I have to find the fourier transform for $$ {1\over 1+16t^4} $$ I guess going there is a better way to solve it than going throug the integral but I'm not even sure if the factorization i made is correct $$ 1+16t^4=(1+4t^2)^2-8t^2=(4t^2+\sqrt 8t+1)(4t^2-\sqrt 8t+1) $$ Then i find the two couple of complex conjugates as solutions, but they are really ugly how do i find the transform from the factors at this point? Maybe I'm approaching this problem the wrong way?
2026-04-06 11:31:49.1775475109
Fourier transform doubting factorization
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Yes, your factorization is correct and leads in the next step to the correct roots. Their expression is not that horrible.
In another choice of the first factorization, You get from $16t^4=-1$ the reduced equations $4t^2=\pm i$ and from there $2t=(\pm1\pm i)/\sqrt2$ where the sign choices are independent..
You can evaluate the Fourier integrals via residual calculus.