Entire function: Difference between revisions
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===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 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'''. | ||
Let <math>~f~</math> be entire function. Positive number | |||
<math>~\alpha~</math> is called '''order''' of function | |||
<math>~f~</math>, it 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. | |||
===Range of values=== | ===Range of values=== | ||
Revision as of 04:28, 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 functions, 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, inverse 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.
Any entire function can be expanded in every point to the Tailor 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.
Infinitness
Liouville's theorem states: an entire function which is bounded must be constant [1].
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.
Let be entire function. Positive number is called order of function , it 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.
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 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~} and at any contour C evolving 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 with Cauchi 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 }
See also
References
- ↑ 1.0 1.1 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. (1954). Entire Functions. Academic Press. OCLC 847696. Cite error: Invalid
<ref>tag; name "ralph" defined multiple times with different content - ↑ Krantz (1999). Handbook of Complex Variables. Boston, MA: Birkhäuser,. ISBN 0-8176-4011-8.