The expected value of $1/X$ for $X\sim\mathcal{N}(\mu,\sigma^{2})$ is formally written as \begin{equation} \mathbb{E}[X^{-1}]=\frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^{\infty}\frac{1}{x}\exp\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right)\,\mathrm{d}x, \end{equation} which obviously does not converge. I am interested in the Cauchy principal value of this integral, i.e. \begin{equation} \mathbb{E}[X^{-1}]=\frac{1}{\sqrt{2\pi}\sigma}\lim_{\epsilon\nearrow 0}\left(\int_{-\infty}^{-\epsilon}+\int_\epsilon^\infty\right)\frac{1}{x}\exp\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right)\,\mathrm{d}x. \end{equation}
I would have thought that this has been worked out a long time ago but could not find a solution anywhere. Does anyone have a source detailing a solution?
After much looking I found a solution and accompanying proof:
Quenouille, M.H., 1956. Notes on bias in estimation. Biometrika 43, 353–360
For $X\sim\mathcal{N}(\mu,\sigma^{2})$, the Cauchy principal value of the mean is \begin{equation} \mathrm{PV}(\mathbb{E}[X^{-1}])=\frac{\sqrt{2}}{\sigma}\,\mathcal{D}\left(\frac{\mu}{\sqrt{2}\sigma}\right), \end{equation}
where \begin{equation} \mathcal{D}(z)=e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\,\mathrm{d}t, \end{equation} is the Dawson integral. For those interested, I also found a paper that derives $\mathrm{PV}(\mathbb{E}[X^{-1}])$ for the closely related skew-normal distribution:
The first negative moment in the sense of the Cauchy principal value (Chien-Yu Peng)
as well as the skew-t and generalized student-t:
The First Negative Moment of Skew-t and Generalized Student's t-Distributions in the Principal Value Sense (Chien-Yu Peng)