In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
expansion of the
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X\t ...
of an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
.
Statement
Suppose is defined as a function of by an equation of the form
:
where is analytic at a point and
Then it is possible to ''invert'' or ''solve'' the equation for , expressing it in the form
given by a power series
:
where
:
The theorem further states that this series has a non-zero radius of convergence, i.e.,
represents an analytic function of in a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of
This is also called reversion of series.
If the assertions about analyticity are omitted, the formula is also valid for
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
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
where the inverse is a multivalued function.
The theorem was proved by
Lagrange and generalized by
Hans Heinrich Bürmann
Hans Heinrich Bürmann (died 21 June 1817, in Mannheim) was a German mathematician and teacher. He ran an "academy of commerce" in Mannheim since 1795 where he used to teach mathematics. He also served as a censor in Mannheim. He was nominated He ...
, both in the late 18th century. There is a straightforward derivation using
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
and
contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
Contour integration is closely related to the calculus of residues, a method of complex analysis.
...
; the complex formal power series version is a consequence of knowing the formula for
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s, so the theory of
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
s 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
:
with and , then an explicit form of inverse coefficients can be given in term of
Bell polynomial
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno's fo ...
s:
:
where
:
is the
rising factorial
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial
:\begin
(x)_n = x^\underline &= \overbrace^ \\
&= \prod_^n(x-k+1) = \prod_^(x-k) \,.
\e ...
.
When , the last formula can be interpreted in terms of the faces of
associahedra
:
where
for each face
of the associahedron
Example
For instance, the algebraic equation of degree
:
can be solved for by means of the Lagrange inversion formula for the function , resulting in a formal series solution
:
By convergence tests, this series is in fact convergent for
which is also the largest disk in which a local inverse to can be defined.
Sketch of the proof
For simplicity suppose
. We can then compute
:
If we expand the integrand using the geometric series we get
:
where in the last step we used the fact that
has one simple zero.
Finally we can integrate over
taking into account
:
Upon a redefiniton of the summation index we get the stated formula.
Applications
Lagrange–Bürmann formula
There is a special case of Lagrange inversion theorem that is used in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
and applies when
for some analytic
with
Take
to obtain
Then for the inverse
(satisfying
), we have
:
which can be written alternatively as
:
where