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 is defined as a function of by an equation of the form

where is analytic at a point Then it is possible to invert or solve the equation for given by a power series

where

represents an analytic function of in a neighbourhood 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 for any analytic function where the inverse 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 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 polynomials:

where

is the rising factorial.

When , the last formula can be interpreted in terms of the faces of associahedra

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 which is also the largest disk in which a local inverse to can be defined.

Applications

Lagrange–Bürmann formula

There is a special case of Lagrange inversion theorem that is used in combinatorics), we have

which can be written alternatively as

in the Taylor series of a function of .

A generalization of the formula is known as the Lagrange–Bürmann formula:

where is an arbitrary analytic function.

Sometimes, the derivative can be quite complicated. A simpler version of the formula replaces with to get

which involves instead of .

Lambert W function

The Lambert that is implicitly defined by the equation

We may use the theorem to compute the Taylor series Recognizing that

this gives

The radius of convergence (giving the principal branch of the Lambert function).

satisfies the equation

can be expanded into a power series and inverted.

for in the above series. For example, substituting for

Binary trees

Consider of unlabelled binary trees nodes.

yields

is the th 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.

See also

• 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

References

External links

• Bürmann–Lagrange series at Springer EOM