I'm currently investigating non-gaussian pointer states for quantum weak measurements. one of these non-gaussian pointers takes the form of a Laguerre-Gaussian beam mode. Mathematica doesn't seem capable of solving this for me, and I'm unsure on how to do it by hand. Essentially my problem is how to find the analytical solution of the following expression,
$$ \int\int (x + isgn(l)y)^{|l|}e^{\frac{-x^2-y^2}{4\sigma^2}}dxdy$$
where l is the azimuthal index, and $|l| > 1$. Understanding how to do this will allow me to move on to integrating over the probability density, which will be
$$\int\int (x^2 + y^2)^{|l|}e^{\frac{-x^2-y^2}{2\sigma^2}}dxdy$$
i.e. the integral of the expression multiplied by its complex conjugate. I feel as though this should be simple and infact I also believe that the probability density integral may prove easier than the other integral, but I'm having real trouble solving this, and any help would be greatly appreciated. Thanks in advance.