The sinc function
is defined as
sinc(x) = sin(πx)/πx for x different from 0;
sinc(x) = 1 for x = 0.
The motivation for attributing the value 1 to sinc(0) is of course the fact that even if sin(x)/x is not defined at x = 0, limx->0+sinc(x) = limx->0-sinc(x) = 1.
The name "sinc" is shorthand for cardinal sine. The function sin(x)/x (for x different from 0...) is often denoted Sa(x), but is only a scaling of the sinc function.
Sinc(x) has regularly-spaced zeros for each integer value of x. Its maximum is at x = 0, and its full width at half maximum is about 1.2. The rest of the lobes decrease as 1/x (of course), the secondary ones being about 21.7% of the central peak in magnitude.
This is an important function in data processing since it is the Fourier transform of the rectangle (or boxcar) function. It is also the perfect interpolator for a signal sampled above its Nyquist frequency. The cardinal sine also arise in optics, where it is the diffraction pattern for a slit, and Fourier transform spectrometry, where it is the shape of any resolution-limited line.