Calculate Fourier Transform of (1+at^2)^-n

[kbrijesh]

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 ∞

 

[jackmell]

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:

$$\int_{-\infty}^{\infty} \frac{e^{-i\omega t}}{(1+2t^2)^2}dt$$

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.

 

[kbrijesh]

You can do it via contour integration for specific small n but not sure for general n. But first, look at a simple case:

$$\int_{-\infty}^{\infty} \frac{e^{-i\omega t}}{(1+2t^2)^2}dt$$

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.