b (scalar) b (scalar) Y (0, ) (5-60)
&m
However, each component sb calar) may be expressed in terms of spherical harmon-
ics also,
O/lr dQ'hS ly &n (/', '), (5-61)
b (scalar) b (scalar) (0 m / (5-61
where dQ' sin O'dO'd' and hb (scala(0, ) E {h (t r, h tr, 0,),
h, (t, r, 0, <)}.
Now, substituting Eq. (5-61) into Eq. (5-60), we have
bR (scalar) (scalar) (sh s t) (5-62)
Ref. [28] shows that
2f + 1
4-Pe (cos 7)
47T
(5-63)
Y* ( )Y (0, ),
mn
where
cos 7 = coscos 0' + sin 0 sin 0' cos ( ').
(5-64)
However, in the coincidence limit Xa xa, we may have (0, () (7r/2, 0o)
(7/2, *), where = o JrA/f (rc + J2) according to Eq. (4-149). Then, cos 7
above becomes, in the coincidence limit,
cos 7- sin 0' cos(0' 0*) = cos 0,
(5-65)
by the definition of the rotated angles ((, 4) given in Eq. (4-151). By Eqs. (5-63)
and (5-65), hsbscalar) in Eq. (5-62) becomes in the coincidence limit
hb (scalar) + t J ohb(scalar)(, 4))P(os ),
(5-66)
where we rotated the angles in the integral, from d' = sin O'dO'd to dQ =
sin OdOd@. Now, we separate the variables in the integrand hb (scalar) (', 4) and
)
)
)