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.

Statement

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 w=g(z) given by a power series where The theorem further states that this series has a non-zero radius of convergence, i.e., g(z) represents an analytic function of z in a neighbourhood of 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 f'(a)=0, where the inverse g is a multivalued function. The theorem was proved by Lagrange and generalized by Hans Heinrich Bürmann, both in the late 18th century. There is a straightforward derivation using complex analysis and contour integration; 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: where is the rising factorial. When f1 = 1 , the last formula can be interpreted in terms of the faces of associahedra where for each face of the associahedron K_n.

Example

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.

Applications

Lagrange–Bürmann formula

There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when for some analytic \phi(w) with Take a=0 to obtain Then for the inverse g(z) (satisfying ), we have which can be written alternatively as where [w^r] is an operator which extracts the coefficient of w^r 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. Sometimes, the derivative (w) can be quite complicated. A simpler version of the formula replaces (w) with H(w)(1 − (w)/φ(w)) to get which involves (w) instead of (w) .

Lambert W function

The Lambert W function is the function W(z) that is implicitly defined by the equation We may use the theorem to compute the Taylor series of W(z) at z=0. We take f(w) = we^w and a = 0. Recognizing that this gives The radius of convergence of this series is e^{-1} (giving the principal branch of the Lambert function). A series that converges for (approximately ) can also be derived by series inversion. The function satisfies the equation Then can be expanded into a power series and inverted. This gives a series for W(x) can be computed by substituting \ln x - 1 for z in the above series. For example, substituting −1 for z gives the value of

Binary trees

Consider the set \mathcal{B} of unlabelled binary trees. An element of \mathcal{B} is either a leaf of size zero, or a root node with two subtrees. Denote by B_n the number of binary trees on n nodes. Removing the root splits a binary tree into two trees of smaller size. This yields the functional equation on the generating function Letting, one has thus Applying the theorem with yields This shows that B_n is the nth Catalan 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.