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Power series. In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the n th 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.
Differential equations. In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.
e. In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions.
A formal power series is a special kind of formal series, of the form. where the called coefficients, are numbers or, more generally, elements of some ring, and the are formal powers of the symbol that is called an indeterminate or, commonly, a variable. Hence, power series can be viewed as a generalization of polynomials where the number of ...
The Cauchy product of these two power series is defined by a discrete convolution as follows: ... Zizler, Václav (2015), An Introduction to Modern Analysis, Springer ...
Description. Perturbation theory develops an expression for the desired solution in terms of a formal power series known as a perturbation series in some "small" parameter, that quantifies the deviation from the exactly solvable problem. The leading term in this power series is the solution of the exactly solvable problem, while further terms ...
Generating 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. There are various types of generating ...
If a(n) is a discrete function of a positive integer n, then the power series associated to a(n) is the series = where x is a real variable (see Z-transform). Replacing summation over n with integration over t , a continuous version of the power series becomes ∫ 0 ∞ f ( t ) x t d t {\displaystyle \int _{0}^{\infty }f(t)x^{t}\,dt} where the ...