I'm looking for the Fourier transform of $f(t)=\frac{1}{t}\theta(t)$ ($\theta$ is the step function), I know how to do both factors separately but not if they are multiplying. Can someone help me ?
I try to do that in the wolframalpha page but it couldn't (at least by free).
And I'm keen on the Fourier transform of $e^{-\frac{1}{t}}\theta(t)$ or $e^{-\frac{1}{t^2}}\theta(t)$ too, actually I'm more interested in this two last ones than in $\hat{f}$.
Many thanks in advance.
You cannot multiply distributions, you have to define $\theta(t)/t$ separately. If it's defined as $$\left( \frac {\theta(t)} t, \phi \right) = \int_{t > 0} \frac {\phi(t) - \phi(0) [t < 1]} t dt,$$ then you can combine known results for the transforms of $1/t$ and $1/|t|$ to obtain $$\left( \frac {\theta(t)} t, e^{i p t} \right) = -\ln(-i p ) - \gamma.$$
For the other two functions, these integrals exist in the ordinary sense: $$\int_0^\infty (e^{-1/t} - 1) \,e^{i p t} dt = \frac {2 K_1(2 \sqrt{-i p \,})} {\sqrt{-i p \,}} -\frac i p, \\ \int_0^\infty (e^{-1/t^2} - 1) \,e^{i p t} dt = \cases{ \frac i {p \sqrt \pi} G_{0, 3}^{3, 0} \left( -\frac {p^2} 4 \middle| {- \atop 0, \frac 1 2, 1} \right) -\frac i p, & $p < 0$ \\ \\ \frac i {p \sqrt \pi} G_{3, 0}^{0, 3} \left( -\frac 4 {p^2} \middle| {0, \frac 1 2, 1 \atop -} \right) -\frac i p, & $p > 0$},$$ where $K$ is the modified Bessel function and $G$ is the Meijer G-function. Then adding the transform of $\theta(t)$ gives the answer.