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Re: Fourier Transform

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  • Subject: [mg93584] Re: Fourier Transform
  • From: "Nasser Abbasi" <nma at>
  • Date: Sun, 16 Nov 2008 07:03:41 -0500 (EST)
  • References: <gf8rgp$poq$> <gfbur0$43e$> <gfgqhc$dkv$> <gfh3cl$g1l$> <gfimic$g0f$>
  • Reply-to: "Nasser Abbasi" <nma at>

"Nikolaus Rath" <Nikolaus at> wrote in message 
news:gfimic$g0f$1 at

> But what is the point of that? There is nothing special about the
> integral in a Fourier transform, so any functionality that helps to
> evaluate integrals is best added to Integrate[].
>   -Nikolaus

my 2 cents on this good question is the following:

Using FourierTransform and using the equivlant integeration command is not 
always the same.

In[6]:= func = Cos[w0 t];
Integrate[func* Exp[(-I)*w*t],   {t, -Infinity, Infinity}]

Integrate::idiv: Integral of E^(-I t w) Cos[t w0] does not converge on 

FourierTransform[func, t, w]
Sqrt[Pi/2] DiracDelta[w-w0]+Sqrt[Pi/2] DiracDelta[w+w0]

Cos[] is not square integrable (its average power is not zero, the average 
power of cos[] is 1/2)
So, Cos[] does not have a FourierTransform (one of the conditions to have 
FourierTransform is for the function to be square integrable). So, for 
non-square integrable functions, FourierTransform does not exist.

Yet, you see that FourierTransform does generate a FourierTransform for the 

Since there are many useful functions which have infinite energy, but we 
want to find its fourier transform,  a more generalized definition is used 
for the integral which uses a dirac delta function to allow one to integrate 
such functions, and that is why sometimes FourierTransform gives different 
answer than direct use of the Intgeration function.

Delta Functions: An Introduction to Generalised Functions by R. F. Hoskins


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