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Sinc function

2,094 bytes added, 12:46, 4 September 2011
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<math>\operatorname{sinc} x := \left\lbrace \begin{array}{rl} 1, & x = 0 \\ \frac{\sin x}{x} & x \ne 0\end{array}\right.</math>
 
==Key data==
 
{| class="sortable" border="1"
! Item !! Value
|-
| default [[domain]] || all [[real number]]s, i.e., all of <math>\R</math>.
|-
| [[range]] || the [[closed interval]] <math>[\alpha,1]</math> where <math>\alpha</math> is ''approximately'' <math>-2/(3\pi)</math>.
|-
| [[period]] || none; the function is not periodic
|-
| [[local maximum value]]s and points of attainment || The local maxima occur at points <math>x</math> satisfying <math>\tan x = x</math> and <math>x in [2n\pi,(2n + 1)\pi]</math> or <math>x \in [-(2n + 1)\pi,-2n\pi]</math> for <math>n</math> a positive integer.<br> There is an anomalous local maximum at <math>x = 0</math> with value 1. Apart from that, the other local maxima occur at points of the form <math>\pm(2n\pi + \alpha_n)</math> where <math>\alpha_n</math> is fairly close to <math>\pi/2</math> for all <math>n > 0</math>. The local maximum value at this point is slightly more than <math>1/(2n\pi + \pi/2)</math>.
|-
| [[local minimum value]]s and points of attainment || The local minima occur at points <math>x</math> satisfying <math>\tan x = -x</math> and <math>x in [(2n - 1)\pi,2n\pi]</math> or <math>x \in [-2n\pi,-(2n - 1)\pi]</math> for <math>n</math> a positive integer.<br> The local minima occur at points of the form <math>\pm(2n\pi - \alpha_n)</math> where <math>\alpha_n</math> is fairly close to <math>\pi/2</math> for all <math>n > 0</math>. The local minimum value at this point is slightly less than <math>-1/(2n\pi - \pi/2)</math>.
|-
| [[point of inflection|points of inflection]] || {{fillin}}
|-
| [[derivative]] || <math>\operatorname{sinc}'x = \left\lbrace \begin{array}{rl} 0, & x = 0 \\ \frac{x \cos x - \sin x}{x^2} \\\end{array}\right.</math>
|}
==Graph==
 
Below is a graph of the function for the domain restricted to <math>[-3\pi,3\pi]</math>:
 
[[File:Sincfunctionbasic.png|600px]]
 
The picture is a little unclear, so we consider an alternative depiction of the graph where the <math>x</math>-axis and <math>y</math>-axis are scaled differently to make it clearer:
 
[[File:Sincfunctionbasicnottoscale.png|800px]]
==Graph==
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