Difference between revisions of "Weierstrass factorization theorem"
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| + | __NOTOC__ | ||
| + | ==Theorem== | ||
| + | (Weierstrass factorization theorem) Let $f \colon \mathbb{C} \rightarrow \mathbb{C}$ be [[entire]] and let $\{a_n\}$ denote the set of zeros of $f$ repeated according to multiplicity. Suppose that $f$ has a zero at $z=0$ of [[order of zero|order]] $m \geq 0$ (if $0$ is a zero of order $0$ means that $f(0) \neq 0$). Then there exists an entire function $g$ and a sequence of integers $\{p_n\}$ such that | ||
| + | $$f(z) = z^m e^{g(z)} \displaystyle\prod_{k=1}^{\infty} E_{p_n} \left( \dfrac{z}{a_k} \right),$$ | ||
| + | where $E_{p_n}$ denotes a [[Weierstrass elementary factors|Weierstrass elementary factor]]. | ||
| − | = | + | ==Proof== |
| − | + | ||
| + | ==See also== | ||
| + | [[Weierstrass factorization of sine]]<br /> | ||
| + | [[Weierstrass factorization of cosine]]<br /> | ||
| + | [[Weierstrass factorization of sinh]]<br /> | ||
| + | [[Weierstrass factorization of cosh]]<br /> | ||
| + | [[Gamma function Weierstrass product]]<br /> | ||
| + | |||
| + | ==References== | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 19:12, 26 November 2016
Theorem
(Weierstrass factorization theorem) Let $f \colon \mathbb{C} \rightarrow \mathbb{C}$ be entire and let $\{a_n\}$ denote the set of zeros of $f$ repeated according to multiplicity. Suppose that $f$ has a zero at $z=0$ of order $m \geq 0$ (if $0$ is a zero of order $0$ means that $f(0) \neq 0$). Then there exists an entire function $g$ and a sequence of integers $\{p_n\}$ such that $$f(z) = z^m e^{g(z)} \displaystyle\prod_{k=1}^{\infty} E_{p_n} \left( \dfrac{z}{a_k} \right),$$ where $E_{p_n}$ denotes a Weierstrass elementary factor.
Proof
See also
Weierstrass factorization of sine
Weierstrass factorization of cosine
Weierstrass factorization of sinh
Weierstrass factorization of cosh
Gamma function Weierstrass product
