Lagrange inversion theorem

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.

Statement

Suppose z is defined as a function of w by an equation of the form

${\displaystyle z=f(w)}$

where f is analytic at a point a and ${\displaystyle f'(a)\neq 0.}$  Then it is possible to invert or solve the equation for w, expressing it in the form ${\displaystyle w=g(z)}$  given by a power series[1]

${\displaystyle g(z)=a+\sum _{n=1}^{\infty }g_{n}{\frac {(z-f(a))^{n}}{n!}},}$

where

${\displaystyle g_{n}=\lim _{w\to a}{\frac {d^{n-1}}{dw^{n-1}}}\left[\left({\frac {w-a}{f(w)-f(a)}}\right)^{n}\right].}$

The theorem further states that this series has a non-zero radius of convergence, i.e., ${\displaystyle g(z)}$  represents an analytic function of z in a neighbourhood of ${\displaystyle z=f(a).}$  This is also called reversion of series.

If the assertions about analyticity are omitted, 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 ${\displaystyle f'(a)=0,}$  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;[6] the complex formal power series version is 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 some property of the formal residue, and a more direct formal proof is available.

If f is a formal power series, then the above formula does not give the coefficients of the compositional inverse series g directly in terms for the coefficients of the series f. If one can express the functions f and g in formal power series as

${\displaystyle f(w)=\sum _{k=0}^{\infty }f_{k}{\frac {w^{k}}{k!}}\qquad {\text{and}}\qquad g(z)=\sum _{k=0}^{\infty }g_{k}{\frac {z^{k}}{k!}}}$

with f0 = 0 and f1 ≠ 0, then an explicit form of inverse coefficients can be given in term of Bell polynomials:[7]

${\displaystyle g_{n}={\frac {1}{f_{1}^{n}}}\sum _{k=1}^{n-1}(-1)^{k}n^{(k)}B_{n-1,k}({\hat {f}}_{1},{\hat {f}}_{2},\ldots ,{\hat {f}}_{n-k}),\quad n\geq 2,}$

where

{\displaystyle {\begin{aligned}{\hat {f}}_{k}&={\frac {f_{k+1}}{(k+1)f_{1}}},\\g_{1}&={\frac {1}{f_{1}}},{\text{ and}}\\n^{(k)}&=n(n+1)\cdots (n+k-1)\end{aligned}}}

is the rising factorial.

When f1 = 1, the last formula can be interpreted in terms of the faces of associahedra [8]

${\displaystyle g_{n}=\sum _{F{\text{ face of }}K_{n}}(-1)^{n-\dim F}f_{F},\quad n\geq 2,}$

where ${\displaystyle f_{F}=f_{i_{1}}\cdots f_{i_{m}}}$  for each face ${\displaystyle F=K_{i_{1}}\times \cdots \times K_{i_{m}}}$  of the associahedron ${\displaystyle K_{n}.}$

Example

For instance, the algebraic equation of degree p

${\displaystyle x^{p}-x+z=0}$

can be solved for x by means of the Lagrange inversion formula for the function f(x) = xxp, resulting in a formal series solution

${\displaystyle x=\sum _{k=0}^{\infty }{\binom {pk}{k}}{\frac {z^{(p-1)k+1}}{(p-1)k+1}}.}$

By convergence tests, this series is in fact convergent for ${\displaystyle |z|\leq (p-1)p^{-p/(p-1)},}$  which is also the largest disk in which a local inverse to f can be defined.

Applications

Lagrange–Bürmann formula

There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when ${\displaystyle f(w)=w/\phi (w)}$  for some analytic ${\displaystyle \phi (w)}$  with ${\displaystyle \phi (0)\neq 0.}$  Take ${\displaystyle a=0}$  to obtain ${\displaystyle f(a)=f(0)=0.}$  Then for the inverse ${\displaystyle g(z)}$  (satisfying ${\displaystyle f(g(z))\equiv z}$ ), we have

{\displaystyle {\begin{aligned}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{aligned}}}

which can be written alternatively as

${\displaystyle [z^{n}]g(z)={\frac {1}{n}}[w^{n-1}]\phi (w)^{n},}$

where ${\displaystyle [w^{r}]}$  is an operator which extracts the coefficient of ${\displaystyle w^{r}}$  in the Taylor series of a function of w.

A generalization of the formula is known as the Lagrange–Bürmann formula:

${\displaystyle [z^{n}]H(g(z))={\frac {1}{n}}[w^{n-1}](H'(w)\phi (w)^{n})}$

where H is an arbitrary analytic function.

Sometimes, the derivative H(w) can be quite complicated. A simpler version of the formula replaces H(w) with H(w)(1 − φ(w)/φ(w)) to get

${\displaystyle [z^{n}]H(g(z))=[w^{n}]H(w)\phi (w)^{n-1}(\phi (w)-w\phi '(w)),}$

which involves φ(w) instead of H(w).

Lambert W function

The Lambert W function is the function ${\displaystyle W(z)}$  that is implicitly defined by the equation

${\displaystyle W(z)e^{W(z)}=z.}$

We may use the theorem to compute the Taylor series of ${\displaystyle W(z)}$  at ${\displaystyle z=0.}$  We take ${\displaystyle f(w)=we^{w}}$  and ${\displaystyle a=b=0.}$  Recognizing that

${\displaystyle {\frac {d^{n}}{dx^{n}}}e^{\alpha x}=\alpha ^{n}e^{\alpha x},}$

this gives

{\displaystyle {\begin{aligned}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{aligned}}}

The radius of convergence of this series is ${\displaystyle e^{-1}}$  (giving 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 ${\displaystyle f(z)=W(e^{z})-1}$  satisfies the equation

${\displaystyle 1+f(z)+\ln(1+f(z))=z.}$

Then ${\displaystyle z+\ln(1+z)}$  can be expanded into a power series and inverted. This gives a series for ${\displaystyle f(z+1)=W(e^{z+1})-1{\text{:}}}$

${\displaystyle W(e^{1+z})=1+{\frac {z}{2}}+{\frac {z^{2}}{16}}-{\frac {z^{3}}{192}}-{\frac {z^{4}}{3072}}+{\frac {13z^{5}}{61440}}-O(z^{6}).}$

${\displaystyle W(x)}$  can be computed by substituting ${\displaystyle \ln x-1}$  for z in the above series. For example, substituting −1 for z gives the value of ${\displaystyle W(1)\approx 0.567143.}$

Binary trees

Consider the set ${\displaystyle {\mathcal {B}}}$  of unlabelled binary trees. An element of ${\displaystyle {\mathcal {B}}}$  is either a leaf of size zero, or a root node with two subtrees. Denote by ${\displaystyle B_{n}}$  the number of binary trees on 'nodes.

Removing the root splits a binary tree into two trees of smaller size. This yields the functional equation on the generating function ${\displaystyle \textstyle B(z)=\sum _{n=0}^{\infty }B_{n}z^{n}{\text{:}}}$

${\displaystyle B(z)=1+zB(z)^{2}.}$

Letting ${\displaystyle C(z)=B(z)-1}$ , one has thus ${\displaystyle C(z)=z(C(z)+1)^{2}.}$  Applying the theorem with ${\displaystyle \phi (w)=(w+1)^{2}}$  yields

${\displaystyle B_{n}=[z^{n}]C(z)={\frac {1}{n}}[w^{n-1}](w+1)^{2n}={\frac {1}{n}}{\binom {2n}{n-1}}={\frac {1}{n+1}}{\binom {2n}{n}}.}$

This shows that ${\displaystyle B_{n}}$  is the nth 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.

References

1. ^ M. Abramowitz; I. A. Stegun, eds. (1972). "3.6.6. Lagrange's Expansion". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. p. 14.
2. ^ Lagrange, Joseph-Louis (1770). "Nouvelle méthode pour résoudre les équations littérales par le moyen des séries". Histoire de l'Académie Royale des Sciences et Belles-Lettres de Berlin: 251–326. https://archive.org/details/uvresdelagrange18natigoog/page/n13 (Note: Although Lagrange submitted this article in 1768, it was not published until 1770.)
3. ^ 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: Hindenburg, Carl Friedrich, ed. (1798). "Versuch einer vereinfachten Analysis; ein Auszug eines Auszuges von Herrn Bürmann" [Attempt at a simplified analysis; an extract of an abridgement by Mr. Bürmann]. Archiv der reinen und angewandten Mathematik [Archive of pure and applied mathematics]. 2. Leipzig, Germany: Schäferischen Buchhandlung. pp. 495–499.
4. ^ 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.)
5. ^ 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).
6. ^ E. T. Whittaker and G. N. Watson. A Course of Modern Analysis. Cambridge University Press; 4th edition (January 2, 1927), pp. 129–130
7. ^ Eqn (11.43), p. 437, C.A. Charalambides, Enumerative Combinatorics, Chapman & Hall / CRC, 2002
8. ^ Aguiar, Marcelo; Ardila, Federico (2017). "Hopf monoids and generalized permutahedra". arXiv:1709.07504 [math.CO].