Green's Function for 2D Poisson Equation In two dimensions, Poisson's equation has the fundamental solution,
$$G(\mathbf{r},\mathbf{r'}) = \frac{\log|\mathbf{r}-\mathbf{r'}|}{2\pi}. $$
I was trying to derive this using the Fourier transformed equation, and the process encountered an integral that was divergent.  I was able to extract the correct function eventually, but the math was sketchy at best.  I am hoping someone could look at my work and possibly justify it.  Here goes.
First off, make the assumption that $G$ only depends on the difference $\mathbf{v}=\mathbf{r}-\mathbf{r'}$.  Now, let's write $G$ as an inverse Fourier Transform and take the Laplacian,
$$\nabla^2G(\mathbf{v}) = \int\frac{d^2k}{(2\pi)^2}(-k^2)e^{i\mathbf{k} \cdot \mathbf{v}} \hat{G}(\mathbf{k}) = \delta(\mathbf{v}) $$
For this to be a delta function, we require that $\hat{G}(\mathbf{k}) = -1/k^2$.  Now taking the inverse Fourier Transform of $G$...
\begin{align*}
G(\mathbf{v}) &= -\int\frac{d^2k}{(2\pi)^2} \frac{e^{i\mathbf{k}\cdot\mathbf{v}}}{k^2} = -\int\limits_{0}^{\infty} \int\limits_{0}^{2\pi} \frac{dkd\theta}{(2\pi)^2} \frac{e^{i|\mathbf{k}||\mathbf{v}|\cos\theta}}{k}\\
&= - \int\limits_0^{\infty}\frac{dk}{2\pi}\frac{J_0(kv)}{k}
\end{align*}
Here $J_0$ is a Bessel function of the first kind.  This integral is divergent as far as I can tell, but let's continue onward and take a derivative with respect to $|\mathbf{v}|$.
\begin{align*}
\frac{dG}{dv} &= \int\limits_0^{\infty}\frac{dk}{2\pi} J_1(kv)\\
&= \frac{1}{2\pi v}
\end{align*}
Then integrating this and setting the constant to zero we get the desired result...
$$ G(\mathbf{v}) = \frac{\log v}{2\pi} $$
Clearly this was a lot of heuristics, but I am hoping someone could justify some of this with distributions etc...  Could someone tell me what on earth I have done and why it worked?

Strictly speaking the Green function isn't Fourier transferrable as it is not L2 integrable. Any math that attempt to show directly the FT relation is necessarily flimsy. One remedy is to multiply an exponential function that decreases to 0 toward infinity but remain a constant 1 effectively within any finite region of interest. This way you should be able to justify otherwise flimsy math rigorously by carefully consider the FT integral toward infinity. However, the result implies that the Green function is the limit of a sequence of L2 functions whos FT converges to -1/k^2 pointwise but not in L2, as the L2 limit does not exist.
