= Solution
The <Heaviside step function> restricts the integral defining the <Fourier transform> to $t\geq0$, so
$$
\widetilde g(\omega)=\int_0^\infty e^{-(\alpha+i\omega)t}\,dt
=\boxed{\frac1{\alpha+i\omega}},\qquad \operatorname{Re}\alpha>0.
$$
The convergence condition is essential. For real $\alpha$ it means $\alpha>0$, which is the decay assumption used in the subsequent parts. At $\alpha=0$ a <distributional Fourier transform> is needed; for negative real $\alpha$ the defining improper integral diverges.
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