冪級数、べき級数

WordNet

(electronics) connection of components in such a manner that current flows first through one and then through the other; "the voltage divider consisted of a series of fixed resistors"
a periodical that appears at scheduled times (同)serial, serial publication
(mathematics) the sum of a finite or infinite sequence of expressions
(sports) several contests played successively by the same teams; "the visiting team swept the series"
a group of postage stamps having a common theme or a group of coins or currency selected as a group for study or collection; "the Post Office issued a series commemorating famous American entertainers"; "his coin collection included the complete series of
similar things placed in order or happening one after another; "they were investigating a series of bank robberies"
supply the force or power for the functioning of; "The gasoline powers the engines"
(physics) the rate of doing work; measured in watts (= joules/second)
one possessing or exercising power or influence or authority; "the mysterious presence of an evil power"; "may the force be with you"; "the forces of evil" (同)force
possession of controlling influence; "the deterrent power of nuclear weapons"; "the power of his love saved her"; "his powerfulness was concealed by a gentle facade" (同)powerfulness
(often used in combination) having or using or propelled by means of power or power of a specified kind; "powered flight"; "kerosine-powered jet engines"

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(関連のあるもの・同種のものの)(規則的な)『連続』,一続き《+『of』+『名』》 / (同種の貨幣・切手などの)一組(set)《+『of』+『名』》 / (テレビ番組などの)連続物;(出版物の)双書,シリーズ / (電気の)直列 / (電気が)直列の
〈U〉(名身が生れつき備えている,また物事を遂行できる)『能力』,『力』 / 〈U〉(…に対する)権力,支配権《+『over』+『名』》 / 〈C〉(法・官職などによる)権限 / 《複数形で》(生まれながらの)体力,知力,才能 / 〈C〉権力者,実力者;影響力のある集団;強国・大国〈U〉軍事力;国力 / 〈U〉強さ,力 / 〈C〉《通例複数形で》神;悪魔 / 〈U〉動力;工率,仕事率 / 〈U〉(電気などの)エネルギー,力 / 〈C〉(数学で)累乗 / 〈U〉(レンズの)倍率 / 〈U〉《a~》《話》(…の)多数,多量 / 〈機械〉‘に'動力を供給する

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2012/11/05 19:21:59」(JST)

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

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). This series usually arises as the Taylor series of some known function; the Taylor series article contains many examples.

In many situations c is equal to zero, for instance when considering a Maclaurin series. 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). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at ⅟10. In number theory, the concept of p-adic numbers is also closely related to that of a power series.

The exponential function (in blue), and the sum of the first n+1 terms of its Maclaurin power series (in red).

1 Examples
2 Radius of convergence
3 Operations on power series
- 3.1 Addition and subtraction
- 3.2 Multiplication and division
- 3.3 Differentiation and integration
4 Analytic functions
5 Formal power series
6 Power series in several variables
7 Order of a power series
8 See also
9 References
10 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. For instance, the polynomial can be written as a power series around the center as

or around the center as

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

which is valid for , is one of the most important examples of a power series, as are the exponential function formula

and the sine formula

valid for all real x. These power series are also examples of Taylor series.

Negative powers are not permitted in a power series, for instance is not considered a power series (although it is a Laurent series). Similarly, fractional powers such as are not permitted (but see Puiseux series). The coefficients are not allowed to depend on , thus for instance:

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 f(x) in powers of (x-c) will converge at x = c. (The correct value f(c) = a₀ requires interpreting the expression 0⁰ as equal to 1.) If c is not the only convergent point, then 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

or, equivalently,

(this is the Cauchy–Hadamard theorem; see limit superior and limit inferior for an explanation of the notation). A fast way to compute it is

if this limit exists.

The series converges absolutely for |x − c| < r and converges uniformly on every compact subset of {x : |x − c| < r}. That is, the series is absolutely and compactly convergent on the interior of the disc of convergence.

For |x − c| = r, we cannot make any general statement on whether the series converges or diverges. However, for the case of real variables, Abel's theorem states that the sum of the series is continuous at x if the series converges at x. In the case of complex variables, we can only claim continuity along the line segment starting at c and ending at x.

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:

then

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:

The sequence is known as the convolution of the sequences and .

For division, observe:

and then use the above, comparing coefficients.

Differentiation and integration

Once a function is given as a power series, it is differentiable on the interior of the domain of convergence. 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.

Analytic functions

A function f defined on some open subset U of R or C is called analytic if it is locally given by a convergent power series. 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.

Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. All holomorphic functions are complex-analytic. 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 denotes the nth derivative of f at c, and . This means that every analytic function is locally represented by its Taylor series.

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.

If a power series with radius of convergence r is given, one can consider analytic continuations of the series, i.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 − c| = r such that no analytic continuation of the series can be defined at x.

The power series expansion of the inverse function of an analytic function can be determined using the Lagrange inversion theorem.

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. This leads to the concept of formal power series, a concept of great utility in algebraic combinatorics.

Power series in several variables

An extension of the theory is necessary for the purposes of multivariable calculus. 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. For instance, the power series is absolutely convergent in the set between two hyperbolas. (This is an example of a log-convex set, in the sense that the set of points , where 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.

References

Solomentsev, E.D. (2001), "Power series", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=Power_series&oldid=15309

External links

Weisstein, Eric W., "Formal Power Series" from MathWorld.
Weisstein, Eric W., "Power Series" from MathWorld.
Complex Power Series Module by John H. Mathews
Powers of Complex Numbers by Michael Schreiber, Wolfram Demonstrations Project.

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