The Fourier transform of a function $f\in\mathscr L^1(\mathbb R)$ is

$$\widehat f\colon\mathbb R\rightarrow\mathbb C, x\mapsto\int\_{-\infty}^\infty f(t)\exp(-ixt)\,\textrm{d}t$$

When is this indeed a function in $\mathbb C$? Most of calculations you get functions in $\mathbb R$. When in $\mathbb C$?

Add: I know there are results like $\frac{e^{ait}-e^ {-ait}}{2i}=\sin(at)$ multiplied by 'anything', but I am asking for a function which you cannot write as a function in $\mathbb R$.