when Fourier transform function in $\mathbb C$? 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 $\mathbb R$.

There is a basic fact about Fourier transform on the Schwartz space: Let $f\in \mathcal{S}(\mathbb{R})$, then $\widehat{f'} = it\widehat{f}$. Thus, if $\widehat{f}$ is real-valued, then $\widehat{f'}$ is complex-valued.
