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)

    “There must be a profound recognition that parents are the first teachers and that education begins before formal schooling and is deeply rooted in the values, traditions, and norms of family and culture.”
    —Sara Lawrence Lightfoot (20th century)

    “I never saw love as luck, as that gift from the gods which put everything else in place, and allowed you to succeed. No, I saw love as reward. One could find it only after one’s virtue, or one’s courage, or self-sacrifice, or generosity, or loss, has succeeded in stirring the power of creation.”
    —Norman Mailer (b. 1923)

    “The theory of truth is a series of truisms.”
    —J.L. (John Langshaw)

    “Those things which now most engage the attention of men, as politics and the daily routine, are, it is true, vital functions of human society, but should be unconsciously performed, like the corresponding functions of the physical body.”
    —Henry David Thoreau (1817–1862)