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A function also called the Sampling Function and defined by
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(1) |
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(2) |
The sinc function can be written as a complex Integral by noting that
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(3) |
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(4) |
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(5) |
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(6) |
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(7) |
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(8) |
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(9) |
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(10) |
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(11) |
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(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
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(13) |
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(14) |
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(15) |
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(16) |
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(17) |
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(18) |
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(19) |
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(20) |
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(21) |
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(22) |
An interesting property of
is that the set of Local Extrema of
corresponds
to 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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© 1996-9 Eric W. Weisstein