Moments of inverse of a non-central chi distributed variable I have a non-central chi variable $r$ with the distribution,
\begin{align}
p(r) = \frac{r^3\lambda}{(\lambda r)^{3/2}}\exp\left[-0.5(r^2 + \lambda^2)\right]I_{1/2}(r\lambda)
\end{align}
I'm looking for the following expectations.
\begin{align}
E\left[\frac{1}{r}\right]; \quad E\left[\frac{1}{r^2} \right]
\end{align}
In other words, is there a closed-form expressions for the integrals of the form,
\begin{align}
I(m) = \int_{0}^{\infty} \frac{1}{r^m} \frac{r^3\lambda}{(\lambda r)^{3/2}}\exp\left[-0.5(r^2 + \lambda^2)\right]I_{1/2}(r\lambda).
\end{align}
Thanks.
Edit :
$I_v(.)$ is the modified Bessel function of the first kind and order $v$.

The density simplifies to
$$p(r)=\frac{\sqrt{\frac{2}{\pi }} r e^{-(\lambda ^2+r^2)/2} \sinh (\lambda  r)}{\lambda }$$
where $\sinh$ is the hyperbolic sine function.  Integrating with Mathematica results in
$$E(1/r)=\frac{\text{erf}\left(\frac{\lambda }{\sqrt{2}}\right)}{\lambda }$$
$$E(1/r^2)=\frac{\sqrt{2} F\left(\frac{\lambda }{\sqrt{2}}\right)}{\lambda }$$
where $\text{erf}$ is the error function and $F$ is the Dawson F function.
Higher moments don't seem to exist.
