Difference between revisions of "Cosine"
From specialfunctionswiki
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File:Complex cos.jpg|Domain coloring of analytic continuation of $\cos$ to $\mathbb{C}$. | File:Complex cos.jpg|Domain coloring of analytic continuation of $\cos$ to $\mathbb{C}$. | ||
</gallery> | </gallery> | ||
| + | </div> | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> $$\cos(x) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k x^{2k}}{(2k)!}$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> $$\cos(x) = \prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right)$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
</div> | </div> | ||
Revision as of 05:55, 31 October 2014
The cosine function, $\cos \colon \mathbb{R} \rightarrow \mathbb{R}$ is the unique solution of the second order initial value problem $$y=-y;y(0)=1,y'(0)=0.$$
- 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: $$\cos(x) = \prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right)$$
Proof: █
