sinc function

Definition The $\operatorname{sinc}$ -function is the function $\operatorname{sinc}:\sR\to \sR$ defined as \begin{eqnarray*} \operatorname{sinc}(x) &=& \left\{ \begin {array}{ll} \frac{\sin x}{x} & \mbox{when}\,\, x\neq 0, \\ 1 & \mbox{when}\,\, x= 0. \end{array}.

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

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

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 $1$ for $x=0$ is motivated.
• Jordan's inequality implies that $|\operatorname{sinc}(x)|\le 1$ for all $x\in \sR$ . 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 $L^2$ , but not in $L^1$ , and [1] \begin{eqnarray*} \int_{-\infty}^\infty \operatorname{sinc}(x)\,dx &=& \pi,\\ \int_{-\infty}^\infty \operatorname{sinc}^2(x)\,dx &=& \pi, \end{eqnarray*}where the first of these denotes an improper Riemann integral.
• For all $x\in \sR$ , we have [1] \begin{eqnarray*} \operatorname{sinc} (x) &=& \sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k+1)!}, \\ \operatorname{sinc} (x) &=& \prod_{k=1}^\infty \left(1-\frac{x^2}{(k\pi)^2}\right), \\ \operatorname{sinc} (x) &=& \prod_{k=1}^\infty \cos \frac{x}{2^k}. \end{eqnarray*}
• The Fourier transform of $\operatorname{sinc}$ is the box function, i.e.
  
  
• The $\operatorname{sinc}$ function satisfies the differential equation
  
  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.

Use

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).