= Solution
For the stated angular-frequency <Fourier transform> convention, the <convolution theorem> gives
$$
\widetilde{f*g}(k)=\widetilde f(k)\widetilde g(k),
\qquad
(f*g)(x)=\int_{-\infty}^{\infty}f(\xi)g(x-\xi)\,d\xi.
$$
The <Fourier inversion theorem> and the assumed absolute integrability therefore imply
$$
\boxed{h(x)=\int_{-\infty}^{\infty}f(\xi)g(x-\xi)\,d\xi.}
$$
Since $|-i|=1$ and the principal argument is $-\pi/2$,
$$
\operatorname{Log}(-i)=-\frac{\pi i}{2}.
$$
All logarithms are $-\pi i/2+2\pi i n$, $n\in\mathbb Z$.
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