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Question: How do you find Fourier Transform of non-integer power

Question:How do you find Fourier Transform of non-integer power

Nadeem_Malik 20 Maple
fourier inttrans
July 27 2019
0

 

Dear Maple users,

I want to find an expression for the Fourier Transform (FT) of an expression like  f(t)=exp(-t^2)/t^a, where a>0 is a constant.

I note that integer values of a (postive or negative) is ok; but non-integer fails. See sheet attached where I have tried 1 or 2 cases, a=0, 1, 0.3, etc.

So the questions are:

(1) how can I find the FT of the above for typical non-integer values of a>0 ?

(2) how can I find the FT of the above for general a -- i.e. declare a as a parameter?

 

Thanks

Nadeem


 

with(inttrans)

fourier(exp(-t^2), t, w)

exp(-(1/4)*w^2)*Pi^(1/2)

 (1)

fourier(t*exp(-t^2), t, w)

-((1/2)*I)*w*exp(-(1/4)*w^2)*Pi^(1/2)

 (2)

fourier(exp(-t^2)/t, t, w)

-I*Pi*erf((1/2)*w)

 (3)

fourier(t^.3*exp(-t^2), t, w)

fourier(t^(3/10)*exp(-t^2), t, w)

 (4)

fourier(exp(-t^2)/t^.3, t, w)

fourier(exp(-t^2)/t^(3/10), t, w)

 (5)

``


 

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