3.7 Determine and plot the CTFT (continuous time Fourier Transform) for continuous time function

Plot the CTFT \(X(\omega )\) of \(x(t)=3 \sin (t)\). By definition

\begin{align*} X(\omega ) &= \int _{t=-\infty }^{\infty } x(t) e^{-i\omega t} \mathop {dt} \end{align*}

Mathematica

Clear["Global`*"]; 
f = 3*Sin[t]; 
y = FourierTransform[f,t,w,FourierParameters->{1, -1}]
 

Out[138]= -3 I Pi DiracDelta[-1+w]+3 I Pi DiracDelta[1+w]
 

tab = Table[{k,y/.w->k},{k,-3,3}] 
tab = tab/.DiracDelta[0]->1 
tab[[All,2]]=Map[Sign[Im[#]]Abs[#]&,tab[[All,2]]] 
 
ListPlot[tab,PlotStyle->AbsolutePointSize[5], 
  AxesLabel->{"frequency \[CapitalOmega] rad/sec", 
  "|F(\[CapitalOmega]|"}, 
  PlotRange->{All,{-12,12}}, 
  Filling->Axis, 
  FillingStyle->{{Thick,Red}}]
 

Matlab

syms t; 
F=fourier(3*sin(t))
 

F = 
-pi*(dirac(w-1)-dirac(w+1))*3*i
 

Need to do the plot.

Maple

restart; 
transform := expand(inttrans:-fourier(3*sin(t), t, w))
 

gives

transform := 3*I*Pi*Dirac(w + 1) - 3*I*Pi*Dirac(w - 1)