Editing Lagrange inversion theorem

{{short description|Formula for the Taylor series expansion of the inverse function of an analytic function}}
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 {{mvar|z}} is defined as a function of {{mvar|w}} by an equation of the form

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

where {{mvar|f}} is analytic at a point {{mvar|a}} and <math>f'(a)\neq 0.</math> Then it is possible to ''invert'' or ''solve'' the equation for {{mvar|w}}, expressing it in the form <math>w=g(z)</math> given by a [[power series]]<ref>{{cite book |editor=M. Abramowitz |editor2=I. A. Stegun |title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables |chapter=3.6.6. Lagrange's Expansion |place=New York |publisher=Dover |page=14 |year=1972 |url=http://people.math.sfu.ca/~cbm/aands/page_14.htm}}</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 {{mvar|z}} in a [[neighbourhood (mathematics)|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 {{math|''F''(''g''(''z''))}} for any analytic function {{mvar|F}}; and it can be generalized to the case <math>f'(a)=0,</math> where the inverse {{mvar|g}} is a multivalued function.

The theorem was proved by [[Joseph Louis Lagrange|Lagrange]]<ref>{{cite journal |author=Lagrange, Joseph-Louis |year=1770 |title=Nouvelle méthode pour résoudre les équations littérales par le moyen des séries |journal=Histoire de l'Académie Royale des Sciences et Belles-Lettres de Berlin |pages=251–326 |url=http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=02-hist/1768&seite:int=257}} 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: {{cite book |editor=Hindenburg, Carl Friedrich |title=Archiv der reinen und angewandten Mathematik |trans-title=Archive of pure and applied mathematics |location=Leipzig, Germany |publisher=Schäferischen Buchhandlung |year=1798 |volume=2 |chapter=Versuch einer vereinfachten Analysis; ein Auszug eines Auszuges von Herrn Bürmann |trans-chapter=Attempt at a simplified analysis; an extract of an abridgement by Mr. Bürmann |pages=495–499 |chapter-url=https://books.google.com/books?id=jj4DAAAAQAAJ&pg=495}}</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:  [http://gallica.bnf.fr/ark:/12148/bpt6k3217h.image.f22.langFR.pagination "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 [[polynomial]]s, so the theory of [[analytic function]]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 power series#Formal residue|formal residue]], and a more direct formal [[Formal power series#The Lagrange inversion formula|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>{{cite book | last1=Richard | first1=Stanley | title=Enumerative combinatorics. Volume 1. | series =Cambridge Stud. Adv. Math. | volume=49 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-1-107-60262-5 | mr=2868112 }}</ref><ref>{{Citation |last1=Ira|first1=Gessel |date=2016 |title=Lagrange inversion |journal=Journal of Combinatorial Theory, Series A |volume=144 |language=en |pages=212–249 |doi=10.1016/j.jcta.2016.06.018 |arxiv=1609.05988|mr=3534068}}</ref><ref>{{Citation |last1=Surya|first1=Erlang |last2=Warnke |first2=Lutz |date=2023 |title=Lagrange Inversion Formula by Induction |journal=The American Mathematical Monthly |volume=130 |issue=10 |language=en |pages=944–948 |doi=10.1080/00029890.2023.2251344 |arxiv=2305.17576|mr=4669236}}</ref>


If {{mvar|f}} is a formal power series, then the above formula does not give the coefficients of the compositional inverse series {{mvar|g}} directly in terms for the coefficients of the series {{mvar|f}}. If one can express the functions {{mvar|f}} and {{mvar|g}} 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 {{math|1=''f''<sub>0</sub> = 0}} and {{math|''f''<sub>1</sub> ≠ 0}}, then an explicit form of inverse coefficients can be given in term of [[Bell polynomial]]s:<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 {{math|1=''f''<sub>1</sub> = 1}}, the last formula can be interpreted in terms of the faces of [[Associahedron|associahedra]]  <ref>{{cite arXiv|eprint=1709.07504|class=math.CO|title=Hopf monoids and generalized permutahedra|last1=Aguiar|first1=Marcelo|last2=Ardila|first2=Federico|year=2017}}</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 {{mvar|p}}
:<math> x^p - x + z= 0</math>
can be solved for {{mvar|x}} by means of the Lagrange inversion formula for the function {{math|1=''f''(''x'') = ''x'' − ''x''<sup>''p''</sup>}},  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 {{mvar|f}} 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 {{mvar|w}}.

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 {{math|''H''}} is an arbitrary analytic function.

Sometimes, the derivative {{math|''{{prime|H}}''(''w'')}} can be quite complicated. A simpler version of the formula replaces {{math|''{{prime|H}}''(''w'')}} with {{math|''H''(''w'')(1 &minus; ''{{prime|φ}}''(''w'')/''φ''(''w''))}} to get 

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

which involves {{math|''{{prime|φ}}''(''w'')}} instead of {{math|''{{prime|H}}''(''w'')}}.

===Lambert ''W'' function===

{{main|Lambert W function}}
The Lambert {{mvar|W}} 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|<\sqrt{{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>{{cite conference |url=https://dl.acm.org/doi/pdf/10.1145/258726.258783 |title=A sequence of series for the Lambert W function |last1=Corless |first1=Robert M. |last2=Jeffrey |first2= David J.|author-link2=|last3=Knuth|first3=Donald E.|author-link3=Donald E. Knuth|date=July 1997 |book-title=Proceedings of the 1997 international symposium on Symbolic and algebraic computation |pages=197&ndash;204|doi=10.1145/258726.258783 |url-access=subscription }}</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 {{mvar|z}} in the above series. For example, substituting {{math|−1}} for {{mvar|z}} gives the value of <math>W(1) \approx 0.567143.</math>

===Binary trees===

Consider<ref>{{cite book |last1=Harris|first1= John |last2=Hirst |first2= Jeffry L.| last3= Mossinghoff| first3= Michael |date=2008 |title=Combinatorics and Graph Theory  |publisher= Springer |pages=185–189 |isbn=978-0387797113}}</ref> the set <math>\mathcal{B}</math> of unlabelled [[binary tree]]s. 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 {{mvar|n}}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 formula]]

==References==
{{reflist|colwidth=30em}}

==External links==
*{{MathWorld |urlname=BuermannsTheorem |title=Bürmann's Theorem}}
*{{MathWorld |urlname=SeriesReversion |title=Series Reversion}}
*[http://www.encyclopediaofmath.org/index.php/B%C3%BCrmann%E2%80%93Lagrange_series Bürmann–Lagrange series] at [[Encyclopedia of Mathematics|Springer EOM]]

[[Category:Inverse functions]]
[[Category:Theorems in real analysis]]
[[Category:Theorems in complex analysis]]
[[Category:Theorems in combinatorics]]