Weierstrass's theorem - Revision history

Vipul: 3 revisions

2008-05-18T19:19:29Z

3 revisions

← Older revision	Revision as of 19:19, 18 May 2008
(No difference)

Vipul at 00:41, 27 April 2008

2008-04-27T00:41:51Z

← Older revision		Revision as of 00:41, 27 April 2008
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	==Statement==		==Statement==

	Suppose <math>(z_n)</math> is a sequence of (possibly repeating) complex numbers that does not cluster in <math>\mathbb{C}</math>: in other words, <math>\|z_n\| \to \infty</math> if the sequence is infinite. Then, there exists an [[entire function]] <math<f:\mathbb{C} \to \mathbb{C}</math> such that for any <math>z \in \mathbb{C}</math>, the [[order of zero for function at point\|order of zero]] for <math>f</math> at <math>z</math> equals the number of times <math>z</math> occurs in the sequence <math>(z_n)</math>.		Suppose <math>(z_n)</math> is a sequence of (possibly repeating) complex numbers that does not cluster in <math>\mathbb{C}</math>: in other words, <math>\|z_n\| \to \infty</math> if the sequence is infinite. Then, there exists an [[entire function]] <math>f:\mathbb{C} \to \mathbb{C}</math> such that for any <math>z \in \mathbb{C}</math>, the [[order of zero for function at point\|order of zero]] for <math>f</math> at <math>z</math> equals the number of times <math>z</math> occurs in the sequence <math>(z_n)</math>.

	==Importance==		==Importance==

	This solves [[Cousin's multiplicative problem]] in a particular case over <math>\mathbb{C}</math>.		This solves [[Cousin's multiplicative problem]] in a particular case over <math>\mathbb{C}</math>.

Vipul at 00:41, 27 April 2008

2008-04-27T00:41:08Z

← Older revision		Revision as of 00:41, 27 April 2008
Line 4:		Line 4:

	Suppose <math>(z_n)</math> is a sequence of (possibly repeating) complex numbers that does not cluster in <math>\mathbb{C}</math>: in other words, <math>\|z_n\| \to \infty</math> if the sequence is infinite. Then, there exists an [[entire function]] <math<f:\mathbb{C} \to \mathbb{C}</math> such that for any <math>z \in \mathbb{C}</math>, the [[order of zero for function at point\|order of zero]] for <math>f</math> at <math>z</math> equals the number of times <math>z</math> occurs in the sequence <math>(z_n)</math>.		Suppose <math>(z_n)</math> is a sequence of (possibly repeating) complex numbers that does not cluster in <math>\mathbb{C}</math>: in other words, <math>\|z_n\| \to \infty</math> if the sequence is infinite. Then, there exists an [[entire function]] <math<f:\mathbb{C} \to \mathbb{C}</math> such that for any <math>z \in \mathbb{C}</math>, the [[order of zero for function at point\|order of zero]] for <math>f</math> at <math>z</math> equals the number of times <math>z</math> occurs in the sequence <math>(z_n)</math>.

			==Importance==

			This solves [[Cousin's multiplicative problem]] in a particular case over <math>\mathbb{C}</math>.

Vipul: New page: {{basic fact}} ==Statement== Suppose $(z_n)$ is a sequence of (possibly repeating) complex numbers that does not cluster in $\mathbb{C}$ : in other words, $|z_n...$

2008-04-27T00:40:34Z

New page: {{basic fact}} ==Statement== Suppose <math>(z_n)</math> is a sequence of (possibly repeating) complex numbers that does not cluster in <math>\mathbb{C}</math>: in other words, <math>|z_n...

New page

{{basic fact}}

==Statement==

Suppose <math>(z_n)</math> is a sequence of (possibly repeating) complex numbers that does not cluster in <math>\mathbb{C}</math>: in other words, <math>|z_n| \to \infty</math> if the sequence is infinite. Then, there exists an [[entire function]] <math<f:\mathbb{C} \to \mathbb{C}</math> such that for any <math>z \in \mathbb{C}</math>, the [[order of zero for function at point|order of zero]] for <math>f</math> at <math>z</math> equals the number of times <math>z</math> occurs in the sequence <math>(z_n)</math>.

Weierstrass's theorem - Revision history

Vipul: 3 revisions

Vipul at 00:41, 27 April 2008

Vipul at 00:41, 27 April 2008

Vipul: New page: {{basic fact}} ==Statement== Suppose (z_n) is a sequence of (possibly repeating) complex numbers that does not cluster in \mathbb{C}: in other words, |z_n...

Vipul: New page: {{basic fact}} ==Statement== Suppose $(z_n)$ is a sequence of (possibly repeating) complex numbers that does not cluster in $\mathbb{C}$ : in other words, $|z_n...$