Entire function: Difference between revisions
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==Properties== | ==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=== | ===Power series=== | ||
The [[radius of convergence]] of a [[power series]] is always distance until the nearest | The [[radius of convergence]] of a [[power series]] is always distance until the nearest [[singularity(mathematics)|singularity]]. | ||
Therefore, it is infinite for entire functions. | Therefore, it is infinite for entire functions. | ||
Every entire function can be represented as a [[power series]] or [[Tailor expansion]] which [[convergence (series)|converges]] everywhere. | Every entire function can be represented as a [[power series]] or [[Tailor expansion]] 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. | |||
===Infinitness=== | ===Infinitness=== | ||
[[Liouville's theorem]] establishes an important property of entire functions | [[Liouville's theorem]] establishes an important property of entire functions: '''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>. | {{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>. | ||
===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">{{cite book | ||
|first=Ralph P. | |first=Ralph P. | ||
last=Boas | last=Boas | ||
Revision as of 01:54, 17 May 2008
Definition
In the mathematical analysis and, in particular, in the theory of functions of complex variable, The entire function is function that is holomorphic in the whole complex plane [1][2].
Examples
Entires
Examples of entire functions are the polynomials and the exponentials. All sums, products and compositions of these functions also are entire functions.
All the derivatives and some of integrals of entired funcitons, for example erf, Si, , also are entired functions.
Non-entires
In general, neither series nor limit of a sequence of entire funcitons needs to be an entire function.
Inverse of an entire function has no need to be entire function. Usually, invetse of a non-trivial function is not entire. (The inverse of the linear function is entire). In particular, inverse 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 always distance until the nearest singularity. Therefore, it is infinite for entire functions. Every entire function can be represented as a power series or Tailor expansion 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.
Infinitness
Liouville's theorem establishes an important property of entire functions: an entire function which is bounded must be constant[3].
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.
Cauchi integral
Entire function , at any complex and at any contour C evolving point just once, can be expressed with Cauchi theorem
See also
References
- ↑ Conway, John B. (1978). Functions of One Complex Variable I, 2nd edition. Springer. ISBN 0-387-90328-3.
- ↑ 2.0 2.1 Boas, Ralph P.. Entire Functions. Academic Press. OCLC 847696. Cite error: Invalid
<ref>tag; name "ralph" defined multiple times with different content - ↑ Conway, John B. (1978). Functions of One Complex Variable I, 2nd edition. Springer. ISBN 0-387-90328-3.