Power series - Citizendia

In mathematics, a power series (in one variable) is an infinite series of the form

$f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n = a_0 + a_1 (x-c)^1 + a_2 (x-c)^2 + a_3 (x-c)^3 + \cdots$

where a_n represents the coefficient of the nth term, c is a constant, and x varies around c (for this reason one sometimes speaks of the series as being centered at c). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with This series usually arises as the Taylor series of some known function; the Taylor series article contains many examples. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

In many situations c is equal to zero, for instance when considering a Maclaurin series. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In such cases, the power series takes the simpler form

$f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots.$

These power series arise primarily in analysis, but also occur in combinatorics (under the name of generating functions) and in electrical engineering (under the name of the Z-transform). Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n In Mathematics and Signal processing, the Z-transform converts a discrete Time-domain signal which is a Sequence of real The familiar decimal notation for integers can also be viewed as an example of a power series, but with the argument x fixed at 10. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In number theory, the concept of p-adic numbers is also closely related to that of a power series. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897

The exponential function (in blue), and the sum of the first n+1 terms of its Maclaurin power series (in red). The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

1 Examples
2 Radius of convergence
3 Operations on power series
4 Analytic functions
5 Formal power series
6 Power series in several variables
7 Order of a power series
8 External links

Examples

Any polynomial can be easily expressed as a power series around any center c, albeit one with most coefficients equal to zero. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations For instance, the polynomial $f (x) = x 2 + 2 x + 3$ can be written as a power series around the center $c = 0$ as

$f(x) = 3 + 2 x + 1 x^2 + 0 x^3 + 0 x^4 + \cdots \,$

or around the center $c = 1$ as

$f(x) = 6 + 4 (x-1) + 1(x-1)^2 + 0(x-1)^3 + 0(x-1)^4 + \cdots \,$

or indeed around any other center c. One can view power series as being like "polynomials of infinite degree," although power series are not polynomials.

The geometric series formula

$\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + \cdots,$

which is valid for $| x | < 1$ , is one of the most important examples of a power series, as are the exponential function formula

$e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots,$

and the sine formula

$\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}+\cdots,$

valid for all real x. In Mathematics, a geometric series is a series with a constant ratio between successive terms. These power series are also examples of Taylor series. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

Negative powers are not permitted in a power series, for instance $1 + x^{-1} + x^{-2} + \cdots$ is not considered a power series (although it is a Laurent series). In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms Similarly, fractional powers such as $x 1 / 2$ are not permitted (but see Puiseux series). In Mathematics, a Puiseux expansion is a Formal power series expansion of an Algebraic function. The coefficients $a n$ are not allowed to depend on $x$ , thus for instance:

$\sin(x) x + \sin(2x) x^2 + \sin(3x) x^3 + \cdots \,$ is not a power series.

Radius of convergence

A power series will converge for some values of the variable x and may diverge for others. All power series will converge at x = c. There is always a number r with 0 ≤ r ≤ ∞ such that the series converges whenever |x − c| < r and diverges whenever |x − c| > r. The number r is called the radius of convergence of the power series; in general it is given as

$r=\liminf_{n\to\infty} \left|a_n\right|^{-\frac{1}{n}}$

or, equivalently,

$r^{-1}=\limsup_{n\to\infty} \left|a_n\right|^{\frac{1}{n}}$

(see limit superior and limit inferior). In Mathematics, the radius of convergence of a Power series is a non-negative quantity either a real number or \scriptstyle \infty that represents a In Mathematics, the limit inferior and limit superior (also called infimum limit and supremum limit, or liminf and limsup A fast way to compute it is

$r^{-1}=\lim_{n\to\infty}\left|{a_{n+1}\over a_n}\right|$

if this limit exists.

The series converges absolutely for |x - c| < r and converges uniformly on every compact subset of {x : |x − c| < r}. In Mathematics, a series (or sometimes also an Integral) is said to converge absolutely if the sum (or integral of the Absolute value of the In the mathematical field of analysis, uniform convergence is a type of Convergence stronger than Pointwise convergence.

For |x - c| = r, we cannot make any general statement on whether the series converges or diverges. However, Abel's theorem states that the sum of the series is continuous at x if the series converges at x. In Mathematics, Abel's theorem for Power series relates a limit of a power series to the sum of its Coefficients It is named after Norwegian

Operations on power series

Addition and subtraction

When two functions f and g are decomposed into power series around the same center c, the power series of the sum or difference of the functions can be obtained by termwise addition and subtraction. That is, if:

$f(x) = \sum_{n=0}^\infty a_n (x-c)^n$

$g(x) = \sum_{n=0}^\infty b_n (x-c)^n$

then

$f(x)\pm g(x) = \sum_{n=0}^\infty (a_n \pm b_n) (x-c)^n.$

Multiplication and division

With the same definitions above, for the power series of the product and quotient of the functions can be obtained as follows:

$f(x)g(x) = \left(\sum_{n=0}^\infty a_n (x-c)^n\right)\left(\sum_{n=0}^\infty b_n (x-c)^n\right)$

$= \sum_{i=0}^\infty \sum_{j=0}^\infty a_i b_j (x-c)^{i+j}$

$= \sum_{n=0}^\infty \left(\sum_{i=0}^n a_i b_{n-i}\right) (x-c)^n.$

The sequence $m_n = \sum_{i=0}^n a_i b_{n-i}$ is known as the convolution of the sequences $a n$ and $b n$ . In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and

For division, observe:

${f(x)\over g(x)} = {\sum_{n=0}^\infty a_n (x-c)^n\over\sum_{n=0}^\infty b_n (x-c)^n} = \sum_{n=0}^\infty d_n (x-c)^n$

$f(x) = \left(\sum_{n=0}^\infty b_n (x-c)^n\right)\left(\sum_{n=0}^\infty d_n (x-c)^n\right)$

and then use the above, comparing coefficients.

Differentiation and integration

Once a function is given as a power series, it is continuous wherever it converges and is differentiable on the interior of this set. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " It can be differentiated and integrated quite easily, by treating every term separately:

$f^\prime (x) = \sum_{n=1}^\infty a_n n \left( x-c \right)^{n-1}= \sum_{n=0}^\infty a_{n+1} \left(n+1 \right) \left( x-c \right)^{n}$

$\int f(x)\,dx = \sum_{n=0}^\infty \frac{a_n \left( x-c \right)^{n+1}} {n+1} + k = \sum_{n=1}^\infty \frac{a_{n-1} \left( x-c \right)^{n}} {n} + k.$

Both of these series have the same radius of convergence as the original one. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

Analytic functions

A function f defined on some open subset U of R or C is called analytic if it is locally given by power series. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in This means that every a ∈ U has an open neighborhood V ⊆ U, such that there exists a power series with center a which converges to f(x) for every x ∈ V. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space.

Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " All holomorphic functions are complex-analytic. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane Sums and products of analytic functions are analytic, as are quotients as long as the denominator is non-zero.

If a function is analytic, then it is infinitely often differentiable, but in the real case the converse is not generally true. For an analytic function, the coefficients a_n can be computed as

$a_n = \frac {f^{\left( n \right)}\left( c \right)} {n!}$

where $f (n) (c)$ denotes the nth derivative of f at c, and $f (0) (c) = f (c)$ . This means that every analytic function is locally represented by its Taylor series. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

The global form of an analytic function is completely determined by its local behavior in the following sense: if f and g are two analytic functions defined on the same connected open set U, and if there exists an element c∈U such that f⁽ⁿ⁾(c) = g⁽ⁿ⁾(c) for all n ≥ 0, then f(x) = g(x) for all x ∈ U. In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece"

If a power series with radius of convergence r is given, one can consider analytic continuations of the series, i. In Complex analysis, a branch of Mathematics, analytic continuation is a technique to extend the domain of definition of a given Analytic function. e. analytic functions f which are defined on larger sets than { x : |x - c| < r } and agree with the given power series on this set. The number r is maximal in the following sense: there always exists a complex number x with |x - a| = r such that no analytic continuation of the series can be defined at x. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

The power series expansion of the inverse function of an analytic function can be determined using the Lagrange inversion theorem. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In Mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange-Bürmann formula, gives the Taylor series expansion of the Inverse

Formal power series

Main article: Formal power series

In abstract algebra, one attempts to capture the essence of power series without being restricted to the fields of real and complex numbers, and without the need to talk about convergence. In Mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of Power series in settings that do not Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division This leads to the concept of formal power series, a concept of great utility in algebraic combinatorics. In Mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of Power series in settings that do not Algebraic combinatorics is an area of Mathematics that employs methods of Abstract algebra, notably Group theory and Representation theory, in

Power series in several variables

An extension of the theory is necessary for the purposes of multivariable calculus. Multivariable calculus is the extension of Calculus in one Variable to calculus in several variables the functions which are differentiated and integrated involve A power series is here defined to be an infinite series of the form

$f(x_1,\dots,x_n) = \sum_{j_1,\dots,j_n = 0}^{\infty}a_{j_1,\dots,j_n} \prod_{k=1}^n \left(x_k - c_k \right)^{j_k},$

where j = (j₁, . . . , j_n) is a vector of natural numbers, the coefficients a_{(j₁,. . . ,j_n)} are usually real or complex numbers, and the center c = (c₁, . . . , c_n) and argument x = (x₁, . . . , x_n) are usually real or complex vectors. In the more convenient multi-index notation this can be written

$f(x) = \sum_{\alpha \in \mathbb{N}^n} a_{\alpha} \left(x - c \right)^{\alpha}.$

The theory of such series is trickier than for single-variable series, with more complicated regions of convergence. The Mathematical notation of multi-indices simplifies formulae used in Multivariable calculus, Partial differential equations and the theory of distributions For instance, the power series $\sum_{n=0}^\infty x_1^n x_2^n$ is absolutely convergent in the set ${(x 1, x 2): | x 1 x 2 | < 1}$ between two hyperbolae. (This is an example of a log-convex set, in the sense that the set of points $(log | x 1 | ,log | x 2 | )$ , where $(x 1, x 2)$ lies in the above region, is a convex set. More generally, one can show that when c=0, the interior of the region of absolute convergence is always a log-convex set in this sense. ) On the other hand, in the interior of this region of convergence one may differentiate and integrate under the series sign, just as one may with ordinary power series.

Order of a power series

Let α be a multi-index for a power series f(x₁, x₂, …, x_n). The order of the power series f is defined to be the least value |α| such that a_α ≠ 0, or 0 if f ≡ 0. In particular, for a power series f(x) in a single variable x, the order of f is the smallest power of x with a nonzero coefficient. This definition readily extends to Laurent series. In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms

External links

Eric W. Weisstein, Formal Power Series at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Eric W. Weisstein, Power Series at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Complex Power Series Module by John H. Mathews
Powers of Complex Numbers by Michael Schreiber, The Wolfram Demonstrations Project.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents