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I am having trouble getting Mathematica to evaluate a trivial Fourier transform.

I would like to be able to evaluate the Fourier transform of the right-sided exponential decay function as shown below. (The real proof is explored here: fourier.eng.hmc.edu)

$\qquad \mathscr{F}[e^{- t }u(t)] = \frac{1}{1 + j \omega} $

Once I have evaulated the Fourier transform I would like to plot it for $\omega = {1,5}$

I have tried the following:

FourierTransform[Exp[-t], t, w]
Table[%, {w, 5}]
ListLinePlot[Abs[%]]

The output from FourierTransform is my input with some formatting.

My attempt to create the table doesn't work...

Where am I going wrong?

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    $\begingroup$ By using {w,5}, you set w to 1,2,3,4,5. If you want w only be 1 and 5, use {w,{1,5}} $\endgroup$ – Wjx Jun 7 '16 at 0:22
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    $\begingroup$ Documentation Center > FourierTransform > Details and Options first bullet items explains the Fourier integral is taken from -infinity to infinity. Which in turn should explain both what was incorrect, and why the approach of @bills makes sense. $\endgroup$ – Daniel Lichtblau Jun 7 '16 at 12:25
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Probably what you want is:

ft[w_] = FourierTransform[UnitStep[t] Exp[-t], t, w]

Otherwise, for t negative, the integral diverges. Also, you might want to check out the option FourierParameters. To plot:

Plot[Abs[ft[w]], {w, 0, 10}]
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