mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

expansion of the

inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...

of an

analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...

. Lagrange inversion is a special case of the

inverse function theorem In mathematics, the inverse function theorem is a theorem that asserts that, if a real function ''f'' has a continuous derivative near a point where its derivative is nonzero, then, near this point, ''f'' has an inverse function. The inverse fu ...

Statement

Suppose is defined as a function of by an equation of the form :

z = f(w)

where is analytic at a point and

f'(a)\neq 0.

Then it is possible to ''invert'' or ''solve'' the equation for , expressing it in the form

w=g(z)

given by a

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

g(z) = a + \sum_^ g_n \frac,

where :

g_n = \lim_ \frac \left left( \frac \right)^n \right

The theorem further states that this series has a non-zero radius of convergence, i.e.,

g(z)

represents an analytic function of in a

neighbourhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...

z= f(a).

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, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...

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'(a)=0,

where the inverse is a multivalued function. The theorem was proved by

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaHans 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 that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

and

contour integration In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the Residue theorem, calculus of residues, a method of co ...

; the complex formal power series version is a consequence of knowing the formula for

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s, so the theory of

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 most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

is available. In fact, the Lagrange inversion theorem has a number of additional rather different proofs, including ones using tree-counting arguments or induction. 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(w) = \sum_^\infty f_k \frac \qquad \text  \qquad  g(z) = \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_^ (-1)^k n^\overline B_(\hat_1,\hat_2,\ldots,\hat_), \quad n \geq 2,

where :

\begin
  \hat_k &= \frac, \\
   g_1 &= \frac, \text \\
   n^ &= n(n+1)\cdots (n+k-1)
 \end

is the rising factorial. When , the last formula can be interpreted in terms of the faces of associahedra :

g_n = \sum_ (-1)^ 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 (p-1)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 Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

and applies when

f(w)=w/\phi(w)

for some analytic

\phi(w)

with

\phi(0)\ne 0.

Take

a=0

to obtain

f(a)=f(0)=0.

Then for the inverse

g(z)

(satisfying

f(g(z))\equiv z

), we have :

z^n, \end

which can be written alternatively as :

\phi(w)^n,

where

^r /math> 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:
:^n H (g(z)) = \frac^(H' (w) \phi(w)^n) where  is an arbitrary analytic function.

Sometimes, the derivative  can be quite complicated. A simpler version of the formula replaces  with  to get 

:^n H (g(z)) =^n H(w) \phi(w)^ (\phi(w) - w \phi'(w)), which involves  instead of .

Lambert ''W'' function

The Lambert function is the function

W(z)

that is implicitly defined by the equation :

W(z) e^ = z.

We may use the theorem to compute the

W(z)

z=0.

We take

f(w) = we^w

and

a = 0.

Recognizing that :

\frac e^ = \alpha^n e^,

this gives :

\frac \\ &= \sum_^ (-n)^ \frac \\ &= z-z^2+\fracz^3-\fracz^4+O(z^5). \end

The

radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest Disk (mathematics), disk at the Power series, center of the series in which the series Convergent series, converges. It is either a non-negative real number o ...

of this series is

e^

(giving the principal branch of the Lambert function). A series that converges for

, \ln(z)-1, <\sqrt

(approximately

0.0655 < z < 112.63

) can also be derived by series inversion. The function

f(z) = W(e^z) - 1

satisfies the equation :

1 + f(z) + \ln (1 + f(z)) = z.

Then

z + \ln (1 + z)

can be expanded into a power series and inverted. This gives a series for

f(z+1) = W(e^)-1\text

W(e^) = 1 + \frac + \frac - \frac - \frac + \frac -  O(z^6).

W(x)

can be computed by substituting

\ln x - 1

for in the above series. For example, substituting for gives the value of

W(1) \approx 0.567143.

Binary trees

Consider the set

\mathcal

of unlabelled

binary tree In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the ''left child'' and the ''right child''. That is, it is a ''k''-ary tree with . A recursive definition using set theor ...

s. 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(z) = \sum_^\infty B_n z^n\text

B(z) = 1 + z B(z)^2.

Letting

C(z) = B(z) - 1

, one has thus

C(z) = z (C(z)+1)^2.

Applying the theorem with

\phi(w) = (w+1)^2

yields :

(w+1)^ = \frac \binom = \frac \binom.

This shows that

B_n

is the th

Catalan number The Catalan numbers are a sequence of natural numbers that occur in various Enumeration, counting problems, often involving recursion, recursively defined objects. They are named after Eugène Charles Catalan, Eugène Catalan, though they were p ...

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.

References

External links

* *{{MathWorld , urlname=SeriesReversion , title=Series Reversion
Bürmann–Lagrange series
at Springer EOM Inverse functions Theorems in real analysis Theorems in complex analysis Theorems in combinatorics