Some derivation in QFT in Curved SpaceTime by Birrell and Davies

[mad mathematician]

I am trying to derive equation (3.61) below (the attachments are from pages 51-53).
As far as I can tell, one can get (3.61) from (3.59) directly, just plug $x=x'$ and $t-t'=\Delta \tau$, but then there's this remark that "$(1-v^2)^{1/2}$ is absorbed in epsilon". So I think my "direct derivation" must be wrong, something must be missing in their derivation.
Can you help?

 

[renormalize]

As far as I can tell, one can get (3.61) from (3.59) directly, just plug $x=x'$ and $t-t'=\Delta \tau$...

No, because your last equation for $\Delta\tau$ is wrong since it contradicts B&D's eq. (3.57). Remember that $t$ is coordinate time and $\tau$ is proper time. For an inertial trajectory they differ by a factor of $\left(1-v^2\right)^{1/2}$.