Difference between revisions of "Sinc"

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=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 />

Revision as of 05:26, 18 May 2015

The $\mathrm{sinc}$ function (sometimes called the unnormalized $\mathrm{sinc}$ function) is defined by $$\mathrm{sinc}(x) = \left\{ \begin{array}{ll} \dfrac{\sin x}{x} &; x \neq 0 \\ 1 &; x=0. \end{array} \right.$$

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