*Crosspost from math.stackexchange. Have a look at the great answers there, even though they do not quite answer the question completely.*

Define $$G(\theta) = \int\limits_0^\infty \int\limits_0^{2\pi} \frac{r\,e^{-r^2/2t}}{\sqrt{1-(\sin\theta\sin r \cos\varphi + \cos \theta \cos r)^2}} \mathrm{d} r \,\mathrm{d} \varphi$$ Clearly, for $\theta=0$, this does not converge. However, I would like to obtain an asymptotic expansion for $\theta\searrow 0$.

How could I approach this problem? The function $$F(r, \theta) = \int_0^{2\pi} \frac{\mathrm{d}\varphi}{\sqrt{1-(\sin\theta\sin r \cos\varphi + \cos \theta \cos r)^2}}$$ looks somewhat like an elliptic integral, but I have no experience in dealing with these and I have no idea how this could help.

**Any thoughts on how to approach this problem**?