Lagrange inversion theorem

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Template:Short description Template:For 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. Lagrange inversion is a special case of the inverse function theorem.

Statement

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

<math>z = f(w)</math>

where Template:Mvar is analytic at a point Template:Mvar and <math>f'(a)\neq 0.</math> Then it is possible to invert or solve the equation for Template:Mvar, expressing it in the form <math>w=g(z)</math> given by a power series<ref>Template:Cite book</ref>

<math> g(z) = a + \sum_{n=1}^{\infty} g_n \frac{(z - f(a))^n}{n!}, </math>

where

<math> g_n = \lim_{w \to a} \frac{d^{n-1}}{dw^{n-1}} \left[\left( \frac{w-a}{f(w) - f(a)} \right)^n \right]. </math>

The theorem further states that this series has a non-zero radius of convergence, i.e., <math>g(z)</math> represents an analytic function of Template:Mvar in a neighbourhood of <math>z= f(a).</math> 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 Template:Math for any analytic function Template:Mvar; and it can be generalized to the case <math>f'(a)=0,</math> where the inverse Template:Mvar is a multivalued function.

The theorem was proved by Lagrange<ref>Template:Cite journal https://archive.org/details/uvresdelagrange18natigoog/page/n13 (Note: Although Lagrange submitted this article in 1768, it was not published until 1770.)</ref> and generalized by Hans Heinrich Bürmann,<ref>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: Template:Cite book</ref><ref>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.)</ref><ref>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).</ref> both in the late 18th century. There is a straightforward derivation using complex analysis and contour integration;<ref>E. T. Whittaker and G. N. Watson. A Course of Modern Analysis. Cambridge University Press; 4th edition (January 2, 1927), pp. 129–130</ref> 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. In fact, the Lagrange inversion theorem has a number of additional rather different proofs, including ones using tree-counting arguments or induction.<ref>Template:Cite book</ref><ref>Template:Citation</ref><ref>Template:Citation</ref>

If Template:Mvar is a formal power series, then the above formula does not give the coefficients of the compositional inverse series Template:Mvar directly in terms for the coefficients of the series Template:Mvar. If one can express the functions Template:Mvar and Template:Mvar in formal power series as

<math>f(w) = \sum_{k=0}^\infty f_k \frac{w^k}{k!} \qquad \text{and} \qquad g(z) = \sum_{k=0}^\infty g_k \frac{z^k}{k!}</math>

with Template:Math and Template:Math, then an explicit form of inverse coefficients can be given in term of Bell polynomials:<ref>Eqn (11.43), p. 437, C.A. Charalambides, Enumerative Combinatorics, Chapman & Hall / CRC, 2002</ref>

<math> g_n = \frac{1}{f_1^n} \sum_{k=1}^{n-1} (-1)^k n^\overline{k} B_{n-1,k}(\hat{f}_1,\hat{f}_2,\ldots,\hat{f}_{n-k}), \quad n \geq 2, </math>

where

<math>\begin{align}
 \hat{f}_k &= \frac{f_{k+1}}{(k+1)f_{1}}, \\
  g_1 &= \frac{1}{f_{1}}, \text{ and} \\
  n^{\overline{k}} &= n(n+1)\cdots (n+k-1)
\end{align}</math>

is the rising factorial.

When Template:Math, the last formula can be interpreted in terms of the faces of associahedra <ref>Template:Cite arXiv</ref>

<math> g_n = \sum_{F \text{ face of } K_n} (-1)^{n-\dim F} f_F , \quad n \geq 2, </math>

where <math> f_{F} = f_{i_{1}} \cdots f_{i_{m}} </math> for each face <math> F = K_{i_1} \times \cdots \times K_{i_m} </math> of the associahedron <math> K_n .</math>

Example

For instance, the algebraic equation of degree Template:Mvar

<math> x^p - x + z= 0</math>

can be solved for Template:Mvar by means of the Lagrange inversion formula for the function Template:Math, resulting in a formal series solution

<math> x = \sum_{k=0}^\infty \binom{pk}{k} \frac{z^{(p-1)k+1} }{(p-1)k+1} . </math>

By convergence tests, this series is in fact convergent for <math>|z| \leq (p-1)p^{-p/(p-1)},</math> which is also the largest disk in which a local inverse to Template:Mvar can be defined.

Applications

Lagrange–Bürmann formula

There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when <math>f(w)=w/\phi(w)</math> for some analytic <math>\phi(w)</math> with <math>\phi(0)\ne 0.</math> Take <math>a=0</math> to obtain <math>f(a)=f(0)=0.</math> Then for the inverse <math>g(z)</math> (satisfying <math>f(g(z))\equiv z</math>), we have

<math>\begin{align}
 g(z) &= \sum_{n=1}^{\infty} \left[ \lim_{w \to 0} \frac {d^{n-1}}{dw^{n-1}} \left(\left( \frac{w}{w/\phi(w)} \right)^n \right)\right] \frac{z^n}{n!} \\
 {} &= \sum_{n=1}^{\infty} \frac{1}{n} \left[\frac{1}{(n-1)!} \lim_{w \to 0} \frac{d^{n-1}}{dw^{n-1}} (\phi(w)^n) \right] z^n,
\end{align}</math>

which can be written alternatively as

<math>[z^n] g(z) = \frac{1}{n} [w^{n-1}] \phi(w)^n,</math>

where <math>[w^r]</math> is an operator which extracts the coefficient of <math>w^r</math> in the Taylor series of a function of Template:Mvar.

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

<math>[z^n] H (g(z)) = \frac{1}{n} [w^{n-1}] (H' (w) \phi(w)^n)</math>

where Template:Math is an arbitrary analytic function.

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

<math> [z^n] H (g(z)) = [w^n] H(w) \phi(w)^{n-1} (\phi(w) - w \phi'(w)), </math>

which involves Template:Math instead of Template:Math.

Lambert W function

Template:Main The Lambert Template:Mvar function is the function <math>W(z)</math> that is implicitly defined by the equation

<math> W(z) e^{W(z)} = z.</math>

We may use the theorem to compute the Taylor series of <math>W(z)</math> at <math>z=0.</math> We take <math>f(w) = we^w</math> and <math>a = 0.</math> Recognizing that

<math>\frac{d^n}{dx^n} e^{\alpha x} = \alpha^n e^{\alpha x},</math>

this gives

<math>\begin{align}
  W(z) &= \sum_{n=1}^{\infty} \left[\lim_{w \to 0} \frac{d^{n-1}}{dw^{n-1}} e^{-nw} \right] \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).
\end{align}</math>

The radius of convergence of this series is <math>e^{-1}</math> (giving the principal branch of the Lambert function).

A series that converges for <math>|\ln(z)-1|<\sqrtTemplate:4+\pi^2</math> (approximately <math>0.0655 < z < 112.63</math>) can also be derived by series inversion. The function <math>f(z) = W(e^z) - 1</math> satisfies the equation

<math>1 + f(z) + \ln (1 + f(z)) = z.</math>

Then <math>z + \ln (1 + z)</math> can be expanded into a power series and inverted.<ref>Template:Cite conference</ref> This gives a series for <math>f(z+1) = W(e^{z+1})-1\text{:}</math>

<math>W(e^{1+z}) = 1 + \frac{z}{2} + \frac{z^2}{16} - \frac{z^3}{192} - \frac{z^4}{3072} + \frac{13 z^5}{61440} - O(z^6).</math>

<math>W(x)</math> can be computed by substituting <math>\ln x - 1</math> for Template:Mvar in the above series. For example, substituting Template:Math for Template:Mvar gives the value of <math>W(1) \approx 0.567143.</math>

Binary trees

Consider<ref>Template:Cite book</ref> the set <math>\mathcal{B}</math> of unlabelled binary trees. An element of <math>\mathcal{B}</math> is either a leaf of size zero, or a root node with two subtrees. Denote by <math>B_n</math> the number of binary trees on <math>n</math> nodes.

Removing the root splits a binary tree into two trees of smaller size. This yields the functional equation on the generating function <math>\textstyle B(z) = \sum_{n=0}^\infty B_n z^n\text{:}</math>

<math>B(z) = 1 + z B(z)^2.</math>

Letting <math>C(z) = B(z) - 1</math>, one has thus <math>C(z) = z (C(z)+1)^2.</math> Applying the theorem with <math>\phi(w) = (w+1)^2</math> yields

<math> B_n = [z^n] C(z) = \frac{1}{n} [w^{n-1}] (w+1)^{2n} = \frac{1}{n} \binom{2n}{n-1} = \frac{1}{n+1} \binom{2n}{n}.</math>

This shows that <math>B_n</math> is the Template:Mvarth 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

References

Template:Reflist

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