Difference between revisions of "Cosine"
From specialfunctionswiki
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=Properties= | =Properties= | ||
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| − | <strong>Proposition:</strong> | + | <strong>Proposition:</strong> $\cos(x) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k x^{2k}}{(2k)!}$ |
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| − | <strong>Proposition:</strong> | + | <strong>Proposition:</strong> $\cos(x) = \prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right)$ |
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
</div> | </div> | ||
</div> | </div> | ||
Revision as of 05:56, 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: █
