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In [[Mathematics|mathematical]] analysis, and in particular the [[theory of functions of complex variable]],
an '''entire function''' is a [[function (mathematics)|function]] that is [[holomorphic function|holomorphic]] in the whole [[complex plane]]
<ref name="john">{{cite book|first=John B.|last=Conway|authorlink=John B. Conway|year=1978|title=Functions of One Complex Variable I|edition=2nd edition|publisher=Springer|id=ISBN 0-387-90328-3}}</ref> <ref name="ralph">{{cite book|first=Ralph P.|last=Boas |year=1954|title=Entire Functions|publisher=Academic Press|id=OCLC 847696}}</ref>.


In the [[mathematical analysis]] and, in particular, in the [[theory of functions of complex variable]],
==Examples==
'''The entire function''' is [[finction(mathematics)|function]] that is [[holomorphic]] in the whole [[complex plane]].
===Entire functions===
Examples of '''entire functions''' are the [[polynomial]]s and the [[exponential]]s.
Examples of entire functions are [[polynomial]] and [[exponential]] functions.
All [[sum(mathematics)|sum]]s, [[product(mathematics)|product]]s and [[composition(,athematics)|composition]]s of these functions also are '''entire functions'''.
All [[sum (mathematics)|sum]]s, and [[product (mathematics)|product]]s of entire functions are entire, so that the entire functions form a '''C'''-algebra.  Further, [[Function composition|composition]]s of entire functions are also entire.


All the derivatives and some of integrals of entired funcitons, for example [[erf(function)|erf]], [[Integral sinus|Si]],
All the [[derivative]]s and some of the [[integral]]s of entire functions, for example the [[error function]] erf, [[sine integral]] Si and the [[Bessel function]] ''J''<sub>0</sub> are also entire functions.
[[Bessel function|<math>J_0</math>]], also are entired functions.


Every entire function can be represented as a [[power series]] or [[Tailor expansion]] which [[convergence (series)|converges]] everywhere.
===Non-entire functions===
In general, neither [[series(mathematics)|series]] nor [[limit(mathematics)|limit]] of a [[sequence(mathematics)|sequence]] of entire functions need be an entire function.


In general, neither [[series(mathematics)|series]] nor [[limit(mathematics)|limit]] of a [[sequence(mathematics)|sequence]] of entire funcitons needs to  be an entire function.
The inverse of an '''entire function''' has need not be entire.  Usually, inverse of a non-trivial function is not entire.  (The inverse of a [[linear function]] is entire). In particular, inverses of [[trigonometric function]]s are not entire.


Inverse of an '''entire function''' has no need to be entire function.
More non-entire functions: [[rational function]] <math>~f(z)=\frac{a+b x}{c+x}~</math> at any complex
 
Examples of non-entire functions: [[rational function]] <math>~f(z)=\frac{a+b x}{c+x}~</math> at any complex
<math>~a~</math>,
<math>~a~</math>,
<math>~b~</math>,
<math>~b~</math>,
<math>~c~</math> ,
<math>~c~</math> ,
[[square root]], [[logarithm]], [[funciton Gamma]], [[tetration(mathematics)|tetration]].
[[square root]], [[logarithm]], [[function Gamma]], [[tetration]].
 
In particular, non-analytic functions also should be qualified as non-entire:
[[real part|<math>\Re</math>]],
[[imaginary part|<math>\Im</math>]],
[[complex conjugation]],
[[modulus of complex number|modulus]],
[[argument of complex number|argument]],
[[Dirichlet function]].
 
==Properties==
The entire functions have all general properties of other [[analytic functions]], but the infinite [[range of analyticity]]
enhances the set of the properties, making the entire functions especially [[beautiful (mathematics)|beautiful]] and attractive for applications.
===Power series===
The [[radius of convergence]] of a [[power series]] is the distance the nearest [[singularity (mathematics)|singularity]].  Therefore, it is infinite for entire functions.
 
'''Any entire function can be expanded in every point to the [[Taylor series]] which [[convergence (series)|converges]] everywhere'''.
 
This does not mean that one can always use the [[power series]] for precise [[evaluation]] of an entire function,
but helps a lot to [[proof (mathematics)|prove]] the [[theorem]]s.
 
===Unboundedness===
[[Liouville's theorem]] states: '''an entire function which is bounded must be constant''' <ref name="john" />.
 
===Order of an entire function===
As all entire functions (except the constants) are unbounded, they grow as the argument become large, and can be characterised by their growth rate, which is called '''order'''.
 
Let <math>~f~</math> be entire function. Positive number
<math>~\alpha~</math> is called '''order''' of function
<math>~f~</math>, if for all positive numbers
<math>~\beta~</math>, larger than 
<math>~\alpha~</math>, there exist positive number
<math>~\rho~</math> such that  for all complex
<math>~z~</math> such that
<math>~|z|>\rho~</math>, the relation
<math>~|f(z)|<\exp\big(|z|^\beta\big)~</math> holds
<ref name="steven">{{cite book
|firsrt=Steven G.
|last=Krantz  <!-- |author=S.G. Krantz !-->
|title=Handbook of Complex Variables
|publisher= Boston, MA: Birkhäuser
|page=121
|year=1999
|isbn=0-8176-4011-8
}}</ref>.
 
In particular, all polynomials have order 0; the [[exponential]] has order 1; and [[erf]], as the [[Gaussian exponential]], has order 2.
 
===Range of values===
[[Picard theorem|Picard's little theorem]] states: '''a non-constant entire function takes on every complex number as value, except possibly one''' <ref name="ralph" />.
<!-- This property can be used for an elegant proof of the [[fundamental theorem of algebra]]. !-->
 
For example, the [[exponential function|exponential]] never takes on the value 0.


Entire function <math>~f~</math>, at any complex <math>~z~</math> and at any contour '''C ''' evolving point <math>z</math>
===Cauchy integral===
just once, can be expressed with [[Cauchi theorem]]  
<!-- I am not sure if this section should be here. Perhaps, it also should be separted article !-->
<math>  
 
Entire function <math>~f~</math>, at any complex <math>~z~</math> and at any contour '''C ''' enclosing the point <math>z</math> just once, can be expressed the  [[Cauchy theorem|Cauchy's theorem]]  
<math>  
f(x)=\frac{1}{2\pi {\rm i}} \oint_{\mathbf C} \frac{f(t)}{t-z} {\rm d}t
f(x)=\frac{1}{2\pi {\rm i}} \oint_{\mathbf C} \frac{f(t)}{t-z} {\rm d}t
</math>
</math>


<!--
<!--
If the entire function is not a constant, its modulus becomes larger thgan any positive value at some value of the argument.
For non-entire functions, the same formula holds, but the contour '''C''' should avoid singularities; in the case of entire funcitons, the user has no neer to worry about such rubbish things
 
!-->
If the entire function is not a constant, then its set of values covers the whole complex plane.
<!--
 
I am not sure it these are clear sentences. May be it would be better to type the formulas?  
I am not sure it these are clear sentences. May be it would be better to type the formulas?  


Should I sign the stubs I write?  Kouznetsov
Should I sign the stubs I write?  Kouznetsov
!-->
!-->
==See also==
 
*[[Cauchi formula]]
==Attribution==
*[[Tailor series]]
{{WPAttribution}}
 
==References==
==References==
<references/>
<!-- Wikipedia cites
<!-- Wikipedia cites
*Ralph P. Boas (1954). Entire Functions. Academic Press. OCLC 847696.  
*
 
Ralph P. Boas (1954). Entire Functions. Academic Press. OCLC 847696.  
but I wanted to cite more suitable source!-->


{{stub}mathematics}}
but I wanted to cite more suitable source!-->[[Category:Suggestion Bot Tag]]
[[Category:Mathematics]]

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In mathematical analysis, and in particular the theory of functions of complex variable, an entire function is a function that is holomorphic in the whole complex plane [1] [2].

Examples

Entire functions

Examples of entire functions are polynomial and exponential functions. All sums, and products of entire functions are entire, so that the entire functions form a C-algebra. Further, compositions of entire functions are also entire.

All the derivatives and some of the integrals of entire functions, for example the error function erf, sine integral Si and the Bessel function J0 are also entire functions.

Non-entire functions

In general, neither series nor limit of a sequence of entire functions need be an entire function.

The inverse of an entire function has need not be entire. Usually, inverse of a non-trivial function is not entire. (The inverse of a linear function is entire). In particular, inverses of trigonometric functions are not entire.

More non-entire functions: rational function at any complex , , , square root, logarithm, function Gamma, tetration.

In particular, non-analytic functions also should be qualified as non-entire: , , complex conjugation, modulus, argument, Dirichlet function.

Properties

The entire functions have all general properties of other analytic functions, but the infinite range of analyticity enhances the set of the properties, making the entire functions especially beautiful and attractive for applications.

Power series

The radius of convergence of a power series is the distance the nearest singularity. Therefore, it is infinite for entire functions.

Any entire function can be expanded in every point to the Taylor series which converges everywhere.

This does not mean that one can always use the power series for precise evaluation of an entire function, but helps a lot to prove the theorems.

Unboundedness

Liouville's theorem states: an entire function which is bounded must be constant [1].

Order of an entire function

As all entire functions (except the constants) are unbounded, they grow as the argument become large, and can be characterised by their growth rate, which is called order.

Let be entire function. Positive number is called order of function , if for all positive numbers , larger than , there exist positive number such that for all complex such that , the relation holds [3].

In particular, all polynomials have order 0; the exponential has order 1; and erf, as the Gaussian exponential, has order 2.

Range of values

Picard's little theorem states: a non-constant entire function takes on every complex number as value, except possibly one [2].

For example, the exponential never takes on the value 0.

Cauchy integral

Entire function , at any complex and at any contour C enclosing the point just once, can be expressed the Cauchy's theorem


Attribution

Some content on this page may previously have appeared on Wikipedia.

References

  1. 1.0 1.1 Conway, John B. (1978). Functions of One Complex Variable I, 2nd edition. Springer. ISBN 0-387-90328-3. 
  2. 2.0 2.1 Boas, Ralph P. (1954). Entire Functions. Academic Press. OCLC 847696. 
  3. Krantz (1999). Handbook of Complex Variables. Boston, MA: Birkhäuser. ISBN 0-8176-4011-8.