Weierstrass's theorem: Difference between revisions

Latest revision as of 19:19, 18 May 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 $C$ : in other words, $| z_{n} | \to \infty$ if the sequence is infinite. Then, there exists an entire function $f : C \to C$ such that for any $z \in C$ , 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 $C$ .

@@ 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>.