Difficulty in Solution of Poisson's equation using Fourier Transform I wanted to find the potential on an infinite plane due to a point charge located at some point $(x_0,y_0,z_0)$.
So I decided to solve the $2D$ Poisson's equation.
$$\nabla^2 V(x,y)=-\frac{q}{\epsilon}\delta (x-x_0)\delta (y-y_0)$$
Rather than the conventional Green's Function, my method of approach was Fourier transform.  Twice application of FT gave
$$V(k,l)=\frac{q}{2\pi \epsilon}\cdot \frac{\exp\left(ikx_0+ily_0\right)}{k^2+l^2}$$
Then I started doing the inverse transforms. After two inverse fourier transforms, I got the integral
$$V(x,y)=\frac{q}{4\pi \epsilon}\cdot \int_{-\infty}^{\infty} \frac{\exp\left(ik(x-x_0)+k\lvert y-y_0\rvert\right)}{k} dk$$
Now this integral on computation using complex analysis gives $\pi i$. And the potential becomes complex. Whereas it should have been logarithmic.
Can someone tell me where I am mistaken? And how I can better compute the integral?
Or maybe the problem lies in the first few steps??

Your equation for finding this potential is wrong since it is$$\nabla^2 V(x,y)=-\frac{q}{\epsilon}\delta(\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2})$$but i think solving this using the same relation of pointwise electrical charge potential is easier.$$V(x,y,z)=\dfrac{q}{4\pi\epsilon_0\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}}$$
