Integrating with volume element (d^3)x

[captain]

i'm at a loss about how to do this type of integration. can some one show me how to evaluate the integral of (d^3)k exp[ik*(x1-x2)]/[(k^2+m^2)(2pi)^3], where "*" is the dot product between the 3 vector k and (x1-x2), which are both 3 vectors. this come from the energy equation used to get the inverse square law (1/r^2).

 

[G01]

The notation d^3x is usually shorthand for a volume integral with differential elements dx, dy, dz (In Cartesian coordinates.) So:

$$\int_V f(x,y,z) d^3x = \int\int\int f(x,y,z) dxdydz$$

So, in your case, $$d^3k$$ probably stands for $$dk_x dk_y dk_z$$, where k_x, etc. are the components of the k vector. (Again, in Cartesian Coordinates)