Difference between revisions of "Cosine"
From specialfunctionswiki
| Line 19: | Line 19: | ||
<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| − | <strong>Proposition:</strong> $\cos(x) = \prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right)$ | + | <strong>Proposition:</strong> $\cos(x) = \displaystyle\prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right)$ |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
</div> | </div> | ||
</div> | </div> | ||
Revision as of 01:50, 4 November 2014
The cosine function, $\cos \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula $$\cos(z)=\dfrac{e^{iz}-e^{-iz}}{2},$$ where $e^z$ is the exponential function.
- Cosine.png
Graph of $\cos$ on $\mathbb{R}$.
- Complex cos.jpg
Domain coloring of analytic continuation of $\cos$ to $\mathbb{C}$.
Properties
Proposition: $\cos(x) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k x^{2k}}{(2k)!}$
Proof: █
Proposition: $\cos(x) = \displaystyle\prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right)$
Proof: █
