I am trying to solve the following Coulomb integral of two gaussians:
$$ \int_{- \infty}^{ \infty}dx1\int_{- \infty}^{ \infty} \frac{e^{-b1 (x1-c1)^2}e^{-b2 (x2-c2)^2}}{\left | x1-x2 \right |}dx2, $$
where $b1,b2,c1,c2$ are real parameters and $b1$ and $b2$ are positive. Does anyone have an idea how to solve this? I was trying to use the Gaussian product rule, but I could not get an analytic expression in the end.
just as partical answer: Using polar coordinates ($x_1=r cos(\theta), x_2 =r sin(\theta)$, |det(J)|=r) will give you an standard gaussian integral in the radial variable...