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.