Lagrange inversion theorem

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 ${\displaystyle K_{n}.}$

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.

For simplicity suppose ${\displaystyle z=0=f(w=0)}$. We can then compute

${\displaystyle \oint _{w=0}{\frac {dw}{2\pi i}}{\frac {1}{f(w)-z}}=\oint _{w=0}{\frac {dw}{2\pi i}}{\frac {1}{f'(g(z))w+O(w^{2})}}={\frac {1}{f'(g(z))}}=g'(f(w))=g'(z).}$

If we expand the integrand using the geometric series we get

${\displaystyle \oint _{w=0}{\frac {dw}{2\pi i}}{\frac {1}{f(w)-z}}=\sum _{n=0}^{\infty }z^{n}\oint _{w=0}{\frac {dw}{2\pi i}}{\frac {1}{(f(w))^{n+1}}}=\sum _{n=0}^{\infty }z^{n}\oint _{w=0}{\frac {dw}{2\pi i}}{\frac {1}{w^{n+1}}}\left({\frac {w}{f(w)}}\right)^{n+1}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\left.{\frac {d^{n}}{dw^{n}}}\left({\frac {w}{f(w)}}\right)^{n+1}\right|_{w=0},}$

where in the last step we used the fact that ${\displaystyle f(w)}$ has one simple zero.

Finally we can integrate over ${\displaystyle z}$ taking into account ${\displaystyle g(0)=0}$

${\displaystyle g'(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\left.{\frac {d^{n}}{dw^{n}}}\left({\frac {w}{f(w)}}\right)^{n+1}\right|_{w=0}~~\Longrightarrow ~~g(z)=\sum _{n=0}^{\infty }{\frac {z^{n+1}}{(n+1)!}}\left.{\frac {d^{n}}{dw^{n}}}\left({\frac {w}{f(w)}}\right)^{n+1}\right|_{w=0}.}$

Upon a redefiniton of the summation index we get the stated 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.
• Lagrange reversion theorem for another theorem sometimes called the inversion theorem
• Formal power series#The Lagrange inversion formula

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