I got a rather specific question for when calculating Fourier transform with residue, hoping someone understand what I'm looking for and can help me!
My solution is rather long, but it's only the last step I'm not getting correct. Therefore I will not show my whole solution.
So I want to compute Fourier transform of $$f(x)=\frac{1}{x^4+4x^2+4}$$
using Residue Theorem.
So first we assume we have negative $t$ in our transform so nothing explodes.
This is done correctly I get my answer to $$\widehat{f(x)}(t) = \frac{-\pi e^{t \sqrt{2}}}{8}(2t-\sqrt{2}), t \leq 0$$
Since function is real we know that $$\widehat{f(x)}(t) = \overline{\widehat{f(x)}(-t)}$$
So now we want to expand is for the function to be valid with $t \in \mathbb R$.
My answer to this is $$\widehat{f(x)}(t) = \frac{-\pi e^{-t \sqrt{2}}}{8}(-2t-\sqrt{2}), t > 0$$
while the correct answer according to the course literature is $$\widehat{f(x)}(t) = \frac{-\pi e^{t \sqrt{2}}}{8}(-2t-\sqrt{2}), t > 0.$$
I would like to argue that this is wrong, since it would explode as $t \to -\infty$, or am I wrong?