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


)


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