In either case, the value at x = 0 is defined to be the limiting value
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. In both cases, the value of the function at the removable singularity at zero is understood to be the limit value 1. The sinc function is then analytic everywhere and hence an entire function
.
The function has also been called the cardinal sine or sine cardinal function.[3][4] The term sinc/ˈsɪŋk/ 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
Rayleigh's formula) for the zeroth-order spherical Bessel function
of the first kind.
Properties
cosine function
.
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. That is, sin(ξ)/ξ = cos(ξ) for all points ξ where the derivative of sin(x)/x is zero and thus a local extremum is reached. This follows from the derivative of the sinc function:
The first few terms of the infinite series for the x coordinate of the n-th extremum with positive x coordinate are
where
and where odd n lead to a local minimum, and even n to a local maximum. Because of symmetry around the y axis, there exist extrema with x coordinates −xn. In addition, there is an absolute maximum at ξ0 = (0, 1).
The normalized sinc function has a simple representation as the infinite product:
The cardinal sine function sinc(z) plotted in the complex plane from -2-2i to 2+2i
L2(R), with highest angular frequency ωH = π (that is, highest cycle frequency fH = 1/2).
Other properties of the two sinc functions include:
The unnormalized sinc is the zeroth-order spherical Bessel function of the first kind, j0(x). The normalized sinc is j0(πx).
where Si(x) is the
sine integral
,
λ sinc(λx) (not normalized) is one of two linearly independent solutions to the linear ordinary differential equation The other is cos(λx)/x, which is not bounded at x = 0, unlike its sinc function counterpart.
Using normalized sinc,
The following improper integral involves the (not normalized) sinc function:
This is not an ordinary limit, since the left side does not converge. Rather, it means that
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.
Summation
All sums in this section refer to the unnormalized sinc function.
The sum of sinc(n) over integer n from 1 to ∞ equals π − 1/2:
The sum of the squares also equals π − 1/2:[10][11]
When the signs of the
addends
alternate and begin with +, the sum equals 1/2:
The alternating sums of the squares and cubes also equal 1/2:[12]
Series expansion
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):
The series converges for all x. The normalized version follows easily:
Euler famously compared this series to the expansion of the infinite product form to solve the Basel problem.
Higher dimensions
The product of 1-D sinc functions readily provides a
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