How to get inverse Fourier transform and plot it?

Questions
1

Hi everybody.
I have the below function and want to get inverse Fourier transform of it and plot it.

ft = Exp[1 - (1 + \[Omega]^4)^(a/4)]

when I use the below command, I get the same input as output.

ift[t_] = 
 InverseFourierTransform[ft, \[Omega], t, Assumptions -> 0 < a < 2]

Generally, I want to plot the inverse Fourier transform of this function to check its behavior.
How Can I do it?

thanks for your help in advance.

2

You get a plot with numerical integration:

Manipulate[
 Plot[1/Sqrt[2 \[Pi]]
    NIntegrate[Exp[1 - (1 + \[Omega]^4)^(a/4)]*Exp[-I \[Omega] t],
    {\[Omega], -Infinity, Infinity}],
  {t, -2 Pi, 2 Pi}, Frame -> True,
  Axes -> False, PlotRange -> All],
 {{a, 1}, 0, 2}]
3

Thanks dear Gianluca Gorni.

22.png
22.png550×132 3.72 KB

In this picture, it’s clear when I play animation control, it shows aborted. Is there a way to be able to see chart changes simultaneously when I play animation?

4

Try giving Plot an option like PerformanceGoal -> "Speed"

5

I try this:

Manipulate[
 Plot[1/Sqrt[2 \[Pi]] NIntegrate[
    Exp[1 - (1 + \[Omega]^4)^(a/4)]*
     Exp[-I \[Omega] t], {\[Omega], -Infinity, Infinity}], {t, -2 Pi, 
   2 Pi}, Frame -> True, Axes -> False, PlotRange -> All, 
  PerformanceGoal -> "Speed"], {{a, 1}, 0, 2}]

But it shows aborted again.

6

Try this variation, without Manipulate:

With[{a = 1},
 Plot[1/Sqrt[2 \[Pi]] 
   NIntegrate[Exp[1 - (1 + \[Omega]^4)^(a/4)]*Exp[-I \[Omega] t],
    {\[Omega], -Infinity, Infinity}],
  {t, -2 Pi, 2 Pi},
  Frame -> True, Axes -> False, PlotRange -> All]]