[1] In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by

[2] In digital signal processing and information theory, the normalized sinc function is commonly defined for x ≠ 0 by

for all real a ≠ 0 (the limit can be proven using the squeeze theorem).

The normalization causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of π).

As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of x.

The only difference between the two definitions is in the scaling of the independent variable (the x axis) by a factor of π.

[3][4] The term sinc was introduced by Philip M. Woodward in his 1952 article "Information theory and inverse probability in telecommunication", in which he said that the function "occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own",[5] and his 1953 book Probability and Information Theory, with Applications to Radar.

[6][7] The function itself was first mathematically derived in this form by Lord Rayleigh in his expression (Rayleigh's formula) for the zeroth-order spherical Bessel function of the first kind.

The zero crossings of the unnormalized sinc are at non-zero integer multiples of π, while zero crossings of the normalized sinc occur at non-zero integers.

The local maxima and minima of the unnormalized sinc correspond to its intersections with the cosine function.

Because of symmetry around the y axis, there exist extrema with x coordinates −xn.

The normalized sinc function has a simple representation as the infinite product:

and is related to the gamma function Γ(x) through Euler's reflection formula:

The continuous Fourier transform of the normalized sinc (to ordinary frequency) is rect(f):

This Fourier integral, including the special case

This is not an ordinary limit, since the left side does not converge.

for every Schwartz function, as can be seen from the Fourier inversion theorem.

In the above expression, as a → 0, the number of oscillations per unit length of the sinc function approaches infinity.

Nevertheless, the expression always oscillates inside an envelope of ±⁠1/πx⁠, regardless of the value of a.

This complicates the informal picture of δ(x) as being zero for all x except at the point x = 0, and illustrates the problem of thinking of the delta function as a function rather than as a distribution.

A similar situation is found in the Gibbs phenomenon.

All sums in this section refer to the unnormalized sinc function.

When the signs of the addends alternate and begin with +, the sum equals ⁠1/2⁠:

The Taylor series of the unnormalized sinc function can be obtained from that of the sine (which also yields its value of 1 at x = 0):

Euler famously compared this series to the expansion of the infinite product form to solve the Basel problem.

For a non-Cartesian lattice this function can not be obtained by a simple tensor product.

However, the explicit formula for the sinc function for the hexagonal, body-centered cubic, face-centered cubic and other higher-dimensional lattices can be explicitly derived[13] using the geometric properties of Brillouin zones and their connection to zonotopes.

For example, a hexagonal lattice can be generated by the (integer) linear span of the vectors

This construction can be used to design Lanczos window for general multidimensional lattices.

[13] Some authors, by analogy, define the hyperbolic sine cardinal function.