Another derivation from Quantum Fields on curved spacetime

[mad mathematician]

TL;DR

The enquiry is from pages 52-53 from the book Quantum Fields on curved spacetime by Birrell and Davies.

Hi, can anyone tell me how to derive equation eq (3.64) from equation (3.59)?

attached are the relevant pages:

 

[TSny]

From $t = \alpha \sinh(\tau/\alpha)$, we have $z = (t^2 + \alpha^2)^{1/2} = \alpha \cosh(\tau/\alpha)$. Use these to express the denominator of (3.59) in terms of $\tau$ and $\tau'$. Note $\mathbf x = z$ and $\mathbf x' = z'$.

Using identities for the hyperbolic sine and cosine functions, show that the denominator of (3.59) may be written to first order in $\varepsilon$ as $$16\pi^2\alpha^2 \left[\sinh^2\left(\frac{\tau-\tau'}{2\alpha}\right) - \frac{i\varepsilon \Delta t}{2\alpha^2}\right].$$ In the last term, $\Delta t = t - t'$, which is a function of $\tau$ and $\tau'$ that will ultimately be absorbed into $\varepsilon$.

Once you get this far, you can think about how to get the final result (3.64).