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== Energy == | == Energy == | ||
+ | A time shift should not effect the energy of a function. | ||
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<math>u = (t-2)</math> | <math>u = (t-2)</math> | ||
Revision as of 05:46, 5 September 2008
cos(t-2)
Energy
A time shift should not effect the energy of a function.
$ u = (t-2) $
$ E=\int_{-2}^{2\pi-2}{|cos(u)|^2du} $
$ E=\frac{1}{2}\int_{-2}^{2\pi-2}(1+cos(2(u)))du $
$ E=\frac{1}{2}((u+\frac{1}{2}sin(2(u)))|_{u=-2}^{u=2\pi-2} $
$ E=\frac{1}{2}(2\pi-2 + .378 -(-2 - .378)) $
$ E=\pi $
Power
$ E=\frac{1}{2\pi-0}\int_0^{2\pi}{|cos(u)|^2du} $
$ =\frac{1}{2\pi-0} *{\frac{1}{2}}\int_0^{2\pi}(1+cos(2u))du $
$ =\frac{1}{4\pi}((u)+\frac{1}{2}sin(2u))|_{u=-\pi}^{u=2\pi-\pi} $
$ =\frac{1}{4\pi}(\pi+0-(0-\pi)) $
$ =\frac{1}{2} $