It seems I am stuck with a (at a first sight) trivial problem. 

It's from the *"Quarks and Leptons" (Halzen, Martin)* book page $141$, where one considers the following integral: 

$$\tag{1} T_{fi} = -i\int \!d^4x \, A(t,\vec{x})\,B(t,\vec{x})\frac{1}{|\vec{q}|^2}. $$
In equation $(1)$, $A$ and $B$ are two electron currents. 

Now according to the authors, one can rewrite $(1)$ by making use of the Fourier transform 

$$\tag{2} \frac{1}{|q|^2} = \int\! d^3x\, e^{i\vec{q}\cdot\vec{x}}\frac{1}{4\pi|\vec{x}|}, $$
to the following 
$$ \tag{3} T_{fi} = -i\int \!dt\int d^3x_1\int d^3x_2 \, \frac{A(t,\vec{x_1})\,B(t,\vec{x_2})}{4\pi|\vec{x_2}-\vec{x_1}|}. $$

Plugging $(2)$ into $(1)$ is just not enough, there seems to be a delta function missing somewhere. Maybe it's the notation or I have misunderstood their statements, i.e. perhaps $(1)$ and $(2)$ should not be equal to each other, rather just proportional or something? 

Anyone got any idea? This is of course not any homework of any kind, just for fun. 

___
I'm sure I've seen this integral (similar problem) in classical electrodynamics, it looks very familiar.