Difference between revisions of "Sine"
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The sine function $\sin \colon \mathbb{R} \rightarrow \mathbb{R}$ is the unique solution of the second order initial value problem $y''=-y;y(0)=0,y'(0)=1$. | The sine function $\sin \colon \mathbb{R} \rightarrow \mathbb{R}$ is the unique solution of the second order initial value problem $y''=-y;y(0)=0,y'(0)=1$. | ||
| − | + | <div align="center"> | |
| − | + | <gallery> | |
| − | + | File:Sine.png|Graph of $\sin$ on $\mathbb{R}$. | |
| + | File:Complex sin.jpg|[[Domain coloring]] of [[analytic continuation]] of $\sin$. | ||
| + | </gallery> | ||
| + | </div> | ||
=Properties= | =Properties= | ||
Revision as of 05:51, 31 October 2014
The sine function $\sin \colon \mathbb{R} \rightarrow \mathbb{R}$ is the unique solution of the second order initial value problem $y=-y;y(0)=0,y'(0)=1$.
- Sine.png
Graph of $\sin$ on $\mathbb{R}$.
- Complex sin.jpg
Domain coloring of analytic continuation of $\sin$.
Properties
Proposition: $\sin(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kx^{2k+1}}{(2k+1)!}$
Proof: proof goes here █
Proposition: $\sin(x) = x \displaystyle\prod_{k=1}^{\infty} \left( 1 - \dfrac{x^2}{k^2\pi^2} \right)$
Proof: proof goes here █
