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Sinc function

Table of contents: Definitions - Illustrations - Domain - Primary formula - Zeros - Specific values - Functional equations - Derivatives and differential equations - Series and product representations - Representation by special functions - Integral representations - Integrals - Summation - Extreme points and limits - Bounds and inequalities

Definitions

4cbfd4
Symbol: Sinc sinc(z)\operatorname{sinc}(z) Sinc function

Illustrations

639d7b
Image: Plot of sinc(x)\operatorname{sinc}(x) and sinc ⁣(πx)\operatorname{sinc}\!\left(\pi x\right) on x[3π,3π]x \in \left[-3 \pi, 3 \pi\right]
baf960
Image: X-ray of sinc(z)\operatorname{sinc}(z) on z[8,8]+[8,8]iz \in \left[-8, 8\right] + \left[-8, 8\right] i

Domain

a527c4
sinc(z) is holomorphic on zC\operatorname{sinc}(z) \text{ is holomorphic on } z \in \mathbb{C}
89be58
xR        sinc(x)Rx \in \mathbb{R} \;\implies\; \operatorname{sinc}(x) \in \mathbb{R}
2379e6
zC        sinc(z)Cz \in \mathbb{C} \;\implies\; \operatorname{sinc}(z) \in \mathbb{C}
41998e
xR        sinc(x)(0.217234,1]x \in \mathbb{R} \;\implies\; \operatorname{sinc}(x) \in \left(-0.217234, 1\right]

Primary formula

fa9283
sinc(z)=sin(z)z\operatorname{sinc}(z) = \frac{\sin(z)}{z}
b18020
sinc(0)=1\operatorname{sinc}(0) = 1
01422b
sinc(z)={sin(z)z,z01,z=0\operatorname{sinc}(z) = \begin{cases} \frac{\sin(z)}{z}, & z \ne 0\\1, & z = 0\\ \end{cases}

Zeros

af4516
zeroszCsinc(z)={πn:nZandn0}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{sinc}(z) = \left\{ \pi n : n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0 \right\}
1349b5
zeroszCsinc ⁣(πz)={n:nZandn0}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{sinc}\!\left(\pi z\right) = \left\{ n : n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0 \right\}

Specific values

593e63
sinc ⁣(πn)={1,n=00,n0\operatorname{sinc}\!\left(\pi n\right) = \begin{cases} 1, & n = 0\\0, & n \ne 0\\ \end{cases}
fdc94c
sinc ⁣(π2)=2π\operatorname{sinc}\!\left(\frac{\pi}{2}\right) = \frac{2}{\pi}
340936
sinc ⁣(π3)=332π\operatorname{sinc}\!\left(\frac{\pi}{3}\right) = \frac{3 \sqrt{3}}{2 \pi}
c9ead2
sinc ⁣(π4)=22π\operatorname{sinc}\!\left(\frac{\pi}{4}\right) = \frac{2 \sqrt{2}}{\pi}
45740a
sinc ⁣(π6)=3π\operatorname{sinc}\!\left(\frac{\pi}{6}\right) = \frac{3}{\pi}

Functional equations

Even symmetry

f19e0a
sinc ⁣(z)=sinc(z)\operatorname{sinc}\!\left(-z\right) = \operatorname{sinc}(z)

Conjugate symmetry

3a428f
sinc ⁣(z)=sinc(z)\operatorname{sinc}\!\left(\overline{z}\right) = \overline{\operatorname{sinc}(z)}

Multiplication formulas

b41d08
sinc ⁣(iz)=sinh(z)z\operatorname{sinc}\!\left(i z\right) = \frac{\sinh(z)}{z}
d5000a
sinc ⁣(2z)=sinc(z)cos(z)\operatorname{sinc}\!\left(2 z\right) = \operatorname{sinc}(z) \cos(z)

Derivatives and differential equations

First derivatives

768c77
sinc(z)={cos(z)zsin(z)z2,z00,z=0\operatorname{sinc}'(z) = \begin{cases} \frac{\cos(z)}{z} - \frac{\sin(z)}{{z}^{2}}, & z \ne 0\\0, & z = 0\\ \end{cases}
90c66a
sinc(z)={(2z31z)sin(z)2cos(z)z2,z013,z=0\operatorname{sinc}''(z) = \begin{cases} \left(\frac{2}{{z}^{3}} - \frac{1}{z}\right) \sin(z) - \frac{2 \cos(z)}{{z}^{2}}, & z \ne 0\\-\frac{1}{3}, & z = 0\\ \end{cases}

Linear ordinary differential equations

c6e6b2
zsinc(z)+2sinc(z)+zsinc(z)=0z \operatorname{sinc}''(z) + 2 \operatorname{sinc}'(z) + z \operatorname{sinc}(z) = 0
aa15f0
zf(z)+2f(z)+A2zf(z)=0   where f(z)=C1sinc ⁣(Az)+C2cos ⁣(Az)zz f''(z) + 2 f'(z) + {A}^{2} z f(z) = 0\; \text{ where } f(z) = {C}_{1} \operatorname{sinc}\!\left(A z\right) + {C}_{2} \frac{\cos\!\left(A z\right)}{z}

Higher derivatives

1c3766
sinc(n)(0)={(1)n/21n+1,n even0,n odd{\operatorname{sinc}}^{(n)}(0) = \begin{cases} {\left(-1\right)}^{\left\lfloor n / 2 \right\rfloor} \frac{1}{n + 1}, & n \text{ even}\\0, & n \text{ odd}\\ \end{cases}
ff5e82
z(n2+5n+6)an+3+(n2+5n+6)an+2+zan+1+an=0   where an=sinc(n)(z)n!z \left({n}^{2} + 5 n + 6\right) {a}_{n + 3} + \left({n}^{2} + 5 n + 6\right) {a}_{n + 2} + z {a}_{n + 1} + {a}_{n} = 0\; \text{ where } {a}_{n} = \frac{{\operatorname{sinc}}^{(n)}(z)}{n !}

Series and product representations

4f9844
sinc(z)=n=0(1)nz2n(2n+1)!\operatorname{sinc}(z) = \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n} {z}^{2 n}}{\left(2 n + 1\right)!}
f64eef
sinc ⁣(πz)=n=1(1z2n2)\operatorname{sinc}\!\left(\pi z\right) = \prod_{n=1}^{\infty} \left(1 - \frac{{z}^{2}}{{n}^{2}}\right)
24c17b
sinc(z)=n=1cos ⁣(z2n)\operatorname{sinc}(z) = \prod_{n=1}^{\infty} \cos\!\left(\frac{z}{{2}^{n}}\right)

Representation by special functions

d16cb4
sinc ⁣(πz)=1Γ ⁣(1+z)Γ ⁣(1z)\operatorname{sinc}\!\left(\pi z\right) = \frac{1}{\Gamma\!\left(1 + z\right) \Gamma\!\left(1 - z\right)}
e2878f
sinc(z)=0F1 ⁣(32,z24)\operatorname{sinc}(z) = \,{}_0F_1\!\left(\frac{3}{2}, -\frac{{z}^{2}}{4}\right)
50f72f
sinc(z)=z30F1 ⁣(52,z24)\operatorname{sinc}'(z) = -\frac{z}{3} \,{}_0F_1\!\left(\frac{5}{2}, -\frac{{z}^{2}}{4}\right)
19d7d9
sinc(z)=(2zπ)1/2J1/2 ⁣(z)\operatorname{sinc}(z) = {\left(\frac{2 z}{\pi}\right)}^{-1 / 2} J_{1 / 2}\!\left(z\right)

Integral representations

6e4f58
sinc(z)=01cos ⁣(zx)dx\operatorname{sinc}(z) = \int_{0}^{1} \cos\!\left(z x\right) \, dx
e2c10d
sinc ⁣(az)=1a0acos ⁣(zx)dx\operatorname{sinc}\!\left(a z\right) = \frac{1}{a} \int_{0}^{a} \cos\!\left(z x\right) \, dx
729c78
sinc ⁣(πz)=01cos ⁣(πzx)dx\operatorname{sinc}\!\left(\pi z\right) = \int_{0}^{1} \cos\!\left(\pi z x\right) \, dx
08583a
sinc(z)=1211eizxdx\operatorname{sinc}(z) = \frac{1}{2} \int_{-1}^{1} {e}^{i z x} \, dx
b1d132
sinc ⁣(az)=12aaaeizxdx\operatorname{sinc}\!\left(a z\right) = \frac{1}{2 a} \int_{-a}^{a} {e}^{i z x} \, dx
99ad29
sinc ⁣(πz)=1/21/2e2πizxdx\operatorname{sinc}\!\left(\pi z\right) = \int_{-1 / 2}^{1 / 2} {e}^{2 \pi i z x} \, dx
45f05f
1sinc ⁣(πz)=01xz+1dx\frac{1}{\operatorname{sinc}\!\left(\frac{\pi}{z}\right)} = \int_{0}^{\infty} \frac{1}{{x}^{z} + 1} \, dx

Integrals

Sine integral

122e3d
0zsinc(x)dx=Si(z)\int_{0}^{z} \operatorname{sinc}(x) \, dx = \operatorname{Si}(z)
d6c29e
absinc(x)dx=Si(b)Si(a)\int_{a}^{b} \operatorname{sinc}(x) \, dx = \operatorname{Si}(b) - \operatorname{Si}(a)
8ef3d7
zsinc(x)dx=Si(z)+π2\int_{-\infty}^{z} \operatorname{sinc}(x) \, dx = \operatorname{Si}(z) + \frac{\pi}{2}
2b7b1d
zsinc(x)dx=π2Si(z)\int_{z}^{\infty} \operatorname{sinc}(x) \, dx = \frac{\pi}{2} - \operatorname{Si}(z)
f0f0a6
sinc(x)dx=π\int_{-\infty}^{\infty} \operatorname{sinc}(x) \, dx = \pi

Gibbs constant

81f531
0πsinc(x)dx=1.85193705198246617036105337016  (nearest 30 digits)\int_{0}^{\pi} \operatorname{sinc}(x) \, dx = 1.85193705198246617036105337016 \;\, {\scriptstyle (\text{nearest } 30 \text{ digits})}

Integrals on the real line

a0b0b3
0sinc(x)dx=+\int_{0}^{\infty} \left|\operatorname{sinc}(x)\right| \, dx = +\infty
cb152f
0sinc(x)dx=π2\int_{0}^{\infty} \operatorname{sinc}(x) \, dx = \frac{\pi}{2}
1a7e22
0sinc2 ⁣(x)dx=π2\int_{0}^{\infty} \operatorname{sinc}^{2}\!\left(x\right) \, dx = \frac{\pi}{2}
be0f54
0sinc3 ⁣(x)dx=3π8\int_{0}^{\infty} \operatorname{sinc}^{3}\!\left(x\right) \, dx = \frac{3 \pi}{8}
1596d2
0sincn ⁣(x)dx=π2n(n1)!k=0n/2(1)k(nk)(n2k)n1\int_{0}^{\infty} \operatorname{sinc}^{n}\!\left(x\right) \, dx = \frac{\pi}{{2}^{n} \left(n - 1\right)!} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose k} {\left(n - 2 k\right)}^{n - 1}
af8328
0k=0nsinc ⁣(x2k+1)dx={π2,n{0,1,,6}467807924713440738696537864469467807924720320453655260875000π2,n=7\int_{0}^{\infty} \prod_{k=0}^{n} \operatorname{sinc}\!\left(\frac{x}{2 k + 1}\right) \, dx = \begin{cases} \frac{\pi}{2}, & n \in \{0, 1, \ldots, 6\}\\\frac{467807924713440738696537864469}{467807924720320453655260875000} \frac{\pi}{2}, & n = 7\\ \end{cases}
3fe2b0
sinc ⁣(x+a)sinc ⁣(x+b)dx=πsinc ⁣(ab)\int_{-\infty}^{\infty} \operatorname{sinc}\!\left(x + a\right) \operatorname{sinc}\!\left(x + b\right) \, dx = \pi \operatorname{sinc}\!\left(a - b\right)
108daa
sinc ⁣(ax)sinc ⁣(bx)dx=π2a+babab\int_{-\infty}^{\infty} \operatorname{sinc}\!\left(a x\right) \operatorname{sinc}\!\left(b x\right) \, dx = \frac{\pi}{2} \frac{\left|a + b\right| - \left|a - b\right|}{a b}
f5887b
sinc ⁣(x+πn)sinc ⁣(x+πm)dx={π,n=m0,nm\int_{-\infty}^{\infty} \operatorname{sinc}\!\left(x + \pi n\right) \operatorname{sinc}\!\left(x + \pi m\right) \, dx = \begin{cases} \pi, & n = m\\0, & n \ne m\\ \end{cases}

Integral transforms

2a69ce
eiaxsinc(x)dx=cos ⁣(ax)sinc(x)dx={π,a<1π2,a=10,a>1\int_{-\infty}^{\infty} {e}^{i a x} \operatorname{sinc}(x) \, dx = \int_{-\infty}^{\infty} \cos\!\left(a x\right) \operatorname{sinc}(x) \, dx = \begin{cases} \pi, & \left|a\right| < 1\\\frac{\pi}{2}, & \left|a\right| = 1\\0, & \left|a\right| > 1\\ \end{cases}
38dc04
0eaxsinc(x)dx=acot(a)\int_{0}^{\infty} {e}^{-a x} \operatorname{sinc}(x) \, dx = \operatorname{acot}(a)
78fca3
0eax2sinc(x)dx=π2erf ⁣(12a)\int_{0}^{\infty} {e}^{-a {x}^{2}} \operatorname{sinc}(x) \, dx = \frac{\pi}{2} \operatorname{erf}\!\left(\frac{1}{2 \sqrt{a}}\right)

Other definite integrals

c2976e
0π/21sinc(x)dx=2G\int_{0}^{\pi / 2} \frac{1}{\operatorname{sinc}(x)} \, dx = 2 G
4a5b9a
0π/2xsinc(x)dx=2πG7ζ(3)2\int_{0}^{\pi / 2} \frac{x}{\operatorname{sinc}(x)} \, dx = 2 \pi G - \frac{7 \zeta(3)}{2}
dad27b
0π/21sinc2 ⁣(x)dx=πlog(2)\int_{0}^{\pi / 2} \frac{1}{\operatorname{sinc}^{2}\!\left(x\right)} \, dx = \pi \log(2)
5c9675
0π/41sinc2 ⁣(x)dx=πlog(2)4+Gπ216\int_{0}^{\pi / 4} \frac{1}{\operatorname{sinc}^{2}\!\left(x\right)} \, dx = \frac{\pi \log(2)}{4} + G - \frac{{\pi}^{2}}{16}

Summation

Infinite series

2a8ec9
({k=0Nak<2π,N=0k=0Nak2π,N1)    (n=k=0Nsinc ⁣(akn)=k=0Nsinc ⁣(akx)dx)\left(\begin{cases} \sum_{k=0}^{N} {a}_{k} < 2 \pi, & N = 0\\\sum_{k=0}^{N} {a}_{k} \le 2 \pi, & N \ge 1\\ \end{cases}\right) \implies \left(\sum_{n=-\infty}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} n\right) = \int_{-\infty}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} x\right) \, dx\right)
005478
({k=0Nak<2π,N=0k=0Nak2π,N1)    (n=1k=0Nsinc ⁣(akn)=12+0k=0Nsinc ⁣(akx)dx)\left(\begin{cases} \sum_{k=0}^{N} {a}_{k} < 2 \pi, & N = 0\\\sum_{k=0}^{N} {a}_{k} \le 2 \pi, & N \ge 1\\ \end{cases}\right) \implies \left(\sum_{n=1}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} n\right) = -\frac{1}{2} + \int_{0}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} x\right) \, dx\right)
4d5410
n=sinc(n)=π\sum_{n=-\infty}^{\infty} \operatorname{sinc}(n) = \pi
1f9beb
n=sinc2 ⁣(n)=π\sum_{n=-\infty}^{\infty} \operatorname{sinc}^{2}\!\left(n\right) = \pi
8814ad
n=sinc3 ⁣(n)=3π4\sum_{n=-\infty}^{\infty} \operatorname{sinc}^{3}\!\left(n\right) = \frac{3 \pi}{4}
49514d
n=sinc4 ⁣(n)=2π3\sum_{n=-\infty}^{\infty} \operatorname{sinc}^{4}\!\left(n\right) = \frac{2 \pi}{3}
0c847f
n=sinc5 ⁣(n)=115π192\sum_{n=-\infty}^{\infty} \operatorname{sinc}^{5}\!\left(n\right) = \frac{115 \pi}{192}
b894a3
n=sinc6 ⁣(n)=11π20\sum_{n=-\infty}^{\infty} \operatorname{sinc}^{6}\!\left(n\right) = \frac{11 \pi}{20}
4a1b00
n=sinc7 ⁣(n)=129423π201684π2+144060π354880π4+11760π51344π6+64π723040\sum_{n=-\infty}^{\infty} \operatorname{sinc}^{7}\!\left(n\right) = \frac{129423 \pi - 201684 {\pi}^{2} + 144060 {\pi}^{3} - 54880 {\pi}^{4} + 11760 {\pi}^{5} - 1344 {\pi}^{6} + 64 {\pi}^{7}}{23040}

Extreme points and limits

Extreme points

632d1c
maxxRsinc(x)=1\mathop{\max}\limits_{x \in \mathbb{R}} \operatorname{sinc}(x) = 1
b1a260
arg max*xRsinc(x)=0\mathop{\operatorname{arg\,max*}}\limits_{x \in \mathbb{R}} \operatorname{sinc}(x) = 0
1e6344
arg minxRsinc(x)={a,a}   where a=j3/2,1\mathop{\operatorname{arg\,min}}\limits_{x \in \mathbb{R}} \operatorname{sinc}(x) = \left\{-a, a\right\}\; \text{ where } a = j_{3 / 2,1}
da7fb1
minxRsinc(x)=sinc ⁣(j3/2,1)\mathop{\min}\limits_{x \in \mathbb{R}} \operatorname{sinc}(x) = \operatorname{sinc}\!\left(j_{3 / 2,1}\right)
95c04c
arg min*x(0,]sinc(x)=4.49340945790906417530788092728  (nearest 30 digits)\mathop{\operatorname{arg\,min*}}\limits_{x \in \left(0, \infty\right]} \operatorname{sinc}(x) = 4.49340945790906417530788092728 \;\, {\scriptstyle (\text{nearest } 30 \text{ digits})}
2ac5eb
minxRsinc(x)=0.217233628211221657408279325562  (nearest 30 digits)\mathop{\min}\limits_{x \in \mathbb{R}} \operatorname{sinc}(x) = -0.217233628211221657408279325562 \;\, {\scriptstyle (\text{nearest } 30 \text{ digits})}

Limits at infinity

5e0c58
sinc()=limxsinc(x)=0\operatorname{sinc}(\infty) = \lim_{x \to \infty} \operatorname{sinc}(x) = 0
a2f5a9
sinc ⁣()=limxsinc(x)=0\operatorname{sinc}\!\left(-\infty\right) = \lim_{x \to -\infty} \operatorname{sinc}(x) = 0
2f09ad
sinc()=limxsinc ⁣(ai+x)=0\operatorname{sinc}(\infty) = \lim_{x \to \infty} \operatorname{sinc}\!\left(a i + x\right) = 0
6c28fa
sinc ⁣()=limxsinc ⁣(ai+x)=0\operatorname{sinc}\!\left(-\infty\right) = \lim_{x \to -\infty} \operatorname{sinc}\!\left(a i + x\right) = 0
aa404c
sinc ⁣(i)=limxsinc ⁣(ix)=\operatorname{sinc}\!\left(i \infty\right) = \lim_{x \to \infty} \operatorname{sinc}\!\left(i x\right) = \infty
088fdb
sinc ⁣(i)=limxsinc ⁣(ix)=\operatorname{sinc}\!\left(-i \infty\right) = \lim_{x \to -\infty} \operatorname{sinc}\!\left(i x\right) = \infty
f4fd7d
sinc ⁣(eiθ)=limxsinc ⁣(eiθx)={0,eiθ{1,1},otherwise\left|\operatorname{sinc}\!\left({e}^{i \theta} \infty\right)\right| = \lim_{x \to \infty} \left|\operatorname{sinc}\!\left({e}^{i \theta} x\right)\right| = \begin{cases} 0, & {e}^{i \theta} \in \left\{-1, 1\right\}\\\infty, & \text{otherwise}\\ \end{cases}

Bounds and inequalities

Real variable

20f069
sinc(x)1\left|\operatorname{sinc}(x)\right| \le 1
4d3f04
sinc(x)>0.217234\operatorname{sinc}(x) > -0.217234
f0325d
sinc(x)1x\left|\operatorname{sinc}(x)\right| \le \frac{1}{\left|x\right|}
d8d286
sinc(x)1+x1+x2\left|\operatorname{sinc}(x)\right| \le \frac{1 + \left|x\right|}{1 + {x}^{2}}
a934d1
sinc(x)<asinh(x)x\left|\operatorname{sinc}(x)\right| < \frac{\operatorname{asinh}(x)}{x}
351d87
sinc(x)<2xtanh ⁣(x2)\left|\operatorname{sinc}(x)\right| < \frac{\sqrt{2}}{x} \tanh\!\left(\frac{x}{\sqrt{2}}\right)
5d16e5
sinc(n)(x)1\left|{\operatorname{sinc}}^{(n)}(x)\right| \le 1

Complex variable

c7c483
sinc(z)exp ⁣(Im(z))\left|\operatorname{sinc}(z)\right| \le \exp\!\left(\left|\operatorname{Im}(z)\right|\right)
51f9b4
sinc(z)sinc ⁣(iz)\left|\operatorname{sinc}(z)\right| \le \operatorname{sinc}\!\left(i \left|z\right|\right)

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-11-11 15:50:15.016492 UTC