I have a complex variable $Z = X + i Y$ where $X$ and $Y$ are Gaussian iid with zero mean and $\sigma^2$ variance.
I am interested in ${\mathbb E} ( X |Z|^p )$. Is there a known distribution for this?
I know that $|Z|$ is Rayleigh distributed. However, for my expression I would have ${\mathbb E} (X+iY)|Z|^p = {\mathbb E} X|Z|^p + i {\mathbb E} Y|Z|^p = (1+i) {\mathbb E} X|Z|^p$.
Write $Z=R\exp i\Theta$, where in requiring $\Theta\in [0,\,2\pi]$ we impose a uniform distribution on $\Theta$. You're studying $R^{p+1}\cos\Theta$, a product of two independent variables. In particular, the mean is $0$ because the cosine averages to $0$.