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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]].
| | A person whom is obese is unable to work effectively.The bodily processes are furthermore not carried properly and hence the person gets subjected to other illnesses too. With increasing fat, the immune program of the individual begins to weaken and he is unable to fight the germs plus bacteria attacking them.<br><br>When I'm struggling to get rid of fat, I eat my last meal at 5:30pm thus, by the time 6pm rolls around, I've finished eating for the day. Stopping eating at 6pm offers the body time to burn off the calories you've consumed throughout the day. But, should you eat following 6pm, nearly all of those calories are not burned off and will be turned into fat while we sleep. If you stop eating at 6pm, you'll even discover you are able to eat slightly more during the day plus nevertheless lose weight fast.<br><br>Avoid consuming cheese and butter. We will be amazed to learn the amount of calories that these aspects have, even if they are consumed in smaller quantities. Even peanut butter is the most harmful thing for we if you would like to lose weight. Instead try opting for healthier choices like low calorie butter plus this method you will consume lesser calories.<br><br>Have breakfast. Skipping breakfast leads to low blood sugar mid morning that results inside cravings for sugary foods to pump the blood glucose back up. A vicious cycle results and continues throughout the day. Your breakfast must comprise of whole grains, a protein source and fruit. It doesn't need to be fancy. A cut of whole wheat toast spread with a tablespoon of low fat peanut butter and an apple will do merely fine. Or we can mix 1/2 cup of nonfat cottage cheese with chopped fresh peaches and a couple of whole wheat crackers. Another possibility is a scrambled egg served on an English muffin with a glass of orange juice.<br><br>1 Practice controlled part by eating little balanced meals every time each day. Preferably [http://safedietplansforwomen.com/how-to-lose-weight-fast lose weight] go for only lean protein plus those foods which are low inside fat. You should not turn to food for emotional comfort.<br><br>A friend who's usually dieting says, "Will the pleasure of eating it be much better than the pleasure of installing into my clothing?" It's mind over matter, in other words. These temptations pass-if I just give me a small time to think.<br><br>Be sure to analysis the diet you choose before selecting any diet. As constantly, before trying any of these or other fast weight reduction diets, it happens to be extremely important to see the doctor thus that you can be sure you're healthy enough to diet in this means. |
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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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A person whom is obese is unable to work effectively.The bodily processes are furthermore not carried properly and hence the person gets subjected to other illnesses too. With increasing fat, the immune program of the individual begins to weaken and he is unable to fight the germs plus bacteria attacking them.
When I'm struggling to get rid of fat, I eat my last meal at 5:30pm thus, by the time 6pm rolls around, I've finished eating for the day. Stopping eating at 6pm offers the body time to burn off the calories you've consumed throughout the day. But, should you eat following 6pm, nearly all of those calories are not burned off and will be turned into fat while we sleep. If you stop eating at 6pm, you'll even discover you are able to eat slightly more during the day plus nevertheless lose weight fast.
Avoid consuming cheese and butter. We will be amazed to learn the amount of calories that these aspects have, even if they are consumed in smaller quantities. Even peanut butter is the most harmful thing for we if you would like to lose weight. Instead try opting for healthier choices like low calorie butter plus this method you will consume lesser calories.
Have breakfast. Skipping breakfast leads to low blood sugar mid morning that results inside cravings for sugary foods to pump the blood glucose back up. A vicious cycle results and continues throughout the day. Your breakfast must comprise of whole grains, a protein source and fruit. It doesn't need to be fancy. A cut of whole wheat toast spread with a tablespoon of low fat peanut butter and an apple will do merely fine. Or we can mix 1/2 cup of nonfat cottage cheese with chopped fresh peaches and a couple of whole wheat crackers. Another possibility is a scrambled egg served on an English muffin with a glass of orange juice.
1 Practice controlled part by eating little balanced meals every time each day. Preferably lose weight go for only lean protein plus those foods which are low inside fat. You should not turn to food for emotional comfort.
A friend who's usually dieting says, "Will the pleasure of eating it be much better than the pleasure of installing into my clothing?" It's mind over matter, in other words. These temptations pass-if I just give me a small time to think.
Be sure to analysis the diet you choose before selecting any diet. As constantly, before trying any of these or other fast weight reduction diets, it happens to be extremely important to see the doctor thus that you can be sure you're healthy enough to diet in this means.