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 $\mathbb R$.

Anything that is not symmetric, such as

$$f(t) = \begin{cases} \\\ e^{-t} & t>0 \\\ 0 & t< 0 \end{cases}$$

whose FT is

$$\hat{f}(x) = \frac{1}{1+i x}$$