Scalar propagator for lightlike separation

[bnado]

Hello everybody.
I have a free scalar in two dimensions. I know that its propagator will diverge for lightlike separations, that is t= ±x. I have to find the prefactor for this delta function, and I don't know how to do this.
How do I see from, for example, $$\int \frac{dk}{\sqrt{k^2+m^2}} e^{i k x - i \sqrt{k^2+m^2} t}+e^{i k x + i \sqrt{k^2+m^2} t}$$ what I get as a prefactor for my $$\delta (t-x)$$?

Normally when calculating this integral we set either x or t to 0, depending on whether the separation is timelike or spacelike, to then restore Lorentz invariance after the integral is solved. What can I do in the case of lightlike separation?
Thanks

 

[ChrisVer]

For some reason I can't see your formulas...

 

[bnado]

weird. The latex code for the first formula is
\int \frac{dk}{\sqrt{k^2+m^2}} e^{i k x - i \sqrt{k^2+m^2} t}+e^{i k x + i \sqrt{k^2+m^2} t}
and it's just the integral that gives you the propagator in position space.
the second one is just \delta (t-x)