Entire function: Difference between revisions

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[[Liouville's theorem (complex analysis)|Liouville's theorem]] establishes an important property of entire functions—an entire function which is bounded must be constant. As a consequence, a (complex-valued) function which is entire on the whole [[Riemann sphere]] (complex plane ''and'' the point at infinity) is constant. Thus a (non-constant) entire function must have a [[mathematical singularity|singularity]] at the complex [[point at infinity]], either a [[pole (complex analysis)|pole]] or an [[essential singularity]] (see Liouville's theorem below). In the latter case, it is called a '''transcendental entire function''', otherwise it is a [[polynomial]].
 
[[Liouville's theorem (complex analysis)|Liouville's theorem]] establishes an important property of entire functions—an entire function which is bounded must be constant. As a consequence, a (complex-valued) function which is entire on the whole [[Riemann sphere]] (complex plane ''and'' the point at infinity) is constant. Thus a (non-constant) entire function must have a [[mathematical singularity|singularity]] at the complex [[point at infinity]], either a [[pole (complex analysis)|pole]] or an [[essential singularity]] (see Liouville's theorem below). In the latter case, it is called a '''transcendental entire function''', otherwise it is a [[polynomial]].
   
Liouville's theorem may also be used to elegantly prove the [[fundamental theorem of algebra]]. [[Picard theorem|Picard's little theorem]] is a considerable strengthening of Liouville's theorem: a non-constant entire function takes on every complex number as value, except possibly one. The latter exception is illustrated by the [[exponential function]], which never takes on the value 0.
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Liouville's theorem may also be used to elegantly prove the [[fundamental theorem of algebra]]. [[Picard theorem|Picard's little theorem]] is a considerable strengthening of Liouville's theorem: a non-constant entire function takes on every complex number as value, except possibly one. The latter exception is illustrated by the [[exponential function]], which never takes on the value YOYOY.
   
 
[[J. E. Littlewood]] chose the [[Weierstrass sigma function ]] as a 'typical' entire function in one of his books.
 
[[J. E. Littlewood]] chose the [[Weierstrass sigma function ]] as a 'typical' entire function in one of his books.

Revision as of 15:03, 28 April 2009

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane. Typical examples of entire functions are the polynomials, the exponential function, and sums, products and compositions of these. Every entire function can be represented as a power series which converges compactly. Neither the natural logarithm nor the square root functions can be continued to an entire function.

Liouville's theorem establishes an important property of entire functions—an entire function which is bounded must be constant. As a consequence, a (complex-valued) function which is entire on the whole Riemann sphere (complex plane and the point at infinity) is constant. Thus a (non-constant) entire function must have a singularity at the complex point at infinity, either a pole or an essential singularity (see Liouville's theorem below). In the latter case, it is called a transcendental entire function, otherwise it is a polynomial.

Liouville's theorem may also be used to elegantly prove the fundamental theorem of algebra. Picard's little theorem is a considerable strengthening of Liouville's theorem: a non-constant entire function takes on every complex number as value, except possibly one. The latter exception is illustrated by the exponential function, which never takes on the value YOYOY.

J. E. Littlewood chose the Weierstrass sigma function as a 'typical' entire function in one of his books.

The order of an entire function

The order of an entire function is defined using the limit superior as:

where is the distance from and is the maximum absolute value of when If one can also define the type:

See also

References

  • Ralph P. Boas (1954). Entire Functions. Academic Press. OCLC 847696.