Fourier transform for $f(y)=e^{-a|y|}$ and complex $a$ show that it converges for $Re(a)>0$

Question

With Fourier transform:

$\mathcal{F}[f(u)]:=\int^{\infty}_{-\infty}e^{iuy}f(y)dy$

and the example $f(y)=e^{-a|y|}$ with $a$ complex

I need to show that:

$\int^{A}_{-A}e^{iuy}f(y)dy$

converges for $A \rightarrow \infty $ and $Re(a)>0$.

I'm not so good with Fourier transform, can someone help me on how to begin this problem?

Answer 1

Just use $$\Bigl|e^{-a|y|}\Bigr|=e^{-Re(a)|y|}$$ to get absolute convergence, that is, $f\in L^1(\Bbb R)$ for $Re(a)>0$..