Difference between revisions of "Cosine"
From specialfunctionswiki
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| − | The cosine function, $\cos \colon \mathbb{ | + | The cosine function, $\cos \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula |
| − | $$ | + | $$\cos(z)=\dfrac{e^{iz}-e^{-iz}}{2},$$ |
| + | where $e^z$ is the [[exponential function]]. | ||
<div align="center"> | <div align="center"> | ||
Revision as of 14:01, 1 November 2014
The cosine function, $\cos \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula $$\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: $\cos(x) = \prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right)$
Proof: █
