Trouble following Birrel and Davies in acomputing Bogoliuov coefficients

I’m having trouble following a computation from the book "quantum fields in curved spacetime" by Birrel and Davies.
In the chapter Where the authors describe particle creation by a collapsing spherical body they use two different coordinate systems to describe the outside region : $$ds^2= C(r) du dv$$ with $$u = t-r^*-R_{0}^* v= t + r^*-R_{0}^*$$ and $$r*=int{C(r)^{-1}}dr$$ For the inside region They use $$ds^2= A(U,V) dU dV$$
with $$U=tau -r+R_{0}V=tau +r-R_{0}$$
and the relation between $R_{0}$ (which is the starting position of the surface of the collapsing body) and $R_0^*$ is the sae as the one between $r$ and $r^*$

They proceed to define $$U=alpha(u)v=beta(V)$$ and then to compute $$alpha'(u)=dU/du=(1-dot{R})C{[AC(1-dot{R}^2)+dot{R}^2]^{1/2}-dot{R}}^{-1} beta ‘(V) = dv/dV= C^{-1}(1+dot{R})^{-1}{[AC(1-dot{R}^2)+dot{R}^2]^{1/2}+dot{R}}$$ by matching the two coordinate systems at $r=R(tau)$ wich is the collapsing surface. This last step is giving me headaches because I can’t seem to find a good way to tackle it. Can somebody help me?