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Sinc function

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</style><div role="note" class="hatnote navigation-not-searchable">"Sinc" redirects here; not to be confused with <a href="Site_of_Importance_for_Nature_Conservation" class="mw-redirect" title="Site of Importance for Nature Conservation">Site of Importance for Nature Conservation</a>.</div>

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</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e0e0e0;padding:0.15em 0.5em 0.25em;font-weight:bold;">Sinc</th></tr><tr><td colspan="2" class="infobox-image" style="padding-bottom:0.4em;"><div class="infobox-caption">Part of the normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale</div></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">General information</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">General definition</th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinc} x={\begin{cases}{\dfrac {\sin x}{x}},&x\neq 0\\1,&x=0\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} x={\begin{cases}{\dfrac {\sin x}{x}},&x\neq 0\\1,&x=0\end{cases}}}</annotation>

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</math><img src="./558236aa92dca99a986726b2542aa62e2c4e5739.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.296ex; height:8.176ex;" alt="{\displaystyle \operatorname {sinc} x={\begin{cases}{\dfrac {\sin x}{x}},&x\neq 0\\1,&x=0\end{cases}}}" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Motivation of invention</th><td class="infobox-data">Telecommunication</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Date of solution</th><td class="infobox-data">1952</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Fields of application</th><td class="infobox-data">Signal processing, spectroscopy</td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Domain, codomain and image</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Domain_of_a_function" title="Domain of a function">Domain</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle [-0.217234\ldots ,1]}</annotation>

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</math><img src="./8f0d3c8e779da502016a1259f426cdfb7aa130cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.58ex; height:2.843ex;" alt="{\displaystyle [-0.217234\ldots ,1]}" loading="lazy"></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Basic features</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Even_and_odd_functions" title="Even and odd functions">Parity</a></th><td class="infobox-data">Even</td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Specific values</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Y-intercept" title="Y-intercept">At zero</a></th><td class="infobox-data">1</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Value at +∞</th><td class="infobox-data">0</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Value at −∞</th><td class="infobox-data">0</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Maxima</th><td class="infobox-data">1 at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} 'x={\begin{cases}{\dfrac {\cos x-\operatorname {sinc} x}{x}},&x\neq 0\\0,&x=0\end{cases}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \int \operatorname {sinc} x\,dx=\operatorname {Si} (x)+C}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} x=\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k+1)!}}}</annotation>

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In <a href="Mathematics" title="Mathematics">mathematics</a>, <a href="Physics" title="Physics">physics</a> and <a href="Engineering" title="Engineering">engineering</a>, the sinc function (/ˈsɪŋk/ SINK), denoted by sinc(x), is defined as either

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (x)={\frac {\sin x}{x}}.}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (x)={\frac {\sin \pi x}{\pi x}}.}</annotation>

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The only difference between the two definitions is in the scaling of the <a href="Independent_variable" class="mw-redirect" title="Independent variable">independent variable</a> (the <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">x axis</a>) by a factor of π. In both cases, the value of the function at the <a href="Removable_singularity" title="Removable singularity">removable singularity</a> at zero is understood to be the limit value 1. The sinc function is then <a href="Analytic_function" title="Analytic function">analytic</a> everywhere and hence an <a href="Entire_function" title="Entire function">entire function</a>.

The π-normalized sinc function is the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the <a href="Rectangular_function" title="Rectangular function">rectangular function</a> with no scaling. It is used in the concept of <a href="Whittaker%E2%80%93Shannon_interpolation_formula" title="Whittaker–Shannon interpolation formula">reconstructing</a> a continuous bandlimited signal from uniformly spaced <a href="Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">samples</a> of that signal. The <a href="Sinc_filter" title="Sinc filter">sinc filter</a> is used in signal processing.

The function itself was first mathematically derived in this form by <a href="Lord_Rayleigh" class="mw-redirect" title="Lord Rayleigh">Lord Rayleigh</a> in his expression (<a href="Bessel_functions" class="mw-redirect" title="Bessel functions">Rayleigh's formula</a>) for the zeroth-order spherical <a href="Bessel_function" title="Bessel function">Bessel function</a> of the first kind.

<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

The sinc function has two forms, normalized and unnormalized.<a href="#cite_note-dlmf-1">[1]</a>

In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (x)={\frac {\sin x}{x}}.}</annotation>

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Alternatively, the unnormalized sinc function is often called the <a href="Sampling_function" class="mw-redirect" title="Sampling function">sampling function</a>, indicated as Sa(x).<a href="#cite_note-2">[2]</a>

In <a href="Digital_signal_processing" title="Digital signal processing">digital signal processing</a> and <a href="Information_theory" title="Information theory">information theory</a>, the normalized sinc function is commonly defined for x ≠ 0 by

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (x)={\frac {\sin(\pi x)}{\pi x}}.}</annotation>

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In either case, the value at x = 0 is defined to be the limiting value

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (0):=\lim _{x\to 0}{\frac {\sin(ax)}{ax}}=1}</annotation>

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</math> for all real a ≠ 0 (the limit can be proven using the <a href="Squeeze_theorem#Second_example" title="Squeeze theorem">squeeze theorem</a>).

The <a href="Normalizing_constant" title="Normalizing constant">normalization</a> causes the <a href="Integral" title="Integral">definite integral</a> of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of <a href="Pi" title="Pi">π</a>). As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of x.

<div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2></div>

The function has also been called the cardinal sine or sine cardinal function.<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a> The term "sinc" is a contraction of the function's full Latin name, the sinus cardinalis<a href="#cite_note-Poynton-5">[5]</a> and was introduced by <a href="Philip_Woodward" title="Philip Woodward">Philip M. Woodward</a> and I.L Davies in their 1952 article "Information theory and <a href="Inverse_probability" title="Inverse probability">inverse probability</a> 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".<a href="#cite_note-6">[6]</a> It is also used in Woodward's 1953 book Probability and Information Theory, with Applications to Radar.<a href="#cite_note-Poynton-5">[5]</a><a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

The <a href="Zero_crossing" title="Zero crossing">zero crossings</a> of the unnormalized sinc are at non-zero integer multiples of π, 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 <a href="Cosine" class="mw-redirect" title="Cosine">cosine</a> function. That is, <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠sin(ξ)/ξ⁠ = cos(ξ) for all points ξ where the derivative of ⁠sin(x)/x⁠ is zero and thus a local extremum is reached. This follows from the derivative of the sinc function:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\operatorname {sinc} (x)={\begin{cases}{\dfrac {\cos(x)-\operatorname {sinc} (x)}{x}},&x\neq 0\\0,&x=0\end{cases}}.}</annotation>

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The first few terms of the infinite series for the x coordinate of the n-th extremum with positive x coordinate are

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<annotation encoding="application/x-tex">{\displaystyle x_{n}=q-q^{-1}-{\frac {2}{3}}q^{-3}-{\frac {13}{15}}q^{-5}-{\frac {146}{105}}q^{-7}-\cdots ,}</annotation>

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where

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<annotation encoding="application/x-tex">{\displaystyle q=\left(n+{\frac {1}{2}}\right)\pi ,}</annotation>

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and where odd n lead to a local minimum, and even n to a local maximum. Because of symmetry around the y axis, there exist extrema with x coordinates −xn. In addition, there is an absolute maximum at ξ0 = (0, 1).

The normalized sinc function has a simple representation as the <a href="Infinite_product" title="Infinite product">infinite product</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\sin(\pi x)}{\pi x}}=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}}}\right)}</annotation>

</semantics>

</math>

and is related to the <a href="Gamma_function" title="Gamma function">gamma function</a> Γ(x) through <a href="Euler's_reflection_formula" class="mw-redirect" title="Euler's reflection formula">Euler's reflection formula</a>:

<mfrac>

<mrow>

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<mrow>

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</mfrac>

</mrow>

<mfrac>

<mrow>

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</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\sin(\pi x)}{\pi x}}={\frac {1}{\Gamma (1+x)\Gamma (1-x)}}.}</annotation>

</semantics>

</math>

<a href="Euler" class="mw-redirect" title="Euler">Euler</a> discovered<a href="#cite_note-8">[8]</a> that

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\sin(x)}{x}}=\prod _{n=1}^{\infty }\cos \left({\frac {x}{2^{n}}}\right),}</annotation>

</semantics>

</math>

and because of the product-to-sum identity<a href="#cite_note-9">[9]</a>

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</munderover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \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\geq 1,}</annotation>

</semantics>

</math>

Euler's product can be recast as a sum

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\sin(x)}{x}}=\lim _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}\cos \left({\frac {n-1/2}{N}}x\right).}</annotation>

</semantics>

</math>

The <a href="Continuous_Fourier_transform" class="mw-redirect" title="Continuous Fourier transform">continuous Fourier transform</a> of the normalized sinc (to ordinary frequency) is <a href="Rectangular_function" title="Rectangular function">rect</a>(f):

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</msubsup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\operatorname {sinc} (t)\,e^{-i2\pi ft}\,dt=\operatorname {rect} (f),}</annotation>

</semantics>

</math>

where the <a href="Rectangular_function" title="Rectangular function">rectangular function</a> is 1 for argument between −⁠1/2⁠ and ⁠1/2⁠, and zero otherwise. This corresponds to the fact that the <a href="Sinc_filter" title="Sinc filter">sinc filter</a> is the ideal (<a href="Brick-wall_filter" class="mw-redirect" title="Brick-wall filter">brick-wall</a>, meaning rectangular <a href="Frequency_response" title="Frequency response">frequency response</a>) <a href="Low-pass_filter" title="Low-pass filter">low-pass filter</a>.

This Fourier integral, including the special case

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<mfrac>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin(\pi x)}{\pi x}}\,dx=\operatorname {rect} (0)=1}</annotation>

</semantics>

</math>

is an <a href="Improper_integral" title="Improper integral">improper integral</a> (see <a href="Dirichlet_integral" title="Dirichlet integral">Dirichlet integral</a>) and not a convergent <a href="Lebesgue_integral" title="Lebesgue integral">Lebesgue integral</a>, as

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\left|{\frac {\sin(\pi x)}{\pi x}}\right|\,dx=+\infty .}</annotation>

</semantics>

</math>

The normalized sinc function has properties that make it ideal in relationship to <a href="Interpolation" title="Interpolation">interpolation</a> of <a href="Sampling_(signal_processing)" title="Sampling (signal processing)">sampled</a> <a href="Bandlimited" class="mw-redirect" title="Bandlimited">bandlimited</a> functions:

<ul><li>It is an interpolating function, i.e., sinc(0) = 1, and sinc(k) = 0 for nonzero <a href="Number#Integers" title="Number">integer</a> k.</li>

<li>The functions xk(t) = sinc(t − k) (k integer) form an <a href="Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> for <a href="Bandlimited" class="mw-redirect" title="Bandlimited">bandlimited</a> functions in the <a href="Lp_space" title="Lp space">function space</a> L2(R), with highest angular frequency ωH = π (that is, highest cycle frequency fH = ⁠1/2⁠).</li></ul>

Other properties of the two sinc functions include:

<ul><li>The unnormalized sinc is the zeroth-order spherical <a href="Bessel_function" title="Bessel function">Bessel function</a> of the first kind, j0(x). The normalized sinc is j0(πx).</li>

<li>where Si(x) is the <a href="Sine_integral" class="mw-redirect" title="Sine integral">sine integral</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{x}{\frac {\sin(\theta )}{\theta }}\,d\theta =\operatorname {Si} (x).}">

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{x}{\frac {\sin(\theta )}{\theta }}\,d\theta =\operatorname {Si} (x).}</annotation>

</semantics>

</math></li>

<li>λ sinc(λx) (not normalized) is one of two linearly independent solutions to the linear <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x{\frac {d^{2}y}{dx^{2}}}+2{\frac {dy}{dx}}+\lambda ^{2}xy=0.}">

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<annotation encoding="application/x-tex">{\displaystyle x{\frac {d^{2}y}{dx^{2}}}+2{\frac {dy}{dx}}+\lambda ^{2}xy=0.}</annotation>

</semantics>

</math> The other is ⁠cos(λx)/x⁠, which is not bounded at x = 0, unlike its sinc function counterpart.</li>

<li>Using normalized sinc, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{2}(\theta )}{\theta ^{2}}}\,d\theta =\pi \quad \Rightarrow \quad \int _{-\infty }^{\infty }\operatorname {sinc} ^{2}(x)\,dx=1,}">

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{2}(\theta )}{\theta ^{2}}}\,d\theta =\pi \quad \Rightarrow \quad \int _{-\infty }^{\infty }\operatorname {sinc} ^{2}(x)\,dx=1,}</annotation>

</semantics>

</math></li>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin(\theta )}{\theta }}\,d\theta =\int _{-\infty }^{\infty }\left({\frac {\sin(\theta )}{\theta }}\right)^{2}\,d\theta =\pi .}</annotation>

</semantics>

</math><img src="./1ce8b52f28fd9d0bbbb42890a8626513f28d5323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.888ex; height:6.676ex;" alt="{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin(\theta )}{\theta }}\,d\theta =\int _{-\infty }^{\infty }\left({\frac {\sin(\theta )}{\theta }}\right)^{2}\,d\theta =\pi .}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{3}(\theta )}{\theta ^{3}}}\,d\theta ={\frac {3\pi }{4}}.}</annotation>

</semantics>

</math><img src="./152b26f32bdfe8b2ca4e5b04e040f8d778311a36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.249ex; height:6.509ex;" alt="{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{3}(\theta )}{\theta ^{3}}}\,d\theta ={\frac {3\pi }{4}}.}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{4}(\theta )}{\theta ^{4}}}\,d\theta ={\frac {2\pi }{3}}.}</annotation>

</semantics>

</math><img src="./c4a8a572139a3530b8f8b0f99daa8dfb4e7ea74d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.249ex; height:6.509ex;" alt="{\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{4}(\theta )}{\theta ^{4}}}\,d\theta ={\frac {2\pi }{3}}.}" loading="lazy"></li>

<li>The following improper integral involves the (not normalized) sinc function: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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}})}}.}">

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<annotation encoding="application/x-tex">{\displaystyle \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}})}}.}</annotation>

</semantics>

</math></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Relationship_to_the_Dirac_delta_distribution">Relationship to the Dirac delta distribution</h2></div>

The normalized sinc function can be used as a <a href="Dirac_delta_function#Representations_of_the_delta_function" title="Dirac delta function">nascent delta function</a>, meaning that the following <a href="Weak_convergence_(Hilbert_space)" title="Weak convergence (Hilbert space)">weak limit</a> holds:

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<annotation encoding="application/x-tex">{\displaystyle \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).}</annotation>

</semantics>

</math>

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

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<annotation encoding="application/x-tex">{\displaystyle \lim _{a\to 0}\int _{-\infty }^{\infty }{\frac {1}{a}}\operatorname {sinc} \left({\frac {x}{a}}\right)\varphi (x)\,dx=\varphi (0)}</annotation>

</semantics>

</math>

for every <a href="Schwartz_space" title="Schwartz space">Schwartz function</a>, as can be seen from the <a href="Fourier_inversion_theorem" title="Fourier inversion theorem">Fourier inversion theorem</a>.

In the above expression, as a → 0, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±⁠1/πx⁠, regardless of the value of a.

This complicates the informal picture of δ(x) as being zero for all x except at the point 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 <a href="Gibbs_phenomenon" title="Gibbs phenomenon">Gibbs phenomenon</a>.

We can also make an immediate connection with the standard Dirac representation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \delta (x)}</annotation>

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</math><img src="./4457507451c205a7e6adda92d919ee4c4a369cea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.188ex; height:2.843ex;" alt="{\displaystyle \delta (x)}" loading="lazy"> by writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=1/a}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=1/a}</annotation>

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</math><img src="./d224eee681f771f7e5247878d0a486b2bf2b694b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.651ex; height:2.843ex;" alt="{\displaystyle b=1/a}" loading="lazy"> and

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<annotation encoding="application/x-tex">{\displaystyle \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^{ikx}dk={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ikx}dk=\delta (x),}</annotation>

</semantics>

</math>

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

<div class="mw-heading mw-heading2"><h2 id="Summation">Summation</h2></div>

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

The sum of sinc(n) over integer n from 1 to ∞ equals ⁠π − 1/2⁠:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }\operatorname {sinc} (n)=\operatorname {sinc} (1)+\operatorname {sinc} (2)+\operatorname {sinc} (3)+\operatorname {sinc} (4)+\cdots ={\frac {\pi -1}{2}}.}</annotation>

</semantics>

</math>

The sum of the squares also equals ⁠π − 1/2⁠:<a href="#cite_note-10">[10]</a><a href="#cite_note-BBB-11">[11]</a>

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<annotation encoding="application/x-tex">{\displaystyle \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}}.}</annotation>

</semantics>

</math>

When the signs of the <a href="Addend" class="mw-redirect" title="Addend">addends</a> alternate and begin with +, the sum equals ⁠1/2⁠:

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \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}}.}</annotation>

</semantics>

</math>

The alternating sums of the squares and cubes also equal ⁠1/2⁠:<a href="#cite_note-FWFS-12">[12]</a>

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<annotation encoding="application/x-tex">{\displaystyle \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}},}</annotation>

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</math>

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<annotation encoding="application/x-tex">{\displaystyle \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}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Series_expansion">Series expansion</h2></div>

The <a href="Taylor_series" title="Taylor series">Taylor series</a> of the unnormalized sinc function can be obtained from that of the sine (which also yields its value of 1 at x = 0):

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<annotation encoding="application/x-tex">{\displaystyle {\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 }</annotation>

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The series converges for all x. The normalized version follows easily:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\sin \pi x}{\pi x}}=1-{\frac {\pi ^{2}x^{2}}{3!}}+{\frac {\pi ^{4}x^{4}}{5!}}-{\frac {\pi ^{6}x^{6}}{7!}}+\cdots }</annotation>

</semantics>

</math>

<a href="Leonhard_Euler" title="Leonhard Euler">Euler</a> famously compared this series to the expansion of the infinite product form to solve the <a href="Basel_problem" title="Basel problem">Basel problem</a>.

<div class="mw-heading mw-heading2"><h2 id="Higher_dimensions">Higher dimensions</h2></div>

The product of 1-D sinc functions readily provides a <a href="Multivariable_calculus" title="Multivariable calculus">multivariate</a> sinc function for the square Cartesian grid (<a href="Lattice_graph" title="Lattice graph">lattice</a>): sincC(x, y) = sinc(x) sinc(y), whose <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> is the <a href="Indicator_function" title="Indicator function">indicator function</a> of a square in the frequency space (i.e., the brick wall defined in 2-D space). The sinc function for a non-Cartesian <a href="Lattice_(group)" title="Lattice (group)">lattice</a> (e.g., <a href="Hexagonal_lattice" title="Hexagonal lattice">hexagonal lattice</a>) is a function whose <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> is the <a href="Indicator_function" title="Indicator function">indicator function</a> of the <a href="Brillouin_zone" title="Brillouin zone">Brillouin zone</a> of that lattice. For example, the sinc function for the hexagonal lattice is a function whose <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> is the <a href="Indicator_function" title="Indicator function">indicator function</a> of the unit hexagon in the frequency space. For a non-Cartesian lattice this function can not be obtained by a simple <a href="Tensor_product" title="Tensor product">tensor product</a>. However, the explicit formula for the sinc function for the <a href="Hexagonal_lattice" title="Hexagonal lattice">hexagonal</a>, <a href="Body-centered_cubic" class="mw-redirect" title="Body-centered cubic">body-centered cubic</a>, <a href="Face-centered_cubic" class="mw-redirect" title="Face-centered cubic">face-centered cubic</a> and other higher-dimensional lattices can be explicitly derived<a href="#cite_note-multiD-13">[13]</a> using the geometric properties of Brillouin zones and their connection to <a href="Zonohedron" title="Zonohedron">zonotopes</a>.

For example, a <a href="Hexagonal_lattice" title="Hexagonal lattice">hexagonal lattice</a> can be generated by the (integer) <a href="Linear_span" title="Linear span">linear span</a> of the vectors

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<annotation encoding="application/x-tex">{\displaystyle \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}}.}</annotation>

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Denoting

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<annotation encoding="application/x-tex">{\displaystyle {\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}},}</annotation>

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</math>

one can derive<a href="#cite_note-multiD-13">[13]</a> the sinc function for this hexagonal lattice as

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation>

</semantics>

</math>

This construction can be used to design <a href="Lanczos_window" class="mw-redirect" title="Lanczos window">Lanczos window</a> for general multidimensional lattices.<a href="#cite_note-multiD-13">[13]</a>

<div class="mw-heading mw-heading2"><h2 id="Sinhc">Sinhc</h2></div>

Some authors, by analogy, define the hyperbolic sine <a href="Cardinal_function" title="Cardinal function">cardinal function</a>.<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a><a href="#cite_note-16">[16]</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {sinhc} (x)={\begin{cases}{\displaystyle {\frac {\sinh(x)}{x}},}&{\text{if }}x\neq 0\\{\displaystyle 1,}&{\text{if }}x=0\end{cases}}}</annotation>

</semantics>

</math><img src="./b48817ba5c10bae44d57bc9cb06754734932a24a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:32.858ex; height:8.509ex;" alt="{\displaystyle \mathrm {sinhc} (x)={\begin{cases}{\displaystyle {\frac {\sinh(x)}{x}},}&{\text{if }}x\neq 0\\{\displaystyle 1,}&{\text{if }}x=0\end{cases}}}" loading="lazy"></dd></dl>

<ul><li><a href="Anti-aliasing_filter" title="Anti-aliasing filter">Anti-aliasing filter</a> – Mathematical transformation reducing the damage caused by aliasing</li>

<li><a href="Borwein_integral" title="Borwein integral">Borwein integral</a> – Type of mathematical integrals</li>

<li><a href="Dirichlet_integral" title="Dirichlet integral">Dirichlet integral</a> – Integral of sin(x)/x from 0 to infinity</li>

<li><a href="Lanczos_resampling" title="Lanczos resampling">Lanczos resampling</a> – Technique in signal processing</li>

<li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of mathematical functions</a></li>

<li><a href="Shannon_wavelet" title="Shannon wavelet">Shannon wavelet</a></li>

<li><a href="Sinc_filter" title="Sinc filter">Sinc filter</a> – Ideal low-pass filter or averaging filter</li>

<li><a href="Sinc_numerical_methods" title="Sinc numerical methods">Sinc numerical methods</a></li>

<li><a href="Trigonometric_functions_of_matrices" title="Trigonometric functions of matrices">Trigonometric functions of matrices</a> – Important functions in solving differential equations</li>

<li><a href="Trigonometric_integral" title="Trigonometric integral">Trigonometric integral</a> – Special function defined by an integral</li>

<li><a href="Whittaker%E2%80%93Shannon_interpolation_formula" title="Whittaker–Shannon interpolation formula">Whittaker–Shannon interpolation formula</a> – Signal (re-)construction algorithm</li>

<li><a href="Winkel_tripel_projection" title="Winkel tripel projection">Winkel tripel projection</a> – Pseudoazimuthal compromise map projection (cartography)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFAinslie2010" class="citation book cs1">Ainslie, Michael (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EqDnP-lAw40C&pg=PA636">Principles of Sonar Performance Modelling</a>. Springer. p. 636. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783540876625</bdi>.</cite>

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<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFGünter2012" class="citation book cs1">Günter, Peter (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8QTpCAAAQBAJ&pg=PA258">Nonlinear Optical Effects and Materials</a>. Springer. p. 258. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783540497134</bdi>.</cite>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFSchächter2013" class="citation book cs1">Schächter, Levi (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jQb9CAAAQBAJ&pg=PA241">Beam-Wave Interaction in Periodic and Quasi-Periodic Structures</a>. Springer. p. 241. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783662033982</bdi>.</cite>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFStenger1993" class="citation book cs1">Stenger, Frank (1993). Numerical Methods Based on Sinc and Analytic Functions. Springer Series on Computational Mathematics. Vol. 20. Springer-Verlag New York, Inc. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-2706-9">10.1007/978-1-4612-2706-9</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781461276371</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SincFunction.html">"Sinc Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li></ul></div><div><div class="zim-footer">

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