# Sinc function

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{{short description|Special mathematical function defined as sin(x)/x}}
{{Redirect|Sinc}}
{{Use American English|date = March 2019}}
{{Infobox mathematical function
| name = Sinc
| image = Si sinc.svg
| imagesize = 350px
| imagealt = Part of the normalized and unnormalized sinc function shown on the same scale
| caption = Part of the normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale
| general_definition = <math>\operatorname{sinc}x = \begin{cases} \dfrac{ \sin x } x, & x \ne 0 \\ 1, & x = 0\end{cases}</math>
| fields_of_application = Signal processing, spectroscopy
| domain = <math>\mathbb{R}</math>
| range = <math>[-0.217234\ldots, 1]</math>
| parity = Even
| zero = 1
| plusinf = 0
| minusinf = 0
| max = 1 at <math>x = 0</math>
| min = <math>-0.21723\ldots</math> at <math>x = \pm 4.49341\ldots</math>
| root = <math>\pi k, k \in \mathbb{Z}_{\neq 0}</math>
| reciprocal = <math>\begin{cases} x \csc x, & x \ne 0 \\ 1, & x = 0 \end{cases}</math>
| derivative = <math>\operatorname{sinc}'x = \begin{cases} \dfrac{\cos x - \operatorname{sinc} x}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases}</math>
| antiderivative = <math>\int \operatorname{sinc} x\,dx = \operatorname{Si}(x) + C</math>
| taylor_series = <math>\operatorname{sinc}x = \sum_{k=0}^\infty \frac{(-1)^k x^{2k}}{(2k + 1)!}</math>
}}

In [mathematics](/source/mathematics), [physics](/source/physics) and [engineering](/source/engineering), the '''sinc function''' ({{IPAc-en|ˈ|s|ɪ|ŋ|k}} {{respell|SINK}}), denoted by {{math|sinc(''x'')}}, is defined as either
<math display="block">\operatorname{sinc}(x) = \frac{\sin x}{x}.</math>
or
<math display="block">\operatorname{sinc}(x) = \frac{\sin \pi x}{\pi x},</math>

the latter of which is sometimes referred to as the '''normalized sinc function'''. The only difference between the two definitions is in the scaling of the [independent variable](/source/independent_variable) (the [{{mvar|x}} axis](/source/Cartesian_coordinate_system)) by a factor of {{pi}}. In both cases, the value of the function at the [removable singularity](/source/removable_singularity) at zero is understood to be the limit value 1. The sinc function is then [analytic](/source/Analytic_function) everywhere and hence an [entire function](/source/entire_function).

The normalized sinc function is the [Fourier transform](/source/Fourier_transform) of the [rectangular function](/source/rectangular_function) with no scaling. It is used in the concept of [reconstructing](/source/Whittaker%E2%80%93Shannon_interpolation_formula) a continuous bandlimited signal from uniformly spaced [samples](/source/Nyquist%E2%80%93Shannon_sampling_theorem) of that signal. The [sinc filter](/source/sinc_filter) is used in signal processing.

The function itself was first mathematically derived in this form by [Lord Rayleigh](/source/Lord_Rayleigh) in his expression ([Rayleigh's formula](/source/Bessel_functions)) for the zeroth-order spherical [Bessel function](/source/Bessel_function) of the first kind.

The ''sinc'' function is also called the '''cardinal sine''' function.

==Definitions==
thumb|The sinc function as audio, at 2000 Hz (±1.5 seconds around zero)

The sinc function has two forms, normalized and unnormalized.<ref name="dlmf">{{dlmf|title=Numerical methods|id=3.3}}.</ref>

In mathematics, the historical '''unnormalized sinc function''' is defined for {{math|''x'' ≠ 0}} by
<math display="block">\operatorname{sinc}(x) = \frac{\sin x}{x}.</math>

Alternatively, the unnormalized sinc function is often called the [sampling function](/source/sampling_function), indicated as Sa(''x'').<ref>{{cite book |title=Communication Systems, 2E |edition=illustrated |first1=R. P. |last1=Singh |first2=S. D. |last2=Sapre |publisher=Tata McGraw-Hill Education |year=2008 |isbn=978-0-07-063454-1 |page=15 |url=https://books.google.com/books?id=WkOPPEhK7SYC}} [https://books.google.com/books?id=WkOPPEhK7SYC&pg=PA15 Extract of page 15]</ref>

In [digital signal processing](/source/digital_signal_processing) and [information theory](/source/information_theory), the '''normalized sinc function''' is commonly defined for {{math|''x'' ≠ 0}} by
<math display="block">\operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x}.</math>

In either case, the value at {{math|1=''x'' = 0}} is defined to be the limiting value
<math display="block">\operatorname{sinc}(0) := \lim_{x \to 0}\frac{\sin(a x)}{a x} = 1</math> for all real {{math|''a'' ≠ 0}} (the limit can be proven using the [squeeze theorem](/source/Squeeze_theorem)).

The [normalization](/source/Normalizing_constant) causes the [definite integral](/source/integral) of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of [{{pi}}](/source/pi)). As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of {{mvar|x}}.

==Etymology==
The function has also been called the '''cardinal sine''' or '''sine cardinal''' function.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Sinc Function |url=https://mathworld.wolfram.com/SincFunction.html |access-date=2023-06-07 |website=mathworld.wolfram.com |language=en}}</ref><ref>{{Cite journal |last=Merca |first=Mircea |date=2016-03-01 |title=The cardinal sine function and the Chebyshev–Stirling numbers |url=https://www.sciencedirect.com/science/article/pii/S0022314X15002863 |journal=Journal of Number Theory |language=en |volume=160 |pages=19–31 |doi=10.1016/j.jnt.2015.08.018 |s2cid=124388262 |issn=0022-314X|url-access=subscription }}</ref> The term "sinc" is a contraction of the function's full Latin name, the {{lang|la|sinus cardinalis}}<ref name=Poynton /> and was introduced by [Philip M.&nbsp;Woodward](/source/Philip_Woodward) and I.L Davies in their 1952 article "Information theory and [inverse probability](/source/inverse_probability) in telecommunication", saying "This function occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own".<ref>{{cite journal |last1=Woodward |first1=P. M. |last2=Davies |first2=I. L. |url=http://www.norbertwiener.umd.edu/crowds/documents/Woodward52.pdf |title=Information theory and inverse probability in telecommunication |journal=Proceedings of the IEE - Part III: Radio and Communication Engineering |volume=99 |issue=58 |pages=37–44 |date= March 1952 |doi=10.1049/pi-3.1952.0011}}</ref> It is also used in Woodward's 1953 book ''Probability and Information Theory, with Applications to Radar''.<ref name="Poynton">{{Cite book |first=Charles A. |last=Poynton |title=Digital video and HDTV  |page=147 |publisher=Morgan Kaufmann Publishers |year=2003 |isbn=978-1-55860-792-7}}</ref><ref>{{cite book |first=Phillip M. |last=Woodward |title=Probability and information theory, with applications to radar|page=29 |location=London |publisher=Pergamon Press |year=1953 |oclc=488749777 |isbn=978-0-89006-103-9}}</ref>

== Properties ==

thumb|350px|right|The local extrema (small white dots) of the unnormalized, red sinc function correspond to its intersections with the blue cosine function.
The [zero crossing](/source/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 extrema](/source/local_extrema) of the unnormalized sinc correspond to its intersections with the [cosine](/source/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](/source/infinite_product):
<math display="block">\frac{\sin(\pi x)}{\pi x} = \prod_{n=1}^\infty \left(1 - \frac{x^2}{n^2}\right)</math>
alt=The cardinal sine function sinc(z) plotted in the complex plane from -2-2i to 2+2i|thumb|The cardinal sine function {{math|sinc(''z'')}} plotted in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}
and is related to the [gamma function](/source/gamma_function) {{math|Γ(''x'')}}, as well as to Gauss' [Pi function](/source/Gamma_function), through [Euler's reflection formula](/source/Euler's_reflection_formula):
<math display="block">\frac{\sin(\pi x)}{\pi x} = \frac{1}{\Gamma(1 + x)\Gamma(1 - x)} = \frac{1}{\Pi(x)\Pi(-x)}.</math>

[Euler](/source/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](/source/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](/source/continuous_Fourier_transform) of the normalized sinc (to ordinary frequency) is {{math|[rect](/source/rectangular_function)(''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](/source/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](/source/sinc_filter) is the ideal ([brick-wall](/source/brick-wall_filter), meaning rectangular [frequency response](/source/frequency_response)) [low-pass filter](/source/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](/source/improper_integral) (see [Dirichlet integral](/source/Dirichlet_integral)) and not a convergent [Lebesgue integral](/source/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](/source/interpolation) of [sampled](/source/sampling_(signal_processing)) [bandlimited](/source/bandlimited) functions:
* It is an interpolating function, i.e., {{math|1=sinc(0) = 1}}, and {{math|1=sinc(''k'') = 0}} for nonzero [integer](/source/Number) {{math|''k''}}.
* The functions {{math|1=''x<sub>k</sub>''(''t'') = sinc(''t'' − ''k'')}} ({{mvar|k}} integer) form an [orthonormal basis](/source/orthonormal_basis) for [bandlimited](/source/bandlimited) functions in the [function space](/source/Lp_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](/source/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](/source/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](/source/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>

== Relationship to the Dirac delta distribution ==

The normalized sinc function can be used as a ''[nascent delta function](/source/Dirac_delta_function)'', meaning that the following [weak limit](/source/weak_convergence_(Hilbert_space)) holds:

<math display="block">\lim_{a \to 0} \frac{\sin\left(\frac{\pi x}{a}\right)}{\pi x} = \lim_{a \to 0}\frac{1}{a} \operatorname{sinc}\left(\frac{x}{a}\right) = \delta(x).</math>

This is not an ordinary limit, since the left side does not converge. Rather, it means that

<math display="block">\lim_{a \to 0}\int_{-\infty}^\infty \frac{1}{a} \operatorname{sinc}\left(\frac{x}{a}\right) \varphi(x) \,dx = \varphi(0)</math>

for every [Schwartz function](/source/Schwartz_space), as can be seen from the [Fourier inversion theorem](/source/Fourier_inversion_theorem).
In the above expression, as {{math|''a'' → 0}}, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of {{math|±{{sfrac|1|π''x''}}}}, regardless of the value of {{mvar|a}}.

This complicates the informal picture of {{math|''δ''(''x'')}} as being zero for all {{mvar|x}} except at the point {{math|1=''x'' = 0}}, and illustrates the problem of thinking of the delta function as a function rather than as a distribution. A similar situation is found in the [Gibbs phenomenon](/source/Gibbs_phenomenon).

We can also make an immediate connection with the standard Dirac representation of <math>\delta(x)</math> by writing <math> b=1/a </math> and 

<math display="block">\lim_{b \to \infty} \frac{\sin\left(b\pi x\right)}{\pi x} = \lim_{b \to \infty} \frac{1}{2\pi} \int_{-b\pi}^{b\pi} e^{ik x}dk= \frac{1}{2\pi} \int_{-\infty}^\infty e^{i k x} dk=\delta(x),</math>

which makes clear the recovery of the delta as an infinite bandwidth limit of the integral.

== Summation ==
All sums in this section refer to the unnormalized sinc function.

The sum of {{math|sinc(''n'')}} over integer {{mvar|n}} from 1 to {{math|∞}} equals {{math|{{sfrac|{{pi}} − 1|2}}}}:

<math display="block">\sum_{n=1}^\infty \operatorname{sinc}(n) = \operatorname{sinc}(1) + \operatorname{sinc}(2) +  \operatorname{sinc}(3) + \operatorname{sinc}(4) +\cdots = \frac{\pi - 1}{2}.</math>

The sum of the squares also equals {{math|{{sfrac|{{pi}} − 1|2}}}}:<ref>{{cite journal | title = Advanced Problem 6241 | journal = American Mathematical Monthly | date = June–July 1980 | volume = 87 | issue = 6 | pages = 496–498 | publisher = [Mathematical Association of America](/source/Mathematical_Association_of_America) | location = Washington, DC | doi = 10.1080/00029890.1980.11995075}}</ref><ref name="BBB">{{cite journal | author1=Robert Baillie | author2-link=David Borwein | author2=David Borwein | author3=Jonathan M. Borwein | author3-link=Jonathan M. Borwein | title=Surprising Sinc Sums and Integrals | journal=American Mathematical Monthly | date=December 2008 | volume=115 | issue=10 | pages=888–901 | jstor = 27642636 | doi=10.1080/00029890.2008.11920606 | hdl=1959.13/940062 | s2cid=496934 | hdl-access=free}}</ref>

<math display="block">\sum_{n=1}^\infty \operatorname{sinc}^2(n) = \operatorname{sinc}^2(1) + \operatorname{sinc}^2(2) + \operatorname{sinc}^2(3) + \operatorname{sinc}^2(4) + \cdots = \frac{\pi - 1}{2}.</math>

When the signs of the [addend](/source/addend)s alternate and begin with +, the sum equals {{sfrac|1|2}}:
<math display="block">\sum_{n=1}^\infty (-1)^{n+1}\,\operatorname{sinc}(n) = \operatorname{sinc}(1) - \operatorname{sinc}(2) + \operatorname{sinc}(3) - \operatorname{sinc}(4) + \cdots = \frac{1}{2}.</math>

The alternating sums of the squares and cubes also equal {{sfrac|1|2}}:<ref name="FWFS">{{cite arXiv |last=Baillie |first=Robert |eprint=0806.0150v2 |class=math.CA |title=Fun with Fourier series |date=2008}}</ref>
<math display="block">\sum_{n=1}^\infty (-1)^{n+1}\,\operatorname{sinc}^2(n) = \operatorname{sinc}^2(1) - \operatorname{sinc}^2(2) + \operatorname{sinc}^2(3) - \operatorname{sinc}^2(4) + \cdots = \frac{1}{2},</math>

<math display="block">\sum_{n=1}^\infty (-1)^{n+1}\,\operatorname{sinc}^3(n) = \operatorname{sinc}^3(1) - \operatorname{sinc}^3(2) + \operatorname{sinc}^3(3) - \operatorname{sinc}^3(4) + \cdots = \frac{1}{2}.</math>

== Series expansion ==
The [Taylor series](/source/Taylor_series) of the unnormalized {{math|sinc}} function can be obtained from that of the sine (which also yields its value of 1 at {{math|1=''x'' = 0}}):
<math display="block">\frac{\sin x}{x} = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n+1)!} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots</math>

The series converges for all {{mvar|x}}. The normalized version follows easily:
<math display="block">\frac{\sin \pi x}{\pi x} = 1 - \frac{\pi^2x^2}{3!} + \frac{\pi^4x^4}{5!} - \frac{\pi^6x^6}{7!} + \cdots</math>

[Euler](/source/Leonhard_Euler) famously compared this series to the expansion of the infinite product form to solve the [Basel problem](/source/Basel_problem).

== Higher dimensions ==
The product of 1-D sinc functions readily provides a [multivariate](/source/multivariable_calculus) sinc function for the square Cartesian grid ([lattice](/source/Lattice_graph)): {{math|sinc<sub>C</sub>(''x'', ''y'') {{=}} sinc(''x'') sinc(''y'')}}, whose [Fourier transform](/source/Fourier_transform) is the [indicator function](/source/indicator_function) of a square in the frequency space (i.e., the brick wall defined in 2-D space). The sinc function for a non-Cartesian [lattice](/source/Lattice_(group)) (e.g., [hexagonal lattice](/source/hexagonal_lattice)) is a function whose [Fourier transform](/source/Fourier_transform) is the [indicator function](/source/indicator_function) of the [Brillouin zone](/source/Brillouin_zone) of that lattice. For example, the sinc function for the hexagonal lattice is a function whose [Fourier transform](/source/Fourier_transform) is the [indicator function](/source/indicator_function) of the unit hexagon in the frequency space. For a non-Cartesian lattice this function can not be obtained by a simple [tensor product](/source/tensor_product). However, the explicit formula for the sinc function for the [hexagonal](/source/hexagonal_lattice), [body-centered cubic](/source/body-centered_cubic), [face-centered cubic](/source/face-centered_cubic) and other higher-dimensional lattices can be explicitly derived<ref name="multiD">{{cite journal |last1=Ye |first1= W. |last2=Entezari |first2= A. |title=A Geometric Construction of Multivariate Sinc Functions |journal=IEEE Transactions on Image Processing |volume=21 |issue=6 |pages=2969–2979 |date=June 2012 |doi=10.1109/TIP.2011.2162421 |pmid=21775264 |bibcode=2012ITIP...21.2969Y|s2cid= 15313688 }}</ref> using the geometric properties of Brillouin zones and their connection to [zonotopes](/source/zonohedron).

For example, a [hexagonal lattice](/source/hexagonal_lattice) can be generated by the (integer) [linear span](/source/linear_span) of the vectors
<math display="block">
  \mathbf{u}_1 = \begin{bmatrix} \frac{1}{2} \\  \frac{\sqrt{3}}{2} \end{bmatrix} \quad \text{and} \quad
  \mathbf{u}_2 = \begin{bmatrix} \frac{1}{2} \\ -\frac{\sqrt{3}}{2} \end{bmatrix}.
</math>

Denoting
<math display="block">
  \boldsymbol{\xi}_1 =  \tfrac{2}{3} \mathbf{u}_1, \quad
  \boldsymbol{\xi}_2 =  \tfrac{2}{3} \mathbf{u}_2, \quad
  \boldsymbol{\xi}_3 = -\tfrac{2}{3} (\mathbf{u}_1 + \mathbf{u}_2), \quad
          \mathbf{x} = \begin{bmatrix} x \\ y\end{bmatrix},
</math>
one can derive<ref name="multiD" /> the sinc function for this hexagonal lattice as
<math display="block">\begin{align}
  \operatorname{sinc}_\text{H}(\mathbf{x}) = \tfrac{1}{3} \big(
    &      \cos\left(\pi\boldsymbol{\xi}_1\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_2\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_3\cdot\mathbf{x}\right) \\
    & {} + \cos\left(\pi\boldsymbol{\xi}_2\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_3\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_1\cdot\mathbf{x}\right) \\
    & {} + \cos\left(\pi\boldsymbol{\xi}_3\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_1\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_2\cdot\mathbf{x}\right)
  \big).
\end{align}</math>

This construction can be used to design [Lanczos window](/source/Lanczos_window) for general multidimensional lattices.<ref name="multiD" />

== Sinhc ==
Some authors, by analogy, define the [hyperbolic](/source/Hyperbolic_function) sine [cardinal function](/source/cardinal_function).<ref>{{cite book |last=Ainslie |first=Michael |date=2010 |title=Principles of Sonar Performance Modelling |publisher=Springer |isbn=9783540876625 |page=636 |url=https://books.google.com/books?id=EqDnP-lAw40C&pg=PA636}}</ref><ref>{{cite book |last=Günter |first=Peter |date=2012 |title=Nonlinear Optical Effects and Materials |publisher=Springer |isbn=9783540497134 |page=258 |url=https://books.google.com/books?id=8QTpCAAAQBAJ&pg=PA258}}</ref><ref>{{cite book |last=Schächter |first=Levi |date=2013 |title=Beam-Wave Interaction in Periodic and Quasi-Periodic Structures |publisher=Springer |isbn=9783662033982 |page=241 |url=https://books.google.com/books?id=jQb9CAAAQBAJ&pg=PA241}}</ref>

:<math>\mathrm{sinhc}(x) = \begin{cases}
  {\displaystyle \frac{\sinh(x)}{x},} & \text{if }x \ne 0 \\
  {\displaystyle 1,} & \text{if }x = 0
\end{cases}</math>

==See also==

* {{annotated link|Anti-aliasing filter}}
* {{annotated link|Borwein integral}}
* {{annotated link|Dirichlet integral}}
* {{annotated link|Lanczos resampling}}
* {{annotated link|List of mathematical functions}}
* {{annotated link|Shannon wavelet}}
* {{annotated link|Sinc filter}}
* {{annotated link|Sinc numerical methods}}
* {{annotated link|Trigonometric functions of matrices}}
* {{annotated link|Trigonometric integral}}
* {{annotated link|Whittaker–Shannon interpolation formula}}
* {{annotated link|Winkel tripel projection}} (cartography)

== References ==
{{Reflist|30em}}

== Further reading ==

* {{cite book |last=Stenger |first=Frank |date=1993 |title=Numerical Methods Based on Sinc and Analytic Functions |publisher=Springer-Verlag New York, Inc. |series=Springer Series on Computational Mathematics|volume=20|doi=10.1007/978-1-4612-2706-9|isbn=9781461276371}}

== External links ==
* {{MathWorld|title=Sinc Function|urlname=SincFunction}}

Category:Signal processing
Category:Elementary special functions

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Adapted from the Wikipedia article [Sinc function](https://en.wikipedia.org/wiki/Sinc_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Sinc_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.