:
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''). This is also called .
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. If 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 . There is a straightforward derivation using
Complex Analysis and
Contour Integration (the complex formal power series version is clearly a consequence of knowing the formula for
Polynomial s, so the theory of
Analytic Function s may be applied).
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 ''n''th derivative of a composite function.
