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====<math>X(s)</math> s is a complex variable==== | ====<math>X(s)</math> s is a complex variable==== | ||
| − | <math>X(s) = x(s)e^{-st}dt</math> | + | ====<math>X(s) = \int_{-\infty}^{\infty}x(s)e^{-st}dt</math>==== |
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| + | <b>The Laplace Transform <math>X(s)</math> evaluated on the imaginary axis <math>X(j\omega)</math> is equal to the F.T> at <math>\omega</math></b> | ||
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| + | So the F.T. is the restriction of the L.T. on the imaginary axis, <math>s=j\omega</math> | ||
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| + | <math>X(s) = \int_{-\infty}^{\infty}x(s)e^{-st}dt = \int_{-\infty}^{\infty}x(s)e^{-(a+j\omega)t}dt = \int_{-\infty}^{\infty}x(s)e^{-at}+e^{j\omega t}dt = F(x(t)e^{-t})</math> | ||
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| + | Where a is the real part of s. | ||
Latest revision as of 11:51, 24 November 2008
$ X(s) $ s is a complex variable
$ X(s) = \int_{-\infty}^{\infty}x(s)e^{-st}dt $
The Laplace Transform $ X(s) $ evaluated on the imaginary axis $ X(j\omega) $ is equal to the F.T> at $ \omega $
So the F.T. is the restriction of the L.T. on the imaginary axis, $ s=j\omega $
$ X(s) = \int_{-\infty}^{\infty}x(s)e^{-st}dt = \int_{-\infty}^{\infty}x(s)e^{-(a+j\omega)t}dt = \int_{-\infty}^{\infty}x(s)e^{-at}+e^{j\omega t}dt = F(x(t)e^{-t}) $
Where a is the real part of s.
