parameters. The techniques involved in the Legendre polynomial expansions and
the integration over ) are described in detail in Appendices C and D of Ref. [18].
Below in Subsections 4.4.1-4.4.4, we present the key steps of calculating the
regularization parameters Aa, Ba, Ca and Da in Eqs. (4-34)-(4-44).
4.4.1 A,-terms
We take the c-2 term from Eq. (4-159) and define
Qa[e-2] q t to (4-160)
2 po
We may express this in a generic form
k -A k 1-0 o k- Ip \P
Q,[ _2] a k p(,)A- -o) (0(- -)
0Qa 210>k3 2 (4-161)
k=0,1 p=0 90
where A r ro, and akp(a) is the coefficient of each individual term that depends
on k and p as well as on the component index a, with a dimension Rk for a = t, r
and R~k+1 for a = 0, The behavior of Qa[e-2], according to the powers of each
factor in Eq. (4-161), is
Qa -2] p-3A -k o-k p (- ) P, (4-162)
where s = k for a = t, r and s k + 1 for a = 0, 0. We recall from Eqs.
(4-148) and (4-149) that the first of the steps to lead to p2 is replacing 0 Q o
by (0 0') JrA/f(rQ + J2) to eliminate the coupling term A (0 to) in PII.
This makes a sum of independent square forms of each of A and 0 ', which is
a necessary step to induce the Legendre polynomial expansions later. Thus, to be
consistent with this modification made for po, the remaining 0 &o in Eq. (4-162)