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# Sinc function

## Definitions

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

## Illustrations

Image: Plot of $\operatorname{sinc}(x)$ and $\operatorname{sinc}\!\left(\pi x\right)$ on $x \in \left[-3 \pi, 3 \pi\right]$
Image: X-ray of $\operatorname{sinc}(z)$ on $z \in \left[-8, 8\right] + \left[-8, 8\right] i$

## Domain

$\operatorname{sinc}(z) \text{ is holomorphic on } z \in \mathbb{C}$
$x \in \mathbb{R} \;\implies\; \operatorname{sinc}(x) \in \mathbb{R}$
$z \in \mathbb{C} \;\implies\; \operatorname{sinc}(z) \in \mathbb{C}$
$x \in \mathbb{R} \;\implies\; \operatorname{sinc}(x) \in \left(-0.217234, 1\right]$

## Primary formula

$\operatorname{sinc}(z) = \frac{\sin(z)}{z}$
$\operatorname{sinc}(0) = 1$
$\operatorname{sinc}(z) = \begin{cases} \frac{\sin(z)}{z}, & z \ne 0\\1, & z = 0\\ \end{cases}$

## Zeros

$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{sinc}(z) = \left\{ \pi n : n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0 \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{sinc}\!\left(\pi z\right) = \left\{ n : n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0 \right\}$

## Specific values

$\operatorname{sinc}\!\left(\pi n\right) = \begin{cases} 1, & n = 0\\0, & n \ne 0\\ \end{cases}$
$\operatorname{sinc}\!\left(\frac{\pi}{2}\right) = \frac{2}{\pi}$
$\operatorname{sinc}\!\left(\frac{\pi}{3}\right) = \frac{3 \sqrt{3}}{2 \pi}$
$\operatorname{sinc}\!\left(\frac{\pi}{4}\right) = \frac{2 \sqrt{2}}{\pi}$
$\operatorname{sinc}\!\left(\frac{\pi}{6}\right) = \frac{3}{\pi}$

## Functional equations

### Even symmetry

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

### Conjugate symmetry

$\operatorname{sinc}\!\left(\overline{z}\right) = \overline{\operatorname{sinc}(z)}$

### Multiplication formulas

$\operatorname{sinc}\!\left(i z\right) = \frac{\sinh(z)}{z}$
$\operatorname{sinc}\!\left(2 z\right) = \operatorname{sinc}(z) \cos(z)$

## Derivatives and differential equations

### First derivatives

$\operatorname{sinc}'(z) = \begin{cases} \frac{\cos(z)}{z} - \frac{\sin(z)}{{z}^{2}}, & z \ne 0\\0, & z = 0\\ \end{cases}$
$\operatorname{sinc}''(z) = \begin{cases} \left(\frac{2}{{z}^{3}} - \frac{1}{z}\right) \sin(z) - \frac{2 \cos(z)}{{z}^{2}}, & z \ne 0\\-\frac{1}{3}, & z = 0\\ \end{cases}$

### Linear ordinary differential equations

$z \operatorname{sinc}''(z) + 2 \operatorname{sinc}'(z) + z \operatorname{sinc}(z) = 0$
$z f''(z) + 2 f'(z) + {A}^{2} z f(z) = 0\; \text{ where } f(z) = {C}_{1} \operatorname{sinc}\!\left(A z\right) + {C}_{2} \frac{\cos\!\left(A z\right)}{z}$

### Higher derivatives

${\operatorname{sinc}}^{(n)}(0) = \begin{cases} {\left(-1\right)}^{\left\lfloor n / 2 \right\rfloor} \frac{1}{n + 1}, & n \text{ even}\\0, & n \text{ odd}\\ \end{cases}$
$z \left({n}^{2} + 5 n + 6\right) a_{n + 3} + \left({n}^{2} + 5 n + 6\right) a_{n + 2} + z a_{n + 1} + a_{n} = 0\; \text{ where } a_{n} = \frac{{\operatorname{sinc}}^{(n)}(z)}{n !}$

## Series and product representations

$\operatorname{sinc}(z) = \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n} {z}^{2 n}}{\left(2 n + 1\right)!}$
$\operatorname{sinc}\!\left(\pi z\right) = \prod_{n=1}^{\infty} \left(1 - \frac{{z}^{2}}{{n}^{2}}\right)$
$\operatorname{sinc}(z) = \prod_{n=1}^{\infty} \cos\!\left(\frac{z}{{2}^{n}}\right)$

## Representation by special functions

$\operatorname{sinc}\!\left(\pi z\right) = \frac{1}{\Gamma\!\left(1 + z\right) \Gamma\!\left(1 - z\right)}$
$\operatorname{sinc}(z) = \,{}_0F_1\!\left(\frac{3}{2}, -\frac{{z}^{2}}{4}\right)$
$\operatorname{sinc}'(z) = -\frac{z}{3} \,{}_0F_1\!\left(\frac{5}{2}, -\frac{{z}^{2}}{4}\right)$
$\operatorname{sinc}(z) = {\left(\frac{2 z}{\pi}\right)}^{-1 / 2} J_{1 / 2}\!\left(z\right)$

## Integral representations

$\operatorname{sinc}(z) = \int_{0}^{1} \cos\!\left(z x\right) \, dx$
$\operatorname{sinc}\!\left(a z\right) = \frac{1}{a} \int_{0}^{a} \cos\!\left(z x\right) \, dx$
$\operatorname{sinc}\!\left(\pi z\right) = \int_{0}^{1} \cos\!\left(\pi z x\right) \, dx$
$\operatorname{sinc}(z) = \frac{1}{2} \int_{-1}^{1} {e}^{i z x} \, dx$
$\operatorname{sinc}\!\left(a z\right) = \frac{1}{2 a} \int_{-a}^{a} {e}^{i z x} \, dx$
$\operatorname{sinc}\!\left(\pi z\right) = \int_{-1 / 2}^{1 / 2} {e}^{2 \pi i z x} \, dx$
$\frac{1}{\operatorname{sinc}\!\left(\frac{\pi}{z}\right)} = \int_{0}^{\infty} \frac{1}{{x}^{z} + 1} \, dx$

## Integrals

### Sine integral

$\int_{0}^{z} \operatorname{sinc}(x) \, dx = \operatorname{Si}(z)$
$\int_{a}^{b} \operatorname{sinc}(x) \, dx = \operatorname{Si}(b) - \operatorname{Si}(a)$
$\int_{-\infty}^{z} \operatorname{sinc}(x) \, dx = \operatorname{Si}(z) + \frac{\pi}{2}$
$\int_{z}^{\infty} \operatorname{sinc}(x) \, dx = \frac{\pi}{2} - \operatorname{Si}(z)$
$\int_{-\infty}^{\infty} \operatorname{sinc}(x) \, dx = \pi$

### Gibbs constant

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

### Integrals on the real line

$\int_{0}^{\infty} \left|\operatorname{sinc}(x)\right| \, dx = +\infty$
$\int_{0}^{\infty} \operatorname{sinc}(x) \, dx = \frac{\pi}{2}$
$\int_{0}^{\infty} \operatorname{sinc}^{2}\!\left(x\right) \, dx = \frac{\pi}{2}$
$\int_{0}^{\infty} \operatorname{sinc}^{3}\!\left(x\right) \, dx = \frac{3 \pi}{8}$
$\int_{0}^{\infty} \operatorname{sinc}^{n}\!\left(x\right) \, dx = \frac{\pi}{{2}^{n} \left(n - 1\right)!} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose k} {\left(n - 2 k\right)}^{n - 1}$
$\int_{0}^{\infty} \prod_{k=0}^{n} \operatorname{sinc}\!\left(\frac{x}{2 k + 1}\right) \, dx = \begin{cases} \frac{\pi}{2}, & n \in \{0, 1, \ldots, 6\}\\\frac{467807924713440738696537864469}{467807924720320453655260875000} \frac{\pi}{2}, & n = 7\\ \end{cases}$
$\int_{-\infty}^{\infty} \operatorname{sinc}\!\left(x + a\right) \operatorname{sinc}\!\left(x + b\right) \, dx = \pi \operatorname{sinc}\!\left(a - b\right)$
$\int_{-\infty}^{\infty} \operatorname{sinc}\!\left(a x\right) \operatorname{sinc}\!\left(b x\right) \, dx = \frac{\pi}{2} \frac{\left|a + b\right| - \left|a - b\right|}{a b}$
$\int_{-\infty}^{\infty} \operatorname{sinc}\!\left(x + \pi n\right) \operatorname{sinc}\!\left(x + \pi m\right) \, dx = \begin{cases} \pi, & n = m\\0, & n \ne m\\ \end{cases}$

### Integral transforms

$\int_{-\infty}^{\infty} {e}^{i a x} \operatorname{sinc}(x) \, dx = \int_{-\infty}^{\infty} \cos\!\left(a x\right) \operatorname{sinc}(x) \, dx = \begin{cases} \pi, & \left|a\right| < 1\\\frac{\pi}{2}, & \left|a\right| = 1\\0, & \left|a\right| > 1\\ \end{cases}$
$\int_{0}^{\infty} {e}^{-a x} \operatorname{sinc}(x) \, dx = \operatorname{acot}(a)$
$\int_{0}^{\infty} {e}^{-a {x}^{2}} \operatorname{sinc}(x) \, dx = \frac{\pi}{2} \operatorname{erf}\!\left(\frac{1}{2 \sqrt{a}}\right)$

### Other definite integrals

$\int_{0}^{\pi / 2} \frac{1}{\operatorname{sinc}(x)} \, dx = 2 G$
$\int_{0}^{\pi / 2} \frac{x}{\operatorname{sinc}(x)} \, dx = 2 \pi G - \frac{7 \zeta\!\left(3\right)}{2}$
$\int_{0}^{\pi / 2} \frac{1}{\operatorname{sinc}^{2}\!\left(x\right)} \, dx = \pi \log(2)$
$\int_{0}^{\pi / 4} \frac{1}{\operatorname{sinc}^{2}\!\left(x\right)} \, dx = \frac{\pi \log(2)}{4} + G - \frac{{\pi}^{2}}{16}$

## Summation

### Infinite series

$\left(\begin{cases} \sum_{k=0}^{N} {a}_{k} < 2 \pi, & N = 0\\\sum_{k=0}^{N} {a}_{k} \le 2 \pi, & N \ge 1\\ \end{cases}\right) \;\implies\; \left(\sum_{n=-\infty}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} n\right) = \int_{-\infty}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} x\right) \, dx\right)$
$\left(\begin{cases} \sum_{k=0}^{N} {a}_{k} < 2 \pi, & N = 0\\\sum_{k=0}^{N} {a}_{k} \le 2 \pi, & N \ge 1\\ \end{cases}\right) \;\implies\; \left(\sum_{n=1}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} n\right) = -\frac{1}{2} + \int_{0}^{\infty} \prod_{k=0}^{N} \operatorname{sinc}\!\left({a}_{k} x\right) \, dx\right)$
$\sum_{n=-\infty}^{\infty} \operatorname{sinc}(n) = \pi$
$\sum_{n=-\infty}^{\infty} \operatorname{sinc}^{2}\!\left(n\right) = \pi$
$\sum_{n=-\infty}^{\infty} \operatorname{sinc}^{3}\!\left(n\right) = \frac{3 \pi}{4}$
$\sum_{n=-\infty}^{\infty} \operatorname{sinc}^{4}\!\left(n\right) = \frac{2 \pi}{3}$
$\sum_{n=-\infty}^{\infty} \operatorname{sinc}^{5}\!\left(n\right) = \frac{115 \pi}{192}$
$\sum_{n=-\infty}^{\infty} \operatorname{sinc}^{6}\!\left(n\right) = \frac{11 \pi}{20}$
$\sum_{n=-\infty}^{\infty} \operatorname{sinc}^{7}\!\left(n\right) = \frac{129423 \pi - 201684 {\pi}^{2} + 144060 {\pi}^{3} - 54880 {\pi}^{4} + 11760 {\pi}^{5} - 1344 {\pi}^{6} + 64 {\pi}^{7}}{23040}$

## Extreme points and limits

### Extreme points

$\mathop{\max}\limits_{x \in \mathbb{R}} \operatorname{sinc}(x) = 1$
$\mathop{\operatorname{arg\,max*}}\limits_{x \in \mathbb{R}} \operatorname{sinc}(x) = 0$
$\mathop{\operatorname{arg\,min}}\limits_{x \in \mathbb{R}} \operatorname{sinc}(x) = \left\{-a, a\right\}\; \text{ where } a = j_{\frac{3}{2}, 1}$
$\mathop{\min}\limits_{x \in \mathbb{R}} \operatorname{sinc}(x) = \operatorname{sinc}\!\left(j_{\frac{3}{2}, 1}\right)$
$\mathop{\operatorname{arg\,min*}}\limits_{x \in \left(0, \infty\right]} \operatorname{sinc}(x) = 4.49340945790906417530788092728 \;\, {\scriptstyle (\text{nearest } 30 \text{ digits})}$
$\mathop{\min}\limits_{x \in \mathbb{R}} \operatorname{sinc}(x) = -0.217233628211221657408279325562 \;\, {\scriptstyle (\text{nearest } 30 \text{ digits})}$

### Limits at infinity

$\operatorname{sinc}(\infty) = \lim_{x \to \infty} \operatorname{sinc}(x) = 0$
$\operatorname{sinc}\!\left(-\infty\right) = \lim_{x \to -\infty} \operatorname{sinc}(x) = 0$
$\operatorname{sinc}(\infty) = \lim_{x \to \infty} \operatorname{sinc}\!\left(a i + x\right) = 0$
$\operatorname{sinc}\!\left(-\infty\right) = \lim_{x \to -\infty} \operatorname{sinc}\!\left(a i + x\right) = 0$
$\operatorname{sinc}\!\left(i \infty\right) = \lim_{x \to \infty} \operatorname{sinc}\!\left(i x\right) = \infty$
$\operatorname{sinc}\!\left(-i \infty\right) = \lim_{x \to -\infty} \operatorname{sinc}\!\left(i x\right) = \infty$
$\left|\operatorname{sinc}\!\left({e}^{i \theta} \infty\right)\right| = \lim_{x \to \infty} \left|\operatorname{sinc}\!\left({e}^{i \theta} x\right)\right| = \begin{cases} 0, & {e}^{i \theta} \in \left\{-1, 1\right\}\\\infty, & \text{otherwise}\\ \end{cases}$

## Bounds and inequalities

### Real variable

$\left|\operatorname{sinc}(x)\right| \le 1$
$\operatorname{sinc}(x) > -0.217234$
$\left|\operatorname{sinc}(x)\right| \le \frac{1}{\left|x\right|}$
$\left|\operatorname{sinc}(x)\right| \le \frac{1 + \left|x\right|}{1 + {x}^{2}}$
$\left|\operatorname{sinc}(x)\right| < \frac{\operatorname{asinh}(x)}{x}$
$\left|\operatorname{sinc}(x)\right| < \frac{\sqrt{2}}{x} \tanh\!\left(\frac{x}{\sqrt{2}}\right)$
$\left|{\operatorname{sinc}}^{(n)}(x)\right| \le 1$

### Complex variable

$\left|\operatorname{sinc}(z)\right| \le {e}^{\left|\operatorname{Im}(z)\right|}$
$\left|\operatorname{sinc}(z)\right| \le \operatorname{sinc}\!\left(i \left|z\right|\right)$

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

2021-03-15 19:12:00.328586 UTC