Difference between revisions of "Entire function"
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In [[complex analysis]], an '''entire function''' is a [[function]] that is [[analytic]] on the whole [[complex plane]]. | In [[complex analysis]], an '''entire function''' is a [[function]] that is [[analytic]] on the whole [[complex plane]]. | ||
| − | [[category:Complex analysis]] | + | [[category:Complex analysis]]. |
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| + | The main result governing the behavior of entire functions is Liouville's theorem, which states that a bounded entire function is constant. Here an entire function <math>f</math> is said to be bounded if there exists a constant <math>M</math> such that for all <math>z \in \mathbb C</math> the bound <math>f(z)<M</math> holds. Liouville's theorem yields a simple proof of the fundamental theorem of algabra: if <math>p(z)</math> were a polynomial with no roots in the complex plane, then one can prove that <math>1/p(z)</math> would be a bounded entire function, and thus constant. | ||
Revision as of 23:12, January 18, 2009
In complex analysis, an entire function is a function that is analytic on the whole complex plane..
The main result governing the behavior of entire functions is Liouville's theorem, which states that a bounded entire function is constant. Here an entire function <math>f</math> is said to be bounded if there exists a constant <math>M</math> such that for all <math>z \in \mathbb C</math> the bound <math>f(z)<M</math> holds. Liouville's theorem yields a simple proof of the fundamental theorem of algabra: if <math>p(z)</math> were a polynomial with no roots in the complex plane, then one can prove that <math>1/p(z)</math> would be a bounded entire function, and thus constant.