Lagrange inversion theorem: Difference between revisions

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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.

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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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.

==Sketch of the proof==

For simplicity suppose <math>z=0=f(w=0)</math>. We can then compute

:<math>

\oint_{w=0} \frac{d w}{2\pi i} \frac{1}{f(w) -z}

=

\oint_{w=0} \frac{d w}{2\pi i} \frac{1}{f'(g(z)) w + O(w^2)}

=

\frac{1}{f'(g(z))}

=

g'(f(w))

=

g'(z)

.

</math>

If we expand the integrand using the geometric series we get

:<math>

\oint_{w=0} \frac{d w}{2\pi i} \frac{1}{f(w) -z}

=

\sum_{n=0}^\infty

z^n

\oint_{w=0} \frac{d w}{2\pi i} \frac{1}{(f(w))^{n+1}}

=

\sum_{n=0}^\infty

z^n

\oint_{w=0} \frac{d w}{2\pi i} \frac{1}{w^{n+1}} \left(\frac{w}{f(w)}\right)^{n+1}

=

\sum_{n=0}^\infty

\frac{ z^n }{n!}

\left. \frac{d^n}{ d w^n}\left(\frac{w}{f(w)}\right)^{n+1} \right|_{w=0}

,

</math>

where in the last step we used the fact that <math>f(w)</math> has one zero.

Finally we can integrate over <math>z</math> taking into account <math>g(0)=0</math>

:<math>

g'(z) = \sum_{n=0}^\infty

\frac{ z^n }{n!}

\left. \frac{d^n}{ d w^n}\left(\frac{w}{f(w)}\right)^{n+1} \right|_{w=0}

~~\Longrightarrow~~

g(z) = \sum_{n=0}^\infty

\frac{ z^{n+1} }{(n+1)!}

\left. \frac{d^n}{ d w^n}\left(\frac{w}{f(w)}\right)^{n+1} \right|_{w=0}

.

</math>

Upon a redefiniton of the summation index we get the stated formula.

==Applications==