Editing Lagrange inversion theorem (section)

==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> for some analytic <math>\phi(w)</math> with <math>\phi(0)\ne 0.</math> Take <math>a=0</math> to obtain <math>f(a)=f(0)=0.</math> Then for the inverse <math>g(z)</math> (satisfying <math>f(g(z))\equiv z</math>), we have

:<math>\begin{align}
  g(z) &= \sum_{n=1}^{\infty} \left[ \lim_{w \to 0} \frac {d^{n-1}}{dw^{n-1}} \left(\left( \frac{w}{w/\phi(w)} \right)^n \right)\right] \frac{z^n}{n!} \\
  {} &= \sum_{n=1}^{\infty} \frac{1}{n} \left[\frac{1}{(n-1)!} \lim_{w \to 0} \frac{d^{n-1}}{dw^{n-1}} (\phi(w)^n) \right] z^n,
 \end{align}</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 {{mvar|w}}.

A generalization of the formula is known as the '''Lagrange–Bürmann formula''':
:<math>[z^n] H (g(z)) = \frac{1}{n} [w^{n-1}] (H' (w) \phi(w)^n)</math>

where {{math|''H''}} is an arbitrary analytic function.

Sometimes, the derivative {{math|''{{prime|H}}''(''w'')}} can be quite complicated. A simpler version of the formula replaces {{math|''{{prime|H}}''(''w'')}} with {{math|''H''(''w'')(1 &minus; ''{{prime|φ}}''(''w'')/''φ''(''w''))}} to get 

:<math> [z^n] H (g(z)) = [w^n] H(w) \phi(w)^{n-1} (\phi(w) - w \phi'(w)),  </math>

which involves {{math|''{{prime|φ}}''(''w'')}} instead of {{math|''{{prime|H}}''(''w'')}}.

===Lambert ''W'' function===

{{main|Lambert W function}}
The Lambert {{mvar|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) = we^w</math> and <math>a = 0.</math> Recognizing that

:<math>\frac{d^n}{dx^n} e^{\alpha x} = \alpha^n e^{\alpha x},</math>

this gives

:<math>\begin{align}
   W(z) &= \sum_{n=1}^{\infty} \left[\lim_{w \to 0} \frac{d^{n-1}}{dw^{n-1}} 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).
 \end{align}</math>

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>0.0655 < z < 112.63</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>

<math>W(x)</math> can be computed by substituting <math>\ln x - 1</math> for {{mvar|z}} in the above series. For example, substituting {{math|−1}} for {{mvar|z}} gives the value of <math>W(1) \approx 0.567143.</math>

===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 |pages=185–189 |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>
:<math>B(z) = 1 + z B(z)^2.</math>

Letting <math>C(z) = B(z) - 1</math>, one has thus <math>C(z) = z (C(z)+1)^2.</math> Applying the theorem with <math>\phi(w) = (w+1)^2</math> yields
:<math> B_n = [z^n] C(z) = \frac{1}{n} [w^{n-1}] (w+1)^{2n} = \frac{1}{n} \binom{2n}{n-1} =  \frac{1}{n+1} \binom{2n}{n}.</math>

This shows that <math>B_n</math> is the {{mvar|n}}th [[Catalan number]].

=== Asymptotic approximation of integrals===
In the Laplace–Erdelyi theorem that gives the asymptotic approximation for Laplace-type integrals, the function inversion is taken as a crucial step.