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 reversion of series.
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 polynomials, so the theory of analytic functions may be applied).
Faà di Bruno's formula
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 was a favourite of the king, who used frequently to discourse to him on the advantages of perfect regularity of life.
Lagrange, who was present, now discussed the whole subject afresh, and in a letter communicated to the Academy in 1808 explained how, by the variation of arbitrary constants, the periodical and secular inequalities of any system of mutually interacting bodies could be determined.
Lagrange's interests were essentially those of a student of pure mathematics: he sought and obtained far-reaching abstract results, and was content to leave the applications to others.
Lagrange did not explicitly recognize groups, but he obtained implicitly some of the simpler properties, including the theorem known after him, which states that the order of a subgroup is a divisor of the order of the group.
Lagrange did not regard the principle as an axiom but rather as a general expression of the law of equilibrium deduced from the laws of the lever and the composition of forces or, alternatively, from the properties of strings and pulleys.
Lagrange, who was present, now discussed the whole subject afresh, and in a letter communicated to the Academy in 1808 explained how, by the variation of arbitrary constants, the periodical and secular inequalities of any system of mutually interacting bodies could be determined.
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