Solution
= Solution
Split the exponential kernel at zero. For $a>0$,
$$
\widetilde f(k)=\int_0^\infty e^{-(a+ik)x}\,dx+\int_{-\infty}^0e^{(a-ik)x}\,dx
=\frac1{a+ik}+\frac1{a-ik}.
$$
Thus \b[its <Fourier transform> is]
$$
\boxed{\widetilde f(k)=\frac{2a}{a^2+k^2}.}
$$
Both half-line integrals converge absolutely because $a>0$.