According to the powers of each factor on the right hand side of Eq. (4-239),
Q.[e1] behaves like
Qa[e1] ~ p(2n+l)A2n+2-k (- o)k-p (0- (4-240)
where s = k 3 for a = t, r and s = k 2 for a = 0, 0. Similarly to the analysis
given in Subsection 4.4.2, Eq. (4-240) develops into
Qa[e1] Po(2n+1)A2n+2-p-i (sin )P+i' (sin 4)P (cos I)' 7) (4-241)
where a binomial expansion over the index i = 0, 1, k p is assumed, and
s = p + i 3 for a = t,r and s = p + i 2 for a =0, The terms like
O[(x xo)k-p+2] and O[(x xo)p+2] have been disregarded, which originate from
(4 0o) -p and (0 7)P, respectively as we transform them into trigonometric
functions and then rotate the coordinates: when substituted back into Eq. (4-240),
these order terms make (0(3) that would correspond to 0(-6) in Eq. (4-33) and
should vanish when summed over in our final self-force calculation by Eq. (4-32).
Then, Qa[e1] can be categorized into the following cases:
(i) i 2j +1 (j 0,1,2,--.)
The integrand for "()" process, F(4) (sin 4)P (cos )i =- (sin 4)" (cos 4)2j+1
has the properties that F(4 + 7) = -F(4) for p = even integers and that
F(4 + 7/2) = -F(4) for p = odd integers. In either cases
(Qa[e1]) = 0, (4-242)
(ii) i 2j (j 0,1,2,...)
For p = odd integers, the integrand F() = (sin 4)" (cos 4)2j has the property
that F(4 + 7r) F(4), hence
(Qa[e1]) = 0.
(4-243)