In
mathematics, a power series (in one
variable) is an
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
of the form
where ''a
n'' represents the
coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, where they arise as
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of
infinitely differentiable function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s. In fact,
Borel's theorem implies that every power series is the Taylor series of some smooth function.
In many situations, ''c'' (the ''center'' of the series) is equal to zero, for instance when considering a
Maclaurin series. In such cases, the power series takes the simpler form
Beyond their role in mathematical analysis, power series also occur in
combinatorics as
generating functions (a kind of
formal power series) and in electronic engineering (under the name of the
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-tim ...
). The familiar
decimal notation
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
for
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s can also be viewed as an example of a power series, with
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
coefficients, but with the argument ''x'' fixed at . In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, the concept of
''p''-adic numbers is also closely related to that of a power series.
Examples
Polynomial
Any
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
can be easily expressed as a power series around any center ''c'', although all but finitely many of the coefficients will be zero since a power series has infinitely many terms by definition. For instance, the polynomial
can be written as a power series around the center
as
or around the center
as
This is because of the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansion of f(x) around
is
as
and the non-zero derivatives are
, so
and
, a constant.
Or indeed the expansion is possible around any other center ''c''. One can view power series as being like "polynomials of infinite degree," although power series are not polynomials.
Geometric series, exponential function and sine
The
geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each suc ...
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
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
.
On the set of exponents
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
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series
: \begin
x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\
&=x^+ 2x^ + x^ + 2x^ + x^ + ...
). The coefficients
are not allowed to depend on thus for instance:
is not a power series.
Radius of convergence
A power series
is
convergent for some values of the variable , which will always include (as usual,
evaluates as and the sum of the series is thus
for ). The series may
diverge for other values of . If is not the only point of convergence, then there is always a number with such that the series converges whenever and diverges whenever . The number is called the
radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
of the power series; in general it is given as
or, equivalently,
(this is the
Cauchy–Hadamard theorem
In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cau ...
; see
limit superior and limit inferior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
for an explanation of the notation). The relation
is also satisfied, if this limit exists.
The set of the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s such that is called the
disc of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
of the series. The series
converges absolutely
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
inside its disc of convergence, and
converges uniformly on every
compact subset of the disc of convergence.
For , there is no general statement on the convergence of the series. However,
Abel's theorem
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 mathematician Niels Henrik Abel.
Theorem
Let the Taylor series
G (x) = \sum_^\infty a_k x^k
be a pow ...
states that if the series is convergent for some value such that , then the sum of the series for is the limit of the sum of the series for where is a real variable less than that tends to .
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
and
then
It is not true that if two power series
and
have the same radius of convergence, then
also has this radius of convergence. If
and
, then both series have the same radius of convergence of 1, but the series
has a radius of convergence of 3.
The sum of two power series will have, at minimum, a radius of convergence of the smaller of the two radii of convergence of the two series (and it may be higher than either, as seen in the example above).
Multiplication and division
With the same definitions for
and
, the power series of the product and quotient of the functions can be obtained as follows:
The sequence
is known as the
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of the sequences
and
For division, if one defines the sequence
by
then
and one can solve recursively for the terms
by comparing coefficients.
Solving the corresponding equations yields the formulae based on
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
s of certain matrices of the coefficients of
and
Differentiation and integration
Once a function
is given as a power series as above, it is
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
on the
interior of the domain of convergence. It can be
differentiated and
integrated quite easily, by treating every term separately:
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'' that 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 differentiable, but in the real case the converse is not generally true. For an analytic function, the coefficients ''a''
''n'' can be computed as
where
denotes the ''n''th derivative of ''f'' at ''c'', and
. This means that every analytic function is locally represented by its
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
.
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
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
open set ''U'', and if there exists an element such that for all , then for all .
If a power series with radius of convergence ''r'' is given, one can consider
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
s of the series, i.e. analytic functions ''f'' which are defined on larger sets than 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
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
with such that no analytic continuation of the series can be defined at .
The power series expansion of the
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X ...
of an analytic function can be determined using the
Lagrange inversion theorem.
Behavior near the boundary
The sum of a power series with a positive radius of convergence is an analytic function at every point in the interior of the disc of convergence. However, different behavior can occur at points on the boundary of that disc. For example:
# ''Divergence while the sum extends to an analytic function'':
has radius of convergence equal to
and diverges at every point of
. Nevertheless, the sum in
is
, which is analytic at every point of the plane except for
.
# ''Convergent at some points divergent at others'':
has radius of convergence
. It converges for
, while it diverges for
.
# ''Absolute convergence at every point of the boundary'':
has radius of convergence
, while it converges absolutely, and uniformly, at every point of
due to
Weierstrass M-test
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous t ...
applied with the
hyper-harmonic convergent series .
# ''Convergent on the closure of the disc of convergence but not continuous sum'':
Sierpiński gave an example of a power series with radius of convergence
, convergent at all points with
, but the sum is an unbounded function and, in particular, discontinuous. A sufficient condition for one-sided continuity at a boundary point is given by
Abel's theorem
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 mathematician Niels Henrik Abel.
Theorem
Let the Taylor series
G (x) = \sum_^\infty a_k x^k
be a pow ...
.
Formal power series
In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, 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
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather ...
. A power series is here defined to be an infinite series of the form
where is a vector of natural numbers, the coefficients are usually real or complex numbers, and the center and argument are usually real or complex vectors. The symbol
is the
product symbol, denoting multiplication. In the more convenient
multi-index
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
notation this can be written
where
is the set of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
s, and so
is the set of ordered ''n''-
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
s of natural numbers.
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 be a multi-index for a power series . The order of the power series ''f'' is defined to be the least value
such that there is ''a''
''α'' ≠ 0 with
, or
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.
Notes
References
*
External links
*
*
Powers of Complex Numbersby Michael Schreiber,
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
{{DEFAULTSORT:Power Series
Real analysis
Complex analysis
Multivariable calculus
Mathematical series