Sinc Function

Min Max

The sinc function , also called the "sampling function," is a function that arises frequently in signal processing and the theory of Fourier transforms. The full name of the function is "sine cardinal," but it is commonly referred to by its abbreviation, "sinc." There are two definitions in common use. The one adopted in this work defines

where is the sine function, plotted above.

This has the normalization

This function is implemented in the Wolfram Language as Sinc[x].

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When extended into the complex plane, is illustrated above.

An interesting property of is that the set of local extrema of corresponds to its intersections with the cosine function , as illustrated above.

The derivative is given by

and the indefinite integral by

where is the sine integral.

Woodward (1953), McNamee et al. (1971), and Bracewell (1999, p. 62) adopt the alternative definition

The latter definition is sometimes more convenient as a result of its simple normalization,

That variant also satisfies the sum

In addition, the binomial coefficient satisfies

which is essentially a restatement of the reflection relation

of the gamma function (M. Somos, pers. comm., Oct. 26, 2006.)

The sinc function is closely related to the spherical Bessel function of the first kind and, in particular,

and is given in terms of the Meijer G-function as

Let be the rectangle function, then the Fourier transform of is the sinc function

The sinc function therefore frequently arises in physical applications such as Fourier transform spectroscopy as the so-called instrument function, which gives the instrumental response to a delta function input. Removing the instrument functions from the final spectrum requires use of some sort of deconvolution algorithm.

The sinc function can be written as a complex integral by noting that, for ,

and that and the integral both equal 1 for .

The sinc function can also be written as the infinite product

a result discovered in 1593 by Francois Viète (Kac 1959, Morrison 1995) and sometimes known as Euler's formula (Prudnikov et al. 1986, p. 757; Gearhart and Shulz 1990). It is also given by

(Gearhart and Shulz 1990) and

(Prudnikov et al. 1986, p. 757).

Another product is given by

(OEIS A118253; Prudnikov et al. 1986, p. 757), where is the constant from polygon circumscribing.

Sums of powers of over the positive integers include

The remarkable fact that the sums of and are equal appears to have first been published in Baillie (1978). Amazingly, the pattern of these sums being equal to plus a rational multiple of breaks down for the power , where the sum equals

where

The sinc function satisfies the identity

Definite integrals involving the sinc function include

After dividing out the constant factor of , the values are again 1/2, 1/2, 3/8, 1/3, 115/384, 11/40, 5887/23040, 151/630, 259723/1146880, ... (OEIS A049330 and A049331; Grimsey 1945, Medhurst and Roberts 1965). These are all special cases of the amazing general result

where and are positive integers such that , , is the floor function, and is taken to be equal to 1 (Kogan; cf. Espinosa and Moll 2000). This spectacular formula simplifies in the special case when is a positive even integer to

where is an Eulerian number (Kogan; cf. Espinosa and Moll 2000). The solution of the integral can also be written in terms of the recurrence relation for the coefficients

The half-infinite integral of can be derived using contour integration. In the above figure, consider the path . Now write . On an arc, and on the x-axis, . Write

where denotes the imaginary part. Now define

where the second and fourth terms use the identities and . Simplifying,

where the third term vanishes by Jordan's lemma. Performing the integration of the first term and combining the others yield

Rearranging gives

so

The same result is arrived at using the method of complex residues by noting

so

Since the integrand is symmetric, we therefore have

giving the sine integral evaluated at 0 as