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How to get inverse Fourier transform and plot it?

M M
Posted 2 years ago

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.
POSTED BY: M M
5 Replies
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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]]
POSTED BY: Gianluca Gorni

Try giving Plot an option like PerformanceGoal -> "Speed"
POSTED BY: Gianluca Gorni
M M
M M
Posted 2 years ago

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.
POSTED BY: M M

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}]
POSTED BY: Gianluca Gorni
M M
M M
Posted 2 years ago

Thanks dear Gianluca Gorni. enter image description here

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?
POSTED BY: M M
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