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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]]. | |||
==Theorem statement== | |||
Suppose ''z'' is defined as a function of ''w'' by an equation of the form | |||
:<math>f(w) = z\,</math> | |||
where ''f'' is analytic at a point ''a'' and ''f'' '(''a'') ≠ 0. Then it is possible to ''invert'' or ''solve'' the equation for ''w'': | |||
:<math>w = g(z)\,</math> | |||
on a [[neighbourhood (mathematics)|neighbourhood]] of ''f(a)'', where ''g'' is analytic at the point ''f''(''a''). This is also called '''reversion of series'''. | |||
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> | |||
:<math> | |||
g(z) = a | |||
+ \sum_{n=1}^{\infty} | |||
\left( | |||
\lim_{w \to a}\left( | |||
{\frac{(z - f(a))^n}{n!}} | |||
\frac{\mathrm{d}^{\,n-1}}{\mathrm{d}w^{\,n-1}} | |||
\left( \frac{w-a}{f(w) - f(a)} \right)^n\right) | |||
\right). | |||
</math> | |||
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. | |||
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. | |||
== 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> and <math>\phi(0)\ne 0.</math> Take <math>a=0</math> to obtain <math>b=f(0)=0.</math> We have | |||
:<math> | |||
g(z) = | |||
\sum_{n=1}^{\infty} | |||
\left( \lim_{w \to 0} | |||
\left( \frac {\mathrm{d}^{n-1}}{\mathrm{d}w^{n-1}} | |||
\left( \frac{w}{w/\phi(w)} \right)^n | |||
\right) | |||
\frac{z^n}{n!} | |||
\right) | |||
</math> | |||
:<math>= | |||
\sum_{n=1}^{\infty} | |||
\frac{1}{n} | |||
\left( | |||
\frac{1}{(n-1)!} | |||
\lim_{w \to 0} \left( | |||
\frac{\mathrm{d}^{n-1}}{\mathrm{d}w^{n-1}} | |||
\phi(w)^n | |||
\right) | |||
\right) | |||
z^n, | |||
</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 w. | |||
A useful generalization of the formula is known as the '''Lagrange–Bürmann formula''': | |||
:<math>[z^{n+1}] H (g(z)) = \frac{1}{(n+1)} [w^n] (H' (w) \phi(w)^{n+1})</math> | |||
where {{math|''H''}} can be an arbitrary analytic function, e.g. {{math|''H''(''w'') {{=}} ''w''<sup>''k''</sup>}}. | |||
===Lambert W function=== | |||
The [[Lambert 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) = w \mathrm{e}^w</math> and <math>a = b = 0.</math> Recognizing that | |||
:<math> | |||
\frac{\mathrm{d}^n}{\mathrm{d}x^n}\ \mathrm{e}^{\alpha\,x}\,=\,\alpha^n\,\mathrm{e}^{\alpha\,x} | |||
</math> | |||
this gives | |||
:<math> | |||
W(z) = | |||
\sum_{n=1}^{\infty} | |||
\lim_{w \to 0} \left( | |||
\frac{\mathrm{d}^{\,n-1}}{\mathrm{d}w^{\,n-1}}\ \mathrm{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). | |||
</math> | |||
The [[radius of convergence]] of this series is <math>e^{-1}</math> (this example refers to the [[principal branch]] of the Lambert function). | |||
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 | |||
:<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. This gives a series for <math>f(z+1) = W(e^{z+1})-1\,</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} | |||
- \frac{47 z^6}{1474560} | |||
- \frac{73 z^7}{41287680} | |||
+ \frac{2447 z^8}{1321205760} + O(z^9).</math> | |||
<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>. | |||
===Binary trees=== | |||
Consider 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 ''n'' nodes. | |||
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>: | |||
:<math>B(z) = 1 + z B(z)^2.</math> | |||
Now let <math>C(z) = B(z) - 1</math> and rewrite this equation as follows: | |||
:<math>z = \frac{C(z)}{(C(z)+1)^2}.</math> | |||
Now apply the theorem with <math>\phi(w) = (w+1)^2:</math> | |||
:<math> B_n = [z^n] C(z) = \frac{1}{n} [w^{n-1}] (w+1)^{2n} | |||
= \frac{1}{n} {2n \choose n-1} = \frac{1}{n+1} {2n \choose n}.</math> | |||
We conclude that <math>B_n</math> is the [[Catalan number]]. | |||
==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}} | |||
==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]] | |||
Revision as of 10:59, 19 January 2014
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.
Theorem statement
Suppose z is defined as a function of w by an equation of the form
where f is analytic at a point a and f '(a) ≠ 0. Then it is possible to invert or solve the equation for w:
on a neighbourhood of f(a), where g is analytic at the point f(a). This is also called reversion of series.
The series expansion of g is given by[1]
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.
The theorem was proved by Lagrange[2] and generalized by Hans Heinrich Bürmann,[3][4][5] 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 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 just some property of the formal residue, and a more direct formal proof is available.
Applications
Lagrange–Bürmann formula
There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when and Take to obtain We have
which can be written alternatively as
where is an operator which extracts the coefficient of in the Taylor series of a function of w.
A useful generalization of the formula is known as the Lagrange–Bürmann formula:
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Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. can be an arbitrary analytic function, e.g. Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
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Lambert W function
The Lambert W function is the function that is implicitly defined by the equation
We may use the theorem to compute the Taylor series of at We take and Recognizing that
this gives
The radius of convergence of this series is (this example refers to the principal branch of the Lambert function).
A series that converges for larger z (though not for all z) can also be derived by series inversion. The function satisfies the equation
Then can be expanded into a power series and inverted. This gives a series for :
can be computed by substituting for z in the above series. For example, substituting -1 for z gives the value of .
Binary trees
Consider the set of unlabelled binary trees. An element of is either a leaf of size zero, or a root node with two subtrees. Denote by the number of binary trees on n nodes.
Note that removing the root splits a binary tree into two trees of smaller size. This yields the functional equation on the generating function :
Now let and rewrite this equation as follows:
We conclude that is the Catalan number.
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
References
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External links
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang (Note: Although Lagrange submitted this article in 1768, it was not published until 1770.) - ↑ 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: 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 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.)
- ↑ 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).