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## Homework Statement

I'm working through Zee for some self study and I'm trying to do all the problems, which is understandably challenging. Problem 1.3.1 is where I'm currently stuck: Verify that D(x) decays exponentially for spacelike separation.

## Homework Equations

The propagator in question is

$$ D(x) = -i \int \frac{d^3k}{2(2\pi)^3} \frac{e^{-i\boldsymbol{k}\cdot\boldsymbol{x}}}{\sqrt{\boldsymbol{k}^2+m^2}} $$

## The Attempt at a Solution

Presumably, I would have to solve the integral and show that it decays exponentially (the spacelike aspect has already been taken into account for the above integral) and what I did was switch to spherical coordinates and integrated over the azimuthal:

$$ D = \frac{-i}{8\pi^2}\int dr\,d\theta\frac{e^{-irx\cos\theta}}{\sqrt{r^2+m^2}}r\cos\theta $$

This is where I'm stuck. The square root in the denominator suggests this is a branch cut integral but I haven't been able to find a source that explains it sufficiently. If anyone could help me figure this out I'd appreciate it. Thanks.