term. Take this from Eq. (4-159) and define
S[O] 1 aPIv t=t+ 3 [. (2) IV + (oaaPIII) PIII t=to
02 o 4 p
15 ) (4-228)
16 p o
Generically, this can be written as
Qa[ 2n+l k nkp(a)A2n+ -k (0 o)k-p (0 P ( ))
Q.[O 2+l 1- (4-229)
n=l k-0 p=0 0
where A r ro, and ynkp(a) is the coefficient of each individual term that depends
on n, k and p as well as on the component index a, with a dimension Rk-2 for
a = t, r and Rk-1 for a = 0, (.
The behavior of Qa[e0], according to the powers of each factor on the right
hand side of Eq. (4-229), is
Qa[] (2n+l) 2n+-k ( ) k-p ( (4-230)
where s = k 2 for a =t, r and s = k 1 for a = 0, 4. Following the same
procedure as in the analysis given in Subsection 4.4.2, Eq. (4-230) becomes
Qa[GO] 0 (2n+1)A2n+1-p-i (sin O)p+i (sin ))p (cos I)i 7, (4-231)
where a binomial expansion over the index i = 0, 1, k p is assumed, and
s =p + i 2 for a = t,r and s p + i 1 for a = Here we have disregarded
any by-products like O[(x xo)k-p+2] and O[(x xo)p+2], which originate from
( 00)k -p and (0 7) respectively when we transform them into trigonometric
functions and then rotate the coordinates: by putting them back into Eq. (4-230)
we simply obtain 0(92) terms, which would correspond to (0-4) in Eq. (4-33) and
should vanish when summed over in our final self-force calculation by Eq. (4-32).