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]]. | ||
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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 | + | 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 algebra: 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. |
| + | [[Category:Complex Analysis]] | ||
Latest revision as of 01:02, August 14, 2018
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \ } is said to be bounded if there exists a constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \ } such that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \in \mathbb{C} \ } the bound Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z)<M \ } holds. Liouville's theorem yields a simple proof of the fundamental theorem of algebra: if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(z) \ } were a polynomial with no roots in the complex plane, then one can prove that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/p(z) \ } would be a bounded entire function, and thus constant.