Sinc Function

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The sinc function sinc(x), 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 sinx is the sine function, plotted above.

This has the normalization

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

When extended into the complex plane, sinc(z) is illustrated above.

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

The derivative is given by

and the indefinite integral by

where Si(z) 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 j_n(x) and, in particular,

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

Let Pi(x) be the rectangle function, then the Fourier transform of Pi(x) 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 x!=0,

and that sinc(nx) and the integral both equal 1 for x=0.

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 K=8.700... is the constant from polygon circumscribing.

Sums of powers of sinc(k) over the positive integers include

The remarkable fact that the sums of sinc(k) and sinc(k)^2 are equal appears to have first been published in Baillie (1978). Amazingly, the pattern of these sums being equal to -1/2 plus a rational multiple of pi breaks down for the power n=7, 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 pi, 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 amazing general result

where a and b are positive integers such that a>=b>c, c=a-b (mod 2), |_x_| is the floor function, and 0^0 is taken to be equal to 1 (Kogan 1999). This spectacular formula simplifies in the special case when n is a positive even integer to

where <n; k> is an Eulerian number (Butler 1960, Kogan 1999). The solution of the integral can also be written in terms of the recurrence relation for the coefficients

The half-infinite integral of sinc(x) can be derived using contour integration. In the above figure, consider the path gamma=gamma_1+gamma_(12)+gamma_2+gamma_(21). Now write z=Re^(itheta). On an arc, dz=iRe^(itheta)dtheta and on the x-axis, dz=e^(itheta)dR. Write

where I denotes the imaginary part. Now define

where the second and fourth terms use the identities e^(i0)=1 and e^(ipi)=-1. 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

See also

Borwein Integrals, Fourier Transform, Fourier Transform--Rectangle Function, Instrument Function, Jinc Function, Kilroy Curve, Sine, Sine Integral, Sinhc Function, Tanc Function

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References

Baillie, R. "Advanced Problem 6241." Amer. Math. Monthly 85, 828, 1978.Baillie, R.; Henrici, P.; and Johnsonbaugh, R. "Solution to Advanced Problem 6241." Amer. Math. Monthly 87, 496-498, 1980.Bracewell, R. "The Filtering or Interpolating Function, sincx." In The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 62-64, 1999.Brown, J. W. and Churchill, R. V. Fourier Series and Boundary Value Problems, 5th ed. New York: McGraw-Hill, 1993.Butler, R. "On the Evaluation of int_0^infty(sint/t)^mdt by the Trapezoidal Rule." Amer. Math. Monthly 67, 566-569, 1960.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Gearhart, W. B. and Schulz, H. S. "The Function sinx/x." College Math. J. 21, 90-99, 1990. https://doi.org/10.1080/07468342.1990.11973290.Grimsey, A. H. R. "On the Accumulation of Chance Effects and the Gaussian Frequency Distribution." Phil. Mag. 36, 294-295, 1945.Higgins, J. R. "Five Short Stories About the Cardinal Series." Bull. Amer. Math. Soc. 12, 45-89, 1985.Kac, M. Statistical Independence in Probability, Analysis and Number Theory. Washington, DC: Math. Assoc. Amer., 1959.Kogan, S. "A Note on Definite Integrals Involving Trigonometric Functions." Unpublished manuscript, 1999. https://web.archive.org/web/19990417085530/http://www.mathsoft.com/asolve/constant/pi/sin/sin.html.McNamee, J.; Stenger, F.; and Whitney, E. L. "Whittaker's Cardinal Function in Retrospect." Math. Comput. 25, 141-154, 1971.Medhurst, R. G. and Roberts, J. H. "Evaluation of the Integral I_n(b)=(2/pi)int_0^infty(sinx/x)^ncos(bx)dx." 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. Integrals and Series, Vol. 1: Elementary Functions. New York: Gordon & Breach, 1986.Sloane, N. J. A. Sequences A049330, A049331, and A118253 in "The On-Line Encyclopedia of Integer Sequences."Stenger, F. Numerical Methods Based on Sinc and Analytic Functions. New York: Springer-Verlag, 1993.Woodward, P. M. Probability and Information Theory with Applications to Radar. New York: McGraw-Hill, 1953.Zimmermann, P. "Int(sin(x)^k/x^k,x=0..infinity)." math-fun@cs.arizona.edu posting, 4 Mar 1997.

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Sinc Function

Cite this as:

Weisstein, Eric W. "Sinc Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SincFunction.html

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