Difference between revisions of "Sinc"

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The $\mathrm{sinc}$ function (sometimes called the unnormalized $\mathrm{sinc}$ function) is defined by
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The $\mathrm{sinc}$ function (sometimes called the "unnormalized" $\mathrm{sinc}$ function) is defined by
$$\mathrm{sinc}(x) = \left\{ \begin{array}{ll}
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$$\mathrm{sinc}(z) = \left\{ \begin{array}{ll}
\dfrac{\sin x}{x} &; x \neq 0 \\
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\dfrac{\sin z}{z} &; z \neq 0 \\
1 &; x=0.
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1 &; z=0.
 
\end{array} \right.$$
 
\end{array} \right.$$
 +
It appears in the definition of the [[Sine integral]] function.
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Sinc.png|Plot of $\mathrm{sinc}$ on $[-15,15]$.
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File:Sincplot.png|Plot of $\mathrm{sinc}$ on $[-15,15]$.
File:Complex sinc.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{sinc}$ on $[-15,15] \times [-15,15] \subset \mathbb{C}$.
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File:Complexsincplot.png|[[Domain coloring]] of $\mathrm{sinc}$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
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[[Sum of values of sinc]]<br />
<strong>Theorem:</strong> The following formula holds:
 
$$\mathrm{sinc}(x)=\displaystyle\prod_{k=1}^{\infty} \cos \left( \dfrac{x}{2^k} \right).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
{{:Sum of values of sinc}}
 
 
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
$$\displaystyle\sum_{k=1}^{\infty} (-1)^{k+1}\mathrm{sinc}(k)=\dfrac{1}{2}.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
 
=Videos=
 
=Videos=
 
[https://www.youtube.com/watch?v=xEFi0xQRCKI Infinite Product Evaluation with the Sinc Function]<br />
 
[https://www.youtube.com/watch?v=xEFi0xQRCKI Infinite Product Evaluation with the Sinc Function]<br />
 +
[https://www.youtube.com/watch?v=sW9Sw0G8KQ4 (The Sinc Function) Inverse Fourier Transform of Sinc & Fourier Transform of Sinc]<br />
 
[https://www.youtube.com/watch?v=ORTQTh4uh7A Fourier Transform of a Sinc Function (or Inverse Fourier Transform of a Low Pass Filter)]<br />
 
[https://www.youtube.com/watch?v=ORTQTh4uh7A Fourier Transform of a Sinc Function (or Inverse Fourier Transform of a Low Pass Filter)]<br />
 +
[https://youtu.be/3Sjn3XLo5XE?t=306 Discrete-Time Signals and Systems Introduction (4/6): Special Functions]<br />
 +
[https://www.youtube.com/watch?v=xx2AQz_ZyC0 Integrating the sinc function]<br />
 +
 +
=See also=
 +
[[Normalized sinc]]<br />
 +
 +
{{:*-c functions footer}}
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 +
[[Category:SpecialFunction]]

Latest revision as of 02:19, 16 September 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.

Properties

Sum of values of sinc

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

See also

Normalized sinc

$*$-c functions