Plot Fourier transform of $\sin (2 t)$

Question

I'm trying to plot the Fourier transform of $\sin (2 t)$.

I tried using the FourierTransform Function (I'm expecting a peak at w = 2) but it gives me undefined at w = 2 because the DiracDelta function is not defined at DiracDelta[0]. Is there any way around this? I used

Plot[FourierTransform[Sin[2*t], t, w], {w, -3, 3}, PlotRange -> Full]

which has gaps at 2 and -2, but no height. How do I get a peak with some height there?

Answer 1

Since the transform has complex delta functions, you have to create a representation by hand.

FourierTransform[Sin[2*t], t, w]
(*
  I Sqrt[\[Pi]/2] DiracDelta[-2 + w] - I Sqrt[\[Pi]/2] DiracDelta[2 + w]
*)

Here's one way that includes marking the axes by hand.

Plot[UnitBox[100 (2 - w)] - UnitBox[100 (2 + w)],
 {w, -3, 3}, Exclusions -> {-2, 2}, PlotStyle -> Thick,
 Ticks -> {Automatic, {{-1, Row[{-I Sqrt[Pi/2], Infinity}]}, {1, 
     Row[{I Sqrt[Pi/2], Infinity}]}}}]