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| 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]]. | | Good to satisfy you, I am Paul Spratt but I in no way seriously preferred that title. In my specialist lifetime I am an administrative assistant and I will not feel I will transform it at any time quickly. As a person what I really like is drawing but I struggle to uncover time for it. I have normally liked residing in Virgin Islands. See what's new on my site here: http://imleme.antalyakobi.com/story.php?title=briefs-underwear-connected-posts<br><br> |
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| ==Theorem statement==
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| Suppose ''z'' is defined as a function of ''w'' by an equation of the form
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| :<math>f(w) = z\,</math>
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| where ''f'' is analytic at a point ''a'' and ''f'' '(''a'') ≠ 0. Then it is possible to ''invert'' or ''solve'' the equation for ''w'':
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| :<math>w = g(z)\,</math>
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| on a [[neighbourhood (mathematics)|neighbourhood]] of ''f(a)'', where ''g'' is analytic at the point ''f''(''a''). This is also called '''reversion of series'''.
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| The series expansion of ''g'' is given by<ref>{{cite book |editors=M. Abramowitz, 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>
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| :<math>
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| g(z) = a
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| + \sum_{n=1}^{\infty}
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| \left(
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| \lim_{w \to a}\left(
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| {\frac{(z - f(a))^n}{n!}}
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| \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}w^{\,n-1}}
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| \left( \frac{w-a}{f(w) - f(a)} \right)^n\right)
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| \right).
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| </math>
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| 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 ''F''(''g''(''z'')) for any analytic function ''F'', and it can be generalized to the case ''f'' '(''a'') = 0, where the inverse ''g'' is a multivalued function.
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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=Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin |volume=24 |pages=251–326 |url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=41070}} (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 |chapterurl=http://books.google.com/books?id=jj4DAAAAQAAJ&pg=495#v=onepage&q&f=false}}</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]]; the complex formal power series version is clearly 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 just 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.
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| == Applications ==
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| ===Lagrange–Bürmann formula===
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| There is a special case of Lagrange inversion theorem that is used in [[combinatorics]] and applies when <math>f(w)=w/\phi(w)</math> and <math>\phi(0)\ne 0.</math> Take <math>a=0</math> to obtain <math>b=f(0)=0.</math> We have
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| :<math>
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| g(z) =
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| \sum_{n=1}^{\infty}
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| \left( \lim_{w \to 0}
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| \left( \frac {\mathrm{d}^{n-1}}{\mathrm{d}w^{n-1}}
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| \left( \frac{w}{w/\phi(w)} \right)^n
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| \right)
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| \frac{z^n}{n!}
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| \right)
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| </math>
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| :<math>=
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| \sum_{n=1}^{\infty}
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| \frac{1}{n}
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| \left(
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| \frac{1}{(n-1)!}
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| \lim_{w \to 0} \left(
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| \frac{\mathrm{d}^{n-1}}{\mathrm{d}w^{n-1}}
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| \phi(w)^n
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| \right)
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| \right)
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| z^n,
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| </math>
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| which can be written alternatively as
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| :<math>[z^n] g(z) = \frac{1}{n} [w^{n-1}] \phi(w)^n,</math>
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| 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 w.
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| A useful generalization of the formula is known as the '''Lagrange–Bürmann formula''':
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| :<math>[z^{n+1}] H (g(z)) = \frac{1}{(n+1)} [w^n] (H' (w) \phi(w)^{n+1})</math>
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| where {{math|''H''}} can be an arbitrary analytic function, e.g. {{math|''H''(''w'') {{=}} ''w''<sup>''k''</sup>}}.
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| ===Lambert W function===
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| The [[Lambert W function]] is the function <math>W(z)</math> that is implicitly defined by the equation
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| :<math> W(z) e^{W(z)} = z.\,</math>
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| We may use the theorem to compute the [[Taylor series]] of <math>W(z)</math> at <math>z=0.</math>
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| We take <math>f(w) = w \mathrm{e}^w</math> and <math>a = b = 0.</math> Recognizing that
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| :<math>
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| \frac{\mathrm{d}^n}{\mathrm{d}x^n}\ \mathrm{e}^{\alpha\,x}\,=\,\alpha^n\,\mathrm{e}^{\alpha\,x}
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| </math>
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| this gives
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| :<math>
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| W(z) =
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| \sum_{n=1}^{\infty}
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| \lim_{w \to 0} \left(
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| \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}w^{\,n-1}}\ \mathrm{e}^{-nw}
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| \right)
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| { \frac{z^n}{n!}}\,=\, \sum_{n=1}^{\infty}
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| (-n)^{n-1}\, \frac{z^n}{n!}=z-z^2+\frac{3}{2}z^3-\frac{8}{3}z^4+O(z^5).
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| </math>
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| The [[radius of convergence]] of this series is <math>e^{-1}</math> (this example refers to the [[principal branch]] of the Lambert function).
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| A series that converges for larger ''z'' (though not for all ''z'') can also be derived by series inversion. The function <math>f(z) = W(e^z) - 1\,</math> satisfies the equation
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| :<math>1 + f(z) + \ln (1 + f(z)) = z.\,</math>
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| Then <math>z + \ln (1 + z)\,</math> can be expanded into a power series and inverted. This gives a series for <math>f(z+1) = W(e^{z+1})-1\,</math>:
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| :<math>W(e^{1+z}) = 1 + \frac{z}{2} + \frac{z^2}{16}
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| - \frac{z^3}{192}
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| - \frac{z^4}{3072}
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| + \frac{13 z^5}{61440}
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| - \frac{47 z^6}{1474560}
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| - \frac{73 z^7}{41287680}
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| + \frac{2447 z^8}{1321205760} + O(z^9).</math>
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| <math>W(x)\,</math> can be computed by substituting <math>\ln x - 1\,</math> for ''z'' in the above series. For example, substituting -1 for ''z'' gives the value of <math>W(1) = 0.567143\,</math>.
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| ===Binary trees===
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| Consider the set <math>\mathcal{B}</math> of unlabelled [[binary tree]]s.
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| 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 ''n'' nodes.
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| Note that removing the root splits a binary tree into two trees of smaller size. This yields the functional equation on the generating function <math>B(z) = \sum_{n=0}^\infty B_n z^n</math>:
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| :<math>B(z) = 1 + z B(z)^2.</math>
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| Now let <math>C(z) = B(z) - 1</math> and rewrite this equation as follows:
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| :<math>z = \frac{C(z)}{(C(z)+1)^2}.</math>
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| Now apply the theorem with <math>\phi(w) = (w+1)^2:</math>
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| :<math> B_n = [z^n] C(z) = \frac{1}{n} [w^{n-1}] (w+1)^{2n}
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| = \frac{1}{n} {2n \choose n-1} = \frac{1}{n+1} {2n \choose n}.</math>
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| We conclude that <math>B_n</math> is the [[Catalan number]].
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| ==See also==
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| *[[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.
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| *[[Lagrange reversion theorem]] for another theorem sometimes called the inversion theorem
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| *[[Formal_power_series#The_Lagrange_inversion_formula]]
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| == References ==
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| {{reflist}}
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| ==External links==
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| *{{MathWorld |urlname=BuermannsTheorem |title=Bürmann's Theorem}}
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| *{{MathWorld |urlname=SeriesReversion |title=Series Reversion}}
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| *[http://www.encyclopediaofmath.org/index.php/B%C3%BCrmann%E2%80%93Lagrange_series Bürmann–Lagrange series] at [[Encyclopedia of Mathematics|Springer EOM]]
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| [[Category:Inverse functions]]
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| [[Category:Theorems in real analysis]]
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| [[Category:Theorems in complex analysis]]
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