Lagrange inversion theorem

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.

StatementEdit

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

 

where f is analytic at a point a and   Then it is possible to invert or solve the equation for w, expressing it in the form   given by a power series[1]

 

where

 

The theorem further states that this series has a non-zero radius of convergence, i.e.,   represents an analytic function of z in a neighbourhood of   This is also called reversion of series.

If the assertions about analyticity are omitted, 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   where the inverse g is a multivalued function.

The theorem was proved by Lagrange[2] and generalized by Hans Heinrich Bürmann,[3][4][5] both in the late 18th century. There is a straightforward derivation using complex analysis and contour integration;[6] the complex formal power series version is a consequence of knowing the formula for polynomials, so the theory of analytic functions 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 f is a formal power series, then the above formula does not give the coefficients of the compositional inverse series g directly in terms for the coefficients of the series f. If one can express the functions f and g in formal power series as

 

with f0 = 0 and f1 ≠ 0, then an explicit form of inverse coefficients can be given in term of Bell polynomials:[7]

 

where

 

is the rising factorial.

When f1 = 1, the last formula can be interpreted in terms of the faces of associahedra [8]

 

where   for each face   of the associahedron  

ExampleEdit

For instance, the algebraic equation of degree p

 

can be solved for x by means of the Lagrange inversion formula for the function f(x) = xxp, resulting in a formal series solution

 

By convergence tests, this series is in fact convergent for   which is also the largest disk in which a local inverse to f can be defined.

ApplicationsEdit

Lagrange–Bürmann formulaEdit

There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when   for some analytic   with   Take   to obtain   Then for the inverse   (satisfying  ), we have

 

which can be written alternatively as

 

where   is an operator which extracts the coefficient of   in the Taylor series of a function of w.

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

 

where H is an arbitrary analytic function.

Sometimes, the derivative H(w) can be quite complicated. A simpler version of the formula replaces H(w) with H(w)(1 − φ(w)/φ(w)) to get

 

which involves φ(w) instead of H(w).

Lambert W functionEdit

The Lambert W function is the function   that is implicitly defined by the equation

 

We may use the theorem to compute the Taylor series of   at   We take   and   Recognizing that

 

this gives

 

The radius of convergence of this series is   (giving the principal branch of the Lambert function).

A series that converges for larger z (though not for all z) can also be derived by series inversion. The function   satisfies the equation

 

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

 

  can be computed by substituting   for z in the above series. For example, substituting −1 for z gives the value of  

Binary treesEdit

Consider the set   of unlabelled binary trees. An element of   is either a leaf of size zero, or a root node with two subtrees. Denote by   the number of binary trees on 'nodes.

Removing the root splits a binary tree into two trees of smaller size. This yields the functional equation on the generating function  

 

Letting  , one has thus   Applying the theorem with   yields

 

This shows that   is the nth Catalan number.

Asymptotic approximation of integralsEdit

In the Laplace-Erdelyi theorem that gives the asymptotic approximation for Laplace-type integrals, the function inversion is taken as a crucial step.

See alsoEdit

ReferencesEdit

  1. ^ M. Abramowitz; I. A. Stegun, eds. (1972). "3.6.6. Lagrange's Expansion". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. p. 14.
  2. ^ Lagrange, Joseph-Louis (1770). "Nouvelle méthode pour résoudre les équations littérales par le moyen des séries". Histoire de l'Académie Royale des Sciences et Belles-Lettres de Berlin: 251–326. https://archive.org/details/uvresdelagrange18natigoog/page/n13 (Note: Although Lagrange submitted this article in 1768, it was not published until 1770.)
  3. ^ 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.
  4. ^ 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.)
  5. ^ A report on Bürmann's theorem by Joseph-Louis Lagrange and Adrien-Marie Legendre appears in: "Rapport sur deux mémoires d'analyse du professeur Burmann," Mémoires de l'Institut National des Sciences et Arts: Sciences Mathématiques et Physiques, vol. 2, pages 13–17 (1799).
  6. ^ E. T. Whittaker and G. N. Watson. A Course of Modern Analysis. Cambridge University Press; 4th edition (January 2, 1927), pp. 129–130
  7. ^ Eqn (11.43), p. 437, C.A. Charalambides, Enumerative Combinatorics, Chapman & Hall / CRC, 2002
  8. ^ Aguiar, Marcelo; Ardila, Federico (2017). "Hopf monoids and generalized permutahedra". arXiv:1709.07504 [math.CO].

External linksEdit