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In
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
expansion of the
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
of an
analytic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.

# Statement

Suppose is defined as a function of by an equation of the form :$z = f\left(w\right)$ where is analytic at a point and $f\text{'}\left(a\right)\neq 0.$ Then it is possible to ''invert'' or ''solve'' the equation for , expressing it in the form $w=g\left(z\right)$ given by a power series :$g\left(z\right) = a + \sum_^ g_n \frac,$ where : The theorem further states that this series has a non-zero radius of convergence, i.e., $g\left(z\right)$ represents an analytic function of in a
neighbourhood A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...
of $z= f\left(a\right).$ This is also called reversion of series. If the assertions about analyticity are omitted, the formula is also valid for
formal power series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
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 $f\text{'}\left(a\right)=0,$ where the inverse is a multivalued function. The theorem was proved by
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaRoman_Forum.html" ;"title="Curia Julia in the Roman Forum">Curia Julia in the Roman Forum A senate is a deliberative assembly, often the upper house or Debating chamber, chamber of a bicame ... and generalized by Hans Heinrich Bürmann, both in the late 18th century. There is a straightforward derivation using
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
and
contour integration In the mathematical field of complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (math ...
; the complex formal power series version is a consequence of knowing the formula for
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... s, so the theory of
analytic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s 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 Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
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 :$f\left(w\right) = \sum_^\infty f_k \frac \qquad \text \qquad g\left(z\right) = \sum_^\infty g_k \frac$ with and , then an explicit form of inverse coefficients can be given in term of Bell polynomials: :$g_n = \frac \sum_^ \left(-1\right)^k n^ B_\left(\hat_1,\hat_2,\ldots,\hat_\right), \quad n \geq 2,$ where :$\begin \hat_k &= \frac, \\ g_1 &= \frac, \text \\ n^ &= n\left(n+1\right)\cdots \left(n+k-1\right) \end$ is the rising factorial. When , the last formula can be interpreted in terms of the faces of Associahedron, associahedra :$g_n = \sum_ \left(-1\right)^ f_F , \quad n \geq 2,$ where $f_ = f_ \cdots f_$ for each face $F = K_ \times \cdots \times K_$ of the associahedron $K_n .$

# Example

For instance, the algebraic equation of degree :$x^p - x + z= 0$ can be solved for by means of the Lagrange inversion formula for the function , resulting in a formal series solution :$x = \sum_^\infty \binom \frac .$ By convergence tests, this series is in fact convergent for $, z, \leq \left(p-1\right)p^,$ which is also the largest disk in which a local inverse to can be defined.

# Applications

## Lagrange–Bürmann formula

There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when $f\left(w\right)=w/\phi\left(w\right)$ for some analytic $\phi\left(w\right)$ with $\phi\left(0\right)\ne 0.$ Take $a=0$ to obtain $f\left(a\right)=f\left(0\right)=0.$ Then for the inverse $g\left(z\right)$ (satisfying $f\left(g\left(z\right)\right)\equiv z$), we have :$\begin g\left(z\right) &= \sum_^ \left\left[ \lim_ \frac \left\left(\left\left( \frac \right\right)^n \right\right)\right\right] \frac \\ &= \sum_^ \frac \left\left[\frac \lim_ \frac \left(\phi\left(w\right)^n\right) \right\right] z^n, \end$ which can be written alternatively as :$\left[z^n\right] g\left(z\right) = \frac \left[w^\right] \phi\left(w\right)^n,$ where $\left[w^r\right]$ is an operator which extracts the coefficient of $w^r$ in the Taylor series of a function of . A generalization of the formula is known as the Lagrange–Bürmann formula: :$\left[z^n\right] H \left(g\left(z\right)\right) = \frac \left[w^\right] \left(H\text{'} \left(w\right) \phi\left(w\right)^n\right)$ where is an arbitrary analytic function. Sometimes, the derivative can be quite complicated. A simpler version of the formula replaces with to get :$\left[z^n\right] H \left(g\left(z\right)\right) = \left[w^n\right] H\left(w\right) \phi\left(w\right)^ \left(\phi\left(w\right) - w \phi\text{'}\left(w\right)\right),$ which involves instead of .

## Lambert ''W'' function

The Lambert function is the function $W\left(z\right)$ that is implicitly defined by the equation :$W\left(z\right) e^ = z.$ We may use the theorem to compute the
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of $W\left(z\right)$ at $z=0.$ We take $f\left(w\right) = we^w$ and $a = 0.$ Recognizing that :$\frac e^ = \alpha^n e^,$ this gives :$\begin W\left(z\right) &= \sum_^ \left\left[\lim_ \frac e^ \right\right] \frac \\ &= \sum_^ \left(-n\right)^ \frac \\ &= z-z^2+\fracz^3-\fracz^4+O\left(z^5\right). \end$ The radius of convergence of this series is $e^$ (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 $f\left(z\right) = W\left(e^z\right) - 1$ satisfies the equation :$1 + f\left(z\right) + \ln \left(1 + f\left(z\right)\right) = z.$ Then $z + \ln \left(1 + z\right)$ can be expanded into a power series and inverted. This gives a series for $f\left(z+1\right) = W\left(e^\right)-1\text$ :$W\left(e^\right) = 1 + \frac + \frac - \frac - \frac + \frac - O\left(z^6\right).$ $W\left(x\right)$ can be computed by substituting $\ln x - 1$ for in the above series. For example, substituting for gives the value of $W\left(1\right) \approx 0.567143.$

## Binary trees

Consider the set $\mathcal$ of unlabelled binary trees. An element of $\mathcal$ is either a leaf of size zero, or a root node with two subtrees. Denote by $B_n$ the number of binary trees on $n$ nodes. Removing the root splits a binary tree into two trees of smaller size. This yields the functional equation on the generating function $\textstyle B\left(z\right) = \sum_^\infty B_n z^n\text$ :$B\left(z\right) = 1 + z B\left(z\right)^2.$ Letting $C\left(z\right) = B\left(z\right) - 1$, one has thus $C\left(z\right) = z \left(C\left(z\right)+1\right)^2.$ Applying the theorem with $\phi\left(w\right) = \left(w+1\right)^2$ yields :$B_n = \left[z^n\right] C\left(z\right) = \frac \left[w^\right] \left(w+1\right)^ = \frac \binom = \frac \binom.$ This shows that $B_n$ 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.