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
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:
If the formal power series g with g(0) = 0 is given implicitly by the equation
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)
“The formal Washington dinner party has all the spontaneity of a Japanese imperial funeral.”
—Simon Hoggart (b. 1946)
“The dignity and stability of government in all its branches, the morals of the people, and every blessing of society, depend so much upon an upright and skilful administration of justice, that the judicial power ought to be distinct from both the legislative and executive, and independent upon both, that so it may be a check upon both, as both should be checks upon that.”
—Thomas Jefferson (1743–1826)
“The woman’s world ... is shown as a series of limited spaces, with the woman struggling to get free of them. The struggle is what the film is about; what is struggled against is the limited space itself. Consequently, to make its point, the film has to deny itself and suggest it was the struggle that was wrong, not the space.”
—Jeanine Basinger (b. 1936)
“Mark the babe
Not long accustomed to this breathing world;
One that hath barely learned to shape a smile,
Though yet irrational of soul, to grasp
With tiny finger—to let fall a tear;
And, as the heavy cloud of sleep dissolves,
To stretch his limbs, bemocking, as might seem,
The outward functions of intelligent man.”
—William Wordsworth (1770–1850)