Lagrange inversion theorem: Difference between revisions

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{{shortShort description|Formula for theinverting a Taylor series expansion of the inverse function of an analytic function}}

{{for|the formal power series expansion of certain implicitly defined functions|Lagrange reversion 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]]. Lagrange inversion is a special case of the [[inverse function theorem]].

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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.

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

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==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''^''p''}}, resulting in a formal series solution

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:<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.

==Derivation==

We can use Cauchy Integral theorem:

<math>

f^{-1}(z)

=

\frac{1}{2\pi i} \oint_{f(C)} \frac{f^{-1}(\xi)}{\xi - z}d\xi

</math>

and substitute:

<math>

\xi=f(\omega)

</math>

<math>

d\xi=f'(\omega)d\omega

</math>

<math>

f(C)\rightarrow C

</math>

<math>

f^{-1}(z)

=

\frac{1}{2\pi i} \oint_{C} \frac{\omega}{f(\omega) - z} f'(\omega) d\omega

</math>

using geometric series:

<math>

\frac{1}{f(\omega) - z}

=

\frac{1}{f(\omega) - f(a) - z + f(a) }

=

\frac{1}{f(\omega) - f(a) }\frac{1}{1 - \frac{z - f(a) }{f(\omega) - f(a)} }

=

\frac{1}{f(\omega) - f(a) }\sum_{n=0}^\infty

\left(\frac{z - f(a) }{f(\omega) - f(a)}\right)^{n}

</math>

<math>

f^{-1}(z)

=

\frac{1}{2\pi i}\sum_{n=0}^\infty

\left({z - f(a) }\right)^{n}

\oint_{C} \frac{ \omega f'(\omega)}{(f(\omega) - f(a))^{n+1} } d\omega

</math>

now by integration by parts: <math>

u

=

\omega

</math> and <math>

dv

=

\frac{f'(\omega) d\omega}{(f(\omega) - f(a))^{n+1} }

</math> where <math>

uv

=

\frac{-1 }{n} \frac{e^{i\theta} }{(f(e^{i\theta} ) - f(a))^{n}}\Biggl|_{0}^{2\pi }

=

0

</math> we get:

<math>

f^{-1}(z)

=

\frac{1}{2\pi i}\sum_{n=0}^\infty

\frac{({z - f(a) })^{n}}{n}

\oint_{C} \frac{1}{(f(\omega) - f(a))^{n} } d\omega

</math>

by residue theorem:

<math>

f^{-1}(z)

=

\sum_{n=0}^\infty

\frac{({z - f(a) })^{n}}{n}

\operatorname{Res}(\frac{1}{(f(\omega) - f(a))^{n} }, w=a)

</math>

finally:

<math>

f^{-1}(z)

=

\sum_{n=0}^\infty

\frac{({z - f(a) })^{n}}{n!}

\lim_{ \omega \to a} \frac{d^{n-1}}{d\omega ^{n-1}} \left(\frac{\omega - a }{f(\omega) - f(a)}\right)^{n}

</math>

''In fact, integration by parts doesn't work for:''

<math>

n=0\rightarrow

\frac{1}{2\pi i}

\oint_{C} \frac{ \omega f'(\omega)}{ f(\omega) - f(a) } d\omega =

\frac{1}{2\pi i}

\oint_{f(C)} \frac{ f^{-1}(u)}{ u - f(a) } du=

f^{-1}(f(a))=a

</math>

''Fortunately, writing the integral as a residue in the last line fixes the problem.''

==Applications==

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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>20.58\ldots \cdot 10^{-6}0655 < z < 2112.869\ldots \cdot 10^663</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>

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===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 |pagepages=185-189185–189 |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>

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=== Asymptotic approximation of integrals===

In the Laplace-ErdelyiLaplace–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

*[[{{Section link|Formal power series#|The Lagrange inversion formula]]}}

==References==