In Mathematical and particularly in analyzes, a whole series is a series of functions of the form

$\backslash \; sum\_\; \{N\; \backslash \; geq\; 0\}\; a\_nz^n$

where the coefficients a_n form a real or complex continuation. The series is known as whole owing to the fact that it utilizes whole powers.

The whole series have remarkable properties of convergence, which are expressed for the majority using a size associated with the series, its ray of convergence R . On the disc of convergence (open disc of center 0 and ray R ), the function summons series can be indefinitely derived term in the long term.

Reciprocally, certain indefinitely derivable functions can be written in the vicinity of one their points C like summons of a whole series of the variable z-c : this one is then them Série of Taylor. One speaks in this case about developable functions in whole series at the point C . When a function is developable in whole series of each one of its points, it is known as analytical.

The whole series appear in analyzes, but also in Combinatoire as a generating Fonction and spreads in the concept of formal Série. In the Theory of the numbers, the concept of Nombre p-adic is close to that of whole series.

Definitions

In what follows, the variable Z is real or complex.

Whole series

A whole series of variable $z$, is a series of general term $a\_n\; \backslash ,\; z^n$, where N is a natural entirety, and $\{(a\_n)\}\_\; \{N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\}$ is a succession of real numbers or complexes. The use wants that one adopts the notation $\backslash \; sum\; a\_nz^n$ or $\backslash \; sum\_\; \{N\; \backslash \; geq\; 0\}\; a\_nz^n$ to speak about a whole series, while one will write $\backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \backslash \; infty\}\; a\_nz^n$ for his possible sum, in the event of convergence, for a Z given.

Ray of convergence

See also: Ray of convergence

A good part of the properties of convergence of the series can be expressed using the following quantity, called ray of convergence of the series

$R\; =\; \backslash \; sup\; \backslash \; left\; \backslash \; \{r\_0\}\; \backslash \; right)\; ^n$.

The first of the terms of this product is limited, the second forms a geometrical Série of reason strictly lower than 1. By comparison of series to positive terms, the conclusion follows.}}

Consequently, it is possible to specify the mode of convergence of this Série of functions

• the whole series converges absolutely for very complex Z of module strictly lower than the ray. The open disc of center 0 and ray R is called open disc of convergence .
• the series coarsely diverges (i.e. the general term does not converge towards 0) for very complex Z from module strictly higher than the ray.
• For any reality R strictly lower than the ray, there is normal convergence on the closed disc of center 0 and R .

If the variable $x$ is real, one still speaks about open disc of convergence, although that indicates a real line interval ($]\; -\; R;\; +R$

When the ray is infinite, the open disc of convergence is the whole of the complex plan (or the real line). On the other hand there is a priori normal convergence only on the closed discs of finished ray. A null ray means on the other hand which there is divergence in any point other than z=0 , as it for example for the series $\backslash \; textstyle\; \backslash \; sum\_\; \{N\; \backslash \; Ge\; 0\}\; \{N\; is\; the\; case!\; \backslash ,\; z^n\}$.

These properties do not settle all the questions of convergence. In particular, at the points of module R , there can be convergence or not, and convergence with or without absolute convergence. For example, the whole series $\backslash \; sum\; \backslash \; frac\; \{1\}\; \{n^2\}\; \backslash ,\; z^n$, $\backslash \; sum\; \backslash \; frac\; \{1\}\; \{N\}\; \backslash ,\; z^n$ and $\backslash \; sum\; z^n$ have as a ray of convergence 1, the whole series $\backslash \; sum\; \backslash \; frac\; \{1\}\; \{n^2\}\; \backslash ,\; z^n$ converges absolutely in any point of module 1 whereas $\backslash \; sum\; \backslash \; frac\; \{1\}\; \{N\}\; \backslash ,\; z^n$ does not converge absolutely in any point of module 1 but converges in any point other than 1 and the whole series $\backslash \; sum\; z^n$ does not converge in any point of module 1.

Calculation of the ray of convergence

The formula of Hadamard gives the expression of the ray of convergence in term of higher Limite

$\backslash \; frac\; 1R\; =\; \backslash \; limsup\_\; \{N\; \backslash \; to\; \backslash \; infty\}\; \backslash \; left\; (|a\_n|^\; \{1/n\}\; \backslash \; right)$.

This formula rises from the application of the Règle of Cauchy.

In practice, if the $a\_n$ are nonnull, it is sometimes possible to apply the Règle of Alembert:

If $\backslash \; lim\_\; \{N\; \backslash \; to\; +\; \backslash \; infty\}\; \backslash \; left|\backslash \; frac\; \{a\_\; \{n+1\}\}\; \{a\_\; \{N\}\}\; \backslash \; right|\; =\; L\; \backslash ,$ (possibly infinite limit), then the ray of convergence is equal to 1/L .

For example, the whole series $\backslash \; textstyle\; \backslash \; sum\_\; \{N\; \backslash \; geq\; 0\}\; n2^n\; \backslash ,\; z^n$ admits a ray of convergence equal to $\backslash \; tfrac12$.

But it is often more effective to employ the properties of convergence to give other characterizations of the ray of convergence. For example, the ray is the upper limit of the modules of the complexes Z for which the continuation of general term $a\_nz^n$ converges towards 0.

Function summons

If $\{(a\_n)\}\_\; \{N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\}$ is a complex continuation such as the whole series $\backslash \; sum\_\; \{N\; \backslash \; geq\; 0\}\; a\_nz^n$ admits a ray of convergence R strictly positive, one can then define his function summons , in any convergence point, by

$f\; (Z)\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \{\backslash \; infty\}\}\; a\_nz^n$

This function is in particular defined on the disc of convergence $D\; (0,\; R)$.

There exists a large variety of possible behaviors for the series and the function summons at the edge of the field of definition. In particular, the divergence of the series in a point of module R is not incompatible with the existence of a limit in R for the function. Thus by nap of a geometrical Series,

$\backslash \; forall\; X\; \backslash \; in]\; -\; 1,1\; \backslash \; frac1\; \{1+x^2\}\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \backslash \; infty\}\; (-\; 1)\; ^nx^\; \{2n\}.$ The function is prolonged by continuity into -1 and 1 which is however values for which the series diverges.

Examples

A Fonction polynomial real or complex is a whole series of infinite ray of convergence.

The geometrical Série $\backslash \; sum\_\; \{N\; \backslash \; Ge\; 0\}\; \{z^n\}$ has as a ray of convergence 1 and its function sum is worth $\backslash \; frac\; \{1\}\; \{1-z\}$ on the open disc D (0,1).

The whole series $\backslash \; sum\_\; \{N\; \backslash \; Ge\; 0\}\; \backslash \; frac\; \{z^n\}\; \{N!\}$ has an infinite ray of convergence. Its function summons, defined in all the complex plan, is called complex function Exponentielle. It is from it that the functions sine and cosine are analytically defined.

The series whole $\backslash \; sum\_\; \{N\; \backslash \; Ge\; 1\}\; \{\backslash \; frac\; \{z^n\}\; \{N\}\}$ has a ray of convergence equal to 1. It constitutes a determination of the Logarithme complexes, i.e. reciprocal of a restriction of exponential complex.

Operations on the whole series

The properties which follow will be stated for two whole series $\backslash \; sum\_\; \{N\; \backslash \; Ge\; 0\}\; a\_n\; z^n$ and $\backslash \; sum\_\; \{N\; \backslash \; Ge\; 0\}\; b\_n\; z^n$, of respective rays of convergence R and R′ , and whose functions summons are written

$f\; (Z)\; =\; \backslash \; sum\; \_\; \{n=0\}\; ^\; \{+\; \backslash \; infty\}\; a\_nz^n,\; \backslash \; qquad\; G\; (Z)\; =\; \backslash \; sum\; \_\; \{n=0\}\; ^\; \{+\; \backslash \; infty\}\; b\_nz^n$

Summon and produced

The nap of the whole series F and G is a whole series. If R and R′ is distinct, its ray is the minimum of R and R′ . If they are equal, it has a ray equal to or higher than this common value.

One can form the produced of the two whole series, by using the properties of the Produit of Cauchy series with complex terms. Thus the series produced is calculated by the formula

$\backslash \; left\; (\backslash \; sum\; \_\; \{n=0\}\; ^\; \{+\; \backslash \; infty\}\; a\_nz^n\; \backslash \; right)\; \backslash \; cdot\; \backslash \; left\; (\backslash \; sum\; \_\; \{n=0\}\; ^\; \{+\; \backslash \; infty\}\; b\_nz^n\; \backslash \; right)\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; left\; (\backslash \; sum\_\; \{k=0\}\; ^n\; a\_k\; b\_\; \{n-k\}\; \backslash \; right)\; z^n.$

She at least admits a ray of convergence higher or equal of the two rays.

Substitution

Under certain conditions, it is possible to carry out the substitution of a whole series in another, which results in composing the functions sums.

The composition is possible if the rays of convergence of the two series are nonnull, and if the coefficient $a\_0=f\; (0)$ is null. The series obtained by substitution is of strictly positive ray. On a sufficiently small disc included in the disc of convergence, the sum of the series is made up the $g\; \backslash \; circ\; f$.

Substitution can in particular be used for the calculation, when it is possible, of reverse of a whole series, then quotient of two whole series.

Derivation

The series $\backslash \; sum\_\; \{N\; \backslash \; geq\; 0\}\; a\_\; \{n+1\}\; \backslash ,\; (n+1)\; \backslash ,\; z^\; \{N\}$ is called series derived from the series $\backslash \; sum\_\; \{N\; \backslash \; geq\; 0\}\; a\_nz^n$. A series admits the same ray of convergence as its derivative, and if this common value is strictly positive, it is possible in the long term to derive term the series in the disc from convergence

For a whole series of the real variable, the function summons associated is thus Dérivable on ] - R, +R and even of class $\backslash \; mathcal\; \{C\}\; ^\; \{\backslash \; infty\}$, since it is possible to carry out in the long term '' p '' derivation successive term, all the successive series derived having even ray from convergence.

For a series of the complex variable, the derivative is to be taken with the direction also complexes, i.e. the function sum is Holomorphe in the disc of convergence.

Developable function in whole series

A function F of the real or complex variable, defined in the vicinity of a point C , is known as developable in whole series in the vicinity of C if there exists a whole series $\backslash \; sum\_\; \{N\; \backslash \; Ge\; 0\}\; a\_nz^n$ of ray R strictly positive such as

$\backslash \; forall\; Z\; \backslash \; in\; D\; (C,\; R),\; \backslash \; qquad\; F\; (Z)\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \{\backslash \; infty\}\}\; a\_n\; (z-c)\; ^n$.

About the existence and unicity of the development

A function F developable in whole series is necessarily of class $\backslash \; mathcal\; \{C\}\; ^\; \{\backslash \; infty\}$ in the vicinity of C . The coefficient of index N of the development is given by the formula

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\},\; \backslash ,\; a\_n=\; \{f^\; \{(N)\}(c)\; \backslash \; over\; \{N!\}\}$

This shows that if the development in whole series exists, it single, and is given by the Série of Taylor of the function to the point C .

It is not enough that a function is $\backslash \; mathcal\; \{C\}\; ^\; \{\backslash \; infty\}$ so that it is developable in whole series.

One can give like counterexample the function defined on the real line by $f\; (X)\; =e^\; \{-\; 1/x^2\}$, prolonged by continuity by F (0) =0 . Indeed this function is derivable with any order, of derivative being worth 0 in the beginning. Its Taylor series in 0 is the null series. She admits an infinite ray of convergence, but does not have as a nap F (X) in any point other than 0.

Usual developments in whole series

These usual developments are often very useful in the calculation of integrals. They are given here with indication of the ray of convergence in the complex or real field.

1. $\backslash \; forall\; Z\; \backslash \; in\; \backslash \; mathbb\; \{C\},\; \backslash ,\; e^z=\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \{\backslash \; infty\}\}\; \{\backslash \; frac\; \{z^n\}\; \{N!\}\}.$
   
   
2. $\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; \backslash ,\; \backslash \; cos$

x= \ sum_ {n=0} ^ {+ {\ infty}} (- 1) ^n \, {\ frac {x^ {2 \, N}} {(2 \, N)!}}.

1. $\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; \backslash ,\; \backslash \; sin\; x=\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \{\backslash \; infty\}\}\; (-\; 1)\; ^n\; \backslash ,\; \{\backslash \; frac\; \{x^\; \{2\; \backslash ,\; n+1\}\}\; \{(2\; \backslash ,\; n+1)!\}\}.$
   
   
2. $\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; \backslash ,\; \backslash \; operatorname\; \{CH\}\; \backslash ,\; x=\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \{\backslash \; infty\}\}\; \{\backslash \; frac\; \{x^\; \{2\; \backslash ,\; N\}\}\; \{(2\; \backslash ,\; N)!\}\}.$
   
   
3. $\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; \backslash ,\; \backslash \; operatorname\; \{HS\}\; \backslash ,\; x=\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \{\backslash \; infty\}\}\; \{\backslash \; frac\; \{x^\; \{2\; \backslash ,\; n+1\}\}\; \{(2\; \backslash ,\; n+1)!\}\}.$
   
   
4. $\backslash \; forall\; Z\; \backslash \; in\; D\; (0,1),\; \backslash ,\; \{1\; \backslash \; over\; \{1-z\}\}\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \{\backslash \; infty\}\}\; \{z^n\}.$
   
   
5. $\backslash \; forall\; X\; \backslash \; in]\; -\; 1,1],\; \backslash ,\; \backslash \; ln\; (1+x)\; =\; \backslash \; sum\_\; \{n=1\}\; ^\; \{+\; \{\backslash \; infty\}\}\; (-\; 1)\; ^\; \{n+1\}\; \{x^\; \{N\}\; \backslash \; over\; \{N\}\}.$
   
   
6. $\backslash \; forall\; X\; \backslash \; in,\; \backslash ,\; \backslash \; operatorname\; \{Arctan\}\; \backslash ,\; x=\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \{\backslash \; infty\}\}\; (-\; 1)\; ^n\; \backslash ,\; \{\backslash \; frac\; \{x^\; \{2\; \backslash ,\; n+1\}\}\; \{2\; \backslash ,\; n+1\}\}\; \backslash ;$, and in particular, $\backslash \; pi=4\; \backslash ,\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \{\backslash \; infty\}\}\; \{\backslash \; frac\; \{(-\; 1)\; ^\; \{N\}\}\; \{2\; \backslash ,\; n+1\}\}$.
   
   
7. $\backslash \; forall\; X\; \backslash \; in\; \backslash ,]\; -\; 1,1\; \backslash \; forall\; \backslash \; alpha\; \backslash ,\; \backslash \; not\; \backslash \; in\; \backslash ,\; \backslash \; mathbb\; \{NR\},\; \backslash ,\; (1+x)\; ^\; \backslash \; alpha\; \backslash ,\; =1\; \backslash ;\; +\; \backslash ;\; \backslash \; sum\_\; \{n=1\}\; ^\; \{+\; \{\backslash \; infty\}\}\; \{\backslash \; frac\; \{\backslash \; alpha\; \backslash ,\; (\backslash \; alpha-1)\; \backslash \; ldots\; (\backslash \; alpha-n+1)\}\{N!\}\backslash ,\; x^n\}.$
   
   #$\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; \backslash ,\; \backslash \; forall\; \backslash \; alpha\; \backslash ,\; \backslash \; in\; \backslash ,\; \backslash \; mathbb\; \{NR\},\; \backslash ,\; (1+x)\; ^\; \backslash \; alpha\; \backslash ,\; =1\; \backslash ;\; +\; \backslash ;\; \backslash \; sum\_\; \{n=1\}\; ^\; \{+\; \{\backslash \; infty\}\}\; \{\backslash \; frac\; \{\backslash \; alpha\; \backslash ,\; (\backslash \; alpha-1)\; \backslash \; ldots\; (\backslash \; alpha-n+1)\}\{N!\}\backslash ,\; x^n\}\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \{\backslash \; alpha\}.$
   
   
8. $\backslash \; forall\; X\; \backslash \; in]\; -\; 1,1\; \backslash \; operatorname\; \{Argth\}\; \backslash ,\; x=\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \{\backslash \; infty\}\}\; \backslash ,\; \{\backslash \; frac\; \{x^\; \{2\; \backslash ,\; n+1\}\}\; \{2\; \backslash ,\; n+1\}\}.$
   
   #
   
   #
   Remarque: one can also write $a\_n=\; \{\{\{2\; \backslash ,\; N\}\; \backslash \; choose\; N\}\; \backslash \; over\; \{4^n\}\}\; =\; \{\{(2\; \backslash ,\; N)!\}\backslash \; over\; \{(N!\; \backslash ,\; 2^n)\; ^2\}\}\; =\; \{1.3\; \backslash \; ldots\; (2\; \backslash ,\; n-1)\; \backslash \; over\; \{2.4\; \backslash \; ldots\; (2\; \backslash ,\; N)\}\}$
   
   #$\backslash \; forall\; X\; \backslash \; in\; \backslash ,\; \backslash \; left]\; -\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\},\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}\; \backslash \; right\; \backslash \; tan\; x=\; \backslash \; frac\; \{2\}\; \{\backslash \; pi\}\; \backslash ,\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \{\backslash \; infty\}\}\; \backslash ,\; \{\backslash \; left\; (\{\backslash \; frac\; \{X\}\; \{\backslash \; pi\}\}\; \backslash \; right)\}^\; \{2\; \backslash ,\; n+1\}\; (2^\; \{2\; \backslash ,\; n+2\}\; -\; 1)\; \backslash ;\; \backslash \; zeta\; (2\; \backslash ,\; n+2)\; \backslash \; quad\; with\; \backslash ;\; \backslash \; forall\; p>1,\; \backslash ,\; \backslash \; zeta\; (p)\; =\; \backslash \; sum\_\; \{n=1\}\; ^\; \{+\; \{\backslash \; infty\}\}\; \backslash ,\; \backslash \; frac\; \{1\}\; \{n^p\}$ ([[Function Zeta of Riemann|function Zeta] of Riemann, which one knows, for all p whole par - not no one - an expression clarifies in the form of the product of a rational by an even power of π).

Analytical functions

See also: analytical Function

A function of the real or complex variable, definite on open a U , is known as analytical on U when she admits a development in whole series in any point of U .

The function summons F of a whole series of ray of convergence R strictly positive is itself analytical on its open disc of convergence D (0, R) . This means that one can change origin for the development into whole series: precisely, if Z ₀ is a complex of module strictly lower than R , then F is developable in whole series on the disc of center Z ₀ and $R-|z\_0|$.

The analytical functions enjoy remarkable properties. According to the “principle of the zeros isolated”, the points of cancellation of such a function are isolated points. The “principle of the analytical prolongation” indicates that, if two analytical functions are defined on open a related U and coincide on a part has included in U presenting at least a not accumulation, then they coincide on U .

In Analyze complexes, it is established that all holomorphic Fonction (i.e. derivable with the direction complexes) on open a U is indefinitely derivable in any point compared to the complex variable and is even analytical. On the contrary in real Analysis, there exist many functions $\{\backslash \; mathcal\; C\}\; ^\; \backslash \; infty$ nonanalytical.

Behavior at the edge of the field of convergence

Uniform theorem of convergence of Abel

See also: Theorem of Abel (analyzes)

The theorem of Abel gives a property of continuity partial of the function summons when there is convergence of the whole series in a point of his circle of convergence.

Precisely, \ sum_ {N \ Ge 0} a_nz^n is $a\; whole\; series\; of\; ray\; of\; convergence$ R strictly positive finished. It is supposed that in a point Z ₀ of module R , the series is convergent. One considers a triangle T having for tops Z ₀ on the one hand and two points of module strictly lower than R on the other hand. Then the series converges uniformly on T .

In particular, there is uniform convergence on the segment $$. This particular case is called theorem of radial Abel.

Singular and regular points

See also: analytical Prolongation

Either $\backslash \; sum\_\; \{N\; \backslash \; Ge\; 0\}\; a\_nz^n$ a whole series of ray of convergence R strictly positive finished, and F the function summons. A point Z ₀ of module R is known as regular if there exists an open disc D centered in this point such as F is prolonged in an analytical function with $D\; \backslash \; cup\; D\; (0,\; R)$. In the contrary case, the point is known as singular .

Among the complexes of module R , there exists always a singular point.

See too

Related headings

• Any function developable in whole series is a function of class $\backslash \; mathrm\; \{C\}\; ^\; \{\backslash \; infty\}$.

• a analytical Fonction is a developable function in whole series in the vicinity of any point.
• the concepts of analytical Function complex and holomorphic Fonction coincide.
• These concepts require some knowledge in topology, concerning the open and the Connexité.
• See also the developments eulériens

Random links:

Melting point | Dromaiidé | Vault-Guillaume | Piri Weepu | Rue de la Ville- the Bishop