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Lagrange inversion theorem: Difference between revisions

Revision as of 15:43, 15 June 2024

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 invertorsolve 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 neighbourhoodof${\displaystyle z=f($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.

Iff 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]

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

where

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

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) = x − x^p, 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.

Derivation

We can use Cauchy's integral formula:

${\displaystyle f^{-1}($z)={\frac {1}{2\pi i}}\oint _{f(C)}{\frac {f^{-1}(\xi )}{\xi -z}}d\xi }

and substitute:

${\displaystyle \xi =f(\omega )}$

${\displaystyle d\xi =f'(\omega )d\omega }$

${\displaystyle f($C)\rightarrow C}

${\displaystyle f^{-1}($z)={\frac {1}{2\pi i}}\oint _{C}{\frac {\omega }{f(\omega )-z}}f'(\omega )d\omega }

using geometric series:

${\displaystyle {\frac {1}{f(\omega )-z}}={\frac {1}{f(\omega )-f($a)-z+f(a)}}={\frac {1}{f(\omega )-f(a)}}{\frac {1}{1-{\frac {z-f(a)}{f(\omega )-f(a)}}}}={\frac {1}{f(\omega )-f(a)}}\sum _{n=0}^{\infty }\left({\frac {z-f(a)}{f(\omega )-f(a)}}\right)^{n}}

${\displaystyle f^{-1}($z)={\frac {1}{2\pi i}}\sum _{n=0}^{\infty }\left({z-f(a)}\right)^{n}\oint _{C}{\frac {\omega f'(\omega )}{(f(\omega )-f(a))^{n+1}}}d\omega }

now by integration by parts: ${\displaystyle u=\omega }$ and ${\displaystyle dv={\frac {f'(\omega )d\omega }{(f(\omega )-f($a))^{n+1}}}} where ${\displaystyle uv={\frac {-1}{n}}{\frac {e^{i\theta }}{(f(e^{i\theta })-f($a))^{n}}}{\Biggl |}_{0}^{2\pi }=0} we get:

${\displaystyle f^{-1}($z)={\frac {1}{2\pi i}}\sum _{n=0}^{\infty }{\frac {({z-f(a)})^{n}}{n}}\oint _{C}{\frac {1}{(f(\omega )-f(a))^{n}}}d\omega }

by the residue theorem:

${\displaystyle f^{-1}($z)=\sum _{n=0}^{\infty }{\frac {({z-f(a)})^{n}}{n}}\operatorname {Res} ({\frac {1}{(f(\omega )-f(a))^{n}}},w=a)}

finally:

${\displaystyle f^{-1}($z)=\sum _{n=0}^{\infty }{\frac {({z-f(a)})^{n}}{n!}}\lim _{\omega \to a}{\frac {d^{n-1}}{d\omega ^{n-1}}}\left({\frac {\omega -a}{f(\omega )-f(a)}}\right)^{n}}

In fact, integration by parts doesn't work for:

${\displaystyle n=0\rightarrow {\frac {1}{2\pi i}}\oint _{C}{\frac {\omega f'(\omega )}{f(\omega )-f($a)}}d\omega ={\frac {1}{2\pi i}}\oint _{f(C)}{\frac {f^{-1}(u)}{u-f(a)}}du=f^{-1}(f(a))=a}

Fortunately, writing the integral as a residue in the last line fixes the problem.

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 seriesof${\displaystyle W($z)} at${\displaystyle z=0.}$ We take ${\displaystyle f($w)=we^{w}} and ${\displaystyle a=0.}$ 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 ${\displaystyle |\ln($z)-1|<{4+\pi ^{2}}} (approximately ${\displaystyle 2.58\ldots \cdot 10^{-6}<z<2.869\ldots \cdot 10^{6}}$) 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.^[9] 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^[10] 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 ${\displaystyle n}$ 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 nthCatalan 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
• Formal power series#The Lagrange inversion formula

References

External links

• Weisstein, Eric W. "Bürmann's Theorem". MathWorld.
• Weisstein, Eric W. "Series Reversion". MathWorld.
• Bürmann–Lagrange seriesatSpringer EOM