Lagrange inversion theorem

Template:Short description 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. Lagrange inversion is a special case of the inverse function theorem.

Statement

Suppose Template:Mvar is defined as a function of Template:Mvar by an equation of the form

$z=f(w)$

where Template:Mvar is analytic at a point Template:Mvar and $f^{\prime }(a)\ne 0.$ Then it is possible to invert or solve the equation for Template:Mvar, expressing it in the form $w=g(z)$ given by a power series

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

where

$g_n=\underset{w\to a}{\lim }\frac{d^{n-1}}{d{w}^{n-1}}\left[{\left(\frac{w-a}{f(w)-f(a)}\right)}^n\right].$

The theorem further states that this series has a non-zero radius of convergence, i.e., $g(z)$ represents an analytic function of Template:Mvar in a neighbourhood of $z=f(a).$ 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 Template:Math for any analytic function Template:Mvar; and it can be generalized to the case $f^{\prime }(a)=0,$ where the inverse Template:Mvar is a multivalued function.

The theorem was proved by Lagrange and generalized by Hans Heinrich 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 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. In fact, the Lagrange inversion theorem has a number of additional rather different proofs, including ones using tree-counting arguments or induction.

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

$f(w)=\sum _{k=0}^{\mathrm{\infty }}{f}_k\frac{w^k}{k!}\text{and}g(z)=\sum _{k=0}^{\mathrm{\infty }}{g}_k\frac{z^k}{k!}$

with Template:Math and Template:Math, then an explicit form of inverse coefficients can be given in term of Bell polynomials:

$g_n=\frac{1}{f_1^n}\sum _{k=1}^{n-1}(-1{)}^k{n}^{\bar{k}}{B}_{n-1,k}({\hat{f}}_1,{\hat{f}}_2,\dots ,{\hat{f}}_{n-k}),n\ge 2,$

where

$\begin{array}{rl}{\hat{f}}_k& =\frac{f_{k+1}}{(k+1)f_1},\\ g_1& =\frac{1}{f_1},\text{ and}\\ n^{\bar{k}}& =n(n+1)\cdots (n+k-1)\end{array}$

is the rising factorial.

When Template:Math, the last formula can be interpreted in terms of the faces of associahedra

$g_n=\sum _{F\text{ face of }{K}_n}(-1{)}^{n-\dim F}{f}_F,n\ge 2,$

where $f_F=f_{i_1}\cdots f_{i_m}$ for each face $F=K_{i_1}\times \cdots \times K_{i_m}$ of the associahedron $K_n.$

Example

For instance, the algebraic equation of degree Template:Mvar

$x^p-x+z=0$

can be solved for Template:Mvar by means of the Lagrange inversion formula for the function Template:Math, resulting in a formal series solution

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

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

Applications

Lagrange–Bürmann formula

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

$\begin{array}{rl}g(z)& =\sum _{n=1}^{\mathrm{\infty }}\left[\underset{w\to 0}{\lim }\frac{d^{n-1}}{d{w}^{n-1}}\left({\left(\frac{w}{w/\varphi (w)}\right)}^n\right)\right]\frac{z^n}{n!}\\ & =\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}[\frac{1}{(n-1)!}\underset{w\to 0}{\lim }\frac{d^{n-1}}{d{w}^{n-1}}(\varphi (w{)}^n)]z^n,\end{array}$

which can be written alternatively as

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

where $[w^r]$ is an operator which extracts the coefficient of $w^r$ in the Taylor series of a function of Template:Mvar.

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

$[z^n]H(g(z))=\frac{1}{n}[w^{n-1}](H^{\prime }(w)\varphi (w{)}^n)$

where Template:Math is an arbitrary analytic function.

Sometimes, the derivative Template:Math can be quite complicated. A simpler version of the formula replaces Template:Math with Template:Math to get

$[z^n]H(g(z))=[w^n]H(w)\varphi (w{)}^{n-1}(\varphi (w)-w{\varphi }^{\prime }(w)),$

which involves Template:Math instead of Template:Math.

Lambert W function

The Lambert Template:Mvar function is the function $W(z)$ that is implicitly defined by the 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 $f(w)=w{e}^w$ and $a=0.$ Recognizing that

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

this gives

$\begin{array}{rl}W(z)& =\sum _{n=1}^{\mathrm{\infty }}\left[\underset{w\to 0}{\lim }\frac{d^{n-1}}{d{w}^{n-1}}{e}^{-nw}\right]\frac{z^n}{n!}\\ & =\sum _{n=1}^{\mathrm{\infty }}(-n{)}^{n-1}\frac{z^n}{n!}\\ & =z-z^2+\frac{3}{2}{z}^3-\frac{8}{3}{z}^4+O(z^5).\end{array}$

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

A series that converges for $|\ln (z)-1|<\sqrt{4+{\pi }^2}$ (approximately $0.0655<z<112.63$) can also be derived by series inversion. The function $f(z)=W(e^z)-1$ satisfies the equation

$1+f(z)+\ln (1+f(z))=z.$

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

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

$W(x)$ can be computed by substituting $\ln x-1$ for Template:Mvar in the above series. For example, substituting Template:Math for Template:Mvar gives the value of $W(1)\approx 0.567143.$

Binary trees

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

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

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

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

$B_n=[z^n]C(z)=\frac{1}{n}[w^{n-1}](w+1{)}^{2n}=\frac{1}{n}(\genfrac{}{}{0ex}{}{2n}{n-1})=\frac{1}{n+1}(\genfrac{}{}{0ex}{}{2n}{n}).$

This shows that $B_n$ is the Template:Mvarth 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.

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
• Template:Section link

External links

• Bürmann–Lagrange series at Springer EOM