I am doing some analytical work in the field of wind turbine wakes. The wake is often described as a 2D Gaussian function. When I integrate this 2D Gaussian function over a circle of an arbitrary size and location, the following integration emerges. $$G(r,\sigma) = \int\limits_0^1 \eta e^{-\eta^2/2\sigma^2} I_0(r\eta/\sigma^2)~d\eta,$$ where $I_0$ is the modified Bessel function of order $0$. This is a famous integration that does not have an analytical solution. In fact, it can appear in many applications like diffusion problems and signal processing. I tried to play around with the integration and ended up with an expression that contains the generating function $$F(r,\sigma) = \sum\limits_{k\ge 0} r^k I_k(r/\sigma^2).$$ I checked with Wolfram Alpha and it says that the summation does not converge even though when plotted against $k$ it seems to do down to $0$ as $k$ increases. I am also confused by this as $G(r,\sigma)$ should be a finite function similar to its improper counterpart $\hat{G}(r,\sigma)$ $$\hat{G}(r,\sigma) = \int\limits_0^{\infty} \eta e^{-\eta^2/2\sigma^2} I_0(r\eta/\sigma^2)~d\eta = \sigma^2 e^{r^2/2\sigma^2}$$ given that the integrand is continous and positive for $\eta\ge0$. I expect $F$ and hence $G$ to be exponentially increasing functions in $r$ just like $\hat{G}$. Can anyone please help with an idea to evaluate $F$?
Thanks a lot.