The Fourier transform of the function (1+at^2)-n can be computed using contour integration techniques, particularly for small values of n (specifically n=2, 3, and 4). The integral of interest is ∫((1+at^2)-n) * exp(-jωt) dt, evaluated from -∞ to ∞. The Residue Theorem is a key tool for solving these integrals, starting with simpler cases before generalizing to larger n values. A systematic approach involves mastering the simpler cases first before attempting to derive a general expression.
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kbrijesh said: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 said: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.