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. Suppose the dependence between the variables w and z is implicitly defined by an equation of the form

$f(w)=z\,$

where f is analytic at a point a and f '(a) ≠ 0. Then it is possible to invert or solve the equation for w:

$w=g(z)\,$

where g is analytic at the point b = f(a). This is also called reversion of series.

The series expansion of g is given by

$\left.g(z)=a+\sum _{n=1}^{\infty }{\frac {\mathrm {d} ^{\,n-1}}{\mathrm {d} w^{\,n-1}}}\left({\frac {w-a}{f(w)-b}}\right)^{n}\right|_{w\,=\,a}{\frac {(z-b)^{n}}{n!}}.$

This formula can for instance be used to find the Taylor series of the Lambert W function (by setting f(w) = w exp(w) and a = b = 0).

The formula is also valid for formal power series and can be generalized in various ways. If 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 [1][2] and generalized by Bürmann, 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).

Example calculation: Lambert W function

The Lambert W function is the function $W(z)$ that satisfies the implicit equation

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

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

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

this gives

$\left.W(z)=\sum _{n=1}^{\infty }{\frac {\mathrm {d} ^{\,n-1}}{\mathrm {d} w^{\,n-1}}}\ \mathrm {e} ^{-nw}\right|_{w\,=\,0}{\frac {z^{n}}{n!}}\,=\,\sum _{n=1}^{\infty }(-n)^{n-1}\,{\frac {z^{n}}{n!}}=z-z^{2}+{\frac {3}{2}}z^{3}-{\frac {8}{3}}z^{4}+O(z^{5})$

The radius of convergence of this series is $e^{-1}$ (this example refers to the principal branch of the Lambert function).

Special case

There is a special case of the theorem that is used in combinatorics and applies when $f(w)=w/\phi (w)$ and $\phi (0)\neq 0.$ Take $a=0$ to obtain $b=f(0)=0.$ We have

$g(z)=\sum _{n=1}^{\infty }\left.{\frac {\mathrm {d} ^{n-1}}{\mathrm {d} w^{n-1}}}\left({\frac {w}{w/\phi (w)}}\right)^{n}\right|_{w\,=\,0}{\frac {z^{n}}{n!}}$

or

$g(z)=\left.\sum _{n=1}^{\infty }{\frac {1}{n}}\left({\frac {1}{(n-1)!}}{\frac {\mathrm {d} ^{n-1}}{\mathrm {d} w^{n-1}}}\phi (w)^{n}\right|_{w=0}\right)z^{n},$

which can be written alternatively as

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

where $[w^{r}]$ is an operator which extracts the coefficient of $w^{r}$ in what follows it.

Example calculation: binary trees

Consider the set ${\mathcal {B}}$ of unlabelled binary trees. An element of ${\mathcal {B}}$ is either a leaf of size zero, or a root node with two subtrees (planar, i.e. no symmetry between them). The Fundamental theorem of combinatorial enumeration (unlabelled case) applies.

The group acting on the two subtrees is $E_{2}$, which contains a single permutation consisting of two fixed points. The set ${\mathcal {B}}$ satisfies

${\mathcal {B}}=1+{\mathcal {Z}}{\mathfrak {S}}_{2}({\mathcal {B}}).$

This yields the functional equation of the OGF $B(z)$ by the number of internal nodes:

$B(z)=1+zB(z)^{2}{\mbox{ or }}z={\frac {B(z)-1}{B(z)^{2}}}.$

Let $B_{\geq 1}(z)=B(z)-1={\frac {1-2z-{\sqrt {1-4z}}}{2z}}$ to obtain

$z={\frac {B_{\geq 1}(z)}{(B_{\geq 1}(z)+1)^{2}}}.$

Now apply the theorem with $\phi (w)=(w+1)^{2}:$

$[z^{n}]B_{\geq 1}(z)={\frac {1}{n}}[w^{n-1}](w+1)^{2n}={\frac {1}{n}}{2n \choose n-1}={\frac {1}{n+1}}{2n \choose n},$

the Catalan numbers.

Faà di Bruno's formula

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.

References

[1] Lagrange, Joseph-Louis "Addition au mémoire sur la résolution des équations numériques, imprimé dans le volume de 1767," Histoire de l'Académie Royale des Sciences et des Belles Lettres de Berlin, vol. 24, pages 111-180 (1768). (Available on-line at: http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=41060& )

[2] Lagrange, Joseph-Louis "Nouvelle méthode pour résoudre les problèmes indéterminés en nombres entiers," Histoire de l'Académie Royale des Sciences et des Belles Lettres de Berlin, vol. 24, pages 181-250 (1768). (Available on-line at: http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=41060& )

See also

• Lagrange reversion theorem for another theorem sometimes called the inversion theorem

External links

• Lagrange expansion on MathWorld
• Series Reversion on MathWorld
• Bürmann-Lagrange series