Transfer function reduction math

The "Lecture Notes in Control and Information Sciences" can be a fascinating book to page through, particularly if you can get your hands on a hard copy.
The particular one I'm looking through is from the early 1980s and there's a lot of interesting material related to optimization problems here, like "Optimal turning strategy for a supercruiser" (aircraft) and "Optimal maintenance policy and sale date for a machine with random deterioration and subject to random catastrophic failure"...
A fair bit of the mathematics assumes a certain baseline knowledge of the field of systems optimization/linear programming/etc and I don't easily follow most papers, but there are some papers of interest to electrical engineering, e.g. one about reduction of order of transfer functions using a minimum-phase approximation, higher order transfer functions sometimes contain more information than you need for a restricted bandwidth.
Unfortunately partly due to the pre-Latex typesetting e I'm unclear what this one is saying exactly:
<https://imgur.com/a/9HIKOEN>
  H(s) is just a regular s-domain transfer function with polynomials top and bottom, so they decompose it into odd and even parts and set it equal to...what's tanh phi(s) supposed to mean? Tanh(s)phi(s)? Tanh(phi(s))?
Seems like they're doing some kind of tanh interpolation but it's not entirely obvious to me how they get from equation (3) to the expression in (5).