Everything is in the title. I've successfully computed that $$ \forall t\in \mathbb{N}^*, \mathcal{F}^{-1}(\bigg(\frac{\sin{k}}{t}\bigg)^t) = \frac{\sqrt{2\pi}(-1)^t}{4(2t-2)!!}\sum_{j=0}^t\begin{pmatrix}t\\j\end{pmatrix}(x-t+2j)^{t-1}(-1)^j \text{sgn}(x-t+2j) $$
But i've been unable to correctly compute it for $t\in \mathbb{R}^+$. I thought I could use the asymptotic development of $(1+x)^t$ with $x=e^{-2 i k }$, but yet, I fail to compute $$\mathcal{F}^{-1} \frac{e^{ik(t-2j)}}{k^t}$$
I'm totally new to Fourier Analysis and I've never had any course on it, so i'm sorry if this is an easy question