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.
Suppose is defined as a function of by an equation of the form
z=f(w)
where is analytic at a point and
f'(a) ≠ 0.
w=g(z)
g(z)=a+
infty | |
\sum | |
n=1 |
gn
(z-f(a))n | |
n! |
,
gn=\limw
dn-1 | |
dwn-1 |
\left[\left(
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)
z=f(a).
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 for any analytic function ; and it can be generalized to the case
f'(a)=0,
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 is a formal power series, then the above formula does not give the coefficients of the compositional inverse series directly in terms for the coefficients of the series . If one can express the functions and in formal power series as
f(w)=
infty | |
\sum | |
k=0 |
fk
wk | |
k! |
and g(z)=
infty | |
\sum | |
k=0 |
gk
zk | |
k! |
with and, then an explicit form of inverse coefficients can be given in term of Bell polynomials:[7]
gn=
1 | ||||||
|
n-1 | |
\sum | |
k=1 |
(-1)kn\overline{k}Bn-1,k(\hat{f}1,\hat{f}2,\ldots,\hat{f}n-k), n\geq2,
where
\begin{align} \hat{f}k&=
fk+1 | |
(k+1)f1 |
,\\ g1&=
1 | |
f1 |
,and\\ n\overline{k
When, the last formula can be interpreted in terms of the faces of associahedra [8]
gn=
\sum | |
FfaceofKn |
(-1)n-\dimfF, n\geq2,
where
fF=
f | |
i1 |
…
f | |
im |
F=
K | |
i1 |
x … x
K | |
im |
Kn.
For instance, the algebraic equation of degree
xp-x+z=0
x=
infty | |
\sum | |
k=0 |
\binom{pk}{k}
z(p-1)k+1 | |
(p-1)k+1 |
.
By convergence tests, this series is in fact convergent for
|z|\leq(p-1)p-p/(p-1),
There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when
f(w)=w/\phi(w)
\phi(w)
\phi(0)\ne0.
a=0
f(a)=f(0)=0.
g(z)
f(g(z))\equivz
\begin{align} g(z)&=
infty | |
\sum | |
n=1 |
\left[\limw
dn-1 | |
dwn-1 |
\left(\left(
w | |
w/\phi(w) |
\right)n\right)\right]
zn | |
n! |
\\ {}&=
infty | |
\sum | |
n=1 |
1 | \left[ | |
n |
1 | |
(n-1)! |
\limw
dn-1 | |
dwn-1 |
(\phi(w)n)\right]zn, \end{align}
which can be written alternatively as
[zn]g(z)=
1 | |
n |
[wn-1]\phi(w)n,
where
[wr]
wr
A generalization of the formula is known as the Lagrange–Bürmann formula:
[zn]H(g(z))=
1 | |
n |
[wn-1](H'(w)\phi(w)n)
where is an arbitrary analytic function.
Sometimes, the derivative can be quite complicated. A simpler version of the formula replaces with to get
[zn]H(g(z))=[wn]H(w)\phi(w)n-1(\phi(w)-w\phi'(w)),
which involves instead of .
See main article: Lambert W function. The Lambert function is the function
W(z)
W(z)eW(z)=z.
We may use the theorem to compute the Taylor series of
W(z)
z=0.
f(w)=wew
a=0.
dn | |
dxn |
e\alpha=\alphane\alpha,
this gives
\begin{align} W(z)&=
infty | |
\sum | |
n=1 |
\left[\limw
dn-1 | |
dwn-1 |
e-nw\right]
zn | |
n! |
\\ {}&=
infty | |
\sum | |
n=1 |
(-n)n-1
zn | |
n! |
\\ {}&=
| ||||
z-z |
| ||||
z |
z4+O(z5). \end{align}
The radius of convergence of this series is
e-1
A series that converges for
|ln(z)-1|<{4+\pi2}
2.58\ldots ⋅ 10-6<z<2.869\ldots ⋅ 106
f(z)=W(ez)-1
1+f(z)+ln(1+f(z))=z.
Then
z+ln(1+z)
f(z+1)=W(ez+1)-1:
W(e1+z)=1+
z | |
2 |
+
z2 | |
16 |
-
z3 | |
192 |
-
z4 | |
3072 |
+
13z5 | |
61440 |
-O(z6).
W(x)
lnx-1
W(1) ≈ 0.567143.
Consider[10] the set
l{B}
l{B}
Bn
n
Removing the root splits a binary tree into two trees of smaller size. This yields the functional equation on the generating function
styleB(z)=
infty | |
\sum | |
n=0 |
Bnzn:
B(z)=1+zB(z)2.
Letting
C(z)=B(z)-1
C(z)=z(C(z)+1)2.
\phi(w)=(w+1)2
Bn=[zn]C(z)=
1 | |
n |
[wn-1](w+1)2n=
1 | |
n |
\binom{2n}{n-1}=
1 | |
n+1 |
\binom{2n}{n}.
This shows that
Bn
In the Laplace–Erdelyi theorem that gives the asymptotic approximation for Laplace-type integrals, the function inversion is taken as a crucial step.