Content deleted Content added
m disambig function |
removed incorrect information the function was first defined far before 1952 before anyone even though of telecom |
||
| (591 intermediate revisions by more than 100 users not shown) | |||
Line 1:
{{short description|Special mathematical function defined as sin(x)/x}}
{{Redirect-distinguish|Sinc|Site of Importance for Nature Conservation}}
{{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]], [[physics]] and [[engineering]], the '''sinc function''' ({{IPAc-en|ˈ|s|ɪ|ŋ|k}} {{respell|SINK}}), denoted by {{math|sinc(''x'')}}, is defined as either
or
<math display="block">\operatorname{sinc}(x) = \frac{\sin \pi x}{\pi x}.</math>
The only difference between the two definitions is in the scaling of the [[independent variable]] (the [[Cartesian coordinate system|{{mvar|x}} axis]]) by a factor of {{pi}}. In both cases, the value of the function at the [[removable singularity]] at zero is understood to be the limit value 1. The sinc function is then [[Analytic function|analytic]] everywhere and hence an [[entire function]].
The {{pi}}-normalized sinc function is the [[Fourier transform]] of the [[rectangular function]] with no scaling. It is used in the concept of [[Whittaker–Shannon interpolation formula|reconstructing]] a continuous bandlimited signal from uniformly spaced [[Nyquist–Shannon sampling theorem|samples]] of that signal. The [[sinc filter]] is used in signal processing.
The function itself was first mathematically derived in this form by [[Lord Rayleigh]] in his expression ([[Bessel functions#Rayleigh's formulas|Rayleigh's formula]]) for the zeroth-order spherical [[Bessel function]] of the first kind.
==Definitions==
[[File:Sinc.wav|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]], 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]] and [[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#Second example|squeeze theorem]]).
The [[Normalizing constant|normalization]] causes the [[integral|definite 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|{{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/ |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 Woodward|Philip M. Woodward]] and I.L Davies in their 1952 article "Information theory and [[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 ==
[[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>
== Relationship to the Dirac delta distribution ==
The normalized sinc function can be used as a ''[[Dirac delta function#Representations of the delta function|nascent delta function]]'', meaning that the following [[weak convergence (Hilbert space)|weak limit]] 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 space|Schwartz function]], as can be seen from the [[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]].
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]] | ___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]]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]] 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>
[[Leonhard Euler|Euler]] famously compared this series to the expansion of the infinite product form to solve the [[Basel problem]].
== Higher dimensions ==
The product of 1-D sinc functions readily provides a [[multivariable calculus|multivariate]] sinc function for the square Cartesian grid ([[Lattice graph|lattice]]): {{math|sinc<sub>C</sub>(''x'', ''y'') {{=}} sinc(''x'') sinc(''y'')}}, whose [[Fourier transform]] is the [[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 (group)|lattice]] (e.g., [[hexagonal lattice]]) is a function whose [[Fourier transform]] is the [[indicator function]] of the [[Brillouin zone]] of that lattice. For example, the sinc function for the hexagonal lattice is a function whose [[Fourier transform]] is the [[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]]. However, the explicit formula for the sinc function for the [[hexagonal lattice|hexagonal]], [[body-centered cubic]], [[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 [[zonohedron|zonotopes]].
For example, a [[hexagonal lattice]] can be generated by the (integer) [[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]] for general multidimensional lattices.<ref name="multiD" />
== Sinhc ==
Some authors, by analogy, define the hyperbolic sine [[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]]
| |||