Let $f \in C^\infty_{c}(\mathbb{R})$ such that $\mathrm{supp}(f) \subset B(0,R)$. Prove that
$$ \widehat{f}(\xi)=\int_{-R}^R e^{i x \xi} f(x) dx $$ with $\xi \in \mathbb{C}$ is a holomorphic function in $\mathbb{C}$. Moreover, $|\widehat{f}(\xi)| \leq C e^{R |\mathrm{Im} \xi}|$. In particular, $\widehat{f}(\xi)$, $\xi \in \mathbb{R}$, is real analytic and $\mathrm{supp}(\widehat{f})$ is not compact unless $\widehat{f}(\xi)=0$.
is a bit of time that I do not touch complex analysis, can you help me?
thank you
Hint: Differentiation under the integral sign. Check that the integrand is analytic and that the assumptions of the theorem on differentiation under the integral sign are satisfied.. For your estimate just estimate the modulus of the integral by the integral of the modulus of the integrand.