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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 (m ...
, 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 serie ...
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 X\t ...
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 an ...
.


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 : 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 (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of 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 sum ...
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 and generalized by
Hans Heinrich Bürmann Hans Heinrich Bürmann (died 21 June 1817, in Mannheim) was a German mathematician and teacher. He ran an "academy of commerce" in Mannheim since 1795 where he used to teach mathematics. He also served as a censor in Mannheim. He was nominated He ...
, 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 Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
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 calculus of residues, a method of complex analysis. ...
; the complex formal power series version is a consequence of knowing the formula for
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s, so the theory of
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 an ...
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 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(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 polynomial In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno's fo ...
s: : g_n = \frac \sum_^ (-1)^k n^ 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 In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...
. 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.


Sketch of the proof

For simplicity suppose z=0=f(w=0). We can then compute : \oint_ \frac \frac = \oint_ \frac \frac = \frac = g'(f(w)) = g'(z) . If we expand the integrand using the geometric series we get : \oint_ \frac \frac = \sum_^\infty z^n \oint_ \frac \frac = \sum_^\infty z^n \oint_ \frac \frac \left(\frac\right)^ = \sum_^\infty \frac \left. \frac\left(\frac\right)^ \_ , where in the last step we used the fact that f(w) has one simple zero. Finally we can integrate over z taking into account g(0)=0 : g'(z) = \sum_^\infty \frac \left. \frac\left(\frac\right)^ \_ ~~\Longrightarrow~~ g(z) = \sum_^\infty \frac \left. \frac\left(\frac\right)^ \_ . 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 Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
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 :\begin g(z) &= \sum_^ \left \lim_ \frac \left(\left( \frac \right)^n \right)\right\frac \\ &= \sum_^ \frac \left frac \lim_ \frac (\phi(w)^n) \rightz^n, \end which can be written alternatively as : ^ng(z) = \frac ^\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: : ^nH (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 : ^nH (g(z)) = ^nH(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
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 serie ...
of W(z) at z=0. We take f(w) = we^w and a = 0. Recognizing that :\frac e^ = \alpha^n e^, this gives :\begin W(z) &= \sum_^ \left lim_ \frac e^ \right\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 at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
of this series is e^ (giving the
principal branch In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane. Examples Trigonometric inverses Principal branches are used ...
of the Lambert function). A series that converges for larger (though not for all ) 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 k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary t ...
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 : B_n = ^nC(z) = \frac ^(w+1)^ = \frac \binom = \frac \binom. This shows that B_n is the th
Catalan number In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Cata ...
.


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.


See also

*
Faà di Bruno's formula Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after , although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French ...
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 ''n''th derivative of a composite function. *
Lagrange reversion theorem In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions. Let ''v'' be a function of ''x'' and ''y'' in terms of another fu ...
for another theorem sometimes called the inversion theorem * Formal power series#The Lagrange inversion formula


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