= Solution
The <Fourier transform of a derivative> gives
$$
\widetilde{p'}(k)=ik\widetilde p(k).
$$
Since $p'=g$,
$$
\boxed{\widetilde p(k)=\frac{\widetilde g(k)}{ik}},
\qquad k\neq0.
$$
An arbitrary additive constant in $p$ affects only the zero-frequency <Dirac delta function> when transforms are interpreted as distributions.
The values of $\log i$ are $i(\pi/2+2\pi n)$. Hence
$$
i^{-2i}=\exp\{-2i\log i\}
=\exp((4n+1)\pi),\qquad n\in\mathbb Z.
$$
These are infinitely many distinct positive real numbers.
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