y = \dfrac{\sin x}{x}

Graph of the Sinc Function $y = \dfrac{\sin x}{x}$

$y = \dfrac{\sin x}{x}$ is known as the sinc function, the sine divided by $x$ ¹. The unnormalized form is treated here.

Definition and singularity

\operatorname{sinc} x = \frac{\sin x}{x}

On the face of it the denominator vanishes at $x = 0$ , but the following limit exists.

\lim_{x \to 0} \frac{\sin x}{x} = 1

So $x = 0$ is a removable singularity. Defining the value there to be $1$ makes the graph continuous, passing through the point $(0, 1)$ .

Domain, range and symmetry

With the value at $x = 0$ filled in, the domain is all real numbers. Both $\sin x$ and $x$ are odd, so their quotient is even and the graph is symmetric about the $y$ -axis. The maximum value is $1$ , attained at $x = 0$ .

Zeros

The function vanishes where $\sin x = 0$ with $x \neq 0$ , that is at $x = n\pi$ for $n \neq 0$ . Note that $x = 0$ is not a zero but the point where the maximum is attained.

Decay and the envelope

Since $|\sin x| \leq 1$ , we have $\left|\dfrac{\sin x}{x}\right| \leq \dfrac{1}{|x|}$ , so the oscillation decays along the envelope $y = \pm\dfrac{1}{x}$ . As $x \to \pm\infty$ the values approach $0$ , so the $x$ -axis is a horizontal asymptote.

Extrema

The extrema sit at the solutions of $\tan x = x$ , and they shrink along the envelope as $x$ moves away from the origin.

$x$	$\dfrac{\sin x}{x}$
$0$	$1$
$\approx 4.4934$	$\approx -0.2172$
$\approx 7.7253$	$\approx 0.1284$
$\approx 10.9041$	$\approx -0.0913$

The minimum is about $-0.2172$ , so the values always lie between roughly $-0.217$ and $1$ .

Integral

The antiderivative cannot be written with elementary functions, but the integral out to infinity has a closed form².

\int_0^{\infty} \frac{\sin x}{x}\,dx = \frac{\pi}{2}

This is the Dirichlet integral. The integral of $\left|\dfrac{\sin x}{x}\right|$ diverges, so the convergence is conditional: the positive and negative swings cancel to leave a finite value.

Applications

The Fourier transform of a rectangular pulse
The interpolation kernel of the sampling theorem, an ideal low-pass filter
The intensity pattern of light diffracted by a single slit

The form $\dfrac{\sin(\pi x)}{\pi x}$ is called the normalized sinc, adjusted so that its zeros fall exactly on the integers. In the sampling theorem it is the interpolation function that reconstructs a continuous signal from discrete samples³.

Sinc function, Wikipedia
Dirichlet integral, Wikipedia
Nyquist-Shannon sampling theorem, Wikipedia

Graph of the Sinc Function y=sin⁡xxy = \dfrac{\sin x}{x}y=xsinx​