Understanding the Delta Function: Integrating from -∞ to ∞

[aaaa202]

My book loves to represent the delta function as:

δ(r-r')=∫_-∞^∞exp(i(r-r')k)dk

Now I can understand this formula if the integration was over the unit circle since. But this is an integration for which the antiderivative as no meaningful limit as x->±∞

 

[pwsnafu]

The integral written is only true as formality. What's really going on is that Dirac is a tempered distribution, and hence has a Fourier transform. It can be shown that $\hat\delta = 1$. What the equation is trying to say is that the Inverse Fourier of the constant 1 is Dirac delta.