Rectangular function: Difference between revisions

Revision as of 16:41, 12 May 2006

Revision as of 16:44, 12 May 2006

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The rectangular function is normalized:

:<math>\int_{-\infty}^\infty \textrmmathrm{rect}(x)\,dx=1</math>

The [[continuous Fourier transform|Fourier transform]] of the rectangular function is

:<math>\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \textrmmathrm{rect}(xt)e^{-i \omega t} \, dt

=\frac{\textrmmathrm{sinc}(\omega/2)}{\sqrt{2\pi}}</math>

where "sinc" is the [[sinc function]]. Viewing the rectangular function as a [[probability distribution]] function, its [[characteristic function (probability theory)|characteristic function]] is therefore written

:<math>\varphi(k)=\textrmmathrm{sinc}(k/2)\,</math>

and its [[moment generating function]] is:

:<math>M(k)=\frac{\textrmmathrm{sinh}(k/2)}{k/2}\,</math>

where "sinh" is the [[hyperbolic sine]] function.