Lagrange inversion theorem: Information from Answers.com

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.

1 Theorem statement
2 Applications
3 See also
4 References
5 External links

Theorem statement

Suppose the dependence between the variables w and z is implicitly defined by an equation of the form

$f(w) = z\,$

where f is analytic at a point a and f '(a) ≠ 0. Then it is possible to invert or solve the equation for w:

$w = g(z)\,$

where g is analytic at the point b = f(a). This is also called reversion of series.

The series expansion of g is given by

$g(z) = a + \sum_{n=1}^{\infty} \left( \lim_{w \to a}\left( \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}w^{\,n-1}} \left( \frac{w-a}{f(w) - b} \right)^n\right) {\frac{(z - b)^n}{n!}} \right).$

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 F(g(z)) for any analytic function F, and it can be generalized to the case f '(a) = 0, where the inverse g is a multivalued function.

The theorem was proved by Lagrange^[1] and generalized by Hans Heinrich Bürmann,^[2]^[3] both in the late 18th century. There is a straightforward derivation using complex analysis and contour integration (the complex formal power series version is clearly a consequence of knowing the formula for polynomials, so the theory of analytic functions may be applied).

Applications

Lagrange–Bürmann formula

There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when $f (w) = w / φ(w)$ and $\phi(0)\ne 0.$ Take $a = 0$ to obtain $b = f (0) = 0.$ We have

$g(z) = \sum_{n=1}^{\infty} \left. \frac{\mathrm{d}^{n-1}}{\mathrm{d}w^{n-1}} \left( \frac{w}{w/\phi(w)} \right)^n \right| _{w\,=\,0} \frac{z^n}{n!}$

$= \sum_{n=1}^{\infty} \frac{1}{n} \left( \left. \frac{1}{(n-1)!} \frac{\mathrm{d}^{n-1}}{\mathrm{d}w^{n-1}} \phi(w)^n \right| _{w = 0} \right) z^n,$

which can be written alternatively as

$[z^n] g(z) = \frac{1}{n} [w^{n-1}] \phi(w)^n,$

where $[w r]$ is an operator which extracts the coefficient of $w r$ in what follows it.

A useful generalization of the formula is known as the Lagrange–Bürmann formula:

$[z^{n+1}] H (g(z)) = \frac{1}{(n+1)} [w^n] (H' (w) \phi(w)^{n+1})$

where $H$ can be an arbitrary analytic function, e.g. $H (w) = w k$ .

Lambert W function

The Lambert W function is the function $W (z)$ that satisfies the implicit equation

$W(z) e^{W(z)} = z\,$

We may use the theorem to compute the Taylor series of $W (z)$ at $z = 0.$ We take $f (w) = w e w$ and $a = b = 0.$ Recognizing that

$\frac{\mathrm{d}^n}{\mathrm{d}x^n}\ \mathrm{e}^{\alpha\,x}\,=\,\alpha^n\,\mathrm{e}^{\alpha\,x}$

this gives

$W(z) = \sum_{n=1}^{\infty} \left. \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}w^{\,n-1}}\ \mathrm{e}^{-nw} \right|_{w\,=\,0} {\frac{z^n}{n!}}\,=\, \sum_{n=1}^{\infty} (-n)^{n-1}\, \frac{z^n}{n!}=z-z^2+\frac{3}{2}z^3-\frac{8}{3}z^4+O(z^5).$

The radius of convergence of this series is $e - 1$ (this example refers to the principal branch of the Lambert function).

Binary trees

Consider the set $\mathcal{B}$ of unlabelled binary trees. An element of $\mathcal{B}$ is either a leaf of size zero, or a root node with two subtrees (planar, i.e. no symmetry between them). The fundamental theorem of combinatorial enumeration (unlabelled case) applies.

The group acting on the two subtrees is $E 2$ , which contains a single permutation consisting of two fixed points. The set $\mathcal{B}$ satisfies

$\mathcal{B} = 1 + \mathcal{Z}\mathfrak{S}_2(\mathcal{B}).$

This yields the functional equation of the OGF $B (z)$ by the number of internal nodes:

$B(z) = 1 + z B(z)^2 \mbox{ or } z = \frac{B(z)-1}{B(z)^2}.$

Let $B_{\ge 1}(z) = B(z) - 1 = \frac{1-2z-\sqrt{1-4z}}{2z}$ to obtain

$z = \frac{B_{\ge 1}(z)}{(B_{\ge 1}(z)+1)^2}.$

Now apply the theorem with $φ(w) = (w + 1) 2 :$

$[z^n] B_{\ge 1}(z) = \frac{1}{n} [w^{n-1}] (w+1)^{2n} = \frac{1}{n} {2n \choose n-1} = \frac{1}{n+1} {2n \choose n}$

the Catalan numbers.

References

^ Lagrange, Joseph-Louis (1768). "Nouvelle méthode pour résoudre les équations littérales par le moyen des séries". Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin 24: 251-326. http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=41070.
^ Bürmann, Hans Heinrich, “Essai de calcul fonctionnaire aux constantes ad-libitum,” submitted in 1796 to the Institut National de France. For a summary of this article, see: Hindenburg, Carl Friedrich, ed (1798). "Versuch einer vereinfachten Analysis; ein Auszug eines Auszuges von Herrn Bürmann [Attempt at a simplified analysis; an extract of an abridgement by Mr. Bürmann]". Archiv der reinen und angewandten Mathematik [Archive of pure and applied mathematics]. 2. Leipzig, Germany: Schäferischen Buchhandlung. pp. 495-499. http://books.google.com/books?id=jj4DAAAAQAAJ&pg=RA1-PA495.
^ Bürmann, Hans Heinrich, "Formules du développement, de retour et d'integration," submitted to the Institut National de France. Bürmann's manuscript survives in the archives of the École Nationale des Ponts et Chaussées [National School of Bridges and Roads] in Paris. (See ms. 1715.)

External links

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Lagrange inversion theorem

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