(a) $z \in \mathbb{C}$. Let: $$\hat{f}(z)=\int_{-\infty}^\infty f(t)e^{-2\pi izt}dt$$ Observe $\hat{f}$ is an entire function, and using integration by parts, show that: $$|\hat{f}(x+iy)|≤\frac{A}{1+x^2}$$ if $|y|≤a$, for any fixed $a≥0$.
I've tried to use the hint to this problem, setting $f(t)$ as my $u$, and the $e$ term as my $v$ and then integrate by parts. However, I haven't been able to get useful results. I've tried to split it up by real and imaginary components first as well, but that also hasn't been promising. I'm not certain what I'm missing. Any suggestions are welcome. Thanks in advance.
Edit: I forgot to mention that $f$ is an arbitrary function which has compact support and is smooth (class $C^2$).
Hint: Since $f$ is continuously differentiable with compact support, $$\int f(t)e^{-2\pi izt}\,dt=\frac1{2\pi iz}\int f'(t)e^{-2\pi izt}\,dt.$$Do that one more time and it follows that $$|\hat f(z)|\le\frac{||f''||_1}{|2\pi z|^2}.$$