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.
Suppose z is defined as a function of w by an equation of the form where f is analytic at a point a and
Then it is possible to invert or solve the equation for w, expressing it in the form
given by a power series[1] where The theorem further states that this series has a non-zero radius of convergence, i.e.,
represents an analytic function of z in a neighbourhood of
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
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.
In fact, the Lagrange inversion theorem has a number of additional rather different proofs, including ones using tree-counting arguments or induction.
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 with f0 = 0 and f1 ≠ 0, then an explicit form of inverse coefficients can be given in term of Bell polynomials:[10] where is the rising factorial.
When f1 = 1, the last formula can be interpreted in terms of the faces of associahedra [11] where
For instance, the algebraic equation of degree p can be solved for x by means of the Lagrange inversion formula for the function f(x) = x − xp, 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 f can be defined.
There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when
is an operator which extracts the coefficient of
in the Taylor series of a function of w. A generalization of the formula is known as the Lagrange–Bürmann formula: where H is an arbitrary analytic function.
A simpler version of the formula replaces H′(w) with H(w)(1 − φ′(w)/φ(w)) to get which involves φ′(w) instead of H′(w).
that is implicitly defined by the equation We may use the theorem to compute the Taylor series of
Recognizing that this gives The radius of convergence of this series is
(giving the principal branch of the Lambert function).
) can also be derived by series inversion.
can be expanded into a power series and inverted.
of unlabelled binary trees.
is either a leaf of size zero, or a root node with two subtrees.
the number of binary trees on
Removing the root splits a binary tree into two trees of smaller size.
is the nth Catalan number.
In the Laplace–Erdelyi theorem that gives the asymptotic approximation for Laplace-type integrals, the function inversion is taken as a crucial step.