Editing Sinc function (section)

== Properties ==

[[File:Si cos.svg|thumb|350px|right|The local maxima and minima (small white dots) of the unnormalized, red sinc function correspond to its intersections with the blue [[cosine function]].]]
The [[zero crossing]]s of the unnormalized sinc are at non-zero integer multiples of {{pi}}, while zero crossings of the normalized sinc occur at non-zero integers.

The local maxima and minima of the unnormalized sinc correspond to its intersections with the [[cosine]] function. That is, {{math|1={{sfrac|sin(''ξ'')|''ξ''}} = cos(''ξ'')}} for all points {{mvar|ξ}} where the derivative of {{math|{{sfrac|sin(''x'')|''x''}}}} is zero and thus a local extremum is reached. This follows from the derivative of the sinc function:
<math display="block">\frac{d}{dx}\operatorname{sinc}(x) = \begin{cases} \dfrac{\cos(x) - \operatorname{sinc}(x)}{x}, & x \ne 0 \\0, & x = 0\end{cases}.</math>

The first few terms of the infinite series for the {{mvar|x}} coordinate of the {{mvar|n}}-th extremum with positive {{mvar|x}} coordinate are {{Citation needed|date=January 2025}}
<math display="block">x_n = q - q^{-1} - \frac{2}{3} q^{-3} - \frac{13}{15} q^{-5} - \frac{146}{105} q^{-7} - \cdots,</math>
where
<math display="block">q = \left(n + \frac{1}{2}\right) \pi,</math>
and where odd {{mvar|n}} lead to a local minimum, and even {{mvar|n}} to a local maximum. Because of symmetry around the {{mvar|y}} axis, there exist extrema with {{mvar|x}} coordinates {{math|−''x<sub>n</sub>''}}. In addition, there is an absolute maximum at {{math|1=''ξ''<sub>0</sub> = (0, 1)}}.

The normalized sinc function has a simple representation as the [[infinite product]]:
<math display="block">\frac{\sin(\pi x)}{\pi x} = \prod_{n=1}^\infty \left(1 - \frac{x^2}{n^2}\right)</math>
[[File:The cardinal sine function sinc(z) plotted in the complex plane from -2-2i to 2+2i.svg|alt=The cardinal sine function sinc(z) plotted in the complex plane from -2-2i to 2+2i|thumb|The cardinal sine function sinc(z) plotted in the complex plane from -2-2i to 2+2i]]
and is related to the [[gamma function]] {{math|Γ(''x'')}} through [[Euler's reflection formula]]:
<math display="block">\frac{\sin(\pi x)}{\pi x} = \frac{1}{\Gamma(1 + x)\Gamma(1 - x)}.</math>

[[Euler]] discovered<ref>{{cite arXiv |last=Euler |first=Leonhard |title=On the sums of series of reciprocals |year=1735 |eprint=math/0506415}}</ref> that
<math display="block">\frac{\sin(x)}{x} = \prod_{n=1}^\infty \cos\left(\frac{x}{2^n}\right),</math>
and because of the product-to-sum identity<ref>{{cite journal |author1=Sanjar M. Abrarov |author2=Brendan M. Quine |title=Sampling by incomplete cosine expansion of the sinc function: Application to the Voigt/complex error function |year=2015 |journal=Appl. Math. Comput. |volume=258 |issue= |pages=425–435 |doi=10.1016/j.amc.2015.01.072 |arxiv=1407.0533 |bibcode=|url=https://www.sciencedirect.com/science/article/pii/S0096300315001046 |hdl-access= }}</ref>
[[File:Sinc cplot.svg|thumb|[[Domain coloring]] plot of {{math|1=sinc ''z'' = {{sfrac|sin ''z''|''z''}}}}]]
<math display="block">\prod_{n=1}^k \cos\left(\frac{x}{2^n}\right) = \frac{1}{2^{k-1}} \sum_{n=1}^{2^{k-1}} \cos\left(\frac{n - 1/2}{2^{k-1}} x \right),\quad \forall k \ge 1,</math>
Euler's product can be recast as a sum
<math display="block">\frac{\sin(x)}{x} = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N \cos\left(\frac{n - 1/2}{N} x\right).</math>

The [[continuous Fourier transform]] of the normalized sinc (to ordinary frequency) is {{math|[[rectangular function|rect]](''f'')}}:
<math display="block">\int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f),</math>
where the [[rectangular function]] is 1 for argument between −{{sfrac|1|2}} and {{sfrac|1|2}}, and zero otherwise. This corresponds to the fact that the [[sinc filter]] is the ideal ([[brick-wall filter|brick-wall]], meaning rectangular [[frequency response]]) [[low-pass filter]].

This Fourier integral, including the special case
<math display="block">\int_{-\infty}^\infty \frac{\sin(\pi x)}{\pi x} \, dx = \operatorname{rect}(0) = 1</math>
is an [[improper integral]] (see [[Dirichlet integral]]) and not a convergent [[Lebesgue integral]], as
<math display="block">\int_{-\infty}^\infty \left|\frac{\sin(\pi x)}{\pi x} \right| \,dx = +\infty.</math>

The normalized sinc function has properties that make it ideal in relationship to [[interpolation]] of [[sampling (signal processing)|sampled]] [[bandlimited]] functions:
* It is an interpolating function, i.e., {{math|1=sinc(0) = 1}}, and {{math|1=sinc(''k'') = 0}} for nonzero [[Number#Integers|integer]] {{math|''k''}}.
* The functions {{math|1=''x<sub>k</sub>''(''t'') = sinc(''t'' − ''k'')}} ({{mvar|k}} integer) form an [[orthonormal basis]] for [[bandlimited]] functions in the [[Lp space|function space]] {{math|'''''L'''''<sup>2</sup>('''R''')}}, with highest angular frequency {{math|1=''ω''<sub>H</sub> = π}} (that is, highest cycle frequency {{math|1=''f''<sub>H</sub> = {{sfrac|1|2}}}}).

Other properties of the two sinc functions include:
* The unnormalized sinc is the zeroth-order spherical [[Bessel function]] of the first kind, {{math|''j''<sub>0</sub>(''x'')}}. The normalized sinc is {{math|''j''<sub>0</sub>(π''x'')}}.
* where {{math|Si(''x'')}} is the [[sine integral]], <math display="block">\int_0^x \frac{\sin(\theta)}{\theta}\,d\theta = \operatorname{Si}(x).</math>
* {{math|''λ'' sinc(''λx'')}} (not normalized) is one of two linearly independent solutions to the linear [[ordinary differential equation]] <math display="block">x \frac{d^2 y}{d x^2} + 2 \frac{d y}{d x} + \lambda^2 x y = 0.</math> The other is {{math|{{sfrac|cos(''λx'')|''x''}}}}, which is not bounded at {{math|1=''x'' = 0}}, unlike its sinc function counterpart.
* Using normalized sinc, <math display="block">\int_{-\infty}^\infty \frac{\sin^2(\theta)}{\theta^2}\,d\theta = \pi \quad \Rightarrow \quad \int_{-\infty}^\infty \operatorname{sinc}^2(x)\,dx = 1,</math>
* <math>\int_{-\infty}^\infty \frac{\sin(\theta)}{\theta}\,d\theta = \int_{-\infty}^\infty \left( \frac{\sin(\theta)}{\theta} \right)^2 \,d\theta = \pi.</math>
* <math>\int_{-\infty}^\infty \frac{\sin^3(\theta)}{\theta^3}\,d\theta = \frac{3\pi}{4}.</math>
* <math>\int_{-\infty}^\infty \frac{\sin^4(\theta)}{\theta^4}\,d\theta = \frac{2\pi}{3}.</math>
* The following improper integral involves the (not normalized) sinc function: <math display="block">\int_0^\infty \frac{dx}{x^n + 1} = 1 + 2\sum_{k=1}^\infty \frac{(-1)^{k+1}}{(kn)^2 - 1} = \frac{1}{\operatorname{sinc}(\frac{\pi}{n})}.</math>