I've problems to evaluate the following integral
$$\int_{\mathbb{R}^3} dx \, dy \, dz \, \frac{e^{-i \Gamma |\vec{r}-\vec{r_0}|}}{|\vec{r}-\vec{r_0}|} \frac{e^{i \Upsilon |\vec{r}|}}{|\vec{r}|} e^{-\frac{|\vec{r}|^2}{2\sigma ^2}}$$
Where $\vec{r}_0$ is a fixed point in the space and $\Gamma,\Upsilon$ and $\sigma$ are real parameters
It should be integrable as the varoious factors in the integrand are integrable (also the text where I found it says it is). I tried to use elliptic coordinates $\mu,\nu,\phi$ in three dimensions (see the paragraph "Alternative definition" in the following page from wikipedia http://en.wikipedia.org/wiki/Prolate_spheroidal_coordinates) but doing so i found the expression
$$\int_0 ^{2\pi} d\phi \int_1 ^\infty d\mu\int_{-1} ^1 d\nu e^{-i(\mu-\nu) \Gamma |\vec{r_0}|/2 } e^{i (\mu+\nu)\Upsilon|\vec{r_0}|/2} e^{-\frac{|\vec{r_0}|^2}{8\sigma^2}(\mu+\nu)^2}$$
Ignoring the integration over $\phi$ and other constant coefficients, doing a rotation of the coordinates $\mu,\nu\rightarrow s,t$ I obtained
$$\int_0 ^\infty ds \int_0 ^\infty dt e^{-it\Gamma |\vec{r_0}|/2 } e^{i s\Upsilon|\vec{r_0}|/2} e^{-\frac{|\vec{r_0}|^2}{8\sigma^2}s^2} $$
It's evident that the integration over $t$ brings to the evaluation of an oscillating function to the infinity, giving so an undetermined result. I thought that I mistaked in the change of coordinates system, maybe it can't be done for some reason I ignore. I'd like to know if the method I used is correct and if there is some other way to calculate such an integral.