Difference between revisions of "Weierstrass factorization theorem"
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| − | <strong>Theorem:</strong> (Weierstrass factorization theorem) Let $f \colon \mathbb{C} \rightarrow \mathbb{C}$ be an [[entire function]] and let $\{ | + | <strong>Theorem:</strong> (Weierstrass factorization theorem) Let $f \colon \mathbb{C} \rightarrow \mathbb{C}$ be an [[entire function]] 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 $0$, then $f(0) \neq 0$). Then there exists an entire function $g$ and a sequence of integers $\{p_n\}$ such that |
$$f(z) = z^m e^{g(z)} \displaystyle\prod_{k=1}^{\infty} E_{p_n} \left( \dfrac{z}{a_k} \right).$$ | $$f(z) = z^m e^{g(z)} \displaystyle\prod_{k=1}^{\infty} E_{p_n} \left( \dfrac{z}{a_k} \right).$$ | ||
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Revision as of 22:07, 28 April 2016
Define the notation $$E_n(z)=\left\{ \begin{array}{ll} 1-z &; n=0 \\ (1-z)e^{z+\frac{z^2}{2}+\frac{z^3}{3}+\ldots+\frac{z^n}{n}} &; \mathrm{otherwise} \end{array} \right.$$
Theorem: (Weierstrass factorization theorem) Let $f \colon \mathbb{C} \rightarrow \mathbb{C}$ be an entire function 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 $0$, then $f(0) \neq 0$). Then there exists an entire function $g$ and a sequence of integers $\{p_n\}$ such that $$f(z) = z^m e^{g(z)} \displaystyle\prod_{k=1}^{\infty} E_{p_n} \left( \dfrac{z}{a_k} \right).$$
Proof: █
Contents
Examples of Weierstrass factorizations
Theorem
The following formula holds: $$\sin(z) = z \displaystyle\prod_{k=1}^{\infty} \left( 1 - \dfrac{z^2}{k^2\pi^2} \right),$$ where $\sin$ denotes the sine function and $\pi$ denotes pi.
Proof
References
Theorem
The Weierstrass factorization of $\cos(x)$ is $$\cos(x) = \displaystyle\prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right).$$
Proof
References
Theorem
The Weierstrass factorization of $\sinh(x)$ is $$\sinh(x)=x\displaystyle\prod_{k=1}^{\infty} 1 + \dfrac{x^2}{k^2\pi^2}.$$
Proof
References
Theorem
The Weierstrass factorization of $\cosh(x)$ is $$\cosh x = \displaystyle\prod_{k=1}^{\infty} 1 + \dfrac{4x^2}{(2k-1)^2\pi^2}.$$
