Sinc

From specialfunctionswiki
Revision as of 05:10, 18 May 2015 by Tom (talk | contribs)
Jump to: navigation, search

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