Lagrange inversion theorem
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 the dependence between the variables
w and
z is implicitly defined by an equation of the form
where
f is analytic at a point
a and
f '(
a) ≠ 0. Then it is possible to
invert or
solve the equation for
w:
where
g is analytic at the point
b =
f(
a). The series expansion of
g is given by
This formula can for instance be used to find the Taylor series of the
Lambert W function (by setting
f(
w) =
w exp(
w) and
a=
b=0).
The formula is also valid for formal power series and can be generalized in various ways. It 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 Bürmann, both in the late 18th century.