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In [[mathematical analysis]] and, in particular, in the [[theory of functions of complex variable]],
{{TOC|right}}
'''The entire function''' is a [[function(mathematics)|function]] that is [[holomorphic]] in the whole [[complex plane]]
In [[Mathematics|mathematical]] analysis, and in particular the [[theory of functions of complex variable]],
<ref name="john">
an '''entire function''' is a [[function (mathematics)|function]] that is [[holomorphic function|holomorphic]] in the whole [[complex plane]]
{{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
<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>.
|first=Ralph P.
|last=Boas  
|year=1954
|title=Entire Functions
|publisher=Academic Press
|id=OCLC 847696
}}</ref>.


==Examples==
==Examples==
===Entires===
===Entire functions===
Examples of '''entire functions''' are the [[polynomial]] and the [[exponential]] fucntions.
Examples of entire functions are [[polynomial]] and [[exponential]] functions.
All [[sum(mathematics)|sum]]s, [[product(mathematics)|product]]s and [[composition(mathematics)|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 [[derivative]]s and some of the [[integral]]s of entired functions, 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 entire functions.


===Non-entire functions===
===Non-entire functions===
In general, neither [[series(mathematics)|series]] nor [[limit(mathematics)|limit]] of a [[sequence(mathematics)|sequence]] of entire functions needs to be an entire function.
In general, neither [[series(mathematics)|series]] nor [[limit(mathematics)|limit]] of a [[sequence(mathematics)|sequence]] of entire functions need be an entire function.


The inverse of an '''entire function''' has no need to be entire function. Usually, inverse of a non-trivial function is not entire.
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.
(The inverse of the [[linear function]] is entire). In particular, inverse of [[trigonometric function]]s are not entire.


More non-entire functions: [[rational function]] <math>~f(z)=\frac{a+b x}{c+x}~</math> at any complex
More non-entire functions: [[rational function]] <math>~f(z)=\frac{a+b x}{c+x}~</math> at any complex
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<math>~b~</math>,
<math>~b~</math>,
<math>~c~</math> ,
<math>~c~</math> ,
[[square root]], [[logarithm]], [[function Gamma]], [[tetration(mathematics)|tetration]].
[[square root]], [[logarithm]], [[function Gamma]], [[tetration]].


In particular, non-analytic functions also should be qualified as non-entire:
In particular, non-analytic functions also should be qualified as non-entire:
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==Properties==
==Properties==
The entire functions have all general properties of other [[analytic functions]], but the infinite [[range of analyticity]]  
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.
enhances the set of the properties, making the entire functions especially [[beautiful (mathematics)|beautiful]] and attractive for applications.
===Power series===
===Power series===
The [[radius of convergence]] of a [[power series]] is always distance until the nearest [[singularity(mathematics)|singularity]].
The [[radius of convergence]] of a [[power series]] is the distance the nearest [[singularity (mathematics)|singularity]]. Therefore, it is infinite for entire functions.
Therefore, it is infinite for entire functions.


'''Any entire function can be expanded in every point to the [[Tailor series]] which [[convergence (series)|converges]] everywhere'''.
'''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,
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.
but helps a lot to [[proof (mathematics)|prove]] the [[theorem]]s.


===Infinitness===
===Unboundedness===
[[Liouville's theorem]] states: '''an entire function which is bounded must be constant''' <ref name="john">
[[Liouville's theorem]] states: '''an entire function which is bounded must be constant''' <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>.


===Order of an entire function===
===Order of an entire function===
As all the entire function (except constant) are not bounded, they grow as the argument become large, and can be characterised with the growth rate, which is called '''order'''.  
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  
Let <math>~f~</math> be entire function. Positive number  
Line 78: Line 67:
}}</ref>.
}}</ref>.


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


===Range of values===
===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">{{cite book
[[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" />.
|first=Ralph P.
last=Boas
|uear=1954
|title=Entire Functions
|publisher=Academic Press
|id=OCLC 847696
}}</ref>.
<!-- This property can be used for an elegant proof of the [[fundamental theorem of algebra]]. !-->
<!-- 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.
For example, the [[exponential function|exponential]] never takes on the value 0.
===Cauchi integral===
 
===Cauchy integral===
<!-- I am not sure if this section should be here. Perhaps, it also should be separted article !-->
<!-- I am not sure if this section should be here. Perhaps, it also should be separted article !-->


Entire function <math>~f~</math>, at any complex <math>~z~</math> and at any contour '''C ''' evolving point <math>z</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]]  
just once, can be expressed with [[Cauchi theorem]]  
<math>  
<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>
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!-->
!-->


==See also==
==Attribution==
*[[Cauchi formula]]
{{WPAttribution}}
*[[Tailor series]]
 
==References==
==References==
<references/>
<references/>
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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!-->
but I wanted to cite more suitable source!-->[[Category:Suggestion Bot Tag]]
 
{{stub|mathematics}}
[[Category:Mathematics]]
[[Category:Functions]]

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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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f(z)=\frac{a+b x}{c+x}~} at any complex , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~b~} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~c~} , square root, logarithm, function Gamma, tetration.

In particular, non-analytic functions also should be qualified as non-entire: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Re} , , 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f~} be entire function. Positive number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\alpha~} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} just once, can be expressed the Cauchy's theorem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\frac{1}{2\pi {\rm i}} \oint_{\mathbf C} \frac{f(t)}{t-z} {\rm d}t }


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. 
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