decompose it into zonal harmonic components

 h)b (scalar)() .T(')P (cOS 0). (5-67)

Then, substituting Eq. (5-67) into Eq. (5-66) and using the identity from Ref. [28]

 I dXP,(X)Pj(X) 2_ + 1' (5-68)

where X cos 0, we can rewrite Eq. (5-66) as


 hsb(scalar) "~- /J 2F -(). (5-69)

This is simply

 hb (scalar) mxa-E> (h(I)) (5-70)

using the notation "()" for the integration-averaging process. Then, we find our
f-mode singular field (scalar) from Eq. (5-70),


 ahb(scalar) (, ((I)) (5-71)

where F~(() is the f-th component of the Legendre polynomial expansions of

hb (scalar) (O, )) as in Eq. (5-67), and hb (scalar) (, ) is obtained from the scalar
source functions, {hS (t, r, 0, s) (t, r, 0, Q) hs (t, r, 0, ))}, via the coordinate

rotation.

 To c0-order, hsb(scalar) has the structure


 ab (scalar) +3
 Po Po Pj
 (5-72)

where x" denotes the Schwarzschild coordinates (t, r, 0, 0). From this we may

develop structure analyses for ce1-term and co-term similar to those in Section 4.4
to find the non-vanishing regularization parameters via Eq. (5-71).

 The results of the calculations are