F(4 + 7/2) = -F(4) for p = odd integers. Thus
(Qa-1) = 0, (4-194)
(ii) i 2j (j 0,1,2,...)
For p = odd integers, the integrand F(4) (sin )"P (cos ))2j has the property
that F(4 + 7r) F(1), hence
(Qa[-) = 0. (4-195)
For p = even integers, using Eqs. (4-155) and (4-157), we can express
(sin O)P+ in Eq. (4-193) above in terms of po and A via a binomial expansion
(sin o)P+2 [2 (1 cos O)]p/2+ + O[(x o)p+2j+2]
p/2+j
= dpj qA P+j-2q + O[(x xo)p+2j2
q=0
S-(/2+j)p2qAP+2J-2q/Rp+2J + O[(x xo)p+2j+2],(4-196)
where q 0,1, ... p + j is the index for a binomial expansion, and the
coefficients dpjq X-(p/2+j) Rp+2j. When Eq. (4-196) is substituted into
Eq. (4-193), the contribution from O[(x Xo)p+2j+2] can be disregarded since
it would correspond to c1 terms or 0(C3) again. Then, we have
Qa [e-] ~ (sin )"p (cos 4)2j X-(P/2+j)0 2(n-q)-1A2(n-q)~s, (4-197)
where s = -1 for a = t, r and s = 0 for a = 0, <, and we can guarantee that
n-q > 0 ahv-,- since 0 < q < p+j= (p+i), < i < k-p and p < k < 2n.
Then, Eq. (4-197) can be subcategorized into the following two cases;
(ii-1) n q> 1