Lagrange inversion theorem

Template:Short description Template:For 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

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

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

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

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

Template:Main 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

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

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 $W(1)\approx \mathrm{0.567143.}$

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

• Template:MathWorld
• Template:MathWorld
• Bürmann–Lagrange series at Springer EOM