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Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center c is equal to zero, for instance for Maclaurin series.
The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any unital Banach algebra B. In this setting, e 0 = 1, and e x is invertible with inverse e −x for any x in B.
The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions include: The partial sums (the Taylor polynomials) of the series can be used as approximations of the function ...
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
Formal power series are used in combinatorics to describe and study sequences that are otherwise difficult to handle, for example, using the method of generating functions. The Hilbert–Poincaré series is a formal power series used to study graded algebras.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
The formula for the exponential results from reducing the powers of G in the series expansion and identifying the respective series coefficients of G 2 and G with −cos(θ) and sin(θ) respectively. The second expression here for e Gθ is the same as the expression for R ( θ ) in the article containing the derivation of the generator , R ( θ ...
A power series with coefficients in the field of algebraic numbers = =! ¯ [[]]is called an E-function [1] if it satisfies the following three conditions: . It is a solution of a non-zero linear differential equation with polynomial coefficients (this implies that all the coefficients c n belong to the same algebraic number field, K, which has finite degree over the rational numbers);