I am trying to express the following integral of the Modified Bessel function either in closed form or even using other special functions. Any ideas ?
$$ \int_{0}^{b}x\exp\left(-\,{x^{2} + z^{2} \over 2\sigma^2}\right) {\rm I}_{0}\left(\vphantom{\large A}xz \over \sigma^{2}\right)\,{\rm d}x $$
Note that the integration to infinity is given to be equal to one (the inside function is a probability function), i.e., :
$$ \int_{0}^{\infty}x\exp\left(-\,{x^{2} + z^{2} \over 2\sigma^2}\right) {\rm I}_{0}\left(\vphantom{\large A}xz \over \sigma^{2}\right)\,{\rm d}x=1 $$
Thank you for your time.
With respect