Difference between revisions of "Weierstrass factorization theorem"
From specialfunctionswiki
(Created page with " =Examples of Weierstrass factorizations= {{:Weierstrass factorization of sine}}") |
|||
| Line 2: | Line 2: | ||
=Examples of Weierstrass factorizations= | =Examples of Weierstrass factorizations= | ||
{{:Weierstrass factorization of sine}} | {{:Weierstrass factorization of sine}} | ||
| + | {{:Weierstrass factorization of cosine}} | ||
Revision as of 06:17, 20 March 2015
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 $$\cos(x) = \displaystyle\prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right).$$
