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Sinc function

315 bytes added, 12:54, 4 September 2011
Key data
| [[period]] || none; the function is not periodic
| [[horizontal asymptote]]s || <math>y = 0</math>, i.e., the <math>x</math>-axis. This is because <math>\lim_{x \to \pm \infty} \frac{\sin x}{x} = 0</math>, which in turn can be deduced from the fact that the numerator is bounded while the magnitude of the denominator approaches <math>\infty</matH>.
| [[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>.
| [[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} , & x \ne 0\\\end{array}\right.</math>
| [[antiderivative]] || the [[sine integral]] (''this is defined as the antiderivative of the sinc function that takes the value 0 at 0'')

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