Difference between revisions of "Sinc"
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| − | The $\mathrm{sinc}$ function (sometimes called the unnormalized $\mathrm{sinc}$ function) is defined by | + | The $\mathrm{sinc}$ function (sometimes called the "unnormalized" $\mathrm{sinc}$ function) is defined by |
| − | $$\mathrm{sinc}( | + | $$\mathrm{sinc}(z) = \left\{ \begin{array}{ll} |
| − | \dfrac{\sin | + | \dfrac{\sin z}{z} &; z \neq 0 \\ |
| − | 1 &; | + | 1 &; z=0. |
\end{array} \right.$$ | \end{array} \right.$$ | ||
It appears in the definition of the [[Sine integral]] function. | It appears in the definition of the [[Sine integral]] function. | ||
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<gallery> | <gallery> | ||
File:Sinc.png|Plot of $\mathrm{sinc}$ on $[-15,15]$. | File:Sinc.png|Plot of $\mathrm{sinc}$ on $[-15,15]$. | ||
| − | File: | + | File:Complexsincplot.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{sinc}$ on $[-15,15] \times [-15,15] \subset \mathbb{C}$. |
</gallery> | </gallery> | ||
</div> | </div> | ||
Revision as of 05:45, 17 May 2016
The $\mathrm{sinc}$ function (sometimes called the "unnormalized" $\mathrm{sinc}$ function) is defined by $$\mathrm{sinc}(z) = \left\{ \begin{array}{ll} \dfrac{\sin z}{z} &; z \neq 0 \\ 1 &; z=0. \end{array} \right.$$ It appears in the definition of the Sine integral function.
Plot of $\mathrm{sinc}$ on $[-15,15]$.
Domain coloring of analytic continuation of $\mathrm{sinc}$ on $[-15,15] \times [-15,15] \subset \mathbb{C}$.
Properties
Theorem: The following formula holds: $$\mathrm{sinc}(x)=\displaystyle\prod_{k=1}^{\infty} \cos \left( \dfrac{x}{2^k} \right).$$
Proof: █
Theorem
The following formula holds: $$\displaystyle\sum_{k=1}^{\infty} \mathrm{sinc}(k) = \dfrac{\pi-1}{2},$$ where $\mathrm{sinc}$ denotes the $\mathrm{sinc}$ function and $\pi$ denotes pi.
Proof
References
Theorem: The following formula holds: $$\displaystyle\sum_{k=1}^{\infty} (-1)^{k+1}\mathrm{sinc}(k)=\dfrac{1}{2}.$$
Proof: █
Videos
Infinite Product Evaluation with the Sinc Function
(The Sinc Function) Inverse Fourier Transform of Sinc & Fourier Transform of Sinc
Fourier Transform of a Sinc Function (or Inverse Fourier Transform of a Low Pass Filter)
Discrete-Time Signals and Systems Introduction (4/6): Special Functions
Integrating the sinc function
