Long story short:

https://i.imgur.com/6pXoUTj.png
Near the bottom it says "The wavefucntions Y'v and Yv' of the two resulting states,

*according to perturbation theory*, the following mixtures blah blah"

Nothing I have been able to find on perturbation theory, undergrad and grad textbooks, that would lead me to those equations.

All I see is Y'0 = Y0 + a1*Y1 + ...

Where a1*Y1 = <Y1|H'|Y0>*Y1 / (E2 - E1)

For example here is a page showing it:

Perturbation Theory
I just have no clue how the book made that jump. With the Hamiltonian and Wavefunctions I'm dealing with, the integrals are incredibly nasty. Trying to calculate the new Wavefunction the way the link shows gives equations that Mathcad is having trouble computing, so I can't really verify if I get their answer if I do it the hard way. This book was written in 1986, so they must have some sort of trick.

Thanks