For the following function $f$, function from real number to real number, with $\mu$ real, $k$ real, $\sigma$ real strictly positif, defined by: \begin{equation} f(x)=cos(k x ) e^{\frac{-(x-\mu)^2}{2\sigma^2}} \end{equation} The Hilbert transformation $\mathcal{H}$ is defined by: \begin{equation} \displaystyle [\mathcal{H}(g)](x) \stackrel{\text{def}}{=} \text{p.v.} \frac{1}{\pi} \int_{- \infty}^{\infty} \frac{g(t)}{x - t} ~ d{x} \end{equation}
What is the expression of $H(f)(x)$?