Fourier Integral Examples

  FourierTransform[UnitStep[t+1]-UnitStep[t-1],t,w]

Recall that C(w) is 1/2 F(w).

  f[t_]:=1/Pi Integrate[Sin[w]/w Exp[I w t],{w,-Infinity,Infinity}]

  f[t]

Do not be thrown off by the output. Look at the graph.

  Plot[Evaluate[f[t]], {t,-2,2}];

Does the fourier transform in Mathematica converge to the proper value? (Ignore the warning messages you see.)

  f[-1]
  f[1]

Let's look at the Fourier Integral representation of the function that is sin(t) from -3Pi to Pi and zero elsewhere.

  FourierTransform[(UnitStep[t+3 Pi]-UnitStep[t-Pi])Sin[t],t,w]

Yuck! There is a valuable lesson to be learned here. It may be better to evaluate the Fourier transform from the definition instead of using the Mathematica command.

  F[w_]:=Integrate[Sin[t] Exp[- I w t],{t,-3 Pi, Pi}]
  F[w]

  Simplify[ComplexExpand[F[w]]]

Recall that C(w) is 1/2 F(w).

  g[t_]:=1/Pi Integrate[1/2 F[w] Exp[I w t],{w,-Infinity,Infinity}]

  Plot[Evaluate[g[t]], {t,-6 Pi,3 Pi}];

Another valuable lesson. Look at the scale of your graph before you judge!

  Plot[Evaluate[g[t]], {t,-6 Pi,3 Pi},PlotRange->{-2,2}];

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