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 sinc 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
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where 
is the sine function, plotted above.
This has the normalization
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This function is implemented in the Wolfram Language as Sinc[x].
When extended into the complex plane, 
is illustrated above.
An interesting property of 
is that the set of local
extrema of 
for real 
corresponds to its intersections with the cosine function
, as illustrated above.
The derivative is given by
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for 
, with 
, and the indefinite
integral by
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where 
is the sine integral.
Woodward (1953), McNamee et al. (1971), and Bracewell (1999, p. 62) adopt the alternative definition
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The latter definition is sometimes more convenient as a result of its simple normalization,
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That variant also satisfies the sum
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In addition, the binomial coefficient satisfies
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which is essentially a restatement of the reflection relation
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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,
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and is given in terms of the Meijer G-function as
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Let 
be the rectangle function. The Fourier
transform of 
is the sinc function
=sinc(pik).](/images/equations/SincFunction/NumberedEquation11.svg) |
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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 integral of a complex function by noting that, for 
and 
,
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 |  | ![1/(2inx)[e^(itx)]_(-n)^n](/images/equations/SincFunction/Inline30.svg) |
(16)
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and that 
and the integral both equal 1 for 
.
The sinc function can also be written as the infinite product
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whose special case 
gives the infinite product for 
discovered by Francois Viète (Morrison 1995). The
general identity is sometimes known as Euler's formula
(Prudnikov et al. 1986, p. 757; Gearhart and Shultz 1990). It is also
given by
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(Gearhart and Shultz 1990) and
![sinc(x)=product_(k=1)^infty[1-4/3sin^2(x/(3^k))]](/images/equations/SincFunction/NumberedEquation14.svg) |
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(Prudnikov et al. 1986, p. 757).
Another infinite product is given by
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(OEIS A118253; Prudnikov et al. 1986, p. 757), where 
is the constant from polygon
circumscribing.
Sums of powers of 
over the positive integers
include
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The remarkable fact that the sums of 
and 
are equal appears to have first been published in
Baillie (1978). The pattern of these sums being equal to
plus a rational multiple of 
breaks down for the power 
, where the sum
equals
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where
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The sinc function satisfies the identity
![int_(-infty)^inftysinc[pi(x-y)]sinc(piy)dy=sinc(pix).](/images/equations/SincFunction/NumberedEquation17.svg) |
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Definite integrals involving the sinc function include
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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, Butler 1960, Medhurst
and Roberts 1965, Finch 2003). These are all special cases of the general result
![int_0^infty(sin^ax)/(x^b)dx=(pi^(1-c)(-1)^(|_(a-b)/2_|))/(2^(a-c)(b-1)!)sum_(k=0)^(|_a/2_|-c)(-1)^k(a; k)(a-2k)^(b-1)[ln(a-2k)]^c,](/images/equations/SincFunction/NumberedEquation18.svg) |
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where 
and 
are positive integers satisfying 
, 
satisfies 
, and 
is the floor function.
The logarithm factor is omitted when 
, and the convention 
is used (Kogan 1999). This formula simplifies in the special
case when 
is a positive integer to
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where 
is an Eulerian number (Butler 1960, Kogan 1999).
For positive integers 
and 
of the same parity satisfying 
, define 
. These coefficients
satisfy the recurrence relation
![c(a,b)={pi/(2^a)(a-1; 1/2(a-1)) for b=1; pi/(2^(a-1))(a-2; 1/2(a-2)) for b=2; (a[(a-1)c(a-2,b-2)-a·c(a,b-2)])/((b-1)(b-2)) for b>=3.](/images/equations/SincFunction/NumberedEquation20.svg) |
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The half-infinite integral of 
can be derived using contour
integration. In the above figure, let 
with 
. The small semicircle 
is traversed clockwise and the
large semicircle 
counterclockwise in the upper
half-plane. The two remaining segments lie on the real
line. The origin lies outside this closed contour,
so
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Define the Cauchy principal value of the line integral using the combined limit
![J=lim_(R_1->0^+)lim_(R_2->infty)[int_(-R_2)^(-R_1)(e^(ix))/xdx+int_(R_1)^(R_2)(e^(ix))/xdx].](/images/equations/SincFunction/NumberedEquation22.svg) |
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On the small semicircle, 
, so its integral
tends to 
.
The integral on the large semicircle
tends to 0 by Jordan's lemma. Consequently,
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and therefore
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Taking the imaginary part gives the ordinary improper integral
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The same Cauchy principal value follows from the half-complex residue of 
at the origin,
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Since the integrand is an even function,
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giving the sine integral evaluated at 0 as
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." In 
by the Trapezoidal Rule." Amer.
Math. Monthly 67, 566-569, 1960.Finch, S. R. 
." College Math. J. 21, 90-99, 1990.
." Math.
Comput. 19, 113-117, 1965.Morrison, K. E. "Cosine
Products, Fourier Transforms, and Random Sums." Amer. Math. Monthly 102,
716-724, 1995.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev,
O. I.