Search results
Results From The WOW.Com Content Network
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.
In a formal power series, the powers of the variable are used only as position-holders for the coefficients, so that the coefficient of is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for ...
In mathematics, a power series (in one variable) is an infinite series of the form = = + + + … where represents the coefficient of the nth term and c is a constant called the center of the series. Power series are useful in mathematical analysis , where they arise as Taylor series of infinitely differentiable functions .
One can also consider a strictly larger ring, by defining as a generalized polynomial an infinite (or finite) formal sum of monomials with a bounded degree. This ring is larger than the usual polynomial ring, as it includes infinite sums of variables. However, it is smaller than the ring of power series in infinitely many variables.
Each type of convergence corresponds to a different metric for the space of functions that are added together in the series, and thus a different type of limit. The Weierstrass M-test is a useful result in studying convergence of function series.
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 reversion theorem for another theorem sometimes called the inversion theorem; Formal power series#The Lagrange inversion ...
The simplest example is the additive formal group law F(x, y) = x + y. The idea of the definition is that F should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the identity of the Lie group is the origin.