Eq. (4-188), according to the powers of each factor, is
Qa[e-1 ~ (2n+"+'A2n-k (- o) k-p (0 P ,
2
(4-189)
where s k- 1 fora = t,r and s = kfor a = 0, Further,
S 0/) 1JrA k-p
Sf(r +2)
S( i) Ak-p-i/ k-p-i (4-190)
S(sin 0)' (cos )' Ak-- p/k-p-i + O[(x xo)k-p+2], (4-191)
where a binomial expansion over the index i
Eq. (4-190), and
(- -
12
0,1, .. k p is assumed in
(sin Oe) (sin 4)" + O[(x- xo)p+2]
(4-192)
as already shown in the analysis of Subsection 4.4.1. Using these, the behavior of
Q[c-1] in Eq. (4-189) looks like
Qa [e1] ~ p-(2n+l)A2n-p-i (sin e)p+i (sin 4))p (cos I)i ) ",
(4-193)
where s = p + i 1 for a = t, r and s p + i for a = 0, Here we have disregarded
any contributions from O[(x- Xo)k-p+2] in Eq. (4-191) and O[(x xo)+2] in Eq.
(4-192): when these are substituted into Eq. (4-189), we obtain e1 terms or 0(c3),
which would correspond to -2v/2Da/(2 1)(2f + 3) or O(-6) in Eq. (4-33).3
Qa[-l1] then can be categorized into the following cases:
(i) i 2j +1 (j0 0,1,2,...)
The integrand for "()" process, F(4)) (sin 4)" (cos I)i = (sin 4)" (cos 4)2j+1
has the properties that F(4 + 7r) = -F() for p = even integers and that
3 While 0(-6) is considered to be completely v i1-l1Iir:. these e1 terms are not
neglected and will be incorporated into the calculations of Da-terms later.