It is known that the Cauchy distribution has undefined moments, and that the expectation has a principal Cauchy value $\operatorname{PV}\left( \mathbb{E} [X] \right)$ of zero.
I wonder if $\operatorname{PV}\left( \mathbb{E} \left[ (X - \operatorname{PV}\left( \mathbb{E} [X] \right))^n \right] \right)$ exists for all $n$, or if there is a highest principal Cauchy moment for the Cauchy distribution.
Do all the higher moment expectations have defined principal Cauchy values for the Cauchy distribution?
The first simplification of the above is to note that $\operatorname{PV} \left(\mathbb{E}[X]\right) = 0 \implies \operatorname{PV}\left( \mathbb{E} \left[ (X - \operatorname{PV}\left( \mathbb{E} [X] \right))^n \right] \right) = \operatorname{PV}\left( \mathbb{E} \left[ X^n \right] \right)$.
From the definition of the Cauchy distribution and the Cauchy principal value we have
$\lim_{\epsilon \rightarrow 0^+} \left[ \int_{-\infty}^{0-\epsilon} x^n \frac{1}{\pi \gamma \left[ 1 + \left(\frac{x-x_0}{\gamma} \right)^2 \right]}dx + \int_{0+\epsilon}^{\infty} x^n \frac{1}{\pi \gamma \left[ 1 + \left(\frac{x-x_0}{\gamma} \right)^2 \right]}dx \right].$
So my question reduces to whether the above expression is defined.
Wolfram Alpha tells me that
$\int x^n \frac{1}{\pi \gamma \left[ 1 + \left(\frac{x-x_0}{\gamma} \right)^2 \right]}dx = \frac{\gamma x^{n+1}}{2 \pi \sqrt{- \gamma^2} (n+1)} \left( \frac{_2F_1 \left(1, n+1;n+2;\frac{x}{m - \sqrt{- \gamma^2}} \right)}{m-\sqrt{- \gamma^2}} - \frac{_2F_1 \left(1, n+1;n+2;\frac{x}{m + \sqrt{- \gamma^2}} \right)}{m + \sqrt{- \gamma^2}} \right) + C$
where $_2F_1 (\cdot, \cdot ; \cdot ; \cdot )$ is a hypergeometric function.