Weierstrass's theorem: Difference between revisions

Revision as of 00:41, 27 April 2008

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

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

Importance

This solves Cousin's multiplicative problem in a particular case over $\mathbb {C}$ .

@@ Line 3: / Line 3: @@
 ==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==
 This solves [[Cousin's multiplicative problem]] in a particular case over <math>\mathbb{C}</math>.