# Why doesn't Mathematica evaluate FourierTransform[Exp[-t], t, w]?

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?

• 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}} – Wjx Jun 7 '16 at 0:22
• 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. – Daniel Lichtblau Jun 7 '16 at 12:25

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}]