where Fe(4) and ge(4) are the f-th components of the expansions of
F(e, ) sin cos 4/(1- X2) and G(0, 4) sin sin /(1 X2) in dP (X)/dX, as in
Eqs. (5-104) and (5-105), respectively, and F(O, 4) and G(H, 4) are obtained from
Eqs. (5-99) and (5-100) together with the vector source functions, {hs (t, r, 0, ),
hs (t, r, 0, h) (t, r, 0, (h) h, (t, r, 0, 0)}, via the coordinate rotation.
To the first two highest orders, hS has the structure
S const -3 linear polynomials in x + cubic polynomials in x" (5- )
at6 6 3 -5 e 3 (55-111)
where x" denotes the Schwarzschild coordinates (t, r, 0, 0). Eq. (5-111) has the
expansion basis dPe(X)/dX, and c-3-term and C-2-term here would correspond
to C-l-term and co-term of the expansion with the basis Pe(X) as in Eq. (5-72),
since the weighting function 1 X2 = sin2 p1 .1ing into Eqs. (5-101), (5-102) and
(5-103) makes up for C2 in the limit 0 0. Structure analyses for C-3-term and
e-2-term may be developed in a similar manner to that in Section 4.4 so that we
can find the non-vanishing regularization parameters via Eqs. (5-108) and (5-109).
The other cases for hs he and hf can be treated in the same manner as
above. The results of the calculations are
he (Bv od) + (Cv + Cod) + O(-2), (5-112)
S = (B + B d) + (ce +c) +O(-2), (5-113)
he = (B + Bod) e d) + (2), (5-114)
hS = (BJ + Bd) + (Cev + Cod) + O(+-2), (5-115)
where
Bv = B d = 0,
(5-116)