PlanetMath: sinc function

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sinc function

(Definition)

Definition The $\operatorname{sinc}$ -function is the function $\operatorname{sinc}:\mathbb{R}\to \mathbb{R}$ defined as

$\displaystyle \operatorname{sinc}(x)$

$\displaystyle =$

$\displaystyle \left\{ \begin {array}{ll} \frac{\sin x}{x} & \mbox{when}\,\, x\neq 0, \ 1 & \mbox{when}\,\, x= 0. \end{array} \right.$

In some situations, it is more convenient to work with an alternative "normalized variant," in which for $x\neq 0$ we redefine the function as

$\displaystyle \operatorname{sinc}(x)=\frac{\sin(\pi x)}{\pi x}.$

The remainder of this entry deals with the initial definition, though most properties can clearly be suitably modified for the normalized version.

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Properties

Using a Taylor expansion of $\sin$ , one can show that $\operatorname{sinc}$ is infinitely many times differentiable. In particular, $\operatorname{sinc}$ is continuous. In this sense, the value for is motivated.
Jordan's inequality implies that $\vert\operatorname{sinc}(x)\vert\le 1$ for all $x\in \mathbb{R}$ . More generally, one can also show that all derivatives of $\operatorname{sinc}$ are bounded by 1. See this entry.
$\operatorname{sinc}$ is an even function.
The zeros of $\operatorname{sinc}$ are $x=\pm \pi, \pm 2\pi,\ldots$ .
$\operatorname{sinc}$ is in , but not in , and [1]

$\displaystyle \int_{-\infty}^\infty \operatorname{sinc}(x)\,dx$ $\displaystyle =$ $\displaystyle \pi,$

$\displaystyle \int_{-\infty}^\infty \operatorname{sinc}^2(x)\,dx$ $\displaystyle =$ $\displaystyle \pi,$

where the first of these denotes an improper Riemann integral.
For all $x\in \mathbb{R}$ , we have [1]

$\displaystyle \operatorname{sinc} (x)$ $\displaystyle =$ $\displaystyle \sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k+1)!},$

$\displaystyle \operatorname{sinc} (x)$ $\displaystyle =$ $\displaystyle \prod_{k=1}^\infty \left(1-\frac{x^2}{(k\pi)^2}\right),$

$\displaystyle \operatorname{sinc} (x)$ $\displaystyle =$ $\displaystyle \prod_{k=1}^\infty \cos \frac{x}{2^k}.$
The Fourier transform of $\operatorname{sinc}$ is the box function, i.e.

$\displaystyle \operatorname{sinc}(x) = {1\over 2} \int_{-1}^1 e^{ixt} dt.$
The $\operatorname{sinc}$ function satisfies the differential equation

$\displaystyle x \operatorname{sinc}'' + 2 \operatorname{sinc}' + x \operatorname{sinc} = 0$

This is a consequence of a comment in the sine integral entry.
There is no known simple expression for the integral of sinc. However, this function is known as the sine integral.

Synonym and Etymology

The $\operatorname{sinc}$ function is also called sine cardinal or cardinal sine.

The sinc function is relevant in several fields. For one, its Fourier transform is a box, so it is the frequency respose of a perfect on/off sampling device, and therefore often the correct way to interpolate between frequencies in a sampled signal. The resulting function is in fact analytic on the entire complex plane.

Bibliography

1: W.B. Gearhart, H.S.Shultz, The Function $\frac{\sin x}{x}$ , The College Mathematics Journal, March 1990, Volume 21, Number 2, pp. 90-99. (online).