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## Sinc function

, 12:46, 4 September 2011
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$\operatorname{sinc} x := \left\lbrace \begin{array}{rl} 1, & x = 0 \\ \frac{\sin x}{x} & x \ne 0\end{array}\right.$

==Key data==

{| class="sortable" border="1"
! Item !! Value
|-
| default [[domain]] || all [[real number]]s, i.e., all of $\R$.
|-
| [[range]] || the [[closed interval]] $[\alpha,1]$ where $\alpha$ is ''approximately'' $-2/(3\pi)$.
|-
| [[period]] || none; the function is not periodic
|-
| [[local maximum value]]s and points of attainment || The local maxima occur at points $x$ satisfying $\tan x = x$ and $x in [2n\pi,(2n + 1)\pi]$ or $x \in [-(2n + 1)\pi,-2n\pi]$ for $n$ a positive integer.<br> There is an anomalous local maximum at $x = 0$ with value 1. Apart from that, the other local maxima occur at points of the form $\pm(2n\pi + \alpha_n)$ where $\alpha_n$ is fairly close to $\pi/2$ for all $n > 0$. The local maximum value at this point is slightly more than $1/(2n\pi + \pi/2)$.
|-
| [[local minimum value]]s and points of attainment || The local minima occur at points $x$ satisfying $\tan x = -x$ and $x in [(2n - 1)\pi,2n\pi]$ or $x \in [-2n\pi,-(2n - 1)\pi]$ for $n$ a positive integer.<br> The local minima occur at points of the form $\pm(2n\pi - \alpha_n)$ where $\alpha_n$ is fairly close to $\pi/2$ for all $n > 0$. The local minimum value at this point is slightly less than $-1/(2n\pi - \pi/2)$.
|-
| [[point of inflection|points of inflection]] || {{fillin}}
|-
| [[derivative]] || $\operatorname{sinc}'x = \left\lbrace \begin{array}{rl} 0, & x = 0 \\ \frac{x \cos x - \sin x}{x^2} \\\end{array}\right.$
|}
==Graph==

Below is a graph of the function for the domain restricted to $[-3\pi,3\pi]$:

[[File:Sincfunctionbasic.png|600px]]

The picture is a little unclear, so we consider an alternative depiction of the graph where the $x$-axis and $y$-axis are scaled differently to make it clearer:

[[File:Sincfunctionbasicnottoscale.png|800px]]
==Graph==
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