rel="nofollow" href="#ulink_7ee717f1-6116-52c5-9438-65d66a2c965a">Eq. (2.472) is multiplied by itself – using x and y at a time as arguments, one gets
(2.527)
in which case the two fractions may be collapsed to produce
Eq. (2.528) may be rewritten as
with the aid of Eq. (2.15). After splitting the right‐hand side, Eq. (2.529) turns to
(2.530)
– which can be condensed to
at the expense of Eq. (2.473). Departure from Eq. (2.473) applied to x and y will instead lead to
(2.532)
where multiplication of the first by the second factor in the right‐hand side generates
after lumping exponential functions in the usual fashion, Eq. (2.533) transforms to
Algebraic reorganization of Eq. (2.534) gives rise to
(2.535)
which is equivalent to
due to Eq. (2.473). Ordered addition of Eq. (2.531) to Eq. (2.536) unfolds
(2.537)
that breaks down to
after condensation of similar terms; while ordered subtraction of Eq. (2.536) from Eq. (2.531) yields
(2.539)
– where negatives may be taken of both sides to get
Except for the change in sign, Eq. (2.538) matches Eq. (2.325), and Eq. (2.540) likewise matches Eq. (2.321).
Insertion of Eqs. (2.524) and (2.538) transforms Eq. (2.482) to
(2.541)
with division of both numerator and denominator by cosh x cosh y allowing simplification to (the more usual) form
again with the aid of Eq. (2.482); whereas divison of Eq. (2.526) by Eq. (2.540) supports
(2.543)
with division of both numerator and denominator by cosh x and cosh y justifying reformulation to
The similarity of Eqs. (2.542) and (2.544) to Eqs. (2.336) and (2.333), respectively, is again noteworthy – except for the change of sign in denominator.
A common form of Eq. (2.542) arises after setting y = x, viz.
(2.545)
