In _{n}'' represents the coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in

_{''n''} can be computed as
$$a\_n\; =\; \backslash frac$$
where $f^(c)$ denotes the ''n''th derivative of ''f'' at ''c'', and $f^(c)\; =\; f(c)$. This means that every analytic function is locally represented by its

_{''α''} ≠ 0 with $r\; =\; ,\; \backslash alpha,\; =\; \backslash alpha\_1\; +\; \backslash alpha\_2\; +\; \backslash cdots\; +\; \backslash alpha\_n$, or $\backslash infty$ 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

Powers of Complex Numbers

by Michael Schreiber,

mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...

, a power series (in one variable) is an infinite series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of the form
$$\backslash sum\_^\backslash infty\; a\_n\; \backslash left(x\; -\; c\backslash right)^n\; =\; a\_0\; +\; a\_1\; (x\; -\; c)\; +\; a\_2\; (x\; -\; c)^2\; +\; \backslash cdots$$
where ''amathematical analysis
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...

, where they arise as Taylor series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of infinitely differentiable function
is a smooth function with compact support.
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered " ...

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 seriesMaclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin (1 ...

. In such cases, the power series takes the simpler form
$$\backslash sum\_^\backslash infty\; a\_n\; x^n\; =\; a\_0\; +\; a\_1\; x\; +\; a\_2\; x^2\; +\; \backslash cdots.$$
Beyond their role in mathematical analysis, power series also occur in combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other area ...

as generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers (''a'n'') by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinar ...

s (a kind of formal power series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

) and in electronic engineering (under the name of the Z-transform
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

). The familiar decimal notation
The decimal numeral system
A numeral system (or system of numeration) is a writing system
A writing system is a method of visually representing verbal communication
Communication (from Latin ''communicare'', meaning "to share") is t ...

for real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s can also be viewed as an example of a power series, with integer 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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777– ...

, the concept of p-adic number
group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analys ...

s is also closely related to that of a power series.
Examples

The _(in_blue),_and_the_sum_of_the_first_''n''_+_1_terms_of_its_Maclaurin_series">Maclaurin_power_series_(in_red)..html" ;"title="aclaurin_series.html" ;"title="exponential function (in blue), and the sum of the first ''n'' + 1 terms of its Maclaurin series">Maclaurin power series (in red).">aclaurin_series.html" ;"title="exponential function (in blue), and the sum of the first ''n'' + 1 terms of its Maclaurin series">Maclaurin power series (in red). Any polynomial 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 $f(x)\; =\; x^2\; +\; 2x\; +\; 3$ can be written as a power series around the center $c\; =\; 0$ as $$f(x)\; =\; 3\; +\; 2\; x\; +\; 1\; x^2\; +\; 0\; x^3\; +\; 0\; x^4\; +\; \backslash cdots$$ or around the center $c\; =\; 1$ as $$f(x)\; =\; 6\; +\; 4(x\; -\; 1)\; +\; 1(x\; -\; 1)^2\; +\; 0(x\; -\; 1)^3\; +\; 0(x\; -\; 1)^4\; +\; \backslash cdots$$ 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. Thegeometric series
In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series
:\frac \,+\, \frac \,+\, ...

formula
$$\backslash frac\; =\; \backslash sum\_^\backslash infty\; x^n\; =\; 1\; +\; x\; +\; x^2\; +\; x^3\; +\; \backslash cdots,$$
which is valid for $,\; x,\; <\; 1$, is one of the most important examples of a power series, as are the exponential function formula
$$e^x\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; 1\; +\; x\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots,$$
and the sine formula
$$\backslash sin(x)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; x\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots,$$
valid for all real ''x''.
These power series are also examples of Taylor series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

.
On the set of exponents

Negative powers are not permitted in a power series; for instance, $1\; +\; x^\; +\; x^\; +\; \backslash cdots$ is not considered a power series (although it is aLaurent series
(Holomorphic functions are analytic, analytic).
In mathematics, the Laurent series of a complex function ''f''(''z'') is a representation of that function as a power series which includes terms of negative degree. It may be used to express compl ...

). Similarly, fractional powers such as $x^\backslash frac$ are not permitted (but see Puiseux series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

). The coefficients $a\_n$ are not allowed to depend on thus for instance:
$$\backslash sin(x)\; x\; +\; \backslash sin(2x)\; x^2\; +\; \backslash sin(3x)\; x^3\; +\; \backslash cdots$$
is not a power series.
Radius of convergence

A power series $\backslash sum\_^\backslash infty\; a\_n(x-c)^n$ isconvergent
Convergent is an adjective for things that wikt:converge, converge. It is commonly used in mathematics and may refer to:
*Convergent boundary, a type of plate tectonic boundary
* Convergent (continued fraction)
* Convergent evolution
* Convergent s ...

for some values of the variable , which will always include (as usual, $(x-c)^0$ evaluates as and the sum of the series is thus $a\_0$ 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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of the power series; in general it is given as
$$r\; =\; \backslash liminf\_\; \backslash left,\; a\_n\backslash ^$$
or, equivalently,
$$r^\; =\; \backslash limsup\_\; \backslash left,\; a\_n\backslash ^\backslash frac$$
(this is the Cauchy–Hadamard theoremIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

; see limit superior and limit inferior
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

for an explanation of the notation). The relation
$$r^\; =\; \backslash lim\_\backslash left,\; \backslash $$
is also satisfied, if this limit exists.
The set of the complex number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s such that is called the disc of convergence
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of the series. The series converges absolutely inside its disc of convergence, and converges uniformly on every compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British N ...

subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are u ...

of the disc of convergence.
For , there is no general statement on the convergence of the series. However, Abel's theorem
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 $$f(x)\; =\; \backslash sum\_^\backslash infty\; a\_n\; (x\; -\; c)^n$$ and $$g(x)\; =\; \backslash sum\_^\backslash infty\; b\_n\; (x\; -\; c)^n$$ then $$f(x)\; \backslash pm\; g(x)\; =\; \backslash sum\_^\backslash infty\; (a\_n\; \backslash pm\; b\_n)\; (x\; -\; c)^n.$$ It is not true that if two power series $\backslash sum\_^\backslash infty\; a\_n\; x^n$ and $\backslash sum\_^\backslash infty\; b\_n\; x^n$ have the same radius of convergence, then $\backslash sum\_^\backslash infty\; \backslash left(a\_n\; +\; b\_n\backslash right)\; x^n$ also has this radius of convergence. If $a\_n\; =\; (-1)^n$ and $b\_n\; =\; (-1)^\; \backslash left(1\; -\; \backslash frac\backslash right)$, then both series have the same radius of convergence of 1, but the series $\backslash sum\_^\backslash infty\; \backslash left(a\_n\; +\; b\_n\backslash right)\; x^n\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; x^n$ has a radius of convergence of 3.Multiplication and division

With the same definitions for $f(x)$ and $g(x)$, the power series of the product and quotient of the functions can be obtained as follows: $$\backslash begin\; f(x)g(x)\; \&=\; \backslash left(\backslash sum\_^\backslash infty\; a\_n\; (x-c)^n\backslash right)\backslash left(\backslash sum\_^\backslash infty\; b\_n\; (x\; -\; c)^n\backslash right)\; \backslash \backslash \; \&=\; \backslash sum\_^\backslash infty\; \backslash sum\_^\backslash infty\; a\_i\; b\_j\; (x\; -\; c)^\; \backslash \backslash \; \&=\; \backslash sum\_^\backslash infty\; \backslash left(\backslash sum\_^n\; a\_i\; b\_\backslash right)\; (x\; -\; c)^n.\; \backslash end$$ The sequence $m\_n\; =\; \backslash sum\_^n\; a\_i\; b\_$ is known as theconvolution
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of the sequences $a\_n$ and
For division, if one defines the sequence $d\_n$ by
$$\backslash frac\; =\; \backslash frac\; =\; \backslash sum\_^\backslash infty\; d\_n\; (x\; -\; c)^n$$
then
$$f(x)\; =\; \backslash left(\backslash sum\_^\backslash infty\; b\_n\; (x\; -\; c)^n\backslash right)\backslash left(\backslash sum\_^\backslash infty\; d\_n\; (x\; -\; c)^n\backslash right)$$
and one can solve recursively for the terms $d\_n$ by comparing coefficients.
Solving the corresponding equations yields the formulae based on determinant
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s of certain matrices of the coefficients of $f(x)$ and $g(x)$
$$d\_0=\backslash frac$$
$$d\_n=\backslash frac\; \backslash begin\; a\_n\; \&b\_1\; \&b\_2\; \&\backslash cdots\&b\_n\; \backslash \backslash \; a\_\&b\_0\; \&b\_1\; \&\backslash cdots\&b\_\backslash \backslash \; a\_\&0\; \&b\_0\; \&\backslash cdots\&b\_\backslash \backslash \; \backslash vdots\; \&\backslash vdots\&\backslash vdots\&\backslash ddots\&\backslash vdots\; \backslash \backslash \; a\_0\; \&0\; \&0\; \&\backslash cdots\&b\_0\backslash end$$
Differentiation and integration

Once a function $f(x)$ is given as a power series as above, it isdifferentiable
In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

on the interior
Interior may refer to:
Arts and media
* Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* The Interior (novel) ...

of the domain of convergence. It can be and integrated
Integration may refer to:
Biology
*Modular integration, where different parts in a module have a tendency to vary together
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome ...

quite easily, by treating every term separately:
$$\backslash begin\; f\text{'}(x)\; \&=\; \backslash sum\_^\backslash infty\; a\_n\; n\; (x\; -\; c)^\; =\; \backslash sum\_^\backslash infty\; a\_\; (n\; +\; 1)\; (x\; -\; c)^n,\; \backslash \backslash \; \backslash int\; f(x)\backslash ,dx\; \&=\; \backslash sum\_^\backslash infty\; \backslash frac\; +\; k\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; +\; k.\; \backslash end$$
Both of these series have the same radius of convergence as the original one.
Analytic functions

A function ''f'' defined on someopen subset
Open or OPEN may refer to:
Music
* Open (band)
Open is a band.
Background
Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...

''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
A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically ...

''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
Interior may refer to:
Arts and media
* Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* The Interior (novel) ...

of its region of convergence. All holomorphic function
A rectangular grid (top) and its image under a conformal map ''f'' (bottom).
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...

s 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''Taylor series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

.
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 such that for all , then for all .
If a power series with radius of convergence ''r'' is given, one can consider analytic continuation
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic can also have the following meanings:
Natural sciences Chemistry
* ...

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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

with such that no analytic continuation of the series can be defined at .
The power series expansion of the inverse function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of an analytic function can be determined using the Lagrange inversion theorem
In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (numbe ...

.
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'': $\backslash sum\_^z^n$ has radius of convergence equal to $1$ and diverges at every point of $,\; z,\; =1$. Nevertheless, the sum in $,\; z,\; <1$ is $\backslash frac$, which is analytic at every point of the plane except for $z=1$. # ''Convergent at some points divergent at others'': $\backslash sum\_^(-1)^\backslash frac$ has radius of convergence $1$. It converges for $z=1$, while it diverges for $z=-1$ # ''Absolute convergence at every point of the boundary'': $\backslash sum\_^\backslash frac$ has radius of convergence $1$, while it converges absolutely, and uniformly, at every point of $,\; z,\; =1$ due to Weierstrass M-test applied with the hyper-harmonic convergent series $\backslash sum\_^\backslash frac$. # ''Convergent on the closure of the disc of convergence but not continuous sum'': gave an example of a power series with radius of convergence $1$, convergent at all points with $,\; z,\; =1$, but the sum is an unbounded function and, in particular, discontinuous. A sufficient condition for one-sided continuity at a boundary point is given byAbel's theorem
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
Formal power series

Inabstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...

, one attempts to capture the essence of power series without being restricted to the field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

s of real and complex numbers, and without the need to talk about convergence. This leads to the concept of formal power series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, a concept of great utility in algebraic combinatorics
Algebraic combinatorics is an area of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (m ...

.
Power series in several variables

An extension of the theory is necessary for the purposes ofmultivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with function of several variables, functions of several variables: the Differential calculus, different ...

. A power series is here defined to be an infinite series of the form
$$f(x\_1,\; \backslash dots,\; x\_n)\; =\; \backslash sum\_^\backslash infty\; a\_\; \backslash prod\_^n\; (x\_k\; -\; c\_k)^,$$
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 $\backslash Pi$ 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
Multivariable calculus (also known as multivariate calculus) is the extension of calculus
Calculus, originally called infinitesimal calcu ...

notation this can be written
$$f(x)\; =\; \backslash sum\_\; a\_\backslash alpha\; (x\; -\; c)^\backslash alpha.$$
where $\backslash N$ is the set of natural number
File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...)
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...

s, and so $\backslash N^n$ is the set of ordered ''n''-tuple
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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 $\backslash sum\_^\backslash infty\; x\_1^n\; x\_2^n$ is absolutely convergent in the set $\backslash $ between two hyperbolas. (This is an example of a ''log-convex set'', in the sense that the set of points $(\backslash log\; ,\; x\_1,\; ,\; \backslash log\; ,\; x\_2,\; )$, 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 be a multi-index for a power series . The order of the power series ''f'' is defined to be the least value $r$ such that there is ''a''Laurent series
(Holomorphic functions are analytic, analytic).
In mathematics, the Laurent series of a complex function ''f''(''z'') is a representation of that function as a power series which includes terms of negative degree. It may be used to express compl ...

.
Notes

References

*External links

* *Powers of Complex Numbers

by Michael Schreiber,

Wolfram Demonstrations Project
File:Legal cases tree (Wolfram Demonstrations Project).jpeg, 150px, Legal structures.
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are mea ...

.
{{DEFAULTSORT:Power Series
Real analysis
Complex analysis
Multivariable calculus
Mathematical series