Lagrange inversion theorem

Template:Short description 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

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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

• 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 nth derivative of a composite function.
• Lagrange reversion theorem for another theorem sometimes called the inversion theorem
• Formal power series#The Lagrange inversion formula

External links

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:BuermannsTheorem%7CBuermannsTheorem.html}} |title = Bürmann's Theorem |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:SeriesReversion%7CSeriesReversion.html}} |title = Series Reversion |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• Bürmann–Lagrange series at Springer EOM