees on formula_59 nodes.
Removing the root splits a binary tree into two trees of smaller size. This yields the functional equation on the generating function formula_60
Letting formula_62, one has thus formula_63 Applying the theorem with formula_64 yields
This shows that formula_58 is the th Catalan number.
Asymptotic approximation of integrals.
In the Laplace-Erdelyi theorem that gives the asymptotic approximation for Laplace-type integrals, the function inversion is taken as a crucial step.