Fourier transform

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how to get the fourier transform of (1+at^2)^-n ? n is a natural number such that (n>1) and a is any positive number.

i.e. ∫((1+at^2)^-n)*exp(-jωt)dt; limits of integration goes from -∞ to ∞
 

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  • #2
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how to get the fourier transform of (1+at^2)^-n ? n is a natural number such that (n>1) and a is any positive number.

i.e. ∫((1+at^2)^-n)*exp(-jωt)dt; limits of integration goes from -∞ to ∞
You can do it via contour integration for specific small n but not sure for general n. But first, look at a simple case:

[tex]\int_{-\infty}^{\infty} \frac{e^{-i\omega t}}{(1+2t^2)^2}dt[/tex]

Now that can be solved by the Residue Theorem. First get that one straight, then go on to n=3, maybe 4, then try and come up with an expression for the general case.
 
  • #3
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You can do it via contour integration for specific small n but not sure for general n. But first, look at a simple case:

[tex]\int_{-\infty}^{\infty} \frac{e^{-i\omega t}}{(1+2t^2)^2}dt[/tex]

Now that can be solved by the Residue Theorem. First get that one straight, then go on to n=3, maybe 4, then try and come up with an expression for the general case.


Yes, you are true. Even using contour integration we can do it upto n=3. But I am y\trying to get a generalized solution.

Anyway, thanks for your reply.
 

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