Difference between revisions of "Sinc"

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File:Sinc.png|Plot of $\mathrm{sinc}$ on $[-15,15]$.
 
File:Sinc.png|Plot of $\mathrm{sinc}$ on $[-15,15]$.
File:Complexsincplot.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 $\mathrm{sinc}$.
 
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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.

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

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