Editing Pascal's triangle (section)

=== Fourier transform of sin(''x'')<sup>''n''+1</sup>/''x'' ===

As stated previously, the coefficients of (''x''&nbsp;+&nbsp;1)<sup>''n''</sup> are the nth row of the triangle. Now the coefficients of (''x''&nbsp;−&nbsp;1)<sup>''n''</sup> are the same, except that the sign alternates from +1 to −1 and back again. After suitable normalization, the same pattern of numbers occurs in the [[Fourier transform]] of sin(''x'')<sup>''n''+1</sup>/''x''. More precisely: if ''n'' is even, take the [[real part]] of the transform, and if ''n'' is odd, take the [[imaginary part]]. Then the result is a [[step function]], whose values (suitably normalized) are given by the ''n''th row of the triangle with alternating signs.<ref>For a similar example, see e.g. {{citation|title=Solvent suppression in Fourier transform nuclear magnetic resonance|first=P. J.|last=Hore|journal=Journal of Magnetic Resonance|year=1983|volume=55|issue=2|pages=283–300|doi=10.1016/0022-2364(83)90240-8|bibcode=1983JMagR..55..283H}}.</ref> For example, the values of the step function that results from:

:<math>\mathfrak{Re}\left(\text{Fourier} \left[  \frac{\sin(x)^5}{x}   \right]\right)</math>

compose the 4th row of the triangle, with alternating signs. This is a generalization of the following basic result (often used in [[electrical engineering]]):

:<math>\mathfrak{Re}\left(\text{Fourier} \left[ \frac{\sin(x)^1}{x}\right] \right)</math>

is the [[boxcar function]].<ref>{{citation|title=An Introduction to Digital Signal Processing|first=John H.|last=Karl|publisher=Elsevier|year=2012|isbn=9780323139595|page=110|url=https://books.google.com/books?id=9Dv1PClLZWIC&pg=PA110}}.</ref>  The corresponding row of the triangle is row 0, which consists of just the number&nbsp;1.

If n is [[congruence relation|congruent]] to 2 or to 3 mod 4, then the signs start with&nbsp;−1. In fact, the sequence of the (normalized) first terms corresponds to the powers of [[imaginary unit|i]], which cycle around the intersection of the axes with the unit circle in the complex plane: <math display="block">  +i,-1,-i,+1,+i,\ldots  </math>