In

Bürmann–Lagrange series

at Encyclopedia of Mathematics, Springer EOM Inverse functions Theorems in real analysis Theorems in complex analysis Theorems in combinatorics

mathematical analysis
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, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series
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expansion of the inverse function
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of an analytic function
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.
Statement

Suppose is defined as a function of by an equation of the form :$z\; =\; f(w)$ where is analytic at a point and $f\text{'}(a)\backslash neq\; 0.$ Then it is possible to ''invert'' or ''solve'' the equation for , expressing it in the form $w=g(z)$ given by a power series :$g(z)\; =\; a\; +\; \backslash sum\_^\; g\_n\; \backslash frac,$ where :$g\_n\; =\; \backslash lim\_\; \backslash frac\; \backslash left;\; href="/html/ALL/s/left(\_\backslash frac\_\backslash right)^n\_\backslash right.html"\; ;"title="left(\; \backslash frac\; \backslash right)^n\; \backslash right">left(\; \backslash frac\; \backslash right)^n\; \backslash right$ The theorem further states that this series has a non-zero radius of convergence, i.e., $g(z)$ represents an analytic function of in aneighbourhood
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 $z=\; f(a).$ This is also called reversion of series.
If the assertions about analyticity are omitted, the formula is also valid for formal power series
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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 $f\text{'}(a)=0,$ 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
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and generalized by Hans Heinrich Bürmann, both in the late 18th century. There is a straightforward derivation using complex analysis
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and contour integration
In the mathematical field of complex analysis
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; the complex formal power series version is a consequence of knowing the formula for polynomial
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s, so the theory of analytic function
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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
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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
:$f(w)\; =\; \backslash sum\_^\backslash infty\; f\_k\; \backslash frac\; \backslash qquad\; \backslash text\; \backslash qquad\; g(z)\; =\; \backslash sum\_^\backslash infty\; g\_k\; \backslash frac$
with and , then an explicit form of inverse coefficients can be given in term of Bell polynomials:
:$g\_n\; =\; \backslash frac\; \backslash sum\_^\; (-1)^k\; n^\; B\_(\backslash hat\_1,\backslash hat\_2,\backslash ldots,\backslash hat\_),\; \backslash quad\; n\; \backslash geq\; 2,$
where
:$\backslash begin\; \backslash hat\_k\; \&=\; \backslash frac,\; \backslash \backslash \; g\_1\; \&=\; \backslash frac,\; \backslash text\; \backslash \backslash \; n^\; \&=\; n(n+1)\backslash cdots\; (n+k-1)\; \backslash end$
is the rising factorial.
When , the last formula can be interpreted in terms of the faces of Associahedron, associahedra
:$g\_n\; =\; \backslash sum\_\; (-1)^\; f\_F\; ,\; \backslash quad\; n\; \backslash geq\; 2,$
where $f\_\; =\; f\_\; \backslash cdots\; f\_$ for each face $F\; =\; K\_\; \backslash times\; \backslash cdots\; \backslash times\; K\_$ 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 :$x\; =\; \backslash sum\_^\backslash infty\; \backslash binom\; \backslash frac\; .$ By convergence tests, this series is in fact convergent for $,\; z,\; \backslash leq\; (p-1)p^,$ 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 and applies when $f(w)=w/\backslash phi(w)$ for some analytic $\backslash phi(w)$ with $\backslash phi(0)\backslash ne\; 0.$ Take $a=0$ to obtain $f(a)=f(0)=0.$ Then for the inverse $g(z)$ (satisfying $f(g(z))\backslash equiv\; z$), we have :$\backslash begin\; g(z)\; \&=\; \backslash sum\_^\; \backslash left[\; \backslash lim\_\; \backslash frac\; \backslash left(\backslash left(\; \backslash frac\; \backslash right)^n\; \backslash right)\backslash right]\; \backslash frac\; \backslash \backslash \; \&=\; \backslash sum\_^\; \backslash frac\; \backslash left[\backslash frac\; \backslash lim\_\; \backslash frac\; (\backslash phi(w)^n)\; \backslash right]\; z^n,\; \backslash end$ which can be written alternatively as :$[z^n]\; g(z)\; =\; \backslash frac\; [w^]\; \backslash phi(w)^n,$ where $[w^r]$ 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: :$[z^n]\; H\; (g(z))\; =\; \backslash frac\; [w^]\; (H\text{'}\; (w)\; \backslash phi(w)^n)$ where is an arbitrary analytic function. Sometimes, the derivative can be quite complicated. A simpler version of the formula replaces with to get :$[z^n]\; H\; (g(z))\; =\; [w^n]\; H(w)\; \backslash phi(w)^\; (\backslash phi(w)\; -\; w\; \backslash phi\text{'}(w)),$ 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 theTaylor series
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of $W(z)$ at $z=0.$ We take $f(w)\; =\; we^w$ and $a\; =\; 0.$ Recognizing that
:$\backslash frac\; e^\; =\; \backslash alpha^n\; e^,$
this gives
:$\backslash begin\; W(z)\; \&=\; \backslash sum\_^\; \backslash left[\backslash lim\_\; \backslash frac\; e^\; \backslash right]\; \backslash frac\; \backslash \backslash \; \&=\; \backslash sum\_^\; (-n)^\; \backslash frac\; \backslash \backslash \; \&=\; z-z^2+\backslash fracz^3-\backslash fracz^4+O(z^5).\; \backslash end$
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 $f(z)\; =\; W(e^z)\; -\; 1$ satisfies the equation
:$1\; +\; f(z)\; +\; \backslash ln\; (1\; +\; f(z))\; =\; z.$
Then $z\; +\; \backslash ln\; (1\; +\; z)$ can be expanded into a power series and inverted. This gives a series for $f(z+1)\; =\; W(e^)-1\backslash text$
:$W(e^)\; =\; 1\; +\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; O(z^6).$
$W(x)$ can be computed by substituting $\backslash ln\; x\; -\; 1$ for in the above series. For example, substituting for gives the value of $W(1)\; \backslash approx\; 0.567143.$
Binary trees

Consider the set $\backslash mathcal$ of unlabelled binary trees. An element of $\backslash mathcal$ 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 $\backslash textstyle\; B(z)\; =\; \backslash sum\_^\backslash infty\; B\_n\; z^n\backslash text$ :$B(z)\; =\; 1\; +\; z\; B(z)^2.$ Letting $C(z)\; =\; B(z)\; -\; 1$, one has thus $C(z)\; =\; z\; (C(z)+1)^2.$ Applying the theorem with $\backslash phi(w)\; =\; (w+1)^2$ yields :$B\_n\; =\; [z^n]\; C(z)\; =\; \backslash frac\; [w^]\; (w+1)^\; =\; \backslash frac\; \backslash binom\; =\; \backslash frac\; \backslash binom.$ 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.See also

*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 ''n''th derivative of a composite function. *Lagrange reversion theorem for another theorem sometimes called the inversion theorem *Formal power series#The Lagrange inversion formulaReferences

External links

* *{{MathWorld , urlname=SeriesReversion , title=Series ReversionBürmann–Lagrange series

at Encyclopedia of Mathematics, Springer EOM Inverse functions Theorems in real analysis Theorems in complex analysis Theorems in combinatorics