Heat Equation with Singular Initial Condition

Question

I'm trying to solve the 2D heat equation in cartesian coordinates with initial condition of the form $1/r$. The problem is as follows:

$$ \begin{equation} \begin{cases} \frac{\partial u}{\partial t} = \alpha \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right), & x \in \mathbb{R}, y \in \mathbb{R}, t > 0 \\ u(x, y, 0) = \frac{-y}{x^2 + y^2}, & x \in \mathbb{R}, y \in \mathbb{R}. \end{cases} \end{equation} $$

Separating variables doesn't seem to be the best approach here and I couldn't find the Fourier transform of this singular initial condition. So I'm trying to solve it via convolution:

$u(x, y, t) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} G(x - \xi, y - \eta, t) \frac{-\eta}{\xi^2 + \eta^2} \, d\xi \, d\eta$

where: $G(x, y, t) = \frac{1}{4 \pi \alpha t} e^{-\frac{x^2 + y^2}{4 \alpha t}}$

The problem is that the integral for $u(x,y,t)$ doesn't seem to have an analytical solution. Is that really that case? Can this PDE only be solved numerically?

Edit: The Fourier Transform of the initial condition can be found explicitly with Mathematica:

initialFourierTransform = -((I ky (HeavisideTheta[-kx] + HeavisideTheta[kx]))/(kx^2 + ky^2))

However the evaluation of the inverse Fourier transform to find the solution is taking a lifetime and I suspect there might be an error.

uHat[kx_, ky_, t_] := 
 initialFourierTransform*Exp[-alpha*(kx^2 + ky^2)*t]
(Compute the inverse Fourier transform to find u(x,y,t))
u[x_, y_, t_] := 
 InverseFourierTransform[uHat[kx, ky, t], {kx, ky}, {x, y}]