| 释义 |
Sinc FunctionA function also called the Sampling Function and defined by
 | (1) |
 is the Sine function. Let  be the Rectangle Function, then the Fourier Transform of is the sinc function
 | (2) |
 as the so-called Instrument Function, which gives the instrumentalresponse to a Delta Function input. Removing the instrument functions from the final spectrum requires use of somesort of Deconvolution algorithm.
The sinc function can be written as a complex Integral by noting that
The sinc function can also be written as the Infinite Product
 | (4) |
These are all special cases of the amazing general result
 | (10) |
 and  are Positive integers such that  ,  ,  is the Floor Function,and  is taken to be equal to 1 (Kogan). This spectacular formula simplifies in the special case when  is a PositiveEven integer to
 | (11) |
 is an Eulerian Number (Kogan). The solution of the integral can also be written in terms of theRecurrence Relation for the coefficients
 | (12) |
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
 | (13) |
 denotes the Imaginary Point. 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 theothers yield
 | (16) |
 | (17) |
 | (18) |
so
 | (20) |
 | (21) |
 | (22) |
An interesting property of is that the set of Local Extrema of correspondsto its intersections with the Cosine function  , as illustrated above. See also Fourier Transform, Fourier Transform--Rectangle Function,Instrument Function, Jinc Function, Sine, Sine Integral
References
Kogan, S. ``A Note on Definite Integrals Involving Trigonometric Functions.'' http://www.mathsoft.com/asolve/constant/pi/sin/sin.html.Morrison, K. E. ``Cosine Products, Fourier Transforms, and Random Sums.'' Amer. Math. Monthly 102, 716-724, 1995.
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