Formal Power Series - Interpreting Formal Power Series As Functions

Interpreting Formal Power Series As Functions

In mathematical analysis, every convergent power series defines a function with values in the real or complex numbers. Formal power series can also be interpreted as functions, but one has to be careful with the domain and codomain. If f = ∑an Xn is an element of R], S is a commutative associative algebra over R, I is an ideal in S such that the I-adic topology on S is complete, and x is an element of I, then we can define

f(X) = \sum_{n\ge 0} a_n X^n.

This latter series is guaranteed to converge in S given the above assumptions on X. Furthermore, we have

and

Unlike in the case of bona fide functions, these formulas are not definitions but have to be proved.

Since the topology on R] is the (X)-adic topology and R] is complete, we can in particular apply power series to other power series, provided that the arguments don't have constant coefficients (so that they belong to the ideal (X)): f(0), f(X2−X) and f( (1 − X)−1 − 1) are all well defined for any formal power series f∈R].

With this formalism, we can give an explicit formula for the multiplicative inverse of a power series f whose constant coefficient a = f(0) is invertible in R:

f^{-1} = \sum_{n \ge 0} a^{-n-1} (a-f)^n.

If the formal power series g with g(0) = 0 is given implicitly by the equation

f(g) = X \,

where f is a known power series with f(0) = 0, then the coefficients of g can be explicitly computed using the Lagrange inversion formula.

Read more about this topic:  Formal Power Series

Famous quotes containing the words interpreting, formal, power, series and/or functions:

    “Drawing is a struggle between nature and the artist, in which the better the artist understands the intentions of nature, the more easily he will triumph over it. For him it is not a question of copying, but of interpreting in a simpler and more luminous language.”
    —Charles Baudelaire (1821–1867)

    “On every formal visit a child ought to be of the party, by way of provision for discourse.”
    —Jane Austen (1775–1817)

    “The farmer imagines power and place are fine things. But the President has paid dear for his White House. It has commonly cost him all his peace, and the best of his manly attributes. To preserve for a short time so conspicuous an appearance before the world, he is content to eat dust before the real masters who stand erect behind the throne.”
    —Ralph Waldo Emerson (1803–1882)

    “As Cuvier could correctly describe a whole animal by the contemplation of a single bone, so the observer who has thoroughly understood one link in a series of incidents should be able to accurately state all the other ones, both before and after.”
    —Sir Arthur Conan Doyle (1859–1930)

    “Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
    —Ralph Waldo Emerson (1803–1882)